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Unformatted text preview: cen58933_ch13.qxd 9/9/2002 9:57 AM Page 667 CHAPTER H E AT E X C H A N G E R S eat exchangers are devices that facilitate the exchange of heat between two fluids that are at different temperatures while keeping them from mixing with each other. Heat exchangers are commonly used in practice in a wide range of applications, from heating and air-conditioning systems in a household, to chemical processing and power production in large plants. Heat exchangers differ from mixing chambers in that they do not allow the two fluids involved to mix. In a car radiator, for example, heat is transferred from the hot water flowing through the radiator tubes to the air flowing through the closely spaced thin plates outside attached to the tubes. Heat transfer in a heat exchanger usually involves convection in each fluid and conduction through the wall separating the two fluids. In the analysis of heat exchangers, it is convenient to work with an overall heat transfer coefficient U that accounts for the contribution of all these effects on heat transfer. The rate of heat transfer between the two fluids at a location in a heat exchanger depends on the magnitude of the temperature difference at that location, which varies along the heat exchanger. In the analysis of heat exchangers, it is usually convenient to work with the logarithmic mean temperature difference LMTD, which is an equivalent mean temperature difference between the two fluids for the entire heat exchanger. Heat exchangers are manufactured in a variety of types, and thus we start this chapter with the classification of heat exchangers. We then discuss the determination of the overall heat transfer coefficient in heat exchangers, and the LMTD for some configurations. We then introduce the correction factor F to account for the deviation of the mean temperature difference from the LMTD in complex configurations. Next we discuss the effectiveness–NTU method, which enables us to analyze heat exchangers when the outlet temperatures of the fluids are not known. Finally, we discuss the selection of heat exchangers. H 13 CONTENTS 13–1 Types of Heat Exchangers 668 13–2 The Overall Heat Transfer Coefficient 671 13–3 Analysis of Heat Exchangers 678 13–4 The Log Mean Temperature Difference Method 680 13–5 The Effectiveness–NTU Method 690 13–6 Selection of Heat Exchangers 700 667 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 668 668 HEAT TRANSFER 13–1 I TYPES OF HEAT EXCHANGERS Different heat transfer applications require different types of hardware and different configurations of heat transfer equipment. The attempt to match the heat transfer hardware to the heat transfer requirements within the specified constraints has resulted in numerous types of innovative heat exchanger designs. The simplest type of heat exchanger consists of two concentric pipes of different diameters, as shown in Figure 13–1, called the double-pipe heat exchanger. One fluid in a double-pipe heat exchanger flows through the smaller pipe while the other fluid flows through the annular space between the two pipes. Two types of flow arrangement are possible in a double-pipe heat exchanger: in parallel flow, both the hot and cold fluids enter the heat exchanger at the same end and move in the same direction. In counter flow, on the other hand, the hot and cold fluids enter the heat exchanger at opposite ends and flow in opposite directions. Another type of heat exchanger, which is specifically designed to realize a large heat transfer surface area per unit volume, is the compact heat exchanger. The ratio of the heat transfer surface area of a heat exchanger to its volume is called the area density . A heat exchanger with 700 m2/m3 23 (or 200 ft /ft ) is classified as being compact. Examples of compact heat exchangers are car radiators ( 1000 m2/m3), glass ceramic gas turbine 2 3 heat exchangers ( 6000 m /m ), the regenerator of a Stirling engine 20,000 m2/m3). Compact heat ( 15,000 m2/m3), and the human lung ( exchangers enable us to achieve high heat transfer rates between two fluids in T T Ho t fl Hot flui Cold d Co ld f FIGURE 13–1 Different flow regimes and associated temperature profiles in a double-pipe heat exchanger. Cold in Hot out Cold in (a) Parallel flow luid fluid Cold out Hot in uid Hot out Hot in Cold out (b) Counter flow cen58933_ch13.qxd 9/9/2002 9:57 AM Page 669 669 CHAPTER 13 a small volume, and they are commonly used in applications with strict limitations on the weight and volume of heat exchangers (Fig. 13–2). The large surface area in compact heat exchangers is obtained by attaching closely spaced thin plate or corrugated fins to the walls separating the two fluids. Compact heat exchangers are commonly used in gas-to-gas and gas-toliquid (or liquid-to-gas) heat exchangers to counteract the low heat transfer coefficient associated with gas flow with increased surface area. In a car radiator, which is a water-to-air compact heat exchanger, for example, it is no surprise that fins are attached to the air side of the tube surface. In compact heat exchangers, the two fluids usually move perpendicular to each other, and such flow configuration is called cross-flow. The cross-flow is further classified as unmixed and mixed flow, depending on the flow configuration, as shown in Figure 13–3. In (a) the cross-flow is said to be unmixed since the plate fins force the fluid to flow through a particular interfin spacing and prevent it from moving in the transverse direction (i.e., parallel to the tubes). The cross-flow in (b) is said to be mixed since the fluid now is free to move in the transverse direction. Both fluids are unmixed in a car radiator. The presence of mixing in the fluid can have a significant effect on the heat transfer characteristics of the heat exchanger. FIGURE 13–2 A gas-to-liquid compact heat exchanger for a residential airconditioning system. Cross-flow (unmixed) Cross-flow (mixed) Tube flow (unmixed) (a) Both fluids unmixed Tube flow (unmixed) (b) One fluid mixed, one fluid unmixed FIGURE 13–3 Different flow configurations in cross-flow heat exchangers. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 670 670 HEAT TRANSFER Tube outlet FIGURE 13–4 The schematic of a shell-and-tube heat exchanger (one-shell pass and one-tube pass). Shell inlet Baffles Front-end header Rear-end header Tubes Shell Shell outlet Shell-side fluid In Tube-side fluid Out In Out (a) One-shell pass and two-tube passes Shell-side fluid In Out Tubeside fluid In Out (b) Two-shell passes and four-tube passes FIGURE 13–5 Multipass flow arrangements in shelland-tube heat exchangers. Tube inlet Perhaps the most common type of heat exchanger in industrial applications is the shell-and-tube heat exchanger, shown in Figure 13–4. Shell-and-tube heat exchangers contain a large number of tubes (sometimes several hundred) packed in a shell with their axes parallel to that of the shell. Heat transfer takes place as one fluid flows inside the tubes while the other fluid flows outside the tubes through the shell. Baffles are commonly placed in the shell to force the shell-side fluid to flow across the shell to enhance heat transfer and to maintain uniform spacing between the tubes. Despite their widespread use, shelland-tube heat exchangers are not suitable for use in automotive and aircraft applications because of their relatively large size and weight. Note that the tubes in a shell-and-tube heat exchanger open to some large flow areas called headers at both ends of the shell, where the tube-side fluid accumulates before entering the tubes and after leaving them. Shell-and-tube heat exchangers are further classified according to the number of shell and tube passes involved. Heat exchangers in which all the tubes make one U-turn in the shell, for example, are called one-shell-pass and twotube-passes heat exchangers. Likewise, a heat exchanger that involves two passes in the shell and four passes in the tubes is called a two-shell-passes and four-tube-passes heat exchanger (Fig. 13–5). An innovative type of heat exchanger that has found widespread use is the plate and frame (or just plate) heat exchanger, which consists of a series of plates with corrugated flat flow passages (Fig. 13–6). The hot and cold fluids flow in alternate passages, and thus each cold fluid stream is surrounded by two hot fluid streams, resulting in very effective heat transfer. Also, plate heat exchangers can grow with increasing demand for heat transfer by simply mounting more plates. They are well suited for liquid-to-liquid heat exchange applications, provided that the hot and cold fluid streams are at about the same pressure. Another type of heat exchanger that involves the alternate passage of the hot and cold fluid streams through the same flow area is the regenerative heat exchanger. The static-type regenerative heat exchanger is basically a porous mass that has a large heat storage capacity, such as a ceramic wire mesh. Hot and cold fluids flow through this porous mass alternatively. Heat is transferred from the hot fluid to the matrix of the regenerator during the flow of the hot fluid, and from the matrix to the cold fluid during the flow of the cold fluid. Thus, the matrix serves as a temporary heat storage medium. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 671 671 CHAPTER 13 FIGURE 13–6 A plate-and-frame liquid-to-liquid heat exchanger (courtesy of Trante Corp.). The dynamic-type regenerator involves a rotating drum and continuous flow of the hot and cold fluid through different portions of the drum so that any portion of the drum passes periodically through the hot stream, storing heat, and then through the cold stream, rejecting this stored heat. Again the drum serves as the medium to transport the heat from the hot to the cold fluid stream. Heat exchangers are often given specific names to reflect the specific application for which they are used. For example, a condenser is a heat exchanger in which one of the fluids is cooled and condenses as it flows through the heat exchanger. A boiler is another heat exchanger in which one of the fluids absorbs heat and vaporizes. A space radiator is a heat exchanger that transfers heat from the hot fluid to the surrounding space by radiation. 13–2 I Cold fluid Hot fluid Heat transfer Ti Hot fluid Ai hi THE OVERALL HEAT TRANSFER COEFFICIENT A heat exchanger typically involves two flowing fluids separated by a solid wall. Heat is first transferred from the hot fluid to the wall by convection, through the wall by conduction, and from the wall to the cold fluid again by convection. Any radiation effects are usually included in the convection heat transfer coefficients. The thermal resistance network associated with this heat transfer process involves two convection and one conduction resistances, as shown in Figure 13–7. Here the subscripts i and o represent the inner and outer surfaces of the Cold fluid Wall Ao ho To Ti Ri = –1 –– hi A i Rwall Ro = –1 –– ho Ao FIGURE 13–7 Thermal resistance network associated with heat transfer in a double-pipe heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 672 672 HEAT TRANSFER inner tube. For a double-pipe heat exchanger, we have Ai DiL and Ao DoL, and the thermal resistance of the tube wall in this case is Rwall R D Di o Outer tube Outer fluid Inner fluid (13-1) where k is the thermal conductivity of the wall material and L is the length of the tube. Then the total thermal resistance becomes L Heat transfer ln (Do /Di ) 2 kL Inner tube Ao = π Do L Ai = π Di L FIGURE 13–8 The two heat transfer surface areas associated with a double-pipe heat exchanger (for thin tubes, Di Do and thus Ai Ao). Rtotal Ri Rwall Ro ln (Do /Di ) 2 kL 1 hi Ai 1 ho Ao (13-2) The Ai is the area of the inner surface of the wall that separates the two fluids, and Ao is the area of the outer surface of the wall. In other words, Ai and Ao are surface areas of the separating wall wetted by the inner and the outer fluids, respectively. When one fluid flows inside a circular tube and the other outside DiL and Ao DoL (Fig. 13–8). of it, we have Ai In the analysis of heat exchangers, it is convenient to combine all the thermal resistances in the path of heat flow from the hot fluid to the cold one into a single resistance R, and to express the rate of heat transfer between the two fluids as · Q T R UA T Ui Ai T Uo Ao T (13-3) where U is the overall heat transfer coefficient, whose unit is W/m2 · °C, which is identical to the unit of the ordinary convection coefficient h. Canceling T, Eq. 13-3 reduces to 1 UAs 1 Ui Ai 1 Uo Ao R 1 hi Ai Rwall 1 ho Ao (13-4) Perhaps you are wondering why we have two overall heat transfer coefficients Ui and Uo for a heat exchanger. The reason is that every heat exchanger has two heat transfer surface areas Ai and Ao, which, in general, are not equal to each other. Note that Ui Ai Uo Ao, but Ui Uo unless Ai Ao. Therefore, the overall heat transfer coefficient U of a heat exchanger is meaningless unless the area on which it is based is specified. This is especially the case when one side of the tube wall is finned and the other side is not, since the surface area of the finned side is several times that of the unfinned side. When the wall thickness of the tube is small and the thermal conductivity of the tube material is high, as is usually the case, the thermal resistance of the tube is negligible (Rwall 0) and the inner and outer surfaces of the tube are almost identical (Ai Ao As). Then Eq. 13-4 for the overall heat transfer coefficient simplifies to 1 U 1 hi 1 ho (13-5) where U Ui Uo. The individual convection heat transfer coefficients inside and outside the tube, hi and ho, are determined using the convection relations discussed in earlier chapters. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 673 673 CHAPTER 13 The overall heat transfer coefficient U in Eq. 13-5 is dominated by the smaller convection coefficient, since the inverse of a large number is small. When one of the convection coefficients is much smaller than the other (say, hi ho), we have 1/hi 1/ho, and thus U hi. Therefore, the smaller heat transfer coefficient creates a bottleneck on the path of heat flow and seriously impedes heat transfer. This situation arises frequently when one of the fluids is a gas and the other is a liquid. In such cases, fins are commonly used on the gas side to enhance the product UAs and thus the heat transfer on that side. Representative values of the overall heat transfer coefficient U are given in Table 13–1. Note that the overall heat transfer coefficient ranges from about 10 W/m2 · °C for gas-to-gas heat exchangers to about 10,000 W/m2 · °C for heat exchangers that involve phase changes. This is not surprising, since gases have very low thermal conductivities, and phase-change processes involve very high heat transfer coefficients. When the tube is finned on one side to enhance heat transfer, the total heat transfer surface area on the finned side becomes As Atotal Afin Aunfinned (13-6) where Afin is the surface area of the fins and Aunfinned is the area of the unfinned portion of the tube surface. For short fins of high thermal conductivity, we can use this total area in the convection resistance relation Rconv 1/hAs since the fins in this case will be very nearly isothermal. Otherwise, we should determine the effective surface area A from As Aunfinned fin Afin (13-7) TABLE 13–1 Representative values of the overall heat transfer coefficients in heat exchangers Type of heat exchanger Water-to-water Water-to-oil Water-to-gasoline or kerosene Feedwater heaters Steam-to-light fuel oil Steam-to-heavy fuel oil Steam condenser Freon condenser (water cooled) Ammonia condenser (water cooled) Alcohol condensers (water cooled) Gas-to-gas Water-to-air in finned tubes (water in tubes) Steam-to-air in finned tubes (steam in tubes) *Multiply the listed values by 0.176 to convert them to Btu/h · ft2 · °F. † Based on air-side surface area. ‡ Based on water- or steam-side surface area. U, W/m2 · °C* 850–1700 100–350 300–1000 1000–8500 200–400 50–200 1000–6000 300–1000 800–1400 250–700 10–40 3060† 400–850† 30–300† 400–4000‡ cen58933_ch13.qxd 9/9/2002 9:57 AM Page 674 674 HEAT TRANSFER where fin is the fin efficiency. This way, the temperature drop along the fins 1 for isothermal fins, and thus Eq. 13-7 is accounted for. Note that fin reduces to Eq. 13-6 in that case. Fouling Factor The performance of heat exchangers usually deteriorates with time as a result of accumulation of deposits on heat transfer surfaces. The layer of deposits represents additional resistance to heat transfer and causes the rate of heat transfer in a heat exchanger to decrease. The net effect of these accumulations on heat transfer is represented by a fouling factor Rf , which is a measure of the thermal resistance introduced by fouling. The most common type of fouling is the precipitation of solid deposits in a fluid on the heat transfer surfaces. You can observe this type of fouling even in your house. If you check the inner surfaces of your teapot after prolonged use, you will probably notice a layer of calcium-based deposits on the surfaces at which boiling occurs. This is especially the case in areas where the water is hard. The scales of such deposits come off by scratching, and the surfaces can be cleaned of such deposits by chemical treatment. Now imagine those mineral deposits forming on the inner surfaces of fine tubes in a heat exchanger (Fig. 13–9) and the detrimental effect it may have on the flow passage area and the heat transfer. To avoid this potential problem, water in power and process