Ind. Eng. Chem. Fundam. 1986,2 5 , 577-581
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Optimal Pinch Approach Temperature in Heat-Exchanger Networks Yourun Ll and R. L. Motard” Department of Chemical Engineering, Washington University, St. Louis, Missouri 63 130
Calculation of the optimal pinch approach temperature in heat-exchanger networks has two major aspects. Before the network is designed, one may balance the capital cost of area against the cost of heating and cooling utility savings. An initial optimal pinch approach temperature equation is derived by maximizing the annual net cost saving. After that, further economic savings may be realized as one explores reducing the minimum number of exchangers through changing of ATpinchto make the heat loads of the streams linearly dependent separately above and below the pinch.
Introduction The pinch approach temperature provides a limit to feasible heat recovery in a heat-exchanger network design. Its properties were first described by Linnhoff and Flower (1978) and Umeda et al. (1978). The pinch divides a network into two parts: the hot end is above the pinch and requires only utility heating; the cold end is below the pinch and requires only utility cooling. There is no heat transfer across the pinch. Before conceiving a network, a preanalysis is needed to determine the minimum utilities consumption, the minimum number of exchangers, and the minimum heattransfer area. The pinch approach temperature has a great influence on these targets. As the pinch approach temperature is varied, the utility consumption, the heattransfer area, and possibly the structure of the network will vary. Therefore, in the synthesis of heat-exchanger networks, one first needs to determine a suitable or initial optimal pinch approach temperature. Usually, the pinch approach temperature is determined through the engineer’s experience, such as “for a general carbon steel network, ATpinch is equal to 20 O C and for a refrigerated system it is 5 “C”, and so on. This is a rough estimate. It cannot be always correct or exact. Some papers have proposed obtaining the optimal pinch approach temperature by searching methods. Floudas et al. (1986) applied the “golden section” search method in an outer loop of their proposed synthesis procedure. For every iteration in which a value for the pinch approach temperature was assumed, a total network synthesis calculation was performed. Linnhoff and Hindmarsh (1983) searched for the optimal pinch approach temperature using the pinch design method by costing several designs. Those methods are tedious and time-consuming. In fact, they cannot be used to determine the optimal pinch approach temperature before planning the network. Ghamarian et al. (1985) presented an incremental heat flux method to predict the optimal approach temperature for a single heat exchanger.
In order to get an analytical solution, some assumptions are needed: 1. Separately substitute straight lines for the hot and cold composite curves above and below the pinch. 2. Calculate the heat-transfer area by the countercurrent heat-transfer equation. 3. Assume separately that the number of heat recovery exchangers will be the minimum number of exchangers, above and below the pinch. Figure 1 can be used to illustrate the relationships between the network recovery heat load, the heat-transfer area, and the pinch approach temperature. We suppose that the pinch temperature on the hot side, Tz, is fixed. While moving the cold-stream line a short distance to the left, the pinch approach temperature is reduced by dT. The average heat-transfer driving force decreases slightly, and the heat-transfer area increases by dAl above the pinch and dAz below the pinch. The network heat recovery increases by dQ. The heating and cooling utility costs decrease by amoiints depending on dQ also. Therefore, there is a trade-off between the capital cost increase corresponding to dA and the heating and cooling utilities savings corresponding to dQ. Optimization determines the pinch approach temperature at which the net annual cost saving is maximum. The following equations are used LO describe the heatexchanger network system. Economic equation: ANS = Q ( C h + C,) - N u ( A / N ) ~ R , (1) where ANS is the annual net savings (dollars/year), Q is exchanger duty, the total heat recovtred (kW), Ch is the heating utility COSC (dollars/(kW year)), C, is the cooling utility cost (dollars/(kW year)), N is the minimum number of exchangers, a, b a r e coefficients, A is the total exchanger area (m2),and R, is the annual rate of return on investment (year-’). Heat-transfer and thermodynamic equations: Above the pinch
