chapter_11 - 11 HEAT TRANSFER AND ENERGY CONSERVATION...

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Unformatted text preview: 11 HEAT TRANSFER AND ENERGY CONSERVATION Example 11.1 Optimizing Recovery of Waste Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 11.2 Optimal Shell-and-Ihbe Heat Exchanger Design . . . . . . . . . . . . . . . . . . . . . 422 11.3 Optimization of a Multi-Effect Evaporator . . . . . . . . . . . . . . . . . . . . . . . . . .430 11.4 Boiler/Furbo-Generator System Optimization . . . . . . . . . . . . . . . . . . . . . . . 435 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Supplementary References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 417 418 PART 111: Applications of Optimization A VARIETY OF AVAILABLE energy conservation measures can be adopted to opti- mize energy usage throughout a chemical plant or refinery. The following is a rep- resentative list of design or operating factors related to heat transfer and energy use that can involve optimization: 1. Fired heater combustion controls . Heat recovery from stack gases . Fired heater convection section cleaning Heat exchanger network configuration A Extended surface heat exchanger tubing to improve heat transfer Scheduling of heat exchanger cleaning Air cooler performance Fractionating towers: optimal reflux ratio, heat exchange, and so forth Instrumentation for monitoring energy usage 10. Reduced leakage in vacuum systems and pressure lines and condensers ll. Cooling water savings 12. Efficient water treatment for steam raising plants 13. Useful work from steam pressure reduction 14. Steam traps, tracing, and condensate recovery 15. CO boilers on catalytic cracking units 16. Electrical load leveling 17. Power factor improvement 18. Power recovery from gases or liquids 19. Loss control in refineries 20. Catalyst improvements pwsoweww Many of the conservation measures require detailed process analysis plus opti- mization. For example, the efficient firing of fuel (category 1) is extremely impor- tant in all applications. For any rate of fuel combustion, a theoretical quantity of air (for complete combustion to carbon dioxide and water vapor) exists under which the most efficient combustion occurs. Reduction of the amount of air available leads to incomplete combustion and a rapid decrease in efficiency. In addition, car— bon particles may be formed that can lead to accelerated fouling of heater tube sur- faces. To allow for small variations in fuel composition and flow rate and in the air flow rates that inevitably occur in industrial practice, it is usually desirable to aim for operation with a small amount of excess air, say 5 to 10 percent, above the the— oretical amount for complete combustion. Too much excess air, however, leads to increased sensible heat losses through the stack gas. In practice, the efficiency of a fired heater is controlled by monitoring the oxygen concentration in the combustion products in addition to the stack gas temperature. Dampers are used to manipulate the air supply. By tying the measuring instruments into a feedback loop with the mechanical equipment, optimization of operations can take place in real time to account for variations in the fuel flow rate or heating value. As a second example (category 4), a typical plant contains large numbers of heat exchangers used to transfer heat from one process stream to another. It is important to continue to use the heat in the streams efficiently throughout the process. Incoming crude oil is heated against various product and reflux streams CHAPTER 11: Heat Transfer and Energy Conservation 419 before entering a fired heater in order to be brought to the desired fractionating col- umn flash zone temperature. Among the factors that must be considered in design or retrofit are 1. What should be the configuration of flows (the order of heat exchange for the crude oil)? 2. How much heat exchange surface should be supplied within the chosen con- figuration? Additional heat exchange surface area leads to improved heat recovery in the crude oil unit but increases capital costs so that increasing the heat transfer surface area soon reaches diminishing returns. The optimal configuration and areas selected, of course, are strongly dependent on fuel costs. As fuel costs rise, existing plants can usually profit from the installation of additional heat exchanger surface in circum- stances previously considered only marginally economic. As a final example (category 6), although heat exchangers may be very effec- tive when first installed, many such systems become dirty in use and heat transfer rates deteriorate significantly. It is therefore often useful to establish optimal heat exchanger cleaning schedules. Although the schedules can be based on observa— tions of the actual deterioration of the overall heat transfer of the