663 TRANSIENT HEAT CONDUCTION IN SEWZINFINITE SOLIDS5 NITH vi1LA TEMPERATURE DEPENDENT PROPERTZES(U) RMY BALLISTIC
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US ARMYMATER1,-
COMMA;.D
TECHNICAL REPORT BRL-TR-2720 DTICELECTEAPR 15
TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS WITH
€ TEMPERATURE DEPENDENT PROPERTIES
Lang-Mann Chang
March 1986
-.J
APPROVED FOR PULUC RELEASE; DISTRBUTION UNUMITED.
US ARMY BALLISTIC RESEARCH LABORATORYABERDEEN PROVING GROUND, MARYLAND
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The use of trade names or manufacturers, names in this reportdoes not constitute indorsement of any commercial product.
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Technical Report BRL-TR--27204. TI TL E (And S-btttlo) TYPE OF REPOCIT & PERIOD CCvERED
Transient Heat Conduction in Semi-Infinite Solids TcnclRprwith Temperature-Dependent Properties 6 PERFORMING ORG. REPORT N...'BER
7. AUTI4OR(e) 11. CONITRACT OR GRANT NUMBER(a)
Lang-Mann Chang
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* IS. SUPPLEMENTARY NOTES
1 KEY WORDS (Cotinue on reverse side it nesaary and Identify by' block number)
Transient Heat Conduction Similarity TransformationSemi-infinite Solid Similarity Solution.Temperature- dependent Properties,'Time.tvarying Boundary Condition,
*2.A9ST- ACT O'Ciak t~s, 'eveee oe* f necea Me idetity by block number)V
Similarity solutions are obtained for the transient heat conduction in a semi-infinite solid with temperature-dependent thermal properties. The surface ofthe solid is considered subjected to a time-var~ing boundary condition, either .-
* heat flux Qw as at or temperature T - T (1 +'t ). Solutions for the tern-
perature distribution and the local wheat 0flux are presented. In particular,type 420 stainless steel, whose thermal properties are strong functions oftemperature, was chosen for analysis to demonstrate the significance of the
DD 147373 M ~ i~.O My0 SOSLT UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PA-,E rg~ert Dare Fwtrepd)
e-e_. . e
UNCLASSIFIED
SECURITY CLASSIFICATION OF' TIS PAGE(Whan Date Entered)
20. ABSTRACT (continued)
effect of the temperature dependence of thermal properties on heat transfercalculations. ;,
a* %
"* .
-"-
. .
4..
IJL
UNCLASSIFTED)SECURITY CLASSIFICATION OF T4IS PAGE'9' L)Rfs EF".rd
7. n
TABLE OF CONTENTS
Pag e
LIST OF FIGURES ............................................................. 5
LIST OF TABLES ................... o.........................................o.7
I. INTRODUCTION ....... o..................................................9
II. FORMULATION OF PROBLEM ........ o.......................o......o................
bA. Prescribed Beat Flux Q -at.....o.......................12B. Prescribed Surface Temperature T~ - T (1 + ..... ..... 13P V 0
III. METHOD OF SOLUTIO1N........o................ o......................... 13
A. Prescribed Heat Flux Qw= atb .......................... o......... 13
B. Prescribed Surface Temperature Tw TOO1 + tb*.................. 16
IV. SAMPLE CALCULATIONS AND DISCUSSION .................................. 17
A. Prescribed Heat Flux Qw= atb ....................................17B. Prescribed Surface Temperature Tw TOO1 +e t~) .................. 25
V. SUMMARY AND CONCLUSIONS ..........o...................................29
REFERENCES ........................... o.............................. 31
NOMENCLATURE ........................................................ 33
DISTRIBUTION LIST ....... o.....o...................................... 35
Accesion ForNTIS CRA&IDTIC TABunannounced E
Justification
By.- . .
