ON THE CALCULATION OF COMPLEX HEAT EXCHANGERS

G. I. Fuks

1

We shall call a complex heat exchanger a system that is composed of an arbitrary number of simple heat exchangers. The individual heat exchangers or parts of the system may be of different types. Series and parallel connection of parts of the system is possible. With series connection, the general direction of the heat-exchanging fluid may be in parallel flow or in counterflow. With parallel connection, one of the heat-exchanging fluids passes through the individual heat exchangers in series, while the other is branched among them into parallel streams. The heat-transfer coefficient and the heat capacities of the fluids for the individual heat exchangers of the system may be different.

The calculation of complex heat exchangers, both direct and checking calculations, is usually associated with cumbersome computations. These calculations can be simplified if the complex heat exchanger is replaced by an equivalent one. The equivalent heat exchanger of the system must give the same thermal effect as the system, i.e., the same heat transfer, and have the same value of the heating surface. Below, methods are considered for selecting an equivalent heat exchanger for a system.

2

The scheme of a system of series heat exchangers with a common parallel flow of the heat-exchanging fluids is given in Fig. 1. The individual heat exchangers \(a, b, c\) may be of any type. For each of the heat exchangers, the dependence \(p = f(\varepsilon, R)\) is known [1]. The cooling criterion of the hot fluid for the whole system is

\[ p = \frac{t_1' - t_1''}{t_1' - t_2'} \tag{a} \]

For the individual heat exchangers \(a, b, c \ldots\), the heat-transfer coefficients, water equivalents, and surfaces are respectively equal to \(k_a, k_b \ldots W_{1a}, W_{2a} \ldots F_a, F_b \ldots\). Then for heat exchangers \(a\) and \(b\)

\[ \varepsilon_a = \frac{k_a F_a}{W_{1a}}, \tag{b} \]

\[ R_a=\frac{W_{1a}}{W_{2a}}\cdot \frac{t_{2ab}-t'_2}{t'_1-t_{1ab}}, \tag{c} \]

\[ p_a=\frac{t'_1-t_{1ab}}{t'_1-t'_2}, \tag{d} \]

\[ R_b=\frac{W_{1b}}{W_{2b}}\cdot \frac{t_{2bc}-t_{2ab}}{t_{1ab}-t_{1bc}}, \tag{c'} \]

Fig. 1.

\[ p_b=\frac{t_{1ab}-t_{1bc}}{t_{1ab}-t_{2ab}}, \tag{d'} \]

\[ \varphi_b=\frac{\kappa_b F_b}{W_{1b}}. \tag{b} \]

For heat exchanger \(b\), from \((d')\), \((d)\), and \((c)\) we have

\[ p_b= \frac{\dfrac{t'_1-t_{1bc}}{t'_1-t'_2}-p_a} {1-p_a-p_aR_a}. \tag{e} \]

If the complex heat exchanger consists of two, \(a\) and \(b\), then \(t_{1bc}=t_1\), and the fraction in the numerator of the right-hand side of \((e)\) is equal to \(p\). Therefore

\[ p_{(2)}=p_a+p_b-p_a(1+R_a). \tag{1} \]

An analogous calculation gives, for a heat exchanger consisting of three parts (Fig. 1):

\[ p_{(3)}=(p_a+p_b+p_c)-[p_a(1+R_a)p_b+p_a(1+R_a)p_c+p_b(1+R_b)p_c]+ \]

\[ +\,p_a(1+R_a)p_b(1+R_b)p_c. \tag{2} \]

In general, for a series system composed of \(n\) simple heat exchangers, with an overall parallel flow, we have

\[ p_{(n)}=(p_a+p_b+\ldots+p_n)-[p_a(1+R_a)p_b+p_a(1+R_a)p_c+\ldots+ \]

\[ +p_{n-1}(1+R_{n-1})p_n]+[p_a(1+R_a)p_b(1+R_b)p_c+p_a(1+R_a)\cdot \]

\[ p_b(1+R_b)p_c+\ldots+p_{n-2}(1+R_{n-2})p_{n-1}(1+R_{n-1})p_n]- \]

\[ -[p_a(1+R_a)p_b(1+R_b)p_c(1+R_c)p_d+\ldots+p_{n-3}(1+ \]

\[ +R_{n-3})p_{n-2}(1+R_{n-2})p_{n-1}(1+R_{n-1})p_n]+\ldots+ \]

\[ +(-1)^{n-1}p_a(1+R_a)p_b(1+R_b)\ldots p_{n-1}(1+R_{n-1})p_n . \tag{3} \]

The construction of relation (3) is clear. In the first brackets stands the sum of the values \(p_a, p_b\ldots\) for all heat exchangers. The number of terms in this bracket is equal to \(C_n^1\cdot n\). The number of terms in the second bracket is \(C_n^2\), and so on. Relations (1) and (2) can be obtained as particular cases for \(n=2\) and \(n=3\), respectively.

