On the Electrical-Energy Utilization of Low-Potential Secondary Energy Resources in Industry

S. V. Polozhii

The cycles of vacuum power installations, depending on the heat source, may be closed or open. In the first case, the source of low-potential heat is the cooling water of production units (blast furnaces and open-hearth furnaces, condensers of steam turbines, etc.). The essence of the closed cycle is that the heat-transfer medium, removing heat from the production unit, is used repeatedly, circulating through a closed loop: production unit, evaporator of the vacuum installation, production unit.

In the second case, the heat source is either natural warm waters (water of tropical seas, hot springs), or heated waters—production waste. At present, vacuum power installations operating according to a closed scheme are of the greatest interest to us, owing to the presence of substantial secondary energy resources in industry and the relative simplicity of their utilization.

Fig. 1

Figure 1 shows the basic closed scheme of a vacuum power installation. It consists of a heat source—the industrial unit—1, an evaporator—2, a condensing vacuum turbogenerator—3, a surface or mixing condenser—4; pumps: for pumping the heat-transfer medium—5, circulation pump—6, feed or condensate pump—7, air pump—8, and a source of cooling water—9.

The utilization of heat removed from the cooled surfaces of a production unit for generating electric power in a vacuum installation changes the nature of cooling of the heat-transfer medium. In this case the evaporator is functionally connected with the heat source. In some cases we can vary the temperature of the cooling water of the heat source arbitrarily—schemes of vacuum installations with a free circulation ratio; in other cases we do not have this possibility—schemes with a forced circulation ratio. In a scheme with a free circulation ratio, for one and the same amount of heat removed from the production unit, the weight quantity of heat-transfer medium circulating in the system may be different. It depends on the temperature difference worked out in the evaporator and is expressed by the formula

\[ G=\frac{Q}{c\,(t_0-t_1)}, \]

where

\(Q\) is the thermal capacity of the heat source, kcal/hour;
\(t_0\) is the water temperature at the outlet from the heat source;
\(t_1\) is the temperature of the water leaving the evaporator, approximately equal to the water temperature \(t_3\) at the inlet to the heat source;
\(c\) is the heat capacity of the liquid (for water at low temperatures \(c=1\)).

The feature of the theoretical and actual closed cycle of vacuum installations is determined to a considerable extent by the presence of a structurally embodied heat source. The process of steam generation in a closed cycle is carried out in two stages.

1. Heating of the heat-transfer medium in the industrial unit—the heat source.
2. Production of steam from the heat-transfer medium in the evaporator.

The connection between the first and second stages is determined from the equality

\[ Q=G(t_0-t_3)\simeq G(t_0-t_1), \]

\[ Q=G\Delta t_{ht}\simeq G\Delta t_w . \]

This equality is possible when the addition of heat-transfer medium is small and the difference between the temperature at the inlet to the heat source \(t_3\) and in the evaporator \(t_1\) is small, which is practically the case.

\[ D=\frac{Q}{i_1-t_1}=\frac{G(t_0-t_1)}{r_1}\ \text{kg/hour}, \]

where

\(D\) is the quantity of steam obtained in the evaporator,
\(r_1\) is the heat of vaporization at the saturation temperature in the evaporator.

This quantity of steam from the evaporator is directed to the vacuum turbine. The unevaporated heat-transfer medium at temperature \(t_1\) and the turbine condensate at temperature \(t_2\), or an addition of cold water, are returned to the heat source. Condensation of steam in the vacuum turbine may be carried out in surface or mixing condensers. Cooling water from the source of this water is supplied to the condenser by a circulating pump. Air extraction is performed by an air pump.

Thermodynamic Foundations of Vacuum Installations

In ordinary steam-power installations it is customary to consider the theoretical cycle without taking into account the energy expenditure for carrying out the cycle, and to take these expenditures into account only when determining the net efficiency of the actual installation. Such consideration simplifies the study of the process and makes it possible to establish more clearly the influence of the main factors on the operation of the cycle. However,

it is possible only with an insignificant expenditure of energy for the installation’s own needs and in the absence of any substantial connection between the energy consumption for carrying out the cycle and the parameters of the working fluid.

Extending the adopted method for studying steam-power cycles to the study of cycles of vacuum installations is not only fundamentally incorrect, but also leads to practically erroneous results.

