A mathematical model for the heat transfer within the electronics package of a Chaparral missile was performed. The Grashof number for this configuration was less than 2000 which indicated that the primary mode of heat transfer was conduction. The Vodicka theory for heat conduction in laminated composite media was utilized to obtain the solution for the model

Huddleston, H. P., Jr.

Mulholland, G. P.

English

NASA Technical Reports Server

N79-19043
Paper No. 35
EVALUATION OF THERMAL SOAK TIMES
J
G. P. Mulholland, New Mexico State University, Las Cru_s, New Mexico
H. P. Huddleston, Jr., White Sands Missile Range, New Mex/c.o
ABSTRACT
/
A mathematical model for the heat traztsfer within the
electronics package of a Chaparral missile has been
performed. The Grasbof number for this configuration
was less than 2000 which indicated that the primary mode
of heat transfer was conduction. The Vodicka theory for
heat conduction in laminated composite media was utilized
to obtain the solution for the model.
INTRODUCTION
Modern missiles and various items of equipment required to
support them are composed of many complex electrical and elec-
tronic systems. Many of these systems, if improperly insulated, .'_
can malfunction if exposed to excessively high or low tempera-
tures. The thermal performance of these systems must man)- times
Le predicted by a mathematical model.
, The purpose of the present investigation is to construct a
mathematical model for the heat transfer within the electronics
package of a Chaparral missile. For this particular configura-
tion, the Crasher number was less than 2000 which indicated
that the heat transfer is essentially all conduction. To model
this system, the Vodicka theory for the heat conduction in
laminated composite media was utilized. This method enables one
to predict both the temperature of and the heat flux through
the electronics package of the missile given ei her the surface
temperature of the missile or the environmental temperature.
To check the validity of the theory an experimental test
was conducted in which the electronics section of a Chaparral
missile was placed in the thermal chamber at White Sands Missile
Range. The electronics package was instrumented with thermo-
couples and temperatures were recorded as the chamber temperature
was varied. The experimental results shot_ good agreement with
the mathematical modei indicating that the original assumptions
are valid for this situation. This agreement between the
experimental results and the theory enables one no'. only to
predict potentially harmful situations to the electronics package
due to fluctuations in environmental temperature but to also
435
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1979010842-424
https://ntrs.nasa.gov/search.jsp?R=19790010872 2020-03-20T18:11:39+00:00Z
design thermal tests of reasonable duration.
m
SYMBOLS
Ain , Bin, Ai, Bi constant
• specific heat at constant pressure in the
CPX ith section
D.2 thermal diffusivity in the ith section
I
Jo zero order Bessel function of the first
kind
th
K. thermal conductivity in the i section
i
L(r) function of r defined by Eq. (8)
gml' Nm constants defined by Eqs (5) and (6)
r spatial coordinate
T.(r.t) temperature in the i section1
T_(t) surface temperature
t time
V initial temperature distribution
X. (r) eigenfunctLon
in
Y zero order Sessel function o" the secand
o kind
Yn eigcnvalue
o i density in the i th section
ANALYSIS
Experimental evidence [1,2] indicates that the heat transfer
in enclosed air spaces is primarily due to conduction if the
Grashof number is less th_n 201)0 for vertical plates and less
than 1700 for horizontal plate_. The particular problem modeled
in this paper is this type of situatxon where free convection is
retarded and the he_*t transfer through the region can be
simulated by the heat transfer through a laminated _o_posite.
The heat conduction equation without heat generation for
the i th section of k solidly joined cylinders is
436
0
1979010842-425
p
r
I _Ti (r,t) I _ _)T (r.t)
=,_ m r m (1)2 _t r _r arD.
1
The boundary, internal and initial conditions for the composite
are taken to be
aT 1 (rl,t) = 0
b) Tk(Vk+l, t) = TA(t)
T i . < r < i - 1 2, k (2)c) (r,o) _ V, r I _ ri. I ......
., d) Ti(ri - Ti_l(ri
3Ti (ri+l,t) 3Ti+l (r1 |e) Ki -_'- = Ki*l 3r
• )undary condition (2b) assum2s the wall temperature is known
while (2a) assumes layer 1 is selid. For the actual .-ituation,
the first layer is hollow; but for practical applications there
is no convenient method for monitoring the temperature at this
