Accuracy issues with the calculation of equivalent conductances and ScriptFij

Group-to-group equivalent conductances through each of the conductive mechanisms (radiative, conductive, and convective) are calculated from the heat flows between two groups with the following expression:

where

• G_BAtype is the equivalent conductance for a particular heat transfer mechanism (conductive, radiative, etc.).
• Q_BAtype is the heat flow through that mechanism.
• T_A, T_B are the average group temperatures for groups B and A.

A typical printout on the report log file looks like this:

                Group-to-Group Heat Flow Report:
                
                           Temp         Q             Cap      Description
                
                Groupi:    54.73   5.8073E+02    3.2361E+00    Group B
                Groupj:    25.00   1.0243E+02    1.0000E+30    Group A
                
                Type              Cond. value  Heat Flow i->j  ScriptFij  Black Body Vfij
                
                Conduction         0.0000E+00    0.0000E+00
                Radiation          1.9522E+01    5.8039E+02       0.85     1.000004E+00
                Linear Th. coupl.  1.1484E-02    3.4140E-01
                Total              1.9534E+01    5.8073E+02
        

A group’s temperature is calculated by averaging. Since this definition is arbitrary, and is only accurate if the thermal gradients within a group are small, and conductances are computed from group temperatures, the conductances will be accurate only if the gradients within a group are small. If the local gradients within a group are large, conductances are not accurate, it is even possible to get negative conductance values for positive heat flows.

ScriptFij is an alternate convention to the gray body view factor for defining the heat flow through a radiative conductance:

ScriptF_ij is a property that depends only on element surface properties and areas, not on the heat flows between groups. In the above output, ScriptF_ij is accurately calculated from optical and surface properties, not from heat flows between groups using the following algorithm:

where:

• T_jko is the temperature of the Oppenheim element of the k^th element of group j, calculated under the following conditions:
  • All elements in group i are set to their correct temperatures.
  • All other non-Oppenheim temperatures are set to absolute zero.
  • All non-Oppenheim temperatures are sinks.

• T_ko is the temperature of the Oppenheim element of the k^th element of the group that includes all elements except the elements of group i, calculated under the above conditions.

However, because of the arbitrariness of the method for calculating group temperatures, the above equation in general will not be match the radiative heat flow calculation between groups, unless thermal gradients within a group are small.