plants is extensively treated and its solid contents are removed before it is allowed to circulate through the system. The solid ash particles in the flue gases accumulating on the surfaces of air preheaters create similar problems. Another form of fouling, which is common in the chemical process industry, is corrosion and other chemical fouling. In this case, the surfaces are fouled by the accumulation of the products of chemical reactions on the surfaces. This form of fouling can be avoided by coating metal pipes with glass or using plastic pipes instead of metal ones. Heat exchangers may also be fouled by the growth of algae in warm fluids. This type of fouling is called biological fouling and can be prevented by chemical treatment. In applications where it is likely to occur, fouling should be considered in the design and selection of heat exchangers. In such applications, it may be FIGURE 13–9 Precipitation fouling of ash particles on superheater tubes (from Steam, Its Generation, and Use, Babcock and Wilcox Co., 1978). cen58933_ch13.qxd 9/9/2002 9:57 AM Page 675 675 CHAPTER 13 necessary to select a larger and thus more expensive heat exchanger to ensure that it meets the design heat transfer requirements even after fouling occurs. The periodic cleaning of heat exchangers and the resulting down time are additional penalties associated with fouling. The fouling factor is obviously zero for a new heat exchanger and increases with time as the solid deposits build up on the heat exchanger surface. The fouling factor depends on the operating temperature and the velocity of the fluids, as well as the length of service. Fouling increases with increasing temperature and decreasing velocity. The overall heat transfer coefficient relation given above is valid for clean surfaces and needs to be modified to account for the effects of fouling on both the inner and the outer surfaces of the tube. For an unfinned shell-and-tube heat exchanger, it can be expressed as 1 UAs 1 Ui Ai 1 Uo Ao R 1 hi Ai Rf, i Ai ln (Do /Di ) 2 kL Rf, o Ao 1 ho Ao (13-8) where Ai Di L and Ao Do L are the areas of inner and outer surfaces, and Rf, i and Rf, o are the fouling factors at those surfaces. Representative values of fouling factors are given in Table 13–2. More comprehensive tables of fouling factors are available in handbooks. As you would expect, considerable uncertainty exists in these values, and they should be used as a guide in the selection and evaluation of heat exchangers to account for the effects of anticipated fouling on heat transfer. Note that most fouling factors in the table are of the order of 10 4 m2 · °C/W, which is equivalent to the thermal resistance of a 0.2-mm-thick limestone layer (k 2.9 W/m · °C) per unit surface area. Therefore, in the absence of specific data, we can assume the surfaces to be coated with 0.2 mm of limestone as a starting point to account for the effects of fouling. EXAMPLE 13–1 Overall Heat Transfer Coefficient of a Heat Exchanger Hot oil is to be cooled in a double-tube counter-flow heat exchanger. The copper inner tubes have a diameter of 2 cm and negligible thickness. The inner diameter of the outer tube (the shell) is 3 cm. Water flows through the tube at a rate of 0.5 kg/s, and the oil through the shell at a rate of 0.8 kg/s. Taking the average temperatures of the water and the oil to be 45°C and 80°C, respectively, determine the overall heat transfer coefficient of this heat exchanger. SOLUTION Hot oil is cooled by water in a double-tube counter-flow heat exchanger. The overall heat transfer coefficient is to be determined. Assumptions 1 The thermal resistance of the inner tube is negligible since the tube material is highly conductive and its thickness is negligible. 2 Both the oil and water flow are fully developed. 3 Properties of the oil and water are constant. Properties The properties of water at 45°C are (Table A–9) k 990 kg/m3 0.637 W/m · °C Pr 3.91 / 0.602 10 6 m2/s TABLE 13–2 Representative fouling factors (thermal resistance due to fouling for a unit surface area) (Source: Tubular Exchange Manufacturers Association.) Fluid Distilled water, sea water, river water, boiler feedwater: Below 50°C Above 50°C Fuel oil Steam (oil-free) Refrigerants (liquid) Refrigerants (vapor) Alcohol vapors Air Rf , m2 · °C/W 0.0001 0.0002 0.0009 0.0001 0.0002 0.0004 0.0001 0.0004 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 676 676 HEAT TRANSFER The properties of oil at 80°C are (Table A–16). 852 kg/m3 0.138 W/m · °C k Hot oil 0.8 kg/s 2 cm 1 hi 1 U 3 cm FIGURE 13–10 Schematic for Example 13–1. 490 37.5 10 6 m2/s Analysis The schematic of the heat exchanger is given in Figure 13–10. The overall heat transfer coefficient U can be determined from Eq. 13-5: Cold water 0.5 kg/s Pr 1 ho where hi and ho are the convection heat transfer coefficients inside and outside the tube, respectively, which are to be determined using the forced convection relations. The hydraulic diameter for a circular tube is the diameter of the tube itself, Dh D 0.02 m. The mean velocity of water in the tube and the Reynolds number are · m Ac m · m (1 4 D2) 0.5 kg/s (990 kg/m3)[1 (0.02 m)2] 4 1.61 m/s and m Dh Re (1.61 m/s)(0.02 m) 0.602 10 6 m2/s 53,490 which is greater than 4000. Therefore, the flow of water is turbulent. Assuming the flow to be fully developed, the Nusselt number can be determined from hDh k Nu 0.023 Re0.8 Pr 0.4 0.023(53,490)0.8(3.91)0.4 240.6 Then, h k Nu Dh 0.637 W/m · °C (240.6) 0.02 m 7663 W/m2 · °C Now we repeat the analysis above for oil. The properties of oil at 80°C are 852 kg/m3 0.138 W/m · °C k 37.5 490 Pr 10 6 m2/s The hydraulic diameter for the annular space is Dh TABLE 13–3 Do Di 0.03 0.02 0.01 m The mean velocity and the Reynolds number in this case are Nusselt number for fully developed laminar flow in a circular annulus with one surface insulated and the other isothermal (Kays and Perkins, Ref. 8.) Di /Do Nui Nuo 0.00 0.05 0.10 0.25 0.50 1.00 — 17.46 11.56 7.37 5.74 4.86 3.66 4.06 4.11 4.23 4.43 4.86 m · m Ac [1 4 · m 2 (Do 0.8 kg/s D i2)] (852 kg/m )[1 4 3 (0.032 0.022)] m2 2.39 m/s and Re m Dh (2.39 m/s)(0.01 m) 37.5 10 6 m2/s 637 which is less than 4000. Therefore, the flow of oil is laminar. Assuming fully developed flow, the Nusselt number on the tube side of the annular space Nui corresponding to Di /Do 0.02/0.03 0.667 can be determined from Table 13–3 by interpolation to be Nu 5.45 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 677 677 CHAPTER 13 and k Nu Dh ho 0.138 W/m · °C (5.45) 0.01 m 75.2 W/m2 · °C Then the overall heat transfer coefficient for this heat exchanger becomes U 1 1 hi 1 1 ho 1 7663 W/m2 · °C 1 75.2 W/m2 · °C 74.5 W/m2 · °C Discussion Note that U ho in this case, since hi ho. This confirms our earlier statement that the overall heat transfer coefficient in a heat exchanger is dominated by the smaller heat transfer coefficient when the difference between the two values is large. To improve the overall heat transfer coefficient and thus the heat transfer in this heat exchanger, we must use some enhancement techniques on the oil side, such as a finned surface. EXAMPLE 13–2 Effect of Fouling on the Overall Heat Transfer Coefficient A double-pipe (shell-and-tube) heat exchanger is constructed of a stainless steel (k 15.1 W/m · °C) inner tube of inner diameter Di 1.5 cm and outer diameter Do 1.9 cm and an outer shell of inner diameter 3.2 cm. The convection heat transfer coefficient is given to be hi 800 W/m2 · °C on the inner surface of the tube and ho 1200 W/m2 · °C on the outer surface. For a fouling factor of Rf, i 0.0004 m2 · °C/ W on the tube side and Rf, o 0.0001 m2 · °C/ W on the shell side, determine (a) the thermal resistance of the heat exchanger per unit length and (b) the overall heat transfer coefficients, Ui and Uo based on the inner and outer surface areas of the tube, respectively. SOLUTION The heat transfer coefficients and the fouling factors on the tube Cold fluid and shell sides of a heat exchanger are given. The thermal resistance and the overall heat transfer coefficients based on the inner and outer areas are to be determined. Assumptions The heat transfer coefficients and the fouling factors are constant and uniform. Analysis (a) The schematic of the heat exchanger is given in Figure 13–11. The thermal resistance for an unfinned shell-and-tube heat exchanger with fouling on both heat transfer surfaces is given by Eq. 13-8 as Outer layer of fouling 1 UAs 1 Ui Ai 1 Uo Ao Ai Ao R Di L Do L 1 hi Ai Rf, i Ai ln (Do /Di ) 2 kL Rf, o Ao where (0.015 m)(1 m) (0.019 m)(1 m) 0.0471 m2 0.0597 m2 Substituting, the total thermal resistance is determined to be 1 ho Ao Tube wall Hot fluid Inner layer of fouling Cold fluid Hot fluid Di = 1.5 cm hi = 800 W/m2·°C Rf, i = 0.0004 m2·°C/ W Do = 1.9 cm ho = 1200 W/ m2·°C Rf, o = 0.0001 m2·°C/ W FIGURE 13–11 Schematic for Example 13–2. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 678 678 HEAT TRANSFER 1 (800 W/m2 · °C)(0.0471 m2) ln (0.019/0.015) 2 (15.1 W/m · °C)(1 m) R 0.0004 m2 · °C/ W 0.0471 m2 0.0001 m2 · °C/ W 1 0.0597 m2 (1200 W/m2 · °C)(0.0597 m2) (0.02654 0.00849 0.0025 0.00168 0.01396)°C/ W 0.0532°C/ W Note that about 19 percent of the total thermal resistance in this case is due to fouling and about 5 percent of it is due to the steel tube separating the two fluids. The rest (76 percent) is due to the convection resistances on the two sides of the inner tube. (b) Knowing the total thermal resistance and the heat transfer surface areas, the overall heat transfer coefficient based on the inner and outer surfaces of the tube are determined again from Eq. 13-8 to be Ui 1 RAi 1 (0.0532 °C/ W)(0.0471 m2) 399 W/m2 · °C Uo 1 RAo 1 (0.0532 °C/ W)(0.0597 m2) 315 W/m2 · °C and Discussion Note that the two overall heat transfer coefficients differ significantly (by 27 percent) in this case because of the considerable difference between the heat transfer surface areas on the inner and the outer sides of the tube. For tubes of negligible thickness, the difference between the two overall heat transfer coefficients would be negligible. 13–3 I ANALYSIS OF HEAT EXCHANGERS Heat exchangers are commonly used in practice, and an engineer often finds himself or herself in a position to select a heat exchanger that will achieve a specified temperature change in a fluid stream of known mass flow rate, or to predict the outlet temperatures of the hot and cold fluid streams in a specified heat exchanger. In upcoming sections, we will discuss the two methods used in the analysis of heat exchangers. Of these, the log mean temperature difference (or LMTD) method is best suited for the first task and the effectiveness–NTU method for the second task as just stated. But first we present some general considerations. Heat exchangers usually operate for long periods of time with no change in their operating conditions. Therefore, they can be modeled as steady-flow devices. As such, the mass flow rate of each fluid remains constant, and the fluid properties such as temperature and velocity at any inlet or outlet remain the same. Also, the fluid streams experience little or no change in their velocities and elevations, and thus the kinetic and potential energy changes are negligible. The specific heat of a fluid, in general, changes with temperature. But, in cen58933_ch13.qxd 9/9/2002 9:57 AM Page 679 679 CHAPTER 13 a specified temperature range, it can be treated as a constant at some average value with little loss in accuracy. Axial heat conduction along the tube is usually insignificant and can be considered negligible. Finally, the outer surface of the heat exchanger is assumed to be perfectly insulated, so that there is no heat loss to the surrounding medium, and any heat transfer occurs between the two fluids only. The idealizations stated above are closely approximated in practice, and they greatly simplify the analysis of a heat exchanger with little sacrifice of accuracy. Therefore, they are commonly used. Under these assumptions, the first law of thermodynamics requires that the rate of heat transfer from the hot fluid be equal to the rate of heat transfer to the cold one. That is, · Q · mcCpc(Tc, out Tc, in) (13-9) · Q · mhCph(Th, in Th, out) (13-10) and where the subscripts c and h stand for cold and hot fluids, respectively, and ·· mc, mh Cpc, Cph Tc, out, Th, out Tc, in, Th, in mass flow rates specific heats outlet temperatures inlet temperatures · Note that the heat transfer rate Q is taken to be a positive quantity, and its direction is understood to be from the hot fluid to the cold one in accordance with the second law of thermodynamics. In heat exchanger analysis, it is often convenient to combine the product of the mass flow rate and the specific heat of a fluid into a single quantity. This quantity is called the heat capacity rate and is defined for the hot and cold fluid streams as Ch · mhCph and Cc · mcCpc (13-11) The heat capacity rate of a fluid stream represents the rate of heat transfer needed to change the temperature of the fluid stream by 1°C as it flows through a heat exchanger. Note that in a heat exchanger, the fluid with a large heat capacity rate will experience a small temperature change, and the fluid with a small heat capacity rate will experience a large temperature change. Therefore, doubling the mass flow rate of a fluid while leaving everything else unchanged will halve the temperature change of that fluid. With the definition of the heat capacity rate above, Eqs. 13-9 and 13-10 can also be expressed as · Q Cc(Tc, out Tc, in) T Hot fluid ∆T1 Ch ∆T ∆T2 Cold fluid Cc = Ch ∆T = ∆T1 = ∆T2 = constant (13-12) and · Q x Ch(Th, in Th, out) (13-13) That is, the heat transfer rate in a heat exchanger is equal to the heat capacity rate of either fluid multiplied by the temperature change of that fluid. Note that the only time the temperature rise of a cold fluid is equal to the temperature drop of the hot fluid is when the heat capacity rates of the two fluids are equal to each other (Fig. 13–12). Inlet Outlet FIGURE 13–12 Two fluids that have the same mass flow rate and the same specific heat experience the same temperature change in a well-insulated heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 680 680 HEAT TRANSFER Two special types of heat exchangers commonly used in practice are condensers and boilers. One of the fluids in a condenser or a boiler undergoes a phase-change process, and the rate of heat transfer is expressed as · Q T (13-14) · where m is the rate of evaporation or condensation of the fluid and hfg is the enthalpy of vaporization of the fluid at the specified temperature or pressure. An ordinary fluid absorbs or releases a large amount of heat essentially at constant temperature during a phase-change process, as shown in Figure 13–13. The heat capacity rate of a fluid during a phase-change process must approach infinity since the temperature change is practically zero. That is, · · · C mCp → when T → 0, so that the heat transfer rate Q mCp T is a finite quantity. Therefore, in heat exchanger analysis, a condensing or boiling fluid is conveniently modeled as a fluid whose heat capacity rate is infinity. The rate of heat transfer in a heat exchanger can also be expressed in an analogous manner to Newton’s law of cooling as Condensing fluid . Q Cold fluid · Q Inlet · mhfg Outlet (a) Condenser (Ch → ) T Hot fluid . Q UAs Tm (13-15) where U is the overall heat transfer coefficient, As is the heat transfer area, and Tm is an appropriate average temperature difference between the two fluids. Here the surface area As can be determined precisely using the dimensions of the heat exchanger. However, the overall heat transfer coefficient U and the temperature difference T between the hot and cold fluids, in general, are not constant and vary along the heat exchanger. The average value of the overall heat transfer coefficient can be determined as described in the preceding section by using the average convection coefficients for each fluid. It turns out that the appropriate form of the mean temperature difference between the two fluids is logarithmic in nature, and its determination is presented in Section 13–4. Boiling fluid Inlet Outlet (b) Boiler (Cc → ) FIGURE 13–13 Variation of fluid temperatures in a heat exchanger when one of the fluids condenses or boils. 