The advantages and possible pitfalls of using a generalized method of measuring and, based on these measurements, predicting the transient or steady-state thermal response characteristics of an electronic equipment designed to operate in a space environment are reviewed. The method requires generation of a set of thermal influence coefficients by test measurement in vacuo. A implified thermal mockup isused in this test. Once this data set is measured, temperatures resulting from arbitrary steady-state or time varying power profiles can be economically calculated with the aid of a digital computer

Gale, E. H.

English

NASA Technical Reports Server

N81-14161
THE USE OF THE PRINCIPLE OF SUPERPOSITION IN MEASURING AND PREDICTING
THE THERMAL CHARACTERISTICS OF AN ELECTRONIC EQUIPMENT OPERATED IN A
SPACE ENVIRONMENT
Dr. Earl H. Gale*
ABSTRACT
This paper reviews the advantages and possible pitfalls of using a gener-
alized method of measuring and, based on these measurements, predicting the
transient or steady-state thermal response characteristics of an electronic
equipment designed to operate in a space environment. The method requires
generation of a set of thermal influence coefficients by test measurement in
vacuo. A simplified thermal mockup is used in this test. Once this data set
has been measured, temperatures resulting from arbitrary steady-state or time
varying power profiles can be economically calculated with the aid of a digi-
tal computer.
INTRODUCTION
The two most common methods of predicting the thermal performance of an
electronic equipment are to:
I) Make a thermal mathematical model and solve the resulting network
equations using finite difference or finite element _thods with the
aid of a computer
2) Build a thermal mockup of the equipment which approximates the geome-
try and the dissipation profile of the equipment under design and di-
rectly measure the thermal performance in a simulated environment.
This paper describes a middle road to the above methods utilizing both
measurements to initially thermally chqracterize an equipment and then a simple
computer program to accurately and economically predict either the transient
or steady-state thermal response of the equipment for any power versus time
or steady-state profiles of interest. The method is of great value when the
thermal performance with more than one power profile (duty cycle) is required.
The method requires the construction of a thermal mockup of similar con-
figuration to the equipment under design with a unit power resistor attached
within the equipment at each location of an electronic component having a sig-
nificant heat dissipation. The unit resistors may be all of the same wattage
rating and dissipation despite the fact they are located at positions where
varying amounts of heat are to be dissipated. A thermocouple is installed in
the vicinit) of each of the unit resistors as wel] as at all other locations
within the equipment at which temperature information is desired.
A single set of measurements in which each unit resistor is powered, one
at a time, and the resulting temperature-time profiles at all points of inter-
est are measured and recorded. This data bank allows the late_ prediction of
temperatures using any power profile accurately, quickly and economically uti-
lizing an easy to write computer program. This simple method is accurate only
*Ceneral Electric Company, Aircraft Equipment Division, Utica, New York.
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1981005626-300
https://ntrs.nasa.gov/search.jsp?R=19810005649 2020-03-21T14:30:11+00:00Z
I
in systems whose characteristic underlying differential equations are linear
such as the equations describing the heat conduction within an equipment in a
space (no air) environment.
This paper describes how the method was legitimately and successiully
used to predict the steady-state thermal perfarmance of an electrenic equip-
ment having various duty cycles despite the fact that the equipment case-to-
surroundings heat transfer within a spacecraft was by thermal radiation, a
mode of heat transfer described by nonlinear equations. A discussion of when
it would be permissible to include such nonlinear surface heat transfer is
presented.
NOMENCLATURE
A radiating area of case
C thermal capacity of conducting material
E constapt which is function of emissivities of equipment case
finish and s_l'oud finish in temperature-vacuum chamber
k thermal conduct ivit _,
q" heat flu._ per unit area
q"' heat generation per u,it time per unit volume
T absolute temperature
w specific weight of conductivity material
x,y,;: spatial coordinates
o Stefan-Bol tzmann constant
Subscripts
c cold
h hot
i index number of test measurement of temperature field resulting
from single source
in mean
t total
