PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV
Vol. 66, No. 2, 1948

CALCULATION OF HORIZONTAL IMBALANCE IN WATER HEAT NETWORKS

V. T. YURINSKY

By horizontal imbalance of a water heat network one should understand changes in flow distribution as compared with the normal one, manifested as a result of switchovers in the network, disconnection of individual consumers, connection of new consumers, transition to emergency supply schemes, etc. At the same time, in all sections of the network that remain connected, the pressure-flow pattern is assumed to be preserved, i.e., any section may be considered as a pipeline filled with water.

The problem of calculating the horizontal imbalance of a network has to be solved rather often in design and operating practice. In the present work, its solution is considered only as applied to radial networks with one source producing the acting head. The method set forth may also be extended to looped networks, provided that the looped network can be decomposed into several radial ones.

The basis of the method is formed by new concepts concerning the types of horizontal imbalance and the concepts of the reduced resistance of a branched network and its parts.

Reduced Resistance of a Network

Let there be a branched radial heat network (Fig. 1). For any section of this heat network one may write

\[ \Delta H_i = s_i G_i^2, \tag{1} \]

where \(\Delta H_i\)—head loss in the section . . . . . m w.c.,
\(\qquad G_i\)—water flow rate in the section . . . . . t/hour,
\(\qquad s_i\)—resistance of the section . . . . . h\(^2\)/m\(^2\).

The water flow rate in any of the sections may be expressed as a fraction of the flow rate in the main section, i.e., for each of the sections one may write the relation

\[ G_i = x_i G_0. \tag{2} \]

Here \(G_0\)—water flow rate in the main section,
\(\qquad x_i\)—fractional coefficient of the given section.

From equations (1), (2) we obtain:

\[ \Delta H_i = s_i x_i^2 G_0^2. \tag{3} \]

The total head loss along any of the circulation loops of the heat carrier is equal to the head \(H_n\) produced by the network pump. Consequently,

\[ \left. \begin{aligned} H_n &= \Delta H_0 + \Delta H_1 + \ldots + \Delta H_8 \\ \cdot\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \\ H_n &= \Delta H_0 + \Delta H_{10} \end{aligned} \right\} \tag{4} \]

After substitution from (3),

\[ \left. \begin{aligned} H_n &= s_0\left(1+\frac{s_1}{s_0}x_1^2+\ldots+\frac{s_8}{s_0}x_8^2\right)G_0^2 \\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \\ H_n &= s_0\left(1+\frac{s_{10}}{s_0}x_{10}^2\right)G_0^2, \end{aligned} \right\} \tag{5} \]

The quantity

\[ k_0=\left(1+\frac{s_1}{s_0}x_1^2+\ldots+\frac{s_8}{s_0}x_8^2\right)=\ldots=1+\frac{s_{10}}{s_0}x_{10}^2 \tag{6} \]

we shall call the reduction coefficient.

Then, for a branched network, instead of the group of equations (5) one may write

\[ H_n=k_0s_0G_0^2=sG_0^2. \tag{7} \]

In this equation, \(s=k_0s_0\) is the resistance of the branched network reduced to its head section. As we see, a complex branched network can be replaced by a simple one having a single section. The water flow rate in this simple network is equal to the flow rate in the head section of the branched network; the resistance of the simple network is equal to the resistance \(s_0\) of the head section of the branched network, multiplied by the reduction coefficient \(k_0\).

Expression (7) is the equation of a parabola passing through the origin of the coordinates \(H, G_0\). For given values of \(s_0, s_1,\ldots\), the reduction coefficient and the reduced resistance of the network have quite definite values.

The reduction coefficient depends on the hydraulic characteristics of the sections and on the fractional distribution of flow in the network, i.e., it is determined by the network scheme at the given moment. All kinds of switching operations in the network affect, through a change in the reduction coefficient, the value \(s\) of the reduced resistance of the network, which in the coordinates \(H, G_0\) is expressed by a steeper or flatter position of the reduced characteristic of the simple network replacing the branched one. Having found the value of \(s\), one can construct the resistance parabola; the point of intersection of this curve with the characteristic of the network pump will make it possible to determine the total head loss in the network and the water flow rate in the head section.