13–4 I THE LOG MEAN TEMPERATURE DIFFERENCE METHOD Earlier, we mentioned that the temperature difference between the hot and cold fluids varies along the heat exchanger, and it is convenient to have a · mean temperature difference Tm for use in the relation Q UAs Tm. In order to develop a relation for the equivalent average temperature difference between the two fluids, consider the parallel-flow double-pipe heat exchanger shown in Figure 13–14. Note that the temperature difference T between the hot and cold fluids is large at the inlet of the heat exchanger but decreases exponentially toward the outlet. As you would expect, the temperature of the hot fluid decreases and the temperature of the cold fluid increases along the heat exchanger, but the temperature of the cold fluid can never exceed that of the hot fluid no matter how long the heat exchanger is. Assuming the outer surface of the heat exchanger to be well insulated so that any heat transfer occurs between the two fluids, and disregarding any cen58933_ch13.qxd 9/9/2002 9:57 AM Page 681 681 CHAPTER 13 changes in kinetic and potential energy, an energy balance on each fluid in a differential section of the heat exchanger can be expressed as · Q · mh Cph dTh T Th, in . δ Q = U (Th – Tc ) d As Th (13-16) ∆T ∆T1 and · Q · mc Cpc dTc ˙ Q ˙ m hCph dTh (13-18) ∆T2 Th, out Tc, out dTc (13-17) That is, the rate of heat loss from the hot fluid at any section of a heat exchanger is equal to the rate of heat gain by the cold fluid in that section. The temperature change of the hot fluid is a negative quantity, and so a negative · sign is added to Eq. 13-16 to make the heat transfer rate Q a positive quantity. Solving the equations above for dTh and dTc gives dTh . δQ Tc Tc, in 1 Hot fluid ∆T1 = Th, in – Tc, in ∆T2 = Th, out – Tc, out 2 dAs Tc, out dAs As Th, out Th, in and ˙ Q ˙ mc Cpc dTc (13-19) Taking their difference, we get dTh dTc d(Th · 1 Q· m h Cph Tc) 1 · m c Cpc (13-20) The rate of heat transfer in the differential section of the heat exchanger can also be expressed as · Q U(Th Tc) dAs (13-21) Substituting this equation into Eq. 13-20 and rearranging gives d(Th Th Tc) Tc 1 U dAs · m h Cph 1 · m c Cpc (13-22) Integrating from the inlet of the heat exchanger to its outlet, we obtain ln Th, out Th, in Tc, out Tc, in 1 UAs · m h Cph 1 · m c Cpc (13-23) · · Finally, solving Eqs. 13-9 and 13-10 for mcCpc and mhCph and substituting into Eq. 13-23 gives, after some rearrangement, · Q UAs Tlm (13-24) T1 T2 ln ( T1/ T2) (13-25) where Tlm is the log mean temperature difference, which is the suitable form of the average temperature difference for use in the analysis of heat exchangers. Here T1 and T2 represent the temperature difference between the two fluids Cold fluid Tc, in FIGURE 13–14 Variation of the fluid temperatures in a parallel-flow double-pipe heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 682 682 HEAT TRANSFER Tc, out ∆T2 Hot fluid Th,in ∆T1 Th,out Cold fluid Tc, in ∆T1 = Th,in – Tc, in ∆T2 = Th,out – Tc, out (a) Parallel-flow heat exchangers Cold fluid Tc, in ∆T2 Hot fluid Th,in Th,out ∆T1 Tc, out at the two ends (inlet and outlet) of the heat exchanger. It makes no difference which end of the heat exchanger is designated as the inlet or the outlet (Fig. 13–15). The temperature difference between the two fluids decreases from T1 at the inlet to T2 at the outlet. Thus, it is tempting to use the arithmetic mean T2) as the average temperature difference. The temperature Tam 1 ( T1 2 logarithmic mean temperature difference Tlm is obtained by tracing the actual temperature profile of the fluids along the heat exchanger and is an exact representation of the average temperature difference between the hot and cold fluids. It truly reflects the exponential decay of the local temperature difference. Note that Tlm is always less than Tam. Therefore, using Tam in calculations instead of Tlm will overestimate the rate of heat transfer in a heat exchanger between the two fluids. When T1 differs from T2 by no more than 40 percent, the error in using the arithmetic mean temperature difference is less than 1 percent. But the error increases to undesirable levels when T1 differs from T2 by greater amounts. Therefore, we should always use the logarithmic mean temperature difference when determining the rate of heat transfer in a heat exchanger. ∆T1 = Th,in – Tc, out ∆T2 = Th,out – Tc, in (b) Counter-flow heat exchangers FIGURE 13–15 The T1 and T2 expressions in parallel-flow and counter-flow heat exchangers. Counter-Flow Heat Exchangers The variation of temperatures of hot and cold fluids in a counter-flow heat exchanger is given in Figure 13–16. Note that the hot and cold fluids enter the heat exchanger from opposite ends, and the outlet temperature of the cold fluid in this case may exceed the outlet temperature of the hot fluid. In the limiting case, the cold fluid will be heated to the inlet temperature of the hot fluid. However, the outlet temperature of the cold fluid can never exceed the inlet temperature of the hot fluid, since this would be a violation of the second law of thermodynamics. The relation above for the log mean temperature difference is developed using a parallel-flow heat exchanger, but we can show by repeating the analysis above for a counter-flow heat exchanger that is also applicable to counterflow heat exchangers. But this time, T1 and T2 are expressed as shown in Figure 13–15. For specified inlet and outlet temperatures, the log mean temperature difference for a counter-flow heat exchanger is always greater than that for a Tlm, PF, and thus a smaller parallel-flow heat exchanger. That is, Tlm, CF surface area (and thus a smaller heat exchanger) is needed to achieve a specified heat transfer rate in a counter-flow heat exchanger. Therefore, it is common practice to use counter-flow arrangements in heat exchangers. In a counter-flow heat exchanger, the temperature difference between the hot and the cold fluids will remain constant along the heat exchanger when the heat capacity rates of the two fluids are equal (that is, T constant · · T2, and the last log when Ch Cc or mhCph mcCpc). Then we have T1 mean temperature difference relation gives Tlm 0 , which is indeterminate. 0 It can be shown by the application of l’Hôpital’s rule that in this case we have T1 T2, as expected. Tlm A condenser or a boiler can be considered to be either a parallel- or counterflow heat exchanger since both approaches give the same result. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 683 683 CHAPTER 13 Multipass and Cross-Flow Heat Exchangers: Use of a Correction Factor T Th, in The log mean temperature difference Tlm relation developed earlier is limited to parallel-flow and counter-flow heat exchangers only. Similar relations are also developed for cross-flow and multipass shell-and-tube heat exchangers, but the resulting expressions are too complicated because of the complex flow conditions. In such cases, it is convenient to relate the equivalent temperature difference to the log mean temperature difference relation for the counter-flow case as Tlm F Tlm, CF t2 T1 t1 t1 Th ∆T Cold fluid Tc, in Th, in T2 t1 Th, out Tc, out FIGURE 13–16 The variation of the fluid temperatures in a counter-flow double-pipe heat exchanger. Cold Tc, in fluid Th,in T1 t2 (13-28) Cold fluid Hot fluid Hot fluid · (m Cp)tube side · (m C ) Th, out Tc, in (13-27) and R Tc (13-26) where F is the correction factor, which depends on the geometry of the heat exchanger and the inlet and outlet temperatures of the hot and cold fluid streams. The Tlm, CF is the log mean temperature difference for the case of a counter-flow heat exchanger with the same inlet and outlet temperatures and is determined from Eq. 13-25 by taking Tl Th, in Tc, out and T2 Th, out Tc, in (Fig. 13–17). The correction factor is less than unity for a cross-flow and multipass shelland-tube heat exchanger. That is, F 1. The limiting value of F 1 corresponds to the counter-flow heat exchanger. Thus, the correction factor F for a heat exchanger is a measure of deviation of the Tlm from the corresponding values for the counter-flow case. The correction factor F for common cross-flow and shell-and-tube heat exchanger configurations is given in Figure 13–18 versus two temperature ratios P and R defined as P Hot fluid Tc, out ∆T1 Cross-flow or multipass shell-and-tube heat exchanger ∆T2 Th,out Tc, out p shell side where the subscripts 1 and 2 represent the inlet and outlet, respectively. Note that for a shell-and-tube heat exchanger, T and t represent the shell- and tube-side temperatures, respectively, as shown in the correction factor charts. It makes no difference whether the hot or the cold fluid flows through the shell or the tube. The determination of the correction factor F requires the availability of the inlet and the outlet temperatures for both the cold and hot fluids. Note that the value of P ranges from 0 to 1. The value of R, on the other hand, ranges from 0 to infinity, with R 0 corresponding to the phase-change (condensation or boiling) on the shell-side and R → to phase-change on the tube side. The correction factor is F 1 for both of these limiting cases. Therefore, the correction factor for a condenser or boiler is F 1, regardless of the configuration of the heat exchanger. Heat transfer rate: . Q = UAsF ∆Tlm, CF where ∆Tlm, CF = ∆T1 – ∆T2 ln(∆T1/∆T2) ∆T1 = Th,in – Tc,out ∆T2 = Th,out – Tc,in and F = … (Fig. 13–18) FIGURE 13–17 The determination of the heat transfer rate for cross-flow and multipass shell-and-tube heat exchangers using the correction factor. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 684 684 HEAT TRANSFER Correction factor F 1.0 T1 t2 t1 0.9 0.8 R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 T2 0.2 0.7 T1 – T2 R = ——– t2 – t1 0.6 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (a) One-shell pass and 2, 4, 6, etc. (any multiple of 2), tube passes t2 – t1 P = ——– T1 – t1 Correction factor F 1.0 T1 0.9 t2 0.8 R = 4.0 3.0 2.0 1.5 t1 1.0 0.8 0.6 0.4 0.2 0.7 T2 0.6 0.5 T1 – T2 R = ——– t2 – t1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t2 – t1 P = ——– T1 – t1 (b) Two-shell passes and 4, 8, 12, etc. (any multiple of 4), tube passes Correction factor F 1.0 T1 0.9 0.8 R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 0.2 t1 0.7 T1 – T2 R = ——– t2 – t1 0.6 0.5 0 0.1 0.2 T2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (c) Single-pass cross-flow with both fluids unmixed Correction factor F 1.0 FIGURE 13–18 Correction factor F charts for common shell-and-tube and cross-flow heat exchangers (from Bowman, Mueller, and Nagle, Ref. 2). t2 t2 – t1 P = ——– T1 – t1 T1 0.9 0.8 R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 t1 0.2 t2 0.7 0.6 0.5 0 T1 – T2 R = ——– t2 – t1 0.1 0.2 T2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (d ) Single-pass cross-flow with one fluid mixed and the other unmixed t2 – t1 P = ——– T1 – t1 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 685 685 CHAPTER 13 EXAMPLE 13–3 The Condensation of Steam in a Condenser Steam in the condenser of a power plant is to be condensed at a temperature of 30°C with cooling water from a nearby lake, which enters the tubes of the condenser at 14°C and leaves at 22°C. The surface area of the tubes is 45 m2, and the overall heat transfer coefficient is 2100 W/m2 · °C. Determine the mass flow rate of the cooling water needed and the rate of condensation of the steam in the condenser. SOLUTION Steam is condensed by cooling water in the condenser of a power plant. The mass flow rate of the cooling water and the rate of condensation are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 There is no fouling. 5 Fluid properties are constant. Properties The heat of vaporization of water at 30°C is hfg 2431 kJ/kg and the specific heat of cold water at the average temperature of 18°C is Cp 4184 J/kg · °C (Table A–9). Analysis The schematic of the condenser is given in Figure 13–19. The condenser can be treated as a counter-flow heat exchanger since the temperature of one of the fluids (the steam) remains constant. The temperature difference between the steam and the cooling water at the two ends of the condenser is T1 T2 Th, in Th, out Tc, out Tc, in (30 (30 22)°C 14)°C Steam 30°C Cooling water 14°C 8°C 16°C That is, the temperature difference between the two fluids varies from 8°C at one end to 16°C at the other. The proper average temperature difference between the two fluids is the logarithmic mean temperature difference (not the arithmetic), which is determined from Tlm T2 T1 ln ( T1/ T2) 8 16 ln (8/16) 11.5°C 30°C This is a little less than the arithmetic mean temperature difference of 1 (8 16) 12°C. Then the heat transfer rate in the condenser is determined 2 from · Q UAs Tlm (2100 W/m2 · °C)(45 m2)(11.5°C) 1.087 106 W 1087 kW Therefore, the steam will lose heat at a rate of 1,087 kW as it flows through the condenser, and the cooling water will gain practically all of it, since the condenser is well insulated. The mass flow rate of the cooling water and the rate of the condensation of the · · · steam are determined from Q [m Cp (Tout Tin)]cooling water (m hfg)steam to be · m cooling water · Q Cp (Tout Tin) 1,087 kJ/s (4.184 kJ/kg · °C)(22 22°C 14)°C 32.5 kg/s FIGURE 13–19 Schematic for Example 13–3. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 686 686 HEAT TRANSFER and · Q hfg · m steam 1,087 kJ/s 2431 kJ/kg 0.45 kg/s Therefore, we need to circulate about 72 kg of cooling water for each 1 kg of steam condensing to remove the heat released during the condensation process. EXAMPLE 13–4 Heating Water in a Counter-Flow Heat Exchanger A counter-flow double-pipe heat exchanger is to heat water from 20°C to 80°C at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160°C at a mass flow rate of 2 kg/s. The inner tube is thin-walled and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the heat exchanger is 640 W/m2 · °C, determine the length of the heat exchanger required to achieve the desired heating. Hot geothermal 160°C water 2 kg/s Cold water 20°C 1.2 kg/s 80°C D = 1.5 cm SOLUTION Water is heated in a counter-flow double-pipe heat exchanger by geothermal water. The required length of the heat exchanger is to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 There is no fouling. 5 Fluid properties are constant. Properties We take the specific heats of water and geothermal fluid to be 4.18 and 4.31 kJ/kg · °C, respectively. Analysis The schematic of the heat exchanger is given in Figure 13–20. The rate of heat transfer in the heat exchanger can be determined from · Q · [m Cp(Tout Tin)]water (1.2 kg/s)(4.18 kJ/kg · °C)(80 20)°C 301 kW Noting that all of this heat is supplied by the geothermal water, the outlet temperature of the geothermal water is determined to be · Q · [m Cp(Tin Tout)]geothermal → Tout Tin · Q ·C mp 160°C FIGURE 13–20 Schematic for Example 13–4. 301 kW (2 kg/s)(4.31 kJ/kg · °C) 125°C Knowing the inlet and outlet temperatures of both fluids, the logarithmic mean temperature difference for this counter-flow heat exchanger becomes T1 T2 Th, in Th, out Tc, out Tc, in (160 (125 80)°C 20)°C 80°C 105°C and Tlm T1 T2 ln ( T1/ T2) 80 105 ln (80/105) 92.0°C Then the surface area of the heat exchanger is determined to be · Q UAs Tlm → As · Q U Tlm 301,000 W (640 W/m2 · °C)(92.0°C) 5.11 m2 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 687 687 CHAPTER 13 To provide this much heat transfer surface area, the length of the tube must be As DL → L As D 5.11 m2 (0.015 m) 108 m Discussion The inner tube of this counter-flow heat exchanger (and thus the heat exchanger itself) needs to be over 100 m long to achieve the desired heat transfer, which is impractical. In cases like this, we need to use a plate heat exchanger or a multipass shell-and-tube heat exchanger with multiple passes of tube bundles. EXAMPLE 13–5 Heating of Glycerin in a Multipass Heat Exchanger A 2-shell passes and 4-tube passes heat exchanger is used to heat glycerin from 20°C to 50°C by hot water, which enters the thin-walled 2-cm-diameter tubes at 80°C and leaves at 40°C (Fig. 13–21). The total length of the tubes in the heat exchanger is 60 m. The convection heat transfer coefficient is 25 W/m2 · °C on the glycerin (shell) side and 160 W/m2 · °C on the water (tube) side. Determine the rate of heat transfer in the heat exchanger (a) before any fouling occurs and (b) after fouling with a fouling factor of 0.0006 m2 · °C/ W occurs on the outer surfaces of the tubes. SOLUTION Glycerin is heated in a 2-shell passes and 4-tube passes heat exchanger by hot water. The rate of heat transfer for the cases of fouling and no fouling are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Heat transfer coefficients and fouling factors are constant and uniform. 5 The thermal resistance of the inner tube is negligible since the tube is thin-walled and highly conductive. Analysis The tubes are said to be thin-walled, and thus it is reasonable to assume the inner and outer surface areas of the tubes to be equal. Then the heat transfer surface area becomes As DL (0.02 m)(60 m) 3.77 m2 The rate of heat transfer in this heat exchanger can be determined from · Q UAs F Tlm, CF where F is the correction factor and Tlm, CF is the log mean temperature difference for the counter-flow arrangement. These two quantities are determined from T1 T2 Tlm, CF Th, in Tc, out (80 50)°C Th, out Tc, in (40 20)°C T1 T2 30 20 ln ( T1/ T2) ln (30/20) 30°C 20°C 24.7°C Cold glycerin 20°C 40°C Hot water 80°C 50°C FIGURE 13–21 Schematic for Example 13–5. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 688 688 HEAT TRANSFER and P t2 T1 t1 t1 40 20 80 80 0.67 R T1 t2 T2 t1 20 40 50 80 0.75 uF 0.91 (Fig. 13–18b) (a) In the case of no fouling, the overall heat transfer coefficient U is determined from 1 U 1 hi 1 1 160 W/m2 · °C 1 ho 1 25 W/m2 · °C 21.6 W/m2 · °C Then the rate of heat transfer becomes · Q UAs F Tlm, CF (21.6 W/m2 · °C)(3.77m2)(0.91)(24.7°C) 1830 W (b) When there is fouling on one of the surfaces, the overall heat transfer coefficient U is U 1 hi 1 1 ho Rf 1 160 W/m2 · °C 1 1 25 W/m2 · °C 0.0006 m2 · °C/ W 21.3 W/m2 · °C The rate of heat transfer in this case becomes · Q UAs F Tlm, CF (21.3 W/m2 · °C)(3.77 m2)(0.91)(24.7°C) 1805 W Discussion Note that the rate of heat transfer decreases as a result of fouling, as expected. The decrease is not dramatic, however, because of the relatively low convection heat transfer coefficients involved. EXAMPLE 13–6 Cooling of an Automotive Radiator A test is conducted to determine the overall heat transfer coefficient in an automotive radiator that is a compact cross-flow water-to-air heat exchanger with both fluids (air and water) unmixed (Fig. 13–22). The radiator has 40 tubes of internal diameter 0.5 cm and length 65 cm in a closely spaced plate-finned matrix. Hot water enters the tubes at 90°C at a rate of 0.6 kg/s and leaves at 65°C. Air flows across the radiator through the interfin spaces and is heated from 20°C to 40°C. Determine the overall heat transfer coefficient Ui of this radiator based on the inner surface area of the tubes. SOLUTION During an experiment involving an automotive radiator, the inlet and exit temperatures of water and air and the mass flow rate of water are measured. The overall heat transfer coefficient based on the inner surface area is to be determined. Assumptions 1 Steady operating conditions exist. 