Efficiency and Extraction Pressures of a Regenerative Cycle

G. I. Fuks

Professor, Doctor of Technical Sciences

Efficiency

In the paper “Mean Thermodynamic Temperature,” a general formula was obtained for calculating the efficiency of a regenerative cycle

\[ \eta_p = 1 - \frac{g_k}{g_1}\,\frac{T_k \Delta S_k}{T' \Delta S'} . \tag{1} \]

It was further shown that this expression can be written as follows:

\[ \eta_p = 1 - \frac{i_{01}-i'_{01}}{i_1-i'_{01}}\, \frac{i_{02}-i'_{02}}{i_{01}-i'_{02}}\, \cdots \frac{i_{0n}-i'_{0n}}{i_{0(n-1)}-i'_{0n}}\, \frac{i_k-i'_k}{i_{0n}-i'_k}, \tag{2} \]

\[ \eta_p = 1 - (1-\eta_1)(1-\eta_2)\cdots(1-\eta_n)(1-\eta_{(n+1)}). \tag{3} \]

The following notation is adopted here:

\(g_k\) and \(g_1\) are the quantities of steam going to the condenser and generated in the boiler (per 1 kg of fuel);

\(T_k\) and \(\Delta S_k\) are the mean thermodynamic temperature and the change in entropy of the steam in the condenser; \(T'\) and \(\Delta S'\) are the same quantities for the boiler plant; \(i_1, i'_1, i_k, i'_k\) are the enthalpies of steam and water in the boiler and condenser. The enthalpies of steam and water at the extractions are denoted by the symbol 0 with a digit indicating the extraction number (counting from the boiler). Thus, \(i_{02}\) and \(i'_{03}\) denote the enthalpy of steam at the pressure of the 2nd extraction and the enthalpy of water at the pressure of the 3rd extraction.

Formulas (2) and (3) were derived under the following restrictions:

a) adiabatic expansion of the steam in the machine;

b) the use of mixing heaters.

Fig. 1

It is not difficult, however, to show that these formulas can be generalized. With mixing heaters and any line of steam expansion \(1K\) (see Fig. 1), we have the balance equations of the heaters:

\[ (g_1-g_2)(i_{01}-i'_{01}) = g_2(i'_{01}-i'_{02}), \]

\[ (g_2-g_3)(i_{02}-i_{02}')=g_3(i_{02}'-i_{03}), \tag{4} \]

\[ \cdot \]

\[ \cdot \]

\[ \cdot \]

\[ (g_n-g_k)(i_{0n}-i_{0n}')=g_k(i_{0n}'-i_k'), \]

whence

\[ \frac{g_k}{g_1} = \frac{i_{01}-i_{01}'}{i_{01}-i_{02}}\, \frac{i_{02}-i_{02}'}{i_{02}-i_{03}}\, \cdots\, \frac{i_{0n}-i_{0n}'}{i_{0n}-i_k'}. \tag{5} \]

Since, according to the definition of the mean thermodynamic temperature,

\[ T_k \Delta S_k = i_k - i_k' \]

and

\[ T'\Delta S' = i_1 - i_{01}', \tag{6} \]

then from (1), (5), and (6) one obtains

\[ \eta_p = 1- \frac{i_{01}-i_{01}'}{i_1-i_{01}'}\, \frac{i_{02}-i_{02}'}{i_{01}-i_{02}}\, \cdots\, \frac{i_{0n}-i_{0n}'}{i_{0(n-1)}-i_{0n}'}\, \frac{i_k-i_k'}{i_{0n}-i_k'}. \tag{2} \]

On the other hand, for each of the cycles \(1\,01\,01'\,1'\,1\), \(01\,02\,02'\,01'\,01\), etc., we have:

Fig. 2

\[ \eta_1=\frac{i_1-i_{01}}{i_1-i_{01}'}, \tag{7} \]

\[ \eta_2=\frac{i_{01}-i_{02}}{i_{01}-i_{02}'}. \]

From (2) and (7) relation (3) is obtained.

If the installation is provided not with mixing heaters but with surface heaters, and the drain (condensate) from each extraction is delivered by a pump into the feed line near the heater (it is immaterial whether before or after the heater), then in each heater the heat of steam condensation and the heat of the condensate are used, as in mixing heaters. The difference is that here there will inevitably be an underheating of the water to the extraction temperature \(\varepsilon'\), \(\varepsilon''\ldots\), etc. (see Fig. 2). Thus, the state of the water after the heaters will be represented by the points \(a'\), \(a''\ldots\) (Fig. 2).

The balance equations of the heaters in this case are written as follows:

\[ (g_1-g_2)(i_{01}-i_a')=g_2(i_a'-i_a''), \]

\[ (g_2-g_3)(i_{02}-i_a'')=g_3(i_a''-i_a'''), \tag{8} \]

\[ \cdot \]

\[ \cdot \]

\[ \cdot \]

\[ (g_n-g_k)(i_{0n}-i_{a(n)}')=g_k(i_{a(n)}'-i_k'), \]

whence

\[ \frac{g_k}{g_1} = \frac{i_{01}-i_a''}{i_{01}-i_a'}\, \frac{i_{02}-i_a'''}{i_{02}-i_a''}\, \cdots\, \frac{i_{0n}-i_{a(n)}'}{i_{0n}-i_k'}. \tag{9} \]

Further, obviously,

\[ \begin{aligned} T_k \Delta S_k &= i_k - i'_k,\\ T' \Delta S' &= i_1 - i''_a . \end{aligned} \tag{10} \]

From (1), taking account of (9) and (10),

\[ \eta_p = \frac{i_{01}-i'_a}{i_1-i'_a}\, \frac{i'_0-i''_a}{i_{01}-i''_a}\cdots \frac{i_k-i'_k}{i_{0n}-i'_k} \tag{11} \]

Further, it is obvious that relation (3) is applicable in the present case as well, since for the individual cycles

\[ \eta_1=\frac{i_1-i_{01}}{i_1-i''_a}, \tag{12} \]

\[ \eta_2=\frac{i_{01}-i_{02}}{i_{01}-i''_a} \]

and so on.

Finally, relations (3) and (11) are in principle also applicable when the heaters are connected in cascade, with all drains discharged to any point on the feed line or to the next heater in order. In these cases the calculation must be carried out using the “conditional” terminal temperature differences that would be obtained if the drain were supplied to the feed line near the heater itself. The conditional terminal temperature differences are calculated depending on the features of the given heater-connection arrangement. For the arrangement in Fig. 3, according to which the drain from all heaters is discharged to the feed line after the last heater or directly to the condenser, the conditional terminal temperature differences are calculated from the following considerations.

Fig. 3

For the given steam extractions, the final result of feed-water heating does not depend on the point at which the drains are introduced into the feed line, i.e., the actual and conditional terminal temperature difference of the first heater are the same, or

\[ y'=\varepsilon'. \tag{13} \]

To calculate the conditional terminal temperature difference of the 2nd heater we write the equations.

1. Balance equation of the first heater

\[ (g_1-g_2)T'_p\Delta S'_p = g_1(\Delta q'-\varepsilon'+\varepsilon''). \tag{14} \]

1. Conditional balance equation of the 1st heater (drawn up for the same quantities \(g_1\) and \(g_2\) as actually occur, and with the drain supplied directly near the 1st heater)

\[ (g_1-g_2)(T'_p\Delta S'_p+\varepsilon') = g_2(\Delta q'-\varepsilon'+y''). \tag{15} \]

Eliminating \(g_1\) and \(g_2\) from (14) and (15), we obtain

\[ y''=\frac{\varepsilon''+\Delta q' a_1}{1-a_1}, \tag{16} \]

where

\[ a_1=\frac{\Delta q'-\varepsilon'+\varepsilon''}{T'_p\Delta S'_p} =\frac{\Delta q_1}{T'_p\Delta S'_p}, \tag{17} \]

or

\[ a_1=\frac{i'_a-i''_a}{i_{01}-i'_{01}} \tag{17'} \]

(see Fig. 2).