It is not difficult to see that for the reduction coefficient one may also write another expression,

\[ k_0=\left(1+\frac{\Delta H_1}{\Delta H_0}+\ldots+\frac{\Delta H_8}{\Delta H_0}\right)=\ldots=1+\frac{\Delta H_{10}}{\Delta H_0}. \]

The latter form of the equation for determining \(k_0\) is sometimes more convenient.

In a number of cases it may be necessary to replace some branched part of the network by a simple network. For example, with reference to Fig. 1, one may replace the entire part of the network located to the right of node IV by a simple network consisting of a single section. We may speak of a partial reduced resistance, partial reduction coefficients, and partial fractional coefficients. For the part of the network located to the right of node IV (Fig. 1), one may write

\[ H_{1V}=s_4\left(1+\frac{s_5}{s_4}y_5^2+\ldots+\frac{s_8}{s_4}y_8^2\right)G_4^2= \]

\[ =\ldots=s_4\left(1+\frac{s_{50}}{s_4}y_{50}^2\right)G_4^2=k_4s_4G_4^2. \]

Here \(k_4\) is a partial reduction coefficient,

\[ y_5=\frac{G_5}{G_4}, \]

\[ y_8=\frac{G_8}{G_4}, \]

\[ \left\}\ \text{partial fractional coefficients.}\right. \]

Fig. 1.

The concepts of partial reduction coefficients and partial fractional coefficients are necessary for calculating horizontal maladjustment in highly branched networks.

Types of horizontal maladjustment

Suppose that, as a result of certain switching operations in the network, the water flow rates in all sections without exception have changed, and moreover in such a way that, in the terminal sections, the deviations have occurred in one and the same direction (for example, toward an increase). A maladjustment of this kind may be called coordinated. If, during switching operations, the flow rates in some terminal sections, deviating from their normal values, increase, while the flow rates in other terminal sections, on the contrary, decrease, then the network is in a state of uncoordinated maladjustment.

A typical example of the manifestation of coordinated maladjustment in a network is the behavior of a branched network when one or several consumers are completely disconnected. In this case, the water flow rates in all terminal sections that remain connected change in the direction of an increase in their absolute values. A state of uncoordinated maladjustment will be observed in the network when the water supply to some consumers is partially throttled. In the systems of throttled consumers the water flow rates will decrease; in the remaining systems the water flow rates will increase.

In the present work, the calculation of coordinated maladjustment is considered, since it is precisely this type of maladjustment that is usually meant when calculations for start-up regulation are required, when determining the “extreme” positions of piezometric lines, in calculations for emergency operating conditions, etc. Uncoordinated maladjustment can be calculated by the same method, but provided that additional data are available on the change in the resistances of the throttled sections.

With coordinated maladjustment of a network, there may be two varieties of it: proportional and nonproportional.

Under proportional deregulation, for any pair of completely arbitrarily chosen sections, the constant ratio of the water flow rates in these sections is preserved. If before the disturbing action was introduced into the operation of the network the flow rates in the sections were \(G_0, G_1, G_2,\ldots\), and after the introduction of the disturbing action they became \(G_0', G_1', G_2',\ldots\), then

\[ \frac{G_1'}{G_2'}=\frac{G_1}{G_2};\quad \frac{G_0'}{G_2'}=\frac{G_0}{G_2}\quad \text{and so on.} \]

Consequently, under proportional deregulation the flow rates \(G_i\) change in absolute magnitude, but the values of the share coefficients \(x_i\) are preserved. Under nonproportional deregulation, however, not only do the values of \(G_i\) deviate from the calculated values, but at the same time the share coefficients \(x_i\) also change.

Proportional deregulation in a network arises in the case when only the head at the station node changes, with no switching in the network. In this case the share flow distribution in the network does not change. Therefore, knowing the share coefficients for the normal value of the head of the network pump, one may limit oneself to finding only the new value \(G_0'\) of the water flow rate in the main section in order to determine the water flow rates in all the remaining sections of the branched network. The reduction coefficient \(k_0\) in this case does not change, which leads, as is easy to see, to the well-known simplest relation

\[ \frac{H_n'}{H_n}=\left(\frac{G_0'}{G_0}\right)^2=A^2, \]

where \(A\) is the coefficient of water flow rate in the network.

