2.1.1. Flow Mixing and Splitting

When, in an arbitrary mesh node, streams *q*1, *q*2, ... , *qm* are mixed into a single stream *j*, we can write

$$\sum\_{q \in \{q\_1, q\_2, \dots, q\_m\}} \dot{m}\_q \mathbf{c}\_{\mathbf{P}, \mathbf{q}} \{T\_j - T\_q\} = 0. \tag{1}$$

Here, . *mq* denotes the mass flow rate of the *q*th stream, *c*p,*<sup>q</sup>* the specific heat capacity, and *Tj* and *Tq* the corresponding stream temperatures. Each specific heat capacity should be taken as the mean value obtained for the corresponding temperature range [*Tj*, *Tq*].

If, on the other hand, a single stream *j* is split into streams *r*1, *r*2, ... , *rn*, the outflow temperature is the same for all these streams, and the respective *n* equations are

$$T\_r = T\_{\bar{\jmath}}, \ r \in \{r\_1, r\_2, \dots, r\_n\}. \tag{2}$$

In some systems, there may be blind edges with zero mass flow rate. The temperatures in the nodes of these edges are calculated as if the edges were of the regular type featuring outflow (see also the schematic in Figure 4).

**Figure 4.** Internal flows in a system vs. flow mixing; there are 4 locations (the T-joints) for which the flow mixing/splitting equations must be present in the final linear system.

In a general case with streams *q*1, *q*2, ... , *qm* being mixed and then split into streams *r*1, *r*2, ... , *rn*, one will get one Equation (1) governing the resulting outflow temperature *Tj* and (*n* − 1) Equation (2), that is, (*n* − 1) identities for the remaining outflow temperatures. The total number of equations governing the mixing/splitting in the node will, therefore, be equal to number of outflow streams.