For a thermodynamic analysis of the process of converting thermal energy into mechanical energy, it is necessary to consider the change of the working fluid in the cycle of a vacuum installation. Let us suppose that the heat source contains \(m_1\) kg of water at temperature \(t_1\). This quantity of water is heated in the heat source to such a temperature \(t_0\) that in the evaporator 1 kg of saturated steam at temperature \(t_1\) is obtained from it. The quantity \(m_1\), i.e., the amount of heated water required to obtain 1 kg of steam in the cycle, we shall call the evaporation ratio. The unevaporated water of the same temperature \(t_1\), with the addition of 1 kg of liquid, is fed by a pump back to the heat source. The process is then repeated. To carry out the part of the cycle under consideration, work must be expended on pumping the water through the heat source and the evaporator. Since the energy consumption for pumping the heat-transfer medium existed before the industrial unit was included in the installation cycle, we shall take into account the energy consumption for the “feed” pump only in the event that it differs from the previously existing one.

The working fluid in this part of the cycle undergoes the following changes. In the heat source, heat is supplied to the working fluid and its temperature is raised from \(t_1\) to \(t_0\) without a change in the state of aggregation. In the vacuum evaporator, during boiling without the supply of heat from outside, steam is formed and the heat-transfer medium is cooled from \(t_0\) to \(t_1\)

\[ m_1=\frac{r_1 x_1}{t_0-t_1}, \]

where \(r_1\) is the heat of evaporation at the pressure in the evaporator,
\(x_1\) is the dryness fraction of the steam.

The change in the state of the working fluid in the condenser is associated with the expenditure of energy for supplying cooling water and removing air. The quantity of cooling water \(m_2\) required for condensing one kilogram of steam is determined from the equation for the circulation ratio

\[ m_2=\frac{r_2 x_2}{t_2-t'_2}, \]

where \(x_2\) is the dryness fraction of the steam,
\(r_2\) is the heat of condensation at the pressure in the condenser,
\(t_2\) and \(t'_2\) are the temperature of the water at the inlet and outlet of the condenser.

As is easy to see, \(m_1\) and \(m_2\) are functions of the temperature difference, increasing as this difference decreases.

In accordance with \(m_1\) and \(m_2\), there is an energy expenditure for the feed and circulating pumps of the installation. When 1 kg of steam is condensed, the air introduced into the evaporator by \(m_1\) kilograms of water must be removed. In the case of mixing condensation, the air dissolved in \(m_2\) kilograms of cooling water must also be removed.

Because of the special dependence of the operation of the cycle and its individual elements on the temperature in the evaporator, the high values of \(m_1\) and \(m_2\), and the low capacity for work of a kilogram of steam, it is necessary to consider the theoretical—and still more the real—cycle of a vacuum installation not in isolation, but in the interrelation and interdependence of the operation of all elements of the cycle, taking into account the expenditure of energy for carrying out the cycle. An ideal vacuum installation operating according to the theoretical cycle shall be taken to mean an installation in

in which there is no difference in elevation levels and no distance between the heat source and the cooling water from the unit, and in which there are also no losses in the cycle (losses of heat, heat-transfer medium, etc.), there is no air infiltration into the system, steam humidity, etc. In the actual cycle of vacuum installations, the real conditions of the installation arrangement are taken into account (the distance and difference in elevation between the unit and the heat source and the cooling), the losses in the cycle, and the actual operating conditions of the installation (steam humidity, air infiltration, etc.).

The operation of a vacuum installation, in general form, is expressed by the dependence

\[ AW_{o}=h_{o}-AW_{on}-AW_{ok}-AW_{oэk}\ldots(a) \text{—for the theoretical cycle,} \]

\[ AW=h-AW_{n}-AW_{k}-AW_{эk}-AW_{nn}\ldots(в) \text{—for the real cycle,} \]

where

\[ h_{0},h \]
— the heat drop in the vacuum turbine of the theoretical and real cycles, kcal/kg,

\[ AW_{on},AW_{n} \]
— the theoretical and actual work of the feed pump (for pumping the heat-transfer medium), kcal/kg,

\[ AW_{ok},AW_{k} \]
— the same for the circulating pump (for pumping the cooling water), kcal/kg,

\[ AW_{oэk},AW_{эk} \]
— the same for the air pump (for removing air from the condenser), kcal/kg,

\[ AW_{nn} \]
— the work of the make-up pump, kcal/kg.