If it is assumed that the water equivalents of the individual heat exchangers are identical, i.e.

\[ \begin{array}{cccc} W_{1a} & W_{1b} & \ldots & W_1,\\ W_{2a} & W_{2b} & \ldots & W_2, \end{array} \]

then

\[ \begin{array}{cccc} R_a & R_b & \ldots & R, \end{array} \]

where \(W_1\), \(W_2\), and \(R\) are the corresponding values for the whole heat exchanger. Then

\[ p_{(n)}=\sum\left[p-(1+R)\,p\cdot p+(1+R)^2p\cdot p\cdot p\right. \]

\[ \left.-(1+R)^3p\cdot p\cdot p+\ldots+(-1)^{n-1}(1+R)^{n-1}p\cdot p\ldots p\cdot p\right]. \tag{4} \]

Here \(\sum p=p_a+p_b+\ldots+p_n;\quad \sum p\cdot p=p_ap_b+p_ap_c+\ldots\)

The number of such terms will be equal to \(C_n^1\), \(C_n^2\), and so on. In the last term stands the product \(p_ap_b\ldots p_n\).

The criteria of cooling of the hot liquid \(p\) and heating of the cold liquid \(p'\) are related to one another by the relation \(p=p'R'\), where

\[ R'=\frac{1}{R}\quad [1]. \]

Therefore we also have

\[ p'_{(n)}=\sum\left[p'-(1+R')p'\cdot p' +(1+R')^2p'\cdot p'\cdot p'-(1+ \]

\[ +R')^3p'\cdot p'\cdot p'\cdot p'+\ldots+(-1)^{n-1}(1+R')^{n-1}p'\cdot p'\ldots p'\cdot p'\right]. \tag{5} \]

From (4) and (5) it is seen that the result of operation of a system of heat exchangers, i.e. the relative cooling of the hot liquid or the relative heating of the cold liquid, does not change if the individual heat exchangers are interchanged. In doing so, the overall motion of the liquid in the system as a parallel flow must be preserved. For the calculation of intermediate temperatures, each of the heat exchangers must be put “in its own place” in the scheme. Of course, these consequences are valid only under the condition of constant or only slightly varying heat capacities of the heat-exchanging liquids.

If the system consists of individual heat exchangers identical in character, with identical values of the heating-surface criterion \(z_a, z_b,\ldots\), then

\[ \begin{array}{cccc} p_a & p_b & \ldots & p_i,\\ p'_a & p'_b & \ldots & p'_i. \end{array} \]

for the entire system:

\[ p_{(n)} = n p_i - C_n^2(1+R)p_i^2 + C_n^3(1+R)^2p_i^3 - \]
\[ {} - C_n^4(1+R)^3p_i^4 + \ldots + (-1)^{n-1}(1+R)^{n-1}p_i^n; \tag{6} \]

\[ p'_{(n)} = n p'_i - C_n^2(1+R')p_i'^2 + C_n^3(1+R')^2p_i'^3 - C_n^4(1+ \]
\[ {} + R')^3p_i'^4 + \ldots + (-1)^{n-1}(1+R')^{n-1}p_i'^n . \tag{7} \]

By means of simple transformations, from (6) and (7) one can obtain:

\[ p_{(n)}=\frac{1}{1+R}\, \frac{\left[1-p_i(1+R)\right]^n}{1+R}, \tag{8} \]

\[ p_i=\frac{1-\sqrt[n]{\,1-p_{(n)}(1+R)\,}}{1+R}, \tag{9} \]

\[ p'_{n}=\frac{1}{1+R'}\, \frac{\left[1-p'_i(1+R')\right]^n}{1+R'}, \tag{8'} \]

\[ p'_i=\frac{1-\sqrt[n]{\,1-p'_{(n)}(1+R')\,}}{1+R'}. \tag{9'} \]

In this case, the heating surfaces of the individual heat exchangers are related by

\[ k_a F_a = k_b F_b = \ldots = k_i F_i . \]

If the heat-transfer coefficients are identical, then the heating surfaces of the individual heat exchangers are also equal.

3

The scheme of a system consisting of two simple heat exchangers with overall counterflow is shown in Fig. 2. Heat exchangers \(a\) and \(b\) may be of arbitrary type. For the entire heat exchanger:

\[ p'=\frac{t_2''-t_2'}{t_1'-t_2'} . \tag{a} \]

Fig. 2.