exchanger in ques— tion, it is also possible to optimize the details of the cleaning schedules depending on an economic assessment of each exchanger. In this chapter we illustrate the application of various optimization techniques to heat-transfer—system design. First we show how simple rules of thumb on boiler temperature differences can be derived (Example 11.1). Then a more complicated design of a heat exchanger is examined (Example 11.2), leading to a constrained optimization problem involving some discrete-valued variables. Example 11.3 dis- cusses the use of optimization in the design and operation of evaporators, and we conclude this chapter by demonstrating how linear programming can be employed to optimize a steam/power system (Example 11.4). For optimization of heat exchanger networks by mathematical programming methods, refer to Athier et a1. (1997), Briones and Kokossis (1996), and Zamora and Grossmann (1998). EXAMPLE 11.1 OPTIMIZING RECOVERY OF WASTE HEAT A variety of sources of heat at elevated temperatures exist in a typical chemical plant that may be economically recoverable for production of power using steam or other working fluids, such as freon or light hydrocarbons. Figure E11.1 is a schematic of such a system. The system power output Can be increased by using larger heat exchanger surface areas for both the boiler and the condenser. However, there is a trade-off between power recov— ery and capital cost of the exchangers. Jegede and Polley (1992), Reppich and Zager— mann (1995), Sama (1983), Swearingen and Ferguson (1984), and Steinmeyer (1984) have proposed some simple rules based on analytical optimization of the boiler A]? In a power system, the availability expended by any exchanger is equal to the net work that could have been accomplished by having each stream exchange heat with the surroundings through a reversible heat engine or heat pump. In the boiler in Fig- ure E11.1, heat is transferred at a rate Q (the boiler load) from the average hot fluid 420 PART 111: Applications of Optimization Heat source Boiler (7;) Expander turbine FIGURE E11.1 Schematic of power system. temperature Ts to the working fluid at TH. The working fluid then exchanges heat with the condenser at temperature T2. If we ignore mechanical friction and heat leaks, the reversible work available from Q at temperature T5 with the condensing (cold-side) temperature at T2 is T _ T2 W = s 1 Q< Ts > The reversible work available from the condenser using the working fluid temperature TH (average value) and the heat sink temperature T2 is T—T W2=Q< HTH 2) Hence the ideal power available from the boiler can be found by subtracting W2 from W1 (a) (b) T T WZ—W1=AW=Q<T—2-?2> (c) H s In this expression TS and T2 are normally specified, and TH is the variable to be adjusted. If Q is expressed in Btu/h, and the operating cost is Cop, then the value of the available power is T2 T2 Cop = CHTIYQ<— _ _) TH TS (d) CHAPTER 11: Heat Transfer and Energy Conservation 421 where 1; = overall system efficiency (0.7 is typical) y = number of hours per year of operation CH amalgamates the value of the power in $/kWh and the necessary conversion factors to have a consistent set of units You can see, using Equation (d) only, that C01) is minimized by setting TH = Ts (infinitesimal boiler AT). However, this outcome increases the required boiler heat transfer area to an infinite area, as can be noted from the calculation for the area _ Q A ‘ Um — TH) @ (In Equation (e) an average value for the heat transfer coefficient U is assumed, ignor- ing the effect of pressure drop. U depends on the working fluid and the operating tem- perature.) Let the cost per unit area of the exchanger be C A and the annualization fac— tor for capital investment be denoted by r. Then the annualized capital cost for the boiler is CAQr C = -— C _ TH) (f) Finally, the objective function to be minimized with respect to TH, the working fluid temperature, is the sum of the operating cost and surface area costs: _ 2 _ E i f — CanyQ<TH TS) + Um _ TH) (g) To get an expression for the minimum of f, we differentiate Equation (g) with respect to TH and equate the derivative to zero to obtain __Ti i = CHWQ( T13) + U(T, — TH)2 0 (h) To solve the quadratic equation for TH, let 0‘1 = CHTIYTZU a2 = CAr Q cancels in both terms. On rearrangement, the resulting quadratic equation is (all — a2)T,3 — 20¢;er + 04le = 0 (i) The solution to (i) for TH < TS is Ta = T, —_— (j) 422 PART III: Applications of Optimization For a system with CA = $25/ft2, a power cost of $0.06/kWh (CH = 1.76 X 10‘5). U = 95 Btu/(h)(°R)(ft2), y = 8760 h/year, r = 0.365, 17 = 0.7, T2 = 600°R, and TS = 790°R, the optimal value TH is 760.7°R, giving a AT of 29.3°R. Swearingen and Fer- guson showed that Equation (h) can be expressed implicitly as 1/2 AT = T, — TH = Tfl(fl> (k) a2 In this form, it appears that