A state of proportional deregulation in the network may also arise if the resistances in all sections are changed simultaneously, and in such a measure that the share coefficients do not change. If only some subscribers or groups of subscribers are disconnected, then the network enters a state of nonproportional deregulation, characterized by a change in the share coefficients \(x_i\). Nevertheless, even in this case it is possible to find zones of partial proportional deregulation in the network. In these zones the values of the partial share coefficients \(y\), expressing the flow rate in a given section as a fraction of the flow rate in some section of the network zone under consideration, will be preserved. Therefore, for a zone of partial proportional deregulation, it is sufficient to find the water flow rate in its main section. After that it will not be difficult to find the flow rates in all the remaining sections of this network zone.

Calculation of Horizontal Deregulation

We shall show, by concrete examples, the calculation method based on the ideas formulated above.

Example 1. A network is given, represented in the form of the calculation diagram in Fig. 1. For the calculation diagram of the network, assuming that all branches are included, i.e., for the normal flow distribution in the network, the following are known:

flow rates in all sections \(G_0, G_1, G_2,\ldots\),

share coefficients \(x_0=1, x_1, x_2,\ldots\),

available heads at the nodes \(H_{st}, H_I, H_{II},\ldots\),

head losses in the sections \(\Delta H_0, \Delta H_1, \Delta H_2,\ldots\),

resistances of the sections \(s_0, s_1, s_2,\ldots\).

It is required to find the flow distribution in the network when branch \(40\) is disconnected.

After branch 40 is disconnected, both the absolute values of the flows and the share coefficients on all sections change. Only on the head section will the share coefficient retain its value \(x'_0=x_0=1\).

In the zone of the network to the right of node IV, a partial proportional deregulation will occur. The entire network will be deregulated consistently, nonproportionally. The heads at all nodes, including node IV, will change.

Let us make the deliberately incorrect assumption that, after branch 40 is disconnected, the head at node IV will remain unchanged,

\[ H'_{IV}=H_{IV}. \]

Under this assumption, the flows on all sections in the zone of the network located to the right of node IV will remain unchanged not only with respect to the partial share distribution, but also in absolute magnitude. Therefore, without considering this part of the network, we may state that on the head section 4 the water flow after branch 40 is disconnected is \(G'_4=G_4\). The quantity \(G_4\) is known for the normal flow distribution.

For section 3 (Fig. 1)

\[ G'_3=G'_4=G_4, \]

\[ \Delta H'_3=s_3G_4'^2 \]

Next,

\[ H'_{III}=H'_{IV}+\Delta H'_3=H_{IV}+\Delta H'_3, \]

\[ G'_{30}=\sqrt{\frac{H'_{III}}{s_{30}}}, \]

\[ G'_2=G'_{30}+G'_3, \]

\[ \Delta H'_2=s_2G_2'^2, \]

\[ H'_{II}=H'_{III}+\Delta H'_2, \]

\[ G'_{20}=\sqrt{\frac{H'_{II}}{s_{20}}}, \]

then, analogously, we determine

\[ G'_1,\Delta H'_1,H'_I,G'_{10},G'_0,\Delta H'_0 \text{ and } H'_{\text{st}}. \]

Thus, by successively solving the simplest equations, we determine \(H'_{\text{st}}\), i.e., the head at the station node under the condition that at node IV the same head is maintained, \(H'_{IV}=H_{IV}\), as before branch 40 was disconnected. At the same time, the absolute flows \(G'_0,G'_1,G'_2\) on all sections will be determined.

