On the Choice of 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 for 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 the heat \(Q\) corresponding to an arbitrary outdoor temperature \(t_n\), the heat supply is regulated. At present, in the operating practice of heat networks, three methods of regulating heat supply have mainly found application: 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 heating consumers changes. As \(t_n\) increases, \(t_{gor}\) decreases. The latter makes it possible to use extraction steam of lower pressure for heating the water, which leads to a greater generation of electrical energy on 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 \(t_n\) increases, the quantity of heat carrier \(G\) decreases. A decrease in \(G\) requires a smaller expenditure of power for pumping the heat carrier. Taking this into account, in many heat networks a stepped regulation schedule is used in order to save electrical energy for pumping water.

In our opinion, when choosing 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:

\[ N_t=\frac{D_0\cdot h_0\cdot \eta_{oe}}{860}\ \text{kW}, \tag{3} \]

where: \(D_0\) is the quantity of extraction steam, kg/h;
\(h_0\) is the available heat drop 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:

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

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

With qualitative regulation, for a given design heat demand,

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

With quantitative regulation, \(h_0\) and \(t_k\) remain unchanged over the entire range of variation of \(t_n\). Therefore the generation for heat demand, \(N''_m\), under 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 \(t_n\). The extraction pressure \(P_{\text{ext}}\) is determined from the saturation temperature at this pressure:

\[ t_s=t_{\text{sup}}+\Delta t, \tag{5} \]

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

From the heat-transfer equation in the heating devices, for an arbitrary \(t_n\),

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

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

\(K\) and \(K'\) are the heat-transfer coefficients at \(t_n\) and \(t'_n\),

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

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

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

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

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

\[ \Delta t_{\text{av}}=\frac{t_{\text{sup}}-t_{\text{ret}}}{2}-t_v \tag{8} \]

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

\[ \varphi=\frac{t_{\text{sup}}-t_{\text{ret}}}{t'_{\text{sup}}-t'_{\text{ret}}}. \tag{9} \]

Solving equations 7, 8, and 9 simultaneously, we obtain for \(t_{\text{sup}}\)

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

where \(\Delta t'_c=t'_{\text{sup}}-t'_{\text{ret}}\) is the design temperature difference of the heat carrier. Consequently,

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

In Fig. 1, according to equation (11), the dependence \(t_s=f(t_n)\) is shown for a particular example in which \(\Delta t=5^\circ\mathrm{C}\), \(t'_{2op}=130^\circ\mathrm{C}\), \(t'_{obr}=70^\circ\mathrm{C}\), \(t'_n=-40^\circ\mathrm{C}\). In the same Fig. 1 the dependence \(t_{2op}=f(t_n)\), 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 bleed. Within the limits from \(t_{n2}\) to \(t'_n\), the temperature \(t_s\) also does not vary and corresponds to the saturation temperature at the maximum pressure in the bleed.

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

From \(t_s\) we determine the pressure in the bleed for each \(t_n\) and the available heat drop to the bleed. In Fig. 1, for the previously analyzed example, the dependences \(P_{otb}=f(t_n)\) and \(h_0=f(t_n)\) are plotted for qualitative regulation. With quantitative regulation, both the pressure in the bleed and the heat drop to the bleed will be unchanged over the entire interval of variation of the outdoor-air temperature \(t_n\). In the same Fig. 1, the dependence \(P_{otb}=f(t_n)\) and \(h_0=f(t_n)\) for quantitative regulation for the same example is shown by dashed lines.

Using the graph in Fig. 1, formulas (2) and (4) make it possible to determine the change in the power generated by extraction steam, for qualitative and quantitative control, as a function of \(t_n\). Since for quantitative control \(h_o = const\) and \(t_k = const\), the dependence of \(N_m^n\) on \(t_n\) is expressed by the law of a straight line.

In Fig. 2 a graphical solution of equation (4) is given for qualitative and quantitative control at

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

From Fig. 2 it is seen that at all outdoor temperatures \(t_n > t_{\mathrm{CHP}}\), where \(t_{\mathrm{CHP}}\) is the design temperature of the CHP plant, \(N'_m > N''_m\). In Fig. 3, from the curves of Fig. 2, the dependence is plotted

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

Fig. 2

We proceed to determine the power expended on pumping water. Under qualitative control the amount of water \(G = const\); therefore the power, in kW, for pumping the heat-transfer medium is

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

Under quantitative control,

\[ N''_n = \frac{S_o a^3 (G^1)^3}{3.67 \cdot 10^5 \cdot \eta_n} \ \text{kW}. \tag{13} \]

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

\(\alpha = \dfrac{G}{G'}\) represents the ratio of the water flow rate at an arbitrary temperature \(t_n\) to the design quantity of water, and may be called the coefficient of relative water flow rate.

Fig. 3

Visible labels in the figure: vertical axis \(\Delta N_m; \Delta N_h\), kW; horizontal axis \(t_n^\circ\mathrm{C}\). Curves and annotations: “quantity regulation”; “quality regulation”; \(\Delta N_m = N_m' - N_m'' = f(t_n)\); \(\Delta N_h = N_h' - N_h'' = f(t_n)\).

The value of the coefficient \(\alpha\) can be determined from the following relations. For quantity regulation, the relative heat load is

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

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

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

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

The difference in the powers for pumping water under quality and quantity regulation will be

\[ \Delta N_h = N_h' - N_h'' = (1-\alpha^3)\cdot \frac{S_0 G^{\prime 3}} {3.67\cdot 10^5 \cdot \eta_h} \]

If \(\Delta N_t > \Delta N_h\), then the additional power generated for heat consumption under quality regulation is greater than the reduction in power for pumping water due to the decrease in the quantity of heat carrier under quantity regulation. Therefore, for these conditions, quality regulation is more advantageous. If \(\Delta N_t < \Delta N_h\), then quantity regulation is more advantageous, since in this case the reduction in power for pumping water pre-

gives an advantage over 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 considered 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 expedient 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 carried out. Usually the change in the quantity of water is stepwise in character, depending on the number and characteristics of the pumps. However, this does not change the conclusions obtained. The transition from one water flow rate to another is expedient only in the interval where \(\Delta N_{h}>\Delta N_{m}\).

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

1. The higher the design temperature of the CHP plant \(t_{\mathrm{CHP}}\), the wider the range of application of quantitative regulation, and vice versa. If \(t_{\mathrm{CHP}}=t'_{h}\), then, at 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.