Heat loss

Three types of heatlosses in the pipeline are considered, in accordance to the NEN-EN 13941+A1. These are heat loss through:

  • Tube wall

  • Subsoil

  • Though neighboring pipeline

Visualized schematically, these are heat losses/fluxes as follows:

../_images/pipeline_heatloss.png

If we write it in a set of equations (which can be formulated as constraints), we get the following equation for the temperature loss inside a pipe from its in- to its outport:

(1)\[{\left( {{c_p}\dot mT} \right)_{in}} - {\left( {{c_p}\dot mT} \right)_{out}} = {Q_{loss}}\]

In which the heat loss \(Q_{loss}\) is equal to:

(2)\[{Q_{loss}} = L\left( {{U_1} - {U_2}} \right)\left( {{T_h} - {T_g}} \right) + L{U_2}\left( {{T_h} - {T_c}} \right)\]

In which:

\(L\): Length of pipeline [m]

\(T_h\): Temperature in hot (feed) pipeline [K]

\(T_c\): Temperature in cold (return) pipeline [K]

\(T_g\): Temperature at ground temperature [K]

The values for \(U_1\) and \(U_2\) follow from the following set of equations:

(3)\[{U_1} = \frac{{{R_g} + {R_{iso}}}}{{{{\left( {{R_g} + {R_{iso}}} \right)}^2} - R_m^2}}\]
(4)\[{U_2} = \frac{{{R_m}}}{{{{\left( {{R_g} + {R_{iso}}} \right)}^2} - R_m^2}}\]

In which:

\(R_g\): Subsoil heat resistance [mK/W]

\(R_{iso}\): Insulation heat resistance [mK/W]

\(R_m\): Heat resistance due to neighboring pipeline [mK/W]

As the description above shows, \(U_1\) and \(U_2\) are constant values based on type, placement en dimensions of the pipelines.

For the MILP formulation of HeatMixin, the hot and cold temperature lines are fixed. The heat loss of a pipe in the MILP formulation is therefore not dependent on the flow rate. For the NLP formulation of QTHMixin, the temperature of a pipe is the average of its in- and outgoing temperatures. This means that the heat loss is dependent on flow rate.