Proceedings
of the Tomsk Order of the Red Banner of Labor Polytechnic
Institute named after S. M. Kirov

Volume 89　　　　　　　　　　　　　　　　　　　　　　　　　1957

On the Question of Selecting a Method for Regulating Heat Supply in Heat Networks

E. N. Shadrin

The quantity of heat \(Q\) required for heating consumers at an arbitrary outdoor temperature \(t_n\) is equal to

\[ Q = \varphi \cdot Q', \tag{1} \]

where \(Q'\) is the design quantity of heat required by consumers at the design outdoor temperature \(t'_n\), and \(\varphi\) is the relative heat load. Its value is determined as

\[ \varphi = \frac{t_v - t_n}{t_v - t'_n} \tag{2} \]

Here \(t_v\) is the temperature inside the heated premises.

To supply consumers with heat \(Q\), corresponding to an arbitrary outdoor temperature \(t_n\), regulation of the heat supply is carried out. At present, in the practice of operating heat networks, three methods of regulating heat supply have found application, mainly: qualitative, quantitative, and mixed, which is a combination of qualitative and quantitative regulation.

With qualitative regulation, depending on the outdoor-air temperature \(t_n\), the temperature \(t_{gor}\) of the hot water going to the heat consumers changes. As \(t_n\) rises, \(t_{gor}\) decreases. The latter makes it possible to use extraction steam of lower pressure for heating the water, which results in greater generation of electric energy in heat consumption.

With quantitative regulation, \(t_{gor}\) remains unchanged throughout the entire heating period and is equal to the design temperature \(t'_{gor}\). The change in heat supply in this case is achieved by changing the quantity of heat carrier. As \(t_n\) rises, the quantity of heat carrier \(G\) decreases. A decrease in \(G\) requires less power expenditure for pumping the heat carrier. Taking this into account, in many heat networks, in order to save electric energy for pumping water, a stepped regulation schedule is used.

In our opinion, when selecting one or another method of regulating heat supply, it is necessary to take into account the energy efficiency of the methods. For this purpose it is necessary to consider the joint operation of the turbine and the heat network.

The power generated by extraction steam is expressed by the following formula:

\[ N_t = \frac{D_o \cdot h_o \cdot \eta_{oe}}{860}\ \text{kW}, \tag{3} \]

where: \(D_o\) is the quantity of extraction steam, kg/hour;
\(h_o\) is the available heat drop up to the pressure in the extraction, kcal/kg;
\(\eta_{oe}\) is the relative electrical efficiency.

Expressing \(N_m\) in terms of the relative heat load, we obtain:

\[ N_m=\frac{\varphi \cdot Q' \cdot h_0 \cdot \eta_{0э}}{860(i_0-t_k-h_0\cdot \eta_{0i})}\ \text{kW}. \tag{4} \]

Here \(i_0\) is the heat content of the steam before the turbine, kcal/kg;
\(t_k\) is the temperature of the condensate of the extraction steam, \({}^{\circ}\mathrm{C}\);
\(\eta_{0i}\) is the relative internal efficiency of the steam flow going to extraction.

With qualitative regulation, for a specified design heat consumption,

\[ N'_m=f(\varphi,h_0,Q',t_k). \]

With quantitative regulation, \(h_0\) and \(t_k\) remain unchanged over the entire range of variation of \(t_h\). Therefore the generation on heat consumption \(N''_m\) for quantitative regulation depends only on \(\varphi\) and \(Q'\).

To find \(h_0\) and \(t_k\), it is necessary to determine the extraction pressure for various \(t_h\). The extraction pressure \(P_{\mathrm{ext}}\) is determined from the saturation temperature at this pressure:

\[ t_s=t_{\mathrm{hot}}+\Delta t, \tag{5} \]

where \(\Delta t\) is the temperature head in the heaters, in \({}^{\circ}\mathrm{C}\). The temperature \(t_{\mathrm{hot}}=f(\varphi)\) and can be determined by jointly solving the equation of heat transfer in the heating devices and the equation of heat transport.

From the equation of heat transfer in the heating devices, for an arbitrary \(t_h\),

\[ \varphi=\frac{F\cdot K\cdot \Delta t_{\mathrm{av}}}{F\cdot K'\cdot \Delta t'_{\mathrm{av}}}. \tag{6} \]

where: \(\Delta t_{\mathrm{av}}\) and \(\Delta t'_{\mathrm{av}}\) are the mean temperature heads in the devices at \(t_h\) and \(t'_h\), \({}^{\circ}\mathrm{C}\);

\(K\) and \(K'\) are the heat-transfer coefficients at \(t_h\) and \(t'_h\),

\[ \frac{\text{kcal}}{\text{m}^2\text{ h }{}^{\circ}\mathrm{C}}, \]

\(F\) is the surface area of the heating devices, \(\mathrm{m}^2\).