Calculation of an Optimal Pinch Temperature Difference In this paper, an optimal pinch approach temperature equation is derived from economic, thermodynamic, and heat-transfer formulas. The problem is divided into two parts by the pinch. The functional relations between the heat-transfer area, the heat recovery duty, and ATpinchare predicted separately above and below the pinch. The optimal pinch approach temperature corresponds to a maximum annual cost saving that is the difference between the heat recovery saving and the increase in annualized exchanger capital cost. 0196-4313/86/1025-0577$01.50/0CZ 1986 American Chemical Society
(3) (4)
(5)
578
Ind. Eng. Chem. Fundam., Vol, 25, No. 4, 1986 Table I. Data for Example 1 stream FCp, kW/K Cl 24.79 7.91 hl 5.80 h2 2.37 h3 31.65 h4 6.33 h5 65.94 h6 CW F
’ --$Qc dQ+ -----
-
02
-
01
---A
---Q H, Enthalpy, kw Figure 1. Heat-exchanger network T-H diagram ~
-
Oh
Ti,,,K
T””,.K
288.9 630.5 583.3 555.5 494.4 477.8 422.2 300.0 700.0
650.0 338.9 505.5 319.4 447.2 311.1 383.3 333.3 700.0
+
~
where U is the overall heat-transfer coefficient, T D = T2 - t,, and FCp is the heat capacity flow rate. The subscripts denote the following: 1, above the pinch; 2, below the pinch; h, hot stream; c, cold stream. Assuming x = ST2/TI,,eq 6 and 7 give 1 - (-!.T3/TD) = Rmln2(1- X) (8) where Rmln2= (FCp),2/(FCp)h2,the ratio of the lower to the higher heat capacity flow rate (RmlnZ< 1). Then eq 5 can be written as In [ ( I - (1 - ~)Rmin2)/xl Q2 A? = (9) UTD (1 - x)(I - RminJ ~
From eq 6 we get
I 1
0
1
1
4000 Q2 ti, Enthalpy, kw
1
1
,
I
8000 QI
Figure 2. Example 1. T-H diagram.
x=l--
Q2
(10)
(FCp)c2TD Substituting eq 10 into 9, we get
Differentiating eq 11 with respect to
Q2
leads to
Above the pinch, T,, Ti, and Ql are constants. Equations 3 and 4 give (13) S T , = (Ti - TZI(1 - Rmlnl)+ where R,,,,l = (FCp)hl/(FCp),, (Rmml< 1). Equation 2 can be written as A, =
3001
(FCp)hl In [‘(I - R m i n l )
{
(TI - TA(1 - R m i n l ) / T D
____
X
+x
1
(14) Differentiating eq 14 and 10 with respect to Q2,we obtain
The differentiation of eq 1 with respect to heat duty leads to the equation
The annual net savings is maximum when the derivatives of ANS with respect to Q is zero. Because Q1 is constant, dQ is equal to dQ2. Solving eq 16 and 17, we obtain the optimal pinch approach temperature, x. Example 1,taken from Floudas et al. (1986),consists of six hot streams, one cold stream, and one hot and one cold utility. The problem data are shown in Table I. The problem T-H diagram is shown in Figure 2. From the “problem table” or the T-H diagram, we know that the pinch on the hot-stream side is 494.4 K and estimate the equivalent composite curve flow rate capacities, (FCP)hl.h2,cl,c2. The parameters in eq 16 and 17 are the following: Ch = 174 dollars/(kW year), C, = 4.63 dollars/(kW year), cj = 0.4 kW/(m2 K), a = 1300, b = 0.6, R, = 1, TI,= T, - t l = 494.4 - 288.9 = 205.5 K, TI = 630.5 K, T2 = 494.4 K, t , = 288.9 K, N = (4 - 1) + (6 - 1) = 8 (not including the heater and the cooler, (FCp),, = (FCp),, = 24.795 kW/K, (FCp)hl = 12.295 5W/K, (FCp),, = 49.04 kW/K, Rminl= 12.295124.795 = 0.4959, and Rmh2 24.795149.04 = 0.505. By setting eq 17 to zero, we get
Substituting data into eq 18 and solving for x yield an optimal pinch approach temperature of 5.92 K. In Floudas’ work, optimization of the minimum temperature approach was performed applying the “golden
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
579
I.2
t
-
h n u a l net savina
0.4
0.0
0.8
U, Heat Transfer Coefficient, kw/m
0
K
Figure 5. Relationship between the heat-transfer coefficient and the optimal pinch approach temperature.