2 Changes in the kinetic and potential energies of fluid streams are negligible. 3 Fluid properties are constant. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 689 689 CHAPTER 13 90°C Air flow (unmixed) 20°C 40°C 65°C Water flow (unmixed) Properties The specific heat of water at the average temperature of (90 65)/ 2 77.5°C is 4.195 kJ/kg · °C. Analysis The rate of heat transfer in this radiator from the hot water to the air is determined from an energy balance on water flow, · Q · [mCp (Tin Tout)]water (0.6 kg/s)(4.195 kJ/kg · °C)(90 65)°C 62.93 kW The tube-side heat transfer area is the total surface area of the tubes, and is determined from Ai n Di L (40) (0.005 m)(0.65 m) 0.408 m2 Knowing the rate of heat transfer and the surface area, the overall heat transfer coefficient can be determined from · Q Ui Ai F Tlm, CF → Ui · Q Ai F Tlm, CF where F is the correction factor and Tlm, CF is the log mean temperature difference for the counter-flow arrangement. These two quantities are found to be T1 T2 Tlm, CF Th, in Tc, out (90 40)°C Th, out Tc, in (65 20)°C T1 T2 50 45 ln ( T1/ T2) ln (50/45) 50°C 45°C 47.6°C and P t2 T1 t1 t1 65 20 90 90 0.36 R T1 t2 T2 t1 20 65 40 90 0.80 uF 0.97 (Fig. 13–18c) Substituting, the overall heat transfer coefficient Ui is determined to be Ui · Q Ai F Tlm, CF 62,930 W (0.408 m2)(0.97)(47.6°C) 3341 W/m2 · °C Note that the overall heat transfer coefficient on the air side will be much lower because of the large surface area involved on that side. FIGURE 13–22 Schematic for Example 13–6. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 690 690 HEAT TRANSFER 13–5 I THE EFFECTIVENESS–NTU METHOD The log mean temperature difference (LMTD) method discussed in Section 13–4 is easy to use in heat exchanger analysis when the inlet and the outlet temperatures of the hot and cold fluids are known or can be determined from an energy balance. Once Tlm, the mass flow rates, and the overall heat transfer coefficient are available, the heat transfer surface area of the heat exchanger can be determined from · Q UAs Tlm Therefore, the LMTD method is very suitable for determining the size of a heat exchanger to realize prescribed outlet temperatures when the mass flow rates and the inlet and outlet temperatures of the hot and cold fluids are specified. With the LMTD method, the task is to select a heat exchanger that will meet the prescribed heat transfer requirements. The procedure to be followed by the selection process is: 1. Select the type of heat exchanger suitable for the application. 2. Determine any unknown inlet or outlet temperature and the heat transfer rate using an energy balance. 3. Calculate the log mean temperature difference Tlm and the correction factor F, if necessary. 4. Obtain (select or calculate) the value of the overall heat transfer coefficient U. 5. Calculate the heat transfer surface area As . The task is completed by selecting a heat exchanger that has a heat transfer surface area equal to or larger than As . A second kind of problem encountered in heat exchanger analysis is the determination of the heat transfer rate and the outlet temperatures of the hot and cold fluids for prescribed fluid mass flow rates and inlet temperatures when the type and size of the heat exchanger are specified. The heat transfer surface area A of the heat exchanger in this case is known, but the outlet temperatures are not. Here the task is to determine the heat transfer performance of a specified heat exchanger or to determine if a heat exchanger available in storage will do the job. The LMTD method could still be used for this alternative problem, but the procedure would require tedious iterations, and thus it is not practical. In an attempt to eliminate the iterations from the solution of such problems, Kays and London came up with a method in 1955 called the effectiveness–NTU method, which greatly simplified heat exchanger analysis. This method is based on a dimensionless parameter called the heat transfer effectiveness , defined as · Q Qmax Actual heat transfer rate Maximum possible heat transfer rate (13-29) The actual heat transfer rate in a heat exchanger can be determined from an energy balance on the hot or cold fluids and can be expressed as · Q Cc(Tc, out Tc, in) Ch(Th, in Th, out) (13-30) cen58933_ch13.qxd 9/9/2002 9:57 AM Page 691 691 CHAPTER 13 · · where Cc mcCpc and Ch mcCph are the heat capacity rates of the cold and the hot fluids, respectively. To determine the maximum possible heat transfer rate in a heat exchanger, we first recognize that the maximum temperature difference in a heat exchanger is the difference between the inlet temperatures of the hot and cold fluids. That is, Tmax Th, in Tc, in (13-31) The heat transfer in a heat exchanger will reach its maximum value when (1) the cold fluid is heated to the inlet temperature of the hot fluid or (2) the hot fluid is cooled to the inlet temperature of the cold fluid. These two limiting conditions will not be reached simultaneously unless the heat capacity rates of the hot and cold fluids are identical (i.e., Cc Ch). When Cc Ch, which is usually the case, the fluid with the smaller heat capacity rate will experience a larger temperature change, and thus it will be the first to experience the maximum temperature, at which point the heat transfer will come to a halt. Therefore, the maximum possible heat transfer rate in a heat exchanger is (Fig. 13–23) · Q max Cmin(Th, in Tc, in) · where Cmin is the smaller of Ch mhCph and Cc clarified by the following example. EXAMPLE 13–7 20°C 25 kg/s Cold water (13-32) · mcCpc. This is further Hot oil 130°C 40 kg/s . Upper Limit for Heat Transfer in a Heat Exchanger Cold water enters a counter-flow heat exchanger at 10°C at a rate of 8 kg/s, where it is heated by a hot water stream that enters the heat exchanger at 70°C at a rate of 2 kg/s. Assuming the specific heat of water to remain constant at Cp 4.18 kJ/kg · °C, determine the maximum heat transfer rate and the outlet temperatures of the cold and the hot water streams for this limiting case. SOLUTION Cold and hot water streams enter a heat exchanger at specified temperatures and flow rates. The maximum rate of heat transfer in the heat exchanger is to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Heat transfer coefficients and fouling factors are constant and uniform. 5 The thermal resistance of the inner tube is negligible since the tube is thin-walled and highly conductive. Properties The specific heat of water is given to be Cp 4.18 kJ/kg · °C. Analysis A schematic of the heat exchanger is given in Figure 13–24. The heat capacity rates of the hot and cold fluids are determined from Ch · m hCph (2 kg/s)(4.18 kJ/kg · °C) 8.36 kW/°C Cc · m cCpc (8 kg/s)(4.18 kJ/kg · °C) 33.4 kW/°C Cc = mcCpc = 104.5 kW/°C . Ch = mcCph = 92 kW/°C Cmin = 92 kW/°C ∆Tmax = Th, in – Tc, in = 110°C . Qmax = Cmin ∆Tmax = 10,120 kW FIGURE 13–23 The determination of the maximum rate of heat transfer in a heat exchanger. 10°C 8 kg/s Cold water Hot water 70°C 2 kg/s and FIGURE 13–24 Schematic for Example 13–7. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 692 692 HEAT TRANSFER Therefore Cmin Ch 8.36 kW/°C which is the smaller of the two heat capacity rates. Then the maximum heat transfer rate is determined from Eq. 13-32 to be · Q max Cmin(Th, in Tc, in) (8.36 kW/°C)(70 10)°C 502 kW That is, the maximum possible heat transfer rate in this heat exchanger is 502 kW. This value would be approached in a counter-flow heat exchanger with a very large heat transfer surface area. The maximum temperature difference in this heat exchanger is Tmax Th, in Tc, in (70 10)°C 60°C. Therefore, the hot water cannot be cooled by more than 60°C (to 10°C) in this heat exchanger, and the cold water cannot be heated by more than 60°C (to 70°C), no matter what we do. The outlet temperatures of the cold and the hot streams in this limiting case are determined to be · Q Cold fluid Hot fluid . mh ,Cph . . Q = mh Cph ∆Th . = mc Cpc ∆Tc If . . mc Cpc = mh Cph Tc, in) → Tc, out Tc, in · Q . mc ,Cpc Cc(Tc, out Ch(Th, in Th, out) → Th, out Th, in · Q Cc · Q Ch 10°C 502 kW 33.4 kW/°C 25°C 70°C 502 kW 8.38 kW/°C 10°C Discussion Note that the hot water is cooled to the limit of 10°C (the inlet temperature of the cold water stream), but the cold water is heated to 25°C only when maximum heat transfer occurs in the heat exchanger. This is not surprising, since the mass flow rate of the hot water is only one-fourth that of the cold water, and, as a result, the temperature of the cold water increases by 0.25°C for each 1°C drop in the temperature of the hot water. You may be tempted to think that the cold water should be heated to 70°C in the limiting case of maximum heat transfer. But this will require the temperature of the hot water to drop to 170°C (below 10°C), which is impossible. Therefore, heat transfer in a heat exchanger reaches its maximum value when the fluid with the smaller heat capacity rate (or the smaller mass flow rate when both fluids have the same specific heat value) experiences the maximum temperature change. This example explains why we use Cmin in the evaluation of · Q max instead of Cmax. We can show that the hot water will leave at the inlet temperature of the cold water and vice versa in the limiting case of maximum heat transfer when the mass flow rates of the hot and cold water streams are identical (Fig. 13–25). We can also show that the outlet temperature of the cold water will reach the 70°C limit when the mass flow rate of the hot water is greater than that of the cold water. then ∆Th = ∆Tc FIGURE 13–25 The temperature rise of the cold fluid in a heat exchanger will be equal to the temperature drop of the hot fluid when the mass flow rates and the specific heats of the hot and cold fluids are identical. · The determination of Q max requires the availability of the inlet temperature of the hot and cold fluids and their mass flow rates, which are usually specified. Then, once the effectiveness of the heat exchanger is known, the actual · heat transfer rate Q can be determined from · Q · Q max Cmin(Th, in Tc, in) (13-33) cen58933_ch13.qxd 9/9/2002 9:57 AM Page 693 693 CHAPTER 13 Therefore, the effectiveness of a heat exchanger enables us to determine the heat transfer rate without knowing the outlet temperatures of the fluids. The effectiveness of a heat exchanger depends on the geometry of the heat exchanger as well as the flow arrangement. Therefore, different types of heat exchangers have different effectiveness relations. Below we illustrate the development of the effectiveness relation for the double-pipe parallel-flow heat exchanger. Equation 13-23 developed in Section 13–4 for a parallel-flow heat exchanger can be rearranged as ln Th, out Th, in Tc, out Tc, in UAs 1 Cc Cc Ch (13-34) Tc, in) (13-35) Also, solving Eq. 13-30 for Th, out gives Th, out Cc (T Ch c, out Th, in Substituting this relation into Eq. 13-34 after adding and subtracting Tc, in gives Th, in Tc, in Tc, in ln Cc (T Ch c, out Tc, out Th, in Tc, in) UAs 1 Cc Tc, in Cc Ch which simplifies to ln 1 1 Cc Tc, out Ch Th, in Tc, in Tc, in UAs 1 Cc Cc Ch (13-36) We now manipulate the definition of effectiveness to obtain · Q · Q max Cc(Tc, out Cmin(Th, in Tc, in) Tc, in) → Tc, out Th, in Tc, in Tc, in Cmin Cc Substituting this result into Eq. 13-36 and solving for gives the following relation for the effectiveness of a parallel-flow heat exchanger: 1 UAs 1 Cc exp parallel flow Cc Ch (13-37) Cc Cmin Ch Cc 1 Taking either Cc or Ch to be Cmin (both approaches give the same result), the relation above can be expressed more conveniently as 1 UAs 1 Cmin exp parallel flow 1 Cmin Cmax Cmin Cmax (13-38) Again Cmin is the smaller heat capacity ratio and Cmax is the larger one, and it makes no difference whether Cmin belongs to the hot or cold fluid. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 694 694 HEAT TRANSFER Effectiveness relations of the heat exchangers typically involve the dimensionless group UAs /Cmin. This quantity is called the number of transfer units NTU and is expressed as NTU UAs Cmin UAs ·C ) (m p min (13-39) where U is the overall heat transfer coefficient and As is the heat transfer surface area of the heat exchanger. Note that NTU is proportional to As . Therefore, for specified values of U and Cmin, the value of NTU is a measure of the heat transfer surface area As . Thus, the larger the NTU, the larger the heat exchanger. In heat exchanger analysis, it is also convenient to define another dimensionless quantity called the capacity ratio c as Cmin Cmax c (13-40) It can be shown that the effectiveness of a heat exchanger is a function of the number of transfer units NTU and the capacity ratio c. That is, function (UAs /Cmin, Cmin /Cmax) function (NTU, c) Effectiveness relations have been developed for a large number of heat exchangers, and the results are given in Table 13–4. The effectivenesses of some common types of heat exchangers are also plotted in Figure 13–26. More TABLE 13–4 Effectiveness relations for heat exchangers: NTU UAs /Cmin and · · c Cmin/Cmax (m Cp)min/(m Cp)max (Kays and London, Ref. 5.) Heat exchanger type 1 Double pipe: Parallel-flow Counter-flow 2 Shell and tube: One-shell pass 2, 4, . . . tube passes Effectiveness relation 1 exp [ NTU(1 c )] 1c 1 exp [ NTU(1 c )] 1 c exp [ NTU(1 c )] 21 c 1 c2 1 1 exp [ NTU exp [ NTU 1 1 3 Cross-flow (single-pass) Both fluids unmixed 1 Cmax mixed, Cmin unmixed 1 c (1 Cmin mixed, Cmax unmixed 1 exp 1 exp( NTU) 4 All heat exchangers with c 0 exp NTU0.22 [exp ( c NTU0.78) c exp {1 c[1 1 c [1 exp ( NTU)]}) exp ( c NTU)] 1] c 2] c 2] 1 9:57 AM Page 695 695 CHAPTER 13 100 100 5 0.71.00 / mi n 5 5 0.2 C Tube fluid 60 =0 0.5 0 Effectiveness ε, % mi n ax Cm 80 0.25 0.50 0.75 1.00 60 40 =0 / ax Cm C Effectiveness ε, % 80 Shell fluid 40 Tube fluid 20 20 Shell fluid 0 3 4 5 1 2 Number of transfer units NTU = AsU/Cmin (a) Parallel-flow (b) Counter-flow 100 n Shell fluid Effectiveness ε, % 40 C 60 80 0.50 0.75 1.00 m /C 100 =0 0.25 x ma m i 80 3 4 5 1 2 Number of transfer units NTU = AsU/Cmin C 0 Effectiveness ε, % =0 x C ma 25 / 0. 0 in 0.5 0.75 1.00 60 Shell fluid 40 20 20 Tube fluid Tube fluid 1 2 3 4 0 5 Number of transfer units NTU = AsU/Cmin (c) One-shell pass and 2, 4, 6, … tube passes 60 =0 5 0.2 0 0.5 5 0.7 00 1. 80 5 d xe mi un , =0 0.25 4 0.5 2 0.75 1.33 1 Hot fluid C mi xe Cold fluid 40 4 (d ) Two-shell passes and 4, 8, 12, … tube passes Effectiveness ε, % mi n / ax Cm 3 2 Number of transfer units NTU = AsU/Cmin 100 100 80 1 d /C 0 C 9/9/2002 Effectiveness ε, % cen58933_ch13.qxd 60 Mixed fluid 40 20 20 0 0 Unmixed fluid 1 2 3 4 5 Number of transfer units NTU = AsU/Cmin (e) Cross-flow with both fluids unmixed 1 2 3 4 5 Number of transfer units NTU = AsU/Cmin ( f ) Cross-flow with one fluid mixed and the other unmixed FIGURE 13–26 Effectiveness for heat exchangers (from Kays and London, Ref. 5). cen58933_ch13.qxd 9/9/2002 9:57 AM Page 696 696 HEAT TRANSFER extensive effectiveness charts and relations are available in the literature. The dashed lines in Figure 13–26f are for the case of Cmin unmixed and Cmax mixed and the solid lines are for the opposite case. The analytic relations for the effectiveness give more accurate results than the charts, since reading errors in charts are unavoidable, and the relations are very suitable for computerized analysis of heat exchangers. We make these following observations from the effectiveness relations and charts already given: 1. The value of the effectiveness ranges from 0 to 1. It increases rapidly 1 Counter-flow ε Cross-flow with both fluids unmixed 0.5 Parallel-flow (for c = 1) 0 0 1 2 3 4 NTU = UAs /Cmin 5 FIGURE 13–27 For a specified NTU and capacity ratio c, the counter-flow heat exchanger has the highest effectiveness and the parallel-flow the lowest. ε 0 ε = 1 – e– NTU (All heat exchangers with c = 0) 0 1 2 3 4 NTU = UAs /Cmin FIGURE 13–28 The effectiveness relation reduces to 1 exp( NTU) for all max heat exchangers when the capacity ratio c 0. max 1 exp( NTU) (13-41) regardless of the type of heat exchanger (Fig. 13–28). Note that the temperature of the condensing or boiling fluid remains constant in this case. The effectiveness is the lowest in the other limiting case of c Cmin/Cmax 1, which is realized when the heat capacity rates of the two fluids are equal. 1 0.5 with NTU for small values (up to about NTU 1.5) but rather slowly for larger values. Therefore, the use of a heat exchanger with a large NTU (usually larger than 3) and thus a large size cannot be justified economically, since a large increase in NTU in this case corresponds to a small increase in effectiveness. Thus, a heat exchanger with a very high effectiveness may be highly desirable from a heat transfer point of view but rather undesirable from an economical point of view. 2. For a given NTU and capacity ratio c Cmin /Cmax, the counter-flow heat exchanger has the highest effectiveness, followed closely by the cross-flow heat exchangers with both fluids unmixed. As you might expect, the lowest effectiveness values are encountered in parallel-flow heat exchangers (Fig. 13–27). 3. The effectiveness of a heat exchanger is independent of the capacity ratio c for NTU values of less than about 0.3. 