ON THE CALCULATION OF HEAT EXCHANGERS

G. I. Fuks

1.

For the calculation of heat exchangers by the relation

\[ Q = kF\Delta t \tag{1} \]

it is necessary to determine the temperature head \(\Delta t\). With the exception of the simplest cases, the calculation of \(\Delta t\) is carried out by complicated formulas.

In practice it is customary to compute the dimensionless parameters \(\dfrac{\Delta t}{t'_1 - t'_2}\) [1] or \(\dfrac{\Delta t}{\Delta t_{pr}}\) [2], [3], which are expressed as functions of two dimensionless temperature ratios. Usually graphs are used, from which the temperature head \(\Delta t\) is determined, and then the heating surface is calculated from (1).

Thus, when the corresponding graphs are available, carrying out a direct (design) calculation of a heat exchanger presents no particular difficulties.

Carrying out the inverse (checking) calculation of a heat exchanger on the basis of the indicated graphs is possible only by trial, i.e., by repeated calculations. The use of the indicated graphs is also inconvenient for calculations of complex heat exchangers, which may be composed of simple heat exchangers.

2.

Boshnyakovich [2] pointed out that the calculation of a heat exchanger can be carried out without computing the temperature head. From the general equation of the heat balance of a heat exchanger

\[ W_1\left(t'_1 - t''_1\right) = W_2\left(t''_2 - t'_2\right) = kF\Delta t \tag{1'} \]

one can obtain the criterion of the heating surface

\[ \varphi = \frac{kF}{W_1}, \tag{2} \]

determination of which makes it possible to calculate the heating surface \(F\) directly. For each heat exchanger the criterion \(\varphi\) can be found as a function of two others. Boshnyakov proposed using the criteria:

1) the criterion of water equivalents

\[ R=\frac{W_1}{W_2}=\frac{t_2''-t_2'}{t_1'-t_1''}; \tag{3} \]

2) the criterion of cooling of the hot liquid

\[ p=\frac{t_1'-t_1''}{t_1'-t_2'}. \tag{4} \]

In some cases the following system of criteria is more convenient:

\[ \varphi'=\frac{kF}{W_2}, \tag{5} \]

\[ R'=\frac{W_2}{W_1}=\frac{t_1'-t_1''}{t_2''-t_2'}, \tag{6} \]

and

\[ p'=\frac{t_2''-t_2'}{t_1'-t_2'}. \tag{7} \]

This system differs from the preceding one only in that the “scale” in it is the change in temperature of the cold (heated) liquid. Obviously:

\[ \varphi=\varphi' R', \tag{8} \]

\[ \varphi'=\varphi R, \tag{8'} \]

\[ p=p'R', \tag{9} \]

\[ p'=pR, \tag{9'} \]

\[ RR'=1. \tag{10} \]

The existence of these relations makes it possible to use any of the criteria given, which may provide certain simplifications. Using the heat-balance equation of the heat exchanger and the definitions of the criteria, one can obtain the auxiliary relations:

\[ \frac{t_2''-t_2'}{t_1'-t_2'}=p'=pR, \tag{7'} \]

\[ \frac{t_1'-t_1''}{t_1'-t_2'}=p=p'R', \tag{4'} \]

\[ \frac{t_1''-t_2''}{t_1'-t_2'}=1-(1+R)p=1-(1+R')p', \tag{11} \]

\[ \frac{t_1''-t_2'}{t_1'-t_2'}=1-p=1-p'R', \tag{12} \]

\[ \frac{t_1'-t_1''}{t_1'-t_2'}=\frac{p}{1-p}=\frac{p'R'}{1-p'R'}, \tag{12'} \]

\[ \frac{t_1'-t_2''}{t_1'-t_2'}=1-pR=1-p', \tag{13} \]

\[ \frac{t_2''-t_2'}{t_1'-t_1''}=\frac{p'}{1-p'}=\frac{pR}{1-pR}, \tag{13'} \]

\[ \frac{t_1'-t_2'}{t_1'-t_1''}=\frac{1}{p}-R=\frac{1}{R'}\left(\frac{1}{p'}-1\right), \tag{14} \]

\[ \frac{t_1''-t_2'}{t_2''-t_2'}=\frac{1}{p'}-R'=\frac{1}{R}\left(\frac{1}{p}-1\right). \tag{15} \]

The formulas for calculating \(\varphi\) or \(\varphi'\) in most cases have a simpler form than the relation for calculating the temperature head or \(\dfrac{\Delta t}{\Delta t_{pr}}\). The presence of a direct relationship between \(\varphi, p\), and \(R\) (or \(\varphi', p'\), and \(R'\)) makes it equally simple to determine the heating surface and to carry out a checking calculation. The problem of calculating a complex heat exchanger is also solved more simply.

