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 stefanboltzman 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}^4T_{l}^4).
Node Temperatures and Energy Balance
In the case of regular, nonboundary 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 nonboundary
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' + (1R)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 nonboundary 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 nonboundary 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. Workaround 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.