Dist 1butior,
Availability Codes
Dit Avail a7!d IorDit sp, Cidl
3
Y'~~'%, '. 1Y Y~~~X Y.T~p.., -., r ri -w
LIST OF FIGURES
Figure Page
1. Thermal Properties of Type 4130 Steel and Type 420 WStainless Steel (reproduced from Ref. 2) ............................ 10
2. Semi-infinite Solid with Temperature-Dependent ThermalProperties ........................... ......................... ............ 11
3. Thermal Properties of Type 420 Stainless Steel (normalized) ......... 18b)~t .... 24. Profiles of Temperature Function (boundary condition Qw - atb)......20
5. Profiles of Derivative of Temperature Function (boundary condition
Q atb) .......................................................... 21
6. Temperature Distribution for Constant Properties (boundary conditionQw= constant) ...................................................... 22
7. Surface Temperature vs. Time (boundary condition Qw constant) . 23
8. Surface Temperature vs. Time (boundary Condition Qw - atb) .......... 24
9. Profiles of Temperature Function (boundary conditionTw - TO(O + t )) ................................................... 26
10. Profiles of Derivative of Temperature Function (boundary condition
Tw - TOOl + tb*)) .................................................. 27
11. Temperature Distribution and Heat Flux (boundary conditionT = constant) ...................................................... 28
5
I A
,.-.z._..
* LIST OF TABLES
*Table Page
1. Solutions of Temperature Function f and Its Derivative f' atSurface 0ri )...................................................19 6
2. Values of Temperature Function f and Its Derivative f'Inside the Solid for T1, Constant .................................29
61'
7
I. INTRODUCTION
The transient heat conduction in semi-infinite solids is an importantheat transfer problem. Typical examples are the heating by propellant gas oflarge caliber gun barrels, impingement heating on a ship deck during missile -launching, and solar heating of the earth surfa e. Solutions to these problemsare well known when constant thermal properties are assumed. Some materials,such as type 420 stainless steel and type 4130 steel, however, possess therma.
*T properties which are strong functions of temperature as shown in Figure 1.Neglect of this fact may lead to significant errors in heat transfercalculations. This is especially the case when high heating rates areinvolved.
When the temperature dependence of thermal properties is accounted for,
the heat equation becomes nonlinear and its exact solution is unattainable.In the relatively simple case in which the thermal conductivity is
temperature-dependent and the surface temperature is constant, Schubert, eta ob t a in e d a similarity solution for their one-dimensional solid state
. model of the oceanic lithosphere and asthenosphere. However, solutions for asolid with both thermal conductivity and specific heat as functions oftemperature and subjected to a time-varying boundary condition have not beenreported.
In this study we consider a semi-infinite homogeneous solid with thermalproperties which have a power law dependence on temperature. Two types of
. time-de endent boundary conditions are p iscribed at the surface: a heat fluxQw at0 and a temperature T+ To(1+ tt ). The power law representations
are often useful in many engineering applications. Utilizing the method ofsimilarity transformation via one-parameter groups, we transformed the non-
.d linear governing partial differential equation into an ordinary differential Iequation that can be integrated numerically by using any one of the severalordinary differential equation packages. This method is simple and, above all,
the solution obtained in terms of similarity variables provides a broadrepresentation. Such an advantage is usually not obtained when the governingpartial differential equation is directly solved by numerical methods.
Using type 420 stainless steel properties for sample calculations,solutions obtained are the local temperature and local heat flux at anydistance from the surface of the medium. In the limiting case of constanttemperature or constant heat flux applied to the solid with constant therma"properties, a comparison is made with the available exact solution.-
Meanwhile, the significance of the temperature dependence of thermalproperties for heat transfer calculations is demonstrated.
1 H. Carslaw and J. Jaeger, Conduction of Heat in Solids, Oxford Univerity
Press, London, 1959, Chapter 2.
. Aerospace Structural Metals Handbook, U.S. Army Materials and MechanicsResearch Center, Watertown, MA, 1980.
3 G. Schubert, C. Froidevaux, and D.A. Yuen, "Oceanic Lithosphere andAsthenosphere: Thermal and Mechanical Structure," J. Geophys. Res. 81, No. 20,1976, pp. 3525-3540.