For heat exchanger \(a\), the heat-transfer coefficient, heating surface, and water equivalents are respectively equal to \(k_a\), \(F_a\), \(W_{1a}\) and

\(W_{2a}\), i.e. \(R'_a=\dfrac{W_{1a}}{W_{2a}}\) and \(\varphi'_a=\dfrac{\kappa_a F_a}{W_{2a}}\) [1]. The temperature of the hot liquid at the outlet is equal to \(t_{1ab}\), and that of the liquid being heated at the inlet is \(t_{2ab}\). Then, according to [1],

\[ t'_2=t_{2ab}-\frac{p'_a}{1-p'_a}\,(t'_1-t''_2), \tag{b} \]

\[ t'_1=t_{1ab}-\frac{p'_a R'_a}{1-p'_a}\,(t'_1-t''_2). \tag{c} \]

For heat exchanger \(b\):

\[ p'_b=\frac{t_{2ab}-t'_2}{t_{1ab}-t'_2} =\frac{(t'_2-t'_b)-(t'_2-t_{2ab})}{(t'_1-t'_2)-(t'_1-t_{1ab})}. \]

Taking into account \((a)\), \((b)\), and \((c)\), this gives

\[ p'_{(2)}=\frac{p'_a+p'_b-p'_a(1+R'_a)p'_b} {1-R'_a p'_a p'_b}. \tag{10} \]

For a system consisting of three simple heat exchangers \(a\), \(b\), \(c\), an analogous calculation gives

\[ p'_{(3)}= \frac{ (p'_a+p'_b+p'_c)-[p'_a(1+R'_a)p'_b+p'_a(1+R'_a)p'_c+p'_b(1- } { 1-[p'_a R'_a p'_b+p'_a R'_a p'_c } \]

\[ \frac{ +R'_b)p'_c]+p'_a p'_b p'_c(1+R'_a+R'_a R'_b) } { +p'_b R'_b p'_c]+p'_a p'_b p'_c R'_a(1+R'_b) }. \tag{11} \]

For a system of \(n\) simple heat exchangers with a common counterflow of the heat-exchanging liquids, one obtains

\[ p'_{(n)}=\frac{A}{B}, \tag{12} \]

where

\[ \begin{aligned} A={}&(p'_a+p'_b+\cdots+p'_n) -\bigl[p'_a(1+R'_a)p'_b+p'_a(1+R'_a)p'_c+\cdots \\ &\quad +p'_{n-1}(1+R'_{n-1})p'_n\bigr] -\bigl[p'_a p'_b p'_c(1+R'_a+R'_a R'_b)+\cdots \\ &\quad \cdots+p'_{n-2}p'_{n-1}p'_n(1+R'_{n-2}+R'_{n-2}R'_{n-1})\bigr]- \\ &\quad -\bigl[p'_a p'_b p'_c p'_d(1+R'_a+R'_a R'_b+R'_a R'_b R'_c)+\cdots \\ &\quad \cdots+p'_{n-3}p'_{n-2}p'_{n-1}p'_n (1+R'_{n-3}+R'_{n-3}R'_{n-2}+\cdots \\ &\quad \cdots+R'_{n-3}R'_{n-2}R'_{n-1})\bigr] +\cdots+(-1)^{\,n-1}p'_a p'_b\cdots p'_n(1+ \\ &\quad +R'_a+R'_a R'_b+\cdots+R'_a R'_b\cdots R'_{n-1}). \end{aligned} \tag{13} \]

\[ \begin{aligned} B={}&1-\left[p'_a R'_a p'_b+p'_a R'_a p'_c-\cdots+p'_{n-1}R'_{n-1}p'_n\right]+\\ &+\left[p'_a p'_b p'_c R'_a(1+R'_b)+\cdots+p'_{n-2}p'_{n-1}p'_n R'_{n-2}(1+\right.\\ &\left.\quad +R'_{n-1})\right]-\left[p'_a p'_b p'_c p'_d R'_a(1+R'_b+R'_bR'_c)+\cdots+p'_{n-3}p'_{n-2}\right.\\ &\left.\quad p'_{n-1}p'_n R'_{n-3}(1+R'_{n-2}+R'_{n-2}R'_{n-1})\right]+\cdots+(-1)^{n-1}p'_a p'_b p'_c\cdots\\ &\quad \cdots p'_n R'_a(1+R'_b+R'_bR'_c+\cdots+R'_bR'_c\cdots R'_{n-1}). \end{aligned} \tag{14} \]

The number of terms in the second term in the numerator and denominator of (12) is equal to \(C_n^2\), in the third—to \(C_n^3\), and so on. The previously obtained (10) and (11) are particular cases of the latter for \(n=2\) and \(n=3\).

