Journal of Mathematical Sciences, Vol. 79, No. 6, 1996
THE TEMPERATURE FIELD IN A HEAT-SENSITIVE WITH A CUBIC CAVITY SUBJECT TO HEAT FLUX
SPACE
Yu. M. K o l y a n o , E. G. I v a n i k , a n d O. V. S i k o r a
UDC 539.377:536.12
We obtain the solution of the stationary heat-conduction problem for a space with a cubic cavity under the assumption that the coe~cient of thermal conductivity depends on the temperature. The surfaces of the cavity are subject to heat flux. By using the Kirchhoff variable and the method of continuation of functions We reduce the problem to a linear differential equation with singular coefficients on the right-hand side. We conduct a numerical study of the temperature distribution as a function of the spatial coordinate and the Kirpichev criterion that characterizes the thermal flux density.
We consider a heat-sensitive space containing a cubic cavity w = {(x, y, z) : txl _< a, lYl -< a, Iz ! < a}. We assume that a heat flux of power q is given on the surface of the cavity, i.e.
[A(t)~q-q]N*(y,z)=0 when x = q - a (xyz).
(i)
where A(t) is the coefficient of thermal conductivity, (xyz) denotes cyclic permutation of x, y, and z, N * ( a , Z) = N ( a ) Y ( ~ ) , g ( a ) = S + ( a + a) - S _ ( a - a), S • are asymmetric unit functions [3], and the temperature and heat flux vanish at infinity: hi
ot
t)~xx I~1--.or
=
lyl~
:
Izl---.~ = O,
.Izl,lyl,l~l---zr
= 0.
(21
The steady-state temperature field is determined from the heat equation of [6]:
o-7
~
+
~(t)
+ ~
1/:
Using the Kirchhoff variable
t~ = -~o
~(t)
= 0
(3)
(4)
A(t) dt,
we can reduce the boundary-value problem (1)-(3) to the form
~
= 0,
~+V ~*(y,z)=O for 0z9
0z9
lxl-~ = ~
0z9
M-~ = ~
(~) x=•
I~l-.~ = o,
(xyz), ~]l~t,l~.,~l-~-" O.
(6) (7)
In (4)-(7) we use the following notation: Q = )-q~o' A is the Laplacian. and ~o is the base coefficient of thermal conductivity. The unknown function zg(x, y, z) is defined in the domain f2 = ~2 \ ~. To have the possibility of applying an integral transform on the spatial coordinates, the definition of this function must be extended to the entire domain of variation of the coordinates. To do this we extend the function z9 to all of ]R3 as follows [1, 2]: = ~ M ( x , y, z), (8) Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 94-100. Original article submitted October 20, 1989. 1472
1072-3374/96/7906-1472515.00 9
Plenum Publishing Corporation
where M(.r. y, z) = 1 - N(x)N(y)N(z). Multiplying each term of Eq. (5) by the funct{on M(x,y, z) while taking account of the boundary conditions (6) and the symmetry of the boundary-value problem (5)-(7) with respect to the coordinates z, y, z, after certain transformations we obtain the following differential equation with singular coefficients for determining 0: ao = -[ N* ( y , z ) ~ ~) + N*(x,z)~,~) N'(~,y)~)], (9) where ~ ) = QL+(a) + Oi =~+0L~_(a), and L+(a) = 6+(c~ + a) + 6_(a - a). We represent the unknown values of Oa=a+0, c~ = x, y, z, on the square surfaces of the cubic cavity by a double functional series of Fourier type:
v~Iz=a+o = ~_~ ~ a~mcosAnycosAmz n----Om=O
(10)
(xyz),
where Aj = @, j = n. to. The coefficients a~,,, which are as yet unknown, are determined as follows:
anm
lff
a2
a
(11)
/)1 x=a+0 COSA.y COSA~ Z dy dz. a
Eq. (9) transforms into
Then
(12)
AO =-[N*(y,z)~(o~)+ N*(x,z)~(o y) + N'(x,y)~(o~)]. Here ~(o~) = QL+(x) + L_(x) ~ n=O
~ an,~cOsAnyCOSA~Z (xyz). rn=O
After applying the Fourier transform on the variables x and y to the differential equation (12) we have d2#
dz 2
~2~ = - F ( ~ , r~, ~),
(13)
where
fF
~ = .~_~1 oo~ F(~,7?,z) =
1{
~ O(x, y, z)e i~z+i'Ty dx dy,
N(z)[d12(~7)+dle(rl,~)]+2QL+(z)
sin~asinrja
~r]
72 = ~2 + rl2, ~
~
~
+ L'_(z)~ ~
)
a,~mdo(GAn)do(rLAm) ,
n=Om=O
d12 (~, r/) = dl(~, r]) - 42 (~, r~) dl (~, r/) -= 2r]- t Q cos ~a sin r]a,
d:(~,,) = ~sin~a ~
~~ anmcos~mzd0(,,~),
do(~,~ ) - sin(~n - ~)a + sin(An + ~)a
. = o m=o
M - ~
M +
Using the Fourier inversion formula to solve Eq. (13), after some transformations we obtain the following expression for the unknown function 0:
O(x, y, z) = QF(x, y, z).
