DEGREE OF PERFECTION OF HEAT-EXCHANGE APPARATUS

G. I. FUKS

1

To evaluate the degree of perfection of a heat exchanger, two indicators are used: the heat-retention coefficient

\[ \varepsilon=\frac{Q_x}{Q_z} \tag{1} \]

and the efficiency (e.f.) of the heat exchanger

\[ \eta=\frac{Q_x}{Q_p}, \tag{2} \]

where \(Q_x\) and \(Q_z\) are, respectively, the heat supplied to the cold (heated) liquid and the heat removed from the hot (heating) liquid; \(Q_p\) is the available heat that can be removed from the hot liquid when it is cooled to the maximum possible extent under the given conditions.

In accordance with these concepts, two expressions may be written for the heat balance of the heat exchanger:

\[ Q_z = Q_x + Q_n \tag{3} \]

and

\[ Q_p = Q_x + Q_y + Q_n, \tag{4} \]

where \(Q_n\) and \(Q_y\) are the heat lost to the surrounding medium and with the hot liquid leaving the heat exchanger, respectively.

Evaluation of the degree of perfection by the heat-retention coefficient \(\varepsilon\) is meaningful in the case when the amount of heat that is to be transferred to the cold liquid, as well as the heat content of the hot liquid at the inlet to the heat exchanger, are specified by the operating conditions of the heat exchanger. Such a case occurs, for example, in calculating the water economizer of a steam boiler.

Usually, when designing a heat exchanger, the task is set of achieving in it, all other conditions being equal, the maximum possible transfer of heat from the hot to the cold liquid. In this case the degree of its perfection should be evaluated by the efficiency of the heat exchanger \(\eta\). But in doing so it is necessary to calculate the value of the available heat of the hot liquid \(Q_p\).

For this purpose the following procedure may be applied. The greatest heat transfer can be achieved, as is known, all other conditions being equal, in a counterflow heat exchanger. For a counterflow heat exchanger, in accordance with the notation in Fig. 1, the temperatures at the inlet and

Fig. 1

the outlet sides of the heat exchanger (counting in the direction of motion of the hot liquid) are related by

\[ t_1''-t_2'=(t_1'-t_2'')e^{-k\left(\frac{1}{W_1}-\frac{1}{W_2}\right)F}, \tag{5} \]

and the heat-balance equation has the form

\[ W_1(t_1'-t_1'')=W_2(t_2''-t_2'), \tag{6} \]

where \(W_1\) and \(W_2\) are the water equivalents of the hot and cold liquids, respectively.

In the limiting case of an infinitely large counterflow heat exchanger, two cases are possible:

1. If the water equivalent of the hot liquid \(W_1\) is less than the water equivalent of the cold liquid \(W_2\), then the hot liquid will, in the limit, be cooled to the inlet temperature of the cold liquid

\[ t_1^{pr}=t_2'. \tag{7} \]

This follows directly from relation (5), since the exponent on the right-hand side, as the heating surface increases, grows all the time while remaining negative, and \(t_1'\) is always greater than \(t_2''\).

1. For \(W_1>W_2\), the cold liquid will, in the limit, be heated to the inlet temperature of the hot liquid, but the temperature of the hot liquid at the outlet will be higher than the inlet temperature of the cold liquid. From the heat-balance equation, taking into account that in this case \(t_2''=t_1'\), one obtains

\[ t_1^{pr}=t_1'-\frac{W_2}{W_1}(t_1'-t_2'). \tag{8} \]

In the case of equality of the water equivalents \(W_1\) and \(W_2\), relation (8) gives for the limiting temperature of the hot liquid the same value as relation (7).

The decrease in the temperature of the hot liquid in no case can be lower than \(t_1^{pr}\), since this would be associated with the transfer of heat from a body with a lower temperature to a body with a higher temperature. Therefore, in all cases it is necessary to take

\[ Q_p=W_1(t_1'-t_1^{pr}). \tag{9} \]

2.