Work of one kilogram of steam in a vacuum turbine

The theoretical expansion work of one kilogram of steam in the turbine is determined as the difference between the heat contents of the steam in the evaporator and in the condenser

\[ h_{0}=i_{1}-i_{2}. \]

The change in the theoretical heat drop of a vacuum turbine with the temperature in the evaporator is shown in Fig. 3 and others (curve \(h_{0}\)). To

Fig. 2

obtain the maximum work of steam in the turbine, irrespective of its connection with the other elements of the cycle, it is necessary to bring the temperature in the evaporator \(t_{1}\) close to the initial temperature of the heat-transfer medium \(t_{0}\).

The actual work of one kilogram of steam in the turbogenerator has the value

\[ h=h_{0}\cdot \eta_{oi}\cdot \eta_{м}\cdot \eta_{г}=(i_{1}-i_{2})\cdot \eta_{oэ} \]

\(\eta_{oi}\eta_m\) — the internal relative and mechanical efficiency of the turbine,

\(\eta_g\) — the efficiency of the electric generator,

\(\eta_{oi}\) — is assumed to be small, since vacuum turbines, operating entirely in the region of wet steam, will be made as single- or two-stage units and will have a small unit capacity.

Fig. 3

Operation of the Feed Pump

Since in the ideal cycle of a vacuum installation there are no distance and difference in elevation between the heat source and the evaporator, the theoretical work of the pump consists in overcoming the difference in hydraulic resistances before and after inclusion of the heat source in the cycle, at the same value of \(m_1\):

\[ AW_{on}=AW\Delta H_{it}m_1, \]

\(\Delta H_{it}\) — the difference in the hydraulic resistance of the heat source before and after its inclusion in the cycle, m water column.

The dependence of the evaporation ratio on \(t_1\), for fixed \(t_0\) and \(t_2\), is shown in Fig. 2. In a closed cycle, the work of the feed pump depends on how the inclusion of the heat source in the installation cycle affects the water flow rate. If the previously existing degree of heating \((t_0-t_1)\) decreases, then the flow rate of the heat carrier will increase, and vice versa. Since vacuum installations are connected to already existing industrial units designed for definite operating conditions, the change in the heat-carrier flow rate will be reflected in the hydraulic resistance of the heat source, changing in proportion to the square of the heat-carrier flow rate:

\[ H_{it}=H_o\frac{G_{it}^2}{G_o^2}\ \text{m water column}. \]

\(H_{it}, G_{it}\) — the hydraulic resistance and heat-carrier flow rate through the heat source after its inclusion in the cycle of the vacuum installation,

\(H_o, G_o\) — the design resistance and heat-carrier flow rate through the heat source before inclusion in the cycle of the vacuum installation.

As \(t_1 \to t_0\), \(G_{it}\) and \(H_{it}\) tend to infinity. Accordingly, the work of the feed pump will increase.

The actual work of the feed pump depends on the distance and elevation of the installation relative to the heat source, on the efficiency of the pump unit, and also on heat losses (heat losses during transport of the stream and in the evaporator)

\[ AW_{u}=\frac{A(\Delta H_{ut}+H)m_{1}}{\eta_{n}\cdot\eta_{pot}}, \]

\(H\) — hydraulic resistance of the water path, m water column,
\(\eta_{n}\) — efficiency of the pump unit,
\(\eta_{pot}\) — efficiency of the heat stream during its transport within the installation, taking into account the increase in the evaporation ratio due to heat losses in the circuit.

Operation of the Circulating Pump

The theoretical work of the pump for removing cooling water from the condenser in the case of mixing condensation is determined from the equation:

\[ AW_{ok}=A(P_{0}-P_{2})\cdot m_{2} \]

\(P_{0}\) — barometric pressure, m water column.
\(P_{2}\) — pressure in the condenser, m water column.

The theoretical work of the pump for supplying cooling water to the condenser in the case of surface condensation

\[ AW_{ok}=AH_{k}\cdot m_{2}, \]

\(H_{k}\) — hydraulic resistance of the condenser, m water column.

In Figs. 4 and 5 (curve \(\varphi_{1}\)) the change is shown in the relative work of the circulating pump

\[ \left(\varphi_{1}=\frac{AW_{ok}}{h_{0}}\right) \]

as a function of the temperature in the evaporator for mixing condensation.

Such a dependence of the relative work of the circulating pump limits the reduction of the temperature in the evaporator at fixed \(t_{0}\) and \(t_{2}\).

Under actual conditions, the work of the pump for pumping cooling water depends on the distance and elevation of the installation above the source of cooling water. Owing to the large flow rate of cooling water, these factors are of decisive importance and in many cases will limit the use of vacuum installations.