To determine the conditional underheating of the 3rd heater, the balance equation of the 2nd heater is written as

\[ (g_2-g_3)T''_p\Delta S''_p =g_1(\Delta q''-\varepsilon''+\varepsilon''') \tag{18} \]

and the conditional balance equation of this same heater is

\[ (g_2-g_3)(T''_p\Delta S''_p+y'') =g_3(\Delta q''-y''+y'''). \tag{19} \]

Eliminating \(g_2\) and \(g_3\) from (18) and (19), and taking account of (16) and (17), we obtain

\[ y'''=\frac{\varepsilon'''+(\Delta q'+\Delta q'')a_1+\Delta q''a_2}{1-(a_1+a_2)}, \tag{20} \]

where

\[ a_2=\frac{\Delta q''-\varepsilon''+\varepsilon'''}{T''_p\Delta S''_p} =\frac{\Delta q_2}{T''_p\Delta S''_p}, \tag{21} \]

or

\[ a_2=\frac{i''_a-i'''_a}{i_{02}-i'_{02}}. \tag{21'} \]

For the conditional underheating of the fourth heater, we obtain from the balance equations of the 3rd heater

\[ y^{IV}= \frac{\varepsilon^{IV}+(\Delta q'+\Delta q''+\Delta q''')a_1 +(\Delta q''+\Delta q''')a_2+\Delta q'''a_3} {1-(a_1+a_2+a_3)}, \tag{22} \]

where

\[ a_3=\frac{i'''_a-i^{IV}_a}{i_{03}-i'_{03}}. \tag{23} \]

In general, for the conditional underheating of the \(n\)-th heater, from the balance equations of the \((n-1)\)-st heater one obtains

\[ y^{(n)}= \frac{ \varepsilon^{(n)} +(\Delta q'+\Delta q''+\ldots+\Delta q^{(n-1)})a_1 +(\Delta q''+\Delta q'''+\ldots+\Delta q^{(n-1)})a_2 +\ldots+\Delta q^{(n-1)}a_{n-1} }{ 1-(a_1+a_2+\ldots+a_{n-1}) }, \tag{24} \]

where

\[ a_{n-1}= \frac{i^{(n-1)}_a-i^{(n)}_a}{i_{0(n-1)}-i'_{0(n-1)}}. \tag{25} \]

With the underheatings calculated in this way, the general relations (11) and (3) may be used to compute the efficiency.

With the cascade connection of heaters, with the drains led into the next heater according to the diagram of Fig. 4, for calculating the underheatings of the 1st and 2nd heaters the same relations (13) and (16) are used as in the preceding case. The balance equations of the 2nd heater are written as follows:

1) Actual:

\[ (g_2-g_3)T_p''\Delta S_p''+(g_1-g_2)\Delta q'=g_1(\Delta q''-\varepsilon'+\varepsilon''). \tag{26} \]

2) Conditional:

\[ (g_2-g_3)(T_p''\Delta S_p''+y'')=g_3(\Delta q''-y''+y'''). \tag{19} \]

From (26) and (19), taking into account (13) and (16), we obtain

\[ y'''=\frac{\varepsilon'''+\Delta q''[a_2+(1-a')a_1]}{1-[a_2+(1-a')a_1]}. \tag{27} \]

Similarly, for the conditional underheating of the 4th heater, we obtain

\[ y^{\mathrm{IV}}= \frac{\varepsilon^{\mathrm{IV}}+\Delta q''' \{a_3+(1-a'')[a_2+(1-a')a_1]\}} {1-\{a_3+(1-a'')[a_2+(1-a')a_1]\}} \tag{28} \]

and so on.

Here, in addition to the notation indicated above, the following has been adopted:

\[ a'=\frac{\Delta q'}{T_p''\Delta S_p''}; \]

\[ a''=\frac{\Delta q''}{T_p'''\Delta S_p'''} \tag{29} \]

and so on.

After calculating the conditional underheatings \(y'\), \(y''\), \(y'''\ldots\), the efficiency can be calculated from (3) or (11).

Obviously, the indicated method of calculating conditional underheatings can serve to reduce schemes with any method of connecting heaters to the simplest one. This means that relations (3) and (11) can be applied for calculating the efficiency in a wide variety of cases. But, of course, it is possible to avoid calculating the conditional underheatings \(y'\), \(y''\ldots\), and directly use relation (1) for calculating the efficiency, having written a formula for calculating \(\dfrac{g_k}{g_1}\).

For the scheme in Fig. 3, from the actual balance equations of the heaters (14), (18), and the following, we have

Fig. 4

\[ \frac{g_2}{g_1}=1-a_1, \]

\[ \frac{g_3}{g_1}=1-(a_1+a_2), \]

\[ \cdot \]

\[ \cdot \]

\[ \cdot \]

\[ \frac{g_k}{g_1}=1-(a_1+a_2+\ldots+a_n) \tag{30} \]

Substitution into (1), taking into account (10), gives

\[ \eta_p=1-\left[1-(a_1+a_2+\ldots+a_n)\right]\frac{i_k-i_k'}{i_1-i'_{01}}. \tag{31} \]

The use of relation (31), however, is not as simple as it may seem at first glance, since it also requires preliminary calculations associated with determining

\[ a_n=\frac{\Delta q^{(n)}-\left(\varepsilon^{(n)}+\alpha\right)} {T_p^{(n)}\Delta S_p^{(n)}} . \tag{32} \]

Here \(\alpha\) is the heating of the condensate from the condenser due to mixing with the drain (point \(m\) in the diagram of Fig. 3). From the mixing equation at the drain-discharge point,

\[ (g_1-g_2)(\Delta q' + \Delta q''+\cdots+\Delta q^{(n)}-\alpha) +(g_2-g_3)(\Delta q''+\Delta q'''+\cdots+ \Delta q^{(n)}-\alpha)+\cdots+(g_n-g_k)(\Delta q^{(n)}-\alpha)=g_k\alpha \tag{33} \]

taking (30) into account, one can obtain

\[ \alpha = \frac{ (\Delta q'+\Delta q''+\cdots+\Delta q^{(n)})a_1 +(\Delta q''+\Delta q'''+\cdots+\Delta q^{(n)})a_2 +\Delta q^{(n)} \frac{\Delta q^{(n)}-\varepsilon^{(n)}}{T_p^{(n)}\Delta S_p^{(n)}} }{ 1+\frac{\Delta q^{(n)}}{T_p^{(n)}\Delta S_p^{(n)}} }. \tag{34} \]

For the cascade arrangement according to Fig. 4, from the balance equations of the heaters (14), (26), and analogous equations, we obtain:

\[ \frac{g_2}{g_1}=1-a_1, \]

\[ \frac{g_3}{g_1}=1-\left[a_2+(1-a')a_1\right], \tag{35} \]

\[ \frac{g_2}{g_1}=1-\left\{a_3+(1-a'')\left[a_2+(1-a')a_1\right]\right\}, \]

\[ \frac{g_k}{g_1} = 1-\left\{\left\{a_n+(1-a^{(n-1)})\left|a_{n-1}\right| +(1-a^{(n-2)})\left\{a_{n-2}+\cdots+(1-a')a_1\right\}\right\}\right\}. \]

From (35) and (1), taking (10) into account, we obtain

\[ \eta_p = 1- \left\{ \left[ 1- \left\{ \left\{ a_n+(1-a^{(n-1)})\left|a_{n-1}\right| +(1-a^{(n-2)})\left\{a_{n-2}+\cdots+ (1-a')\right\} \right\} \right\} \right] \frac{i_k-i_k'}{i_1-i_{01}'} \right\}. \tag{36} \]

Calculation by this formula likewise requires a preliminary determination of \(\alpha\) in order to compute \(a_n\) by (32). Writing the mixing equation for the point at which the drain is supplied to the feed line (point \(m\) in the diagram of Fig. 4), taking (27) into account, we obtain

\[ \alpha = \frac{ \Delta q^{(n)} \left\{ \frac{\Delta q^{(n)}-\varepsilon^{(n)}}{T_p^{(n)}\Delta S_p^{(n)}}+ \right. }{ 1+ } \]

\[ \frac{ \left. +(1-a^{(n-1)})\left[ a_{n-1} +(1-a^{(n-2)})\left\{a_{n-2}+\cdots+(1-a')a_1\right\} \right] \right\} }{ +\frac{\Delta q^{(n)}}{T_p^{(n)}\Delta S_p^{(n)}} }. \tag{37} \]

For a direct calculation of the efficiency at specified extraction pressures, the use of relations (31)—(34) and (36)—(37) is simpler than the preliminary calculation of the fictitious underheatings, although in terms of the amount of computational work the difference is small. But the fundamental possibility of reducing the calculation of the efficiency in all cases to the use of a single-type relation (11) is important because it makes it possible to outline a simple solution of certain general questions, in particular the question of selecting the most advantageous extraction pressures.