3

For a parallel-flow heat exchanger, the relation between the final and initial temperatures has the form:

\[ t_1''-t_2''=(t_1'-t_2')e^{-kF\left(\frac{1}{W_1}+\frac{1}{W_2}\right)}. \]

Simple transformations give:

\[ \frac{t_1''-t_2''}{t_1'-t_2'}= e^{-\frac{kF}{W_1}\left(1+\frac{W_1}{W_2}\right)}. \]

Taking into account (2), (3), and (11), we have:

\[ 1-(1+R)p=e^{-\varphi(1+R)}, \]

whence

\[ \varphi= \frac{\ln \dfrac{1}{1-p(1+R)}}{1+R}. \tag{16} \]

\[ p=\frac{1-e^{-\varphi(1+R)}}{1+R}. \tag{17} \]

Replacing \(\varphi, p\), and \(R\) by \(\varphi', p'\), and \(R'\) according to (8), (9), and (10) gives:

\[ \varphi'= \frac{\ln \dfrac{1}{1-p'(1+R')}}{1+R'}, \tag{18} \]

\[ p'=\frac{1-e^{-\varphi'(1+R')}}{1+R'}. \tag{19} \]

The calculation of the heating surface, i.e., of the quantities \(\varphi\) or \(\varphi'\) by (16) and (18), is in any case no more complicated than the calculation of the temperature head. The problem of a checking calculation can be solved from relation (17) or (19).

Fig. 1.

Visible labels in the figure:

\[ p=f(\varphi,R), \qquad p=\frac{t_1-t_1''}{t_1-t_2'}, \qquad \varphi=\frac{KF}{W_1}, \qquad R=\frac{W_1}{W_2}, \]

\[ p'=f(\varphi',R'), \qquad p'=\frac{t_2''-t_2'}{t_1-t_2'}, \qquad \varphi'=\frac{KF}{W_2}, \qquad R'=\frac{W_2}{W_1}. \]

All calculations are simplified if the dependence between the given criteria is represented in the form of a graph (Fig. 1). The values \(p\) and \(R\) are plotted on the axes; the quantity \(\varphi\) is a parameter and is indicated on the curves. In accordance with (16), (17), (18), and (19), the values \(p, R\), and \(\varphi\) on this graph may be replaced respectively by the values \(p', R'\), and \(\varphi'\). Therefore on the abscissa axis the value \(R\) is given within the limits from 0 to 1, since for \(R>1\) one can pass to

\[ R'=\frac{1}{R}<1, \]

accordingly changing the other designations as well.

Analogous conclusions for a counterflow heat exchanger give the relations:

\[ \varepsilon = \frac{ \ln \dfrac{1-pR}{1-p} }{ 1-R }, \tag{20} \]

\[ p = \frac{1-e^{-\varepsilon(1-R)}}{1-Re^{-\varepsilon(1-R)}} , \tag{21} \]

and also

\[ \varepsilon' = \frac{ \ln \dfrac{1-p'R'}{1-p'} }{ 1-R' }, \tag{22} \]

\[ p' = \frac{1-e^{-\varepsilon'(1-R')}}{1-R'e^{-\varepsilon'(1-R')}} . \tag{23} \]

The corresponding calculation graph is given in Fig. 2.

In the figure:

\[ p=f(\varphi,R), \qquad p=\frac{t_1'-t_2''}{t_1'-t_2'}, \qquad \varphi=\frac{kF}{W_1}, \qquad R=\frac{W_1}{W_2}, \]

\[ p'=f(\varphi',R'), \qquad p'=\frac{t_2''-t_2'}{t_1'-t_2'}, \qquad \varphi'=\frac{kF}{W_2}, \qquad R'=\frac{W_2}{W_1}. \]

Fig. 2.

4.

The derivation of the relation \(\varepsilon=f(p,R)\) for a cross-flow heat exchanger with mixing of the hot liquid was carried out by the method set forth in [1]. As a result, the following was obtained:

\[ \varepsilon = \frac{ \ln \dfrac{1}{1+R\ln(1-p)} }{ R }. \tag{24} \]

\[ p = 1 - e^{-\frac{1}{R}\left(1-e^{-\varphi R}\right)} . \tag{2} \]

A graph for calculations using these relations is given in Fig. 3. Taking into account relations (8), (9), and (10), we also have

\[ \varphi'=\ln \frac{R'}{R' + \ln(1-p'R')}, \tag{26} \]

\[ p'=\frac{1-e^{-R'(1-e^{-\varphi'})}}{R'} . \tag{27} \]

Fig. 3.

