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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
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 Spring '08
 Dunia
 Heat Transfer

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