The Most Advantageous Extraction Pressures

The question of selecting the most advantageous pressures of the steam extractions in the regenerative process has been the subject of a considerable number of investigations¹. An exact solution of this problem, however, has not yet been obtained.

In the case of an installation with mixing heaters, this question is solved as follows.

The balance equations of the heaters in this case may be written as:

\[ (g_1-g_2)T'_p \Delta S'_p = g_2 \Delta q', \]

\[ (g_2-g_3)T''_p \Delta S''_p = g_3 \Delta q'', \]

\[ \vdots \]

\[ (g_n-g_k)T^{(n)}_p \Delta S^{(n)}_p = g_k \Delta q^{(n)} \tag{38} \]

(see Fig. 1),

whence

\[ \frac{g_k}{g_1} = \frac{T^{(n)}_p \Delta S^{(n)}_p}{T^{(n)}_p \Delta S^{(n)}_p+\Delta q^{(n)}} \cdots \frac{T''_p \Delta S''_p}{T''_p \Delta S''_p+\Delta q''} \frac{T'_p \Delta S'_p}{T'_p \Delta S'_p+\Delta q'} . \tag{39} \]

From (39) and (1) we have

\[ \eta_p = 1 - \]

\[ - \frac{T_k \Delta S_k}{T' \Delta S'} \frac{T^{(n)}_p \Delta S^{(n)}_p}{T^{(n)}_p \Delta S^{(n)}_p+\Delta q^{(n)}} \cdots \frac{T''_p \Delta S''_p}{T''_p \Delta S''_p+\Delta q''} \frac{T'_p \Delta S'_p}{T'_p \Delta S'_p+\Delta q'} . \tag{40} \]

With an unchanged state of the steam entering the condenser, the maximum of the efficiency \((\eta_p)\) coincides with the maximum of the function:

\[ U= \frac{T^{(n)}_p \Delta S^{(n)}_p+\Delta q^{(n)}}{T^{(n)}_p \Delta S^{(n)}_p} \cdots \frac{T''_p \Delta S''_p+\Delta q''}{T'' \Delta S''_p} \frac{T'_p \Delta S'_p+\Delta q'}{T'_p \Delta S'_p} \, T' \Delta S' . \tag{41} \]

Introduce the notation:

\[ T'_p \Delta S'_p=b_1, \]

\[ T''_p \Delta S''=b_2, \]

\[ \vdots \]

\[ T^{(n)}_p \Delta S^{(n)}_p=b_n . \tag{42} \]

The quantity \(T'\Delta S'\) (Fig. 1) represents the amount of heat supplied to 1 kg of steam in the boiler plant. It consists of the heat of vaporization of the liquid and the superheating of the steam

\[ r+\Delta i_n=b_0, \]

which remains unchanged for unchanged steam parameters, and the heat for heating the liquid

\[ \Delta q_0=i'_1-i'_{01}, \]

which changes if the pressure in the first extraction changes. In general,

\[ T'\Delta S' = b_0 + \Delta q_0. \tag{43} \]

Further, we denote (see Fig. 1):

\[ \begin{aligned} \Delta q^{(n)} &= i_{0n}' - i_k' = X,\\ \Delta q^{(n-1)} &= i_{0(n-1)}' - i_{0n}' = Y - X,\\ &\vdots\\ \Delta q'' &= i_{02}' - i_{03}' = W - V,\\ \Delta q' &= i_{01}' - i_{02}' = Z - W,\\ \Delta q_0 &= i_1' - i_{01}' = A - Z. \end{aligned} \tag{44} \]

Obviously, \(A\) represents the total amount of heat required to heat 1 kg of water from the state in the condenser to the beginning of boiling in the boiler. For given parameters of the installation, \(A\) is a constant quantity.

Taking into account the notation introduced, expression (41) may be rewritten as:

\[ U = \left(1+\frac{X}{b_n}\right) \left(1+\frac{Y-X}{b_{n-1}}\right) \cdots \left(1+\frac{W-V}{b_2}\right) \left(1+\frac{Z-W}{b_1}\right) \left(1+\frac{A-Z}{b_0}\right). \tag{45} \]

Here \(X, Y, \ldots, V, W, Z\) are independent variables, since the extraction pressures may be chosen arbitrarily. The quantities \(b_n, b_{n-1}, \ldots, b_2, b_1\) must likewise be regarded as variables, with

\[ \begin{aligned} b_n &= \varphi_n(X),\\ b_{n-1} &= \varphi_{n-1}(Y),\\ &\vdots\\ b_2 &= \varphi_2(W),\\ b_1 &= \varphi_1(Z). \end{aligned} \tag{46} \]

This follows from the fact that the quantities \(X, Y, \ldots, W, Z\), in the sense of the notation (44) and Fig. 1, each represent the amount of heat required to heat 1 kg of water from the state in the condenser to the boiling state in the corresponding heater, i.e., for example,

\[ W = i_{02}' - i_k', \]

\[ Z = i_{01}' - i_k', \]

and so on. For a given character of the expansion curve, the value \(X\) determines \(b_n\), \(Y\)—\(b_{n-1}\), …, \(Z\)—\(a_1\), and so on, which is expressed symbolically by relation (46).

The conditions for the maximum of (45) will be

\[ \left(\frac{\partial U}{\partial X}\right)=0;\quad \left(\frac{\partial U}{\partial Y}\right)=0;\quad \ldots\quad \frac{\partial U}{\partial Z}=0. \tag{47} \]

Taking (46) into account, this may be written as:

\[ \frac{U}{1+\dfrac{X}{b_n}}\, \frac{b_n - X\,\dfrac{db_n}{dX}}{b_n^2} = \frac{U}{1+\dfrac{Y-X}{b_{n-1}}}\, \frac{1}{b_{n-1}}, \]

\[ \frac{U}{1+\dfrac{Y-X}{b_{n-1}}}\, \frac{b_{n-1}-(Y-X)\dfrac{db_{n-1}}{dY}}{b_{n-1}^{2}} = \frac{U}{1+\dfrac{P-Y}{b_{n-2}}}\,\frac{1}{b_{n-2}}, \]

\[ \frac{U}{1-\dfrac{Z-W}{b_{1}}}\, \frac{b_{1}-(Z-W)\dfrac{db_{1}}{dZ}}{b_{1}^{2}} = \frac{U}{1-\dfrac{A-Z}{b_{0}}}\,\frac{1}{b_{0}}. \tag{48} \]

After simple transformations it is easy to obtain

\[ \frac{b_n+X}{1-\dfrac{X}{b_n}\dfrac{db_n}{dX}}+X=b_{n-1}+Y, \]

\[ \frac{b_{n-1}+(Y-X)}{1-\dfrac{Y-X}{b_{n-1}}\dfrac{db_{n-1}}{dY}}+Y=b_{n-2}+P, \tag{49} \]

\[ \vdots \]

\[ \frac{b_1+(Z-W)}{1-\dfrac{(Z-W)}{b_1}\dfrac{db_1}{dZ}}+Z=A+b_0. \]

The obtained equations (49), whose number is equal to the number of unknowns \(X, Y, \ldots Z\), constitute the fundamental solution of the problem posed. It is expedient first of all to consider certain approximate solutions.