The indicators of the degree of perfection of a heat exchanger, \(\varepsilon\) and \(\eta\), take into account only the quantitative side of the processes in the heat exchanger, but do not take into account its qualitative side. Heat transfer in a heat exchanger always proceeds, in all its parts, from a body with a higher temperature to a body with a lower temperature. If the heat exchanger is part of a power installation, then, as a result of heat transfer, a decrease is obtained in the working capacity of the working body, even in the absence of losses. It is quite natural in this case to consider the energy balance of the heat exchanger and to introduce a special energetic coefficient of efficiency of the heat exchanger for assessing the degree of its perfection [1]. But the loss caused by a decrease in the potential of thermal energy occurs in all heat exchangers. To judge the magnitude of this loss, it is necessary to compare the result of the operation of the heat exchanger with the result of the operation of an ideal heat exchanger, in which only equilibrium processes are carried out. In doing this we proceed analogously to the case of assessing the degree of perfection of the process of steam expansion in a steam-

... engine by comparing its result with the result of ideal isentropic expansion.

Reversible processes in the transfer of heat from one body to another can in principle be carried out with the aid of thermocompressors, for example, of the mechanical or chemical type. If only equilibrium processes are carried out, their result must be the same, independently of the type of thermocompressor; otherwise this would inevitably lead to a contradiction with the law of conservation and transformation of energy. Therefore, for calculating the result of the operation of an ideal heat exchanger, any type of thermocompressor may be adopted.

The simplest calculations are made for a thermocompressor of the mechanical type. Let the process of heat removal from the hot body be represented on the \(TS\) diagram by curve 1—2, Fig. 2, and the process of heating the cold body by curve 3—4. Then \(Q_z=\) area 12 \(ab\), and \(Q_x=\) area 34 \(cd\) \({}^{1}\). For calculations it is convenient to introduce the mean thermodynamic temperatures [2] of heat removal \(T'\) and heat supply \(T''\), defined by the relations

Fig. 2.

\[ T'=\frac{Q_z}{\Delta S_z}, \tag{10} \]

\[ T''=\frac{Q_x}{\Delta S_x}. \tag{11} \]

By means of the heat removed from the hot body, in the presence of only equilibrium processes, work can be obtained

\[ A L_z=(T'-T_0)\Delta S_z, \tag{12} \]

where \(T_0\) is the temperature of the surrounding medium (air). If this work is expended on actuating an ideal heat pump that removes heat from the surrounding medium at \(T_0\) and transfers this heat to the body being heated in process 34, i.e., at the mean thermodynamic temperature \(T''\), then the cold body, in the presence of only reversible processes, will receive heat in the amount

\[ Q_x^{m}=T''\Delta S_x^{m}, \tag{13} \]

\({}^{1}\) The \(TS\)-diagram is constructed conditionally not for 1 kg, but for the total flow quantity of the hot and cold body.

where \(\Delta S_x^m\) is the increase in the entropy of the heated body when heat is supplied to it. This quantity can be calculated from the relation

\[ (T' - T_0)\Delta S_x = (T'' - T_0)\Delta S_x^m . \tag{14} \]

The heat actually transferred to the cold body is

\[ Q_x = T''\Delta S_x \tag{15} \]

less than \(Q_x^m\), which it can receive as a result of reversible processes in the heat exchanger. Let us call

\[ \varepsilon_{\mathrm e}=\frac{Q_x}{Q_x^m} \tag{16} \]

the energetic coefficient of heat retention. Using relations (1), (10), (11), (12), (13), and (14), one can obtain:

\[ \varepsilon_{\mathrm e}=\varepsilon\, \frac{1-\dfrac{T_0}{T''}} {1-\dfrac{T_0}{T'}} \tag{17} \]

and

\[ \varepsilon_{\mathrm e}= \frac{\varepsilon Q_c - T_0\Delta S_x} {Q_c - T_0\Delta S_c} \tag{18} \]

or

\[ \varepsilon_{\mathrm e}= \frac{Q_x - T_0\Delta S_x} {Q_c - T_0\Delta S_c}. \tag{18'} \]

It follows from relation (16) that the energetic coefficient of heat retention \(\varepsilon_{\mathrm e}\) is the ratio of the heat transferred to the cold body to the limiting amount of heat that can be transferred to the cold body in the same temperature interval of heating \(T_3—T_4\) and with the same removal of heat from the hot body \(Q_c\) in a perfect heat exchanger without losses. Since \(T' > T''\), it is evident from relation (17) that \(\varepsilon_{\mathrm e}<\varepsilon\). Even in the limiting case of absence of losses to the surroundings \((\varepsilon=1)\), the energetic coefficient of heat retention is less than unity, which characterizes the loss due to the irreversibility of the real heat-exchange process.

