Thermal modeling

This document describes the basic concepts and equations involved in using Rob Boyle's (Goddard Space Flight Center) thermal modeling Excel spreadsheet. It also describes specifically how to use the spreadsheet and documents several attempts to match models with laboratory runs.

Basic concepts behind modeling process

Model the thermal system as a circuit made up of conductors and nodes. The conductors represent thermal paths. The nodes represent isothermal surfaces. Each conductor is connected to two nodes. Each node is connected to one or more conductors. Conductors model both radiative and conductive thermal paths. Nodes are either standard nodes or boundary nodes. Boundary nodes are those which represent thermal sources or sinks such as cryogen reservoirs.

Conductance

The heat flow across a conductor is given by
Q(Watt) = G(T_h-T_l).
where T_h and T_l are the temperatures of the hot and cold boundaries respectively. The conductance, G is calculated for conductive thermal paths as
G(Watt/K) = kA/L
where k (W/cm K) is the thermal conductivity, A(cm^2) is the cross sectional area of the conductor and L(cm) is the conductor's length. The thermal conductivity k depends on the material and the temperature range spanned by the conductor.

The conductance, G is calculated for radiative thermal paths as

G(Watt/K) = seA(T_h+T_l)(T_h^2+T_l^2)
where s is the stefan-boltzman constant, e is the effective emissivity and A is the effective surface area. The effective emissivity depends on the emissivities of the surfaces and the number of MLI layers. The effective area depends on the view factor of the two surfaces. This calculation of the conductance, when inserted into the calculation of heat flow given above is equivalent to the more familiar calculation of radiative heat flow
Q=seA(T_h^4-T_l^4).

Node Temperatures and Energy Balance

In the case of regular, non-boundary nodes, the heat flowing into the node should equal the heat flowing out of the node. This condition will not be met immediately. Depending upon the specific model, many iterations can be needed. The energy balance for the ith node, given by
Ei = [heating + sum(GjTk) + mdot*cp*Tv] - Ti[sum(Gj) + mdot*cp]
is a good diagnostic to check the progress of the model. The heating term allows for the direct input of heat. The sum over GjTk includes all conductors attached to the ith node and the temperature of the other node they're attached to. The mdot terms account for vapor cooling (where mdot is the rate at which the cryogen is vaporizing in g/s and cp is the heat capacity of the gas).

If an iterative solution wasn't necessary, the temperature for each non-boundary node could be calculated by setting Ei = 0 and solving for Ti as follows:

Ti' = [heating + sum(Gj)Tk + mdot*cp*Tv] / [sum(Gj) + mdot*cp].
In order to assure converging iterations, however, the temperature for regular nodes is calculated as
Ti = R*Ti' + (1-R)Ti(last)
where R, the relaxation factor, can be set between 0 and 1, and Ti(last)is the value of Ti from the previous iteration.

Material Conductivities

The material conductivities (k) are calculated for each conductor based on the temperatures at the end points of the conductor and a cubic fit of the material's properties. The file "conducti.xls" or "conductivity coeff" contains the coefficients for the cubic fit.

Effective emissivities, effective areas and MLI

Coming Soon . . .

Fudge Factors

Emissivities and contact resistances are, in general, poorly known. They are highly dependent upon the specifics of the surface and joint in question. Thus the only way to definitively know these properties is to individually test each surface and joint. Unfortunately, it is rare to have the opportunity to perform these tests systematically. As a result, it is quite useful to include a safety factor when using these quantites in a model. In order to successfully model a cryogenic run of the lear jet dewar (without cables or drive shafts), I found it necessary to introduce 3 different fudge factors: one for emissivities, one for MLI emissivity and one for contact resistances. In order to match the measured LHe and LN2 hold times, I found it necessary to make the MLI 3 times less efficient and the contact resistance between the LN2 reservoir and LN2 shield 3 times more efficient. The bare metal emissivities worked well as quoted. The initial MLI calculation used neglected conduction through the MLI surfaces.

