Chemical Engineering Science, Vol. 46, No. 2, pp. 451458,
Printed in Great Britain.
ANALYSIS FOR HEAT EXCHANGER NETWORKS+
R. RATNAM Chemical
cam-2509/91 1E3.w l 0.00 0 1990 Pergamon Press plc
(Received 15 February
and V. S. PATWARDHAN$ National
11 June 1990)
Abstract-The sensitivity of a heat exchanger network (HEN) refers to the deviations in network temperature as a result of changes in operating parameters such as source temperatures, UA values and Row rates. Generating such information by varying one parameter at a time and solving the HEN all over again is straightforward but time consuming. In this study a new approach is developed which is based on the inherently linear nature of HENS. This approach eliminates the need for repeated matrix inversIons and leads to an efficient computational procedure for generating the entire sensitivity information. Illustrative examples are also presented which demonstrate the computational efficiency and indicate that the procedures developed can be applied even when the deviations in operating parameters are large and simultaneous.
The design often been
of heat exchanger networks (HENS) has dominated by two important considera-
tions, namely the utility requirement and the capital cost, mainly because both can be quantified in terms of money. In contrast, the flexibility and the operability of HENS have received attention only rather recently (for example see Saboo and Morari, 1983; Saboo et al., 1985; Marselle et al., 1982; Calandranis and Stephanopoulos, 1986; Halemane and Grossmann, 1983; Swaney and Grossmann, 1985a, b). Kotjabasakis and Linnhoff (1986) have described a procedure for considering flexibility right at the design stage which establishes a trade off between flexibility and the total cost of an HEN. Their procedure makes extensive use of what they termed as sensitivity tables. The sensitivity of an HEN can be characterized by the deviations in network temperatures (global output as well as intermediate temperatures) resulting from given changes in operating conditions such as source temperatures, UA values and stream flow rates. Such information is obviously important in designing HENS which are expected to deal with planned process changes
as well as random
size, the amount of information to be generated would increase very rapidly indeed. In this paper we address the problem of generating such information in a computationally efficient manner by taking advantage of the linearity that is inherent in HENS. An obvious, brute force method for generating sensitivity information is to perturb one parameter at a time, solve the resulting network as a fresh problem and compare the resulting temperatures with the base case values. This involves repetitive application of the same solution procedure, which becomes time consuming in view of the large number of cases that need to be considered. In this paper, the HEN problem is formulated as a linear problem. It is then shown that once the base case solution is obtained, the effect of variation in any parameter (or any arbitrary combination of parameters) can be calculated with only a minor additional computational effort by making use of the base case results already available. The procedure is illustrated with examples and the results are compared with those reported in the literature. It is also shown that the present procedure is applicable even for large deviations in operational parameters.
in designing control schemes. It is worthwhile looking at the size of the problem that one has to tackle for generating complete information regarding the sensitivity of an HEN. In the illustrative example considered later, there are four process streams, 11 unknown stream temperatures and seven heat exchangers. Thus one has to look at the deviations in 11 temperatures for given changes in each of the 15 operating parameters (four source temperatures, four heat capacity flowrates and seven UA values). In a practical problem which could be of a much larger
‘NCL Communication No. 4852. ‘To whom correspondence should be addressed. 451
Consider the counterflow heat exchanger Fig. 1. The heat balance equations
and the design
equation 4 = (UAW,,
can be algebraically manipulated to relate the output temperature to the input temperatures as -T2+~1T1+(l-a,)T,=0
RATNAM and CP
Fig. 1. Schematic
diagram of a countercurrent changer.
where a1 =(P-R)exp(P)/[Pexp(P)-Rexp(R)] 0~~= (R - P) exp (R)/[R
(R)- P exp (P)]
P = (UA)/F,
R = ( UA)/Fi.
It is easily shown
that a, and txZ are related
P(1 - a,) = R(1 - az).