4. The value of the capacity ratio c ranges between 0 and 1. For a given NTU, the effectiveness becomes a maximum for c 0 and a minimum for c 1. The case c Cmin /Cmax → 0 corresponds to Cmax → , which is realized during a phase-change process in a condenser or boiler. All effectiveness relations in this case reduce to 5 Once the quantities c Cmin /Cmax and NTU UAs /Cmin have been evaluated, the effectiveness can be determined from either the charts or (preferably) the effectiveness relation for the specified type of heat exchanger. Then · the rate of heat transfer Q and the outlet temperatures Th, out and Tc, out can be determined from Eqs. 13-33 and 13-30, respectively. Note that the analysis of heat exchangers with unknown outlet temperatures is a straightforward matter with the effectiveness–NTU method but requires rather tedious iterations with the LMTD method. We mentioned earlier that when all the inlet and outlet temperatures are specified, the size of the heat exchanger can easily be determined using the LMTD method. Alternatively, it can also be determined from the effectiveness–NTU method by first evaluating the effectiveness from its definition (Eq. 13-29) and then the NTU from the appropriate NTU relation in Table 13–5. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 697 697 CHAPTER 13 TABLE 13–5 NTU relations for heat exchangers NTU UAs /Cmin and c · · (m Cp )min/(m Cp )max (Kays and London, Ref. 5.) Heat exchanger type NTU relation 1 Double-pipe: Parallel-flow NTU Counter-flow NTU Cmin /Cmax 2 Shell and tube: One-shell pass 2, 4, . . . tube passes 1 c NTU Cmin mixed, Cmax unmixed 4 All heat exchangers with c 0 (1 c 1 NTU 3 Cross-flow (single-pass) Cmax mixed, Cmin unmixed ln [1 1 ln ln 1 1 1 c 1 1 c )] c 2 ln ln (1 NTU 1 1 c c 1 1 c2 c2 c) c ln [c ln (1 c ln(1 ) NTU 2/ 2/ ) 1] Note that the relations in Table 13–5 are equivalent to those in Table 13–4. Both sets of relations are given for convenience. The relations in Table 13–4 give the effectiveness directly when NTU is known, and the relations in Table 13–5 give the NTU directly when the effectiveness is known. EXAMPLE 13–8 Using the Effectiveness–NTU Method Repeat Example 13–4, which was solved with the LMTD method, using the effectiveness–NTU method. SOLUTION The schematic of the heat exchanger is redrawn in Figure 13–29, and the same assumptions are utilized. Analysis In the effectiveness–NTU method, we first determine the heat capacity rates of the hot and cold fluids and identify the smaller one: Ch Cc · mhCph · mcCpc (2 kg/s)(4.31 kJ/kg · °C) 8.62 kW/°C (1.2 kg/s)(4.18 kJ/kg · °C) 5.02 kW/°C Cold water 20°C 1.2 kg/s Hot geothermal 160°C brine 2 kg/s 80°C D = 1.5 cm Therefore, Cmin Cc FIGURE 13–29 Schematic for Example 13–8. 5.02 kW/°C and c Cmin /Cmax 5.02/8.62 0.583 Then the maximum heat transfer rate is determined from Eq. 13-32 to be · Q max Cmin(Th, in Tc, in) (5.02 kW/°C)(160 702.8 kW 20)°C cen58933_ch13.qxd 9/9/2002 9:57 AM Page 698 698 HEAT TRANSFER That is, the maximum possible heat transfer rate in this heat exchanger is 702.8 kW. The actual rate of heat transfer in the heat exchanger is · Q · [mCp(Tout Tin)]water (1.2 kg/s)(4.18 kJ/kg · °C)(80 20)°C 301.0 kW Thus, the effectiveness of the heat exchanger is · Q · Q max 301.0 kW 702.8 kW 0.428 Knowing the effectiveness, the NTU of this counter-flow heat exchanger can be determined from Figure 13–26b or the appropriate relation from Table 13–5. We choose the latter approach for greater accuracy: NTU 1 c 1 ln c 1 1 1 0.583 1 ln 0.428 1 0.428 0.583 0.651 1 Then the heat transfer surface area becomes NTU UAs Cmin → As NTU Cmin U (0.651)(5020 W/°C) 640 W/m2 · °C 5.11 m2 To provide this much heat transfer surface area, the length of the tube must be As DL → L As D 5.11 m2 (0.015 m) 108 m Discussion Note that we obtained the same result with the effectiveness–NTU method in a systematic and straightforward manner. EXAMPLE 13–9 150°C Hot oil is to be cooled by water in a 1-shell-pass and 8-tube-passes heat exchanger. The tubes are thin-walled and are made of copper with an internal diameter of 1.4 cm. The length of each tube pass in the heat exchanger is 5 m, and the overall heat transfer coefficient is 310 W/m2 · °C. Water flows through the tubes at a rate of 0.2 kg/s, and the oil through the shell at a rate of 0.3 kg/s. The water and the oil enter at temperatures of 20°C and 150°C, respectively. Determine the rate of heat transfer in the heat exchanger and the outlet temperatures of the water and the oil. Oil 0.3 kg/s 20°C Water 0.2 kg/s FIGURE 13–30 Schematic for Example 13–9. Cooling Hot Oil by Water in a Multipass Heat Exchanger SOLUTION Hot oil is to be cooled by water in a heat exchanger. The mass flow rates and the inlet temperatures are given. The rate of heat transfer and the outlet temperatures are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 The thickness of the tube is negligible since it is thin-walled. 4 Changes in the kinetic and potential energies of fluid streams are negligible. 5 The overall heat transfer coefficient is constant and uniform. Analysis The schematic of the heat exchanger is given in Figure 13–30. The outlet temperatures are not specified, and they cannot be determined from an cen58933_ch13.qxd 9/9/2002 9:57 AM Page 699 699 CHAPTER 13 energy balance. The use of the LMTD method in this case will involve tedious iterations, and thus the –NTU method is indicated. The first step in the –NTU method is to determine the heat capacity rates of the hot and cold fluids and identify the smaller one: Ch Cc · mhCph · mcCpc (0.3 kg/s)(2.13 kJ/kg · °C) (0.2 kg/s)(4.18 kJ/kg · °C) 0.639 kW/°C 0.836 kW/°C Therefore, Cmin Ch 0.639 kW/°C and c Cmin Cmax 0.639 0.836 0.764 Then the maximum heat transfer rate is determined from Eq. 13-32 to be · Q max Cmin(Th, in Tc, in) (0.639 kW/°C)(150 20)°C 83.1 kW That is, the maximum possible heat transfer rate in this heat exchanger is 83.1 kW. The heat transfer surface area is As n( DL) 8 (0.014 m)(5 m) 1.76 m2 Then the NTU of this heat exchanger becomes NTU UAs Cmin (310 W/m2 · °C)(1.76 m2) 639 W/°C 0.853 The effectiveness of this heat exchanger corresponding to c NTU 0.853 is determined from Figure 13–26c to be 0.764 and 0.47 We could also determine the effectiveness from the third relation in Table 13–4 more accurately but with more labor. Then the actual rate of heat transfer becomes · Q · Q max (0.47)(83.1 kW) 39.1 kW Finally, the outlet temperatures of the cold and the hot fluid streams are determined to be · Q · Q Cc(Tc, out Tc, in) → Tc, out Tc, in Cc 39.1 kW 66.8°C 20°C 0.836 kW/°C · Q · Q Ch(Th, in Th, out) → Th, out Th, in Ch 150°C 39.1 kW 0.639 kW/°C 88.8°C Therefore, the temperature of the cooling water will rise from 20°C to 66.8°C as it cools the hot oil from 150°C to 88.8°C in this heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 700 700 HEAT TRANSFER 13–6 I SELECTION OF HEAT EXCHANGERS Heat exchangers are complicated devices, and the results obtained with the simplified approaches presented above should be used with care. For example, we assumed that the overall heat transfer coefficient U is constant throughout the heat exchanger and that the convection heat transfer coefficients can be predicted using the convection correlations. However, it should be kept in mind that the uncertainty in the predicted value of U can even exceed 30 percent. Thus, it is natural to tend to overdesign the heat exchangers in order to avoid unpleasant surprises. Heat transfer enhancement in heat exchangers is usually accompanied by increased pressure drop, and thus higher pumping power. Therefore, any gain from the enhancement in heat transfer should be weighed against the cost of the accompanying pressure drop. Also, some thought should be given to which fluid should pass through the tube side and which through the shell side. Usually, the more viscous fluid is more suitable for the shell side (larger passage area and thus lower pressure drop) and the fluid with the higher pressure for the tube side. Engineers in industry often find themselves in a position to select heat exchangers to accomplish certain heat transfer tasks. Usually, the goal is to heat or cool a certain fluid at a known mass flow rate and temperature to a desired temperature. Thus, the rate of heat transfer in the prospective heat exchanger is · Q max · mCp(Tin Tout) which gives the heat transfer requirement of the heat exchanger before having any idea about the heat exchanger itself. An engineer going through catalogs of heat exchanger manufacturers will be overwhelmed by the type and number of readily available off-the-shelf heat exchangers. The proper selection depends on several factors. Heat Transfer Rate This is the most important quantity in the selection of a heat exchanger. A heat exchanger should be capable of transferring heat at the specified rate in order to achieve the desired temperature change of the fluid at the specified mass flow rate. Cost Budgetary limitations usually play an important role in the selection of heat exchangers, except for some specialized cases where “money is no object.” An off-the-shelf heat exchanger has a definite cost advantage over those made to order. However, in some cases, none of the existing heat exchangers will do, and it may be necessary to undertake the expensive and time-consuming task of designing and manufacturing a heat exchanger from scratch to suit the needs. This is often the case when the heat exchanger is an integral part of the overall device to be manufactured. The operation and maintenance costs of the heat exchanger are also important considerations in assessing the overall cost. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 701 701 CHAPTER 13 Pumping Power In a heat exchanger, both fluids are usually forced to flow by pumps or fans that consume electrical power. The annual cost of electricity associated with the operation of the pumps and fans can be determined from Operating cost (Pumping power, kW) (Hours of operation, h) (Price of electricity, $/kWh) where the pumping power is the total electrical power consumed by the motors of the pumps and fans. For example, a heat exchanger that involves a 1-hp pump and a 1 -hp fan (1 hp 0.746 kW) operating 8 h a day and 5 days 3 a week will consume 2017 kWh of electricity per year, which will cost $161.4 at an electricity cost of 8 cents/kWh. Minimizing the pressure drop and the mass flow rate of the fluids will minimize the operating cost of the heat exchanger, but it will maximize the size of the heat exchanger and thus the initial cost. As a rule of thumb, doubling the mass flow rate will reduce the initial cost by half but will increase the pumping power requirements by a factor of roughly eight. Typically, fluid velocities encountered in heat exchangers range between 0.7 and 7 m/s for liquids and between 3 and 30 m/s for gases. Low velocities are helpful in avoiding erosion, tube vibrations, and noise as well as pressure drop. Size and Weight Normally, the smaller and the lighter the heat exchanger, the better it is. This is especially the case in the automotive and aerospace industries, where size and weight requirements are most stringent. Also, a larger heat exchanger normally carries a higher price tag. The space available for the heat exchanger in some cases limits the length of the tubes that can be used. Type The type of heat exchanger to be selected depends primarily on the type of fluids involved, the size and weight limitations, and the presence of any phasechange processes. For example, a heat exchanger is suitable to cool a liquid by a gas if the surface area on the gas side is many times that on the liquid side. On the other hand, a plate or shell-and-tube heat exchanger is very suitable for cooling a liquid by another liquid. Materials The materials used in the construction of the heat exchanger may be an important consideration in the selection of heat exchangers. For example, the thermal and structural stress effects need not be considered at pressures below 15 atm or temperatures below 150°C. But these effects are major considerations above 70 atm or 550°C and seriously limit the acceptable materials of the heat exchanger. A temperature difference of 50°C or more between the tubes and the shell will probably pose differential thermal expansion problems and needs to be considered. In the case of corrosive fluids, we may have to select expensive cen58933_ch13.qxd 9/9/2002 9:57 AM Page 702 702 HEAT TRANSFER corrosion-resistant materials such as stainless steel or even titanium if we are not willing to replace low-cost heat exchangers frequently. Other Considerations There are other considerations in the selection of heat exchangers that may or may not be important, depending on the application. For example, being leak-tight is an important consideration when toxic or expensive fluids are involved. Ease of servicing, low maintenance cost, and safety and reliability are some other important considerations in the selection process. Quietness is one of the primary considerations in the selection of liquid-to-air heat exchangers used in heating and air-conditioning applications. EXAMPLE 13–10 Installing a Heat Exchanger to Save Energy and Money In a dairy plant, milk is pasteurized by hot water supplied by a natural gas furnace. The hot water is then discharged to an open floor drain at 80°C at a rate of 15 kg/min. The plant operates 24 h a day and 365 days a year. The furnace has an efficiency of 80 percent, and the cost of the natural gas is $0.40 per therm (1 therm 105,500 kJ). The average temperature of the cold water entering the furnace throughout the year is 15°C. The drained hot water cannot be returned to the furnace and recirculated, because it is contaminated during the process. In order to save energy, installation of a water-to-water heat exchanger to preheat the incoming cold water by the drained hot water is proposed. Assuming that the heat exchanger will recover 75 percent of the available heat in the hot water, determine the heat transfer rating of the heat exchanger that needs to be purchased and suggest a suitable type. Also, determine the amount of money this heat exchanger will save the company per year from natural gas savings. SOLUTION A water-to-water heat exchanger is to be installed to transfer energy Hot 80°C water Cold water 15°C FIGURE 13–31 Schematic for Example 13–10. from drained hot water to the incoming cold water to preheat it. The rate of heat transfer in the heat exchanger and the amount of energy and money saved per year are to be determined. Assumptions 1 Steady operating conditions exist. 2 The effectiveness of the heat exchanger remains constant. Properties We use the specific heat of water at room temperature, Cp 4.18 kJ/ kg · °C (Table A–9), and treat it as a constant. Analysis A schematic of the prospective heat exchanger is given in Figure 13–31. The heat recovery from the hot water will be a maximum when it leaves the heat exchanger at the inlet temperature of the cold water. Therefore, · Q max · mhCp(Th, in Tc, in) 15 kg/s (4.18 kJ/kg · °C)(80 60 67.9 kJ/s 15)°C That is, the existing hot water stream has the potential to supply heat at a rate of 67.9 kJ/s to the incoming cold water. This value would be approached in a counter-flow heat exchanger with a very large heat transfer surface area. A heat exchanger of reasonable size and cost can capture 75 percent of this heat cen58933_ch13.qxd 9/9/2002 9:57 AM Page 703 703 CHAPTER 13 transfer potential. Thus, the heat transfer rating of the prospective heat exchanger must be · Q · Q max (0.75)(67.9 kJ/s) 50.9 kJ/s That is, the heat exchanger should be able to deliver heat at a rate of 50.9 kJ/s from the hot to the cold water. An ordinary plate or shell-and-tube heat exchanger should be adequate for this purpose, since both sides of the heat exchanger involve the same fluid at comparable flow rates and thus comparable heat transfer coefficients. (Note that if we were heating air with hot water, we would have to specify a heat exchanger that has a large surface area on the air side.) The heat exchanger will operate 24 h a day and 365 days a year. Therefore, the annual operating hours are Operating hours (24 h/day)(365 days/year) 8760 h/year Noting that this heat exchanger saves 50.9 kJ of energy per second, the energy saved during an entire year will be Energy saved (Heat transfer rate)(Operation time) (50.9 kJ/s)(8760 h/year)(3600 s/h) 1.605 109 kJ/year The furnace is said to be 80 percent efficient. That is, for each 80 units of heat supplied by the furnace, natural gas with an energy content of 100 units must be supplied to the furnace. Therefore, the energy savings determined above result in fuel savings in the amount of Fuel saved Energy saved Furnace efficiency 19,020 therms/year 1.605 109 kJ/year 1 therm 0.80 105,500 kJ Noting that the price of natural gas is $0.40 per therm, the amount of money saved becomes Money saved (Fuel saved) (Price of fuel) (19,020 therms/year)($0.40/therm) $7607/ year Therefore, the installation of the proposed heat exchanger will save the company $7607 a year, and the installation cost of the heat exchanger will probably be paid from the fuel savings in a short time. SUMMARY Heat exchangers are devices that allow the exchange of heat between two fluids without allowing them to mix with each other. Heat exchangers are manufactured in a variety of types, the simplest being the double-pipe heat exchanger. In a parallel-flow type, both the hot and cold fluids enter the heat exchanger at the same end and move in the same direction, whereas in a counter-flow type, the hot and cold fluids enter the heat exchanger at opposite ends and flow in opposite directions. In