If it is taken into account that for heating devices \(K=C\cdot \Delta t_{\mathrm{av}}^{x}\), then equation (6) is written as

\[ \varphi=\left(\frac{\Delta t_{\mathrm{av}}}{\Delta t'_{\mathrm{av}}}\right)^{1+x}. \tag{7} \]

Denoting the mean air temperature before and after the heating devices by \(t_v\), we obtain for the mean temperature head

\[ \Delta t_{\mathrm{av}}=\frac{t_{\mathrm{hot}}-t_{\mathrm{return}}}{2}-t_v \tag{8} \]

On the other hand, from the equations of heat transport for an arbitrary \(t_h\) for qualitative regulation,

\[ \varphi=\frac{t_{\mathrm{hot}}-t_{\mathrm{return}}}{t'_{\mathrm{hot}}-t'_{\mathrm{return}}}. \tag{9} \]

Solving equations 7, 8, and 9 jointly, we obtain for \(t_{\mathrm{hot}}\)

\[ t_{\mathrm{hot}}=\varphi^{\frac{1}{1+x}}\cdot \Delta t'_{\mathrm{av}}+\frac{\varphi}{2}\Delta t'_c+t_v, \tag{10} \]

where \(\Delta t'_c=t'_{\mathrm{hot}}-t'_{\mathrm{return}}\) is the design temperature difference of the heat carrier.

Consequently,

\[ t_s=\varphi^{\frac{1}{1+x}}\cdot \Delta t'_{\mathrm{av}}+\frac{\varphi}{2}\Delta t'_c+t_v+\Delta t \tag{11} \]

In Fig. 1, according to equation (11), the dependence \(t_s=f(t_n)\) is shown for a particular example, when \(\Delta t=5^\circ\mathrm{C}\), \(t'_{\mathrm{hot}}=130^\circ\mathrm{C}\), \(t'_{\mathrm{return}}=70^\circ\mathrm{C}\), \(t'_n=-40^\circ\mathrm{C}\). On the same Fig. 1 the dependence \(t_{\mathrm{hot}}=f(t_n)\), constructed according to equation (10), is shown.

Fig. 1

Within the limits from \(t_{n1}\) to \(t_{n2}\), the temperature \(t_s\) varies as a function of \(t_n\) in accordance with equation (11). Up to \(t_{n1}\), \(t_s=\mathrm{const}\) and is determined by the minimum pressure in the extraction. Within the limits from \(t_{n2}\) to \(t'_n\), the temperature \(t_s\) likewise does not change and corresponds to the saturation temperature at the maximum pressure in the extraction.

For the example under consideration, the change in pressure in the extraction is taken from \(1.25\ \mathrm{ata}\) to \(2.5\ \mathrm{ata}\). The extraction pressure \(1.25\ \mathrm{ata}\) corresponds to the horizontal segment of the curve \(t_s=f(t_n)\), \(a'b'\); the maximum extraction pressure \(2.5\ \mathrm{ata}\) corresponds to the segment of the curve \(b'2'\). Thus, the available heat drop up to the extraction, as well as the temperature \(t_k\), will change under qualitative regulation only within the range of variation of the outdoor-air temperature from \(t_{n1}\) to \(t_{n2}\).

From \(t_s\) we determine the extraction pressure for each \(t_n\) and the available heat drop up to the extraction. In Fig. 1, for the example considered earlier, the dependences \(P_{\mathrm{ex}}=f(t_n)\) and \(h_0=f(t_n)\) under qualitative regulation are plotted. Under quantitative regulation, both the extraction pressure and the heat drop up to the extraction will be unchanged over the entire interval of variation of the outdoor-air temperature \(t_n\). In the same Fig. 1, the dependences \(P_{\mathrm{ex}}=f(t_n)\) and \(h_0=f(t_n)\) for quantitative regulation for the same example are shown by dashed lines.

Using the graph in Fig. 1, by formulas (2) and (4) one can determine the change in the power produced by the extraction steam for qualitative and quantitative regulation as a function of \(t_n\). Since for quantitative regulation \(h_0=\text{const}\) and \(t_k=\text{const}\), the dependence of \(N'_m\) on \(t_n\) is expressed by the law of a straight line.

Figure 2 gives a graphical solution of equation (4) for qualitative and quantitative regulation at

\[ Q' = 50 \ \frac{\text{Mcal}}{\text{h}} . \]

From Fig. 2 it is seen that, at all outdoor temperatures, \(t_n>t_{\text{TEC}}\), where \(t_{\text{TEC}}\) is the design temperature of the thermal electric station, \(N'_m>N''_m\). In Fig. 3, from the curves of Fig. 2, the dependence

\[ \Delta N_m = N'_m - N''_m \quad \text{on } t_n \]

is plotted.