t
0.4
I-
t
r
0.2
0
Investment cost
8
4
12
1
16
ATpinc h ,O K
Figure 3. Relations between the costs and the 4Tpinch. c
i
1
\
a
I
I
1000 2000 a , Exchanger Cost Coefficient
0
Figure 6. Relationship between the exchanger price and the optimal pinch approach temperature.
t 0
100
200
300
ch, Unit Utility Cost of Heating, $/kw-yr
Figure 4. Relationship between the utility price and the optimal pinch approach temperature. Q
section" search method. The result is 6.38 K. They are very close; the results can be used to illustrate the efficiency of the method in this paper. The heat-transfer area in this method is less than the area of the real synthesis network because of the purely countercurrent heat-transfer calculation. Therefore, the optimal pinch approach temperature in this method will be somewhat smaller than the practical optimal pinch approach temperature. Figure 3 shows the relations between the annual net saving, the heat duty saving, the investment cost, and the pinch approach temperature. When the economic, the thermodynamic, and the heat-transfer parameters vary, the optimal pinch approach temperature will also vary. Figures 4-7 show the influences on the optimal pinch approach temperature of the unit utility cost heating Ch, the heat-transfer coefficient U , the exchanger cost coefficient a, and a variation in the heat capacity flow rate (FCp),, for example 1. From these figures we find that the unit utility cost and exchanger price coefficient have strong influences on the optimal pinch approach temperature; any error in the composite stream curve heat capacity flow rate has only a small influence.
4[
0
I 46
I
I
I
I
48
47
(FCpIh2, Heot Capacity Flux, kw/K
Figure 7. Relationship between the heat capacity flow rate and the optimal pinch approach temperature. Table 11. Data for Example 2 stream cl c2
hl h2
FCp, kW/"C 2.0 4.0 3.0 1.5
T.,
20
Tt, "C 135
80 170 150
140 60 30
O C
Example 2 comes from Linnhoff and Hindmarsh (1983). The stream data are shown in Table 11. The composite curves are shown in Figure 8. For this example, the pinch temperature on the coldstream side is constant a t 80 O C and the temperature on the hot side is varied with the pinch approach temperature,
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Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
14
i'l Y
5
c
C
i l
IO
0 H, Enthalpy, k w
20 40 60 80 100 120 140 160 T, O C
Figure 8. Example 2. T-H diagram
Figure 9. Integrated heat content graph for example 2.
AT2. The heat recovery below the pinch, Qz, is fixed. TI, - tz. The above equations should be rewritten as
Table 111. Example 2 Results
= Ti
ATpmch,
kW Qz, kW Q, kW A,, m2 A 2 , m2 A , m2 QI,
Nmin
Q(c,+ C,)
Na ( A
ANS
The optimal pinch approach temperature in this example is 13.73 "C.