compact heat exchangers, the two fluids move perpendicular to each other, and such a flow configuration is called cross-flow. Other common types of heat exchangers in industrial applications are the plate and the shell-and-tube heat exchangers. Heat transfer in a heat exchanger usually involves convection in each fluid and conduction through the wall separating the two fluids. In the analysis of heat exchangers, it is convenient to cen58933_ch13.qxd 9/9/2002 9:57 AM Page 704 704 HEAT TRANSFER work with an overall heat transfer coefficient U or a total thermal resistance R, expressed as 1 UAs 1 Ui Ai 1 Uo Ao R 1 hi Ai 1 ho Ao Rwall where the subscripts i and o stand for the inner and outer surfaces of the wall that separates the two fluids, respectively. When the wall thickness of the tube is small and the thermal conductivity of the tube material is high, the last relation simplifies to 1 U 1 hi 1 Ui Ai 1 hi Ai 1 R Uo Ao Rf, i ln (Do /Di ) Ai 2 kL where Rf, o Ao is the log mean temperature difference, which is the suitable form of the average temperature difference for use in the analysis of heat exchangers. Here T1 and T2 represent the temperature differences between the two fluids at the two ends (inlet and outlet) of the heat exchanger. For cross-flow and multipass shell-and-tube heat exchangers, the logarithmic mean temperature difference is related to the counter-flow one Tlm, CF as 1 ho Ao Di L and Ao Do L are the areas of the inner and where Ai outer surfaces and Rf, i and Rf, o are the fouling factors at those surfaces. In a well-insulated heat exchanger, the rate of heat transfer from the hot fluid is equal to the rate of heat transfer to the cold one. That is, · · Q mcCpc(Tc, out Tc, in) Cc(Tc, out Tc, in) Tlm · mhCph(Th, in Th, out) Ch(Th, in Th, out) where the subscripts c and h stand for the cold and hot fluids, respectively, and the product of the mass flow rate and the spe· cific heat of a fluid mCp is called the heat capacity rate. Of the two methods used in the analysis of heat exchangers, the log mean temperature difference (or LMTD) method is F Tlm, CF where F is the correction factor, which depends on the geometry of the heat exchanger and the inlet and outlet temperatures of the hot and cold fluid streams. The effectiveness of a heat exchanger is defined as · Q Actual heat transfer rate Qmax Maximum possible heat transfer rate where · Q max and · Q T1 T2 ln ( T1/ T2) Tlm 1 ho where U Ui Uo. The effects of fouling on both the inner and the outer surfaces of the tubes of a heat exchanger can be accounted for by 1 UAs best suited for determining the size of a heat exchanger when all the inlet and the outlet temperatures are known. The effectiveness–NTU method is best suited to predict the outlet temperatures of the hot and cold fluid streams in a specified heat exchanger. In the LMTD method, the rate of heat transfer is determined from · Q UAs Tlm Cmin(Th, in Tc, in) · · and Cmin is the smaller of Ch mhCph and Cc mcCpc. The effectiveness of heat exchangers can be determined from effectiveness relations or charts. The selection or design of a heat exchanger depends on several factors such as the heat transfer rate, cost, pressure drop, size, weight, construction type, materials, and operating environment. REFERENCES AND SUGGESTED READING 1. N. Afgan and E. U. Schlunder. Heat Exchanger: Design and Theory Sourcebook. Washington D.C.: McGrawHill/Scripta, 1974. 4. K. A. Gardner. “Variable Heat Transfer Rate Correction in Multipass Exchangers, Shell Side Film Controlling.” Transactions of the ASME 67 (1945), pp. 31–38. 2. R. A. Bowman, A. C. Mueller, and W. M. Nagle. “Mean Temperature Difference in Design.” Transactions of the ASME 62 (1940), p. 283. 5. W. M. Kays and A. L. London. Compact Heat Exchangers. 3rd ed. New York: McGraw-Hill, 1984. 3. A. P. Fraas. Heat Exchanger Design. 2d ed. New York: John Wiley & Sons, 1989. 6. W. M. Kays and H. C. Perkins. In Handbook of Heat Transfer, ed. W. M. Rohsenow and J. P. Hartnett. New York: McGraw-Hill, 1972, Chap. 7. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 705 705 CHAPTER 13 7. A. C. Mueller. “Heat Exchangers.” In Handbook of Heat Transfer, ed. W. M. Rohsenow and J. P. Hartnett. New York: McGraw-Hill, 1972, Chap. 18. 8. M. N. Özisik. Heat Transfer—A Basic Approach. New , York: McGraw-Hill, 1985. 9. E. U. Schlunder. Heat Exchanger Design Handbook. Washington, D.C.: Hemisphere, 1982. 10. Standards of Tubular Exchanger Manufacturers Association. New York: Tubular Exchanger Manufacturers Association, latest ed. 11. R. A. Stevens, J. Fernandes, and J. R. Woolf. “Mean Temperature Difference in One, Two, and Three Pass Crossflow Heat Exchangers.” Transactions of the ASME 79 (1957), pp. 287–297. 12. J. Taborek, G. F. Hewitt, and N. Afgan. Heat Exchangers: Theory and Practice. New York: Hemisphere, 1983. 13. G. Walker. Industrial Heat Exchangers. Washington, D.C.: Hemisphere, 1982. PROBLEMS* Types of Heat Exchangers 13–1C Classify heat exchangers according to flow type and explain the characteristics of each type. 13–2C Classify heat exchangers according to construction type and explain the characteristics of each type. 13–3C When is a heat exchanger classified as being compact? Do you think a double-pipe heat exchanger can be classified as a compact heat exchanger? 13–4C How does a cross-flow heat exchanger differ from a counter-flow one? What is the difference between mixed and unmixed fluids in cross-flow? 13–10C Under what conditions is the thermal resistance of the tube in a heat exchanger negligible? 13–11C Consider a double-pipe parallel-flow heat exchanger of length L. The inner and outer diameters of the inner tube are D1 and D2, respectively, and the inner diameter of the outer tube is D3. Explain how you would determine the two heat transfer surface areas Ai and Ao. When is it reasonable to assume Ai Ao As? 13–12C Is the approximation hi ho h for the convection heat transfer coefficient in a heat exchanger a reasonable one when the thickness of the tube wall is negligible? 13–5C What is the role of the baffles in a shell-and-tube heat exchanger? How does the presence of baffles affect the heat transfer and the pumping power requirements? Explain. 13–13C Under what conditions can the overall heat transfer coefficient of a heat exchanger be determined from U (1/hi 1/ho) 1? 13–6C Draw a 1-shell-pass and 6-tube-passes shell-and-tube heat exchanger. What are the advantages and disadvantages of using 6 tube passes instead of just 2 of the same diameter? 13–14C What are the restrictions on the relation UAs Ui Ai Uo Ao for a heat exchanger? Here As is the heat transfer surface area and U is the overall heat transfer coefficient. 13–7C Draw a 2-shell-passes and 8-tube-passes shell-andtube heat exchanger. What is the primary reason for using so many tube passes? 13–15C In a thin-walled double-pipe heat exchanger, when is the approximation U hi a reasonable one? Here U is the overall heat transfer coefficient and hi is the convection heat transfer coefficient inside the tube. 13–8C What is a regenerative heat exchanger? How does a static type of regenerative heat exchanger differ from a dynamic type? The Overall Heat Transfer Coefficient 13–9C What are the heat transfer mechanisms involved during heat transfer from the hot to the cold fluid? *Problems designated by a “C” are concept questions, and students are encouraged to answer them all. Problems designated by an “E” are in English units, and the SI users can ignore them. Problems with an EES-CD icon are solved using EES, and complete solutions together with parametric studies are included on the enclosed CD. Problems with a computer-EES icon are comprehensive in nature, and are intended to be solved with a computer, preferably using the EES software that accompanies this text. 13–16C What are the common causes of fouling in a heat exchanger? How does fouling affect heat transfer and pressure drop? 13–17C How is the thermal resistance due to fouling in a heat exchanger accounted for? How do the fluid velocity and temperature affect fouling? 13–18 A double-pipe heat exchanger is constructed of a copper (k 380 W/m · °C) inner tube of internal diameter Di 1.2 cm and external diameter Do 1.6 cm and an outer tube of diameter 3.0 cm. The convection heat transfer coefficient is reported to be hi 700 W/m2 · °C on the inner surface of the tube and ho 1400 W/m2 · °C on its outer surface. For a fouling factor Rf, i 0.0005 m2 · °C/W on the tube side and Rf, o 0.0002 m2 · °C/W on the shell side, determine (a) the thermal resistance of the heat exchanger per unit length and (b) the cen58933_ch13.qxd 9/9/2002 9:57 AM Page 706 706 HEAT TRANSFER overall heat transfer coefficients Ui and Uo based on the inner and outer surface areas of the tube, respectively. 13–19 Reconsider Problem 13–18. Using EES (or other) software, investigate the effects of pipe conductivity and heat transfer coefficients on the thermal resistance of the heat exchanger. Let the thermal conductivity vary from 10 W/m · ºC to 400 W/m · ºC, the convection heat transfer coefficient from 500 W/m2 · ºC to 1500 W/m2 · ºC on the inner surface, and from 1000 W/m2 · ºC to 2000 W/m2 · ºC on the outer surface. Plot the thermal resistance of the heat exchanger as functions of thermal conductivity and heat transfer coefficients, and discuss the results. 13–20 Water at an average temperature of 107°C and an average velocity of 3.5 m/s flows through a 5-m-long stainless steel tube (k 14.2 W/m · °C) in a boiler. The inner and outer diameters of the tube are Di 1.0 cm and Do 1.4 cm, respectively. If the convection heat transfer coefficient at the outer surface of the tube where boiling is taking place is ho 8400 W/m2 · °C, determine the overall heat transfer coefficient Ui of this boiler based on the inner surface area of the tube. 13–21 Repeat Problem 13–20, assuming a fouling factor Rf, i 0.0005 m2 · °C/W on the inner surface of the tube. 13–22 Reconsider Problem 13–21. Using EES (or other) software, plot the overall heat transfer coefficient based on the inner surface as a function of fouling factor Fi as it varies from 0.0001 m2 · ºC/W to 0.0008 m2 · ºC/W, and discuss the results. 13–23 A long thin-walled double-pipe heat exchanger with tube and shell diameters of 1.0 cm and 2.5 cm, respectively, is used to condense refrigerant 134a by water at 20°C. The refrigerant flows through the tube, with a convection heat transfer coefficient of hi 5000 W/m2 · °C. Water flows through the shell at a rate of 0.3 kg/s. Determine the overall heat transfer Answer: 2020 W/m2 · °C coefficient of this heat exchanger. 13–24 Repeat Problem 13–23 by assuming a 2-mm-thick layer of limestone (k 1.3 W/m · °C) forms on the outer surface of the inner tube. 13–25 Reconsider Problem 13–24. Using EES (or other) software, plot the overall heat transfer coefficient as a function of the limestone thickness as it varies from 1 mm to 3 mm, and discuss the results. 13–26E Water at an average temperature of 140°F and an average velocity of 8 ft/s flows through a thin-walled 3 -in.4 diameter tube. The water is cooled by air that flows across the 12 ft/s at an average temperature tube with a velocity of of 80°F. Determine the overall heat transfer coefficient. Analysis of Heat Exchangers 13–27C What are the common approximations made in the analysis of heat exchangers? 13–28C Under what conditions is the heat transfer relation · · · Q mcCpc(Tc, out Tc, in) mhCph (Th, in Th, out) valid for a heat exchanger? 13–29C What is the heat capacity rate? What can you say about the temperature changes of the hot and cold fluids in a heat exchanger if both fluids have the same capacity rate? What does a heat capacity of infinity for a fluid in a heat exchanger mean? 13–30C Consider a condenser in which steam at a specified temperature is condensed by rejecting heat to the cooling water. If the heat transfer rate in the condenser and the temperature rise of the cooling water is known, explain how the rate of condensation of the steam and the mass flow rate of the cooling water can be determined. Also, explain how the total thermal resistance R of this condenser can be evaluated in this case. 13–31C Under what conditions will the temperature rise of the cold fluid in a heat exchanger be equal to the temperature drop of the hot fluid? The Log Mean Temperature Difference Method · 13–32C In the heat transfer relation Q UAs Tlm for a heat exchanger, what is Tlm called? How is it calculated for a parallel-flow and counter-flow heat exchanger? 13–33C How does the log mean temperature difference for a heat exchanger differ from the arithmetic mean temperature difference (AMTD)? For specified inlet and outlet temperatures, which one of these two quantities is larger? 13–34C The temperature difference between the hot and cold fluids in a heat exchanger is given to be T1 at one end and T2 at the other end. Can the logarithmic temperature difference Tlm of this heat exchanger be greater than both T1 and T2? Explain. 13–35C Can the logarithmic mean temperature difference Tlm of a heat exchanger be a negative quantity? Explain. 13–36C Can the outlet temperature of the cold fluid in a heat exchanger be higher than the outlet temperature of the hot fluid in a parallel-flow heat exchanger? How about in a counter-flow heat exchanger? Explain. 13–37C For specified inlet and outlet temperatures, for what kind of heat exchanger will the Tlm be greatest: double-pipe parallel-flow, double-pipe counter-flow, cross-flow, or multipass shell-and-tube heat exchanger? · 13–38C In the heat transfer relation Q UAs F Tlm for a heat exchanger, what is the quantity F called? What does it represent? Can F be greater than one? 13–39C When the outlet temperatures of the fluids in a heat exchanger are not known, is it still practical to use the LMTD method? Explain. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 707 707 CHAPTER 13 13–40C Explain how the LMTD method can be used to determine the heat transfer surface area of a multipass shell-andtube heat exchanger when all the necessary information, including the outlet temperatures, is given. 13–41 Steam in the condenser of a steam power plant is to be 2305 kJ/kg) with condensed at a temperature of 50°C (hfg cooling water (Cp 4180 J/kg · °C) from a nearby lake, which enters the tubes of the condenser at 18°C and leaves at 27°C. The surface area of the tubes is 58 m2, and the overall heat transfer coefficient is 2400 W/m2 · °C. Determine the mass flow rate of the cooling water needed and the rate of condensation of Answers: 101 kg/s, 1.65 kg/s the steam in the condenser. Steam 50°C 27°C 13–45 A test is conducted to determine the overall heat transfer coefficient in a shell-and-tube oil-to-water heat exchanger that has 24 tubes of internal diameter 1.2 cm and length 2 m in a single shell. Cold water (Cp 4180 J/kg · °C) enters the tubes at 20°C at a rate of 5 kg/s and leaves at 55°C. Oil 2150 J/kg · °C) flows through the shell and is cooled (Cp from 120°C to 45°C. Determine the overall heat transfer coefficient Ui of this heat exchanger based on the inner surface area Answer: 13.9 kW/m2 · °C of the tubes. 13–46 A double-pipe counter-flow heat exchanger is to cool 2560 J/kg · °C) flowing at a rate of ethylene glycol (Cp 3.5 kg/s from 80°C to 40°C by water (Cp 4180 J/kg · °C) that enters at 20°C and leaves at 55°C. The overall heat transfer coefficient based on the inner surface area of the tube is 250 W/m2 · °C. Determine (a) the rate of heat transfer, (b) the mass flow rate of water, and (c) the heat transfer surface area on the inner side of the tube. Cold water 20°C Hot glycol 18°C 80°C 3.5 kg/s Water 50°C 40°C FIGURE P13–41 FIGURE P13–46 13–42 A double-pipe parallel-flow heat exchanger is to heat water (Cp 4180 J/kg · °C) from 25°C to 60°C at a rate of 0.2 kg/s. The heating is to be accomplished by geothermal water (Cp 4310 J/kg · °C) available at 140°C at a mass flow rate of 0.3 kg/s. The inner tube is thin-walled and has a diameter of 0.8 cm. If the overall heat transfer coefficient of the heat exchanger is 550 W/m2 · °C, determine the length of the heat exchanger required to achieve the desired heating. 13–43 Reconsider Problem 13–42. Using EES (or other) software, investigate the effects of temperature and mass flow rate of geothermal water on the length of the heat exchanger. Let the temperature vary from 100ºC to 200ºC, and the mass flow rate from 0.1 kg/s to 0.5 kg/s. Plot the length of the heat exchanger as functions of temperature and mass flow rate, and discuss the results. 13–44E A 1-shell-pass and 8-tube-passes heat exchanger is used to heat glycerin (Cp 0.60 Btu/lbm · °F) from 65°F to 140°F by hot water (Cp 1.0 Btu/lbm · °F) that enters the thinwalled 0.5-in.-diameter tubes at 175°F and leaves at 120°F. The total length of the tubes in the heat exchanger is 500 ft. The convection heat transfer coefficient is 4 Btu/h · ft2 · °F on the glycerin (shell) side and 50 Btu/h · ft2 · °F on the water (tube) side. Determine the rate of heat transfer in the heat exchanger (a) before any fouling occurs and (b) after fouling with a fouling factor of 0.002 h · ft2 · °F/Btu occurs on the outer surfaces of the tubes. 13–47 Water (Cp 4180 J/kg · °C) enters the 2.5-cminternal-diameter tube of a double-pipe counter-flow heat exchanger at 17°C at a rate of 3 kg/s. It is heated by steam 2203 kJ/kg) in the shell. If the condensing at 120°C (hfg overall heat transfer coefficient of the heat exchanger is 1500 W/m2 · °C, determine the length of the tube required in order to heat the water to 80°C. 13–48 A thin-walled double-pipe counter-flow heat exchanger is to be used to cool oil (Cp 2200 J/kg · °C) from 150°C to 40°C at a rate of 2 kg/s by water (Cp 4180 J/kg · °C) that enters at 22°C at a rate of 1.5 kg/s. The diameter of the tube is 2.5 cm, and its length is 6 m. Determine the overall heat transfer coefficient of this heat exchanger. 