DETERMINATION OF THE OPTIMAL TEMPERATURE OF THE EVAPORATOR OF A VACUUM UNIT DURING SINGLE-STAGE EVAPORATION

S. V. POLOZHII

The problem of using heated waters for the production of electric power, although it has been subjected in France to considerable theoretical investigations, nevertheless, as regards the determination of the optimal temperature and the corresponding maximum work, these investigations were of an abstract mathematical character and did not take into account a number of phenomena and features inherent in a vacuum unit as a power-generating aggregate. Such investigations did not reflect the actual processes and the possible effect of utilizing heated waters and, in view of this, are insufficient.

There arises the necessity of a comprehensive consideration of the question of the utilization of heated waters, taking into account all the phenomena and features of a vacuum unit. Such a formulation and the correct solution of this question will make it possible to judge the feasibility and economy of using heated waters in each particular case.

The aim of the present work is to determine the optimal evaporation temperature of a vacuum unit and the factors influencing it.

The work of one kilogram of heated water in a vacuum unit depends on the initial temperature of the heated water—$T_0$, the temperature of the cooling water—$T_2$, and the temperature of the water (steam)—$T_1$, leaving the evaporator-boiler of the vacuum unit.

If the first two factors (the temperature of the hot and cold bodies) are common to all heat-engine installations, then the third factor (the temperature of the water leaving the evaporator) is a specific feature of vacuum units. With constant initial and final temperatures of the cycle of the vacuum unit, the work of a kilogram of water will have different values depending on the temperature in the evaporator, having a maximum value only at a quite definite optimal temperature in the latter.

In all other cases, when the temperature in the evaporator is above or below the optimal temperature, the work of a kilogram of water will always be smaller. When the temperature in the evaporator is raised above the optimal temperature, the amount of steam released from a kilogram of water decreases, but the temperature drop falling on the vacuum turbine increases. Conversely, when the temperature in the evaporator is lowered below the optimal temperature, the amount of steam released from a kilogram of water increases, but the temperature drop falls. This process proceeds in such a way that, in its limiting positions, the work of one kilogram of water in both cases is equal to zero.

In the first case, when the temperature in the evaporator is equal to the initial temperature of the heated water, $T_1 = T_0$, the amount of steam is equal to zero.

In the second case, when the temperature in the evaporator is equal to the temperature of the cold source, $T_1 = T_2$, the temperature drop of the vacuum turbine is equal to zero.

The determination of the theoretical optimal temperature of the evaporator of a vacuum unit was developed by Reem [2], who determined the maximum work of one kilogram of heated water and the temperature in the evaporator corresponding to this work analytically, and proposed an empirical equation that agrees very accurately with the theoretical one. In determining the optimal temperature, Reem did not take into account the operation of the vacuum unit as a whole. He did not consider the influence of the evaporator temperature on the energy consumption for pumping heated and cooling water and for air extraction. As a result, the theoretical optimal temperature determined by Reem does not correspond to the temperature that yields the maximum work of an actual vacuum unit.

Before proceeding to consider the influence of the temperature in the evaporator on the operation of a vacuum unit, it is necessary to examine the cycle of the latter. Figure 1 shows a diagram of a vacuum unit.

Heated water at temperature \(T_0\) is fed to the evaporator, in which, owing to a pressure drop below the boiling pressure, steam is formed, by virtue of the latent heat of evaporation, in the amount

\[ \frac{T_0 - T_1}{r_1} \]

kilograms per kilogram of heated water. The steam from the evaporator enters the vacuum turbine, where it expands from the pressure in the evaporator—\(P_1\)—to the pressure in the turbine condenser—\(P_2\).

The process of steam generation in the evaporator and the production of energy in the turbine constitute the main part of the cycle of the vacuum unit.

But in order to carry out the cycle of the vacuum unit it is necessary: for obtaining steam—to supply heated water to the evaporator; for condensing the steam—to supply cooling water to the turbine condenser; and for maintaining the vacuum—to remove air from the condenser.

Fig. 1. 1—evaporator; 2—vacuum turbine; 3—mixing condenser; 4—pumps; 5—air extractor; 6—generator.