Sometimes, when considering the question of determining the optimum extraction pressures, the assumption is made that the amount of heat released during condensation of the steam in all heaters is the same [3], i.e.

\[ b_0=b_1=b_2=\ldots=b_n=b \tag{50} \]

Under this condition equations (49) give

\[ \begin{aligned} 2X&=Y,\\ 2Y-X&=P,\\ &\vdots\\ 2Z-W&=A. \end{aligned} \tag{51} \]

If (44) is taken into account, then the latter conditions may be formulated as follows:

\[ \Delta q_0=\Delta q'=\Delta q''=\ldots=\Delta q^{(n)}=\frac{A}{n+1}. \tag{52} \]

Thus, if the conditions expressed by equalities (50) are accepted, the optimum extraction pressures are determined by equality of the water heating in all heaters.

In its clearest form this solution was given in the works of Ya. M. Rubinshtein [2] and T. I. Petelin [3].

But the conditions (50) can be interpreted differently and another solution of the same problem obtained. Indeed, from (50) it follows that

\[ \frac{db_n}{dX}=\frac{db_{n-1}}{dY}=\ldots=\frac{db_1}{dZ}=0. \tag{53} \]

If we introduce the enthalpy quantities in accordance with Fig. 1, then condition (53) can be written in the general form as

\[ \frac{d(i_0-i'_0)}{d(i_0-i'_k)}=0, \tag{54} \]

where \(i_0\) is the enthalpy of the steam at an arbitrary point on the expansion line \(1K\), and \(i'_0\) is the enthalpy of boiling water at the corresponding pressure. Thus, for any intermediate pressure along the expansion line,

\[ i_0-i'_0=C. \tag{55} \]

The constant of integration is not difficult to determine if equation (55) is applied to the condenser pressure:

\[ C=i_k-i'_k. \tag{56} \]

Consequently,

\[ i_0-i_k=i'_0-i'_k. \tag{57} \]

The latter means that the adoption of condition (53) or (50) is equivalent to assuming a quite definite linear dependence between the changes in enthalpy on the expansion curve and on the left-hand boundary curve. And since condition (50), in seeking the maximum efficiency, led us to equality of the heating of the water in all heaters, it follows also from (57) that the heat drops between the extraction points are equal, i.e.,

\[ i_1-i_{01}=i_{01}-i_{02}=\ldots=i_{0n}-i_k=\frac{i_1-i_k}{n+1}. \tag{58} \]

This solution was first given by Laupichler [8]. In reality, of course, condition (50), and the conditions (53) and (54) following from it, are satisfied only approximately; therefore the selection of optimum extraction pressures according to (52) and (58) gives different results. At ordinary condenser pressure and with not too small a value of the internal relative efficiency of the machine, the increase in the enthalpy of the water, counted from the condenser conditions, is at first slower than the increase in the enthalpy of the steam on the expansion line. At high pressures the opposite phenomenon is observed. In this case, satisfying equality of the water-heating increments (52) will give higher steam-extraction pressures than satisfying equality of the heat drops on the expansion line (58). The actual optimum values of the extraction pressures prove to be intermediate between those determined by the conditions indicated above. Therefore, for example, the method of determining the maximum steam-extraction pressures proposed by Gochstein [4], despite the fact that the derivation of this method is based by the author on a whole series of plainly incorrect assumptions¹, nevertheless gives generally acceptable results. Indeed, if the heat capacity of the water on the left-hand boundary curve is assumed constant, then relation (52) can be rewritten as

\[ T_1-T_{01}=T_{01}-T_{02}=\ldots=T_{0n}-T_k=\frac{T_1-T_k}{n+1}, \tag{59} \]

whence it is easy to obtain the formula for calculating the temperature in heater number \(i\) with \(n\) stages of extraction:

\[ T_i=\frac{(n+1-i)T_1+iT_k}{n+1}. \tag{60} \]

The corresponding formula of Hochstein [4]

\[ T_i=\sqrt[n+1]{T_1^{\,n+1-i}T_k^i}, \tag{61} \]

will obviously, for the same temperature values, give smaller results than (60). But since the difference between the limiting temperatures in the cycle of a steam-power plant is small in comparison with the absolute values of these temperatures, the difference in the results of calculations by (60) and (61) is small.

It follows from this, among other things, that there are no grounds for recommending, for finding the optimal extraction pressures, the method proposed by Hochstein, since it has no advantages in comparison with the simpler method of equal water heating in all stages of heating (52), and calculations by (61) are no simpler than by (52).

The application of the exact relations (49) for calculating the optimal extraction pressures is made difficult by the fact that it does not appear possible to outline an analytical form of the dependence

\[ i_0-i_0'=f(i_0'-i_k'), \]

i.e., it is impossible to obtain expressions for

\[ \frac{db_n}{dX},\quad \frac{db_{n-1}}{dY},\ldots \frac{db_1}{dZ}. \]

It is clear that the form of this dependence will change with a change in the parameters of the machine and in the magnitude of the internal relative efficiency \(\eta_{io}\). Moreover, attempts to substitute even a simple dependence of this function into (49) led to such cumbersome and complicated equations that their solution became possible only by trial.

Meanwhile, the possibility of an exact solution of the problem posed must be regarded as fundamentally important, since:

a) in some cases an exact solution is important because each hundredth of a percent of gain in the efficiency of the installation can give an appreciable amount in fuel savings;

b) the choice of one or another method of approximate solution cannot be made consciously if it is not possible to compare the results of this method with the corresponding results of exact calculations.

At first glance it may seem that, for the latter purpose, one can make do with calculations of \(\eta_p\) at different extraction pressures, then finding the maximum \(\eta_p\) from the graph. However, when the number of steam extractions is \(n \geq 3\), it is necessary to carry out so many calculations to construct the corresponding curves that this method can hardly be recognized as rational.

Proceeding from these considerations, we consider it rational to give a graphical-analytical method for selecting the optimal pressures by (49), with the derivatives appearing in the denominators calculated graphically. In practice, this method, as tests have shown, gives a completely exact solution of the question.

According to the notation in Fig. 1, we have

\[ b_n=i_{0n}-i'_{0n}, \]

\[ X=i'_{0n}-i'_k, \]

Consequently,

\[ \frac{db_n}{dX}=\frac{di_{0n}}{di'_{0n}}-1. \tag{62} \]

Further,

\[ b_{n-1}+Y=i_{0(n-1)}-i'_k. \tag{63} \]

From the first equation of relation (49), taking (62) and (63) into account, we obtain:

\[ i_{0(n-1)}-i'_k = (i_{0n}-i'_{0n}) \frac{i_{0n}-i'_k} {(i_{0n}-i'_k)-(i'_{0n}-i'_k)\dfrac{di_{0n}}{di'_{0n}}} + (i'_{0n}-i'_k). \tag{64} \]

An analogous transformation for the second equation of relation (49) gives

\[ i_{0(n-2)}-i'_k = (i_{0(n-1)}-i'_{0(n-1)}) \frac{(i_{0(n-1)}-i'_{0(n-1)})+(i'_{0(n-1)}-i'_{0n})} {(i_{0(n-1)}-i'_{0n})-(i'_{0(n-1)}-i'_{0n})\dfrac{di_{0(n-1)}}{di'_{0(n-1)}}} + (i'_{0(n-1)}-i'_k). \tag{65} \]

Finally, for the first heater from the boiler one obtains

\[ i_1-i'_k = (i_{01}-i'_{01}) \frac{(i_1-i'_{01})+(i'_{01}-i'_{02})} {(i_{01}-i'_{02})-(i'_{01}-i'_{02})\dfrac{di_{01}}{di'_{01}}} + (i'_{01}-i'_k). \tag{66} \]

More convenient formulas for notation are obtained if the order of numbering of the heaters is changed, i.e., if the low-pressure heater is taken as the first heater and the heater number is placed as a subscript in parentheses. Then, for heater No. 2:

\[ (i-i'_k)_2 = \bigl[(i-i'_k)-(i'-i'_k)\bigr]_1 \frac{(i-i'_k)_1} {(i-i'_k)_1-(i'-i'_k)_1\left(\dfrac{di}{di'}\right)_1} + (i'-i'_k)_1. \tag{67} \]

For heater No. 3:

\[ (i-i'_k)_3 = \bigl[(i-i'_k)-(i'-i'_k)\bigr]_2 \frac{(i-i'_k)_2-(i'-i'_k)_1} {(i-i'_k)_2-(i'-i'_k)_1-\bigl[(i'-i'_k)_2-(i'-i'_k)_1\bigr]\left(\dfrac{di}{di'}\right)_2} + (i'-i'_k)_2. \tag{68} \]

For heater No. 4:

\[ (i-i'_k)_4 = \bigl[(i-i'_k)-(i'-i'_k)\bigr]_3 \frac{(i-i'_k)_3-(i'-i'_k)_2} {(i-i'_k)_3-(i'-i'_k)_2-\bigl[(i'-i'_k)_3-(i'-i'_k)_2\bigr]\left(\dfrac{di}{di'}\right)_3} + (i'-i'_k)_3. \tag{69} \]

On this basis the following method is outlined. From the steam expansion curve in \(is\) coordinates, the corresponding values of enthalpy and pressure \((i_0\) and \(p)\) are read off, and from the pressures, using the steam tables, the corresponding values of the enthalpy of boiling water \((i'_0)\) are found. From these data the curve

\[ i_0-i'_k=f(i'_0-i'_k) \]

is plotted (see Fig. 5).