the allowable AT increases as the working fluid tempera- ture increases. This suggests that the optimum AT for a heat source at 900°R is lower than that for a heat source at 1100°R. In fact, Equation (j) indicates that the optimum AT is directly proportional to TS. Sama argues that this is somewhat counterintuitive because the Carnot “value” of a high-temperature source implies using a smaller AT to reduce lost work. The working fluid must be selected based on the heat source temperature, as discussed by Swearingen and Ferguson. See Sama for a discussion of optimal tem- perature differences for refrigeration systems; use of Equation (k) leads to AT’s ranging from 8 to 10°R. EXAMPLE 11.2 OPTIMAL SHELL-AND-TUBE HEAT EXCHANGER DESIGN In this example we examine a procedure for optimizing the process design of a baf- fled shell-and—tube, single-pass, counterflow heat exchanger (see Figure E11.2a), in which the tube fluid is in turbulent flow but no change of phase of fluids takes place in the shell or tubes. Usually the following variables are specified a priori by the designer: 1. Process fluid rate (the hot fluid passes through the tubes), Wi 2. Process fluid temperature change, T2 — T1 Process fluid (-— Tz (tube) flow rate W, FIGURE E11.23 Process diagram of shell-and—tube counterflow heat exchanger. Key: At, = T1 — t1 cold-end temperature difference; At2 = T2 — t2 warm-end temperature difference. CHAPTER 11: Heat Transfer and Energy Conservation 423 3. Coolant inlet temperature (the coolant flows through the shell), t1 4. Tube spacing and tube inside and outside diameters (Di, D0). \ Conditions 1 and 2 imply the heat duty Q of the exchanger is known. The variables that might be calculated via optimization include Total heat transfer area, A0 Warm-end temperature approach, Atz Number and length of tubes, N, and L Number of baffle spacings, nb Tube-side and shell-side pressure drop Coolant flow, WC P‘WPPP’.‘ Not all of these variables are independent, as shown in the following discussion. In contrast to the analysis outlined in Example 11.1, the objective function in this example does not make use of reversible work. Rather, a cost is assigned to the usage of coolant as well as to power losses because of the pressure drops of each fluid. In addition, annualized capital cost terms are included. The objective function in dollars per year is formulated using the notation in Table E1 1.2A C = Cchy + CAAo + CiEl-AU + CoEvo (a) Suppose we minimize the objective function using the following set of four vari— ables, a set slightly different from the preceding list. 1. Atzz warm-end temperature difference 2. A0: tube outside area 3. hi: tube inside heat transfer coefficient 4. ho: tube outside heat transfer coefficient Only three of the four variables are independent. If A0, hi, and ho are known, then At2 can be found from the heat duty of the exchanger Q: Atz _ Atl Q = F,U A 0 am (I?) F , is unity for a single-pass exchanger. U0 is given by the values of ho, hi, and the foul— ing coefficient hZ as follows: 1 1 1 1 — = — + — + — (6) U0 fAhi ho ht Cichelli and Brinn (1956) showed that the annual pumping loss terms in Equation (a) could be related to hi and ho by using friction factor and j—factor relationships for tube flow and shell flow: E = ¢ih?'5 E0 = ¢oh3'75 (e) 424 PART III: Applications of Optimization TABLE E11.2A Nomenclature for heat exchanger optimization Ah" Log mean of inside and outside tube surface areas A,- Inside tube surface area, ft2 A0 Outside tube surface area, ft2 C Total annual cost, $/year CA Annual cost of heat exchanger per unit outside tube surface area, $/(ft2)(year) Cc Cost of coolant, $/lb mass Ci Annual cost of supplying l(ft)(1bf)/h to pump fluid flowing inside tubes, ($)(h)/(ft)(1br)(year) Annual cost of supplying l(ft)(lbf)/h to pump shell side fluid, ($)(h)/(ft)(lbf)(year) Specific heat at constant pressure, Btu/(lbm)(°F) Tube inside diameter, ft 'Iube outside diameter, ft Power loss inside tubes per unit outside tube area, (ft)(lbf)/(ft2)(h) Power loss outside tubes per unit outside tube area, (ft)(lbf)/(ft2)(h) Friction factor, dimensionless Ai/AD Multipass exchanger factor Conversion factor, (ft)(lbm)/(lbf)(h2) = 4.18 X 108 Fouling coefficient Coefficient of heat transfer inside tubes, Btu/(h)(ft2)(°F) Coefficient of heat transfer outside tubes, Btu/(h)(ft2)(°F) Combined coefficient for tube wall and dirt films, based on tube outside area Btu/(h)(ft2)(°F) QU p ‘3 at) ksun-J “[7; say as :z: b 9? a “k L’A, 1 A, 1 + —— + — k,,A,,, hf, A, hfi, i h, Thermal conductivity, Btu/(h)(ft)(°F) Lagrangian function Length of tubes, ft Thickness of tube wall, ft b4th (continued) The coefficients d),- and d), depend on fluid specific heat c, thermal conductivity k, density p, and viscosity ,u, as well as the tube diameters. (p, is based on either in-line or staggered tube arrangements. If