Consequently, part of the energy obtained from the heated water must inevitably be expended on the supply of heated and cooling water and on the extraction of air, i.e., on the own needs of the vacuum unit. The latter, because of the enormous specific flow rates of cooling and heated water, considerably reduce the possible delivery of energy to external consumers. In addition, the energy consumption for the own needs of the vacuum unit, at constant temperatures of the heated and cooling water and other equal conditions, is a variable quantity depending on the temperature in the evaporator. Therefore, considering the cycle of the vacuum unit as a whole, we come to the conclusion that it is necessary to obtain the maximum power from 1 kg of water for external consumers—net power (power excluding the energy expended by the vacuum unit on itself), and not the potential energy contained in the heated water.

Accordingly, it is necessary to find such a water temperature in the evaporator of the vacuum unit at which it is possible, other conditions being equal, to obtain the maximum net power of the utilized heated water. Before showing the error in applying Reem’s formula for determining the optimal temperature from the standpoint of obtaining the maximum net power of the utilized source of heat—

of heated water, it is necessary to show the influence of the evaporator temperature of the vacuum unit on the quantitative changes in the individual elements of the latter.

For simplicity, we shall carry out this analysis for a particular example, in which it is assumed: \(T_0=303.7^\circ\); \(T_2=279.7\). With a constant temperature of the heated water and a constant temperature of steam condensation, by varying the temperature in the evaporator we obtain the dependence of the separate quantities on the latter.

The quantity of steam formed in the evaporator is calculated from the expression

\[ \alpha=\frac{T_0-T_1}{r_1}\ \text{kg/kg} \]

with the heat capacity of water \(=1\), or by the approximate formula

\[ \alpha=\frac{T_0-T_1}{752.2-0.567T_1}\ \text{kg/kg}, \]

where \(r_1\) is the latent heat of vaporization at the temperature in the evaporator.

The change in the quantity of evaporator steam and in the heat drop of the turbine as a function of the evaporator temperature is shown in Fig. 2.

Fig. 2

The specific steam consumption per kWh of the vacuum turbine is determined by the formula:

\[ d=\frac{860}{\eta_{oe}\cdot h_0}\ \text{kg/kWh}, \]

where \(\eta_{oe}\) is the relative electrical efficiency,

\(h_0\) is the theoretical heat drop of the vacuum turbine, kcal/kg.

The change in the specific steam consumption with the temperature in the evaporator is shown in Fig. 3; the specific steam consumption increases hyperbolically as the temperature in the evaporator decreases. The consumption of heated water per kWh of the vacuum turbine is determined as the quotient obtained by dividing the steam consumption by the turbine by the amount of steam obtained from one kilogram of heated water in the evaporator,

\[ q_1=\frac{d}{\alpha}\ \text{kg/kWh}. \]

The change in the specific consumption of heated water of the vacuum unit with the temperature in the evaporator is shown in Fig. 4, curve \(q_1\). As is evident from the graph, the specific consumption of heated water has a minimum only at a quite definite temperature in the evaporator.

The amount of cooling water per kWh of the vacuum turbine with mixed condensation is determined from the equation

\[ q_2=\frac{d\,(i_2-t_2)}{t_2-t'_2}\ \text{kg/kWh}, \]

where \(i_2\) is the heat content of the steam in the condenser, allowing for losses in the vacuum turbine (at \(\eta_{oi}=0.70\)) and at a pressure corresponding to the condensation temperature \(t_2=T_2-273\), kcal/kg;

\(t_2\) is the temperature of steam condensation in the turbine condenser, °C;

\(t'_2\) is the temperature of the cooling water entering the condenser, °C;

\(d\) is the specific steam consumption by the turbine, kg/kWh;

\(q_2\) is the amount of cooling water, kg/kWh.

The change in the amount of cooling water as a function of the evaporator temperature is shown in Fig. 4, curve \(q_2\). As can be seen from the graph, the amount of cooling water decreases as the temperature in the evaporator rises, and the curve has no pronounced minimum.

According to the consumption of heated and cooling water, the energy consumption for pumping this water is determined.