Finding the Minimum Number of Heat Exchangers For a network with many exchangers, ATplnch also has an influence on the minimum number of exchangers and so it affects the predicted network capital cost. Therefore, in determining the initial optimal pinch approach temperature of a network, one should also consider the influences of ATpinchon the minimum number of exchangers. How does ATpln& influence the minimum number of exchangers? As we know, the minimum number of exchangers in a network, according to Euler's theorem, is Nmin
=
Nstream
- 1
But, if there exist some linear dependencies between the streams, the minimum number will decrease. Linear dependency means that a subset of the hot-stream heat loads is equal in magnitude to a subset of the cold-stream heat loads. For the pinch problem, the minimum number of exchangers should be calculated separately above and below the pinch (Linnhoff and Hindmarsh, 1983). By using the heat load linear-dependence concept, we may change to make the heat loads of the streams linearly dependent above and/or below the pinch separately. This provides an opportunity to reduce the minimum number of exchangers in the network and thus to reduce the capital cost of exchangers. Referring to example 2, we use an "integrated heat content g r a p h to represent the stream conditions and also the heat-transfer matching, as in Figure 9. The vertical axis of the graph indicates the relative magnitudes of heat capacity flow rates of various streams, and the horizontal
10 330 120 450 41.34 13.198 54.54 4 80395 24934 55461
"c
13.73 313.2 120 433.2 33.22 11.09 44.31 5 77393 24067 53326
20 285 120 405 24.388 8.828 33.22 4 72356 18518 53838
axis represents supply and target temperatures. The rectangular areas are proportional to the heat loads. A match between two streams is represented by placing the same number or same label and connecting by an oblique line. The match is thermodynamically feasible when the hot-stream block is on the right of the cold-stream block. An important advantage of this representation is its intuitive guidance. One may quickly judge whether a match is possible without violating heat-transfer feasibility. Figure 9 shows the stream conditions and the pinch position for example 2. The pinch point is at ab (for the cold stream) and cd (for the hot stream). If one moves the cd line right or left, the pinch approach temperature changes. For most values of ijTpinch, the minimum number of heat recovery exchangers above the pinch is 4 - 1 = 3 and below the pinch, 3 - 1 = 2, so the total minimum number is 5. When the temperature at c is 90 " C , that is, ATplnch= 90 - 80 = 10 "c,the heat load of aefg is equal to that of chij, i.e., Qaeo = Qch,, = 240. Thus, above the pinch there will be 2 exchangers for 4 streams and still 2 below the pinch, for a total number of 4 (see Figure 9). When ATpinch = 20 "C, there is another linear dependency below the pinch and the total number is also 4. This means that the capital cost at a of either 10 or 20 will be less than at any nearby ATpinch, because the minimum number of exchangers is one fewer. After obtaining the optimal pinch approach temperature at the normal minimum number of exchangers by the analytical solution, we should compare this result with those costs at the pinch approach temperature that makes the number of exchangers less than the normal number. For example 2, given AT2 = 10 or 20 "C, the total cost can be calculated by solving eq 19, 20, and 1. By comparing the annual net saving at the optimal pinch approach temperature with that at the pinch approach temperatures at which the minimum number of exchangers is less than the others, because of the linear dependency between the stream heat loads, we can choose a real optimal pinch approach temperature. The results are in Table I11 where
Ind. Eng. Chem. Fundam. 1986, 25, 581-588
we find the optimal pinch approach temperature should be lo which is Obtained by 'Onsidering not Only heat-transfer area and heat recovery saving but also the minimum number of exchangers. "'9
581
ented at the Advances in Energy Systems ASME Symposium, Miami, FL, Nov. 1985. Linnhoff, 5,; Flower, J, R, AIChE J , 1978,2 4 , 633, Linnhoff, B.; Hindmarsh, E. Chem. Eng. sci. 1983,35,745. Urneda, T.; Itho, J.; Shiroko, K . Chem. Eng. Prog. 1978, 7 5 , 70.