If the water equivalents of all the separate heat exchangers are the same, i.e.,

\[ \begin{gathered} W_{1a}=W_{1b}=\cdots=W_1,\\ W_{2a}=W_{2b}=\cdots=W_2, \end{gathered} \]

then

\[ R'_a=R'_b=\cdots=R', \]

where \(W_1\), \(W_2\), and \(R'\) are the corresponding quantities for the entire heat exchanger. Then

\[ \begin{aligned} p'_{(n)}={}& \frac{\sum\left[p'-(1+R')p'\cdot p'+(1+R'+R'^2)p'\cdot p'\cdot p'-(1+\right.} {\sum\left[1-R'p'\cdot p'+R'(1+R')p'\cdot p'\cdot p'-R'(1+\right.}\\ &\frac{\left.\left.+R'+R'^2+R'^3\right)p'\cdot p'\cdot p'\cdot p'+\cdots+(-1)^{n-1}(1+\right.} {\left.\left.+R'+R'^2\right)p'\cdot p'\cdot p'\cdot p'+\cdots+(-1)^{n-1}R'(1+\right.}\\ &\frac{\left.\left.+R'+R'^2+\cdots+R'^{\,n-1}\right)p'\cdot p'\cdots p'\right]} {\left.\left.+R'+R'^2+\cdots+R'^{\,n-2}\right)p'\cdot p'\cdots p'\right]} . \end{aligned} \tag{15} \]

In this formula

\[ \sum p' = p'_a+p'_b+\cdots+p'_n, \]

\[ \sum p'\cdot p' = p'_a p'_b+p'_a p'_c+\cdots+p'_{n-1}p'_n, \]

\[ \sum p'\cdot p'\cdot p' = p'_a p'_b p'_c+p'_a p'_b p'_d+\cdots+p'_{n-2}p'_{n-1}p'_n,\quad \text{etc.} \]

It can also be written:

\[ \begin{aligned} p_{(n)}={}& \frac{\sum\left[p-(1+R)p\cdot p+(1+R+R^2)p\cdot p\cdot p-\right.} {\sum\left[1-Rp\cdot p+R(1+R)p\cdot p\cdot p-\right.}\\ &\frac{\left.\left.-(1+R+R^2+R^3)p\cdot p\cdot p\cdot p-\cdots+(-1)^{n-1}(1+\right.} {\left.\left.-R(1+R+R^2)p\cdot p\cdot p\cdot p+\cdots+(-1)^{n-1}R(1+\right.}\\ &\frac{\left.\left.+R+R^2+\cdots+R^n\right)p\cdot p\cdots p\right]} {\left.\left.+R+R^2+\cdots+R^{n-1}\right)p\cdot p\cdots p\right]} . \end{aligned} \tag{16} \]

It follows from (15) and (16) that the relative cooling of the hot liquid and the relative heating of the cold liquid in the system do not change if the separate heat exchangers are interchanged,

while preserving the general counterflow direction. If the individual heat exchangers entering the system are of one and the same type and have identical dimensions and identical heat-transfer coefficients, then \(\varphi'_1,\ \varphi'_2,\ \ldots,\ \frac{\varphi'}{n}\) and \(p'_1,\ p'_2,\ \ldots,\ p'_n\).

Then relation (15) takes the form:

\[ p'_{(n)} = \frac{ n p'_i - (1+R') C_n^2 p_i'^2 + (1+R'+R'^2) C_n^3 p_i'^3 - \cdots }{ 1 - R' C_n^2 p_i'^2 + R'(1+R'+R'^2) C_n^3 p_i'^3 + \cdots } \]
\[ \cdot \frac{ {}+ R'^2 C_n^3 p_i'^3 - (1+R'+R'^2+R'^3) C_n^4 p_i'^4 + \cdots }{ {}- R' C_n^2 p_i'^2 + R'(1+R'+R'^2) C_n^4 p_i'^4 + \cdots } \]
\[ {}+ \frac{ (-1)^{n+1}(1+R'+R'^2+\cdots+R'^{\,n-1}) C_n^n p_i'^n }{ (-1)^{n+1} R'(1+R'+R'^2+\cdots+R'^{\,n-2}) C_n^n p_i'^n }. \tag{17} \]