The actual work of the circulating pump for a mixing condenser is

\[ AW_{k}=\frac{A(P_{0}-P_{2}+\Delta H_{k})\cdot m_{2}}{\eta_{n}}. \]

For a surface condenser this work is equal to

\[ AW_{k}=\frac{A(H_{k}+\Delta H_{k})\cdot m_{2}}{\eta_{n}} \]

\(\Delta H_{k}\) — hydraulic resistance of the water path, taking into account the elevation of the installation above the source of cooling water,
\(\eta_{n}\) — efficiency of the pump unit.

Fig. 4

Visible labels in the figure:

• \(t_0 = 70^\circ\)
• \(t_2 = 10^\circ\)
• \(y_2^{0} = \dfrac{\Delta W_0^{*}}{h_0}\)
• \(y_2 = \dfrac{\Delta W_0}{h_0}\)
• \(h_0\)
• \(\Delta W_0\)
• \(\Delta W_0^{*}\)
• \(W_{\mathrm{c}}\)
• \(t_1^{\mathrm{opt}}\)
• \(t,\ ^\circ\mathrm{C}\)
• \(\dfrac{\mathrm{kcal}}{\mathrm{kg}\,\mathrm{K}}\)

Fig. 5

Visible labels in the figure:

• \(t_0 = 70\)
• \(t_2 = 10\)
• \(Q_{0i} = 0.75;\ \eta_{\mathrm{n}} = 0.6\)
• \(y_2 = \dfrac{\Delta W_1}{h}\)
• \(y_1 = \dfrac{\Delta W_{\eta}}{h}\)
• \(h\)
• \(\Delta W\)
• \(y_1\)
• \(y_2\)
• \(t_1^{\mathrm{opt}}\)
• \(t,\ ^\circ\mathrm{C}\)
• \(\dfrac{\mathrm{kcal}}{\mathrm{kg}\,\mathrm{K}}\)

Operation of the Air Pump

The air entering the condenser with the heated and cooling water (the latter in the case of mixing condensation) is to be removed; the theoretical work of the pump for removing air may be determined from the equation

\[ AW_{\text{air}} = 0.00295\,\frac{P_2}{\gamma_2} \left( \frac{a_1 m_1}{T_0} + \frac{a_2 m_2}{T'_2} - \frac{a_3 P'_2 \cdot m_2}{T_2} \right) \left[ \left(\frac{P_0}{P_2}\right)^{0.285} -1 \right] \]

\(P_0\) — final pressure of air compression, equal to normal barometric pressure,

\(P_2\) — pressure in the condenser,

\(a_1; a_2; a_3\) — coefficients of solubility of air at the temperature of the heated water, the initial temperature of the cooling water, and the temperature of the water at the outlet from the condenser, respectively. In surface condensation the ratio

\[ \frac{a_2 m_2}{T'_2} \quad \text{and} \quad \frac{a_3 P'_2 \cdot m_2}{T_2} =0. \]

The air introduced only by the heated water is subject to removal.

The actual work for removing air depends to a large extent on the efficiency of the extractor and on the method of extraction. Thus, for example, the use of staged extraction reduces the work of air removal by a factor of 2–3. The presence of steam in the extracted air and air leakage into the system increases the work of the pump.

For mixing condensation, the work of a single-stage pump is equal to

\[ AW_{\text{air}} = \frac{ \frac{n}{n-1}\, A P_2 \left[ \frac{0.353}{\gamma_2} \left( \frac{a_1 m_1}{T_0} + \frac{a_2 m_2}{T'_2} - \frac{a_3 P'_2 \cdot m_2}{T_2} + V_n + V_{pr} \right) \right] \left[ \left(\frac{P_0}{P_2}\right)^{\frac{n-1}{n}} -1 \right] }{ \eta_{\text{air}} } \]

\(V_n\) — volume of water vapor removed together with the air,

\(V_{pr}\) — air leakage into the system, \(m^3\),

\(\eta_{\text{air}}\) — efficiency of the extractor,

\(n\) — polytropic exponent.

Work of the Cycle

Substituting the corresponding values of the work of the cycle elements into the general formula (a) or (b), we obtain the total work of the theoretical or actual cycle of vacuum installations. In this case both the theoretical and the actual work of all elements of the vacuum installation and of the entire cycle as a whole are functions of the temperature in the evaporator. Only at a certain temperature in the evaporator \((t_{1,\mathrm{opt}})\) does the theoretical work of the cycle reach its maximum value. At all other temperatures in the evaporator the work of the cycle will have a smaller value, and at an evaporator temperature close to the temperature of the heated water \((t_0)\) or to the steam-condensation temperature \((t_2)\), the work of the cycle may not only be equal to zero, but may also have a negative value. To illustrate the foregoing, Fig. 3 graphically shows the dependence of the theoretical work of a closed cycle on the temperature in the evaporator for a particular example \((t_0 = 70^\circ,\ t_2 = 10^\circ)\).