Knowing the flows on all sections, we find the share coefficients of all sections for the network diagram with branch 40 disconnected.

\[ x'_1=\frac{G'_1}{G'_0}, \]

\[ x'_2=\frac{G'_2}{G'_0}, \]

\[ \ldots \]

The deliberately false assumption that at node IV, after branch 40 is disconnected, the head \(H'_{IV}=H_{IV}\) will be preserved made it possible to find the share flow distribution in the network and the absolute flows on the sections; moreover, at the station node, in order to provide these values of absolute flows, there must exist a head \(H'_{\text{st}}\). In fact, at the station node

we shall have the head \(H_{cm}^{\mathrm{act}}\). As a result, in node IV the available head will be \(H_{IV}^{\mathrm{act}} \ne H_{IV}\). Accordingly, the heads in all the remaining nodes will change, and the flows in the sections will be \(G_0^{\mathrm{act}}, G_1^{\mathrm{act}}, G_2^{\mathrm{act}}, \ldots\). But the fractional distribution of flow in the network will not change when the head in the station node is changed from \(H'_{cm}\) to \(H_{cm}^{\mathrm{act}}\), since in the network with branch 40 switched off this sole action will cause a proportional deregulation. Consequently, the reduced resistance \(s'\) of the network will not change. Determine the value of \(s'\)

\[ s' = s_0 \left( 1 + \frac{s_{10}}{s_0} x_{10}'{}^{2} \right). \]

Let the value \(H_{cm}^{\mathrm{act}}\) be specified at the station node. Then the true water flow in the head section is

\[ G_0^{\mathrm{act}} = \sqrt{\frac{H_{cm}^{\mathrm{act}}}{s'}} . \]

Knowing the flow in the head section, we shall at once find the flows in all the remaining sections

\[ \begin{aligned} G_1^{\mathrm{act}} &= x_1' G_0^{\mathrm{act}},\\ G_2^{\mathrm{act}} &= x_2' G_0^{\mathrm{act}},\\ &\ldots\ldots \end{aligned} \]

After this it will not be difficult, from the system of equations

\[ \begin{aligned} \Delta H_0^{\mathrm{act}} &= s_0 (G_0^{\mathrm{act}})^2,\\ \Delta H_1^{\mathrm{act}} &= s (G_1^{\mathrm{act}})^2,\\ &\ldots\ldots \end{aligned} \]

to find the head losses in all sections and the available heads at the nodes, i.e., all the data necessary for constructing the piezometric lines.

Usually, in all existing calculation methods it is necessary to make the assumption that an unchanged head is maintained at the station node.

The calculation method considered above makes it possible to dispense with this assumption. The problem can also be solved in the case when no regulation of the network pump is performed. For this it is necessary to resort to a generally known graphical construction. In Fig. 2 the \(G\)—\(H\) characteristic of the pump is plotted (curve 1). In the normal network scheme (all branches are switched on), the operating point is point \(B\), belonging to one-

simultaneously with curve 2 (the reduced characteristic of the network in the design operating condition) and the characteristic of pump 1. Curve 2 is constructed from the equation

\[ H=sG_0^2=s_0\left(1+\frac{s_{10}}{s_0}x_{10}^2\right)G_0^2, \]

When branch 40 is disconnected, the reduced characteristic of the network will be curve 3. This curve is easily constructed from the equation

\[ H=s'G_0^2=s_0\left(1+\frac{s_{10}}{s_0}x_{10}^{\prime\,2}\right)G_0^2. \]

The intersection of this curve with the pump characteristic (point \(B_1\)) makes it possible to find the values \(H_{\text{st}}^{\text{int}}\), \(G_0^{\text{int}}\).

Example 2. A network according to Fig. 3 is given. For the normal operating condition, when all consumers are connected, the flows in all sections, the head losses in the sections, and the nodal heads are known.

It is required to find the flow distribution in the network when consumers 60 and 303 are completely disconnected.

Since the normal flows and head losses are specified for all sections, the resistances of the sections \(s_0, s_1, s_2 \ldots\) are given.

Let us consider the network as a whole. After consumers 60 and 303 are disconnected, the network will undergo coordinated nonproportional deregulation. However,

in it two zones can be distinguished: 1) to the right of node IV; 2) downward from node XXXIII, in which partial proportional deregulation will take place. This circumstance can also be used.