The energy consumption for pumping water is determined by the formula:

\[ N=\frac{q\cdot H}{3600\cdot 75\cdot 1.36\cdot \eta_{н}}\ \text{kW/kW}, \]

where \(q\) is the water consumption (respectively heated or cooling), kg/kWh;
\(H\) is the head to be overcome, in meters of water column, of the corresponding water circuit;
\(\eta_{н}\) is the efficiency of the pumping installation.

All other conditions being equal, the energy consumption for pumping water is proportional to the specific consumption of the latter.

The dependence of the energy consumption for pumping heated water (at \(H=10\) m water column) on the temperature in the evaporator is shown in Fig. 5, curve \(N_{н}\). The character of the curve is similar to that of the heated-water consumption curve.

Fig. 3

Fig. 4

The change in the energy consumption for pumping cooling water as a function of the evaporator temperature is presented in the same figure, curve \(N_0\). The curve is similar to the corresponding curve in Fig. 4.

The total energy consumption for pumping heated and cooling water, as a function of the evaporator temperature, is expressed by the curve \(N_{н}+N_0\) in Fig. 5. The amount of dissolved air to be removed from the cooling water by the vacuum turbine, per kWh of the latter, is determined from the expression:

\[ q'_2=\frac{\gamma\cdot K\cdot n\cdot T\cdot q_2}{(T+t'_2)\cdot 1000}\ \text{kg/kWh}. \]

Here: \(\gamma\) is the specific weight of air under normal conditions, in kg/m\(^3\);
\(K\) is the coefficient of solubility of air in water at the temperature of the latter;
\(t'_2\) is the temperature of the cooling water entering the condenser, in °C;
\(n\) is the fraction of air subject to extraction (taken as 0.75);
\(q'_2\) is the specific consumption of cooling water, in kg/kWh.

By this same formula one determines the quantity of air extracted from the heated water entering the evaporator,

\[ q'_1=\frac{\gamma \cdot K \cdot n \cdot T \cdot q_1}{(T-t_0)\,1000}\ \text{kg/kWh} \]

Here: \(t_0\) is the temperature of the heated water entering the evaporator, in °C;
\(K\) is the solubility coefficient at temperature \(t_0\);
\(q'_1\) is the specific consumption of heated water, in kg/kWh.

The change in the quantity of air subject to extraction, as a function of the temperature in the evaporator, is shown in Fig. 6.

From consideration of the formulas for determining the air subject to extraction from the cooling and heated water, and of the curves in Fig. 6, it follows that the change in the quantity of extracted air with a change in the temperature in the evaporator is proportional to the quantity of water for the corresponding temperatures in the evaporator. In this case, the minimum of the total quantity of extracted air is shifted somewhat toward higher temperatures in the evaporator than is the minimum total consumption of cooling and heated water. The latter is due to the increased amount of dissolved air in the cooling water and to its lower temperature.

Fig. 5

Fig. 6

The question of air extraction in vacuum units still requires resolution; therefore, it is difficult to determine precisely the energy consumption for air extraction.

However, if one adopts fractional removal of air by Claude’s method for Academician Rato’s extractor \([1, 3]\), then, for the example under consideration, the actual power expended on air extraction may be determined from the formula

\[ N_v=\frac{0.24}{\eta_{\text{ext}}}\,q_v\ \text{kW/kW} \]

\(\eta_{\text{ext}}\) — efficiency of the extractor (taken as 0.6),
\(q_v\) — total quantity of extracted air, in kg/kWh.

The dependence of the energy consumption for air extraction on the temperature in the evaporator is shown in Fig. 7. In all the figures (Figs. 3–10), the vertical dashed line shows the operating conditions at the temperature in the evaporator calculated by Ray’s empirical formula

\[ T_1 = T_2 + 0.4871\,(T_0 - T_2), \]

which corresponds with sufficient accuracy to the theoretical formula for determining the optimum temperature in the evaporator and the maximum operation of the heated water corresponding to this temperature.

From Figs. 4 and 6 it is evident that the minimum total consumption of heated and cooling water and the minimum amount of air extracted by the vacuum unit do not correspond to the temperature in the evaporator calculated by Ray’s formula, but are considerably higher than it.