From this curve, for a number of points one calculates

\[ \frac{di_0}{di_0'} \sim \frac{\delta i_0}{\delta i_0'} \]

and the values obtained are plotted as a function of the corresponding values of \((i_0' - i_k')\).

Having prepared such a graph, we proceed to the calculation of the optimum extraction pressures. The pressure in the low-pressure heater (No. 1) is specified by eye or from approximate relations. For this pressure, from the steam tables the enthalpy of the liquid \(i_0'\) is read off, and the value \((i_0' - i_k')\) is laid off on the drawing (Fig. 5, point \(a\)). Next, by constructing \(abc\) and \(abdf\), the values \((i_0 - i_k')_1\) and

\[ \left(\frac{di_0}{di_0'}\right)_1 \]

are read off.

Fig. 5

Substituting the quantities obtained into equation (67), we calculate \((i_0 - i_k)_2\), and this value is laid off on the graph (point \(g\)). By the constructions \(gkm\) and \(gknl\) one determines

\[ \left(\frac{di_0}{di_0'}\right)_2 \quad \text{and} \quad (i_0' - i_k')_2, \]

which makes it possible, by (68), to calculate

\[ (i - i_k')_3. \]

If these constructions and calculations are continued, then for the last high-pressure heater one obtains

\[ (i_0 - i_k')_1 = i_1 - i_k', \]

provided that the pressure in the low-pressure heater (No. 1) has been specified correctly. Otherwise, the calculations must be carried out using new values of the pressures in heater No. 1 until agreement is obtained. When it has been reached, the corresponding values of the steam-extraction pressures are read from the steam tables according to \(i_0'\).

Table No. 1 gives the results of the corresponding calculations for the case: boiler pressure \(p_1 = 80\) ata, superheat temperature \(425^\circ\mathrm{C}\), condenser pressure \(p_k = 0.04\) ata, \(\eta_{i0} = 0.82\), with the number of extractions from 1 to 5.

The following notation is adopted:

L — calculation by formula (58) (Laupichler);

\(^1\) Thus, for point \(a'\) (see Fig. 5),

\[ \frac{di_0}{di_0'} \sim \frac{\delta i_0}{\delta i_0'} = \frac{c'd'}{m'd'}. \]

P—calculation by formula (52) (Petelin);
T—calculation by formulas (67), (68), (69) and Fig. 5 (exact method);
G—calculation by formula (61) (Gokhshtein).
In column C the extraction pressures are calculated as the arithmetic means of columns L and P.

Table 1

	L	P	C	G	T
1 extraction point
$P_0$	3,7	8,0	5,8	3,7	5,8
$\eta_p$	36,07	36,11	36,12	36,07	36,12
2 extraction points
$P_{01}$	11,8	21,0	16,4	11,8	16,4
$P_{02}$	0,87	2,25	1,56	1,05	1,50
$\eta_p$	37,00	37,16	37,20	37,19	37,21
3 extraction points
$P_{01}$	19,5	31,0	25,2	20,0	29,0
$P_{02}$	3,70	8,0	5,8	3,7	5,60
$P_{03}$	0,48	1,05	0,76	0,50	0,68
$\eta_p$	37,62	37,60	37,65	37,62	37,66
4 extraction points
$P_{01}$	26,0	39,0	32,5	26,7	34,0
$P_{02}$	7,7	14,7	11,2	7,6	11,3
$P_{03}$	1,70	3,90	2,8	1,72	2,70
$P_{04}$	0,30	0,60	0,45	0,41	0,42
$\eta_p$	37,98	37,93	38,00	37,98	38,00
5 extraction points
$P_{01}$	32,0	45,0	38,5	33,0	40,0
$P_{02}$	12,0	21,0	16,5	11,7	17,0
$P_{03}$	3,7	8,2	6,0	3,7	5,9
$P_{04}$	0,98	2,30	1,64	1,00	1,55
$P_{05}$	0,21	0,41	0,31	0,22	0,282
$\eta_p$	38,20	38,21	38,24	38,19	38,24

From this table it is seen that the efficiency of the regenerative cycle, within those limits of variation of extraction pressures which are obtained by the methods of equal heat drops on the expansion line and equal water heatings in the heaters, changes only slightly. These methods do indeed give limiting values of the extraction pressures. But since beyond this relatively narrow interval of pressures, which these methods give, the efficiency falls sharply, the choice of extraction pressures cannot be arbitrary. It is, of course, desirable to choose the extraction pressures at normal load as close as possible to the optimum ones, since this gives the smallest reduction in the efficiency of the installation under the load fluctuations inevitable in practice.

On the other hand, from Table 1 it is seen that if the selection of pressures according to graph C is applied, then the pressures determined in this way practically do not differ from the optimum ones determined by the exact method. Since the selection of extraction pressures by both limiting methods L and P is very simple, and the pressure by method C is calculated as the arithmetic mean of L and P, the indicated method should be regarded as a very valuable and simple method for selecting optimum extraction pressures. In particular, when selecting optimum pressures by the exact relations, it is recommended first of all to calculate the pressure in the low-pressure heater by method C and to use this value in the first variant of the exact calculation.

Surface Heaters

With surface heaters, with the drain being supplied to the feed line directly near the heater¹, there will be an underheating of the water to the condensation temperature of the extraction steam. Denoting these underheatings, according to Fig. 2, by

\[ \varepsilon' = i_{01}' - i_{a}'', \]

\[ \varepsilon'' = i_{02}' - i_{a}'', \]

\[ \varepsilon^{(n)} = i_{0n}' - i_{a}^{(n)}. \tag{69} \]

Analogously to the preceding case, with the same notation, one may write the expression for the function \(U'\), whose maximum coincides with the maximum of the efficiency, in the form

\[ U' = \left(1+\frac{X-\varepsilon^{(n)}}{b_n+\varepsilon^{(n)}}\right) \left(1+\frac{Y-\varepsilon^{(n-1)}-(X-\varepsilon^{(n)})}{b_{n-1}+\varepsilon^{(n-1)}}\right) \ldots \left(1+ \right. \]

\[ \left. +\frac{Z-\varepsilon'-(W-\varepsilon'')}{b_1+\varepsilon'} \right) \left(1+\frac{A-(Z-\varepsilon')}{b_0}\right). \tag{70} \]

Let us introduce the notation:

\[ b_n+\varepsilon^{(n)} = b'_n, \]

\[ b_{n-1}+\varepsilon^{(n-1)} = b'_{(n-1)}, \tag{71} \]

\[ \cdot \]

\[ \cdot \]

\[ \cdot \]

\[ b_1+\varepsilon' = b'_1, \]

and

\[ \begin{gathered} X-\varepsilon^{(n)}=x',\\ Y-\varepsilon^{(n-1)}=y',\\ \cdot\\ \cdot\\ \cdot\\ Z-\varepsilon'=z'. \end{gathered} \tag{72} \]

Then

\[ U'=\left(1+\frac{x'}{b'_n}\right) \left(1+\frac{y'-x'}{b'_{n-1}}\right)\ldots \left(1+\frac{z'-w'}{b'_1}\right) \left(1+\frac{A-z'}{b_0}\right). \tag{73} \]