we solve for Wc from the energy balance W, — Q — C(Atl _ Atz + T2 _ T1) and substitute for El, E0, and We in Equation (a), the resulting objective function is CcyQ = — + C A + C. .h3'5A + c h4'75A C(At1_ Atz + T2_ T1) A o 1431 l 0 12¢!) a a f CHAPTER 11: Heat Transfer and Energy Conservation 425 TABLE E11.2A (CONTINUED) Nomenclature for heat exchanger optimization nb Number of baffle spacing on shell side = number of baffles plus 1 NC Number of clearances for flow between tubes across shell axis N, Number of tubes in exchanger Apt Pressure drop for flow through tube side, lbf/ft2 Apa Pressure drop for flow through tube side, lbf/ft2 , Q Heat transfer rate in heat exchanger, Btu/h ‘ S0 Minimum cross-sectional area for flow across tubes, ft2 1 T1 Outlet temperature of process fluid, °F ‘ T2 Inlet temperature of process fluid, °F 21 Inlet temperature of coolant, °F t2 Outlet temperature of coolant, °F AT1 T1 - t1, = cold—end temperature difference AT2 T2 —- t2, = warrn—end temperature difference 1 U0 Overall coefficient of heat transfer, based on outside tube area, Btu/(h)(ft2)(°F) vi Average velocity of fluid inside tubes, ft/h v0 Average velocity of fluid outside tubes, ft/h at shell axis W Coolant rate, lb/h W, Flow rate of fluid inside tubes, lbm/h W0 Flow rate of fluid outside tubes, lbm/h y Operating hours per year p, Density of fluid inside tubes, lbm/ft3 [3,, Density of fluid outside tubes, Ibm/ft3 pr. Viscosity of fluid, lbm/(h)(ft) <75,- Factor relating frictionless to h, (#0 Factor relating friction loss to ho to Lagrange multiplier Subscripts c Coolant f Film temperature, midway between bulk fluid and wall temperature 1 Inside the tubes 0 Outside the tubes w Wall To accommodate the constraint ([9), a Lagrangian function L is formed by aug- menting f with Equation (19), using a Lagrange multiplier a) E(At2 * Air) 1 ] (h) L _f+ w[an(At2/At1) Uvo Equation (h) can be differentiated with respect to four variables (hi, ha, Atz, and A0). After some rearrangement, you can obtain a relationship between the optimum ha and hi, namely 0.74C~ . 0‘17 Y h, = h?” (i) 426 PART III: Applications of Optimization This is the same result as derived by McAdams (1942), having the interpretation that the friction losses in the shell and tube sides, and the heat transfer resistances must be balanced economically. The value of hi can be obtained by solving 3.5C,~ ,- hf's CA _ 2-5Ci¢ihi3'5 _ 291(c)(1)o)0'l7(Ci¢ifA)O'83h?'72 _ (Z fA = 0 (j) I The simultaneous solution of Equations (f), (i), and (j) yields another expression: —1+&] (k Atl ) -<1+-—T2‘“>ll(fi> C(CA + CiEi + COEO) — Atz _ Atl n Atl The following algorithm can be used to obtain the optimal values of hi, ho, A0, and Atz without the explicit calculation of w: Solve for h,- from Equation (j) Obtain ho from Equation (1') Calculate U0 from Equation (c) Determine E,- and E, from h,- and h,, using Equations ((1') and (e) and obtain A12 by solving Equation (k) 5. Calculate AU from Equation (b) 6. Find WC from Equation (f) Note that steps 1 to 6 require that several nonlinear equations be solved one at a time. Once these variables are known, the physical dimensions of the heat exchanger can be determined. PPP!‘ 7. Determine the optimal vi and v0 from hi and ho using the appropriate heat trans- fer correlations (see McAdams, 1942); recall that the inside and outside tube diameters are specified a priori. 8. The number of tubes N, can be found from a mass balance: wD? 4 = wt (1) ViNt 9. The length of the tubes L, can be found from A0 = NflTDoL, (m) 10. The number of clearances NC can be found from N,, based on either square pitch or equilateral pitch. The flow area So is obtained from v0 (flow normal to a tube bundle). Finally, baffle spacing (or the number of baffles) is computed from S0, A0, N,, and NC. Having presented the pertinent equations and the procedure for computing the optimum, let us check the approach by computing the degrees of freedom in the design problem. Design Variables Status (number of variables) Wi, T1, T2, t1, tube spacing, Di, D0, Q Given (8) Atz, WC, A0, N,, L,, U0, nb, Apt, Aps, 11,, v0, hi, ha Unspecified (13) Total number of variables = 8 + 13 = 21 CHAPTER 11: Heat Transfer and Energy Conservation 427 Design Relationships Number of Equations 1. Equations (b), (c), (d), (e) (f), (l), (m) 7 2. Heat transfer correlations for hi and he (step 7) 2 3. We = povoso (step 10) 1 Total number of relationships 10 Degrees of freedom for optimization = total number of variables — number of given variables - number of equations =21—8—10=3 Note this result agrees with Equation (h) in that four variables are included in the Lagrangian, but with one constraint corresponding to 3 degrees of freedom. Several simplified cases may be encountered in heat exchanger design. Case 1. U0 is specified and pressure drop costs are ignored in the objective