For the diagram in Fig. 3: mixed hot fluid,

\[ p=f(\varphi,R), \qquad p=\frac{t_1'-t_1''}{t_1'-t_2'}, \qquad \varphi=\frac{kF}{W_1}. \]

A graph for calculation using these relations is given in Fig. 4. For a cross-flow heat exchanger with mixing of the cold fluid, one obtains

\[ \varphi=\ln \frac{R}{R+\ln(1-pR)}, \tag{28} \]

\[ p=\frac{1-e^{-R(1-e^{-\varphi})}}{R}, \tag{29} \]

\[ \varphi'=\frac{\ln \dfrac{1}{1+R'\ln(1-p')}}{R'}, \tag{30} \]

\[ p'=1-e^{-\frac{1-e^{-\varphi'R'}}{R'}} . \tag{31} \]

Comparing relations (24), (25), (26), (27) and (28), (29), (30), and (31) with one another, we find that for this case there is no need to construct new graphs. The graph in Fig. 3 may be regarded as a representation of the dependence \(\varphi' = f(p', R')\) for a cross-flow heat exchanger with mixing of the cold liquid, and the graph in Fig. 4 as a representation of the dependence \(\varphi = f(p, R)\) for the same case.

Fig. 4.

For a cross-flow heat exchanger with simultaneous mixing of the hot and cold liquids, the calculation gives

\[ \frac{1}{p} = \frac{1}{1-e^{-\varphi}} + \frac{R}{1-e^{-\varphi R}} - \frac{1}{\varphi}. \tag{32} \]

and

\[ \frac{1}{p'} = \frac{1}{1-e^{-\varphi'}} + \frac{R'}{1-e^{-\varphi' R'}} - \frac{1}{\varphi'}. \tag{33} \]

The design graph corresponding to these relations is given in Fig. 5.

For a cross-flow heat exchanger in which the heat-exchanging liquids are not mixed, the solution given by Nusselt [5] was used. As a result of numerical integration, the relation \(\varphi=f(p,R)\) was obtained in tabular form. The corresponding graph is given in Fig. 6.

5.

A parallel-mixed heat exchanger with mixing of the hot liquid, the diagram of which is given in Fig. 7, is considered

![Figure 5 diagram not reproduced]

Both fluids are mixed

\[ \rho = f(\varphi, R) \]

\[ \rho = \frac{t_1' - t_1''}{t_1' - t_2'} \]

\[ \varphi = \frac{KF}{W_1} \]

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

\[ \rho' = f(\varphi', R') \]

\[ \rho' = \frac{t_2'' - t_2'}{t_1' - t_2'} \]

\[ \varphi' = \frac{KF}{W_2} \]

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

Fig. 5.

![Figure 6 diagram not reproduced]

\[ \rho = f(\varphi, R) \]

\[ \rho = \frac{t_1' - t_1''}{t_1' - t_2'} \]

\[ \varphi = \frac{KF}{W_1} \]

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

\[ \rho' = f(\varphi', R') \]

\[ \rho' = \frac{t_2'' - t_2'}{t_1' - t_2'} \]

\[ \varphi' = \frac{KF}{W_2} \]

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

Fig. 6.

Underwood (cited from [6]). The calculation gives

\[ \varphi' = \frac{ \ln \dfrac{2-p'\left[1+R'-\sqrt{1-R'^2}\right]} {2-p'\left[1+R'+\sqrt{1-R'^2}\right]} } {\sqrt{1-R'^2}}, \tag{34} \]

\[ p' = \frac{1}{ \dfrac{1+R'+\sqrt{1-R'^2}}{2} - \dfrac{1+R'-\sqrt{1-R'^2}}{2}\, e^{-\varphi'\sqrt{1-R'^2}} }. \tag{35} \]

In the diagram:

\[ p=f(\varphi,R), \qquad p'=f(\varphi',R'), \]

\[ \varphi=\frac{KF}{W_1}, \qquad R=\frac{W_1}{W_2}, \]

\[ \varphi'=\frac{KF}{W_2}, \qquad R'=\frac{W_2}{W_1}. \]

Fig. 7.

Taking (8), (9), and (10) into account, we also have

\[ \varphi = \frac{ \ln \dfrac{2-p\left[1+R-\sqrt{1+R^2}\right]} {2-p\left[1+R+\sqrt{1+R^2}\right]} } {\sqrt{1+R^2}}, \tag{36} \]

and

\[ p = \frac{1}{ \dfrac{1+R+\sqrt{1+R^2}}{2} - \dfrac{1+R-\sqrt{1+R^2}}{2}\, e^{\varphi\sqrt{1+R^2}} }. \tag{37} \]

The graph expressing the dependence \(\varphi=f(p,R)\) or \(\varphi'=f(p',R')\) for this heat exchanger is given in Fig. 7. The same relations and graphs will hold for a two-pass parallel-mixed heat exchanger in which the hot fluid makes two passes.

