The exit temperature TA of the fluid after flowing through a pipe of length l, diameter d and the heat transition coefficient k can be calculated for the static case on assumption of ideal piston current and neglect of the heat production by dissipation as follows:

In the unsteady state case the temperature at the exit from a pipe can be calculated at the time t if the entrance time t-s of the volume element leaving at the end of pipe and the entry temperature T of a volume element which entered from the pipe at the time t – s . κ can be determined:

k is a run time correction factor for the consideration of the heating and/or cooling of the pipe wall and isolation and can be calculated as follows:

with R : Index for material values/measures of the pipe,

I : Index for material values/measures of the isolation,

a, i: Indices for exterior, interior diameters.

In order to determine the speed process of the required entrance time t – s and the entry temperature T(t – s·k) using temporal backward integration, the calculated mass flows and node temperatures of earlier calculation times must be available.

For this, a list is maintained of volume elements which entered relative to the present calculation time for each pipe. For each volume element the entry temperature, the temperature gradient, the retention time and the distance of the pipe beginning are stored.

For each new calculation step a new volume element with the temperature and the temperature upward gradient of the entrance nodes of the pipe is set at the beginning of the volume element list. Since the new volume element has only just entered, the retention time and the place of the volume element are at first zero. All remaining volume elements in the list are moved on by the path travelled in the last calculation step, i.e. by the product of speed at the current calculation time and the time step size. At the same time the retention times of the volume elements are increased by the calculation step size.

The place will exceed the pipe length by at least one volume element. By interpolation between the last volume element whose place the pipe length has not yet exceeded, and the first volume element whose place has exceeded the pipe length, the retention time s of a volume element just at the pipe exit, can be determined.

To consider the heating and/or cooling of the pipe wall/isolation, the entry temperature for a volume element, which entered the pipe at the time t-κ·s is also determine by interpolation. With the interpolation of the entry temperatures, the temperature upward gradients of the corresponding volume elements are considered.

With reversal of the direction of flow during the calculation period the new element must be indexed at the end the list. All previous volume elements are moved backward around the path travelled in the calculation time step, although this time the place of at least one element becomes smaller than zero, i.e. the appropriate volume elements exit at the other side of the pipe. The exit temperature TE(t-k·s) and the retention time s for the calculation of the final temperature are also determined in the same way by interpolation from the retention times and entry temperatures of the volume elements.