THEORETICAl, JUSTIFICATION OF METHOD
Heat transfer in electronic equipments used in space takes place by con-
duction and radiation. It is usual that the method of heat transfer internal
to an equipment, that is, between heat dissipating components and the outer
case of the equipment, is by conduction through structural paths with radiative
heat transfer playing a negliglble role. This usually happens by design as
providing structural paths having high thermal conductances is necessary for
good thermal management. Heat transfer from the outer equipment case may be
by conduction through the equipment mounts, or thermal radiation from the case
to the surroundings or a combination of both of these transfer mechanisms.
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1981005626-301
When heat transfer within the box is only by conduction, the resulting
temperature field must satisfy the general heat conduction equation:
_2T _2T _2T q"' i _T
-- + -- + -- + = (t)
3x 2 _y2 3z 2 k c_ 20
k
= --- (2)
Cw
Since this governing equation is linear, the well-known principle of superpo-
sition can be used in the manner described in this report to predict the time
varying or steady-state temperature field within the equipment. Reference I
states: "The principle of superposition can be applied to all cases in which
the effect of simultaneous superposed actions is the sum of the effects of
each individual action. It can be applied, therefore, to linear system, i.e.,
te physica] systems governed by linear differential or integral equations."
Using this method to predict the thermal performance of a particular equipment
will now be described.
IMPLEMENTATION OF METHOD
The particular equipment for which the author successfully used the analy-
sis method described in this paper had the following characteristics:
I) In application, it was to be operated with a large number of differ-
ent "duty cycles". Each duty cycle was defined by the power versus
time history of each heat dissipating component within the equip-
ment.
2) The equ{pment was radiativelv cooled to its surroundings and, during
the operation, the outer case temperature was approximately uniform.
3) Paths of high thermal conductance between heat dissipating components
and the case were provided by design for heat management purposes;
that is, the unit was designed for efficient internal conduction
cooling.
TESTIN[; PROCEDURE
The electronic equipment chassis that was later used in the construction
of the engineering model was instrumented by the bonding of wire-wound resis-
tors at locations at which there would later be significant thermal dissipations
from Individual electronic components. The_.nocouples were attached at all com-
ponent 1ocatlons at which operating temperature information was required In
this particular test, a total of sixty thermocouples and forty resistor_ _ e
used to characterize the thermal proflle of the unit. A thermocouple was a,-
ways mounted on the chassis structure adjacent to each wlre-wound ¢esistor.
A high wlcuum, 10-5 Tort range, t,st chamber having a temperature con-
trolled shroud was used In the test. lhe temperature of the shroud was held
at that uniform temperature which had bt,cn specified for the design. Each re-
sistor was activated, one at a time, and the resultlng temperature-time history
at all thermocouple 1ocatlons was recorded.
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1981005626-302
DATA PROCESSING
The data was processed in the following manner. The temperature rise
above shroud temperature for each location as a result of powering each re-
sistor was recorded in a matrix format at the end of each ten-minute interval.
Using the shroud temperature as the baseline or reference temperature meant
that the nonlinear radiative equipment-to-surroundings (shroud) thermal resist--
ance was included in the compilation of these thermal "influence coefficients".
The error resulting from the inclusion of tbese radiative resistances w_ll be
discussed below where it will be shown to be:
I) Small
2) On the safe side, i.e., resulting in higher than actual predicted
temperatures.
THE COMPUTER CODE
The concept for the computer code is obvious and simple. A program was
written which simply added the temperature rises at each thermocouple loca-
tion resulting from step infusions of heat at each resistor location at the
end of incremental units of time. The process is one of summation of effects
requiring the direct addition of temperature rises (a noniterative solution)
and the computer time (cost) needed to analyze various duty cycles (power vs
time) is very small.
Had the unit itself been conduction-cooled through its mounts, the above
method would have been completely theoret_cally correct and the following sec-
tion on estimating the upper bound of the error resulting from the nonlinearity