Equations (4)-(g) have been presented earlier (Kern, 1950; Kotjabasakis and Linnhoff, 1986) though in slightly different forms. It may be noted that the sum of the coefficients in eqs(4) and (5) are zero. The parameters P and R (and also the flow ratio Q = FJF, = R/P) are positive. Moreover, both a, and a2 are positive and lie in the range 0 < a, (and G(~)< 1. a, and a2 also have the following physical meaning: 011= (r,
a2 = tT4 - T,)l(T,
The utility of these D: values and their relationship with design parameters of heat exchangers of various geometries has been considered in detail elsewhere (Ratnam and Patwardhan, 1990). Let us now consider the solution procedure for a network of heat exchangers. If equations such as eqs (4) and (5) are written for each heat exchanger in the network of m heat exchangers, we get 2m equations. If there are N global streams then the number of intermediate streams is (2m - N), whose temperatures are unknown. These, together with the N unknown global output temperatures, give a total of 2m unknowns. Thus we have a system of 2m linear equations with 2m unknown temperatures, giving a well formulated problem. Each equation involves three temperatures, at least one of which is unknown. We may also have equations where all three temperatures are unknown. We now transfer all the terms involving known source temperatures to the right hand side to get a system of equations that can be represented as AT=B
where A is a 2m x 2m matrix and T and B are 2mdimensional column vectors. For a given network with known stream properties (source temperatures and heat capacity flow rates) and known UA values, eq. (13) can be solved to give T =A-‘B,
whereby all the temperatures become known. In some cases it may be possible to permute matrix A by row/column operations to a block form which will reduce the computational effort involved in matrix inversion. Consider the case of a network involving stream splits. As far as temperatures are concerned, a splitter does not introduce any new unknown temperatures as all the outlet temperatures in a splitter are equal to the inlet temperature. In the case of a mixer, the outlet temperature is an unknown parameter. However, we get one more equation as the outlet temperature of the mixer can be related to the inlet temperatures and heat capacity flow rates by a heat balance equation. Moreover, this gives an equation that is linear in temperatures. Therefore, the form of eq. (13) remains unchanged even when the network has splitters and/or mixers. The following analysis, which uses eq. (13) as the starting point, is thus applicable in case of splitters and mixers as well, though they are not specifically considered here. SENSITIVITY ANALYSIS
Here we are concerned with the effect of changes in one of the known parameters (source temperatures, heat capacity flow rates and UA values) on the network performance. It may be stressed here that changes in some of the parameters may arise due to fluctuations in the whole plant, or due to causes such as fouling, while some others, such as changes in utility flow rates, can be introduced deliberately. The sensitivity analysis of a network can be a very useful guide in deciding whether the effects of some uncontrollable changes can be nullified by the appropriate manipulation of the controllable factors. The most obvious, brute force method to conduct a sensitivity analysis is to change one parameter at a time, calculate the new A matrix and B vector, and use eq. (14) to calculate the new T vector. Such a procedure involves one matrix inversion for every new value of each parameter, and is computationally lengthy for a complete sensitivity analysis of even medium-sized problems. Such a procedure has been used by Kotjabasakis and Linnhoff (1986), who were mainly interested in illustrating the use of sensitivity analysis in network design. It may be noted that they did not use matrix inversion as such. The example used by them could be solved by a sequential procedure. The entire procedure was used repeatedly by them for calculating the effect of change in each parameter. Here we present a different approach for conducting sensitivity analysis in a computationally efficient manner by avoiding the need for repeated matrix inversions. This is particularly valuable in using sensitivity analysis for network design even for large problems, and where interactive software tools are being used. Let the base case solution be calculated as Tb=A;lh where the subscript
b refers to the base case. A change
Sensitivity analysis for heat exchanger networks in any of the F or UA values changes both A and B while a change in T, values changes only the B vector
(since all the T, terms were transferred to the right hand side). In general, the new problem may be specified as (Ab + AA)(T, + 4) = b
where AA and AB are the changes in A and B resulting from changes in T,, F and WA values, and 4 (a 2m x 1 vector) represents the deviations in different network temperatures (both global outputs and intermediate streams) that we are interested in. For a given situation, the easiest way of calculating AA and AB is to calculate A,,, and B,,, and subtract from them A, and B, respectively. Since A is a sparse matrix and B is a vector, this does not involve much computation. Expanding eq. (16). subtracting the base case equation *bTb
and neglecting the second order term (AA) 4 (which is justified for small deviations; the case of large deviations is considered later), we get A,# + AAT, = AB, which can be rewritten
4 = A; ‘[AB - (AA)T,].
