HEAT AND ENERGY BALANCES OF A THERMAL INSTALLATION

G. I. Fuks

1

The study of the operation of a thermal installation by means of its heat balance is a well-known method. At the same time, the energy balance of an installation has not yet found application in practice, although, as has been pointed out more than once [1], it can give very interesting results, especially in comparison with the heat balance.

The heat balance of an installation as a whole, and of any part of it, is compiled on the basis of the law of conservation of energy (the first law of thermodynamics). On this basis, losses are evaluated in calories, irrespective of the fraction in which these calories can be converted into work in the given installation. A calorie of heat “lost” in the condenser is counted in the heat balance in the same way as a calorie of heat lost in the superheated-steam pipeline from the boiler to the turbine. Meanwhile, as is known, from a calorie of heat in superheated steam one can obtain a certain amount of work, whereas from a calorie of heat given off to the condenser—assuming that it has the lowest possible temperature under the given conditions—no work can be obtained.

The energy balance takes into account not only the quantity but also the “quality” of the losses in the installation, i.e., the amount of work that can be obtained at the expense of the lost heat.

2

The energy balance of an installation and of any part of it can be compiled by various methods. Thus, according to Gouy [2], the loss of work capacity can be calculated from the relation

\[ \Delta A L = T_0 \Delta S, \tag{1} \]

where \(T_0\) is the lowest temperature in the system,
\(\Delta S\) is the increase in entropy.

However, the direct application of this relation for calculation will not always be correct. If, in some part of the installation, irreversible processes occur simultaneously (for example, throttling or heat exchange at a finite temperature difference, etc.) and direct heat transfer to the environment at a lower temperature, then the decrease in the entropy of the system caused by the latter process may partially or completely compensate for the increase in entropy from the former processes. Then, obviously, calculation by (1) will not give correct results¹.

Let there be, in a system with the lowest temperature \(T_0\), a heat exchanger in which the hot liquid undergoes process \(1—2\), see Fig. 1, and the cold liquid—process \(3—4\). The hot liquid gives up heat \(Q_1\), the heated (cold) liquid receives heat \(Q_2\), and \(Q_1 - Q_2 = \Delta Q\) constitutes the loss to the surroundings.

Fig. 1

Using the concepts of the mean thermodynamic temperatures of processes \(1—2\) (\(T'\)) and \(3—4\) (\(T''\)) [4], one can write the following expressions for the available work of processes \(1—2\) and \(3—4\):

\[ AL_1 = Q_1 \left( 1 - \frac{T_0}{T'} \right), \tag{2} \]

\[ AL_2 = Q_2 \left( 1 - \frac{T_0}{T''} \right). \tag{2'} \]

The loss of available work will be

\[ \Delta AL = AL_1 - AL_2 \]

or

\[ \Delta AL = \Delta Q + T_0 \left( \frac{Q_2}{T''} - \frac{Q_1}{T'} \right). \tag{3} \]

From Fig. 1 it is evident that

\[ \frac{Q_2}{T''} = \Delta S_2, \tag{4} \]

\[ \frac{Q_1}{T'} = \Delta S_1. \tag{4'} \]

Consequently, the loss of work capacity in the heat exchanger is

\[ \Delta AL=\Delta Q+T_0\Delta S, \tag{5} \]

where \(\Delta Q=Q_1-Q_2\) gives the direct loss of heat to the environment, while \(\Delta S=\Delta S_2-\Delta S_1\) represents the observed increase of entropy in the heat exchanger. It is clear that direct heat losses always lead to a decrease in the work capacity of the system. The quantities \(\Delta S_1\) and \(\Delta S_2\) are entropy increments of individual bodies in the heat exchanger; therefore their values do not depend on the magnitudes of the zero values of entropy of the individual bodies of the system. For water and steam, for example, the entropy values may be taken directly from tables.