9 .r ,%
• o
0.40
~'30 1 Steel 4130
Steel 4130 1-0.35
S26 p-1-0.30
22
- 0.2518
0.20L)z W-
S14S 0 800 1600 01
TEMPERATURE (OF)
010 800 1600
TEMPERATURE ("F)
0.24 1.18 I 02-Stainless Steel 20
Stainless Steel 420-
S16 c001
>14 01
0.1
S120 800 1600800 ~0.10080 10TEMPERATURE (OF) 080 10
TEMPERATURE (OF)
Figure 1. Thermal Properties of Type 4130 Steel and Type 420Stainless Steel (reproduced from Ref. 2)
10
II. FORMULATION OF PROBLEM
The equation describing the transient beat conduction in a semi-infinitesolid with temperature dependent properties is
aT a aT "Cp(T) - - [K(T)-1. (1)
ai ax ax
Where T, t, and x are the temperature, time, and distance from the surface ofthe solid, respectively. Figure 2 depicts the solid under consideration. Thedensity A is considered to be constant. The thermal conductivity K(T) andthe specific heat C p(T) are assumed to be functions of temperature in thefollowing form:
T-K(T)- KoC [ (2)po TO T ,
T- To nCp(T) C C (3)ii):
where Cm, Cn, m, and n are dimensionless constants which can be determined
from the experimental data of a given material. K0 and Cpo are respectivelythe thermal conductivity and specific heat evaluated at temperature To. It isfurther assumed that the solid is initially at a uniform temperature T o .
+I X
--
0 K K(T
p p
Figure 2. Semi-infinite Solid with Temperature-Dependent Thermal Properties
115
,I
.-....-..._......... ....... ....................-...-........,::,: L'..i.,; '.:_'i . ,.. .., ".... . -.. . .. -. . .. .--.
Eqs. (2) and (3) appear unrealistic when T = T0 since at that temperatureboth K and C become zero. Practically, however, when Kh- C - 0, whichrepresents the case in which the material does not conduct heat away and hasno heat capacity, the temperature will then jump up immediately when thematerial is being heated. Thus, the singular behavior at T - To appears onlyin a very short period of time and will have no significant effect on theoverall heat transfer calculation in most engineering problems.
For the convenience of analysis, we nondimersionalize Eqs. (1) through(3) by introducing the following dimensionless variables:
H[T- T K x C0 - , k = - , s C p - ,."
TO Ko L P Co(4)
t a KoP 1-- , and t- where a --
PO po a
where L is the reference length. Substitution of these variables into Eq. (l)leads to a dimensionless form of the heat equation -
ae c aeem- + mem-l(-)2 9 - 9n " (5)
s 2 as Cm at
We now consider two types of time dependent boundary conditionsprescribed at the surface (i.e., at x s = 0):heat flux Qw atb (6)
and
temperature Tw To(1 + tb* (7)
where a is a constant with units the same as that of heat flux and t is thedimensionless time defined in Equation (4). The powers b and b* aredimensionless constants.
The complete sets of boundary and initial conditions for Eq. (5)corresponding to Equations (6) and (7) are given as follows:
A. Prescribed Heat Flux -. = atb
By replacing the left hand side of Eq. (6) by
8TQw -K(T) - (8)
ax '
. and choosing the reference length
12
:...... '-
4 - - ~ - - ..4.... - .
KoTL 0- (9)
a
we obtain the following conditions for Eq. (5):
-Cmem - = tb at s 0, t > 0 (10a)
6 - 0 at a - , t _ 0 (10b)
G 0 at a > 0, t - 0. (10c) _Lj
B. Prescribed Surface Temperature Tv - To(1 + tb*)
In this case, the boundary and the initial conditions are
.4. b*
0t at s 0, t > 0 (Ila)
0 0 at s , t > 0 (11b)
90 at s 0, t 0 (11c)
in which the reference length L can be chosen to be any constant value.
III. METHOD OF SOLUTION
A. Prescribed Heat Flux Qw atb
We obtained a solution of this problem by using the similaritytransformation via one-parameter groups. 6 After the transformation, thenumber of the independent variables was reduced from two (i.e., t and 9) toone (i.e., 7 ). The resulting ordinary differential equation was thenintegrated numerically.
4 A.J.A. Morgan, "The Reduction by One of the Number of Independent Variablesin Some System of Partial Differential Equations," Quart. J. Math. (Oxford), L2, 1952, p. 250.
5A.G. Hansen, Similarity Analyses of Boundary Value Problems in Engineering,Prentice-Hall, Englewood Cliffs, NJ, 1964.
6 W.F. Ames, Nonlinear Partial Differential Equations in EnRineering, AcademicPress, 1965, pp. 135-141.
13I
In the transformation procedure, we first introduce three variables:
t'-td1 -a and 0- d(12)
* where d is a parameter,0a, and Ylare constants to be determined by usingthe governing heat equation together with the boundary conditions. *-.