Let us make the deliberately false assumption that, at node VI, after consumers 60 and 303 have been disconnected, the head will remain unchanged. Then, in the zone of the network located to the right of node VI, the pattern of distribution of flows and heads will not change. Carrying out the calculation by the method set forth above, we determine the magnitude of the head at node III that must exist in order to preserve at node VI the same head \(H'_{VI}=H_{VI}\) as in the normal network scheme. This calculation will make it possible to find the absolute flows \(G'_3, G'_4, G'_{40}\ldots\) in all sections of the network to the right of node III, as well as the partial fractional coefficients

\[ y_3'=\frac{G_3'}{G_3}=1;\quad y_4'=\frac{G_4'}{G_3'}\quad \text{etc.} \]

Suppose that, as a result of the calculation, the head at node III has been determined as \(H'_{III}\). Let us now assume that at node XXXIII, after disconnection of consumers 60, 303, the value of the head corresponding to the normal network scheme is also preserved. Proceeding from this, one may regard as unchanged the pattern of distribution of flows and heads in the zone of the network located below node XXXIII, and find that value of the head at node III—\(H''_{III}\)—which corresponds to the assumption made.

In the course of the calculation the absolute values of the flows \(G''_{30}, G''_{31}, G''_{301}, \ldots\) will be found on all sections of branch III–34. The partial share coefficients may also be determined:

\[ y''_{30}=1;\quad y''_{31}=\frac{G''_{31}}{G''_{30}};\quad y''_{301}=\frac{G''_{301}}{G''_{30}},\ldots \]

Obviously:

\[ H'_{III}\ne H''_{III}.^{1)} \]

This, however, cannot be so, since node III simultaneously belongs to both of the network zones considered. Hence the assumption of the simultaneous preservation of head at the nodes where consumers 60, 303 are disconnected is incorrect. If the head at node VI remains unchanged, then the head at node XXXIII will change. In this case at node III there will be a head \(H'_{III}\) identical for both branches. Conversely, if the unchanged head is preserved at node XXXIII, then the head at node VI must change; correspondingly, the head at node III will be \(H''_{III}\). Consequently, for one of the two branches the absolute values of the flows in the sections have been found incorrectly. But the ratio of flows, i.e. the partial share flow distribution for either of the network branches considered, has been established correctly, since a change in the head at node III causes in these branches only a proportional deregulation.

Let the head at node III be \(H'_{III}\). Obviously, in this case the water flow in section 30 of branch III–34 will change; its value will be not \(G''_{30}\), but \(G'_{30}\). To determine \(G'_{30}\) we write the equation

\[ H'_{III}=s_{30}\left(1+\frac{s_{301}}{s_{30}}\,y_{301}^{\prime\prime\,2}\right)G_{30}^{\prime\,2}. \]

In the equation written, \(s_{30}, s_{301}, y''_{301}, H'_{III}\) are known, and therefore \(G'_{30}\) can be found from it.

Next we find

\[ G'_{301}=y''_{301}\cdot G'_{30}, \]

\[ G'_{31}=y''_{31}\cdot G'_{30} \]

\[ \ldots\ldots\ldots\ldots \]

The operations performed with respect to branch III–34 may be called reduction to the same head at the common node.

The subsequent course of the calculation is as follows:

Knowing \(G'_{30}, G'_{3}\), we determine \(G'_{2}\) and successively find

\[ \Delta H'_{2}, H'_{II}, G'_{20}, \ldots\ldots G'_{0}, H'_{st}. \]

After this we determine the values of the share coefficients

\[ x'_{0}=1;\quad x'_{1}=\frac{G'_{1}}{G'_{0}};\quad x'_{2}=\frac{G'_{2}}{G'_{0}}\quad \text{and so on.} \]

\(^{1)}\) A coincidental equality is possible.

These fractional coefficients will not change when the head in the station node is changed. If the station maintains the head \(H_{st}^{act}\), then it is easy to find the true value of the absolute flow rate \(G_0^{act}\) in the head section from the equation

\[ H_{st}^{act} = s_0 k_0' \left(G_0^{act}\right)^2; \]

where

\[ k_0' = 1 + \frac{s_{10}}{s_0}\,x_{10}^{2}. \]

The calculation procedure also does not change in calculations for horizontal regulation in highly branched radial networks. The calculation operations retain their simplicity, but their number increases in accordance with the complexity of the diagram being calculated.

Let, for example, the network shown in Fig. 4 be given. It is required to find the new flow distribution when the consumers marked with crosses are disconnected. Let us separate by a dashed line the zones of partial proportional deregulation. We shall make the assumption that the heads in nodes III, IV,

Fig. 4

The disconnected branches are marked with a cross.