The scheme of a parallel-mixed heat exchanger in which the hot fluid is mixed, while the cold fluid passes through 4 pa-

parallel flow, is given in Fig. 8. For the surface element \(df\) one may write the heat-transfer equations

\[ W_1\,d\vartheta_e=-\frac{\kappa\,df}{4}\,(\vartheta-\vartheta_e), \]

\[ -\,W_2\,d\vartheta_m=\frac{\kappa\,df}{4}\,(\vartheta-\vartheta_m), \]

\[ W_2\,d\vartheta_n=\frac{\kappa\,df}{4}\,(\vartheta-\vartheta_n), \]

\[ -\,W_2\,d\vartheta_r=\frac{\kappa\,df}{4}\,(\vartheta-\vartheta_r) \]

and the heat-balance equation

\[ -\,W_1\,d\vartheta = W_2\,(d\vartheta_e-d\vartheta_m+d\vartheta_n-d\vartheta_r). \]

Here \(\vartheta\) is the current temperature of the hot liquid, \(\vartheta_e\) is the current temperature of the cold liquid in the first pass, \(\vartheta_m\) in the second, etc. The origin of coordinates is located at the point where the hot liquid enters.

Fig. 8.

Solving this system of equations, taking into account that on the inlet side \(\vartheta_m=\vartheta_n\), \(\vartheta_e=t'_2\), and \(\vartheta=t'_1\); on the outlet side \(\vartheta_e=\vartheta_m\), \(\vartheta_n=\vartheta_r\), and \(\vartheta=t''_1\), makes it possible to determine all the temperatures \(\vartheta,\vartheta_e,\vartheta_m,\vartheta_n\), and \(\vartheta_r\). To calculate \(p'\) we use relation (7). As a result one obtains

\[ p' = \frac{ 1-e^{-\frac{\varphi'\sqrt{\,1+4R'^2\,}}{2}} }{ \left[ \frac{1+2R'+\sqrt{\,1+4R'^2\,}}{4} -\frac{1}{2(1-e^{-\varphi'})} \right] - \left[ \frac{1+2R'-\sqrt{\,1+4R'^2\,}}{4} +\frac{1}{2(1-e^{-\varphi'})} \right] e^{-\frac{\varphi'\sqrt{\,1+4R'^2\,}}{2}} } \tag{38} \]

The graph for calculation from this dependence is given in Fig. 8. Using relations (8′), (9′), and (10′), one may replace \(p'_1\), \(R'\), and \(\varphi'\) in (38) by \(p\), \(R\), and \(\varphi\). This gives

\[ p = \frac{ 1 - e^{-\frac{\varphi\sqrt{4+R^{2}}}{2}} }{ \left[ \frac{2+R+\sqrt{4+R^{2}}}{4} + \frac{R}{2\left(1+e^{-\frac{\varphi R}{2}}\right)} \right] - \left[ \frac{2+R-\sqrt{4+R^{2}}}{4} + \frac{R}{2\left(1+e^{-\frac{\varphi R}{2}}\right)} \right] e^{-\frac{\varphi\sqrt{4+R^{2}}}{2}} } \tag{39} \]

The corresponding graph is presented in Fig. 9. The same relations for \(p'\) and \(p\) are obtained in the case where the hot liquid flows in the opposite direction.

Fig. 9.

For a four-pass parallel-mixed heat exchanger, in which the hot liquid makes 4 passes, analogous relations are obtained if \(p\), \(\varphi\), and \(R\) are replaced by \(p'\), \(\varphi'\), and \(R'\). Therefore, the same graphs 8 and 9 may be used for their calculation, with the corresponding change of notation.

6.

For a parallel-mixed heat exchanger with 3 passes, whose scheme is shown in Fig. 10, the determination of \(p'\) is carried out by the formula

\[ p'=\frac{y-1}{y-R'}, \tag{40} \]

where

\[ y=\frac{e^{\xi'}-E e^{\eta'}}{1-E} \tag{41} \]

and

\[ E= \frac{1+\gamma}{1+\beta}\, \frac{e^{\xi'}+e^{\eta'}}{e^{\beta'}+e^{\gamma'}} . \tag{42} \]

\(\beta\) and \(\gamma\) are the roots of a quadratic equation and are equal, respectively, to

\[ \beta = -0.5R' - \sqrt{(0.5R')^{2} + \frac{1 - R'}{9}}, \tag{43} \]

\[ \gamma = -0.5R' + \sqrt{(0.5R')^{2} + \frac{1 - R'}{9}} . \tag{44} \]

Fig. 10.

The graph for calculation according to these dependences is given in the same figure. Using relations (8), (9), and (10), we also obtain

\[ p = -\frac{y - 1}{Ry - 1}, \tag{45} \]

where

\[ y = \frac{e^{3\varphi} - E e^{[[unclear: exponent]]}}{1 - E} \tag{46} \]

and

\[ E = \frac{R + \gamma_{1}}{R + \beta_{1}}\, \frac{e^{[[unclear: exponent]]} + e^{R\varepsilon/3}} {e^{\gamma_{1}\varepsilon} + e^{R\varepsilon/3}} . \tag{47} \]

The quantities \(\beta_{1}\) and \(\gamma_{1}\) are calculated from the formulas:

\[ \beta_{1} = -0.5 + \sqrt{ 0.25 - \frac{R(1 - R)}{9} }\, . \tag{48} \]

\[ \gamma_1 = 0.5 - \sqrt{\,0.25 - \frac{R(1-R)}{9}\,}. \tag{49} \]