Substituting these variables into Eqs. (5) and (10a) results in
2a2 m (m18 2e- 22- m) 1 (rn-i) 86 2d el -+md 0
Cn a, i C+1)yj 891- d e~n(13)
02 -(ml)'I -bal b-d C9me.~ =d t* (14)
ax,
To satisfy the invariance requirement of the transformation, the indices ofthe parameter d in each term of the equation are set equal. Then from Eq.
* (13) we have
2a2 ( m + 1)'/1 a 2a02 ( m + 1)YI 01~ (n + 1) v1 (15)
* and from Eq. (14)
2 - +m )i -o (16)
Solving Eqs. (15) and (16) simultaneously gives
1.2b (17)
a2 1 +2b-(m +) b. (8
Let
a2A - - (19)
14
Iand
B--. (20) *
Then from Eqs. (17) and (18), we obtain
A - Ma + 1) - b. (21)
* Now the similarity variables can be formulated as
t IC (22)
We seek temperature solutions 0 of the form
e(s't) t f(77 $3f(O. (23)
*The temperature function f in the equation is a function of the similarity*variable 77 only. In terms of and we obtain the following expressions* through the chain rule.
B B-If A B-3 1 f-at
~B-Af~ (24)as
- ~B-2Af" (4
as2 (4
where the prime denotes the derivative with respect to 77. Substituting Eq.(23) for 0D and its derivatives in Eq. (24) into Eq. (5) leads to thefollowing ordinary differential equation.
f"+f[fI-+ (Cn/Cm)A,7fn- (25
- /CU) flfn-3+l - 0. (25)'
All of the constants C , CU, m, v, A, and B in the equation are known. In thesame manner, the boundary conditions (10a) and (10b) can be transformed to
15
fmf- - _ __ at '7 - 0 (26a)
Cm
f' 0 at '7 -0 (26b)
The initial condition (10c) is not needed for the solution of Eq. (25). Infact, the expression (23) has satisfied the initial condition (10c).
In the simple case that both the thermal properties of the solid and theboundary heat flux are constant (i.e., m n b -0, Cm - - 1), Eqs. (25)and (26) reduce to
f" + (l/2)vf" - (1/2)f" -0 (27)
f -1 at ' 0 (28a)
f 0 at '1) (28b)
B. Prescribed Surface Temperature TV - To(l + tb*)
The heat equation is identical to Eq. (25). In this case we will choose" a different set of expressions for A and B defined in Eqs. (19) and (20) so
that the boundary conditions (la) and (lib) are satisfied. Following theabove procedure, we obtain
A b*(m- n)/2 + 1/2 (29)
and
B = b*. (30)
The boundary conditions for Eq. (25) are
f I at n 10 (31a)
f 0 at ?1in. (31b)
The initial condition is not needed for the solution since the expression(23) has satisfied the initial condition (l1c). "u
In the case that the surface temperature Tw constant, the -.
dimensionless local temperature e given in Eq. (4) has to be redefined as
%..0
4.T e T (32)
16F
. .. "-....
- _ ,_ _' '. _. : _. -- -_ - _ _. . . . . . . . . .... . . .._*° . . . .' " " ." ",. " " " " , _ ,/ . _, ,*° _' 7 " '-. " .
in order to yield the same form of governing equation and boundary conditions
given in Eq. (25) and Eq. (31).
Eq. (25) vi' conditions (26) or (31) is ready to solve for the functionf and its derivative f'. Once the values of f and f" are obtained, the localtemperature in the solid can be calculated from Eq. (23). For the calculationof the local heat flux, we derive that
? T KoT o
Q -K- -Cm(- ) tbfmf: (33)
.L=
In dimensionless form it is
q = -Cm tbfmfr. (34)
IV. SAMPLE CALCULATIONS AND DISCUSSION
We chose type 420 stainless steel properties for sample calculationssince their strong temperature dependence can serve to explain thesignificance of the temperature dependence for heat calculations. Figure 1,
reproduced from Ref. 2, shows the experimental data of the thermalconductivity K and the specific heat C of the steel. Both increasemonotonically with temperature in the temperature range indicated. The datacan be nondimensionalized through the use of the dimensionless variablesdefined in Eq. (4). Taking To w 294 0K (700F) as the reference temperature,the results after nondimensionalization are shown in Figure 3. Theexperimental data can be approximated by the dashed lines which arerepresented by
k Cmm - 00.0 4 727 where Cm - 1 and m 0.04727 (35)
p Cnen - e0.249 1 where Cn = 1 and n - 0.2491. (36)
These approximated curves are much better representations of the thermal
properties as a function of temperature than simply assuming that k and c areconstants. The following will present solutions corresponding to the two typesof boundary conditions given in Eqs. (6) and (7).
