MEREL module information

Automatic Combination

The MEREL module automatically combines all parallel radiative and conductive conductances, and sums all capacitances and heat loads.

Element Merging

Element merging creates a simplified thermal model by merging elements whose temperatures are expected to be similar. Element merging is performed for the following elements:

Elements specified on Card 7
Elements connected with a linear conductance value ≥1.E10
Elements connected by the Card 6e NEARM option
Elements for which an infinite or negative conductance was calculated by the COND module
Reverse side elements created with the Card 9 REVNODE option
Elements created by the Card 5d SPACE

All conductances except the REVNODE and SPACE card follower conductances (MNM = FOL) are written in the MODLCF file to recover the merged element temperatures.

If the Card 9 PARAM NOMRECOV option is specified, no follower conductances are created for the merged elements.

Element merging is identical to element renumbering. When j is merged to i, all references to j are replaced by a reference to i, and parallel linear and radiative conductances, capacitances, and heat loads are combined.

The elements of non-linear conductances, for example hydraulic resistances and convective conductances, are renumbered, but they are not added in parallel with each other.

Elements are renumbered in all references to Card 9 INTERP, PHASE, SINK element, and THERMST cards.

The following type of elements may not be merged with other elements:

Card 5e hydraulic elements
Card 9 MCV elements

However, other elements may be merged to them.

Substructuring

After element merging, substructuring is performed if cards 8 are present or a Card 9 PARAM SUBSTR is present.

Substructuring reduces the thermal model to a smaller mathematically equivalent one by eliminating specified elements. Their temperatures are recovered at each printout interval.

First, a list of elements to be eliminated is created:

Sink elements, MCV elements, phase change elements, hydraulic elements, elements referenced on INTERP or THERMST cards, and elements connected with one-way or non-linear convective conductances are not eliminated.
The elements T1 specified on the PARAM SUBSTR card are not eliminated.
On PARAM SUBSTR GSUM cards elements with Gksum > T3 are eliminated. Gksum is the conductance sum of k.
On PARAM SUBSTR T2 RCMIN card, elements with capacitance/Gksum < T3 are eliminated, including zero-capacitance elements.
On PARAM SUBSTR T2 CMIN cards elements with capacitance values < T3, including zero-capacitance elements, are eliminated.
Card 8 elements are eliminated.
If the PARAM SUBSTR RADNODES option or the equivalent PARAM HYBRID option is used, all elements with radiative conductances are not eliminated.
In case of conflict, the element is not eliminated.

Next, the total conductance G_ksum for each element k to be eliminated is calculated:

Radiative conductances are linearized with the Card 2a TLIN parameter:

where:

TLIN = (Ti⁴ - Tk⁴)/(Ti - Tk) is the estimate specified on Card 2a.
G_ik is the conductance between elements i and k.
Tk is the estimated average absolute temperature of k.
Ti is the estimated average absolute temperature of the elements connected to k.
A_k, A_i are the areas of elements k and i.
E_i, E_k are the emissivities of elementsk and i.
VFG_ik, VFG_ki are the gray body view factors from elements k to i and i to k.
σ is the Stefan-Boltzmann constant from Card 2a.

Next, new equivalent conductances G_ij are calculated between elements i and j connected to element k:

G_ij is set to be a radiative conductance if the smaller of G_ik and G_jk is a radiative conductance, and it is set to be a conductive conductance if the smaller of them is a conductive conductance.

The heat load into element k is redistributed among the connected elements:

where:

Qk(t) is the heat load into element i at time t.
Qi(t) is the new redistributed heat load into element i at time t.

The capacitance of element k is redistributed among the connected elements with:

where:

Cpi is the new additional capacitance redistributed to element i.
Cpk is the capacitance of element k.

As each element k is eliminated, Gksum, the element numbers i of the connected elements, and the connecting conductances Gik are written in the MODLCRF recovery file, in the order the elements k are eliminated. This is the temperature recovery matrix used by the Analyzer to recover the temperatures of the eliminated elements at each printout interval.

The eliminated elements k are written in the MODLCF file as sink SNK elements of constant temperature – 1.E30. This is a flag to the Analyzer that the temperatures of these elements are to be calculated with the recovery process.

Accuracy of the substructuring process

The substructuring process yields a mathematically exact equivalent thermal model for steady state analysis in one of the following cases:

If for each element k, all connected conductances are conductive.
If for each element k, all connected conductances are radiative.

If an element k is connected to a mixture of conductive and radiative conductances, the error introduced by an incorrect estimate of TLIN will be small if the element is either linear conductance or radiative conductance dominated.

An error is introduced when the capacitances are redistributed, since this changes the order of the system of differential equations that describes the model. This error is small if:

where:

dT is the maximum temperature rate of change element k would experience if it were not eliminated.
TM is the maximum temperature change element k would experience if it were not eliminated.

Purpose of substructuring

The purpose of substructuring is to speed up solution time. Substructuring is best used in the following areas:

Eliminating troublesome high-conductance elements that cause ill-conditioning to the conductance matrix and hence slow down iteration.
Eliminating troublesome zero-capacitance elements that cause the transient analysis to slow down, or low-capacitance elements that govern the integration time step. This type of small time constant element is eliminated with the minimum error.

Substructuring may slow down solution time when the number of conductances is increased. To avoid this, in a given region eliminate as many connected elements as possible, and use the PARAM THIN option to eliminate insignificant conductances.

Model thinning

After element merging and substructuring, model thinning is performed if requested on a Card 9 PARAM THIN. This eliminates insignificant conductances from the matrix, in order to speed up solution time. Insignificant conductances may arise from specifying an excessively small Card 2a RK value, from too many odd-shaped solid elements, and as a result of substructuring.

A conductance is considered insignificant if the following two conditions are met:

where:

Gij is the conductance between elements i and j.
T1 is the user-specified thinning parameter.
Gisum is the sum of the conductances to element i.
Gjsum is the sum of the conductances of element j.

Radiative conductances are linearized with TLIN.