An analogous relation can be written for \(p_n\). In a simpler form one obtains

\[ p'_{(n)} = \frac{ 1-\left(\dfrac{1-p'_i}{1-p'_i R'}\right)^n }{ 1+R'\left(\dfrac{1-p'_i}{1-p'_i R'}\right)^n }. \tag{18} \]

and

\[ p_i = \frac{ 1-\sqrt[n]{\dfrac{1-p'_{(n)}}{1-p'_{(n)}R'}} }{ 1-R'\sqrt[n]{\dfrac{1-R'_{(n)}}{1-p'_{(n)}R'}} }. \tag{18'} \]

Analogous relations for \(p_n\) and \(p_i\) have the form:

\[ p_{(n)} = \frac{ 1-\left(\dfrac{1-p_i}{1-p_i R}\right)^n }{ 1+R\left(\dfrac{1-p_i}{1-p_i R}\right)^n }. \tag{19} \]

and

\[ p_i = \frac{ 1-\sqrt[n]{\dfrac{1-P_{(n)}}{1-p_{(n)}R}} }{ 1-R\sqrt[n]{\dfrac{1-p_{(n)}}{1-p_{(n)}R}} }. \tag{19'} \]

4

A system of heat exchangers in which one of the heat-exchanging fluids branches into parallel streams will be called a parallel-mixed system. Let the parallel-mixed system consist of parts \(a\), \(b\), \(c\) of arbitrary type (Fig. 3). The heated fluid passes through \(a\), \(b\), and \(c\) successively; the hot fluid branches into 3 streams in parallel.

According to relation [1]

\[ 1-p'=\frac{t'_1-t''_2}{t'_1-t'_2}. \]

Fig. 3.

for parts \(a\), \(b\), and \(c\), we have:

\[ 1-p'_a=\frac{t'_1-t_{2ab}}{t'_1-t'_2}, \tag{a} \]

\[ 1-p'_b=\frac{t'_1-t_{2bc}}{t'_1-t_{2ab}}, \tag{b} \]

\[ 1-p'_c=\frac{t'_1-t''_2}{t'_1-t_{2bc}}; \tag{c} \]

for the entire heat exchanger:

\[ 1-p'=\frac{t'_1-t''_2}{t'_1-t'_2}. \tag{d} \]

Consequently,

\[ 1-p'_{(3)}=(1-p'_a)(1-p'_b)(1-p'_c). \tag{20} \]

In the general case of a system of \(n\) parts:

\[ 1-p'_{(n)}=(1-p'_a)(1-p'_b)\ldots(1-p'_n). \tag{21} \]

For a system in which the hot liquid flows in series, while the liquid being heated branches into \(n\) streams, an analogous derivation gives

\[ 1-p_{(n)}=(1-p_a)(1-p_b)\ldots(1-p_n). \tag{22} \]

If the individual parts \(a\), \(b\ldots\) are identical, the branching liquid is divided into equal parts, and the heat-transfer coefficients of the individual parts are the same, then relations (21) and (22) become

\[ 1-p'_{(n)}=(1-p'_i)^n, \tag{23} \]

\[ 1-p_{(n)}=(1-p_i)^n. \tag{24} \]

When calculating the values \(\zeta_a\), \(\zeta_b\), etc. from \(p_a\), \(p_b\), …, it is necessary to take into account the actual value of \(R_a\), \(R_b\), i.e., to know the distribution of the branching liquid among the individual heat exchangers.

5

The relations obtained make it possible to calculate certain complex heat exchangers. A cross-flow heat exchanger with 3 passes, the scheme of which is shown in Fig. 4, may be regarded as a system of identical parts \(a\), \(b\), and \(c\), connected in counterflow. We shall assume that the heat-transfer coefficients, water equivalents, and heating surfaces of all parts are identical, i.e.

\[ W_{1a}=W_{1b}=W_{1c}=W_1, \]

\[ W_{2a}=W_{2b}=W_{2c}=W_2. \]

This means that

\[ R_a=R_b=R_c=R, \]

where \(W_1\), \(W_2\), and \(R\) are the corresponding values for the whole heat exchanger. Under this condition

\[ \zeta_a=\zeta_b=\zeta_c=\frac{\zeta}{3}=\frac{kF}{3W_1}=\zeta_i, \]

\[ p_a=p_b=p_c=p_i. \]

Fig. 4.

In a check calculation, the quantities \(\zeta\) and \(R\) may be regarded as known. In the present case, the parts \(a\), \(b\), and \(c\) are cross-flow heat exchangers with mixing of one or both heat-exchanging fluids. From the corresponding relation or graph for this heat exchanger [1], the value \(p_i\) is found. The relative cooling of the hot liquid is determined from (19). Two of the 4 temperatures, \(t_1'\), \(t_1''\), \(t_2'\), and \(t_2''\), must be specified. Using the relations from [1], the remaining temperatures are determined.