The actual work of the cycle of vacuum installations under real conditions includes all the factors that determine the course of the process.

Here, as in the theoretical cycle, the work of each element of the vacuum installation and of the entire cycle as a whole depends on the temperature in the evaporator,

\[ h=f(t_1),\quad W_p=f_2(t_1),\quad W_k=f_3(t_1),\quad W_{\mathrm{ek}}=f_4(t_1),\quad W=F(t_1). \]

It then turns out that the optimum temperature of the real cycle does not coincide with the temperature of the theoretical cycle. The expansion work of 1 kg of steam in the turbogenerator, as compared with the theoretical value, decreases, while the work of the remaining elements of the cycle increases. Under actual conditions it may turn out that

1) \(AW>0\), when \(h>AW_{\mathrm{cn}}\) — positive work of the cycle,

2) \(AW=0\), when \(h=AW_{\mathrm{cn}}\) — zero work of the cycle,

3) \(AW<0\), when \(h<AW_{\mathrm{cn}}\) — negative work of the cycle.

The actual power of the vacuum installation is

\[ N=\frac{AW\cdot D}{860}\ \mathrm{kW}. \]

The power of the turbine of the vacuum installation (for selecting the turbogenerator) is

\[ N_t=\frac{D h_o\cdot \eta_{oi}}{860}=\frac{Dh}{860}\ \mathrm{kW}. \]

The theoretical efficiency of the cycle is

\[ \eta_0=\frac{AW_0}{r_1}. \]

The actual efficiency of vacuum installations is

\[ \eta=\frac{AWD}{Q}=\frac{AW}{x_1 r_1}, \]

where \(Q\) is the heat removed from the industrial unit, kcal/hour;
\(x_1, r_1\) are the dryness fraction of the steam and the heat of vaporization at the pressure in the evaporator, kcal/kg.

In most cases \(r_1\) will lie within the range 560–580.

At the temperature \(t_1=t_{1\mathrm{opt}}\) of the theoretical and actual cycle, the corresponding powers and efficiencies assume their maximum, calculated value.

Figure 4 shows the dependence of the theoretical work of the turbine, feed and circulating pumps, and of the cycle as a whole on the temperature in the evaporator for a heat source with a specified heat-carrier flow rate \((t_0-t_1=58^\circ;\ G_{\mathrm{ht}}=17300\ \mathrm{kg}/\mathrm{m}^2\mathrm{h})\) and hydraulic resistance \((H_{\mathrm{ht}}=1\ \mathrm{m}\) water column) before the heat source is included in the cycle of the vacuum installation. The maximum work of the cycle occurs at a lower temperature in the evaporator \((t_{1\mathrm{opt}}=50^\circ)\) than for the theoretical cycle with variable flow cross sections of the heat source and the condenser of the vacuum turbine, owing to which the hydraulic resistance remains constant as the flow rate of the heat carrier and cooling water changes.

At an evaporator temperature \(t_1=68^\circ\), the work of the theoretical cycle is equal to zero, and at larger \(t_1\) it becomes negative. Taking into account the efficiency of the turbine and pumps, i.e., with some approximation to the real cycle, the maximum work is reduced almost by half and the optimum temperature changes (Fig. 5). At the same time, the region of positive work of the cycle is significantly reduced.

Thermodynamic consideration, analysis of the material and energy structure of the theoretical and actual cycles of vacuum installations reveal their special features and the dependence of the operation of individual elements and of the entire cycle on the temperature in the evaporator. The correct choice of the temperature in the evaporator of a vacuum installation is the principal prerequisite for the efficiency of the cycle, the central issue on which the possibility of utilizing the energy of heated waters and the proper assessment of low-potential resources of industry are based.

For the correct choice of the optimum temperature in the evaporator of a vacuum installation, a graphical-analytical method of calculation may be recommended. According to this method, to calculate the optimum temperature, the operation of the cycle is determined for three or four temperatures in the evaporator. Then, from the maximum of the graphically constructed curve \(AW = F(t_1)\), the optimum cycle temperature corresponding to this work is found.