A. Prescribed Heat Flux Qw atb
With C - Cn - 1, m - 0.04727, and n - 0.2491 as given in Eqs. (35) and(36), the heat equation (25) subjected to the boundary conditions (26) wassolved by using the Adams-Bashfortb method7 for b - 0, 1, 2, and 5. The
7R. Beckett and J. Hurt, Numerical Calculations and Algorithms, McGraw-HillBook Company, New York, 1967, p. 210.
17
-7-
4.01.
3.5 1.2
... .... .. ..
1.0Pj
Approximated: k 0 0.42
c
2.50.
F-0.
~2.0
.1.
0.5.
I*SI
18
second condition, f -0 at 1/- , was numerically satisfied when thecalculation was carried out to 7 - 6. The solutions of the function f and its
. derivative f' are tabulated in Table 1 and plotted in Figures 4 and 5,respectively. For comparison, solutions of f and f" for the case of constantproperties (i.e., Cm = Cn = 1, m - n = 0) are also provided in the figures.
Now the local temperature 0 in the solid can readily be obtained from Eq.(23). Figure 6 presents the result for a simple case of constant properties,i.e. K Ko, C = C and Q - constant. The agreement with the eractsolution from ef. 1 l.s excelrent. The surface temperature as a function oftime is represented by the dashed line in Figure 7. In another case ofconstant properties, when the boundary condition specified at the wallsurface is time dependent heat flux (Qw - atb), the resultant surfacetemperature is shown in Figure 8 (dashed line). These figures show no visibledifference between the present solutions obtained by the similaritytransformation and the exact solutions. When the temperature dependence ofthe thermal properties are accounted for (i.e., variable properties), theexact solutions do not exist and thus no comparison can be made. The solid Llines in Figures 7 and 8 represent the surface temperature for variableproperties. They are considerably lower, especially at large times, than thesurface temperatures for constant properties. This is due to the fact thatheat is absorbed and conducted away faster when the temperature dependence ofthe thermal properties is considered. The third line in Figure 7 shows thediscrepancy of their results for Ow constant. _
Table 1. Solutions of Temperature Function f and ItsDerivative f" at Surface (1 0)
Qw" atb
b fc fc fv fv
0 1.1284 -1.0000 1.2854 -0.98821 0.7523 -1 .0000 0.8977 -1.00512 0.6018 -1.0000 0.7380 -1.0145
5 0.4168 -1.0000 0.5352 -1.0300
Tw = TO(O + tb*)
b* f c fc fv fv
0 1.000 -0.5644 1.000 -0.64271 1.000 -1.1283 1.000 -1.0129
2 1.000 -1.5045 1.000 -1.37435 1.000 -2.2926 1.000 -2.0922
Subscripts c and v denote constant properties andvariable properties, respectively.
19
% ?
1.4
i.. 1.0 -
z
U
~ 0. ______Variable Properties
--- Constant Properties
bt0
0o.4 bs)a
0.2 \\
0.00 1 2 3 4 6
SIMILARITY VARIABLE ('7)
Figure 4. Profiles of Temperature Function(boundary condition Qv atb
20
0.0 --
'
u-4 -0.2
-0.6
'--0.2 / I'8b
-1.2
11/2/
-. 09 up
41.