A graph for calculations by these relations is given in Fig. 11. The calculation of the value \(\rho'\) for a parallel-mixed heat exchanger with five passes of the heated liquid, the diagram of which is shown in Fig. 12, is carried out according to relation (40). The quantity \(y\) is calculated by (41), and the value \(E\) by the formula

\[ E = \frac{1+\gamma}{1+\beta}\, \frac{ 2e^{\varphi'/5}+e^{-\varphi'/5} +e^{\beta\varphi'}\left(1+2e^{-\frac{2}{5}\varphi'}\right) }{ 2e^{\varphi'/5}+e^{-\varphi'/5} +e^{\gamma\varphi'}\left(1+2e^{-\frac{2}{5}\varphi'}\right) }. \tag{50} \]

Fig. 11.

Moreover,

\[ \beta = 0.5R' + \sqrt{\,0.25R'^2 + \frac{1-R'}{25}\,} \tag{51} \]

and

\[ \gamma = 0.5R' - \sqrt{\,0.25R'^2 + \frac{1-R'}{25}\,}. \tag{52} \]

The graph for calculating this heat exchanger is given in the same figure. Relations (45) and (46) may also be used. In this case

\[ E = \frac{R+\gamma_1}{R+\beta_1}\, \frac{ 2e^{\frac{\varphi R}{5}}+e^{-\frac{\varphi R}{5}} +e^{\beta_1\varphi}\left(1+2e^{-\frac{2\varphi R}{5}}\right) }{ 2e^{\frac{\varphi R}{5}}+e^{-\frac{\varphi R}{5}} +e^{\gamma_1\varphi}\left(1+2e^{-\frac{2\varphi R}{5}}\right) }. \tag{53} \]

Figure 12.

Visible labels and formulas in the figure:

\[ \varphi'=\infty \]

Axes:

\[ P' \qquad p' \]

Right-hand notation:

\[ P' = f(\varphi', R') \]

\[ P'=\frac{t_1'-t_2'}{t_1'-t_2''} \]

\[ \varphi'=\frac{kF}{W_2} \]

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

Caption:

Fig. 12.

Figure 13.

Visible labels and formulas in the figure:

Axes:

\[ P \qquad p \]

Right-hand notation:

\[ P=f(\varphi,R) \]

\[ P=\frac{t_1'-t_1''}{t_1'-t_2'} \]

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

\[ \varphi=\frac{kF}{W_1} \]

Caption:

Fig. 13.

\[ \xi_1=-0.5+\sqrt{0.25-\frac{R(1-R)}{25}}, \tag{54} \]

\[ \eta_1=-0.5-\sqrt{0.25-\frac{R(1-R)}{25}}. \tag{55} \]

The graph for calculation by these relations is given in Fig. 13.

7.

The calculation relations presented make it possible to investigate and compare the operation of individual types of heat exchangers. If one of the heat-exchanging fluids maintains a constant temperature, then its water equivalent should be regarded as equal to infinity. Thus, if \(W_2=\infty\), then \(R=0\). Regardless of the nature of the heat exchanger and of the mutual direction of flow of the hot and cold fluids, from (17), (21), (29), (32), (37), (39), (40), and (45) we obtain

\[ p=1-e^{-\varphi} \tag{56} \]

and

\[ \varphi=\ln \frac{1}{1-p}. \tag{57} \]

By these relations, using (2) and (4), one may calculate, for example, the section of the heating surface of a steam boiler in which vaporization occurs. Similarly, for \(R'=0\),

\[ p'=1-e^{-\varphi'}, \tag{56'} \]

\[ \varphi'=\ln \frac{1}{1-p'}. \tag{57'} \]

If the heating surface is increased, then in the limit \(\varphi=\infty\) and \(\varphi'=\infty\). In this case the limiting cooling of the hot fluid and the limiting heating of the cold fluid are attained. The corresponding limiting temperatures may be found from (4) and (7), if \(p_{\text{lim}}\) and \(p'_{\text{lim}}\) are first determined. These quantities may be found from the corresponding relations for the given heat exchanger. Thus, for a counterflow heat exchanger with \(R<1\), from (21) we have \(p_{\text{lim}}=1\). According to (4), this means that \(t_{1\text{lim}}=t'_2\), i.e., the hot fluid will cool to the initial (inlet) temperature of the cold fluid. The cold fluid, however, will not be heated to the initial temperature of the hot fluid. Indeed, in this case \(p'>1\). Therefore relation (23) is rewritten as

\[ p'=\frac{1-e^{\varphi'(R'-1)}}{1-R'e^{\varphi'(R'-1)}}. \tag{23'} \]

Resolving the indeterminacy \((\varphi'=\infty)\), we obtain

\[ p'_{\text{lim}}=\frac{1}{R'}<1. \tag{58} \]

In accordance with (7), this means that

\[ t_{2\text{lim}}<t'_1. \]