Critical Remarks on the Use of Low-Potential Secondary Energy Resources of Industry in a Closed Scheme

The problem of using low-potential heat in our country is being given great importance, and entire teams of scientific workers are working on the solution of this question. The work of the G. M. Krzhizhanovsky Energy Institute of the USSR Academy of Sciences in the field of the energy utilization of the heat of heated industrial waters has been published in two articles by G. I. Rossievsky [1,2].

The method proposed by Rossievsky for investigating the closed cycle of a vacuum installation is recommended for the development and analysis of the electrical-energy utilization of secondary energy resources in a number of branches of industry. However, Rossievsky has not completely freed himself from the errors of the French researchers, whose critique we have already carried out in the press [3,4].

We therefore consider it necessary to dwell on a critique of individual propositions of this study with regard to the use of heat removed from the cooling systems of industrial units.

The principal shortcomings of Rossievsky’s study are:

1. Separate consideration of the theoretical cycle of vacuum installations. The entire cycle is reduced to two elements—the evaporator and the turbine. Excluded from the sphere of joint investigation are the heat source and the evaporator, which together constitute the “boiler” of the vacuum installation, as well as the source of cooling water and the turbine condenser.

2. Likeness of the cycle of vacuum installations to the Rankine cycle and extension of the data from the analysis of steam operation in the turbogenerator to the entire cycle.

3. Neglect of the operation of the auxiliary elements of the theoretical cycle and their influence on the operation of the entire cycle, as well as disregard of certain conditions in the theoretical cycle, for example, the presence of structurally formed industrial units designed for definite operating conditions.

4. Discrepancy between theoretical conclusions and practical application. The theoretical premises are not confirmed by reality, which compels the author to consider actual cycles on another plane and to take into account the influence of certain elements on the operation of the cycle and the temperature in the evaporator. But separate consideration of individual elements outside their general interrelation does not lead to a correct solution of the general question.

5. Separation of heat from the material medium—the heat carrier. The efficiency of the cycle is assessed by the maximum work obtained per unit of heat removed from the source, without taking into account the quantity of heat carrier. The latter leads to the erroneous conclusion regarding the necessity of...

of maximum convergence of the evaporator temperature with the temperature of the heat-transfer medium leaving the industrial unit.

Proceeding from the expressions \(W_a = 10^6 \eta_t\), \(\eta_{\mathrm{inst}}=\eta_t \cdot \eta_{\mathrm{eo}}\) and \(N_{\mathrm{inst}}=1160Q\eta_t\cdot\eta_{\mathrm{eo}}\cdot\eta_p\), the conclusion is drawn that “in closed schemes (as distinct from open ones) any increase in the temperature in the evaporator increases the efficiency and the power obtained,” and further: “for a closed single-stage scheme, for given \(t_1\) and \(t_2\), the efficiency and the power obtained do not depend on the temperature of the heat-transfer medium entering the evaporator.” The latter is readily refuted by our analysis of the closed cycle.

\[ Q=G(t_0-t_1). \]

For unchanged \(t_1\) and \(t_2\), the quantity of heat-transfer medium will depend on \(t_0\), \(G=f(t_0)\) (for a constant quantity of heat removed from the industrial unit, \(Q\)).

As was already established above, with unchanged turbine work \((h_0)\), the work of the cycle may take a zero or negative value depending on the temperature difference \((t_0-t_1)\) falling on the cycle elements: evaporator—heat source.

Thus, for example, for \(1\) megacal of heat removed, at \(t_0-t_1=0.1\), it is necessary to pass through the evaporator an amount of heat-transfer medium equal to

\[ G=\frac{10^6}{0.1}=10^7\ \text{kg/megacal}. \]

But since the heat source is designed to pass a definite quantity of heat-transfer medium, an excessive increase in the flow rate of the heat-transfer medium increases the hydraulic resistance of the heat source.

The influence of the temperature in the evaporator on the operation of the cycle was shown by us above. Here also, at a certain stage of increasing \(t_1\), the work of the cycle may have a zero or negative value as the expansion work of the steam in the turbine increases. Rossievskii recommends determining \(t_1\) in a closed cycle with allowance for factors lying “outside the cycle itself.” Among these factors he includes a number of unquestionable elements of the cycle: the work of the feed pump (pumping the heat-transfer medium through the evaporator and the heat source), the heat-transfer conditions in the heat source (where part of the cycle occurs—the transfer of heat to the heat-transfer medium), and the conditions of steam generation in the evaporator (the connection between the first and second, which we showed above). Such an artificial rupture of the cycle—the separation of its main part, the steam generator—is incorrect, since it does not reflect the actual process and the efficiency of utilization of secondary energy resources.