At the same time, the minimum specific consumption of heated water and of the air extracted from it corresponds to the temperature in the evaporator found from Ray’s formula.

The consumption of cooling water by the vacuum unit, however, and the amount of air subject to extraction at the temperature found from Ray’s formula, are more than twice as great, for the present example, as those calculated by us. The ratio between the consumption of heated and cooling water depends

Fig. 7

Fig. 8

mainly on the temperature difference in the condenser of the vacuum turbine, varying inversely with the difference between the steam condensation temperature and the cooling-water temperature at the condenser inlet, \(t_2 - t'_2\), and may be found from the expression:

\[ \frac{q_2}{q_1} = \frac{\alpha (i_2 - t_2)}{t_2 - t'_2} \]

Figs. 5, 7, and 8, which show the energy consumption for pumping cooling and heated water and for air extraction, are, by the nature of the curves represented in them, similar to the corresponding curves of Figs. 4 and 6.

Figure 8 shows the curves of the total energy consumption for pumping cooling and heated water (curve \(\Sigma N_h\)) and for air extraction (curve \(\Sigma N_b\)), as well as the curve of the overall energy consumption for pumping cooling and heated water and for air extraction, i.e., the energy consumption for the vacuum unit’s own needs per kilowatt-hour of the latter.

The minimum energy consumption for own needs is, first, shifted toward higher temperatures in the evaporator than the temperature found by Ray’s formula and, second, the summation of the energy consumption for pumping

the cooling and heated water with the energy expenditure for air extraction slightly shifts the minimum point of the resultant curve in comparison with the minimum energy expenditure for pumping cooling water, i.e., the curve of energy expenditure for the unit’s own needs may be considered identical to the curve of energy expenditure for water pumping.

In conclusion, Fig. 9 shows the change in the net power of the vacuum unit per gross kilowatt-hour of the latter as a function of the temperature in the evaporator for the example under consideration. As can be seen from this figure, the maximum net power (the power that can be used for external energy consumers) corresponds to a higher temperature in the evaporator than the temperature found by Rey’s formula. Consequently, from the standpoint of obtaining maximum net power, calculation of the vacuum unit by Rey’s formulas is incorrect, and these formulas should not be used for calculating an actual vacuum unit.

From consideration of the curve of the specific consumption of heated water, we see that the temperature corresponding to the minimum total consumption of heated and cooling water corresponds to a greater specific consumption of heated water than the consumption at the temperature in the evaporator found by Rey’s formula (point “a,” Fig. 4). Therefore, taking as the design temperature in the evaporator the temperature found from the minimum consumption of heated and cooling water, which corresponds to the minimum energy expenditure for the unit’s own needs, we cannot obtain the maximum power of the entire source of utilized water, owing to the greater consumption of heated water per kilowatt-hour of the vacuum unit—the consumption corresponding to point “b” in Fig. 4.

Fig. 9

The total power of the vacuum unit is determined as the quotient obtained by dividing the total quantity of utilized heated water \(Q\) by the specific consumption of heated water \(q_1\), namely

\[ W = \frac{Q}{q_1}\ \text{kW}. \]

Thus, it must be assumed that the optimum design temperature of the evaporator lies above the temperature determined by Rey’s formula and below the temperature determined from the minimum energy expenditure for the own needs of the vacuum unit.

In addition to the inconsistency of Rey’s formulas in determining the optimum temperature of a vacuum unit of maximum net output, calculation of a vacuum unit by these formulas places the latter under less favorable operating conditions, owing to the consumption of an increased quantity of cooling water per kilowatt-hour of the vacuum turbine.

This is especially the case, as investigations show, at a small temperature difference of the cooling water in the condenser, which is unavoidable when utilizing heated water of low temperature; thus, for example, in the present example the consumption of cooling water is twice the consumption of heated water.