It is important to note that the inverse appearing in this equation is the base case inverse which is already known. To determine the effect of any parameter change, we just have to calculate corresponding AB and AA, and use eq. (19X which now involves only a matrix multiplication but no additional matrix inversion. It is also worthwhile examining at this point the extent of computation involved. If any of the T, changes, AA = 0 and only two elements of AB are non-zero, which correspond to the equations for the heat exchanger which has the changed T, as an input stream. If any of the UA values change, then at most two rows of AA and at most two elements of AB are non-zero, which correspond to the equations for that particular heat exchanger. If any of the F values change, and if m’ is the number of heat exchangers traversed by that stream, then 2m’ rows of AA and 2m’ elements of AB are non-zero. It is also important to note that any row of AA has at the most three nonzero elements, as is obvious from eqs (4) and (5). These three groups of parameters and their effect on the whole network is considered in detail in the following analysis. Variation in source temperature For this case, AA vanishes and eq. (18) gives I=
of Tsi in these elements.
AB,=pikATsi AB,=&AT,, and the other elements then gives
of AB are zero. Equation
@j = (DjkP:k +Di,Bii)ATsi
where D represents A; 1 (for convenience). The linearity between the deviation in any network temperature and the variation in any Tsi is obvious. It is also easy to show that if several T, values vary simultaneously, their corresponding #j values can be calculated independently and can simply be added up. Variation in UA values When any one or several of the UA values change, both AA and AB are non-zero. An examination of the starting equations, i.e. eqs (4) and (5), indicates that each equation involves only one parameter 01,apart from temperatures. Each of the 2m values depends upon just one UA value, and two flow rates. Consider the case where a particular u, say CL,,,changes. ar appears only in the kth row of A as well as B. It has been mentioned earlier that each equation involves one output and two input temperatures for some heat exchanger. Let TO be the output temperature appearing in the kth equation. Let us define @= -
T; + T;
where T’, is the base case input temperature corresponding to the output temperature T,, and T; is the other base case input temperature. Since the form of the kth equation is given by eq. (4), it is possible to write [AB-(AA)T&=&Aa,.
This equation is valid even when several a values vary simultaneously, since only one a value appears in each equation. Combining this with eq. (19) we get
which can be rewritten
as 4 = SAa
S, = Dij.
The matrix S can be calculated using the base case calculations and source temperatures for a given network. In order to calculate the effect of variation in any or several of the UA values, all that remains to be done is to calculate the new values, evaluate the Aa vector and use eq. (26) to get 4. It may be noted here that S is calculated only once. This leads to a tremendous saving of computation time.
Let us consider the variation of the source temperature TSi. Let k and I be the element numbers of B which involve TSi. Let pa and Be be the corresponding
Variation in only one UA value When only a single UA value, say (UA),, is varied, a further simplification takes place. Let the kth and Ith
and V.S. PATWARDHAN
equations refer to the ith heat exchanger. Then, all elements of Au except the kth and Ith elements are zero. Moreover, since CL* and a, are related by eq. (lo), we get Aa, = QiAa,
where Qi is an appropriately defined flow ratio which is constant. Equation (26) then gives
41 = (Q,Sj,
Variation in F values When any or several of the F values change, both AA and AB are non-zero, in general. As mentioned earlier, each of the 2m a values depends upon two flow rates. A change in any F value changes several a values simultaneously, depending on the heat exchangers traversed by that particular stream. Equations (24)-(27) are applicable in this case as well. Even when only one flow rate changes, at least two a values change. In this case, no linearity such as that given by eq. (28) exists. Therefore, the bj values vary in a complex non-linear manner as flow rates are changed, as shown later in the iilustrative examples. The case of large deviations Equation (18) was obtained from eq. (16) by ignoring the second order terms, which is justified when the deviations are small. If this assumption is removed, then eq. (16) gives b = D(AB - (AA)Cr, + I))
f$(‘) = 0,
4 t+ l) = 7 D,,$,g)(Aa)y).
in 4. It was found that by successive substitu-
r = 0
4(‘+ ‘) = D(AB - (AA)(Tb + &‘))),
r > 0.
(Am),. does not vary from iteration to iteration. Thus, only +y) needs to be calculated in each iteration to evaluate 4 (‘* L ‘) . This was done by using an index table prepared by the preprocessor (described later) to keep track of the stream numbers corresponding to each of the 2m heat exchanger equations.