The expression (5) obtained is suitable not only for a heat exchanger, but also for any part of a thermal installation and for the entire installation as a whole. Thus, for example, if \(G_{\text{hr}}^{k2}\) of steam with parameters \(p_1, t_1, i_1\), and \(s_1\) is supplied through a steam line from the boiler, while steam reaches the turbine with parameters \(p_2, t_2, i_2\), and \(s_2\), which means that the following is lost to the environment

\[ \Delta Q=G(i_1-i_2)\,\frac{\text{kcal}}{\text{hr}} . \]

and steam throttling takes place in the steam line from steam pressure \(p_1\) to \(p_2\), then the loss of work capacity in the steam line will be

\[ \Delta AL=G\left[(i_1-i_2)+T_0(s_2-s_1)\right]\frac{\text{kcal}}{\text{hr}} . \tag{5'} \]

In a wet-steam line, in which the condensate is removed, \(G_2<G_1\) will reach the end of the steam line. The loss of work capacity will be

\[ \Delta AL=(G_1 i_1-G_2 i_2)+T_0(G_2 s_2-G_1 s_2)\frac{\text{kcal}}{\text{hr}} . \tag{5''} \]

The boiler of a steam-power installation as a whole may also be regarded as a heat exchanger. But the heat supplied with the fuel must be taken into account as an equivalent quantity of chemical energy, from which, in the limiting case, maximum work can be obtained in an amount practically equal to the calorific value of the fuel [5]. The loss of work capacity in the boiler will be

\[ \Delta AL=BQ_n^p-G[(i-i_b)-T_0(s-s_b)], \tag{5'''} \]

where \(G\), \(B\), and \(Q_n^p\) are, respectively, the hourly rates of steam and fuel and its calorific value,

\(i\), \(s\), \(i_b\), and \(s_b\) are, respectively, the enthalpies and entropies of steam and water.

As an illustration, an example is given of comparing the heat and energy balances of a condensing steam installation under the following conditions.

The pressure and temperature of the boiler steam are \(p_1=32\ \text{ata}\), \(t_1=420^\circ\text{C}\). Before the turbine \(p_1'=30\ \text{ata}\) and \(t_1'=400^\circ\text{C}\), in the condenser \(p_2=0.05\ \text{ata}\). The efficiencies of the individual elements are: \(\eta_{ku}=0.85\), \(\eta_{oi}=0.80\), \(\eta_m=0.98\), \(\eta_g=0.96\). Power \(N=12000\ \text{kW}\).

The heat balance of the installation according to these data will be¹

Table 1

The energy balance of the installation was calculated from relations (5′), (5″), and (5‴).

Table 2

A comparison of the heat and energy balances is given in Fig. 2. The boiler loss in calories is small. But the steam in the boiler is obtained at a low temperature; therefore its availability amounts to only 31.23% of the availability of the fuel. This would also be the efficiency of the installation if there were no further losses in it during the conversion of energy.

In the heat balance of the installation, the principal loss is the loss of the theoretical cycle (heat rejection to the condenser). In reality this loss is already due to the fact that, when heat was supplied to the working medium in the boiler, irreversible heat transfer took place. The rejection of the corresponding

the amount of heat from the working body thus becomes inevitable for the given character of the theoretical cycle. Heat is removed to the condenser in the theoretical cycle at the lowest temperature in the system,

Balances

a) Thermal

b) Energy

Fig. 2

i.e., the heat is removed which, under the given conditions, cannot be converted into work. Therefore the corresponding loss item is absent in the energy balance.

Thus, the energy balance clearly indicates the weak links in the chain of energy conversion. Therefore it is recommended, along with the usual heat balance, also to compile the energy balance of the installation.

REFERENCES CITED

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

2. Gouy. Sur l’énergie utilisable. Journal de Physique, 1889.

3. A. Keller. Entropie-balance diagramm spotlights steam-power losses. Power Engineering, July 1950.

4. Fuks, G. I. Mean thermodynamic temperatures. Izv. Tomsk Polytechnic Institute, vol. 63, 1944.

5. Fuks, G. I. Degree of perfection of a heat-power installation. Izv. Tomsk Polytechnic Institute, vol. 66, 1948.

6. Vukalovich, M. Thermodynamic properties of water and steam, 1950.

Noted Misprints

1. All calculations were carried out according to M. P. Vukalovich [6]. ↩↩