V, VII, VIII remain unchanged. By successive operations we shall bring all branches of the network to the same head in node I. We shall determine the head \(H_{st}'\) in the station node and the absolute water flow rates in all sections. We shall find the fractional coefficients. We shall determine the reduction coefficient \(k_0'\) and the reduced resistance \(s_0'\) of the network. From the specified value \(H_{st}^{act}\), or graphically, on the basis of the \(G-H\) characteristic of the pump, we shall determine \(G_0^{act}\) and then the distribution of water over the sections.

At the initial and intermediate stages of the calculation, several calculators may participate simultaneously.

Extension of the Concepts of the Network Characteristic and the Pump Characteristic

By the network characteristic is usually understood a curve of the form \(H = sG^2\). It was shown above that, as applied to a branched network, the network characteristic is represented by the curve \(H = k_0 s_0 G_0^2 = sG_0^2\). Depending on the nature of switching operations in the network, the value \(s\) of the reduced resistance of the network changes, i.e., the position of the resistance curve in the coordinates \(G_0-H\) changes. If the valves on all branches of the network are completely open, then the value \(s\) is minimal, and the resistance parabola is located most gently (Fig. 2). When the valves on any of the branches are throttled,

...there occurs a change in \(s\) both due to a change in the fractional flow distribution and due to a change in the resistances in individual sections. The resistance curve of the network changes its parameter and, as it were, slides in the coordinate system \(G_0,H\). At the limit it may coincide with the \(H\) axis. But at any moment there exists a single instantaneous characteristic of the network, which is a parabola whose position is determined by the value \(s'\). From these considerations, the network characteristic should be understood not as a curve and not as a family of curves, but as an area located above the curve \(H=s_{\min}G_0^2\). This area may be regarded as the trace of the motion of the network resistance curve as it is displaced from one limiting position (the ordinate axis) to another \((s=s_{\min})\). Similarly, bearing in mind the possibility of throttling the pump by a gate valve, the pump characteristic should most properly be understood as the area bounded by the coordinate axes and by the limiting \(G-H\) characteristic of the pump. The area indicated by hatching in Fig. 2 represents the possible field of placement of operating points, characterizing the joint operation of the pump with the network, obtained by superposing the network and pump characteristics on one another. If we regard each point of this area as the point of intersection of the instantaneous throttling curve of the pump with the instantaneous reduced characteristic of the network, we can solve the problem of finding the flow distribution in the network under complete disconnections of certain consumers on the basis of the readings of instruments installed at the station. This is possible, of course, if the values \(s_i\) are known for all sections.

Let, for example, the design scheme of the network correspond to Fig. 1. Let consumer 40 be disconnected. Let it be known (Fig. 2) that the total water flow in the network is equal to \(G_0^{\mathrm{meas}}\), and that the available head at the station node (downstream of the pump gate valve) is \(H_{\mathrm{st}}^{\mathrm{meas}}\). The operating point is point \(R\). We write

\[ H_{\mathrm{st}}^{\mathrm{meas}}=s_0 k_0' \left(G_0^{\mathrm{meas}}\right)^2, \]

From this we find the value \(k_0'\). From the expression

\[ k_0'=1+\frac{s_{10}}{s_0}\,x_{10}^{\prime 2} \]

we determine \(x'_{10}\).

Next,

\[ G_{10}'=x_{10}'\cdot G_0^{\mathrm{meas}}, \]

\[ x_1'=1-x_{10}', \]

\[ k_0'=1+\frac{s_1}{s_0}\,x_1^{\prime 2}+\frac{s_{20}}{s_0}\,x_{20}^{\prime 2}. \]

From the last expression we find \(x_{20}'\), and then \(G_{20}'=x_{20}'\cdot G_0^{\mathrm{meas}}\). The further course of the calculation requires no explanation.

The method described above for calculating horizontal deregulation in networks may be applied in a number of cases of heat-engineering practice. A necessary prerequisite for the correctness of the calculation constructions is the absence in the network of pump “boosters” with an independent source of energy supply (centrifugal pumps, propeller pumps of substations). With ejector mixers at consumer inlets, the method of calculation set forth remains valid.

Noted Misprints