Using the concepts developed in the earlier sections, a generalized software was developed for determining the sensitivity of any heat exchanger network with or without loops and recycles. It has a preprocessor at the front end to facilitate data input. Network topology can be input simply by specifying stream numbers for each heat exchanger in a particular order. The preprocessor computes the A matrix and the B vector, which reflect the network topology. In order to illustrate the results we use the example described by Kotjabasakis and Linnhoff (1986) with the stream data given in Table 1 and the network shown in Fig. 2. The UA values for individual heat exchangers were not specified explicitly by them, and were calculated from their Fig. 2, which gives heat loads for all heat exchangers along with the network temperatures at the base case. Table 2 shows the base case results. Global input temperatures are not listed in this table. The column headed “K and L no.” gives the stream numbers used by Kotjabasakis and Linnhoff (1986), which facilitates comparison with their results. Incidentally, there is a small discrepancy between their Table 1 and their Fig. 2 regarding the target temperature of stream no. 4. It has been taken as 240°C as per their Fig. 2, since the temperature of 280°C mentioned in their Table 1 is clearly unattainable using stream at 254°C as specified in their Table 2.
Table 1. Stream data for example Stream “0.
7 13 16 4 10 19
Thus, all deviations in network temperatures are proportional to A\cr,values. It may be mentioned here that as UA varies between 0 and cc, the corresponding a varies from 0 to 1. Therefore, Aa is not proportional to AUA except in close vicinity of the base case. Consequently, ( also varies non-linearly with [IA for large changes in WA. This is quite evident from the sensitivity tables presented by Kotjabasakis and Linnhoff (1986).
which is an implicit equation this can be iteratively solved tion, using
Just two or three iterations were found to give good accuracy even for those parameter changes which were very large from a practical point of view. This is further illustrated later with numerical examples. It is worthwhile examining the computational effort involved in evaluating r#P+l) from eq. (31). Using eq. (24), eq. (31) can be rewritten as
Temperature s0”rl.x target WI
200.0 40.0 140.0
20.0 20.0 254.0
40.0 40.0 254.0
Hot Hot Cold
Cold Cooling water Cooling water Condensing steam
40.0 120.0 70.0
for heat exchanger
Fig. 2. Example
Table 2. Base
Stream no. 2 3 5 6 8 9 11 12 14 15 17 18 20 21
case intermediate temperatures K and L no.+ 1 2 3 4 5 6 7 8 9 10 11
with heat loads and stream
Temperature (“C) 213.3 160.0 80.0 40.0 160.0 71.1 40.0 40.0 140.0 180.0 170.0 213.3 254.0 240.0
+The K and L stream number refers to the corresponding stream number for this example in Kotjabasakis and Linnhoff (1986).
Variation in source temperature The sensitivity of different network temperatures towards different source temperatures is given by the terms in parentheses in eq. (22). The results of this calculation are shown in Table 3. The numbers represent the deviation in network temperature for a 1°C variation in source temperatures. This table matches with the results of Kotjabasakis and Linnhoff (1986), except for the sensitivity of T, L. If we consider a 1°C variation in T,, T,, becomes 213.3 + 0.333 = 213.633”C. With this as the input temperature to the heater, the output temperature T,, from the heater can be easily calculated and turns out to be 24O.llPC. This shows that the sensitivity shown in our Table 2 is the correct sensitivity. For the same reason discrepancies in deviations of this particular temperature have been ignored in subsequent sections.
Variation in UA values Consider a variation in the (UA), value. The variations in network temperature calculated from eq. (26) are shown in Table 4 and are seen to match exactly with the results of Kotjabasakis and LinnhofT (1986, their Table 5b). Figure 3 shows some of these in a graphical form. It is seen that 4 values vary with (UA), value in a monotonic but strongly non-linear manner. Equation(29), however, indicates that #j is directly proportional to Alcc,. Figure 4 shows #j values as a function of Au,. Though linearity between tij and Au, is expected in the vicinity of the base case, the linearity that is seen even for large changes in (Lr.4), is somewhat surprising. This certainly does n’ot follow from eq. (19), which assumes deviations to be small, in some sense. The reason lies in the network structure. The example network in Fig. 2 is acyclic. For such networks, the change in the UA of any exchanger and the resulting changes in the output temperatures of that heat exchanger affects other (downstream) temperatures in an exactly linear manner even for large changes. For cyclic networks, however, the change in the UA value of a heat exchanger could influence the inlet temperatures of the same exchanger (due to information feedback inherent in cyclic networks), in which case, 4 values may not remain linear to Aa for large values of Aa. This would necessitate the use of eqs (30)-(32) developed for large deviations. Variation in F values In this case, eq. (26) has to be applied by actually calculating Aa for the new flow rate values. The results obtained by varying the heat capacity flow rate of source stream 4 are shown in Table 5 and are seen to match the results of Kotjabasakis and Linnhoff (1986). Simultaneous variation of several parameters As an illustration, consider the case used by Kotjabasakis and Linnhoff, where r, is 270°C (in-
and V. S.
