ON THE EFFICIENCY OF VACUUM POWER INSTALLATIONS

S. V. POLOZHII

To evaluate the possibility and effectiveness of using the heat of heated waters for the purpose of producing electrical energy, the correct interpretation of the cycle and the determination of the efficiency of vacuum installations are of considerable importance. On this depends the real assessment of the energy resources of natural waters and secondary energy resources of industry, as well as the possibility of implementing vacuum installations.

The question of determining the efficiency of vacuum installations was studied by Ray \[5\], Shambadal \[2;3\], and Nisolle \[4\].

Ray defined the efficiency as the ratio of the “maximum work” to the potential thermal energy contained in the heated water within the temperature limits of operation of the installation,

\[ \eta=\frac{W_{\max}}{T_0-T_2}. \]

Shambadal proposed a method for estimating the effectiveness of a vacuum installation based on determining the ratio of the work of the installation with a given number of evaporators to the maximum possible work with an infinitely large number of evaporators, in the form of the relation

\[ \eta=\frac{n}{n+1}, \]

where \(n\) is the number of evaporators.

Nisolle arrived at the same result.

Determination of the efficiency of a vacuum installation according to Ray

Ray determined the maximum work of one kilogram of heated water entering the evaporator of a vacuum installation from the approximate expression for the work of a given quantity of steam in a theoretical machine operating according to the Rankine cycle proposed by Rato:

\[ W=\frac{c(T_0-T_1)}{r_1}(T_1-T_2) \left[ \frac{r_1}{T_1}+\frac{T_1-T_2}{T_1+T_2} \right] \ \text{kcal/kg}, \]

where \(T_0\) is the initial absolute temperature of the heated water, in absolute degrees;
\(T_1\) is the temperature of the steam leaving the evaporator, in absolute degrees;
\(T_2\) is the temperature of steam condensation (of the cold body), in absolute degrees;
\(r_1\) is the latent heat of vaporization at temperature \(T_1\), in \(\text{kcal/kg}\);
\(c\) is the heat capacity of water \(=1\).

Taking the maximum of this expression within the limits of the extreme temperatures \(T_0\) and \(T_2\), as a function of the temperature in the evaporator \(T_1\), i.e.,

\[ \frac{dW}{dT_1}=0, \]

Rey obtained a very complex and cumbersome formula, which for practical use he replaced by the simpler empirical formula

\[ W_m = 0.001343\,(T_0 - T_2)^{1.8675}, \]

and obtained the expression for the efficiency

\[ \eta = \frac{W_m}{T_0 - T_2} = \frac{0.001343(T_0 - T_2)^{1.8675}}{T_0 - T_2} = 0.001343\,(T_0 - T_2)^{0.8675}. \]

In reality, as investigations show, this maximum work does not fully characterize the cycle of a vacuum installation, but only part of it—the production of steam and the conversion of the latter into work in a theoretical installation that has no expenditure of energy for carrying out the cycle (pumping of cooling and heated water and extraction of air), which, because of the specific features of vacuum installations, not only under real conditions but also in the theoretical cycle constitutes a significant quantity. Failure to take this circumstance into account leads to incorrect conclusions and to an incorrect understanding of the problem posed under such conditions.

Consideration of Rey’s formula for maximum work shows that even with a small temperature difference \(T_0 - T_2\), heated water gives work; but if one looks at the same time at how much energy is required to carry out the cycle of a vacuum installation, it turns out that the amount of energy produced will be insufficient to carry out the installation cycle, or, in the best case, the installation will be able to serve itself without providing free energy to consumers. In other cases, the energy produced from heated water may be sufficient for the continuous maintenance of the cycle of the vacuum installation, but the latter’s consumption for its own needs will reach such proportions that the useful work of the installation does not justify the investment of funds in the construction and operation of such an installation.

Determination of the Efficiency of a Vacuum Installation According to Shambadal

Shambadal’s determination of the maximum work of a vacuum installation with multistage evaporation was made from the same premises as Rey’s.

As a result of complex mathematical calculations he arrived at the final result for the value of the efficiency of a vacuum installation with multistage evaporation, as the ratio of the maximum work with a given number of evaporators \(\left(W_m^{(n)}\right)\) to the maximum possible work with an infinite number of evaporators \(\left(W_m^{(\infty)}\right)\), in the form of the ratio of the number of evaporators,

\[ \eta = \frac{n}{n+1}. \]

In Fig. 1 this dependence is shown graphically. The efficiency of the installation determined in this way, as is not difficult to see, gives no idea of the effectiveness and possibility of utilizing heated waters for power purposes. Thus, for example, always and in all cases the efficiency of an installation with single-stage evaporation according to Shambadal will be equal to fifty percent

\[ \eta = \frac{n}{n+1} = \frac{1}{1+1} = 0.5. \]

Fig. 1

In essence, this formula represents the theoretical benefit from increasing the number of evaporators, and, moreover, a very incomplete one, since it

Fig. 2

completely disregards the influence of a number of factors in the operation of a theoretical, and still more of a real, installation.