In a direct calculation, the following may be regarded as known:

\[ R=\frac{W_1}{W_2}, \]

and

\[ p_3=\frac{t_1'-t_1''}{t_1'-t_2'}. \]

The value \(p_i\) is determined from (19). From \(p_i\) and \(R\), \(\zeta_i\) is found, and then

\[ F_i=\frac{\zeta_i W_1}{k} \]

and

\[ F=3F_i. \]

Subsequently the calculations may be refined by taking into account the variation of heat capacities and heat-transfer coefficients in the individual parts. This question is considered separately.

When calculating a complex heat exchanger whose scheme is shown in Fig. 5, it may be reduced to an equivalent heat exchanger with the same heat transfer. For this purpose, we “straighten” its scheme along the direction of flow of the hot liquid (Fig. 6). It is seen from the figure that parts \(b\), \(c\), and \(d\) form a counterflow system. If the heat capacity and the heat-transfer coefficient are assumed constant, then, according to (19),

\[ p_{bcd}= \frac{ 1-\left(\dfrac{1-p_i}{1-p_iR}\right)^3 }{ 1-R\left(\dfrac{1-p_i}{1-p_iR}\right)^3 }. \tag{a} \]

Thus, the system of heat exchangers \(b, c, d\) is reduced to the equivalent heat exchanger \(bcd\) with a heating surface of \(3F_i\). Similarly, parts \(e\) and \(f\) constitute a parallel-flow system, for which, by (8),

\[ p_{cf}= \frac{1-\left[1-p_i(1-R)\right]^2}{1-R}. \tag{b} \]

Fig. 5.

Fig. 6.

and the heating surface is equal to \(2F_i\).

As a whole, the heat exchanger under consideration is reduced to a system of three parts according to the scheme given in Fig. 7. Since the parts \(bcd\) and \(ef\) are connected in parallel flow, then, by (4),

\[ p_{bcdef}=p_{bcd}+p_{ef}-(1+R)p_{bcd}p_{ef}. \tag{c} \]

This replaces the parts \(bcd\) and \(ef\) by the equivalent heat exchanger \(bcdef\) with a heating surface of \(5F_i\) (Fig. 8). The entire heat exchanger being calculated may be regarded as consisting of two parts, \(a\) and \(bcdef\), connected in counterflow. Therefore

\[ p= \frac{p_a+p_{bcdef}-(1+R)p_ap_{bcdef}} {1-Rp_ap_{bcdef}}. \tag{d} \]

In a rating calculation the known quantities are \(z_i=\dfrac{kF_i}{W_1}\) and

\[ R=\frac{W_1}{W_2}, \]

from which \(p_i\) is determined. From \((a)\), \((b)\), \((c)\), and \((d)\), the quantities \(p_{bcd}\), \(p_{ef}\), \(p_{bcdef}\), and \(p\) for the entire heat exchanger are determined successively. If, in addition, \(t_1'\) and \(t_2'\) are known, then from

\[ p=\frac{t_1'-t_1''}{t_1'-t_2'} \quad\text{and}\quad R=\frac{t_2''-t_2'}{t_1'-t_1''} \]

one determines \(t_1''\) and \(t_2''\). The intermediate temperatures of the hot and cold fluids between the separate parts can then be determined, and the calculation refined, as shown below.

Fig. 7.

Fig. 8.

In a direct calculation the prescribed quantities are the values of the heat-transfer coefficient \(k_1\), \(W_1\), and \(W_2\), i.e.

\[ R=\frac{W_1}{W_2}, \]

as well as the value

\[ p_{\mathrm{e}}=\frac{t_1'-t_1''}{t_1'-t_2'}. \]

In principle, it is possible, by successively using (a), (b), and (c), to obtain from (d) an equation in which the only unknown quantity will be \(p_i\). From this quantity and \(R\) one can find the values of \(\varepsilon_i\), as indicated in [1], and then \(F_i = \dfrac{\varphi_i W_1}{\kappa}\) and \(F = 6F_i\). But in this case a complicated equation is obtained, which can be solved only by trial.

A more expedient procedure is that of direct trial. Several values of \(\varepsilon_i\) are assigned and the corresponding values \(p_i\) are determined.

Fig. 9.

Then, from (a), (b), (c), and (d), the corresponding values of \(p\) are found. Having plotted the graph \(p = f(\varepsilon_i)\) (Fig. 9), one finds from \(p_0\) the quantity \(\varphi_{i0}\), and from it

\[ F_{i0}=\frac{\varphi_{i0}W_1}{\kappa} \]

and \(F = 6F_i\).