point so (2a) was dee_,led a reasonabla alternative.
The solution to Eq's (i) and (2) has been obtained [31
previously and is given by
( f t dTA (_)
2
Ti(r't) = n_l gn exp ('Yn t) - Znl d--T- exp
0
[-yn_(t-t)]d_ Xin(r)  Li(r)TA(t); r i < r < ri. 1,
i = l, 2..... k _3)
where
k ri+l
I f rVXin(r)dr- _nll'A{0) n = 1, 2, 3 (4)gn • N'- i=lPiCpi .....
n .. t
r.
k l+l
' /• ,_- i_=l piCpi r Li(r)Xin(r)dr, n = 1, 2, 3 .... (5)&hi
n r.
t
437
t
/
t
/,
#
B___I ii
1979010842-426
e rp
_x
k -- ri  1
Nn - i_lOiCpil r[Xin(r)]2dr, n - I, 2, 3, . (6)
-r
1
Xin(r) - A. J r + B. Y rIn o In o , ' i - I, 2..... k
(7)
Lifr}__ = A.,ntr * S.I (8)
The coefficients A. , 3. :or i = I, 2, ., k, n - I, 2, ...
the coefficients A. and1_, f_ i - l, 2..... "k and the eigen-
• • h - Ivalues _, n L, ,, _, ... are obtained in a straightforward
manner b_ applying the boundary and internal conditions to the
assu_ed solution. Details are given in [3].
MISSILE SYSTEM
The electronics system of a Chaparral missile can be
simulated as a five layer composite. Layer one and layer five
are strainless steel, layers two and four are air and layer
three is the e;ectronics package. The physical dimensions and
• thermal properties were taken as
La_Uaz_
g(w/m-*K) 16.264 0.0271 9.08x10 "3 O.0271 16.264
, pfgm/cm3) 7.824 0.00114 0.0818 0.00114 7,824
Cp(cal/gm-eKJ O.ll 0.24 0.45 0.24 O. ll
r I - O, r_ = 1.111 cm, r 3 = 2.85 cm, r 4 3.85 cm, r 5 = 5.97 cm,
r 6 = 6.35 cm
Proceeding in the same manner as outlined previously, we
obtain the solution as
Ti(r't) m_l Vtm exp (-_m2t) - t m T_(t) * Ym tm TA(t)'exp
"t(-_m*t) Xim(r) * TA(tJ (9)
where
Ksr6 dX_m _r6)
= - [10)
_m _m2Nn dr
438
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¢
t
t
t
]9790]0842-427
: REFERENCES
1. M. Jakob, "Free Convection Through Enclosed Plane Gas
Layers," Tran3. ASME, Vol. 68, p. 189 (1946).
2. M. Jakob, Heat Transfer, Vol. 1, John Wiley 6 5ons, Inc.,
New York (1949).
3. G. P. Mulholland and M. H. Cobble, "Diffusion Through
Composite Media," Int. J. Heat Mass Transfer, Vol. 15,
pp. 147-159 (1972).¢
APPENDIX A
The eigenvalues for the electronics secti)n of the missile
• configuration described previously are found by solving the
; matrix equation
_* M01(r2)_M02(r2 ) -N02(r 2)
AINII (r2)-M12(r2)-NI2 (r2)
0 M02(rs) NO2(r3)-M03 (r3) -No3 (r3)
0 0 0 -M03 (r4)-N13 (r3) =0 • °
0 0 0 Mo3(r4) No3(r 4)
0 0 0 A3M13(r4) A3N13(r4)-M14(r4)-N04(r4) 0
, 0 0 0 0 0 M04 (rs) NO4(rs)P °
¢
_l 0 0 0 0 0 A4MI4 (rs)A4NI4(rs)P
where
Mij (r) = Ji ( _. r)
• 3
: Ym
Nij(r) = Yi(_-. r)
J
Ki Di  1
A. =
i Ki+1 D.1
-Mis(rs)Nos(r 6) + Mos(r6)Nis(r 5)
P. =
1 Nos(r 6)
: 439
t
¢
"19790]0842-428
24 r_+ 1N = £_I 2 (p£Cp£ p£+ICp£+I ) X2m _m (r£+l)
2
K£+I
2
+ 2 Ym2 -_r
and where the £ are the roots of the eigenvalue equation given
:, m
in Appendix A.
o_
Figure 1 is a comparison between the experimentally and
anlytically obtained temperatures of the outside surface of
the electronics package during a test conducted in the thermal
ch_ber at White Sands Missile Range. The differences between
the two curves for t > 50 minutes is most likely due to the
inexact values of density, specific hear and the_al conductiv-
ity used in the model. The package _vas ass_ed to be cork at
680F.
Since the package consists of numerous _lectronic items of
different materials, the true evaluation of these properties
would have to be made in a separate test.
The encouraging portion of this figure is for t > 50
minutes. In this region where the effects of density and
specific heat are less important, the two curves converge.
Sin_e the purpose of the study was to evaluate the time
necessary for the electronics package to reach a steady-state
operating condition, it is felt that this model will accomplish
that function. Better temperature profiles can be obtained if
more accurate values for the density and specific heat of the
electronics package are utilized.
CONCLUSIONS
A mathematical model for the temperature distribution and
heat flux across the electronics package of a Chaparral missile
has been obtained. Thir model can be utilized in situations
where natural convection currents are suppressed and the heat
transfer can be simulated by conduction through laminate
composites.
440
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bO
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, \\ \
! Y I _ I --
3o" ll:lll,LV_I'l,lbLl,l.
• 441
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1979010842-430


Evaluation of thermal soak times

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