Fig. 2

We now proceed to determine the power expended on pumping water. Under qualitative regulation the quantity of water is \(G=\text{const}\), therefore the power in kW for pumping the heat-transfer medium is

\[ N'_n=\frac{S_0 G^{1\,3}}{3.67\cdot 10^5 \cdot \eta_n}\ \text{kW}. \tag{12} \]

Under quantitative regulation,

\[ N''_n=\frac{S_0 \alpha^3 G^{1\,3}}{3.67\cdot 10^5 \cdot \eta_n}\ \text{kW}. \tag{13} \]

Here: \(S_0\) is the reduced resistance of the head section of the heat network;
\(G^1\) is the design quantity of heat-transfer medium, kg/h;
\(\eta_n\) is the efficiency of the network pump;

\[ \alpha=\frac{G}{G'} \]

represents the ratio of the water flow rate at an arbitrary temperature \(t_n\) to the calculated quantity of water and may be called the coefficient of relative water flow rate.

Fig. 3. Dependence of \(\Delta N_t\) and \(\Delta N_n\), kW, on the outside-air temperature \(t_n,{}^\circ\mathrm{C}\). The figure indicates the regions of quantitative regulation and qualitative regulation, and the curves \(\Delta N_t = N_t' - N_t'' = f(t_n)\) and \(\Delta N_n = N_n' - N_n'' = f(t_n)\).

The value of the coefficient \(\alpha\) may be determined from the following considerations. For quantitative regulation, the relative heat load is

\[ \varphi= \frac{G(t_{\mathrm{gor}}-t_{\mathrm{obr}})} {G'(t'_{\mathrm{gor}}-t'_{\mathrm{obr}})} = \alpha\cdot \frac{t_{\mathrm{gor}}-t_{\mathrm{obr}}} {t'_{\mathrm{gor}}-t'_{\mathrm{obr}}} \tag{14} \]

Solving equations (7), (8), and (14) simultaneously and taking into account that for quantitative regulation \(t_{\mathrm{gor}}=t'_{\mathrm{gor}}=\mathrm{const}\), we obtain for \(\alpha\):

\[ \alpha= \frac{0.5\varphi\cdot \Delta t'_c} {t'_{\mathrm{gor}}-\varphi^{-\frac{1}{1+x}}\cdot \Delta t'_{\mathrm{sr}}-t_v} \tag{15} \]

From equation (15) it is seen that \(\alpha=f(\varphi)=f'(t_n)\), and therefore the pump power is also \(N_n''=f(\varphi)=f'(t_n)\).

The difference in the powers required for pumping water under qualitative and quantitative regulation will be

\[ \Delta N_n=N_n'-N_n''=(1-\alpha^3)\cdot \frac{S_0G^{\prime 3}} {3.67\cdot 10^5\cdot \eta_n} \]

If \(\Delta N_t>\Delta N_n\), then the additional power generated in heat consumption under qualitative regulation is greater than the reduction in the power required for pumping water due to the decrease in the amount of heat carrier under quantitative regulation. Therefore, for these conditions qualitative regulation is more advantageous. If \(\Delta N_t<\Delta N_n\), then quantitative regulation is more advantageous, since under it the reduction in the power required for pumping water pre-

gives an increase in the power generated by extraction steam under qualitative regulation.

In the same Fig. 3 the dependence \(\Delta N_{h}=f(t_{h})\) is shown for the example analyzed earlier. For the conditions of this example, the curves intersect at two points, at \(t_{h}=-30^\circ\mathrm{C}\) and \(t_{h}=+7^\circ\mathrm{C}\), dividing the entire range of variation of \(t_{h}\) into three intervals, in each of which, from the energy standpoint, it is advisable to apply a definite method of regulating heat supply. If, along with heating heat consumption, there is hot-water supply, then in the region of high outdoor temperatures \(t_{h}\) the transition to quantitative regulation will be determined by the lowest temperature of the water going to hot-water supply.

Under actual operating conditions, ideal quantitative regulation cannot be achieved. Usually the change in the amount of water has a stepwise character, depending on the number and characteristics of the pumps. However, this does not alter the conclusions obtained. The transition from one water flow rate to another is expedient only in that interval where \(\Delta N_{h}>\Delta N_{t}\).

Investigations of this question as applied to other conditions make it possible to draw the following conclusions:

1. The higher the design temperature of the thermal power plant, \(t_{\text{TPP}}\), the wider the range of application of quantitative regulation, and conversely. If \(t_{\text{TPP}}=t'_{h}\), then, for small reduced resistances of the network, in the region of low values of the temperature \(t_{h}\), quantitative regulation may prove inexpedient.

2. The greater the reduced resistance of the network, the wider the range of application of quantitative regulation.

3. The use of high initial parameters at power stations narrows the range of application of quantitative regulation.