Comparing the expression obtained with (45), it is easy to establish their identity. Consequently, the maximum conditions for (73) will also be identical with (45), i.e., they will be

\[ \begin{gathered} \frac{b'_n+x'}{1-\dfrac{x'}{b'_n}\dfrac{db'_n}{dx'}}+x'=b'_{n-1}+y', \\[1.2em] \frac{b'_{n-1}+y'-x'}{1-\dfrac{y'-x'}{b'_{n-1}}\dfrac{db'_{n-1}}{dy'}}+y'=b_{n-2}+p', \\ \cdot\\ \cdot\\ \cdot\\ \frac{b'_1+(z'-w')}{1-\dfrac{z'-w'}{b'_1}\dfrac{db'_1}{dz'}}+z'=A+b_0. \end{gathered} \tag{74} \]

It is expedient, first of all, to derive approximate simple relations for determining the optimum extraction pressures. Assuming, as before, that the heat of condensation of the steam in all heaters is the same, i.e., condition (50), and that the amount by which the water is underheated in all heaters is the same,

\[ \varepsilon'=\varepsilon''=\ldots=\varepsilon^{(n)}=\varepsilon, \tag{75} \]

we obtain, by virtue of (71),

\[ b'_1=b'_2=\ldots=b'_n=b'. \tag{76} \]

Under these conditions equations (74) give:

\[ \begin{gathered} x'=\frac{A-\varepsilon}{n+1} \\[1.2em] y'=\frac{2}{n+1}(A-\varepsilon), \\ \cdot\\ \cdot\\ \cdot\\ z'=\frac{n}{n+1}(A-\varepsilon). \end{gathered} \tag{77} \]

The actual water heatings in the individual heaters are related to the quantities \(x', y', \ldots z'\) in the following way:

\[ \Delta q_1 = z' - w', \]

\[ \vdots \tag{78} \]

\[ \Delta q_{n-1}=y'-x', \]

\[ \Delta q_n=x'. \]

From (77) and (78) one obtains

\[ \Delta q_1=\Delta q_2=\ldots=\Delta q_n=\frac{A-\varepsilon}{n+1}. \tag{79} \]

Thus, under the stated assumptions, the actual water heatings in all heaters turn out to be equal, but smaller than with mixing heaters (cf. Eq. 52).

The extraction pressures are determined by the increase in enthalpy of the boiling water in the heaters. Taking (72) into account, we have

\[ \Delta q^0=\Delta q'=\Delta q''=\ldots=\Delta q^{(n-1)}=\Delta q^{(n-2)}=\frac{A-\varepsilon}{n+1}, \tag{80} \]

\[ \Delta q^{(n)}=\frac{A+n\varepsilon}{n+1}. \tag{81} \]

Further, in the present case, between the heat drops in an individual expansion stage and the enthalpy increase of the boiling water in the heaters there remains the same linear dependence as in the preceding case (57). Consequently, at the optimum extraction pressures,

\[ i_1-i_{01}=i_{01}-i_{02}=\ldots=i_{0(n-1)}-i_{0n} =\frac{i_1-i_k-\varepsilon}{n+1}, \tag{82} \]

and for the heat drop of the last low-pressure stage,

\[ i_{0n}-i_k=\frac{i_1-i_k+n\varepsilon}{n+1}. \tag{83} \]

In the case of unequal water heatings in the different heaters and preservation of condition (50), the quantities \(b_1', b_2', \ldots\) will not be equal to one another, but nevertheless will be constant. In this case, from (74), with allowance for (71) and (78), we obtain:

\[ \Delta q_1= \frac{A-n\varepsilon' + \varepsilon''+\varepsilon''' + \ldots+\varepsilon^{(n)}}{n+1}, \]

\[ \Delta q_2= \frac{A+\varepsilon' - n\varepsilon''+\varepsilon''' + \ldots+\varepsilon^{(n)}}{n+1}, \]

\[ \vdots \tag{84} \]

\[ \Delta q_n= \frac{A+\varepsilon' + \varepsilon''+\varepsilon''' \ldots+\varepsilon^{(n-1)}-n\varepsilon^{(n)}}{n+1}. \]

The extraction pressures are determined from the increase in enthalpy of the boiling water in the heaters:

\[ \Delta q^0=\Delta q_1= \frac{A-n\varepsilon' + \varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}, \]

\[ \Delta q'=\Delta q_2= \frac{A+\varepsilon' - n\varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}, \tag{85} \]

\[ \Delta q^{(n-1)}=\Delta q_n= \frac{A+\varepsilon'+\varepsilon''+\ldots+\varepsilon^{(n-1)}-n\varepsilon^{(n)}}{n+1}. \]

And for the low-pressure stage

\[ \Delta q^{(n)}= \frac{A+\varepsilon'+\varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}. \tag{86} \]

Since in the present case the linear relation is also preserved between the increase in enthalpy of the boiling water in the heaters and the temperature differences (57), then at the optimal extraction pressures the temperature differences between extractions are calculated as follows:

\[ i_1-i_{01}= \frac{i_1-i_k-n\varepsilon' + \varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}, \]

\[ i_{01}-i_{02}= \frac{i_1-i_k+\varepsilon' - n\varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}, \tag{87} \]

\[ i_{0(n-1)}-i_{0n}= \frac{i_1-i_k+\varepsilon'+\varepsilon''\ldots+\varepsilon^{(n-1)}-n\varepsilon^{(n)}}{n+1}, \]

\[ i_{0n}-i_k= \frac{i_1-i_k+\varepsilon'+\varepsilon''+\varepsilon'''+\ldots+\varepsilon^{(n)}}{n+1}. \]

If the underheatings in all stages are identical,

\[ \varepsilon'=\varepsilon''=\ldots=\varepsilon^{(n)}\ne 0, \]

then relation (84) becomes (79), (85) and (86) become (80) and (81), and (87) becomes (82) and (83).

In the limiting case of underheatings equal to zero, we pass to the preceding case of mixing heaters.

An exact solution of the problem posed in the present case, as in the preceding one, is possible by graphical-analytical selection with preliminary construction according to Fig. 5.

From (71) and (72) and Fig. 2 we have

\[ b'_1+z'=b_1+Z=i_{01}-i'_k, \]

\[ \cdot \]

\[ \cdot \]

\[ b'_{n-1}+y'=b_{n-1}+Y=i_{0(n-1)}-i'_k, \]

\[ b'_n+x'=b_n+X=i_{0n}-i'_k . \tag{88} \]

Since the underheatings \(\varepsilon'\), \(\varepsilon''\), \(\ldots\), \(\varepsilon^{(n)}\) must be regarded as prescribed constant quantities, on the basis of (71), (72), and (88)

\[ \frac{db'_1}{dz'}=\frac{db_1}{dZ}=\frac{di_{01}}{di'_{01}}-1, \]

\[ \cdot \]

\[ \cdot \]

\[ \frac{db'_{n-1}}{dy'}=\frac{db_{n-1}}{dY}= \frac{di_{0(n-1)}}{di'_{0(n-1)}}-1, \]

\[ \frac{db'_n}{dx'}=\frac{db_n}{dX}=\frac{di_{0n}}{di'_{0n}}-1 . \tag{89} \]

From (74) we obtain, changing the order of numbering of the heaters (i.e., taking the low-pressure heater as the first heater):

\[ (i_1-i'_k)_2=\bigl[(i-i'_k)-(i'-i'_k)\bigr]_1 \frac{(i-i'_k)_1} {(i-i'_k)_1-\bigl[(i'-i'_k)-\varepsilon\bigr]_1\left(\dfrac{di}{di'}\right)_1} + \bigl[(i'-i'_k)-\varepsilon\bigr]_1, \]

\[ (i-i'_k)_3=\bigl[(i-i'_k)- \]

\[ -(i'-i'_k)\bigr]_2 \frac{(i-i'_k)_2-\bigl[(i'-i'_k)-\varepsilon\bigr]_1} {(i-i'_k)_2-\bigl[(i'-i'_k)-\varepsilon\bigr]_1- \bigl[(i'-i'_k-\varepsilon)_2-(i'-i'_k-\varepsilon)_1\bigr] \left(\dfrac{di}{di'}\right)_2} + \]

\[ +\bigl[i'-i'_k-\varepsilon\bigr]_2, \tag{90} \]

and so on. If the pressure of the first extraction (low pressure) is specified correctly, then for the last heater (high pressure) one should obtain

\[ (i-i'_k)_n=i_1-i'_k . \]

Otherwise the calculation should be repeated until agreement is achieved.

