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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 ShellandIhbe Heat Exchanger Design . . . . . . . . . . . . . . . . . . . . . 422 11.3 Optimization of a MultiEffect Evaporator . . . . . . . . . . . . . . . . . . . . . . . . . .430 11.4 Boiler/FurboGenerator 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 heattransfer—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 discretevalued 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 tradeoff 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 (coldside) 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 hightemperature 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 SHELLANDTUBE HEAT
EXCHANGER DESIGN In this example we examine a procedure for optimizing the process design of a baf
ﬂed shelland—tube, singlepass, 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 shelland—tube counterﬂow heat exchanger. Key: At, = T1 — t1
coldend temperature difference; At2 = T2 — t2 warmend 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 Warmend temperature approach, Atz
Number and length of tubes, N, and L
Number of bafﬂe spacings, nb
Tubeside and shellside 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 + CiElAU + 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 warmend 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 singlepass 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) ksunJ “[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 inline
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 crosssectional 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 _ 25Ci¢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 singlepass design to
a multiplepass 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: LOOin. outside diameter
0.834in. inside diameter
0.25in. clearance
0.083—in. wall thickness
1.25in. 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 linch 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 MULTIEFFECT
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 troublefree 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 multipleeffect evaporation, as shown in Figure E11.3a, the total capacity of
the system of evaporation is no greater than that of a singleeffect 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
Multipleeffect evaporator with forward feed. Steadystate mathematical models of single— and multipleeffect 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 rootﬁnding 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 multieffect evaporator cascades. EXAMPLE 11.4 BOILER/TURBOGENERATOR 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 = pressurereducing 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 Highpressure extraction at 195 psig
Highpressure 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)
Highpressure steam 635 720 1359.8
Mediumpressure steam 195 130 superheat 1267.8
Lowpressure steam 62 130 superheat 1251.4
Feedwater (condensate) 193.0
TABLE 11.4C Demands on the system Resource Demand Mediumpressure steam (195 psig)
Lowpressure 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
Basepurchased 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
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