Table 3. Sensitivity of network temperatures (in “C) for 1°C change in source temperature Source stream number Stream no.
0.3334 0.2929 0.1255 0.0418 0.3333 0.0864 0.0338 0.0338 0.2778 0.308 1 0.5000 0.3333 O.OOW 0.1147
0.3333 0.0909 0.0389 0.0130 0.0000 O.oooO 0.0000 O.OOCKl [email protected]
0.1818 O.CWOO 0.3333 0.0000 0.1147
2 3 5 6 8 9 11 12 14 15 17 18 20 21
13 0.0000 0.1212 0.0519 0.0173 0.0000 0.7408 0.2898 0.2899 0.1666 0.0757 0.0000 O.Mx)o O.QOCQ 0.0000
0.3333 0.4950 0.2122 0.0707 0.6667 0.1728 0.0676 0.0676 0.5556 0.4344 0.5000 0.3333 0.0000 0.1147
Table 4. Sensitivity of network temperatures (in “C) for change in UA of heat exchanger no. 2 Percentage
2 3 5 6 ; 11 12 14 15 17 18 ::
- 12.42 11.68 5.00
1.67 24.8 5 6.44 2.52 2.52 20.71 2.63 - 18.64 ~ 12.42 0.00 - 4.27
- 4.27 4.02 1.72 0.57 8.55 2.22 0.87 0.87 7.12 0.9 1 - 6.41 ~ 4.27 0.00 - 1.47
- 1.85 1.73 0.74 0.25 3.69 0.96 0.37 0.37 3.08 0.39 - 2.77 1.85 0.00 - 0.64
5.00 - 4.70 - 2.01 - 0.67 - 10.00 - 2.59 - 1.01 - 1.01 - x.33 - 1.06 7.50 5.00 0.00 1.72
7.14 - 6.71 - 2.88 - 0.96 - 14.29 - 3.70 - 1.45 - 1.45 - 11.91 - 1.51 10.71 7.14 0.00 2.46
8.26 ~ 7.76 - 3.33 - 1.11 - 16.52 - 4.28 - 1.68 - 1.68 - 13.77 - 1.75 12.39 8.26 0.00 2.84
Fig. 3. Effect of (UA), on several 4 values.
for heat exchanger
stead of the base case value of 3OO”C), T,, is 20°C (instead of 40°C) and F,, is 72 kW/“C (instead of 60 kW/“C). For each of these three changes, they used their sensitivity tables to calculate the effect of each
S -L ff
and then added deviations.
Fig. 4. Variation of Q values with CLvalue.