On the other hand, it follows from relation (18′) that the energetic coefficient of heat retention may also be regarded as the ratio of the working capacities of the cold and hot bodies of the heat exchanger. Consequently, the magnitude of the energetic coefficient of heat retention can be used to evaluate the degree of perfection of a heat exchanger. The quantity \(\varepsilon_{\mathrm e}\), like the quantity \(\varepsilon\), is calculated with respect to the heat \(Q_c\) actually removed from the hot fluid. Therefore it should be applied to evaluate the degree of perfection of a heat exchanger under the condition that the temperature of the hot fluid at the inlet and the amount of heat that can be taken up by the cold fluid are fixed.

When evaluating the degree of perfection of a heat exchanger with respect to the available heat \(Q_p\), it is necessary to introduce the concept of the energetic efficiency coefficient of the heat exchanger. In Fig. 3 are plotted the processes of heat removal from the hot body \(1—2\) to the final temperature \(T_1'\), and the process of its cooling \(1—2_{\mathrm{pr}}\) to the limiting temperature \(T_1^{\mathrm{pr}}\), as well as the process of heating the cold fluid \(3—4\) at the mean thermodynamic temperature \(T^*\). The mean thermodynamic temperature of heat removal from

the hot liquid when cooled to the limiting temperature \(T_1^{pr}\) by

\[ T'_p=\frac{Q_p}{\Delta S_g^{pr}}, \tag{19} \]

Fig. 3.

where \(\Delta S_g^{pr}\) is the change in entropy of the hot body in this process. Removal of the available heat \(Q_p\) from the hot body and its supply to the cold body by reversible processes can give a transfer of heat

\[ Q_x^{pr}=T'\Delta S_x^{pr}, \tag{20} \]

where \(\Delta S_x^{pr}\) is the increase in entropy of the cold body when heat is supplied to it, which is determined from the relation

\[ (T'-T_0)\Delta S_g^{pr}=(T''-T_0)\Delta S_x^{pr} \tag{21} \]

analogous to (14). The quantity

\[ \eta_{\mathrm{e}}=\frac{Q_x}{Q_x^{pr}} \tag{22} \]

is the energetic efficiency of the heat exchanger. Calculations analogous to those that were carried out in determining \(\varepsilon_{\mathrm{e}}\) give:

\[ \eta_{\mathrm{e}}=\eta\, \frac{1-\dfrac{T_0}{T''}}{1-\dfrac{T_0}{T'}} \tag{23} \]

and

\[ \eta_{\mathrm{e}}=\frac{\eta Q_p-T_0\Delta S_x}{Q_p-T_0\Delta S_g} \tag{24} \]

or

\[ \eta_{\mathrm{e}}=\frac{Q_x-T_0\Delta S_x}{Q_p-T_0\Delta S_g}. \tag{24'} \]

According to (22), the energetic efficiency of the heat exchanger is the ratio of the heat transferred to the cold liquid to the limiting amount of heat that the cold liquid can receive when all the available heat is removed from the hot liquid in a perfect heat exchanger without losses, exclusively by equilibrium processes. On the other hand, from relation (24′) it is evident that \(\eta_{\mathrm{e}}\) can also be considered

be taken as the ratio of the work capacities of the heat imparted to the cold body and of the available heat of the hot body.

3.

Of the proposed methods for evaluating the degree of perfection of heat exchangers we shall consider:

1. The method of I. El’perin [3]. To evaluate the degree of perfection of a heat exchanger he proposes using the entropy efficiency of the heat exchanger, which is calculated from the relation

\[ \eta_{\text{entr}}=\frac{|\Delta S_2|}{\Delta S_x}. \tag{25} \]

As justification, the following assertion is given, the essence of which reduces to this. In a heat-exchange apparatus that represents a closed system, heat can pass from the more heated to the colder heat carrier. If these processes occur reversibly, then the absolute change in entropy of the hot heat carrier will be equal to the magnitude of the entropy change of the cold heat carrier, and the degree of perfection according to (25) will be equal to unity. In irreversible processes in the system of heat carriers \(\Delta S_x>|\Delta S_2|\), and consequently \(\eta_{\text{entr}}<1\).