1.2
1.0
0.3
Similarity Solution0 Exact Solution
0.6
0.2
SIMILARITY VARIABLE
Figure 6. Temperature Distribution for Constant Properties-~(boundary condition -w constant)
22
.. .... .r
#I
S20 20
10 10
0 0
0 200 AOO 600 600 1000
TIME (t)
Variable Properties, Similarity Solution--- Constant Properties, Similarity Solution
o Constant Properties, Exact Solution
Figure 7. Surface Temperature vs. Time(boundary condition Qv constant)
23
30
20/
10
012 A a 10
TIME (t)
Variable Properties, Similarity Solution- - Constant Properties, Similarity Solution0 Constant Properties, Exact Solution
Figure 8. Surface Temperature vs. Time(boundary condition = at b)
24
r
B. Prescribed Surface Temperature Tw TO(O + tb*
With the same values of C , C , m, and n used above, the temperaturefunction f and its derivatives I solved from Eq. (25) are tabulated in Table
and plotted in Figures 9 and 10. Using those values, the local temperatureand heat flux in the solid can be calculated from Eq. (23) and Eq. (34),
respectively.
In the case that TV - constant, the temperature 0 should be defined in ,,ethe form of Eq. (32) as explained earlier. As an example, if T V is chosen tobe 10330 K (1400 0 F), the expressions in (35) and (36) are altered to thefollowing:
00.071k-Cem 1.2060 o(37)0
cp -Cnen - 2.4478 00.328 (38)
Since b* -0 in this case, the local temperature is simply given as
e f(17) where ' ,,-. (39)
Figure 11 shows the local temperatures for both constant and variableproperties. The temperature for constant properties, represented by thedashed line, matches precisely the exact solution. The solid line next to itshows a lower temperature for variable properties. As explained in theprevious case, this is due to the fact that heat is absorbed and conducted tothe low temperature region faster when the temperature dependence of thethermal properties is taken into account. Figure 11 also presents acomparison of local heat fluxes versus '7 for the two kinds of thermal -"-
properties. The quantity in the figure is calculated from
qv- qc Cmfvmfv -fc, (40)
qc fc
where qc and qv are heat fluxes for constant properties and variableproperties, respectively. Using the data given in the expressions (37) and(38), the calculated values of fc: fc'I fv' and fv" are listed in Table 2.The similarity variable 17, at a given time, can be treated as the distancefrom the surface of the solid. The result in the figure shows that at thesurface the local heat flux qv for the variable properties is greater than thelocal heat flux qc for the constant properties by 37 percent. It isinteresting to note that the percentage drops to zero at n - 1.4 and thenturns negative as 7 continuously increases. To explain the reason we notethat in Figure 11 the two temperature curves merge at the surface and also faraway from the surface. Consequently, as t7 increases to a certain value, thegradient of the solid line changes from greater to smaller than the gradientof the dashed line. Meanwhile, the thermal conductivity and specific heat inthe expressions (37) and (38) for variable properties decrease withtemperature when ? increases. The combination of these two factors explains
the foregoing change in heat flux.
25
"_:' ,'. :- '* ' -_ . '. - . - . - -' ,' . : ,. . " *- - " , . _ _ . . . . . .- , . : .: - . _ s : l -
. .. . . .
I.A
1.2 '
. 10
z0
0.8Variable Properties
\Constant Properties
=0.6 "
b-0
0.4 b .
0.2
0 1 2 3 4.6
SIMILARITY VARIABLE (,7)
Figure 9. Profiles of Temperature Function(boundary condition Tv - To(0 + tb*))
26 - I-
g '1
. - - -: * -. ,--. .
0.0
boo- 0.4 4
b-1
S-0.8
-1.2
Variable PropertiesI - -- Constant Properties
-.6
<-2.0
-2.4
-2.50 1 2 3 4 5 6
SIMILAR1TY VARIABLE ('1)
- - Figure 10. Profiles of Derivative of Temperature Function(boundary condition T~ T (1 + t b*)
27
ft.
. I
a. a
14 I 0a.
1.2 30
1.0 20 £
S0.3 10
0.6 0E-
0.4 -10
0.2 -20
0.0 -300 12 3 4 5
SIMILARITY VARIABLE (')
Variable Properties, Similarity Solution-- Constant Properties, Similarity Solution0 Constant Properties, Exact Solution
Figure 11. Temperature Distribution and Beat Flux(boundary conditiou T. constant)
-.
-. •a %'
- . .. ..
