Determination of the Optimal Temperature in a Low-Pressure Evaporator Taking into Account Auxiliary Power Consumption

P. S. Stukalov

In a low-pressure evaporator installation operating on a closed cycle, part of the energy obtained from the low-pressure turbogenerator is expended on servicing the auxiliary needs of the evaporator installation.

The auxiliary needs of a low-pressure evaporator installation should include:

1) energy expenditures for supplying water from the low-pressure evaporator to the economizer and for creating excess pressure to prevent the water from boiling during forced operation;
2) energy expenditures for supplying circulating water to the condenser of the turbogenerator;
3) energy expenditures for pumping condensate;
4) energy expenditures for extracting air from the condenser.

Installing an additional heating surface in the form of a low-pressure economizer may also cause an increase in energy expenditures for the operation of draft devices.

However, to take this increase into account in general form is extremely difficult, since it is necessary to know the specific conditions for determining the resistance—the type of surface selected, the number of rows, the diameter and spacing of the heating elements, etc.

Experience in the use of low-pressure economizers in stationary installations has shown that there is almost no increase in energy expenditures for the operation of the smoke exhauster, owing to the reduction in the volumes of the gases being drawn through, the reduction in their velocity and in local resistances, in connection with the lowering of their temperature.

Therefore, when determining the optimal temperature in the evaporator, the influence of draft devices will not be taken into account.

For a closed circuit of a low-pressure evaporator and surface condensation, the following expression is valid, giving the work of 1 kg of steam [2] in heat units:

\[ AW = (i_1 - i_2) - AH_1 m_1 - AH_2 m_2 - 0.00295 \frac{p_a c_1 m_1}{i_2 (T - t_0)} \left[ \left(\frac{p_a}{p_2}\right)^{0.285} - 1 \right], \tag{1} \]

where
\(AW\) — theoretical useful work of 1 kg of steam;
\(i_1\) — heat content of the steam before the turbine, kcal/kg;
\(i_2\) — the same after the turbine, kcal/kg;
\(H_1\) — hydraulic resistance of the heat source, in m water column;
\(m_1\) — evaporation ratio, in kg of water per kg of steam;
\(H_2\) — total hydraulic resistance of the condenser with pipelines, in m water column;
\(m_2\) — cooling ratio, in kg of water per kg of steam;

\(p_2\)—pressure in the condenser, in m water column;

\(\gamma_c\)—specific weight of the mixture of air and steam at \(p_2\), in kg/m\(^3\);

\(a_1\)—coefficient of solubility of air at the temperature of the heated water \(t_0^\circ\)C, kg/kg of water;

\(p_a\)—barometric pressure, in m water column.

Equation (1) cannot be used to determine the optimum temperature in the evaporator with allowance for auxiliary requirements, since the work obtained from 1 kg of steam will increase continuously with increasing steam temperature in the evaporator.

As the initial equation for determining the optimum temperature \(T_1\), it is necessary to take the equation for the work referred to 1 kg of water that has passed through the evaporator.

In this case the work obtained from 1 kg of water will be equal to zero twice, at the values

\[ T_1=T_0 \]

and

\[ T_1=T_2, \]

where

\[ \left. \begin{aligned} T_0&\text{— temperature of the heated water before the evaporator,}\\ T_1&\text{— temperature in the evaporator,}\\ T_2&\text{— temperature of the condensate in the condenser} \end{aligned} \right\} \quad \text{in }^\circ\mathrm{K}. \]

Consequently, between these two temperature values, because of the continuity of the function \(W\), which depends on the temperature \(T_1\), there will at least be one point at which

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

i.e., the function \(W\) will pass through a maximum.

Let us compose the initial equation for the work referred to 1 kg of water that has passed through the evaporator:

\[ W_{\text{net}}=W_{\text{br}}-W_{\text{n.i}}-W_{\text{c.n}}-W_{\text{k.n}}-W_{\text{ej}} \ \frac{\text{kcal}}{\text{kg water}}, \tag{2} \]

where \(W_{\text{br}}\)—work obtained from 1 kg of water, in kcal;

\(W_{\text{n.i}}\)—work of the evaporator pump, in kcal/kg of water;

\(W_{\text{c.n}}\)—work of the circulating pump, in kcal/kg of water;

\(W_{\text{k.n}}\)—work of the condensate pump, in kcal/kg of water;

\(W_{\text{ej}}\)—work of the ejector, in kcal/kg of water.