It is not difficult, however, to see that these considerations are incorrect. The task of a heat exchanger is the removal of heat from the hot body and the transfer of heat to the cold body under specified parameters of the heat carriers. The greater the amount of heat that is transferred to the cold body with the same removal of heat from the hot body, the more perfect, naturally, the heat exchanger should be considered. But an increase in the amount of heat transferred to the cold heat carrier, with the same removal of heat from the hot heat carrier under the indicated conditions, will entail an increase in \(\Delta S_x\), i.e., a decrease in \(\eta_{\text{entr}}\), although from thermodynamic positions the operation of the heat exchanger has improved.

The essence of I. El’perin’s error consists in the fact that in a heat exchanger in which two bodies at different temperatures are present, it is in general impossible to carry out a reversible transfer of heat between these bodies, i.e., it is impossible to obtain \(\eta_{\text{entr}}=1\). Further, a heat exchanger cannot in reality be regarded as an isolated system. With significant heat losses to the surroundings, little heat may be transferred to the heated body, \(\Delta S_x\) will prove to be a small quantity, and \(\eta_{\text{entr}}\) will prove to be greater than unity. Indeed, from (25), taking into account (1), (10), and (11), one may obtain

\[ \eta_{\text{entr}}=\frac{1}{\varepsilon}\,\frac{T''}{T^1}. \tag{26} \]

It follows from this relation that the greater the losses of the heat exchanger to the surroundings, the higher its entropy efficiency. It is clear that this conclusion is essentially incorrect. Consequently, the concept of entropy efficiency should not be applied for evaluating the degree of perfection of a heat exchanger.

1. Professor L. Melent’ev [4], considering the question of evaluating the degree of perfection of a heat exchanger included in the cycle of a steam-power plant, gives for its calculation a formula which, in principle, may be written as follows:

\[ \eta_{A(\text{overall})}=\frac{Q_x}{Q_2+\Delta Q_2}, \tag{27} \]

1) Formula (25) is written in our notation.
2) Our notation has been adopted.

Here \(\Delta Q_2\) represents the additional heat expenditure in the cycle associated with the loss of convertible energy (underutilization of the pressure drop in the turbine), determined from the difference between the heat expenditures of the condensing and district-heating cycles; it is assumed here that the losses of convertible energy of the district-heating cycle are compensated by additional production of energy in the condensing cycle.

First of all, it is necessary to note that the relation we have given for the energetic coefficient of heat retention (16) can be brought to a form analogous to (27). Indeed, from (10), (13), and (14) we have:

\[ Q_x^m = Q_2 \frac{1 - \dfrac{T_0}{T'}} {1 - \dfrac{T_0}{T''}} . \tag{28} \]

Substituting this quantity into (16), after transformations we obtain

\[ e_\theta = \frac{Q_x} { Q_2 + Q_2 \frac{\dfrac{1}{T''} - \dfrac{1}{T'}} {\dfrac{1}{T_0} - \dfrac{1}{T''}} }. \tag{29} \]

The additional term in the denominator of this relation represents the additional heat expenditure in the hot heat carrier which must be expended in the heat exchanger in order to transfer to the cold liquid the heat \(Q_x^m\) instead of \(Q_x\), which is actually transferred. This additional heat expenditure is due to the loss of convertible energy, i.e., it is a consequence of the irreversibility of the processes in the heat exchanger.

At the same time, relation (29), and consequently also relation (16), must be regarded as fundamentally more correct than relation (27) of Prof. L. Melent’ev. The value of the second term in relation (27) depends, for example, on the values of the initial parameters of the steam entering the engine. One and the same heat exchanger, into which a definite quantity of heating steam and heated water with the same parameters enter, will have a different degree of perfection according to (27) if it operates in installations with different initial steam parameters. But the degree of perfection of a heat exchanger cannot depend on quantities that have no relation to its operation. Therefore, assessment of the degree of perfection of a heat exchanger by the energetic coefficient of heat retention, or by the energetic efficiency, is the most acceptable.

References

1. Kirpichev M. Energy balance of thermal installations. Izv. AN SSSR, OTN, No. 12, 1949.

2. Fuks G. Mean thermodynamic temperature. Izv. Tomsk Polytechnic Institute, vol. 63, 1945.

3. El’perin Z. Entropic efficiency of a heat-exchange apparatus. Collection of Scientific Works of the Belorussian State Polytechnic Institute, issue 2, 1952.

4. Melent’ev L. District Heating, part 2, Academy of Sciences of the USSR, 1948.