Table No. 2 gives the results of computations of the extraction pressures and the coefficient of useful action for the same steam parameters as in Table No. 1, with underheating in the high-pressure heater \(\varepsilon'=10\ \text{kcal/kg}\), and in all the others \(6\ \text{kcal/kg}\). Computations by formulas (87) are marked with the letter L, and by formulas (85) and (86) with the letter P.

In column C the extraction pressures are calculated as the arithmetic means of L and P. Finally, in column T the results are given for selecting the extraction pressures by formulas (90) using Fig. 5.

Table 2

	L	P	C	T
1 extraction point
$P_{01}$	4.4	9.4	6.9	6.45
$\eta_p$	35.77	34.36	35.87	35.95
2 extraction points
$P_{01}$	13.1	36.0	24.5	18.0
$P_{02}$	1.15	5.8	3.5	1.80
$\eta_p$	36.71	36.64	36.81	36.88
3 extraction points
$P_{01}$	22.0	34.0	28.0	28.8
$P_{02}$	4.4	9.0	6.7	6.6
$P_{03}$	0.57	1.25	0.90	0.85
$\eta_p$	37.37	37.31	37.37	37.38
4 extraction points
$P_{01}$	29.5	41.0	35.2	37.0
$P_{02}$	8.5	16.0	13.2	12.8
$P_{03}$	2.1	4.4	3.2	3.3
$P_{04}$	0.37	0.75	0.56	0.54
$\eta_p$	37.70	37.09	37.75	37.79
5 extraction points
$P_{01}$	35.0	47.0	41.0	42.0
$P_{02}$	13.2	23.0	18.1	19.0
$P_{03}$	4.4	9.0	6.7	6.6
$P_{04}$	1.15	2.6	1.9	1.85
$P_{05}$	0.26	0.52	0.39	0.38
$\eta_p$	37.80	37.84	37.94	37.95

Table No. 2 confirms the conclusions that were given above on the basis of Table No. 1. The selection of extraction pressures according to graph C in the present case gives results that agree very closely with the results of calculations by the exact relations (T).

For other arrangements of heater connections than the basic ones examined above, it is always possible, as was indicated, to calculate the conditional

underheatings and, in this way, reduce them to the basic scheme. Therefore, in all cases it is possible to use the indicated procedures for determining the optimum extraction pressures. The difficulty encountered here consists in the fact that the magnitude of the conditional underheating is a function of the extraction pressures, which are precisely the quantity sought; see (16), (17), (20), (21), (22), (23), (24), (27), (28), and (29). Therefore, here, when calculating the optimum extraction pressures, one must proceed by successive approximations. First, the optimum extraction pressures are assigned by eye.^1) Then the corresponding conditional underheatings are calculated, and from them, using rule C, the corresponding optimum extraction pressures are also computed. The calculations must be repeated^2) until the assigned and computed optimum extraction pressures agree sufficiently.

Table 3 gives the results of calculations for the plant according to the scheme of Fig. 3; Table 4, for the plant according to the scheme of Fig. 4, with the same plant parameters as those adopted in calculating Table 2. For reliability, some of the results were checked by the exact relations, and in all cases very good agreement was obtained.

Table 3

\(y^{I}\)	\(y^{II}\)	\(y^{III}\)	\(y^{IV}\)	\(P_{01}\)	\(P_{02}\)	\(P_{03}\)	\(P_{04}\)	\(\eta_{p}\)	Note
1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point
10	—	—	—	6,45	—	—	—	35,96	\(\varepsilon' = 10\ \mathrm{kcal/kg}\)
2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points	2 extraction points
10	21,3	—	—	16,0	2,25	—	—	36,58	\(\varepsilon' = 10\ \mathrm{kcal/kg}\) \(\varepsilon'' = 6\ \mathrm{kcal/kg}\)
3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points	3 extraction points
10	17,3	34,5	—	23,2	5,4	1,20	—	36,95	\(\varepsilon' = 10\ \mathrm{kcal/kg}\) \(\varepsilon'' = \varepsilon''' = 6\ \mathrm{kcal/kg}\)
4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points	4 extraction points
10	15,1	28,2	42,2	28,0	8,3	2,6	0,90	37,04	\(\varepsilon' = 10\ \mathrm{kcal/kg}\) \(\varepsilon'' = \varepsilon''' = \varepsilon^{IV} = 6\ \mathrm{kcal/kg}\)

Table 4

\(y^{I}\)	\(y^{II}\)	\(y^{III}\)	\(y^{IV}\)	\(P_{01}\)	\(P_{02}\)	\(P_{03}\)	\(P_{04}\)	\(\eta_{p}\)	Note
1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point	1 extraction point
10	—	—	—	6,45	—	—	—	35,96	\(\varepsilon' = 10\ \mathrm{kcal/kg}\)

^1) In our calculations, we assigned the optimum extraction pressures according to rule C, taking initially the actual underheatings as conditional.

^2) Usually 2–3 times.

Number of extraction points
2 extraction points	10	21.3	—	—	16.0	2.25	—	—	36.58	$\varepsilon' = 10\ \text{kcal/kg}$ $\varepsilon'' = 6$
3 extraction points	10	10.5	23.2	—	24.0	6.0	1.15	—	37.05	$\varepsilon' = 10\ \text{kcal/kg}$ $\varepsilon'' = \varepsilon''' = 6\ \text{kcal/kg}$
4 extraction points	10	13.6	18.9	23.8	31.0	10.5	3.0	0.74	37.29	$\varepsilon' = 10\ \text{kcal/kg}$ $\varepsilon'' = \varepsilon''' = \varepsilon^{\mathrm{IV}} = 6\ \text{kcal/kg}$

From examination of Tables Nos. 3 and 4 and comparison with Table No. 2 it follows:

1. With one extraction point, the method of connecting the heater is of no significance from the thermal standpoint.

2. With two extraction points, connections according to the diagrams of Figs. 3 and 4 give identical results.

3. Particularly disadvantageous from the thermal standpoint is the connection of heaters according to the diagram of Fig. 3 (i.e., the discharge of all drains directly into the condenser) when there is a large number of extraction points.

Since the proposed computational procedure for determining the optimum extraction pressures is more complicated than those proposed earlier, the question naturally arose of comparing the results obtained by these methods. For comparison, the connection diagram of Fig. 3 was taken for three extraction points with equal subcoolings ($\varepsilon' = \varepsilon'' = \varepsilon''' = 6\ \text{kcal/kg}$), for which case formula (37a) is given in the work of I. I. Petelin [3]. As a result of the calculations, the following was obtained:

According to I. I. Petelin:

\[ P_{01} = 22.0;\qquad P_{02} = 8.5;\qquad P_{03} = 2.5\ \text{ata} \]

and the corresponding

\[ \eta_p = 36.83\%. \]

According to the proposed method,

\[ P_{01} = 22.0;\qquad P_{02} = 5.2;\qquad P_{03} = 1.2\ \text{ata} \]

and the corresponding

\[ \eta_p = 36.95\%. \]

From these data it is clear that the proposed method indeed gives more accurate results.

Presence of District-Heating Extractions

The proposed method of calculating the optimum pressures of steam extractions is a general method. This makes it possible to apply it in the most diverse cases. We shall show its application in one particular case.

When regulated extractions for district-heating or process steam are present, the latter are usually also used for regenerative heating of the feed water. But special, unregulated extractions are usually also arranged for the latter purpose.