Similarly, for \(R' < 1\) (\(R > 1\)) we obtain

\[ p'_{\mathrm{lim}} = 1 \tag{59} \]

and

\[ p_{\mathrm{lim}} = \frac{1}{R}. \tag{59'} \]

This means that in this case the cold fluid is heated to the inlet temperature of the hot fluid:

\[ t_{2\mathrm{lim}} = t'_1, \]

and

\[ t_{1\mathrm{lim}} < t'_2. \]

In a parallel-flow heat exchanger, the condition \(\varphi = \infty\) leads, on the basis of relations (17) and (19), to the equalities:

\[ p_{\mathrm{lim}} = \frac{1}{1+R}, \tag{60} \]

\[ p'_{\mathrm{lim}} = \frac{1}{1+R'} = \frac{R}{1+R}. \tag{60'} \]

Taking (3), (4), and (7) into account, this gives

\[ t_{2\mathrm{lim}} = t_{1\mathrm{lim}}. \]

In all other heat exchangers, the limiting degree of cooling of the hot fluid and heating of the cold fluid will be an intermediate value between the corresponding values for counterflow and parallel flow. Thus, for a cross-flow heat exchanger with mixing of the hot fluid, relation (25) gives:

\[ p_{\mathrm{lim}} = 1 - e^{-\frac{1}{R}}. \tag{61} \]

It is easy to show that

\[ 1 > 1 - e^{-\frac{1}{R}} > \frac{1}{1+R}. \]

In a cross-flow heat exchanger with mixing of both fluids, relations (32) and (33) for \(\varphi = \infty\) give

\[ p_{\mathrm{lim}} = \frac{1}{1+R}, \tag{62} \]

\[ p'_{\mathrm{lim}} = \frac{1}{1+R'}. \tag{62'} \]

Thus, the limiting heat transfer in a cross-flow heat exchanger with mixing of both fluids will be the same as in a parallel-flow heat exchanger (for the same \(\varphi\) and \(R\)). If, however, the hot fluid is mixed and the cold fluid is not mixed, then

under the same \(\varphi\) and \(R\), more heat will be transferred. Under these conditions the heat transfer will be still greater in a cross-flow heat exchanger in which the fluids flow without mixing.

In a parallel-mixed heat exchanger with an odd number of passes (see Figs. 10 and 12), the limiting degree of heating of the cold liquid is the same as in a counterflow heat exchanger, i.e., \(p'_{\mathrm{lim}}=1\). Thus, for a three-pass heat exchanger, from relation (42) we have:

\[ E=\frac{1+\gamma}{1+\beta}\, \frac{1+e^{\left(\beta-\frac{1}{3}\right)\varphi'}} {1+e^{\left(\gamma-\frac{1}{3}\right)\varphi'}} . \tag{42'} \]

From (44) it is seen that \(\gamma<0\); consequently, \(\gamma-\frac{1}{3}<0\). Therefore

\[ e^{\left(\gamma-\frac{1}{3}\right)\varphi'} \]

tends to zero as \(\varphi'\to\infty\). From (43) it is seen that for

\[ R'=0 \quad \beta=\frac{1}{3}, \]

and for \(R'=1\), \(\beta=1\). Consequently, for \(0<R'<1\), \(\beta<\frac{1}{3}\).

Therefore \(\beta-\frac{1}{3}<0\), and the quantity

\[ e^{\left(\beta-\frac{1}{3}\right)\varphi'} \]

tends to zero as \(\varphi'\to\infty\). Hence, the limiting value of \(E\) for \(\varphi'=\infty\) will be

\[ E_{\mathrm{lim}}=\frac{1+\gamma}{1+\beta}. \]

Under these conditions, as is seen from (41), \(y_{\mathrm{lim}}=\infty\), and from (40) \(p'_{\mathrm{lim}}=1\). The same result is obtained when considering a heat exchanger with 5 passes.

If the numerical values of \(p'\) from the graphs in Figs. 11 and 12 are compared, it turns out that for \(\varphi'<1\) and equal \(\kappa\) and \(R'\), the heat transfer in three- and five-pass heat exchangers is practically the same. For \(\varphi'>1\), in a three-pass heat exchanger more heat is transferred per unit surface than in a five-pass one.

For a two-pass parallel-mixed heat exchanger (Fig. 7), the limiting degree of heating of the cold liquid is obtained from (35) for \(\varphi'=\infty\):

\[ p'_{\mathrm{lim}}=\frac{2}{1+R'+\sqrt{1+(R')^2}} . \tag{35'} \]

Similarly, for a four-pass heat exchanger (Fig. 8), from (38),

\[ p'_{\mathrm{lim}}=\frac{4}{3+2R'+\sqrt{1+4(R')^2}} . \tag{38'} \]

The value \(p_{\mathrm{lim}}\) from (35′) and (38′) is less than 1 and greater than

\[ \frac{1}{1+R}. \]

Thus, the limiting temperatures in parallel-mixed heat exchangers with an even number of passes will be intermediate between the corresponding values for counterflow and parallel-flow heat exchangers.

