Operation of Looped District Heating Networks

The study presents a methodology for the optimal operation of district heating networks with a circular conduit system. The authors discuss this this topic in an absolutely general form. By a special statement and solution of Kirchhoff Laws, node equations and loop equations, the hydraulic end point of the circle is determined, including the supply ratio of the consumer located at the hydraulic end point. Two objective functions are stated by the authors; one of them for the minimum of flow work, and the other for the minimum of power supplied. It is demonstrated that the objective functions yield different results. Theoretically more economical operation is ensured by the flow pattern resulting from the minimization of the power supplied.


Objective functions and hydraulic equations
shows the topological model (graph) of the pipeline system of a district heating network with a circular conduit system. Consumer hot water flow demands are given and known: n  n  0 1  2  0  1  2 , , , , , , , * * * * . An optimal flow pattern is sought for, with a minimum dissipated energy and / or energy input or pump work. Thus, the problem is examined by solving two objective functions, using two models. In the first task, the objective function is to determine the dissipated energy minimum. This is equivalent to the flow pattern yielded by the solution of Kirchhoff Laws [1].
In the second task, Kirchhoff Law II is allowed to be infringed by placing two pumps at the input point -with right and left side feed. Then the aggregate figures of differential pressures on the right and left side branches are not necessarily equal. The pump works yielded in the two solutions are compared and their differences are evaluated. The network is opened up at the feed point in both models. Figure 2 shows the topology pattern and flow.

Flow pattern at the minimization of dissipated energy [1]
The problem is presented on the network of Fig. 2 by depicting the network of Fig. 1 in a general format, for an n number of consumers. The flow pattern can be sought for by solving the loop equation. In our assumption, the hydraulic end point in the flow is (0), from whose demand is supplied through branch n-0 and through branch 0-n* . Thus, the unknown quantity is proportional factor k.
The objective function: Conditional equations: a) node equation: By further rearrangement:  (1) Constants stated in a closed form: Quadratic equation using the signs: After rearrangement: The equation yields two solutions for proportional factor k; for the most part, only one of them has real physical content, this latter yielding flows  V 01 and  V 01 * and finally, flows  V n and  V n * . In view of these flows, pressure losses on conduit sections can be calculated and the pressure pattern of the network can be set up, simultaneously yielding feed differential pressure and pump delivery head figures. Application of the loop rule also results in the fact that the aggregate of pressure loss figures is equal along routes 0-n and 0-n* , and pumps operate at identical delivery heads.

Flow pattern in the minimization of input power. Separated pumping at the feed location
Let us examine if the quantity of power supplied can be decreased by separated pumping, by opening the loop at the location of pumping.
Objective function to express the minimum of pumping work: This objective function is of different structure compared to the function to express the search for minimum dissipated energy.
The minimum is placed where d dk C = 0 , and Where, using the earlier signs, Equation (8) with the signs introduced: After rearrangement, a quadratic equation is yielded for k: The factor k expressing the division ratio of volumetric flow  V 0 can be determined by the root formula. It can be observed that the expression to determine proportional factor k is not identical with expression (8), therefore results are also different. It is important to decide whether it is economical to invest in two pumps, that is, whether operating cost savings represent real yields in view of investment costs.

Example
Let us perform the hydraulic analysis of the looped district heating network shown in Fig. 3. Let us define the hydraulic end point of the network. Let us open the loop network into a radial network with two feed points. Our example presents calculations for the minimization of both dissipation energy and feed outputs.  Figure 4 illustrates the transformation of the network shown in Fig. 3 into a radial network with two input points.  Figure 4 shows the location of the 11 consumers along the mains conduit with two input points yielded after cutting the loop. The data required for performing the hydraulic analysis -name of section, length of section, caloric output, mass flow and standard pipe diameter -are shown in Table 1. Starting from the presumed end point towards the two inputs produced by cutting, end point consumer demand is supplied in a proportion kV  0 from one direction, and in a proportion 1 0 − ( ) k V  from the opposing direction. The presumed hydraulic end point is consumer 5. For reasons of expanse, results for consumers 4 and 6 as presumed hydraulic end points are not included herein. They represent figures higher than one and lower than zero, respectively, as expected.
Select consumer caloric center 5 as a hydraulic end point.  By substituting the constants yielded in the quadratic correlation as described: This result already corresponds to the expected solution; however, it cannot be stated clearly that this should be the end point of the system before verification thereof by calculations performed for further points. Table 2 shows pressure values at each node and at the feed point. It can be observed that the feed pressure figures required, as calculated in the two directions, are in agreement. It can be observed that the aggregate of the differential pressure figures calculated from the two directions -that is, pump delivery head figures -agree. Δp sz = 5.5920 bar.
The The pressure pattern developed in the network is shown in Fig. 6.   Fig. 6 Illustration of the presumed hydraulic end point A pressure of at least 3 bar must be ensured at pump intake in order to avoid cavitation. Node pressures were modified accordingly. The red line shows pressure in the forward conduit, and the blue line indicates pressure in the return conduit. Green lines represent pressure drops in consumer branches, and black lines are dampers in consumer branches, required to reach the design pressure at each node.