6

A check calculation of a system consisting of an arbitrary number of heat exchangers of different types usually has as its purpose the calculation of the outlet temperatures \(t_1''\) and \(t_2''\), when the inlet temperatures \(t_1'\) and \(t_2'\) are known. If the heat capacities of the heat-exchanging fluids are constant, then the values of the water equivalents \(W_1\) and \(W_2\), and their ratio

\[ R=\frac{W_1}{W_2}, \]

will be the same for the entire system and for the individual heat exchangers. Let the heat-transfer coefficients of the heat exchangers in the system be

\[ \kappa_a,\ \kappa_b,\ldots \]

For each heat exchanger one can determine

\[ \varepsilon_a=\frac{\kappa_a F_a}{W_1}, \]

\[ \varepsilon_b=\frac{\kappa_b F_b}{W_1}. \]

Knowing \(R\), one can, using the equations \(p = f(\varepsilon, R)\) or the corresponding graphs, calculate the values \(p_a, p_b,\ldots\). Depending on the connection scheme of the individual heat exchangers in the system, one can, using (4), (16), (21), or (22), calculate

\[ p=\frac{t_1'-t_i''}{t_i'-t_2'}, \]

which, together with

\[ R=\frac{t_2''-t_2'}{t_1'-t_1''}, \]

makes it possible to determine \(t_1''\) and \(t_2''\).

Since the heat-transfer coefficients \(k_a, k_b,\ldots\) in reality depend on temperature, the calculation can be refined. The values \(p_a, p_b,\ldots\) make it possible to compute the temperatures at the inlet and outlet of each heat exchanger and to refine the values \(k_a, k_b,\ldots\). After this the calculation of the system must be repeated.

With variable heat capacities, the calculation is carried out by successive approximation. As an example, let us consider the counterflow system shown in Fig. 2. For the heat-exchanging liquids the dependences \(i_1=f(t_1)\) and \(i_2=f(t_2)\) are known. First one must specify one of the outlet temperatures, for example \(t'_1\), and find the corresponding value \(t_{2\mathrm{pr}}\). Then, from the heat-balance equation,

\[ G_1\left(i'_1-i'_{1\mathrm{pr}}\right)=G_2\left(i'_{2\mathrm{pr}}-i'_2\right) \tag{25} \]

determine \(i'_{2\mathrm{pr}}\), and from the dependence \(i_2=f(t_2)\)—the corresponding value \(t'_{2\mathrm{pr}}\). The mean values of the water equivalents will be

\[ \overline{W}_1=\frac{G_1\left(i'_1-i'_{1\mathrm{pr}}\right)}{t'_1-t'_2}, \tag{26} \]

\[ \overline{W}_2=\frac{G_2\left(i'_{2\mathrm{pr}}-i'_2\right)}{t'_{2\mathrm{pr}}-t'_2}. \tag{27} \]

From these values one obtains

\[ R'_{\mathrm{pr}}=\frac{\overline{W}_2}{\overline{W}_1}. \tag{a} \]

The first calculation is carried out with \(\overline{W}_{1\mathrm{pr}}\), \(\overline{W}_{2\mathrm{pr}}\), and \(R'_{\mathrm{pr}}\), as indicated above. As a result, the value

\[ p'_{\mathrm{pr}}=\frac{t'_{2}-t'_2}{t'_1-t'_2} \]

is determined, from which a more accurate value of \(t'_2\) can be found and the calculation repeated until sufficient agreement is obtained. The total number of recalculations will not be large, since, as is evident from (26), (27), and (a), the values \(\overline{W}_1\), \(\overline{W}_2\), and \(R'\) will change little during recalculation. The values \(p'_a, p'_b\), and \(t'_2\) from the last calculation are taken as the initial values for further refinement.

Let us now consider heat exchanger \(a\) (Fig. 2). From the relation

\[ 1-p'_a=\frac{t'_1-t'_2}{t'_1-t_{2ab}} \]

[1] \(t_{2ab}\) is determined, and then \(i_{2ab}\). From the heat-balance equations of the heat exchanger,

\[ G_1\left(i'_1-i_{1ab}\right)=G_2\left(i'_2-i_{2ab}\right) \tag{25'} \]

one finds \(i_{1ab}\), and then \(t_{1ab}\). Then

\[ W_{1a}=\frac{G_1\left(i'_1-i_{1ab}\right)}{t'_1-t_{1ab}}, \tag{26'} \]

\[ W_{2a}=\frac{G_2\left(i'_2-i_{2ab}\right)}{t'_2-t_{2ab}}, \tag{27'} \]

\[ R'_a=\frac{W_{2a}}{W_{1a}}. \tag{b} \]

\[ \varphi'_a=\frac{k_a F'_a}{W_{2a}}. \tag{6} \]

From these values of \(R'_a\) and \(\varphi'_a\), a refined value of \(p'_a\) is determined for heat exchanger \(a\), and the calculation is repeated until agreement is reached. As a result of this calculation, \(t'_{1ab}\) and \(t_{2ab}\) are determined before the second heat exchanger of the system, for which the same calculations are carried out, and so on.