We shall express each term on the right-hand side of equation (2) through the principal temperature parameters of the economizer and the low-pressure evaporator—\(T_0\) and \(T_1\).

1. Gross work per 1 kg of water \(W_{\text{br}}\)

The amount of water consumed for the production of 1 kWh will be

\[ q_1=\frac{860}{\eta_{\text{oe}}h_0S} = \frac{860(804.6-0.712T_1)} {\eta_{\text{oe}}(1.43T_1-436)(T_0-T_1)} \ \frac{\text{kg}}{\text{kWh}}, \tag{3} \]

where

\[ h_0=1.43T_1-436\ \frac{\text{kcal}}{\text{kg}}, \]

available temperature difference at a pressure in the condenser \(P_2 = 0.05\) ata

and

\[ g'=\frac{T_0-T_1}{804.6-0.712T_1}\,\frac{\text{kg steam}}{\text{kg water}} \]

the quantity of kg of steam formed from 1 kg of water cooled in the evaporator from temperature \(T_0\) to temperature \(T_1\).

If we pass from kWh to 1 kcal, we shall have

\[ q_1=\frac{1\cdot(804.6-0.712T_1)}{\eta_{oe}(1.43T_1-436)(T_0-T_1)}\,\frac{\text{kg}}{\text{kcal}}. \tag{4} \]

The work performed by the steam obtained from 1 kg of water will be

\[ W_{bp}=\frac{\eta_{oe}(1.43T_1-436)(T_0-T_1)}{804.6-0.712T_1}\,\frac{\text{kcal}}{\text{kg water}}. \tag{5} \]

2. Work of the pump transferring water from the evaporator to the economizer \(W_{n.u.}\)

The hydraulic resistance of the heat source is composed of the pressure difference between the evaporator and the low-pressure economizer and of the hydraulic resistances of the elements of the heating surface of the economizer, the connecting pipelines, headers, and other parts.

At a low velocity of water motion (\(W_v = 0.3\ \text{m/sec}\)), the hydraulic resistances of the elements of the economizer heating surface may be neglected, and only the pressure difference that exists between the evaporator and the economizer may be taken into account.

In the latter, the pressure must be taken with a margin in case of forcing the boiler operating regime, in which the water in the economizer must not reach the boiling temperature.

In the theoretical case, heating of the water in the economizer proceeds along the left limiting curve of the Rankine cycle, for which the relation between pressure and temperature is given in the form of the equation

\[ t_s=100\sqrt[4]{p}, \tag{6} \]

where \(t_s\) is the boiling temperature of water in °C,
\(p\) is the pressure in ata.

In our notation, under the condition of a margin of one atmosphere, this formula will take the form

\[ t_0=100\sqrt[4]{p_0+1}, \tag{7} \]

where \(p_0\) is the pressure in the economizer in ata.

Expressing the pressure in the economizer through the temperature, we obtain

\[ p_0=\left(\frac{t_0}{100}\right)^4-1. \]

From this pressure it is necessary to subtract the pressure in the evaporator in order to obtain the head to be overcome.

The pressure in the evaporator depends on the temperature \(T_1\), and, in view of the fact that the temperature in the evaporator in our case may vary within the limits 40–110°C, the pressure varies as a function of temperature \(t_1\) almost

according to a linear law, which can approximately be expressed by the equation

\[ p_{\text{evap}} = 0.01\, t_1 . \]

Figure 1 graphically presents the dependences of the pressures in the economizer, in the evaporator, and on the boundary curve on the temperatures of the water in the economizer and in the evaporator.

Fig. 1.
Graph of the dependence of pressure on temperature in the evaporator and low-pressure economizer.

The pressure overcome by the pump can be found as the difference

\[ \Delta p = p_0 - p_{\text{evap}} = \left(\frac{t_0}{100}\right)^4 - 0.01\,t_1 - 1 \tag{8} \]

or, converting to meters of water column and to the absolute temperature scale,

\[ H_{\text{l.p.}} = \left(\frac{T_0 - 273}{100}\right)^4 \cdot 10 - 0.1\,T_1 + 17.3 \ \text{m w.c.} \tag{9} \]

Since the calculation is carried out per 1 kg of water, the work consumed by the pump will be expressed by the equation

\[ W_{\text{l.p.}} = H_{\text{l.p.}}\,1\,\frac{A}{\eta_{\text{l.p.}}} = \frac{A}{\eta_{\text{l.p.}}} \left[ \left(\frac{T_0 - 273}{100}\right)^4 \cdot 10 - 0.1\,T_1 + 17.3 \right] \frac{\text{kcal}}{\text{kg water}} . \tag{10} \]

3. Work of the Circulating Pump \(W_{\text{c.p.}}\)

The work of the circulating pump will be proportional to the amount of steam entering the condenser, the cooling ratio, and the head that the pump must overcome.