func- tion. In this case Cl. and C0 can be set equal to zero and Equation (k) can be solved for Atz (see Peters and Timmerhaus (1980) for a similar equation for a condensing vapor). Figure E11.2b shows a solution to Equation (k) (Cichelli and Brinn). Case 2. Coolant flow rate is fixed. Here At2 is known, so the tube side and shell side coefficients and area are optimized. Use Equation (i) and (j) to find no and hi. A0 is then found from Equation (b). In the preceding analysis no inequality constraints were introduced. As a practi- cal matter the following inequality constraints may apply: 10 ~ — Ta \ 7‘ 4‘ “— 3 ‘ fl 8 2 ¥ “ ‘ ‘ 4 At2 E _ AT. 1 2 0.5 ‘ s 0.4 I 0.3 0‘: 0.2 0.23 0 0.1 __|_ 0.1 0.2 0.5 1 2 5 10 CcyU. c(CA + CiEi + C,E,,) FIGURE E11.2b Solution to Equation (k) for the case in which U0 is specified and pressure drop costs are ignored. Construction type Maximum allowable shell diameter, in. 428 PART III: Applications of Optimization TABLE E11.ZB Design specifications for one case of heat exchanger optimization Variables Process fluid Gas Inlet temperature of process fluid, °F 150 Outlet temperature of process fluid, °F 100 Process fluid flow rate, lb/h 20,000 Maximum process fluid velocity, ft/s 160 Minimum process fluid velocity, ft/s 0.001 Utility fluid Water Inlet utility fluid temperature, °F 70 Maximum allowable utility fluid temperature, °F 140 Maximum utility fluid velocity, ft/s 8 Minimum utility fluid velocity, ft/s 0.5 Shell side fouling factor 2000 Tube side fouling factor 1500 Cost of pumping process fluid, $/(ft)(lbf) 0.7533 X 10'8 Cost of pumping utility fluid, $/(ft)(lbf) 0.7533 x 10‘8 Cost of utility fluid, iii/1bm 0.5000 X 10'5 Factor for pressure 1.45 Cost index 1.22 Fractional annual fixed charges 0.20 Fractional cost of installation 0.15 Tube material Steel Type of tube layout Triangular Fixed tube sheet 40 Bypassing safety factor 1.3 Constant for evaluating outside film coat 0.33 Hours operation per year 7000 Thermal conductivity of metal Btu/(h)(ft2)(°F) 26 Number of tube passes Source: Tarrer et a1. (1971). l 1. Maximum velocity on shell or tube side 2. Longest practical tube length 3. Closest practical baffle spacing 4. Maximum allowable pressure drops (shell or tube side) The velocity on the tube side can be modified by changing the single-pass design to a multiple-pass configuration. In this case F t at: 1 in Equation (17). From formulas in McCabe, F , depends on 2:2 (or Atz), hence the necessary conditions derived previously would have to be changed. The fluids could be switched (shell vs. tube side) if con- straints are violated, but there may well be practical limitations such as one fluid being quite dirty or corrosive so that the fluid must flow in the tube side (to facilitate cleaning or to reduce alloy costs). Other practical features that must be taken into account are the fixed and integer lengths of tubes (8, 12, 16, and 20 feet), and the maximum pressure drops allowed. CHAPTER 11: Heat Transfer and Energy Conservation 429 TABLE 11.2C Optimal solution for a heat exchanger involving discrete variables Continuous- Variable . . Optimal Standard Integer Sizes Variables Design 1 2 3 4 Tube length, ft 10.5 8 8 12 12 Number of tubes 66 110 85 64 42 Total area, ft2 193.3 230 178 201 132 Total cost, $/year 734 908 923 738 784 Heat transfer coefficients, Btu/(h)(ft2)(°F) Outside 554 561 649 512 617 Inside 56.2 37.1 45.9 57.4 80.5 Overall 41.0 28.4 34.5 41.5 56.2 Outlet utility fluid temperature (°F) 117.1 102.1 96.5 120.1 112.4 Utility fluid flow rate, lbm/h 5306 7790 9422 4993 5897 Inside pressure drop, psi 0.279 0.086 0.138 0.318 0.701 Outside pressure drop, psi 6.45 5.24 7.91 4.98 9.13 Number of baffle spaces 119 85 79 121 119 Shell diameter, in. 12 16 14 12 10 Tube layout: LOO-in. outside diameter 0.834-in. inside diameter 0.25-in. clearance 0.083—in. wall thickness 1.25-in. pitch Source: Tarrer et al. (1971). Although a 20—psi drop may be typical for liquids such as water, higher values are employed for more Viscous fluids. Exchanging shell sides with tube sides may miti- gate pressure drop restrictions. The tube’s outside diameter is specified a priori in the optimization procedure described earlier; usually %— or l-inch outside diameter (o.d.) tubes are used because of their greater availability and ease of cleaning. Limits on operating variables, such as maximum exit temperature of the coolant, maximum and minimum velocities for both streams, and maximum allowable shell area must be included in the problem specifications along with the number of tube passes. Table 11.2B lists the specifications for a typical exchanger, and Table 11.2C gives the results of optimization for several cases for two standard tube lengths, 8 and 12 ft. The minimum cost occurs for a lZ—ft tube length With 64 tubes (case 3). Many commercial codes exist to carry out heat exchanger design. Search the Web for the most recent versions. 