As a result of the first calculations for the last heat exchanger of the system, it may turn out that the value \(t''_2\) is not equal to the actual one. Consequently, the value of \(t'_1\) will also not be exact. But

\[ p'=\frac{t''_2-t'_2}{t'_1-t'_2} \]

will be closer to the true value than its initially adopted value. Therefore, taking

\[ p'=\frac{t''_1-t'_2}{t'_1-t'_2}, \]

one can find from it a more accurate value of \(t''_2\), by which the calculation is repeated until two successive values of \(t'_2\) coincide.

In these calculations it was assumed that, within each heat exchanger of the system, the heat capacities and heat-transfer coefficients remain unchanged. If the temperatures in a heat exchanger change significantly, then in some cases it may be divided, for purposes of calculation, into separate parts, i.e., considered as a system. This procedure may be applied:

a) for a parallel-flow heat exchanger, since it is obvious that a system composed of consecutively connected parallel-flow heat exchangers with a common parallel flow is completely equivalent to parallel-flow heat exchange with the total heating surface;

b) for a counterflow heat exchanger;

c) for a cross-flow heat exchanger with mixing of one fluid. Thus, if the hot fluid is mixed, the equivalent systems will consist of several heat exchangers connected in parallel with respect to the heated fluid and in series with respect to the hot fluid;

d) for a cross-flow heat exchanger with multiple passes. Its individual parts are connected in parallel flow or counterflow according to the general direction of flow of the heat-exchanging fluids;

e) for any heat exchanger when \(R=0\) (or \(R'=0\)). Indeed, under this condition, as indicated in [1], the dependence \(p=f(\varphi,R)\) is the same for all such heat exchangers. Consequently, for calculation purposes any heat exchanger with \(R\) or \(R'\) equal to zero may be replaced by an equivalent parallel-flow or counterflow exchanger with the same heating surface.

7

When calculating the heating surface of the system, the values of the inlet and outlet temperatures of the heat-exchanging fluids must be specified; from these the actual value

\[ p_0=\frac{t'_1-t''_1}{t'_1-t'_2} \]

(or \(p'_0\)) is determined. If, in addition, the distribution of the heat transferred among the heat exchangers of the system is specified, then the temperatures between the heat exchangers are determined from the heat-balance equations. The heating surface of each heat exchanger of the system can then be calculated. The values of the heat-transfer coefficients are refined in the course of the calculation, as was indicated for the checking calculation.

Sometimes, in calculating a system it is necessary to maintain a specified ratio of the heating surfaces of the parts of the system:

\[ F_a : F_b : \ldots \quad m : n : \ldots \tag{a} \]

If the heat capacities do not change, then the water equivalents of the parts of the system and of the entire system also remain constant, as does the value of \(R\) (or \(R'\)). Several values \(F'_a, F''_a \ldots\) are assigned, and from \((a)\) the corresponding values \(F'_b, F''_b \ldots\), etc., are determined. For each assignment, a check calculation is carried out as indicated above, as a result of which a series of values \(p_1, p_2, \ldots\) (or \(p'_1, p'_2, \ldots\)) is determined.

Having plotted the dependence \(p = f(F_a)\), one finds, from the actual value \(p_d\), the quantity \(F_a\) (Fig. 9), and then from \((a)\) the quantities \(F_b, F_c \ldots\). In the calculation, the values of the heat-transfer coefficients may be refined, as indicated above.

With variable heat capacities, a direct calculation is possible by means of gradual approximation. It is advisable first to carry out the calculation using the mean values \(\overline{W}_1\) and \(\overline{W}_2\), which are determined from (25), (26), and (27). The values \(F_a, F_b \ldots\) obtained in this calculation are taken as the initial values for subsequent refinements, which are carried out in the form of a check calculation, considering the heat exchangers \(a, b, \ldots\) successively, as indicated above. As a result of the calculations, for each assigned value of \(F_a\) a value of \(p\) is obtained. Having plotted the graph (Fig. 9), the required value \(F_a\) is determined from \(p_d\), and then from \((a)\) the values \(F_b, F_c \ldots\).

References

1. Fuks, G. I. “On the Calculation of Heat Exchangers.” Izvestiya TPI, vol. 109, 1960.