Comparing the graphs in Figs. 7 and 8 with each other, we find that for \(\varphi'\ll1\) the value of \(p'\) for two-pass and four-pass

the heat exchangers is practically the same, i.e., in these heat exchangers the heat transfer per unit heating surface (for identical \(k\) and \(R'\)) is one and the same. For \(\varphi' > 1\), in the two-pass heat exchanger the heat transfer per unit heating surface is greater than in the four-pass one.

Finally, comparison of the graphs in Figs. 5, 7, 8, 10, and 12 shows that for \(\varphi' \leq 1\) and identical values of \(k\) and \(R'\), the values of \(p'\) differ by less than \(1\%\), i.e., are practically identical. Therefore, for \(\varphi' \leq 1\), calculations for cross-flow heat exchangers with mixing of both fluids and for parallel-mixed heat exchangers with 2, 3, 4, and 5 passes through which the heated fluid flows may, in practice, be carried out by any of the relations (33), (34), (35), (38), and 40. The most convenient in this case will be the use of relations (34) and (35). It must once again be emphasized that this is permissible only for \(\varphi' \leq 1\).

8.

The relations presented for the calculation of heat exchangers may be used directly, or by means of graphs constructed from them, for direct and inverse calculation.

In a direct calculation, from the specified values of the temperatures and water equivalents, using relations (3), (4), (5), and (6), one can find \(p\) and \(R\) (or \(p'\) and \(R'\)). From the corresponding equation or graph, \(\varphi\) (or \(\varphi'\)) is determined, and then from (2) or (5) the magnitude of the heating surface is calculated.

In the inverse calculation of a heat exchanger there may be different cases. If the heating surface \(F\), the heat-transfer coefficient \(k\), the water equivalents \(W_1\) and \(W_2\), and any two inlet and outlet temperatures are given, then the determination of the other two temperatures is also carried out directly from the corresponding equations or graphs. Thus, for example, let \(k\), \(F\), \(W_1\), \(W_2\), \(t''_1\), and \(t'_2\) be given. Then from (2) and (3) one can find the values \(\varphi\) and \(R\), or from (5) and (6) the values \(\varphi'\) and \(R'\). Then, from the corresponding equation or graph, the value of \(p\) or \(p'\) is determined. Using relations (4) and (3), or (7) and (6), we obtain, for determining the required values \(t'_1\) and \(t''_2\), a system of two equations with two unknowns. In an analogous manner the problem is solved if \(k\), \(F\), and all inlet and outlet temperatures are given and it is required to find the values of the water equivalents of the heat-exchanging fluids. As is known, these cases of inverse calculation are the most frequent.

The need for supplementary construction and for carrying out a calculation by trial may arise in those cases when \(k\), \(F\), one of the water equivalents, and three temperatures are given, and it is required to determine the fourth temperature and the second water equivalent. Thus, for example, let \(k\), \(F\), \(W_1\), \(t'_1\), \(t'_2\), and \(t''_1\) be given, and let it be required to find \(W_2\) and \(t''_2\). From (2) and (4), \(\varphi\) and \(p\) are determined, and then from the corresponding equation or graph—the value of \(R\). Finally, from (3), \(W_2\) and \(t''_2\) are calculated.

When graphs are used for the calculation, a difficulty may arise if in the given case \(R > 1\). Then, on the graph of the dependence \(p = f_1(\varphi, R)\), using the graph \(p' = f_2(\varphi', R')\), one must plot the curve \(p = f(\varphi, R')\) for the given value \(\varphi = \varphi_*\). The values \(R'\) are then laid off on the abscissa axis of the graph \(p = f_1(\varphi, R)\) on the same scale as the values \(R\). The required curve will pass through the origin of coordinates \((R' = 0, R = \infty)\) and through the point \(p\) at \(R = 1\), since in this case \(R' = 1\) and \(p' = p\). The intermediate points may be found in the following way.

Let us specify \(R'=R_1'<1\). Then the corresponding value of \(\varphi_1'\) is found from (8), if one takes into account that, by (10),

\[ R_1=\frac{1}{R_1'}. \]

From the values of \(R_1'\) and \(\varphi_1'\), using the graph \(p'=f_2(\varphi',R')\), we find \(p_1'\); and then, taking (9) into account, the corresponding value \(p\). By \(p\) and \(R'\) we mark an intermediate point of the curve \(m\) (see Fig. 14).

Using this curve, for the specified value \(p_1\), by the construction \(abc\) we determine \(R_d'\), and then from (10) the actual value \(R_d\). After this one may calculate \(t_2''\) and \(W_2\), as indicated above. For reference, the lines \(\rho=f(\varphi,R')\) and \(p'=f(\varphi',R)\) for \(\varphi=0.5\) (\(u z'=0.5\)) are plotted on all the graphs by a dashed line.

Fig. 14.