430 PART III: Applications of Optimization EXAMPLE 11.3 OPTIMIZATION OF A MULTI-EFFECT EVAPORATOR When a process requires an evaporation step, the problem of evaporator design needs serious examination. Although the subject of evaporation and the equipment to carry out evaporation have been studied and analyzed for many years, each application has to receive individual attention. No evaporation configuration and its equipment can be picked from a stock list and be expected to produce trouble-free operation. An engineer working on the selection of optimal evaporation equipment must list What is “known,” “unknown,” and “to be determined.” Such analysis should at least include the following: Known - Production rate and analysis of product 0 Feed flow rate, feed analysis, feed temperature 0 Available utilities (steam, water, gas, etc.) - Disposition of condensate (location) and its purity - Probable materials of construction Unknown - Pressures, temperatures, solids, compositions, capacities, and concentrations 0 Number of evaporator effects - Amount of vapor leaving the last effect 0 Heat transfer surface Features to be determined - Best type of evaporator body and heater arrangement - Filtering characteristics of any solids or crystals ° Equipment dimensions, arrangement - Separator elements for purity of overhead vapors - Materials, fabrication details, instrumentation Utility consumption ' Steam - Electric power - Water ' Air In multiple-effect evaporation, as shown in Figure E11.3a, the total capacity of the system of evaporation is no greater than that of a single-effect evaporator having a heating surface equal to one effect and operating under the same terminal condi- tions. The amount of water vaporized per unit surface area in 11 effects is roughly l/n that of a single effect. Furthermore, the boiling point elevation causes a loss of avail- able temperature drop in every effect, thus reducing capacity. Why, then, are multiple effects often economic? It is because the cost of an evaporator per square foot of sur- face area decreases with total area (and asymptotically becomes a constant value) so that to achieve a given production, the cost of heat exchange surface can be balanced with the steam costs. CHAPTER 11: Heat Transfer and Energy Conservation 431 Cooling water F I (feed) —) -]> 4| —> _|> 4 1’s 71 S l (steam) . A . I i 1' i i/ Product Condensate Condensate Condensate \L FIGURE E11.3a Multiple-effect evaporator with forward feed. Steady-state mathematical models of single— and multiple-effect evaporators involving material and energy balances can be found in McCabe et a1. (1993), Yannio- tis and Pilavachi (1996), and Esplugas and Mata (1983). The classical simplified opti- mization problem for evaporators (Schweyer, 1955) is to determine the most suitable number of effects given (1) an analytical expression for the fixed costs in terms of the number of effects n, and (2) the steam (variable) costs also in terms of n. Analytic dif- ferentiation yields an analytical solution for the optimal 11*, as shown here. Assume we are concentrating an inorganic salt in the range of 0.1 to 1.0 wt% using a plant capacity of 0.1—10 million gallons/day. Initially we treat the number of stages n as a continuous variable. Figure E11.3b shows a single effect in the process. Prior to discussions of the capital and operating costs, we need to define the tem- perature driving force for heat transfer. Examine the notation in Figure E1130; by definition the log mean temperature difference ATlm is T,- — Td AT = —— "“ 1n(T,-/Td) (‘0 Let Ti be equal to constant K for a constant performance ratio P. Because Td = Ti — AT In f M =_A_Tr/"_ (b) 1'“ lntK/K— (Tf/n>)] Let A = condenser heat transfer areas, ft2 cp = liquid heat capacity, 1.05 Btu/(lbm)(°F) CC = cost per unit area of condenser, $6.25/ft2 CE = cost per evaporator (including partitions), $7000/stage 432 PART III: Applications of Optimization qom Steam Condensate qin Steam FIGURE E11.3b Boiling point rise 1;, 7} AT for T (Eff ‘ AT a heater ,~ ective ‘ ) ATf/n (fraction of One stage T, for one stage) Inlet Outlet FIGURE E11.3c C5 = cost of steam, $llb at the brine heater (first stage) Fout = liquid flow out of evaporator, lb/h K = Ti, a constant (Ti = AT — Tl7 at inlet) n = number of stages P = performance ratio, 1b of H20 evaporated/Btu supplied to brine heater Q = heat duty, 9.5 X 108 Btu/h (a constant) qe — total 1b H20 evaporated/h q, total 1b steam used/h r = capital recovery factor S = lb steam supplied/h Tb = boiling point rise, 43°F ATf = flash down range, 250°F U = overall heat transfer coefficient (assumed to be constant), 625 Btu/(ft2)(h)(°F) AH = heat of vaporization of water, about 1000 Btu/1b vap III The optimum number of stages is n*. For a constant performance ratio the total cost of the