To illustrate this, Fig. 2 shows the consumption of heated and cooling water by the installation as a function of the number of evaporators in a particular

Fig. 3

example (initial temperature of the heated water \(90^\circ\mathrm{C}\), temperature of the cooling water \(1.7^\circ\mathrm{C}\), temperature of steam condensation \(6.7^\circ\mathrm{C}\)).

Fig. 3 shows the expenditure of energy for pumping this water per kilowatt-hour of the installation (the head to be overcome is \(10\) m of water). The energy expenditure for pumping water falls especially sharply with two evaporators.

Fig. 4

The influence of the heads to be overcome in the water circuit on the useful work of a kilogram of water (the work produced for external consumers) is shown in

...shown in Fig. 4. Curve 2 shows the useful work of one kilogram of heated water at a head of the water path of 10 m water column; curve 3, at 30 m water column.

Efficiency of Vacuum Installations

The works of French scientists on the use of low-temperature heat resources [2–5] do not correctly reflect either the theoretical or the real possibility and efficiency of using the heat of heated waters, owing to the isolated consideration of the cycle of vacuum installations, the neglect of the work of auxiliary devices in the theoretical cycle, and the assimilation of the cycle of vacuum installations to the Rankine cycle.

The application of the Rankine cycle to vacuum installations cannot be considered correct, because of the peculiar change of the working fluid, the specific dependence between the individual elements of the cycle, and the considerable magnitude of the work of the pumps in comparison with the work of the steam in the engine.

Figure 5 shows diagrams of vacuum power installations in which heated water from a heat source (the heat of natural waters or the secondary heat of water from industrial processes) is directed into a vacuum evaporator, in which a small part of the water is converted into saturated steam as a result of the pressure falling below the boiling temperature. The unevaporated water is delivered back by a pump to the heat source—the closed scheme of the installation (Fig. 5b)—or is discharged—the open scheme (Fig. 5a).

To obtain 1 kg of steam in the evaporator, it is necessary to supply \(m_1\) kilograms of heated water (the evaporation ratio). Approximately, it may be considered that for every 10-degree temperature drop worked off in the evaporator, 55 kg of heated water are required to form 1 kg of steam. Consequently, under the same conditions, the work of supplying water to the evaporator-boiler of the vacuum installation will be \(m_1\) times greater than in the Rankine cycle.

The capacity for work of 1 kg of steam always increases with an increase in the temperature drop falling on the turbine \((t_1 - t_2)\), whereas the maximum work of the cycle depends on the ratio of the heat drops of the evaporator and the turbine, at which it is possible to use most effectively the energy of the heated water contained in the heat source. This is another difference between the cycle of a vacuum installation and the Rankine cycle.

In the cycle of a vacuum installation one cannot neglect the process of condensation of steam in the turbine condenser, as is done when considering the Rankine cycle, because of the dependence of the cooling ratio \(m_2\) on the operating conditions of the cycle (the temperatures of the heated and cooling water, the temperature in the evaporator, the temperature drop in the condenser, and the large specific consumption rates of cooling water). In addition, with mixing condensation there is an expenditure of work for the removal of air, which may reach a considerable magnitude depending on the operating conditions of the cycle.

The low capacity for work of the steam and the large work of the auxiliary units do not allow one to neglect the work of the latter when considering the theoretical cycle of vacuum installations, nor to consider the separate elements of the cycle in isolation, outside their general interrelation.

Proceeding from all these conditions and features of the cycles of vacuum installations, the useful theoretical work of the latter can be determined from the expression:

\[ AW = h_a - AW_u - AW_k - AW_{\mathrm{ek}}\ \text{kcal/kg} \tag{1} \]

where \(h_a\) is the adiabatic heat drop of the vacuum turbine, kcal/kg;
\(\Delta W_i\) is the theoretical work of the pump for pumping the heated water, kcal/kg;
\(\Delta W_k\) is the same for pumping the cooling water, kcal/kg;
\(\Delta W_{\text{ex}}\) is the theoretical work of the air extractor, kcal/kg.

This will be true for ideal vacuum installations in which:
1) there is no distance of the unit from the heat source and the cooling water;
2) there is no difference in the elevation levels of the unit, the heat source,

Fig. 5

1—evaporator; 2—turbine; 3—condenser; 4—heat source; 5—source of cooling water; 6—water pumps; 7—air extractor.

and the cooling water; 3) there are no losses in the cycle; 4) there is no air leakage into the system; 5) the work of the condensate pump is not taken into account.