The temperature \(t_1\) is determined by Rossievskii from the maximum difference

\[ N_{\mathrm{inst}}-N_{\mathrm{aux}}. \]

The net power of the installation is found as the difference

\[ N_{\mathrm{inst}}^{n}=N_{\mathrm{inst}}-\sum N_{\mathrm{aux}}, \]

where \(\sum N_{\mathrm{aux}}\) is the total power of the motors for own needs (circulating pumps of condensers, condensate pumps, air evacuation, etc.).

Included in \(N_{\mathrm{inst}}^{n}\) is not the power \(N_{\mathrm{inst}}^{\max}\), which takes into account the optimum \(\Delta t_p=t_i-t_1\), but the power \(N_{\mathrm{inst}}\), determined by him from the expression:

\[ N_{\mathrm{inst}}=N_e\cdot\eta_p=1160\eta_t\cdot\eta_{\mathrm{eo}}\cdot\eta_p, \]

where

\[ N_e=1160\,Q\eta_t\cdot\eta_{\mathrm{eo}} \]

is the power of the turbogenerator,

\[ \eta_p \]

is a coefficient accounting for heat losses and losses of the heat-transfer medium.

The influence of \(\Sigma N_{s.n.}\) on \(t_1\) and on the maximum work is not taken into account either in the theoretical or in the real cycle, although for the latter the dependence of

\[ \frac{N_{\mathrm{cir}}}{N_{\mathrm{inst}}} \]

on \(t_1\) is given. Rossievskii considers the total power for auxiliary needs \(\Sigma N_{s.n.}\) to be approximately equal to the power consumed by the circulating pump of the turbogenerator \(N_{\mathrm{cir}}\), i.e. \(\Sigma N_{s.n.} \approx N_{\mathrm{cir}}\).

The following particular example of calculating an ideal cycle may serve as an illustration of the foregoing: \(t_0 = 70^\circ\), \(t_2 = 29^\circ\), the temperature drop in the heat source before its inclusion in the cycle of the vacuum installation is \(\Delta t_{\mathrm{sh}} = 41^\circ\), with hydraulic resistance \(H_{\mathrm{sh}} = 10\ \mathrm{m}\) of water column.

According to Rossievskii’s investigations, the theoretical temperature \(t_1\) is determined from the curve \(h_0\) in Fig. 6 and should approach \(t_0\); it may be taken, for example, as \(t_1 = 69^\circ\).

From Rossievskii’s energy-economic calculation of the cycle, the value of the optimal temperature \(t_1\), corresponding to the net installation power \((N_{\mathrm{inst}}^{n})\), corresponds to the temperature \(t_1\) determined from the maximum value of the curve \(h_0\), which has a relative minimum value of

\[ \frac{N_{\mathrm{cir}}}{N_{\mathrm{inst}}}. \]

Fig. 6

Figure labels and legend: vertical axis, kcal/MWh; horizontal axis, \(t_1\). Curves: \(1\ h_0 - AW_{\mathrm{op}}\); \(2\ h_0 - (AW_{\mathrm{op}} + AW_{k}^{40})\); \(3\ h_0 - (AW_{\mathrm{op}} + AW_{k}^{60})\). Other labels: \(h_0\), \(AW_{\mathrm{op}}\), \(AW_{k}^{40}\), \(t_1\), \(t_1^{\mathrm{opt}}\).

Rossievskii has no method for determining \(t_1\) with allowance for the part of the cycle evaporator—heat source; but if one calculates by our method, then \(t_1\) for the theoretical cycle is determined by the maximum of curve 1, and the work of the part of the cycle evaporator—heat source—by the curve \(AW_{\mathrm{op}}\).

The determination of \(t_1\) from the maximum work of the entire cycle is carried out from curves (2), (3)—depending on the hydraulic head loss in the cooling-water circuit. As can be seen from Fig. 6, the maximum of these curves, and consequently the corresponding optimal temperature, shifts in accordance with...

in comparison with the temperature \(t_1\), determined without taking the circulation path into account (curve 1). From the data of this figure one can also see the discrepancy

Fig. 7

Labels in the figure: \(\Delta W_o=0\); “region of positive cycle work”; \(+\Delta W_o\); \(-\Delta W_o\); \(h_0\); \(W_{o n}\); \(\Delta W_o\); \(\Delta W_{o k}\); \(t_o\); \(t_1\); \(t_e\).

between the theoretical work of the turbine–evaporator part of the cycle and the work of the entire cycle. The curve \(\Delta W_{o n}\), at its intersection with the curve \(h_0\), gives the temperature \(t_1\) of zero cycle work—the right-hand boundary of zero work of the theoretical cycle.