13–49 Reconsider Problem 13–48. Using EES (or other) software, investigate the effects of oil exit temperature and water inlet temperature on the overall heat transfer coefficient of the heat exchanger. Let the oil exit temperature vary from 30ºC to 70ºC and the water inlet temperature from 5ºC to 25ºC. Plot the overall heat transfer coefficient as functions of the two temperatures, and discuss the results. 13–50 Consider a water-to-water double-pipe heat exchanger whose flow arrangement is not known. The temperature measurements indicate that the cold water enters at 20°C and leaves at 50°C, while the hot water enters at 80°C and leaves at cen58933_ch13.qxd 9/9/2002 9:57 AM Page 708 708 HEAT TRANSFER 45°C. Do you think this is a parallel-flow or counter-flow heat exchanger? Explain. 13–51 Cold water (Cp 4180 J/kg · °C) leading to a shower enters a thin-walled double-pipe counter-flow heat exchanger at 15°C at a rate of 0.25 kg/s and is heated to 45°C by hot water (Cp 4190 J/kg · °C) that enters at 100°C at a rate of 3 kg/s. If the overall heat transfer coefficient is 1210 W/m2 · °C, determine the rate of heat transfer and the heat transfer surface area of the heat exchanger. 13–52 Engine oil (Cp 2100 J/kg · °C) is to be heated from 20°C to 60°C at a rate of 0.3 kg/s in a 2-cm-diameter thinwalled copper tube by condensing steam outside at a temper2174 kJ/kg). For an overall heat ature of 130°C (hfg transfer coefficient of 650 W/m2 · °C, determine the rate of heat transfer and the length of the tube required to achieve it. Answers: 25.2 kW, 7.0 m Steam 130°C Oil 20°C 0.3 kg/s 60°C 55°C FIGURE P13–52 13–53E Geothermal water (Cp 1.03 Btu/lbm · °F) is to be used as the heat source to supply heat to the hydronic heating system of a house at a rate of 30 Btu/s in a double-pipe counter-flow heat exchanger. Water (Cp 1.0 Btu/lbm · °F) is heated from 140°F to 200°F in the heat exchanger as the geothermal water is cooled from 310°F to 180°F. Determine the mass flow rate of each fluid and the total thermal resistance of this heat exchanger. 13–54 Glycerin (Cp 2400 J/kg · °C) at 20°C and 0.3 kg/s is to be heated by ethylene glycol (Cp 2500 J/kg · °C) at 60°C in a thin-walled double-pipe parallel-flow heat exchanger. The temperature difference between the two fluids is 15°C at the outlet of the heat exchanger. If the overall heat transfer coefficient is 240 W/m2 · °C and the heat transfer surface area is 3.2 m2, determine (a) the rate of heat transfer, (b) the outlet temperature of the glycerin, and (c) the mass flow rate of the ethylene glycol. 13–55 Air (Cp 1005 J/kg · °C) is to be preheated by hot exhaust gases in a cross-flow heat exchanger before it enters the furnace. Air enters the heat exchanger at 95 kPa and 20°C at a rate of 0.8 m3/s. The combustion gases (Cp 1100 J/kg · °C) enter at 180°C at a rate of 1.1 kg/s and leave at 95°C. The product of the overall heat transfer coefficient and the heat transfer surface area is AU 1200 W/°C. Assuming both fluids to be unmixed, determine the rate of heat transfer and the outlet temperature of the air. Air 95 kPa 20°C 0.8 m3/s Exhaust gases 1.1 kg/s 95°C FIGURE P13–55 13–56 A shell-and-tube heat exchanger with 2-shell passes and 12-tube passes is used to heat water (Cp 4180 J/kg · °C) in the tubes from 20°C to 70°C at a rate of 4.5 kg/s. Heat is supplied by hot oil (Cp 2300 J/kg · °C) that enters the shell side at 170°C at a rate of 10 kg/s. For a tube-side overall heat transfer coefficient of 600 W/m2 · °C, determine the heat transAnswer: 15 m2 fer surface area on the tube side. 13–57 Repeat Problem 13–56 for a mass flow rate of 2 kg/s for water. 13–58 A shell-and-tube heat exchanger with 2-shell passes and 8-tube passes is used to heat ethyl alcohol (Cp 2670 J/kg · °C) in the tubes from 25°C to 70°C at a rate of 2.1 kg/s. The heating is to be done by water (Cp 4190 J/kg · °C) that enters the shell side at 95°C and leaves at 45°C. If the overall heat transfer coefficient is 950 W/m2 · °C, determine the heat transfer surface area of the heat exchanger. Water 95°C 70°C Ethyl alcohol 25°C 2.1 kg/s 45°C (8-tube passes) FIGURE P13–58 13–59 A shell-and-tube heat exchanger with 2-shell passes and 12-tube passes is used to heat water (Cp 4180 J/kg · °C) with ethylene glycol (Cp 2680 J/kg · °C). Water enters the tubes at 22°C at a rate of 0.8 kg/s and leaves at 70°C. Ethylene glycol enters the shell at 110°C and leaves at 60°C. If the overall heat transfer coefficient based on the tube side is 280 W/m2 · °C, determine the rate of heat transfer and the heat transfer surface area on the tube side. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 709 709 CHAPTER 13 13–60 Reconsider Problem 13–59. Using EES (or other) software, investigate the effect of the mass flow rate of water on the rate of heat transfer and the tube-side surface area. Let the mass flow rate vary from 0.4 kg/s to 2.2 kg/s. Plot the rate of heat transfer and the surface area as a function of the mass flow rate, and discuss the results. 13–61E Steam is to be condensed on the shell side of a 1-shell-pass and 8-tube-passes condenser, with 50 tubes in each pass at 90°F (hfg 1043 Btu/lbm). Cooling water (Cp 1.0 Btu/lbm · °F) enters the tubes at 60°F and leaves at 73°F. The tubes are thin-walled and have a diameter of 3/4 in. and length of 5 ft per pass. If the overall heat transfer coefficient is 600 Btu/h · ft2 · °F, determine (a) the rate of heat transfer, (b) the rate of condensation of steam, and (c) the mass flow rate of cold water. Steam 90°F 20 lbm/s 73°F 60°F Water 90°F FIGURE P13–61E 13–62E Reconsider Problem 13–61E. Using EES (or other) software, investigate the effect of the condensing steam temperature on the rate of heat transfer, the rate of condensation of steam, and the mass flow rate of cold water. Let the steam temperature vary from 80ºF to 120ºF. Plot the rate of heat transfer, the condensation rate of steam, and the mass flow rate of cold water as a function of steam temperature, and discuss the results. 13–63 A shell-and-tube heat exchanger with 1-shell pass and 20–tube passes is used to heat glycerin (Cp 2480 J/kg · °C) in the shell, with hot water in the tubes. The tubes are thinwalled and have a diameter of 1.5 cm and length of 2 m per pass. The water enters the tubes at 100°C at a rate of 5 kg/s and leaves at 55°C. The glycerin enters the shell at 15°C and leaves at 55°C. Determine the mass flow rate of the glycerin and the overall heat transfer coefficient of the heat exchanger. 13–64 In a binary geothermal power plant, the working fluid isobutane is to be condensed by air in a condenser at 75°C 255.7 kJ/kg) at a rate of 2.7 kg/s. Air enters the con(hfg denser at 21ºC and leaves at 28ºC. The heat transfer surface Air 28°C Isobutane 75°C 2.7 kg/s Air 21°C FIGURE P13–64 area based on the isobutane side is 24 m2. Determine the mass flow rate of air and the overall heat transfer coefficient. 13–65 Hot exhaust gases of a stationary diesel engine are to be used to generate steam in an evaporator. Exhaust gases (Cp 1051 J/kg · ºC) enter the heat exchanger at 550ºC at a rate of 0.25 kg/s while water enters as saturated liquid and evaporates at 200ºC (hfg 1941 kJ/kg). The heat transfer surface area of the heat exchanger based on water side is 0.5 m2 and overall heat transfer coefficient is 1780 W/m2 · ºC. Determine the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water. 13–66 Reconsider Problem 13–65. Using EES (or other) software, investigate the effect of the exhaust gas inlet temperature on the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water. Let the temperature of exhaust gases vary from 300ºC to 600ºC. Plot the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water as a function of the temperature of the exhaust gases, and discuss the results. 13–67 In a textile manufacturing plant, the waste dyeing water (Cp 4295 J/g · ºC) at 75°C is to be used to preheat fresh water (Cp 4180 J/kg · ºC) at 15ºC at the same flow rate in a double-pipe counter-flow heat exchanger. The heat transfer surface area of the heat exchanger is 1.65 m2 and the overall heat transfer coefficient is 625 W/m2 · ºC. If the rate of heat transfer in the heat exchanger is 35 kW, determine the outlet temperature and the mass flow rate of each fluid stream. Fresh water 15°C Dyeing water 75°C Th, out Tc, out FIGURE P13–67 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 710 710 HEAT TRANSFER The Effectiveness–NTU Method 13–68C Under what conditions is the effectiveness–NTU method definitely preferred over the LMTD method in heat exchanger analysis? 13–69C What does the effectiveness of a heat exchanger represent? Can effectiveness be greater than one? On what factors does the effectiveness of a heat exchanger depend? 13–70C For a specified fluid pair, inlet temperatures, and mass flow rates, what kind of heat exchanger will have the highest effectiveness: double-pipe parallel-flow, double-pipe counter-flow, cross-flow, or multipass shell-and-tube heat exchanger? 13–71C Explain how you can evaluate the outlet temperatures of the cold and hot fluids in a heat exchanger after its effectiveness is determined. 13–72C Can the temperature of the hot fluid drop below the inlet temperature of the cold fluid at any location in a heat exchanger? Explain. 13–73C Can the temperature of the cold fluid rise above the inlet temperature of the hot fluid at any location in a heat exchanger? Explain. 13–74C Consider a heat exchanger in which both fluids have the same specific heats but different mass flow rates. Which fluid will experience a larger temperature change: the one with the lower or higher mass flow rate? 13–75C Explain how the maximum possible heat transfer · rate Q max in a heat exchanger can be determined when the mass flow rates, specific heats, and the inlet temperatures of the two · fluids are specified. Does the value of Q max depend on the type of the heat exchanger? 13–76C Consider two double-pipe counter-flow heat exchangers that are identical except that one is twice as long as the other one. Which heat exchanger is more likely to have a higher effectiveness? 13–77C Consider a double-pipe counter-flow heat exchanger. In order to enhance heat transfer, the length of the heat exchanger is now doubled. Do you think its effectiveness will also double? 13–78C Consider a shell-and-tube water-to-water heat exchanger with identical mass flow rates for both the hot and cold water streams. Now the mass flow rate of the cold water is reduced by half. Will the effectiveness of this heat exchanger increase, decrease, or remain the same as a result of this modification? Explain. Assume the overall heat transfer coefficient and the inlet temperatures remain the same. 13–79C Under what conditions can a counter-flow heat exchanger have an effectiveness of one? What would your answer be for a parallel-flow heat exchanger? 13–80C How is the NTU of a heat exchanger defined? What does it represent? Is a heat exchanger with a very large NTU (say, 10) necessarily a good one to buy? 13–81C Consider a heat exchanger that has an NTU of 4. Someone proposes to double the size of the heat exchanger and thus double the NTU to 8 in order to increase the effectiveness of the heat exchanger and thus save energy. Would you support this proposal? 13–82C Consider a heat exchanger that has an NTU of 0.1. Someone proposes to triple the size of the heat exchanger and thus triple the NTU to 0.3 in order to increase the effectiveness of the heat exchanger and thus save energy. Would you support this proposal? 13–83 Air (Cp 1005 J/kg · °C) enters a cross-flow heat exchanger at 10°C at a rate of 3 kg/s, where it is heated by a hot 4190 J/kg · °C) that enters the heat exwater stream (Cp changer at 95°C at a rate of 1 kg/s. Determine the maximum heat transfer rate and the outlet temperatures of the cold and the hot water streams for that case. 13–84 Hot oil (Cp 2200 J/kg · °C) is to be cooled by water (Cp 4180 J/kg · °C) in a 2-shell-pass and 12-tube-pass heat exchanger. The tubes are thin-walled and are made of copper with a diameter of 1.8 cm. The length of each tube pass in the heat exchanger is 3 m, and the overall heat transfer coefficient is 340 W/m2 · °C. Water flows through the tubes at a total rate of 0.1 kg/s, and the oil through the shell at a rate of 0.2 kg/s. The water and the oil enter at temperatures 18°C and 160°C, respectively. Determine the rate of heat transfer in the heat exchanger and the outlet temperatures of the water and the oil. Answers: 36.2 kW, 104.6°C, 77.7°C Oil 160°C 0.2 kg/s Water 18°C 0.1 kg/s (12-tube passes) FIGURE P13–84 13–85 Consider an oil-to-oil double-pipe heat exchanger whose flow arrangement is not known. The temperature measurements indicate that the cold oil enters at 20°C and leaves at 55°C, while the hot oil enters at 80°C and leaves at 45°C. Do you think this is a parallel-flow or counter-flow heat exchanger? Why? Assuming the mass flow rates of both fluids to be identical, determine the effectiveness of this heat exchanger. 13–86E Hot water enters a double-pipe counter-flow waterto-oil heat exchanger at 220°F and leaves at 100°F. Oil enters at 70°F and leaves at 150°F. Determine which fluid has the smaller heat capacity rate and calculate the effectiveness of this heat exchanger. 13–87 A thin-walled double-pipe parallel-flow heat exchanger is used to heat a chemical whose specific heat is 1800 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 711 711 CHAPTER 13 J/kg · °C with hot water (Cp 4180 J/kg · °C). The chemical enters at 20°C at a rate of 3 kg/s, while the water enters at 110°C at a rate of 2 kg/s. The heat transfer surface area of the heat exchanger is 7 m2 and the overall heat transfer coefficient is 1200 W/m2 · °C. Determine the outlet temperatures of the chemical and the water. Chemical 20°C 3 kg/s Hot water 110°C 2 kg/s by hot oil (Cp 2200 J/kg · °C) that enters at 120°C. If the heat transfer surface area and the overall heat transfer coefficients are 6.2 m2 and 320 W/m2 · °C, respectively, determine the outlet temperature and the mass flow rate of oil using (a) the LMTD method and (b) the –NTU method. 13–92 Water (Cp 4180 J/kg · °C) is to be heated by solarheated hot air (Cp 1010 J/kg · °C) in a double-pipe counterflow heat exchanger. Air enters the heat exchanger at 90°C at a rate of 0.3 kg/s, while water enters at 22°C at a rate of 0.1 kg/s. The overall heat transfer coefficient based on the inner side of the tube is given to be 80 W/m2 · °C. The length of the tube is 12 m and the internal diameter of the tube is 1.2 cm. Determine the outlet temperatures of the water and the air. 13–93 FIGURE P13–87 13–88 Reconsider Problem 13–87. Using EES (or other) software, investigate the effects of the inlet temperatures of the chemical and the water on their outlet temperatures. Let the inlet temperature vary from 10ºC to 50ºC for the chemical and from 80ºC to 150ºC for water. Plot the outlet temperature of each fluid as a function of the inlet temperature of that fluid, and discuss the results. 13–89 A cross-flow air-to-water heat exchanger with an effectiveness of 0.65 is used to heat water (Cp 4180 J/kg · °C) with hot air (Cp 1010 J/kg · °C). Water enters the heat exchanger at 20°C at a rate of 4 kg/s, while air enters at 100°C at a rate of 9 kg/s. If the overall heat transfer coefficient based on the water side is 260 W/m2 · °C, determine the heat transfer surface area of the heat exchanger on the water side. Assume Answer: 52.4 m2 both fluids are unmixed. 13–90 Water (Cp 4180 J/kg · °C) enters the 2.5-cminternal-diameter tube of a double-pipe counter-flow heat exchanger at 17°C at a rate of 3 kg/s. Water is heated by steam 2203 kJ/kg) in the shell. If the condensing at 120°C (hfg overall heat transfer coefficient of the heat exchanger is 900 W/m2 · °C, determine the length of the tube required in order to heat the water to 80°C using (a) the LMTD method and (b) the –NTU method. Reconsider Problem 13–92. Using EES (or other) software, investigate the effects of the mass flow rate of water and the tube length on the outlet temperatures of water and air. Let the mass flow rate vary from 0.05 kg/s to 1.0 kg/s and the tube length from 5 m to 25 m. Plot the outlet temperatures of the water and the air as the functions of the mass flow rate and the tube length, and discuss the results. 13–94E A thin-walled double-pipe heat exchanger is to be used to cool oil (Cp 0.525 Btu/lbm · °F) from 300°F to 105°F at a rate of 5 lbm/s by water (Cp 1.0 Btu/lbm · °F) that enters at 70°F at a rate of 3 lbm/s. The diameter of the tube is 1 in. and its length is 20 ft. Determine the overall heat transfer coefficient of this heat exchanger using (a) the LMTD method and (b) the –NTU method. 