Developing a model

Identify nodes.
Identify boundary nodes.
Identify sources of heating and vapor cooling.
Establish initial temperatures.
Identify conductors.
For radiative conductors determine e G(Watt/K) = kA/LFor conductive conductors determine lengths, cross sectional areas and material properties.
Identify connections.
Draw initial heat map.

Anatomy of the spreadsheet

Cryogen and lifetime information

Pasted from cryogen table near bottom of spreadsheet contains temperature, heat capacities, heat of vaporizations calculates mass flow (mdot) and cryogenic lifetimes.

Conductor information

For a radiative conductor - requires effective cross sectional area and effective emissivities.
For a conduction conductor - requires cross sectional area, length and material identification.
For all conductors - requires identification of the nodes on either side of the conductor.

Node connections

Begins with node description. Contains conductances and adjoining nodes for every connection. Connection table is created by assembly macro.

Node parameters

Specifies additional heat inputs, presence of vapor cooling and initial temperatures.
Energy balance (or loads on cryogen reservoirs) and temperatures are calculated.

Reference tables

Cryogen table includes temperatures, heat of vaporization and heat capacties for common cryogens.
Materials table includes thermal conductivities for commonly used materials for several temperature ranges.

Entering model into spreadsheet

Paste cryogen properties from reference table to cryogen and lifetime section.
Create conductors.
Label conductor and type (radiation or conduction).
Specify end point nodes.
Specify relevant area:
- for conductor - cross sectional area
- for radiation - effective surface area
For conduction specify path length.
For conduction, specify material by pasting both the name and the k value from the materials table at bottom of spreadsheet.
For radiation define number of MLI layers and enter effective emissivity.
Name the nodes at top of node connection region.
Specify node properties.
If needed, specify vapor cooling by entering "mdot*cp" into proper row and reference to the upstream node in the cell below.
Specify additional heat input.
Specify initial temperature.
Specify number of nodes and number of conductors.

Running model

Run the Assemble macro to create system of equations.
Set initial values = TRUE.
For boundary nodes set y test each surface and join temperature cell.
If any equations are too long to fit into a single cell, split equation into two cells.
Make sure the energy balance cell is choosing the max energy imbalance from all non-boundary nodes.
Calculate. Set calculate parameters to manual and number of iterations to 100.
Set initial values = FALSE.
Calculate and Iterate.

Convergence methods and criteria

You can determine how long you want the calculations iterate by setting the convergence criteria. A decent convergence criteria is 0.1 times the maximum heatflow across a conductor. After every iteration the convergence criteria is compared to the maximum non-boundary node energy balance. When the energy balance is less than the convergence criteria Excel will stop iterating if Excel's maximum change parameter is set properly (5 works well).

Macro descriptions:

Assemble creates the equations.
Assemble_hide performs the same function as assemble, but without the animated screen display. Runs faster.
Calc_doc is similar to the standard excel calculate now command, but it will only calculate the active document. Normally if you have several spreadsheets open and choose calculate now, all open spreadsheets are recalculated.
Freeze_calc is like calc_doc but without the screen show.

Potential Pitfalls

Large conductors. If you have a conductor that is *much* bigger it will dominate the model. This can lead to an unstable model and/or hide imbalances elsewhere in the model. Work-around suggestion: Nodes connected by huge conductors will be nearly isothermal. For the first model, consider nearly isothermal nodes to be 1 node. Later create another model just of your nearly isothermal nodes and model the rest of the system as boundary conditions.
Fill/vent tubes. When vapor cooled, fill tubes should have several (at least 3) nodes along their length. Otherwise the heat load on the cold bath will be overestimated.

Sensitivity studies

Coming Soon . . .

Heat Flow Checker

Coming Soon . . .

Heat Maps

Coming Soon . . .

Actual Model

Coming Soon . . .

Last revised Feburary 5, 1996. Contact jdotson@nwu.edu.