The problem is solved in an analogous way if the values \(\kappa, F, t_1', t_2', t_2''\), and \(W_2\) are given and it is necessary to determine \(W_1\) and \(t_1''\).

If the values \(\kappa, F, W_1, t_1', t_2'\), and \(t_2''\) are known and it is required to find \(W_2\) and \(t_1''\), then from (2)

\[ \varphi_0=\frac{\kappa F}{W_1} \]

is determined, and from \((7')\)

\[ p_0'=p_0R_0=\frac{t_2''-t_2'}{t_1'-t_2'}. \]

On the line \(\varphi_0\) of the graph \(p=f(\varphi,R)\), a point is selected at which the value \(p_0R_0=p_0'\). Then from (6) \(W_2\) and \(t_1''\) are determined.

If the point for which \(p_0R_0=p_0'\) lies outside the graph, this means that \(R_0>1\). Then the calculation is carried out using the graph

\[ p'=f(\varphi',R') \]

for the given heat exchanger. On the straight line \(p_0'\), the points of intersection with the lines \(\varphi_a', \varphi_b', \ldots\) are marked, and the corresponding values \(R_a', R_b', \ldots\). The sought values \(\varphi_0'\) and \(R_0'\) will correspond to the point for which

\[ \varphi_0'R_0'=\varphi_0. \]

The problem is solved in an analogous way if the values \(\kappa, F, W_2, t_1', t_2''\) are known and it is required to determine \(W_1\) and \(t_2'\).

In the case where the values \(\kappa, F, t_1'', t_2', t_2''\) are known and it is required to find \(W_2\) and \(t_1'\), first, from (2) and (15), \(\varphi_0\) and

\[ \frac{1}{R_0}\left(\frac{1}{p_0}-1\right) \]

are calculated.

On the line \(\varphi_0\) a point with coordinates \(p\) and \(R\) is selected that satisfies the latter condition. Then from (4) and (3) \(t_1'\) and \(W_2\) are determined. For \(R>1\) it is necessary to construct the auxiliary line

\[ p=f(\varphi_0,R'), \]

as indicated above. On this curve a point with coordinates \(p_0'\) and \(R_0'\) is sought that satisfies the condition

\[ R_0'\left(\frac{1}{p_0'}-1\right)=\frac{1}{R_0}\left(\frac{1}{p_0}-1\right). \]

Then from (4) and (6) the required quantities are determined. By analogous methods the problems are also solved for other specifications.

9.

Considering the graphs of the dependences \(p=f(\varphi,R)\) and \(p'=f(\varphi',R')\) for individual heat exchangers, one may note that within the limits \(0<\varphi\leq 1\) and \(0<\varphi'\leq 1\), the curves for the given \(\varphi\) and \(\varphi'\) differ very little from straight lines. Therefore, for the calculation of all the heat exchangers considered above, within the indicated limits of the values of \(\varphi\) \((\varphi')\), approximate dependences may be drawn up:

\[ p=\frac{\varphi}{1-0.58\varphi}\left(1-\frac{a\varphi}{1+b\varphi}R\right), \]

\[ p'=\frac{\varphi'}{1-0.58\varphi'}\left(1-\frac{a\varphi'}{1+b\varphi'}R'\right). \tag{63} \]

The quantities \(a\) and \(b\) for various types of heat exchangers have the values given in this table.

Type of heat exchanger	\(a\)	\(b\)
Parallel-flow	0.42	0.33
Counterflow	0.42	1.00
Cross-flow heat exchangers with mixing of both or one of the heat-exchanging fluids; parallel mixed heat exchangers with 2, 3, 4, and 5 passes	0.40	0.40
Cross-flow heat exchangers in which the heat-exchanging fluids do not mix	0.40	0.60

For a direct calculation, from (63) one obtains:

\[ \varphi=\frac{\alpha+\sqrt{\alpha^{2}-2\beta p}}{\beta}, \tag{64} \]

where

\[ \alpha=(1-0.58p)-bp \tag{65} \]

and

\[ \beta=2\,[aR-(1-0.58p)b]. \tag{66} \]

The real value according to (64), evidently, has only the positive solution.

Check calculations have shown that the error in using the indicated approximate relations does not exceed \(2\%\). It must be emphasized once again that they are suitable only up to values \(\varphi<1\) or \(\varphi'<1\).

REFERENCES

1. Smith. Mean temperature difference in cross flow. Enginneering, 1934.
2. M. A. Mikheev. Fundamentals of Heat Transfer, GEI, 1956.
3. Temperature head. From the draft standards for thermal calculation of a boiler unit. Teploenergetika, 1955.
4. Bosnjakovic. Berechnung der Warmeaustauschern. Schweizerische Bauzetung, 1951.
5. Nusselt W. Der Wärmeübergang im Kreuzstrom Band 55 Nr. 48. V. D. J., 1911.
6. Hausen. Wärmeübertragung in Gegenstrom, Gleichstrom und Kreuzstrom, 1950.