This expression has the form:

\[ AW=(i_1-i_2)-A(p_0-p_1)m_1-A(p_0-p_2)m_2-0.00295\,\frac{p_2}{\gamma_2} \left(\frac{a_1m_1}{T+t_0}+ \frac{a_2m_2}{T+t'_2}-\frac{a_3p'_2m_2}{T+t_2}\right) \times \left[\left(\frac{p_0}{p_2}\right)^{0.285}-1\right] \quad \text{kcal/kg} \tag{2} \]

for an open-cycle arrangement and mixing condensation.

\[ AW=(i_1-i_2)-A(p_0-p_1)m_1-AH_km_2-0.00295\,\frac{p_2}{\gamma_2}\cdot \frac{a_1m_1}{T+t_0} \left[\left(\frac{p_0}{p_2}\right)^{0.285}-1\right] \tag{3} \]

for an open-cycle arrangement and surface condensation.

\[ AW=(i_1-i_2)-AH_{\mathrm{и}}m_1-AH_km_2-0.00295\,\frac{p_2}{\gamma_2}\, \frac{a_1m_1}{T+t_0} \left[\left(\frac{p_0}{p_2}\right)^{0.285}-1\right] \tag{4} \]

for a closed-cycle arrangement and surface condensation; the last term may be neglected because of the repeated deaeration of the water.

Here: \(i_1\) is the heat content of the steam before the turbine, in kcal/kg,
\(i_2\) is the same in the turbine condenser, in kcal/kg,
\(p_0\) is the barometric pressure, in m water column,
\(p_1\) is the pressure in the evaporator, in m water column,
\(m_1\) is the evaporation ratio, kg of water per 1 kg of steam,

\[ m_1=\frac{r_1}{t_0-t_1}, \]

\(r_1\) is the latent heat of vaporization at \(p_1\), kcal/kg,
\(t_0\) is the temperature of the heated water, °C,
\(t_1\) is the temperature in the evaporator, °C,
\(p_2\) is the pressure in the condenser, m water column,
\(p'_2\) is the pressure in the condenser, ata,
\(m_2\) is the cooling ratio, kg of water per 1 kg of steam;

\[ m_2=\frac{i_2-t_2}{t_2-t'_2}=\frac{i_2-t_2}{\Delta t_k}, \]

\(t_2\) is the condensation temperature at \(p_2\), °C,
\(t'_2\) is the temperature of the cooling water at the inlet to the condenser, °C,
\(a_3\) is the coefficient of solubility of air at \(t_2\),
\(\gamma_2\) is the specific weight of air at \(p_2\), kg/m³,
\(a_1\) is the coefficient of solubility of air at \(t_0\), kg/kg,
\(a_2\) is the same at \(t'_2\), kg/kg,
\(H_{\mathrm{и}}\) is the hydraulic resistance of the heat source, m water column,
\(H_k\) is the hydraulic resistance of the surface condenser, m water column.

In all cases (formulas 1–4), at the optimum evaporator temperature \(t_1=t_{1\mathrm{opt}}\), the theoretical useful work of the cycle will have a maximum value.

For the open cycle and for comparative calculations, the optimum evaporator temperature may be determined from Ren’s formulas.

The exact value of \(t_{1\mathrm{opt}}\) for the cycle as a whole is determined by the method proposed by the author [1].

For a closed cycle, the optimum temperature is determined point by point (formula 4) by calculating the cycle.

Figure 6 gives curves of the dependence of the theoretical work of the constituent elements of an ideal open cycle at a temperature

\(t_1=t_{1\mathrm{opt.}}\), calculated by Ray’s formula (pumping of heated and cooling water, air extraction) as a function of the initial temperature of the heated water, in percent of the theoretical work of \(1\ kg\) of steam at the fixed temperature \(t_2=6.7^\circ\mathrm{C}\). As is evident from the character of the curves, the work of the individual elements of the cycle increases sharply as the temperature of the heated water decreases.

Fig. 6

1—pumping of heated water; 2—pumping of cooling water;
3—air extraction.

In Fig. 7, for this same particular example, the theoretical energy expenditure for carrying out the cycle itself \((\Sigma A W_0)\) and the theoretical heat drop of the vacuum turbine for \(1\ kg\) of steam are shown; from these it is evident that the useful work of an ideal installation, at a definite value of \(t_0\) and \(t_2\) (for the present example \(t_0=15^\circ\mathrm{C}\)), is equal to zero.