up the effects to estimate example,
(in “C) for change
2 3 5 6 8 9 11 12 14 15 17 18 20 21
2 1.24 10.89 4.67 1.56 8.42 2.18 0.85 0.85 7.02 14.78 17.37 39.47 0.00 12.88
t 5.29 7.63 3.27 1.09 5.72 1.48 0.58 0.58 4.76 10.51 12.86 29.00 0.00 11.63
6.33 3.04 1.30 0.43 2.17
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 O.CNl
0.22 1.81 4.27 5.47 12.35 0.00 6.50
6. Iterative calculation
in CP of stream
K and L no. 1 2 3
2 3 5
6 8 9 11 12 14 15 17 18 20 21 ‘This
4 5 6 7 8 9 10 11 column
~ 146.3585 - 84.9118 _ 36.3906 - 12.1303 _ 36.0422 18.9642 7.4191 7.42 19 ~ 61.8822 - 107.9672 _ 106.9794 _ 72.6382 - 0.0036 - 41.6373
- 106.9794 - 72.0422 - 0.0031 - 30.7579
the entire network
4.65 2.14 0.92 0.31 1.44 0.37 0.15 0.15 1.20 3.08 4.10 9.39 0.00 - 6.57
8.19 3.72 1.59 0.53 2.45 0.64 0.25 0.25 2.05 5.40 7.26 ~ 16.71 0.00 - 12.71
9.66 4.36 1.87 0.62 2.84 0.74 0.29 0.29 2.38 6.35 8.57 ~ 19.80 0.00 - 15.57
of 4 values (in “C) by eq. (31) tterative
Of the three parameters that are from the base case, two are source temper-
change in T, was - lO.l”C. If the network is solved as a new problem with the changed values of parameters, the change in T, was found to be - 9.816”C. It is interesting to see the results obtained using the equations developed in this study. Application of eq. (19X which covers the effect of all parameter changes simultaneously (but uses only the base case matrix inverse), gives #‘$) = - 9.784”C, which is a much more accurate estimate of the change in T, than that obtained by adding up individual effects. Both these approaches are valid only for small enough changes in parameters. As the changes increase, their accuracy suffers. Application of eq. (31), which represents an iterative approach, for two iterations gives @\I) = - 9.784”C and 4:‘) = - 9.816”C. Thus, only
85.6214 36.6948 12.2317 36.0422 31.1163 12.1731 12.1777 - 75.5532 - 95.7009
with eq. (31)
~ 146.3585 - 84.9118 - 36.3906 - 12.1303 - 36.0422 18.9642 7.4191 7.4219 - 61.8822 - 107.9672 _ 106.9794 _ 72.63X2 - 0.0036 - 41.5497 with new parameter
~ 146.3585 - 84.9118 - 36.3906 - 12.1303 - 36.0422 18.9642 7.4191 7.4219 - 61.8822 ~ 107.9672 _ 106.9794 - 72.63X2 - 0.0036 - 41.6373 values as a new
R. RATNAM and V. S.
of heat exchangers of streams
atures. It has been indicated earlier that when only source temperatures are changed, the 4 values are strictly linear even for large deviations. The third parameter that was changed was one of the heat capacity flow rates which was increased by only 20%. Therefore, the estimate of 45 obtained by using sensitivity tables was quite accurate.
WA)IF, RIP (uA)/‘Fi matrix defined by eq. (27) vector of unknown temperatures
The case of large deviations
eq. (13) source temperature
Let us now consider what happens when both UA and F values are also varied in addition to source temperatures, also by rather large amounts. This will bring out the utility of eq. (3 1) clearly. Consider a case where (UA), is increased by 300%, (UA), and (UA), are reduced by X0”/ each, F4 is increased by 50%, T, is reduced by 50°C and T,, is reduced by 90°C. These are very large parameter changes indeed. Table 6 shows the results obtained both by eq. (15) (i.e. by considering it as a new problem) and by the iterative application of eq. (3 1). First of all, it may be noted that some of the deviations obtained are larger than 100°C due to the large parameter changes. Tn spite of this, it is seen that just two iterations of eq. (31) are enough to calculate deviations correct to four decimal places. The only exception is Tzl, where a third iteration is required to get all four decimals correct. This example, though it involves impractically large parameter changes, clearly brings out the robustness and the inherent convergence possessed by eq. (31). If one attempts to estimate the deviations in this case by using sensitivity tables and adding up the individual deviations, the results are far from satisfactory (for example, #Jo, #+, &Ii and 41s are estimated as ~ 107.77, 24.29, 10.19 and - 100.31”C respectively, which are far different from the exact values shown in Table 6). This is hardly surprising in view of the large changes in parameters. CONCLUSIONS
The partial linearity inherent in heat exchanger networks can be exploited to conduct the complete sensitivity analysis in a very efficient manner. If the effect of simultaneous changes in several parameters (source temperature, UA values and heat capacity flow rates) on network temperatures is to be estimated, one can use eq. (19) or eq. (31) depending upon the magnitude of the changes. Even for very large changes, the iterative procedure of eq. (31) converges rapidly to the correct network temperatures. NOTATION
A B D F
square design matrix ve$x in eq. (13)
in eq. (13)
heat capacity flow rate defined in Fig. 1
temperatures shown in Fig. 1 defined in eq. (23) overall heat transfer coefficient for a heat exchanger
Subscripts b k
base case refers to kth equation log mean value output
1, 2, r Greek El, a2 :
defined by eqs (6) and (7) defined in eq. (21) vector of deviations in network atures defined by eq. (23)
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