Determination of the flow pattern with the minimum of energy input
The calculation principle of the basic data (heat power, mass flow, volumetric flow) coincides with the one presented for flow work minimization. In the calculation based on minimum energy input, the hydraulic end point came to be consumer 7. The data used in the calculation are shown in Table 3. When selecting consumer caloric center 7 as a hydraulic end point, the volumetric flow is  V . The figures of volumetric flows, resistance factors, and differential pressures developed along each section are included in Table 4. The correlations used correspond to those described earlier. On the basis thereof, the value of k can be calculated. k = 0.1208 In case of other nodes, figures lower than 0 or higher than 1 are yielded for k on the basis of this calculation principle as well, therefore only consumer 7 can be the hydraulic end point. Table 5 contains the values of k in case of different presumed hydraulic end points. As regards the calculation principle, it is allowable to infringe Kirchhoff Laws, meaning that the aggregate figures of differential pressures calculated in the two directions are not required to be identical.
Figures of design pressure at each node as well as pump feed differential pressure are included in Table 6.  By breaking up the loop at the pump, the network was converted into a two-feed radial network. In case of an appropriate design, after the correct selection of the hydraulic end point, the pressure pattern would be characterized by the fact that progressing along the mains, the pressure drop of the incoming consumer branch is always smaller at each node, meaning that damping is required on these branches. In this case, however, the design differential pressure at nodes IV, VIII and IX is the pressure drop of the consumer branch. Figure 7 illustrates the differential pressures available along the mains and their values required at critical nodes. The pressure pattern drawn on the mains had to be modified in accordance therewith, meaning that the feed differential pressure had to be increased. In addition to those listed above, critical nodes also include node x because a greater one would be necessary than what is available on the mains. It follows from this that the network was designed improperly. Pipe diameters are not large enough, therefore very high flow rates are produced. Consequently, pressure drops are too large on consumer branches.
In the diagram, light green lines indicate the differential pressures required at critical nodes. Thin red and blue lines show values available between the forward and return sections of the mains, while thick lines show the actual state established.  Figure 8 shows the actual pressure pattern, pressure drops at each consumer branch, and the damping required. The results yielded by the two calculation methods can be compared with the power absorbed by the pumps. P dissipated = 132.67 kW, P input = 108.04 kW.
Output reduction by the minimization of energy input can be calculated as compared to the one calculated for the minimum of dissipated energy. It can be shown that a much more cost-effective operating state can be achieved by minimizing work input than by minimizing flow work. So it is worthwhile to break up the loop at the feed point and to apply two pumps instead of one. Obviously, these savings must be compared to the additional investment cost of the installation of two pumps.

Conclusion
Our study presented a hydraulic analysis method for district heating networks of a circular conduit system with given consumer volumetric flow demands, for both the dissipated energy minimum and the input energy minimum. After stating and arranging a loop equation and the node equations (Kirchhoff Laws I and II), the result is a quadratic equation for the distribution of consumer volumetric flow at the hydraulic end point. This is indicated by the so-called k factor, which is a figure lower than 1. Adoption of the procedure was presented in an example for each of the minimum of dissipated energy and pump work. By comparison of the results yielded, it was demonstrated that circular conduit operation by separated pumping is energetically more advantageous. Obviously, the issue to be examined is whether the installation of two pumping stations represents a better solution in respect of investment costs.