The work of the cycle of vacuum power installations under real conditions includes all the factors that determine the course of the actual process, and significantly reduces the useful work of the latter

\[ A W_g=(i_1-i_2)\eta_{0i}\eta_M\eta_c -\frac{A(p_0-p_1+H'_u)m_1}{\eta_n\eta_M\eta_{mu}} -\frac{A(p_0-p_2+H'_k)m_2}{\eta_n\eta_M\eta_{mk}} -\frac{A H_{kon}}{\eta_M\eta_n} - . \]

\[ --\frac{ \dfrac{n}{n-1} A p'_2 \left[ \frac{0.3529}{\gamma_2} \left( \frac{a_1 m_1}{T+t_0} + \frac{a_2 m_2}{T+t'_2} - \frac{a_3 p''_2 m_2}{T+t_2} \right) +V_n+V_{np} \right] \left[ \left(\frac{p_0}{p'_2}\right)^{\frac{n-1}{n}}-1 \right] }{\eta_{ek}} \tag{5} \]

for an open cycle and mixing condensation.

Here:

\[ \eta_{oi}\eta_M\eta_2=\eta_{oe} \]

is the relative electrical efficiency of the turbogenerator;

\[ \eta_n \]

is the efficiency of the water pumps;

\[ \eta_M \]

is the efficiency of the pump drive;

\[ \eta_{tu} \]

is the thermal efficiency of the evaporator;

\[ \eta_{tk} \]

is the thermal efficiency of the condenser, taking into account the increase in \(m_2\) due to the rise of \(i_2\) under actual conditions;

\[ H'_u \]

is the hydraulic resistance of the evaporator water conduits due to the remoteness of the installation and the difference in elevation levels, taking into account the siphon lowering of the water;

Figure labels: vertical axis \(K\), kcal/kg; horizontal axis: temperature of heated water, \({}^{\circ}C\); \(t_2=6.7^{\circ}C\); \(h_a\); \(\Sigma AW_{cn}\).

Fig. 7

\[ H_{kon} \]

is the hydraulic resistance of the condensate path, m water column;

\[ H'_k \]

is the hydraulic resistance of the cooling-water conduits due to the remoteness of the installation and the difference in elevation levels, m water column;

\[ n \]

is the polytropic exponent;

\[ V_n \]

is the volume of water vapors extracted together with the air, \(\mathrm{m}^3\);

\[ V_{np} \]

is the air leakage, \(\mathrm{m}^3\);

\[ \eta_{ek} \]

is the efficiency of the air extractor, also taking into account the method of air removal;

\[ p'_2 \]

is the partial pressure of air in the condenser, \(\mathrm{kg}/\mathrm{m}^2\).

In analyzing the formula for the work of the cycle of vacuum installations under actual conditions, it may turn out that

\[ A W_d > 0, \quad \text{when } h_0=(i_1-i_2)\eta_{oz} > \sum A W_{sn}, \]

\[ A W_d < 0, \quad \text{when } h_0 < \sum A W_{sn}, \]

\[ A W_d = 0, \quad \text{when } h_0 = \sum A W_{sn}. \]

Consequently, utilization of the energy of heated water for the production of electrical energy can be carried out only in the first case, \(A W_d > 0\), and the useful work must not be below a certain value, determined by a techno-economic calculation, at which the vacuum installation must justify the capital costs of its implementation.

In accordance with the method for determining the work of the cycle of vacuum installations, the efficiency of the installations is determined.

The theoretical efficiency of vacuum installations is

\[ \eta_{aust}=\frac{A W}{(t_0-t_2)m_1}, \]

the actual efficiency of vacuum installations is

\[ \eta_{yst}=\frac{A W_d}{(t_0-t_2)m_{1d}}. \]

When

\[ t_1=t_{1opt}, \]

\[ \eta_{aust} \quad \text{and} \quad \eta_{yst} \]

assume their maximum value. \(^{1)}\)

References

1. S. V. Polozhii. Determination of the optimum evaporator temperature of a vacuum installation with single-stage evaporation, Proceedings of the Tomsk Polytechnic Institute, vol. 66, issue 2, 1949.

2. P. Chambadal. Du fonctionnement eu cerie des bouilleurs servant a’ l’utilisation de l’energie des eaux chaudes. Chaleur et Industrie No. 144, 1932.

3. P. Chambadal. Les transformations d’energie continues et discontinues. La Technique Moderne Nos. 15, 16, 1947.

4. L. Nisolle. Du fonctionnement eu série des bouilleurs servant á l’utilisation des eaux chaudes. Chaleur er Industrie No. 147, 1932.

5. I. Rey. De l’utilisation de l’energie des eaux chaudes industrielles et naturelles, Chaleur et Industrie, mai, 1930.

\(^{1)}\) Editorial note

While fully agreeing with the author that the efficiency of vacuum power installations depends on the expenditure of energy for their own needs, the editors do not concur with him that it is advisable to take this circumstance into account by constructing a special expression for the efficiency of the cycle. Serious objections may be raised against this.