The pressure of the process or district-heating steam is, naturally, specified by the consumer. Thus, the question may arise of the optimum extraction pressures for regenerative heating of the feedwater—

new water in the presence of an extraction (one or several) of fixed pressure. To show the method for solving this question, let us consider the particular case of the process given in Fig. 6. Here extraction is assumed at the fixed pressure \(P_0\), and the possibility of choosing the pressures \(P_{01}\) and \(P_{02}\) in the high-pressure part and \(P_{03}\) and \(P_{04}\)—in the low-pressure region. Surface heaters are assumed, with the drainage discharged into the feed-water line directly near the heater. The magnitude of the underheating of the water in each heater is known.

Fig. 6

Under the stated conditions, for calculating the c.p.d. at known extraction pressures, relation (9) or (11) may be applied; and for the function \(U\), whose maximum coincides with the maximum value of \(\eta_{\mathrm{pr}}\), relation (73), which is rewritten as follows:

\[ U= \left(1+\frac{x'}{b_4'}\right) \left(1+\frac{y'-x'}{b_3'}\right) \left(1+\frac{m'-y'}{n'}\right) \left(1+\frac{w'-m'}{b_2'}\right) \left(1+\frac{z'-w'}{b_1'}\right) \left(1+\frac{A-z'}{b_0}\right). \tag{91} \]

Here, in addition to the notation introduced above, we have

\[ n' = n+\varepsilon^0, \tag{92} \]

where

\[ n=i_0-i_0' \]

and \(\varepsilon^0\) is the underheating of the water in the heater with pressure \(P_c\),

\[ m'=i_a'-i_k'. \tag{93} \]

The maximum condition for (91), with independent variables \(x'\), \(y'\), \(w'\), \(z'\), will be:

\[ \frac{b_4'+x'}{1+\dfrac{x'}{b_4'}\,\dfrac{db_4'}{dx'}}+x'=b_3'+y', \tag{94} \]

\[ \frac{b_3'+(y'-x')}{1+\dfrac{y'-x'}{b_3'}\,\dfrac{db_3'}{dy'}}+y'=m'+n'. \]

and

\[ \frac{b_2' + (w' - m')}{1 + \dfrac{w' - m'}{b_2'} \dfrac{db_2'}{dw'}} + w' = b_i' + z', \]

\[ \frac{b_1' + (z' - w')}{1 + \dfrac{z' - w'}{b_1'} \dfrac{db_1'}{dz'}} + z' = A + b_0 . \tag{95} \]

If the heats of steam condensation and underheating in all heaters are taken to be equal to one another, i.e., conditions (50), (75), and (76), then we obtain

\[ x' = \frac{m'}{3}, \]

\[ y' = \frac{2}{3}\,m', \]

\[ w' = \frac{A + 2m' - \varepsilon}{3}, \]

\[ z' = \frac{2A + m' - 2\varepsilon}{3}, \tag{96} \]

whence, taking (78) into account, we have for the actual heatings

\[ \Delta q_5 = \Delta q_4 = \Delta q_3 = \frac{m'}{3}, \tag{97} \]

\[ \Delta q_2 = \Delta q_1 = \frac{A - m - \varepsilon}{3}, \tag{98} \]

For the enthalpy increment of boiling water we have

\[ \Delta q^0 = \Delta q' = \Delta q'' = \frac{A - m}{3}, \tag{99} \]

\[ \Delta q''' = \Delta q^{\mathrm{IV}} = \frac{m - \varepsilon}{3}, \tag{100} \]

\[ \Delta q^{\mathrm{V}} = \frac{m + 2\varepsilon}{3}, \tag{101} \]

or, through enthalpies,

\[ \Delta q^0 = \Delta q' = \Delta q'' = \frac{i_1' - i_0'}{3}, \tag{99} \]

\[ \Delta q''' = \Delta q^{\mathrm{IV}} = \frac{i_0' - i_k}{3}, \tag{100} \]

\[ \Delta q^{\mathrm{V}} = \frac{i_0' - i_k + 2\varepsilon}{3}. \tag{101} \]

Since relation (57) is preserved here, we have for the heat drops

\[ i_1 - i_{01} = i_{01} - i_{02} = i_{02} - i_0 = \frac{i_1 - i_0}{3}, \tag{102} \]

\[ i_0 - i_{03} = i_{03} - i_{04} = \frac{i_0 - i_k - e}{3}, \tag{103} \]

\[ i_{04} - i_k = \frac{i_0 - i_k + 2e}{3}. \tag{104} \]

As has already been shown, very accurate results are obtained when the calculation is carried out by method C.

In the present case this means that the extraction pressures should be taken as the arithmetic means of those determined from (99)—(102), (100)—(103), and (101)—(104). It presents no fundamental difficulties to find the exact solution of the problem posed, using (94) and (95) and the graph in Fig. 5.

REFERENCES

1. Shinman. Selection of steam extraction points for heating feed water. Bulletin of the Engineering Collective of MOGES, 1928, p. 63.

2. Rubinshtein. Investigation of the regenerative process in steam-turbine installations. Izvestiya VTI, 1929, Nos. 1—9.

3. Petelin. Regenerative heating of feed water. 1932.

4. Gokhshtein. An attempt to use the basic inequality of the second law of thermodynamics in practical heat engineering. Soviet Boiler and Turbine Construction, 1939, No. 9.

5. Othok. The commercial economy of the pressure and high superheat in the central power stations. Power, 1922, pp. 648, 913.

6. Zerkowitz. Das Gegendruckverfahren und seine Anwendung bei der Dampfturbine—III Teil. Das Gegendruckverfahren als Regenerativverfahren. VDI, 1924, S. 1093.

7. Lauffer. Where to bleed a Turbine. Power, 1926, p. 363.

8. Laupichler. Günstigste Anzapfdrücke der Verlustlosen Dampfturbinen bei Dampfentnahme aus beliebig vielen Stufen zur Vorwärmung des Speisewassers. Wärme, 1927, S. 227.

9. Kaizling. Die Wahl des geeigneten Druckes bei Anzapfdampfvorwärmung. Wärme, 1934, S. 433.

10. Chambadal. Le fractionnement du réchauffage de l’eau d’alimentation des chaudiers. Chaleur et Industrie, 1937, p. 279.

11. Kinkeldei. Die Bestimmung der günstigsten Anzapfvorwärmung unter Berücksichtigung des Gutegrades der Dampfturbine. Wärme, 1933, S. 97.

12. Schult. Der Erfolg der Drucksteigerung bei Kondensations und Gegendruck Dampfanlagen. VDJ, 1939, S. 405.

Misprints Noted

Page	Line	Printed	Should be
89	Fig. 3	(Figure inverted)
92	7 from bottom	$o_2$	$-i_{02}$
95	19 from bottom	$Z$	$Z_T$
97			In equation (27), $\alpha$ is the exponent
98	19 from bottom	$Z_m\Delta$	$Z_m\gamma$
101	15 from bottom	surrounding	circumferences
178	17 from bottom	$$s=\frac{n}{n}$$	$$s=\frac{n}{n_0}$$
193	4 from top	$$l_2^2=\frac{E}{1+Z^2}$$	$$l_2^2=\frac{\varepsilon}{1+Z^2}$$
203	12 from top	$$I=I_m\left(1-e^{-\frac{r_M}{L_M}}\right)=\ldots$$	$$I=I_m\left(1-e^{-\frac{r_M}{L_M}t}\right)=\ldots$$
203	17 from bottom	concluding	concluding
214	8 from bottom	(28)	(26)
221	4 from top	$$\ldots \frac{i_0^2-i_a''}{i_{01}-i_a''}$$	$$\ldots \frac{i_{02}-i_a''}{i_{01}-i_a''}$$
318	Caption to Fig. 5	Heat losses of incomplete combustion.	Heat losses from incomplete combustion.
318	14 from bottom	(fig. 3)	(fig. 4)
323	1 from top	ature	temperature
323	1 from top	$r=58$	$r=587$
325	In Fig. 6	$q'_{\mathrm{water}}$ and $q''_{\mathrm{water}}$	$q'_{\mathrm{water}}$ and $q''_{\mathrm{water}}$
326	33 from top	$/2\ 3+25=293\ \mathrm{K},$	$/273+25=293\ \mathrm{K}/,$

1. It is immaterial whether before or after the heater. ↩↩↩