ON THE CHOICE OF STEAM EXTRACTION POINTS IN REGENERATION IN THERMAL POWER INSTALLATIONS

I. N. BUTAKOV

For any \(n\)-th mixing heater one may write the heat balance, in the case of extraction of saturated steam, in the following form

\[ x_n \cdot r_n = y_n\,(T_n - T_{n-1}), \ldots \ldots \ldots \ldots \ldots (1) \]

where \(x_n\) is the amount of extracted steam in kg, \(r_n\) is the latent heat of its vaporization in kcal/kg, \(T_n\) and \(T_{n-1}\) are the absolute temperatures of the water at the outlet from the heater and at the inlet to it, and \(y_n\) is the amount of condensate, in kg, passing through the \(n\)-th heater. It may be assumed that \(y_n\) changes little in passing from one heater to another; therefore, taking \(y_n = 1\), we have

\[ x_n = \frac{T_n - T_{n-1}}{r_n} \tag{2} \]

The work which the extracted steam could perform according to the Rankine cycle is determined by formula (3)

\[ W_n = x_n\,[r_n(1 - T_2/T_n) + T_n - T_2 - T_2 \ln T_n/T_2], \tag{3} \]

where \(T_2\) is the absolute temperature of the cold source. Formula (3) may be represented in the following form \((3')\)

\[ W_n = x_n \cdot r_n(1 - T_2/T_n) \left[ 1 + \frac{T_n - T_2 - T_2 \ln T_n/T_2}{r_n(1 - T_2/T_n)} \right]. \tag{3'} \]

The second term in the square brackets is a small fraction, so that, if it is ignored, we thereby reduce the value \(W_n\). This, however, will not noticeably affect the result, since in fact the steam consumption \(x_n\) will be less than that adopted by formula (2). This is because the main condensate will be supplemented by condensates of the heating steam in gradual order when passing from one heater to another, reaching a full unit only after the last heater. Therefore, with sufficient approximation to reality, one may take

\[ W_n = x_n \cdot r_n(1 - T_2/T_n), \tag{4} \]

or, after substituting here the value of \(x_n\) from expression (2),

\[ W_n = \frac{(T_n - T_{n-1})(T_n - T_2)}{T_n} = T_n - T_{n-1} - T_2 + \frac{T_{n-1}\cdot T_2}{T_n}. \tag{5} \]

If the number of heaters is \(K\), then the total work that could be performed by the extraction steam of all \(K\) heaters will be \(\Sigma W_i\); moreover, for the low-pressure heater \(K\) we have

\[ W_{\kappa}= \frac{(T_{\kappa}-T_2)(T_{\kappa}-T_2)}{T_{\kappa}} = T_{\kappa}-T_2-T_2+\frac{T_2\cdot T_2}{T_{\kappa}}; \]

\[ W_{\kappa-1}= \frac{(T_{\kappa-1}-T_{\kappa})(T_{\kappa-1}-T_2)}{T_{\kappa-1}} = T_{\kappa-1}-T_{\kappa}-T_2+\frac{T_{\kappa}\cdot T_2}{T_{\kappa-1}}; \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \]

\[ W_{II}= \frac{(T_{II}-T_{III})(T_{II}-T_2)}{T_{II}} = T_{II}-T_{III}-T_2+\frac{T_{III}\cdot T_2}{T_{II}}; \]

\[ W_1= \frac{(T_1-T_{II})(T_{II}-T_2)}{T_1} = T_1-T_{II}-T_2+\frac{T_{II}\cdot T_2}{T_1}. \]

Thus,

\[ \Sigma W_i = T_1-(K+1)T_2 + \frac{T_2\cdot T_2}{T_{\kappa}} + \frac{T_{\kappa}\cdot T_2}{T_{\kappa-1}} \ldots + \frac{T_{III}\cdot T_2}{T_{II}} + \frac{T_{II}\cdot T_2}{T_1} \tag{6} \]