evaporator is f1 = C5” + CcA (C) CHAPTER 11: Heat Transfer and Energy Conservation 433 For A we introduce Q A = —— U(AT1m) Then we differentiate f1 in Equation (0) with respect to n and set the resulting expres- sion equal to zero (Q and U are constant): 9 6mm] _ CE + CC U[—an P — 0 (d) With the use of Equation (1)) [am/Am] _ _ 1 _1n<1 — My) an F _ nK(1 — ATf/nK) Arf (8) Substituting Equation (e) into (d) plus introducing the values of Q, U, A7}, CE, and CC, we get 6.25 9.5 x 108 In 1 — AT nK 7000 [( )( )H 1 l ( f/ )]=O 625 nK(1 — ATf/nK) A1} Rearranging (625)(7000)(250) 250 < 250) —— = . = — + — — (6.25)(9.5 x 108) 0184 MC — 250 1“ 1 nK (f) In practice, as the evaporation plant size changes (for constant Q), the ratio of the stage condenser area cost to the unit evaporator cost remains essentially constant so that the number 0.184 is treated as a constant for all practical purposes. Equation (f) can be solved for nK for constant P nK = 590 (8) Next, we eliminate K from Equation (g) by replacing K with a function of P so that n becomes a function of P. The performance ratio (with constant liquid heat capacity at 347°F) is defined as (AHvaque) _ 1000 qe (FoutchA Theater)first stage + K) F out P = (h) The ratio qe/F can be calculated from _ 1.49 qe = 1 _ = 0.31 Font _ where AH (355°F, 143 psi) = 1194 Btu/lb vap AHfiqHzo (350°F) = 322 Btu/lb AHHqHzO (100°F) = 70 Btu/lb 434 PART 111: Applications of Optimization Equations (g) and (h) can be solved together to eliminate K and obtain the desired relation ——4.3=T (i) Equation (i) shows how the boiling point rise (Tb = 43°F) and the number of stages affects the performance ratio. Optimal performance ratio The optimal plant operation can be determined by minimizing the total cost func- tion, including steam costs, with respect to P (liquid pumping costs are negligible) 6A an C — + — + r C aP rCE 6P as —=0 k saP () The quantity for BA/aP can be calculated by using the equations already developed and can be expressed in terms of a ratio of polynomials in P such as a(1 + 1 / P) (1 — bP)2 where a and b are determined by fitting experimental data. The relation for art/8P can be determined from Equation (i). The relation for BS/aP can be obtained from equa— tion (I) P 2 g = qe = 4. Q (AH,,,,)S 10005 or lb 9 s<—>= g h lOOOP or a(8760)qe = —— 1 Sub) lOOOP where a is the fraction of hours per year (8760) during which the system operates. Equation (k), given the costs, cannot be explicitly solved for P*, but P* can be obtained by any effective root-finding technique. If a more complex mathematical model is employed to represent the evaporation process, you must shift from analytic to numerical methods. The material and enthalpy balances become complicated functions of temperature (and pressure). Usu- ally all of the system parameters are specified except for the heat transfer areas in each effect (11 unknown variables) and the vapor temperatures in each effect excluding the last one (It — 1 unknown variables). The model introduces n independent equations that serve as constraints, many of which are nonlinear, plus nonlinear relations among the temperatures, concentrations, and physical properties such as the enthalpy and the heat transfer coefficient. CHAPTER 11: Heat Transfer and Energy Conservation 435 Because the number of evaporators represents an integer—valued variable, and because many engineers use tables and graphs as well as equations for evaporator cal— culations, some of the methods outlined in Chapters 9 and 10 can be applied for the optimization of multi-effect evaporator cascades. EXAMPLE 11.4 BOILER/TURBO-GENERATOR SYSTEM OPTIMIZATION Linear programming is often used in the design and operation of steam systems in the chemical industry. Figure E11.4 shows a steam and power system for a small power house fired by wood pulp. To produce electric power, this system contains two turbo— generators whose characteristics are listed in Table E11.4A. Turbine 1 is a double— extraction turbine with two intermediate streams leaving at 195 and 62 psi; the final stage produces condensate that is used as boiler feed water. Turbine 2 is a single— HPS (635 psig steam) PP(EP) (195 psig steam) (62 psig steam) FIGURE E11.4 Boiler/turbo—generator system. Key: Ii = inlet flow rate for turbine i [lbm/h] HEi = exit flow rate from turbine i to 195 psi header [lbm/h] LE1. = exit flow rate from turbine i to 62 psi header [lbm/h] C = condensate flow rate from turbine 1 [lbm/h] P,- = power generated by turbine 1' [kW] BF 1 bypass flow rate from 635 psi to 195 psi header [lbm/h] BF2 = bypass flow rate from 195 psi to 62 psi header [lbm/h] HPS = flow rate through 635 psi header [lbm/h] MPS = flow rate through 195 psi header [lbm/h] LPS = flow rate through 62 psi header [lbm/h] PP = purchased power [kW] EP = excess power [kW] (difference of purchased power from base power) PRV = pressure-reducing valve 436 PART III: Applications of Optimization extraction turbine with one intermediate stream at 195 psi and an exit stream leaving at 62 psi with no condensate being formed. The first turbine is more efficient due to the energy released from the condensation of steam, but it cannot produce as much power as the second turbine. Excess steam may bypass the turbines to the two levels of steam through pressure—reducing valves. Table E11.4B lists information about the different levels of steam, and Table E11.4C gives the demands on the system. To meet the electric power demand, electric power may be purchased from another producer with a minimum base of 12,000 kW. If the electric power required to meet the system demand is less than this base, the power that is not used will be charged at a penalty cost. Table E11.4D gives the costs of fuel for the boiler and additional electric power to operate the utility system. The system shown in Figure E11.4 may be modeled as linear constraints and com- bined with a linear objective function. The objective is to minimize the operating cost of the system by choice of steam flow rates and power generated or purchased, subject to the demands and restrictions on the system. The following objective function is the cost to operate the system per hour, namely, the sum of steam produced HPS, pur- chased power required PP, and excess power EP: TABLE 11.4A 'lhrbine data Turbine 1 Turbine 2 Maximum generative capacity 6,250 kW Maximum generative capacity 9,000 kW Minimum load 2,500 kW Minimum load 3,000 kW Maximum inlet flow 192,000 lbm/h Maximum inlet flow 244,000 lbm/h Maximum condensate flow 62,000 lbm/h Maximum 62 psi exhaust 142,000 lbm/h Maximum internal flow 132,000 lbm/h High-pressure extraction at 195 psig High-pressure extraction at 195 psig Low—pressure extraction at 62 psig Low—pressure extraction at 62 psig TABLE 11.4B Steam header data Header Pressure (psig) Temperature (°F) Enthalpy (Btu/lbm) High-pressure steam 635 720 1359.8 Medium-pressure steam 195 130 superheat 1267.8 Low-pressure steam 62 130 superheat 1251.4 Feedwater (condensate) 193.0 TABLE 11.4C Demands on the system Resource Demand Medium-pressure steam (195 psig) Low-pressure steam (62 psig) Electric power 271,536 lbm/h 100,623 lbm/h 24,550 kW CHAPTER 11: Heat Transfer and Energy Conservation TABLE 11.4D Energy data Fuel cost $1.68/106 Btu Boiler efficiency 0.75 Steam cost (635 psi) $2.24/106 Btu = $2.24 (1359.8 — 193)/106 = $0.002614/1bm Purchased electric power $0.0239/kWh average Demand penalty $0.009825/kWh Base-purchased power 12,000 kW Minimize: f = 0.00261 HPS + 0.0239 PP + 0.00983 EP The constraints are gathered into the following specific subsets: Turbine 1 P1 5 6250 P1 2 2500 HE S 192,000 C S 62,000 11 — HE S 132,000 Thrbine 2 P2 5 9000 P2 2 3000 12 S 244,000 LE2 S 142,000 Material balances HPS—Il—Iz—BF1=O [1+12+BF1—C—MPS—LPS=0 Il—HEl—LEl—C=O 12—HE2—LE2=0 HE1+ HE; + BFl — BFZ — MPS = O LE1+ LE2 + BFZ - LPS = 0 Power purchased EP + PP 2 12,000 437 (a) (b) (c) (d) (e) 438 PART III: Applications of Optimization Demands MPS 2 271,536 LPS 2 100,623 (f) P1 + P2 + PP 2 24,550 Energy balances 1359.811 — 1267.8HE1 — 1251.4LE1 — 192C —- 3413P] = 0 1359.812 — 1267.812 — 1251.4 LE2 — 3413 P2 = 0 (3) TABLE E11.4E Optimal solution to steam system LP Variable Name Value Status 1 I1 136,329 BASIC 2 [2 244,000 BOUND 3 HEl 128,158 BASIC 4 HE2 143,377 BASIC 5 LEl 0 ZERO 6 LE2 100,623 BASIC 7 C 8,170 BASIC 8 BF, 0 ZERO 9 BF2 O ZERO 10 HPS 380,329 BASIC 11 MPS 271,536 BASIC 12 LPS 100,623 BASIC 13 Pl 6,250 BOUND 14 P2 7,061 BASIC 15 PP 11,239 BASIC 16 EP 761 BASIC Value of objective function = 1268.75 $/h BASIC = basic variable ZERO = 0 BOUND = variable at its upper bound Table E11.4E lists the optimal solution to the linear program posed by Equations (a)—(g). Basic and nonbasic (zero) variables are identified in the table; the minimum cost is $1268.75/h. Note that EP + PP must sum to 12,000 kWh; in this case the excess power is reduced to 761 kWh. REFERENCES Athier, G.; P. Floquet; L. Pibouleau; et a1. “Process Optimization by Simulated Annealing and NLP Procedures. Application to Heat Exchanger Network Synthesis.” Comput Chem Eng 21 (Suppl): 8475—5480 (1997). CHAPTER 11: Heat Transfer and Energy Conservation 439 Briones, V.; and A. Kokossis. “A New Approach for the Optimal Retrofit of Heat Exchanger Networks.” Comput Chem Eng 20 (Suppl): 843—848 (1996). Cichelli, M. T.; and M. S. Brinn. “How to Design the Optimum Heat Exchanger.” Chem Eng 196: May (1956). Esplugas, S.; and J. Mata. “Calculator Design of Multistage Evaporators.” Chem Eng 59 Feb. 7: (1983). Jegede, F. 0.; and G. T. Polley. “Capital Cost Targets for Networks with Non—Uniform Heat Transfer Specifications.” Comput Chem Eng 16: 477 (1992). McAdams, W. H. Heat Transmission. 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