From an energy balance around a node ki the following calculation equation is received on the assumption of an immediate and complete mixing of the entering material flow mij for the node temperature Ti:

with mij : mass flow from the node j to the node i

TijE : node entry temperature of the pipe between the node i and j

Using Gl. 20.39 it is possible to determine the temperature of a node as a function of the temperature of the entering flows. However, the entry temperature of these mass flows into the connections and thus the temperature of the neighbor nodes, from which the node which is to be calculated is fed, must first be known. In order to carry out a thermal calculation of the entire network, a certain calculation sequence of the nodes must therefore be selected. This must guarantee that every nodes to be calculated is only flowed into from nodes that have already been calculated.

• The flow temperature is usually specified by a given heating curve as a function of the outside temperature.
  • In graph theory, such a sequence of the nodes of a graph is called topological order. The production of such a calculation sequence for the node temperatures that are being searched for in a long-distance heating network is possible only if both no thermal meshes are present (i.e., if no circular currents of water are led in the flow or return) and all feeding temperatures are known.
  • If at least one of the two conditions is not fulfilled, then instead of the gradual calculation of the node temperatures, a closed thermal calculation method developed by Icking can also be used, which requires, however, the list of a complex and large set of equations. In addition, each change to the direction of flow of a pipe requires a reconstruction of this set of equations.
  • The entry temperatures TijE Gl. 20.39 can be determined depending on the kind of equipment component between the nodes i and j from the calculation equations Gl. 20.31, Gl. 20.34 and Gl. 20.38.