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 ...
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