The amount of steam entering the condenser per 1 kg of water that has passed through the evaporator will be, as indicated earlier,

\[ g = \frac{T_0 - T_1}{804.6 - 0.712\,T_1} \frac{\text{kg steam}}{\text{kg water}} . \tag{11} \]

Cooling ratio

\[ m_2=\frac{i_2-t_2}{t_2''-t_2'}\,\frac{\text{kg water}}{\text{kg steam}}, \tag{12} \]

where \(i_2\) is the heat content of the steam after the turbine, in \(\text{kcal}/\text{kg}\),
\(t_2\) is the temperature of the condensate in the condenser, in \({}^{\circ}\text{C}\),
\(t_2''\) is the temperature of the cooling water leaving the condenser, in \({}^{\circ}\text{C}\),
\(t_2'\) is the temperature of the intake water (annual average), in \({}^{\circ}\text{C}\).

The value of the heat content \(i_2\) can be expressed as follows:

\[ i_2=i_1-\eta_{0i}h_0, \tag{13} \]

where \(i_1\) is the heat content of the steam before the turbine, in \(\text{kcal}/\text{kg}\),
\(h_0\) is the adiabatic heat drop in the turbine, in \(\text{kcal}/\text{kg}\),
\(\eta_{0i}\) is the internal efficiency of the turbine.

Let us replace \(i_1\) and \(h_0\) in formula (13) in terms of \(T_1\). Then

\[ i_2=t_1+r-\eta_{0i}h_0=T_1-273+804.6-0.712T_1-0.7(1.43T_1-436); \]

\[ i_2=836.6-9.712T_1, \tag{14} \]

where \(r=804.6-0.712T_1\) is the latent heat of vaporization, in \(\text{kcal}/\text{kg}\),
\(h_0=1.43T_1-436\) is the adiabatic heat drop, in \(\text{kcal}/\text{kg}\),
\(\eta_{0i}=0.7\).

Both dependences are valid for the temperatures \(30\text{--}170^{\circ}\text{C}\) adopted by us for the low-pressure evaporator.

Using expressions (12) and (14), we find the cooling ratio

\[ m_2=\frac{i_2-t_2}{t_2''-t_2'}= \frac{836.6-0.712T_1-32.5}{23-9}, \tag{15} \]

where \(t_2=32.5^{\circ}\text{C}\), \(t_2''=23^{\circ}\text{C}\), and \(t_2'=9^{\circ}\text{C}\) are adopted from the operating conditions of naval power plants as applied to the \(n\)-th basin.

The work consumed by the circulating pump of the condenser will be

\[ W_{\text{c.p.}}=\frac{A H_{\text{c.p.}}}{\eta_{\text{c.p.}}}\,m_2 g. \tag{16} \]

The head overcome by the circulating pump may be taken as constant [3] for shipboard conditions:

\[ H_{\text{c.p.}}=8\ \text{m water column}. \]

Substituting the values of \(m_2\) and \(g\) from (11) and (15) into formula (16), we obtain

\[ W_{\text{c.p.}}=\frac{A\cdot 8}{\eta_{\text{c.p.}}}m_2g =\frac{0.57A(T_0-T_1)}{\eta_{\text{c.p.}}}\, \frac{\text{kcal}}{\text{kg water}}. \tag{17} \]

4. Work of the condensate pump \(W_{\text{k.p.}}\)

The condensate pump will return from the condenser to the evaporator \(g\ \text{kg}\) of condensate per \(1\ \text{kg}\) of water that has passed through the evaporator.