13–95 Cold water (Cp 4180 J/kg · °C) leading to a shower enters a thin-walled double-pipe counter-flow heat exchanger at 15°C at a rate of 0.25 kg/s and is heated to 45°C by hot water (Cp 4190 J/kg · °C) that enters at 100°C at a rate of 3 kg/s. If the overall heat transfer coefficient is 950 W/m2 · °C, determine the rate of heat transfer and the heat transfer surface area of the heat exchanger using the –NTU Answers: 31.35 kW, 0.482 m2 method. Cold water 15°C 0.25 kg/s Hot water 13–91 Ethanol is vaporized at 78°C (hfg 846 kJ/kg) in a double-pipe parallel-flow heat exchanger at a rate of 0.03 kg/s 100°C 3 kg/s Oil 120°C 45°C FIGURE P13–95 Ethanol 78°C 0.03 kg/s FIGURE P13–91 13–96 Reconsider Problem 13–95. Using EES (or other) software, investigate the effects of the inlet temperature of hot water and the heat transfer coefficient on the rate of heat transfer and surface area. Let the inlet temperature vary from 60ºC to 120ºC and the overall heat cen58933_ch13.qxd 9/9/2002 9:57 AM Page 712 712 HEAT TRANSFER transfer coefficient from 750 W/m2 · °C to 1250 W/m2 · °C. Plot the rate of heat transfer and surface area as functions of inlet temperature and the heat transfer coefficient, and discuss the results. Steam 30°C 13–97 Glycerin (Cp 2400 J/kg · °C) at 20°C and 0.3 kg/s 2500 J/kg · °C) at is to be heated by ethylene glycol (Cp 60°C and the same mass flow rate in a thin-walled doublepipe parallel-flow heat exchanger. If the overall heat transfer coefficient is 380 W/m2 · °C and the heat transfer surface area is 5.3 m2, determine (a) the rate of heat transfer and (b) the outlet temperatures of the glycerin and the glycol. 13–98 A cross-flow heat exchanger consists of 40 thinwalled tubes of 1-cm diameter located in a duct of 1 m 1 m cross-section. There are no fins attached to the tubes. Cold water (Cp 4180 J/kg · °C) enters the tubes at 18°C with an average velocity of 3 m/s, while hot air (Cp 1010 J/kg · °C) enters the channel at 130°C and 105 kPa at an average velocity of 12 m/s. If the overall heat transfer coefficient is 130 W/m2 · °C, determine the outlet temperatures of both fluids and the rate of heat transfer. 1m Hot air 130°C 105 kPa 12 m/s 1m Water 18°C 3 m/s FIGURE P13–98 13–99 A shell-and-tube heat exchanger with 2-shell passes and 8-tube passes is used to heat ethyl alcohol (Cp 2670 J/kg · °C) in the tubes from 25°C to 70°C at a rate of 2.1 kg/s. The heating is to be done by water (Cp 4190 J/kg · °C) that enters the shell at 95°C and leaves at 60°C. If the overall heat transfer coefficient is 800 W/m2 · °C, determine the heat transfer surface area of the heat exchanger using (a) the LMTD method and (b) the –NTU method. Answer (a): 11.4 m2 13–100 Steam is to be condensed on the shell side of a 1-shell-pass and 8-tube-passes condenser, with 50 tubes in each pass, at 30°C (hfg 2430 kJ/kg). Cooling water (Cp 4180 J/kg · °C) enters the tubes at 15°C at a rate of 1800 kg/h. The tubes are thin-walled, and have a diameter of 1.5 cm and length of 2 m per pass. If the overall heat transfer coefficient is 3000 W/m2 · °C, determine (a) the rate of heat transfer and (b) the rate of condensation of steam. 15°C Water 1800 kg/h 30°C FIGURE P13–100 13–101 Reconsider Problem 13–100. Using EES (or other) software, investigate the effects of the condensing steam temperature and the tube diameters on the rate of heat transfer and the rate of condensation of steam. Let the steam temperature vary from 20ºC to 70ºC and the tube diameter from 1.0 cm to 2.0 cm. Plot the rate of heat transfer and the rate of condensation as functions of steam temperature and tube diameter, and discuss the results. 13–102 Cold water (Cp 4180 J/kg · °C) enters the tubes of a heat exchanger with 2-shell-passes and 13–tube-passes at 20°C at a rate of 3 kg/s, while hot oil (Cp 2200 J/kg · °C) enters the shell at 130°C at the same mass flow rate. The overall heat transfer coefficient based on the outer surface of the tube is 300 W/m2 · °C and the heat transfer surface area on that side is 20 m2. Determine the rate of heat transfer using (a) the LMTD method and (b) the –NTU method. Selection of Heat Exchangers 13–103C A heat exchanger is to be selected to cool a hot liquid chemical at a specified rate to a specified temperature. Explain the steps involved in the selection process. 13–104C There are two heat exchangers that can meet the heat transfer requirements of a facility. One is smaller and cheaper but requires a larger pump, while the other is larger and more expensive but has a smaller pressure drop and thus requires a smaller pump. Both heat exchangers have the same life expectancy and meet all other requirements. Explain which heat exchanger you would choose under what conditions. 13–105C There are two heat exchangers that can meet the heat transfer requirements of a facility. Both have the same pumping power requirements, the same useful life, and the same price tag. But one is heavier and larger in size. Under what conditions would you choose the smaller one? 13–106 A heat exchanger is to cool oil (Cp 2200 J/kg · °C) at a rate of 13 kg/s from 120°C to 50°C by air. Determine the heat transfer rating of the heat exchanger and propose a suitable type. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 713 713 CHAPTER 13 13–107 A shell-and-tube process heater is to be selected to heat water (Cp 4190 J/kg · °C) from 20°C to 90°C by steam flowing on the shell side. The heat transfer load of the heater is 600 kW. If the inner diameter of the tubes is 1 cm and the velocity of water is not to exceed 3 m/s, determine how many tubes need to be used in the heat exchanger. 60°C. If the overall heat transfer coefficient based on the outer surface of the tube is 300 W/m2 · °C, determine (a) the rate of heat transfer and (b) the heat transfer surface area on the outer Answers: (a) 462 kW, (b) 29.2 m2 side of the tube. Hot oil 130°C 3 kg/s Steam 90°C Cold water 20°C 3 kg/s (20-tube passes) 60°C 20°C Water FIGURE P13–107 13–108 Reconsider Problem 13–107. Using EES (or other) software, plot the number of tube passes as a function of water velocity as it varies from 1 m/s to 8 m/s, and discuss the results. 13–109 The condenser of a large power plant is to remove 500 MW of heat from steam condensing at 30°C (hfg 2430 kJ/kg). The cooling is to be accomplished by cooling water 4180 J/kg · °C) from a nearby river, which enters the (Cp tubes at 18°C and leaves at 26°C. The tubes of the heat exchanger have an internal diameter of 2 cm, and the overall heat transfer coefficient is 3500 W/m2 · °C. Determine the total length of the tubes required in the condenser. What type of heat Answer: 312.3 km exchanger is suitable for this task? 13–110 Repeat Problem 13–109 for a heat transfer load of 300 MW. FIGURE P13–113 13–114E Water (Cp 1.0 Btu/lbm · °F) is to be heated by solar-heated hot air (Cp 0.24 Btu/lbm · °F) in a double-pipe counter-flow heat exchanger. Air enters the heat exchanger at 190°F at a rate of 0.7 lbm/s and leaves at 135°F. Water enters at 70°F at a rate of 0.35 lbm/s. The overall heat transfer coefficient based on the inner side of the tube is given to be 20 Btu/h · ft2 · °F. Determine the length of the tube required for a tube internal diameter of 0.5 in. 13–115 By taking the limit as T2 → T1, show that when T2 for a heat exchanger, the Tlm relation reduces to T1 T1 T2. Tlm 13–116 The condenser of a room air conditioner is designed to reject heat at a rate of 15,000 kJ/h from Refrigerant-134a as the refrigerant is condensed at a temperature of 40°C. Air (Cp 1005 J/kg · °C) flows across the finned condenser coils, entering at 25°C and leaving at 35°C. If the overall heat transfer coefficient based on the refrigerant side is 150 W/m2 · °C, determine the heat transfer area on the refrigerant side. Answer: 3.05 m2 R-134a 40°C Review Problems 13–111 Hot oil is to be cooled in a multipass shell-and-tube heat exchanger by water. The oil flows through the shell, with a heat transfer coefficient of ho 35 W/m2 · °C, and the water flows through the tube with an average velocity of 3 m/s. The tube is made of brass (k 110 W/m · °C) with internal and external diameters of 1.3 cm and 1.5 cm, respectively. Using water properties at 25°C, determine the overall heat transfer coefficient of this heat exchanger based on the inner surface. 35°C Air 25°C 13–112 Repeat Problem 13–111 by assuming a fouling factor Rf, o 0.0004 m2 · °C/W on the outer surface of the tube. 13–113 Cold water (Cp 4180 J/kg · °C) enters the tubes of a heat exchanger with 2-shell passes and 20–tube passes at 20°C at a rate of 3 kg/s, while hot oil (Cp 2200 J/kg · °C) enters the shell at 130°C at the same mass flow rate and leaves at 40°C FIGURE P13–116 13–117 Air (Cp 1005 J/kg · °C) is to be preheated by hot exhaust gases in a cross-flow heat exchanger before it enters cen58933_ch13.qxd 9/9/2002 9:57 AM Page 714 714 HEAT TRANSFER the furnace. Air enters the heat exchanger at 95 kPa and 20°C at a rate of 0.8 m3/s. The combustion gases (Cp 1100 J/kg · °C) enter at 180°C at a rate of 1.1 kg/s and leave at 95°C. The product of the overall heat transfer coefficient and the heat transfer surface area is UAs 1620 W/°C. Assuming both fluids to be unmixed, determine the rate of heat transfer. 13–118 In a chemical plant, a certain chemical is heated by hot water supplied by a natural gas furnace. The hot water (Cp 4180 J/kg · °C) is then discharged at 60°C at a rate of 8 kg/min. The plant operates 8 h a day, 5 days a week, 52 weeks a year. The furnace has an efficiency of 78 percent, and the cost of the natural gas is $0.54 per therm (1 therm 100,000 Btu 105,500 kJ). The average temperature of the cold water entering the furnace throughout the year is 14°C. In order to save energy, it is proposed to install a water-to-water heat exchanger to preheat the incoming cold water by the drained hot water. Assuming that the heat exchanger will recover 72 percent of the available heat in the hot water, determine the heat transfer rating of the heat exchanger that needs to be purchased and suggest a suitable type. Also, determine the amount of money this heat exchanger will save the company per year from natural gas savings. 13–119 A shell-and-tube heat exchanger with 1-shell pass and 14-tube passes is used to heat water in the tubes with geothermal steam condensing at 120ºC (hfg 2203 kJ/kg) on the shell side. The tubes are thin-walled and have a diameter of 2.4 cm and length of 3.2 m per pass. Water (Cp 4180 J/kg · ºC) enters the tubes at 22ºC at a rate of 3.9 kg/s. If the temperature difference between the two fluids at the exit is 46ºC, determine (a) the rate of heat transfer, (b) the rate of condensation of steam, and (c) the overall heat transfer coefficient. Steam 120°C 13–121 Air at 18ºC (Cp 1006 J/kg · ºC) is to be heated to 70ºC by hot oil at 80ºC (Cp 2150 J/kg · ºC) in a cross-flow heat exchanger with air mixed and oil unmixed. The product of heat transfer surface area and the overall heat transfer coefficient is 750 W/m2 · ºC and the mass flow rate of air is twice that of oil. Determine (a) the effectiveness of the heat exchanger, (b) the mass flow rate of air, and (c) the rate of heat transfer. 13–122 Consider a water-to-water counter-flow heat exchanger with these specifications. Hot water enters at 95ºC while cold water enters at 20ºC. The exit temperature of hot water is 15ºC greater than that of cold water, and the mass flow rate of hot water is 50 percent greater than that of cold water. The product of heat transfer surface area and the overall heat transfer coefficient is 1400 W/m2 · ºC. Taking the specific heat of both cold and hot water to be Cp 4180 J/kg · ºC, determine (a) the outlet temperature of the cold water, (b) the effectiveness of the heat exchanger, (c) the mass flow rate of the cold water, and (d) the heat transfer rate. Cold water 20°C Hot water 95°C FIGURE P13–122 Computer, Design, and Essay Problems 22°C 14 tubes mass flow rate of geothermal water and the outlet temperatures of both fluids. 120°C Water 3.9 kg/s FIGURE P13–119 13–120 Geothermal water (Cp 4250 J/kg · ºC) at 95ºC is to be used to heat fresh water (Cp 4180 J/kg · ºC) at 12ºC at a rate of 1.2 kg/s in a double-pipe counter-flow heat exchanger. The heat transfer surface area is 25 m2, the overall heat transfer coefficient is 480 W/m2 · ºC, and the mass flow rate of geothermal water is larger than that of fresh water. If the effectiveness of the heat exchanger is desired to be 0.823, determine the 13–123 Write an interactive computer program that will give the effectiveness of a heat exchanger and the outlet temperatures of both the hot and cold fluids when the type of fluids, the inlet temperatures, the mass flow rates, the heat transfer surface area, the overall heat transfer coefficient, and the type of heat exchanger are specified. The program should allow the user to select from the fluids water, engine oil, glycerin, ethyl alcohol, and ammonia. Assume constant specific heats at about room temperature. 13–124 Water flows through a shower head steadily at a rate of 8 kg/min. The water is heated in an electric water heater from 15°C to 45°C. In an attempt to conserve energy, it is proposed to pass the drained warm water at a temperature of 38°C through a heat exchanger to preheat the incoming cold water. Design a heat exchanger that is suitable for this task, and discuss the potential savings in energy and money for your area. 13–125 Open the engine compartment of your car and search for heat exchangers. How many do you have? What type are they? Why do you think those specific types are selected? If cen58933_ch13.qxd 9/9/2002 9:58 AM Page 715 715 CHAPTER 13 you were redesigning the car, would you use different kinds? Explain. 13–126 Write an essay on the static and dynamic types of regenerative heat exchangers and compile information about the manufacturers of such heat exchangers. Choose a few models by different manufacturers and compare their costs and performance. 13–127 Design a hydrocooling unit that can cool fruits and vegetables from 30°C to 5°C at a rate of 20,000 kg/h under the following conditions: The unit will be of flood type that will cool the products as they are conveyed into the channel filled with water. The products will be dropped into the channel filled with water at one end and picked up at the other end. The channel can be as wide as 3 m and as high as 90 cm. The water is to be circulated and cooled by the evaporator section of a refrigeration system. The refrigerant temperature inside the coils is to be –2°C, and the water temperature is not to drop below 1°C and not to exceed 6°C. Assuming reasonable values for the average product density, specific heat, and porosity (the fraction of air volume in a box), recommend reasonable values for the quantities related to the thermal aspects of the hydrocooler, including (a) how long the fruits and vegetables need to remain in the channel, (b) the length of the channel, (c) the water velocity through the channel, (d) the velocity of the conveyor and thus the fruits and vegetables through the channel, (e) the refrigeration capacity of the refrigeration system, and (f) the type of heat exchanger for the evaporator and the surface area on the water side. 13–128 Design a scalding unit for slaughtered chicken to loosen their feathers before they are routed to feather-picking machines with a capacity of 1200 chickens per hour under the following conditions: The unit will be of immersion type filled with hot water at an average temperature of 53°C at all times. Chickens with an average mass of 2.2 kg and an average temperature of 36°C will be dipped into the tank, held in the water for 1.5 min, and taken out by a slow-moving conveyor. Each chicken is expected to leave the tank 15 percent heavier as a result of the water that sticks to its surface. The center-to-center distance between chickens in any direction will be at least 30 cm. The tank can be as wide as 3 m and as high as 60 cm. The water is to be circulated through and heated by a natural gas furnace, but the temperature rise of water will not exceed 5°C as it passes through the furnace. The water loss is to be made up by the city water at an average temperature of 16°C. The ambient air temperature can be taken to be 20°C. The walls and the floor of the tank are to be insulated with a 2.5-cm-thick urethane layer. The unit operates 24 h a day and 6 days a week. Assuming reasonable values for the average properties, recommend reasonable values for the quantities related to the thermal aspects of the scalding tank, including (a) the mass flow rate of the make-up water that must be supplied to the tank; (b) the length of the tank; (c) the rate of heat transfer from the water to the chicken, in kW; (d) the velocity of the conveyor and thus the chickens through the tank; (e) the rate of heat loss from the exposed surfaces of the tank and if it is significant; ( f ) the size of the heating system in kJ/h; (g) the type of heat exchanger for heating the water with flue gases of the furnace and the surface area on the water side; and (h) the operating cost of the scalding unit per month for a unit cost of $0.56 therm of natural gas (1 therm 105,000 kJ). 13–129 A company owns a refrigeration system whose refrigeration capacity is 200 tons (1 ton of refrigeration 211 kJ/min), and you are to design a forced-air cooling system for fruits whose diameters do not exceed 7 cm under the following conditions: The fruits are to be cooled from 28°C to an average temperature of 8°C. The air temperature is to remain above –2°C and below 10°C at all times, and the velocity of air approaching the fruits must remain under 2 m/s. The cooling section can be as wide as 3.5 m and as high as 2 m. Assuming reasonable values for the average fruit density, specific heat, and porosity (the fraction of air volume in a box), recommend reasonable values for the quantities related to the thermal aspects of the forced-air cooling, including (a) how long the fruits need to remain in the cooling section; (b) the length of the cooling section; (c) the air velocity approaching the cooling section; (d) the product cooling capacity of the system, in kg · fruit/h; (e) the volume flow rate of air; and ( f ) the type of heat exchanger for the evaporator and the surface area on the air side. cen58933_ch13.qxd 9/9/2002 9:58 AM Page 716 ...
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