Table 2. Values of Temperature Function f and Its Derivatives f"Inside the Solid for Tw Constant
fc fc, fv f (qv -qc)/qc
0.00 1.00000 -0.5644 1.00000 -0.6427 0.3730.24 0.86519 -0.5563 0.84627 -0.6333 0.3560.48 0.73420 -0.5326 0.69855 -0.5926 0.3080.76 0.59084 -0.4883 0.54251 -0.5172 0.2231.00 0.47931 -0.4394 0.42736 -0.4405 0.1381.24 0.38036 -0.3841 0.33094 -0.3625 0.0521.48 0.29506 -0.3263 0.25269 -0.2900 -0.0281.76 0.21302 -0.2600 0.18189 -0.2173 -0.1072.00 0.15698 -0.2075 0.13597 -0.1664 -0.1612.24 0.11283 -0.1609 0.10100 -0.1258 -0.1992.48 0.07915 -0.1214 0.07467 -0.0942 -0.220
V. SUMMARY AND CONCLUSIONS
Similarity solutions were obtained for the transient heat conduction in asemi-infinite solid with temperature-dependent thermal properties. The methodof similarity transformation via one-parameter groups provides a valuablemeans for the present analysis. The solution procedure is straightforward andthe solutions obtained are highly accurate, which can be used as referencedata for numerical solutions of similar nonlinear problems.
Results of the temperature distribution and the local heat flux in asample solid were obtained for two types of time dependent boundary
, conditions, namely heat flux atb and temperature T - T ( + tb*), In* the limiting case of constant properties, the present resur'ts are in excellent
• agreement with existing exact solutions. A comparison of the results forconstant properties and for variable properties has demonstrated thesignificance of temperature dependence of thermal properties for heat transfercalculations.
L
29"-,
.............................
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",._ ".,'., ," ." ." .- ,'. ". " • .- ,'-,' ,- ,',.-. -. : .. ,. ..-. .-.,. .-... .- ..-......-....-.. ,..- ... . .,-......,..... . ...
REFERENCES
1. H. Carslaw and J. Jaeger, Conduction of Heat in Solids, Oxford University
Press, London, 1959, Chapter 2.
2. Aerospace Structural Metals Handbook, U.S. Army Materials and Mechanics 7AResearch Center, Watertown, MA, 1980.
3. G. Schubert, C. Froidevaux and D.A. Yuen, "Oceanic Lithosphere and
Asthenosphere: Thermal and Mechanical Structure," J. Geophys. Res. 81, No.
20, 1976, pp. 3525-3540.
4. A.J.A. Morgan,"The Reduction by One of the Number of Independent Variables
in Some System of Partial Differential Equations," Quart. J. Math.
(Oxford), 2, 1952, p. 250.
5. A.G. Hansen, Similarity Analyses of Boundary Value Problems in Engineer- C.
iniz,Prentice-Hall, Englewood Cliffs, NJ, 1964.
6. W.F. Ames, Nonlinear Partial Differential Equations in Engineering, Aca-
demic Press, 1965, pp. 135-141.
7. R. Beckett and J. Hurt, Numerical Calculations and .Algorithms, McGraw-
Hill Book Company, New York, 1967, p. 210.
.4*
31
.4-i
rt NOMENCLATURE
*_[ A constant, Eqs. (19) and (21)
a constant, Eq. (6)
B constant, Eq. (20)
b, b* constants, Eqs. (6) and (7)
Cm, Cn constanta, Eqs. (2) and (3)o r
C dimensional specific heat at constant pressurep
pC dimensional specific heat at constant pressure evaluated at
cp dimensionless specific heat at constant pressure
f temperature function, Eq. (23)
" K dimensional thermal conductivity
K dimensional thermal conductivity evaluated at TO
k dimensionless thermal conductivity
- L reference length
m, n constants, Eqs. (2) and (3)
-' Q dimensional heat flux
"Q dimensional heat flux at surface
Qr reference heat flux, dimensional
q dimensionless heat flux
qv dimensionless heat flux at surface
s dimensionless distance, Eq. (4) -
T dimensional temperature
To dimensional initial temperature
Tw dimensional surface temperature i
t dimensional time
.P t dimensionless time
x dimensional coordinate
00 heat diffusivity
33r
• . -..** - .* --. ' X * ~ *
.- . . . . . . . . . .
t , similarity variables, Eq. (22)
e dimensionless temperature, Eq. (4)
dimensionless surface temperature
p dimensionless density
4 dimensional density
large distance
Subscripts-I-,-
c constant propertieso evaluated at To
r reference value
s at surface
v variables (temperature-dependent) properties
33z..c.
.*3 4. --k ri
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