The head to be overcome may be calculated from the approximate formula

\[ p_k=0.01t_1-0.05\ \text{kg}/\text{cm}^2, \]

or, converting to meters of water column and to the absolute temperature scale,

\[ p_k=0.1(T_1-273)-0.5\ \text{m water column}. \]

The work consumed by the pump is

\[ W_{k.n}=\frac{A\cdot p_{k}\cdot g}{\eta_{k.n}} =\frac{A[0.1(T_1-273)-0.5](T_0-T_1)} {\eta_{k.n}(804.6-0.712T_1)} \ \frac{\text{kcal}}{\text{kg water}} . \tag{18} \]

5. Work of the ejector \(W_{\text{ej}}\)

The work expended in removing air will be proportional to the quantity of the steam–air mixture being extracted and to the pressure difference between the condenser and the atmosphere.

The compression of the extracted steam–air mixture will take place along a polytrope, whose exponent may be taken [1] as

\[ n=1.3. \]

The proportionality coefficient between the air and the steam–air mixture by weight is taken [3] as equal to 3, on the basis of data verified by the practice of operating power installations of the marine fleet.

The solubility coefficient of air is selected at the temperature in the condenser [1] and reduced to the dimension kg/kg,

\[ a_1=0.018\,\gamma_{\text{air}}\frac{1}{1000}\cdot\frac{305}{273} =0.000026\ \frac{\text{kg}}{\text{kg water}} . \]

The work consumed by the ejector will be [1]

\[ W_{\text{ej}}=\frac{3n}{n-1}\,p_2\frac{a_1 g}{\gamma_{s.m}} \left[\left(\frac{p_a}{p_2}\right)^{\frac{n-1}{n}}-1\right] \frac{A}{\eta_{\text{ej}}}, \tag{19} \]

where \(p_a=1.033\) ata is the barometric pressure,
\(p_2=0.05\) ata is the pressure in the condenser,
\(p_2=500\ \text{kg}/\text{sq. m}\).

In final form, the expression for the work of the ejector will be

\[ W_{\text{ej}}= \frac{3n}{n-1}\frac{A}{\eta_{\text{ej}}}\cdot p_2 \frac{a_1}{\gamma_{s.m}} \frac{T_0-T_1}{(804.6-0.712T_1)} \left[\left(\frac{p_a}{p_2}\right)^{\frac{n-1}{n}}-1\right] \frac{\text{kcal}}{\text{kg water}} . \tag{20} \]

6. Determination of the optimum temperature in the evaporator

After determining all terms on the right-hand side of equation (2), we write the general expression for the useful work (net)

\[ \begin{aligned} W_{\text{net}}={}&W_{\text{br}}-W_{n.l}-W_{c.n}-W_{k.n}-W_{\text{ej}} \\ ={}& \frac{\eta_{oe}(1.43T_1-436)(T_0-T_1)} {804.6-0.712T_1} -\frac{A}{\eta_{n.l}} \left[\left(\frac{T_0-273}{100}\right)^4\cdot 10-0.1T_1+17.3\right] -\frac{0.57\cdot A\cdot (T_0-T_1)}{\eta_{c.n}} \\ &-\frac{A}{\eta_{k.n}} \frac{[0.1(T_1-273)-0.5](T_0-T_1)} {(804.6-0.712T_1)} -\frac{3n\cdot A}{(n-1)\eta_{\text{ej}}} p_2\frac{a_1}{\gamma_{s.m}} \frac{T_0-T_1}{(804.6-0.712T_1)} \left[\left(\frac{p_a}{p_2}\right)^{\frac{n-1}{n}}-1\right] \frac{\text{kcal}}{\text{kg water}} . \end{aligned} \tag{21} \]

We take the derivative of \(W_{\text{net}}\) with respect to the temperature \(T_1\) and, to determine the maximum, set it equal to zero.

Figure 2 shows two parabolas indicating the optimal relations between the temperatures \(T_0\) and \(T_1\) for the theoretical cycle and for the case of operation taking into account the evaporator’s own requirements.

Figure 3 shows the dependences for \(t_1\) and \(t_0\) on a larger scale; they show that, when the own requirements of the evaporating installation are taken into account, the optimal temperature in the evaporator increases by \(5\text{–}7^\circ\) compared with that obtained from consideration of the theoretical cycle.

LITERATURE

1. Butakov I. N. Use of the heat of circulating water at power plants for the production of electrical energy. Izvestiya TPI, vol. 63, Tomsk, 1945.

2. Polozhii S. V. On the efficiency of vacuum power installations. Izvestiya TPI, vol. 70, Tomsk, 1950.

3. Semeka V. A. Selection of the type of steam-power installation for cargo ships with a carrying capacity of 2000 tons. Proceedings of TsNIIMF, 1948.