The temperature \(T_1\)—the maximum temperature of regenerative heating of the feed water—is usually a value assigned on the basis of techno-economic considerations; therefore the variable quantities in expression (6) are the last terms. It is important to us that the quantity \(\Sigma W_i\) be minimal, i.e., that the loss of work caused by the failure to use the heat drops from the pressures at the steam-extraction points down to the pressure in the condenser be as small as possible. This will occur when the sum of the last terms of expression (6) is minimal. Let us note that the product of these terms is a constant quantity, equal to \(T_2^{k+1}/T_1=\mathrm{const}\). If we have a product of variable quantities \(x,y\), and \(z\) in the form of some constant quantity \(x\cdot y\cdot z=C\), and it is required to find the maximum or minimum of their sum \(\Sigma=x+y+z\), then one takes the partial derivatives with respect to each variable, equates them to zero, and then determines, using the product \(C\), the values of \(x,y\), and \(z\). It turns out that then \(x=y=z=\sqrt[3]{C}\), or, as applied to the particular case under consideration,

\[ \frac{T_2\cdot T_{II}}{T_1} = \frac{T_2\cdot T_{III}}{T_{II}} \ldots = \frac{T_{\kappa}\cdot T_2}{T_{\kappa-1}} = \frac{T_2\cdot T_2}{T_{\kappa}} = \sqrt[k]{\frac{T_2^{k+1}}{T_1}}, \]

or

\[ T_{II}/T_1 = T_{III}/T_{II} \ldots = T_{\kappa}/T_{\kappa-1} = T_2/T_{\kappa} = \sqrt[k]{T_2/T_1}. \tag{7} \]

Thus, in order to obtain the minimum of the work \(\Sigma W_i\) with \(K\) heaters, it is necessary to distribute the absolute temperatures of the extraction steam \(T_{II}, T_{III}\ldots T_{\kappa-1}, T_{\kappa}\) so that they form a geometric progression with common ratio \(\sqrt[k]{T_2/T_1}\).

In the case of the use of surface heaters with the cascade method of routing the condensate line of the heating steam of the extractions, the flow rate of the last \(x_n\), owing to the admission of condensate with a higher temperature from the heater located above, decreases. The same occurs when there are small condensate transfer pumps, and in other cases of routing the condensate lines of the heating steam. Therefore, also in po-

for surface heaters, the same fundamental propositions remain in force as those formulated above for mixing heaters (formulas 4, 5, and 6). The only difference will be that, in surface heaters, there is a temperature difference \(\Delta T\) between the heating steam and the feed water leaving the heater \((5^\circ\text{—}10^\circ\mathrm{C})\). Therefore formulas (4–7) for the case of surface heaters will be rewritten in the following form.

\[ W_n=x_n r_n\left(1-\frac{T_2}{T_n+\Delta T}\right)\ldots \tag{4'} \]

\[ W_n=\frac{T_n-T_{n-1}}{r_n}\cdot r_n \left(1-\frac{T_2}{T_n+\Delta T}\right) = \frac{(T_n-T_{n-1})[(T_n+\Delta T)-T_2]}{T_n+\Delta T}\ldots \tag{5'} \]

The total work will have the form

\[ \sum W_i = T_1-(k+1)T_2 + \frac{T_2(T_{II}+\Delta T)}{T_1+\Delta T} + \frac{T_2(T_{III}+\Delta T)}{T_{II}+\Delta T} +\ldots \]

\[ + \frac{T_2(T_k+\Delta T)}{T_{k-1}+\Delta T} + \frac{T_2(T_2+\Delta T)}{T_k+\Delta T} \ldots \tag{6'} \]

The product of the last terms of expression \((6')\)

\[ \frac{T_2^k(T_2+\Delta T)}{T_1+\Delta T}=\mathrm{const}, \]

therefore

\[ \frac{T_{II}+\Delta T}{T_1+\Delta T} = \frac{T_{III}+\Delta T}{T_{II}+\Delta T} = \ldots = \frac{T_k+\Delta T}{T_{k-1}+\Delta T} = \frac{T_2+\Delta T}{T_k+\Delta T} = \sqrt[k]{\frac{T_2+\Delta T}{T_1+\Delta T}}. \tag{7'} \]

Heaters supplied with bled steam of elevated pressure may receive steam in a superheated state, especially in the case of using steam with high initial parameters. Then in formula \((3')\), within the parentheses, there will appear still another, third term, corresponding to the superheat region of the Rankine diagram. This term will be a small fraction, permitting one to consider the possibility of neglecting it, all the more since in the present case the actual steam consumption \(x_n\) in heaters supplied with superheated steam will be smaller owing to the higher heat content of the heating steam. If, however, we take \(x_n\) according to formula (2), we thereby increase \(x_n\) and, consequently, by this compensate for the decrease of \(W_n\) due to neglecting steam superheat in these heaters. Thus, the presence of superheated steam in some heaters cannot affect the results of selecting the steam extraction points, which in these cases as well will obey the law of a geometric progression of absolute temperatures.

Noted Misprints