Figure 10 shows the influence of different heads of the water circuit on the position of the point of minimum consumption for pumping the cooling and heated water. For heads in the cooling-water circuit that considerably exceed those in the heated-water circuit of the vacuum unit, the minimum energy consumption for pumping the cooling and heated water shifts

toward higher temperatures in the evaporator, for example, point “a,” curve \(N_{5-30}\) (the first subscript is the head overcome in the hot-water circuit, the second in the cooling-water circuit, in m water column). Conversely, when the heads in the hot-water circuit considerably exceed those in the cooling-water circuit, the minimum expenditure for pumping water shifts toward lower temperatures in the evaporator (point “b,” curve \(N_{30-5}\)) relative to the minimum energy expenditure at equal heads in the hot- and cooling-water circuits, for which the optimum temperature was found in the preceding example.

Consequently, the minimum expenditure of energy for pumping the cooling and hot water, and the corresponding optimum temperature of the evaporator, are variable quantities depending on the ratio of the heads to be overcome and of the flow rates in the cooling- and hot-water circuits; and in all cases the minimum expenditure of energy for the vacuum unit’s own needs will correspond to a higher temperature in the evaporator than the temperature found by Ray’s formula.

Fig. 10

For an exact determination of the optimum temperature of the evaporator of a vacuum unit in single-stage evaporation, a new method of calculation is proposed. According to this method of calculation, the optimum temperature of the evaporator is determined in each individual case with allowance for the energy expenditure for its own needs, and with the maximum possible total net power of the vacuum unit thereby obtained.

The discrepancy between Ray’s formulas and the calculation of a vacuum unit, and the application of the proposed calculation method, can readily be traced in the particular example given above.

Table 1 gives the values of the hot-water flow rate, the specific net power, and the quotients obtained by dividing the specific net power by 1000 kilograms of hot-water flow rate for the corresponding temperatures in the evaporator.

Table 1

Using the data in the last column of Table 1, in Fig. 11 a curve of the specific net power per 1000 kg of hot water has been plotted.

At the maximum of the curve is found the optimal temperature corresponding to the maximum net output of the vacuum unit, which for the present example is

\[ t_1^{\mathrm{opt}} = 21.3^\circ \mathrm{C} \]

whereas, according to Rey’s formula, the optimal temperature is

\[ T_1^p = T_2 + 0.4871(T_0 - T_2) = 291.39^\circ \mathrm{K}, \]

or

\[ t_1^p = 18.39^\circ \mathrm{C}. \]

In Fig. 11, point “a” of the specific-power curve corresponds to the optimal temperature. Point “b” corresponds to the temperature calculated by Rey’s formula.

Fig. 11

Fig. 12

The net output at the optimal temperature is

\[ W_n^{\mathrm{opt}} = 0.199 \ \mathrm{kW}/1000 \ \mathrm{kg}. \]

When the temperature is chosen according to Rey,

\[ W_n^p = 0.184 \ \mathrm{kW}/1000 \ \mathrm{kg}. \]

The underproduction of energy by the vacuum unit calculated by Rey’s formula is

\[ \Delta W_n = \frac{0.199 - 0.184}{0.199}\cdot 100 = 7.5\%. \]

The specific consumption of cooling water at the optimal temperature is

\[ q_2^{\mathrm{opt}} = 5000 \ \mathrm{kg}/\mathrm{kW} \]

(point “a,” Fig. 12).

The specific consumption of cooling water when the temperature is chosen according to Rey is

\[ q_2^p = 6100 \ \mathrm{kg}/\mathrm{kW} \]

(point “b,” Fig. 12).

The excess consumption of cooling water by the vacuum unit calculated according to Rey’s formula is

\[ \Delta q_2 = \frac{6100 - 5000}{6100}\cdot 100 = 19\%. \]

References

1. I. N. Butakov—Use of the heat of circulating water at electric power stations for the production of electric energy. Proceedings of the Tomsk Polytechnic Institute, vol. 63, 1944.

2. J. Rey—De l’utilisation de l’énergie des eaux chaudes industrielles et naturelles. “Chaleur et Industrie,” May 1930.

3. M. Chambadal—Le dégazage fractionné des eaux chaudes et froides dans le procédé Claude—Boucherat, “Chaleur et Industrie,” No. 168, 1934.

4. Chambadal—Les transformations d’énergie continues et discontinues, “La Technique Moderne,” vol. XXXIX, Nos. 15–16, 1947.