The intersection of the curve \(h_0\) with \(\Delta W_{o k}\) is also the temperature \(t_1\) of zero cycle work—the left-hand boundary of zero cycle work. Beyond the limits of the temperatures in the evaporator at which the cycle work is zero, toward higher temperatures (from the right-hand boundary) and toward lower temperatures (from the left-hand boundary), there lies the region of negative work of the theoretical cycle (Fig. 7).

Thus, even for the theoretical cycle, Rossevsky’s method does not make it possible to determine correctly the optimum temperature and the maximum work of the cycle.

Fig. 8

Labels in the figure: vertical axis, \(\mathrm{kcal}/\mathrm{kg}\); \(1.\ h-\Delta W_n\); \(2.\ h-(\Delta W_n+\Delta W_k)\); \(h\); \(\Delta W_n\); \(t_1^{\mathrm{opt}}\); \(\Delta W_o\); horizontal axis, \(t_1\). Numeric markings: \(8\cdot 10^1\), 6, 4, 2, 0; 30, 40, 50, 60, 65, 70.

For determining the optimum temperature under real conditions, Rossevsky’s method is still less acceptable.

Under actual operating conditions of the installation, the optimum temperature \(t_1\) does not correspond to the optimum temperature of the theoretical cycle, which emphasizes to an even greater degree the incorrectness of Rossievskii’s method. Thus, for example, in Fig. 8, for the example cited above, the influence of certain actual conditions on the determination of \(t_{1opt}\) and of the maximum work of the installation is shown (internal relative efficiency of the turbine \(\eta_{oi}=0.6\), \(\eta_n=0.6\), hydraulic resistance of the cooling-water path \(10\ m\) water column).

As is seen from curves 1 and 2, \(t_{1opt}\) shifts toward lower temperatures in the evaporator and is a variable quantity.

As studies show for the most typical installations, the theoretical work of the entire cycle is 30–40% less than the theoretical work of the turbine, which Rossievskii takes as the work of the entire cycle.

Under actual conditions, the work of the cycle differs still more from the work of the turbine. For the temperature \(t_1\), determined by Rossievskii’s method, the theoretical work of the entire cycle may have a zero or even negative value.

Rossievskii defines the efficiency of vacuum installations as the ratio

\[ \eta^{a}_{ust}=\frac{860N_a}{10^6Q} \]

or \(\eta^{a}_{ust}=\eta_t\cdot\eta_{eo}\)—for an ideal cycle, and

\[ \eta_{ust}=\frac{860N_e}{10^6Q} \]

or \(\eta_{ust}=\eta_t\cdot\eta_{eo}\cdot\eta_n\)—for an actual cycle.

Considering \(\eta_{eo}\) and \(\eta_n\) to be constant quantities and only slightly dependent on the conditions under which the process proceeds, we obtain

\[ \eta_{ust}=a\cdot\eta_t, \]

i.e., the efficiency of the vacuum power installation is reduced to a single element of the cycle—the vacuum turbine.

Here, too, cases are possible in which, with a relatively high value of the thermal efficiency of the turbine and, consequently, of the efficiency of the installation (according to Rossievskii), the work of the entire cycle will have a zero or negative value.

Such a conception of the cycle of vacuum installations does not provide a correct understanding of the efficiency of utilizing the heat of heated waters; it incorrectly orients industry with regard to the possibility of using the heat of heated waters and considerably overestimates the existing secondary energy resources of industry.

REFERENCES

1. Rossievskii G. I. Electrical-energy utilization of low-potential secondary energy resources of industry in a closed scheme. Izv. AN SSSR, OTN, No. 12, 1948.

2. Rossievskii G. I. Electrical-energy utilization of heat removed from the cooling systems of production units. Za ekonomiyu topliva, No. 5, 1949.

3. Polozhii S. V. Determination of the optimum temperature of the evaporator of a vacuum installation with single-stage evaporation. Izv. Tomskogo politekhnicheskogo instituta, vol. 66, issue 2, 1949.

4. Polozhii S. V. On the efficiency of vacuum power installations. Izv. Tomskogo politekhnicheskogo instituta, vol. 70, issue 1, 1950.

Noted Misprints