Literature Cited Received f o r reoiew June 16, 1986 Accepted July 25, 1986
Floudas. C. A.; Ciric, A. R.; Grossman, I . E. AIChE J . 1986,32, 276. Ghamarian, A.; Thomas, W. R. L.; Sideropoulos, T.: Robertson, J. L. Pres-
Characteristics of a Nickel-Contaminated Cracking Catalyst: Differences in the Reaction and Sintering Characteristics of Each Catalyst Component Tommle R. Davist and Howard F. Rase" Department of Chemical Engineering, The University of Texas, Austin, Texas 78712
The characteristics of a nickel-contaminated fluid-cracking catalyst were examined by sintering and reaction studies. Either methanol or hydrogen treatment produced marked sintering of the nickel, with the degree of sintering on the several catalyst components declining in the order Y zeolite, kaolin, and silica/alumina. Contaminant nickel acted as a promoter for useful methanol reactions of the cracking catalyst, and the promoting effect was not diminished significantly by sintering. By contrast, nickel acted as a catalyst for unwanted dehydrogenation and coke-forming reactions during cumene cracking, but sintering reduced this type of activity markedly.
Economic factors, together with more stringent environmental regulations which limit burning of heavy fuel oils, have placed pressure on refiners to mix heavier petroleum fractions into fluid catalytic cracking (FCC) feeds. Unfortunately, these heavier fractions contain porphyrins-complex, long-chain hydrocarbons with metal ions attached. As the porphyrins are cracked in the reactor, the metals, chiefly nickel and vanadium, are deposited on the catalyst surface. The metals are themselves catalytic. Through their promotion of undesired dehydrogenation reactions, they create many process problems, including increased coking, reduced yield of gasoline, and reduced plant throughput due to limitations in ability to process the extra light gases (Connor et al., 1957; Grane et al., 1961; Habib et al., 1977; Venuto and Habib, 1978). While the problem of metals dontamination is widespread and serious, little has been published on the subject other than process studies outlining the effects of the phenomenon and/or the efficacies of a particular (patented) response to it. Clearly, more fundamental work examining the characteristics of metals-contaminated cracking catalysts is needed. Because of the complex naturc of FCC feed mixtures found in the modern refinery, it is very difficult to gain widely applicable knowledge from the results of refineryscale or pilot-plant tests. Studies involving the reaction of pure single-component feeds are often used to gain more easily interpretable results. Cumene and methanol were selected for this study. Cumene has been widely used for some time as an indicator of cracking activity (Rabo, 1976), and as such, much is known about its reaction over zeolites. The selection of methanol was inspired by its marked Present address: Johnson Matthey Catalytic Systems, 456 Devon Park Dr., Wayne, PA 19087. 0196-4313/86/1025-0581$01.50/0
difference in chemical nature from that of cumene and its recent emergence as a potential source of hydrocarbon fuels by reaction over zeolite catalysts. The single-component approach was also usefully applied to the catalyst itself, a composite of crystalline zeolite and kaolin clay dispersed in amorphous silica/alumina. Behavior of a commercial catalyst CBZ-1 was compared with the three individual components of the catalyst and a physical mixture of the same, for both Ni-contaminated and uncontaminated samples. The experimental program revealed significant differences between methanol and cumene in terms of activity and product distribution when reacted over the nickelcontaminated cracking catalyst. Interesting differences in sintering and reaction behavior of nickel deposits on several components of the catalyst were also observed. Certain of these differences may be used to advantage, even profit. Others are merely suggestive. When considered together, they lead to several useful conclusions concerning the nature of nickel-contaminated cracking catalyst. Literature Background Zeolite Catalysts. There is a rich literature concerning
zeolites and zeolite catalysis. Rabo (1976) has provided a most valuable review. For present purposes it is important to recognize that the typical commercial cracking catalyst is prepared by uniformly dispersing crystalline zeolite particles together with a clay diluent such as kaolin in a separately prepared silica/alumina gel. The mixture is spray-dried to produce the final form of small particles (Venuto and Habib, 1978). The major source of the catalytic activity of zeolites (at least, their carbonium ion activity) appears to be the existence of strongly acidic protons. Various factors have been found to affect catalytic 0 1986 American Chemical
Society