of the radiative heat transfer path between the equipment case and the surround-
ings would not need to be considered.
UPPER BOUND OF NONLINEARITY ERROR
The upper bound of the error resulting from treating the radiative con-
ductance from the case to the shroud as independent of temperature level will
now be dete1_nlned.
The basic equation describing radiative heat transfer from each region
(node) of the equipment case to the shroud may be written in the following form:
q" = EAG(Th 4 - T 4)c (3)
On factoring, equation (3) may be written:
q" = EA_(Th 3 + Th 2To + ThTc2 + Tc3)(Th - Tc ) (4)
References 2 and 3 show that if Th - T c is small compared to (T h - Tc)/2 = Tm,
equation (4) may be replaced by:
q" = 4EAOTm3(T h - Tc) (5)
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1981005626-303
The criterion for the above approximation was always met when measurement
with the individual unit resistors was taking place.
The formula describing the total heat transfer from any case region (or
the entire case if the temperature is uniform) is of the same form as equation
(3):
,, = EAO(Tht 4qT - Tc4) (6)
Rewriting equation (6):
qT EA_(Tht2 + Tc2)(Tht + Tc )(Tht - Tc ) (7)
Solving for (Tnt - Tc ) yields:
I!
qT
(Tht - T ) = 2 2) (8)
c EAo (Tht + T +c (Tht Tc)
(Tht - Tc) found by summing (superposition) using equation (5) is:
_t t!
n n ql qt
(TNt - T ) = _ (rh - Tc) = _ 3 = 3 (9)
c i=l t=l 4EAL_T 4EAoT
m m
The ratio of Tht - Tc calculated by superposition using equation (9) to
the exact total equation (8) has been calculated for a wide range of sink tem-
peratures and temperature differences, and the results are presented in Figure I.
It should be emphasized that this figure does not indicate the actual
error but the upper bound of the actual error as during the unit resistor tests
Tm - T c for each individual test. It should also be realized that this radi-
ative resistance is never the total resistance between source and sink (i.e.,
compou_'nt mounting location and shroud) but m,_rely a series component. Thus,
a lO-percent error in ca]cula[ion of this quantity would result in a lesser
total error.
If the possible upper error bound as shown in Figure | were too large to
be acceptable, the absolute total error could be reduced by including in the
computational program a temperature level correction factor, after equati_ , (5),
to correct the measured ith case-to-shroud temperature difference to a ca_,e
temperature more indicative of the case temperature during operation.
The form of this correction factor would be:
3
Tml
3 (lO)
T
mt
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1981005626-304
where Tmi was the mean absolute temperature resulting from the averaging of
case and shroud temperature during the ith test and Tmt the temperature re-
sulting from the averaging of case and shroud temperatures with the case tem-
perature predicted using the total unit power.
CONCLUSION
The prediction method described in this paper can be an extremely useful
and economical tool for predicting either the steady-state or transient thermal
response. It is of particular value in each of the following situations:
I) Calculating the responses to equipment to be operated with many
different duty cycles, i.e., power time profiles, where it would
be impossible (due to time limitations) or uneconomical to run a
multitude of tests using the actual power profiles
2) The procedure is also a valuable tool for checking the accuracy
of finite element or finite difference programs in which internal
thermal conductances have been determined by calculation rather
than measurement and for which semi-empirlcal verification i_
desired.
REFERENCES
I. Theodore v. Karman and Mauri,:e A. Blot, Mathematical Methods in Engineer-
in___,McGraw-Hill Book Company, Inc., 1940, p. 378.
2. Max Jakob, Heat Transfer, John Wiley, Volume I, 1955, p. 87.
3. Ibld., Volume If, 1957, p. 3.
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1981005626-305
m 1,4 (Tht' + Tc')(Tht + Tc)
o4 R=
"' 4 TC_O
Z
I.U
¢ WHERE: Th = CASE TEMPERATURElu 1.31 50°C
u. Tc = RADIATIVE SINK TEMPERATURE
2S'C
Lu 0"Cm
:_ 25 "CI.,-
< 1.2 50°C
erL.
:E
UJ
I-.-
M. 1.1O
_o
n-
1.0 I 1 J J J
0 5 10 15 20 25 30 35 40
TEMPERATURE DIFFERENCE BETWEEN UNIT AND HEAT SINk ('C)
(Tht" Tc)
Figure I. R-_rlo of Theoretical Haximum Sum of Surface-to-Sink
Temperatures to Actual Surface-to-Sink Temperatur_
Difference at the Same Total Unit Power
297


The use of the principle of superposition in measuring and predicting the thermal characteristics of an electronic equipment operated in a space environment

The use of the principle of superposition in measuring and predicting the thermal characteristics of an electronic equipment operated in a space environment

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