(14)
In (14) we are using the following notation:
F(z, y, z) = 1 [fl+(x.y,z ) _ f~-(x,y,z) + f2+(x,y,z) + f2(x,y,z)] -
-~ ~-= m~--0 a;m
, r i o ~COS/~n'UCOS/~rnv[Fr
U2(x,y,z,u,v)]d~dv 1473
- - ~-'~ Z
7r n=0 rn=0 la+zl
Iz4"al•
fle=(x,y,z) = anm =
f
l (x, y, a) da, f2~(x,y,z) = f
0
o
anrn
Q '
fO(x,
f 2 ( y , z , ot)
/5/5
a~"n
7-1c~176
drl"
f2(y,z, ot)dot, fl(x,y, ot) = f~
ot) + f~
ot),
y, ot) = f l l ( X - , Y-, ot) + f n ( X +, Y - , ot) + f l l ( X - , Y+, ot) + fll(X +, Y+, ot),
= f l l ( Y - , Z-,ot) + fll(Y +, z - , ot) +
f n (u, v, a) = arcsinh ~
,u
X • = la+zl,
fn(Y-, Z+, ot) + Yn(Y+, Z+, a),
Y• = [a•
Fl(~,r/,z) = [~sin~ado(rl,,kn) +
?7sift 0ado(C, Xn)] [)~m sin/~rnaE1 (z) " ~ cos ,~mfiE2(z)dr-2-'/cos/~mzN(z)] ()k 2 +,,/,2)-1 E1 (z) = E _ (z) --1-E+ (z), E+(z) = e -'ylz+~L*, E2(z) = E_(z)sgn_(z - a) - E+(z)sgn+(z + a), sgn+ = 2S+(ot) - 1,
F~(x. y, z, ~, ~) =
Z•
(z•
Z ~, ~ + y ~ ( ~ + ~ ,
Z ~, ~ - y ~ ( ~ - ~ ,
p(~, ~, ~) = ( ~ + ~ + w ~)-3/2,
Z ~, ~ + y ) + p ( ~ - ~ , Z • v-y)],
z + = I~ + ~1.
It follows from (S) that ~91z=a+o = Olz=a+O. Substituting the value of Olz=a+o computed using (14) into (11), we obtain an infinite system of linear algebraic equations to find the unknown coefficients anm that occur in the expression (14) that defines 0, and consequently also the Kirchhoff variable 0. This system has the form OQ
ai~ + Z Z
" = B~j, Anmamn
i , j = 0,...,cx~,
(15)
n=0 rn=0 where A~m = ~
4 /oOf{/o~ o.),-1 cos ~a cos ~?yF1(~, 77,z) d~ dz] + /o~ ocosAnucosAmv[F+(a,y,z,u,v) - F 2 ( a , y , z , u , v ) ] d u d v )
B~j = 7ra1 2
/o~ ~[f2+(a,y,z)
cosAnycosAmzdydz,
+ f 2 ( a , y , z ) - f~(a,y,z)]cosA~ycosXjzdydz.
We note that the expression (14) defines a solution of Eq. (9) and hence a solution of the boundaryvalue problem (5)-(7) without any assumption as to the dimensions of the cubic cavity. The first author and I. I. Verba [2] have studied the case when the size of a square groove 2a is commensurate with the thickness of a thin plate. We assume that in the problem under consideration the size of the cube is also commensurate with the thickness of an imaginary thin plate carved into the body. Therefore instead of (10) we consider the expression
lff
T~ = -~dTa2
~
~zglx=~+~
(xyz),
(16)
i.e., we replace the expression O[~=a+0, ot = x, y, z, on the square surfaces of the cubic cavity by an integral characteristic. Obviously by the symmetry of the problem we can also set T~: = T u = Tz = To. In this case the solution of Eq. (9) has the form
o(z, y, z) = OaF(x, y, ~), where
27rF(x, y, z) = R + (X, Y, Z) - R-{ (X, Y, Z) + ~ E3 (z)[U + (X, Y) + U 1 (X, Y)] + 1474
(17)
f
Values of the coefficients a,~m
7D.
0 1 2 3 4
77,
0
i
2
3
4
0.468802 -0.216030 -0.265640 -0.284657 -0.377100
-0.308930 -0.279177 -0.278788 -0.279498 -0.214400
-0.273374 -0.279465 -0.279499 -0.279501 -0.278300
-0.277250 -0.279500 -0.279501 -0.279420 -0.278940
-0.274940 -0.267050 -0.273830 -0.266830 -0.278720
~[n+(x, Y, z ) + n ; ( x ,
~-~
Y, z)] + To { n ~ ( x , Y, z) - R;~(X, Y, Z)
+ 1 [n2(x, Y, z) - n + ( x , Y,
- Ea(z)]u2+(x,Y)+us Ea(Z)=sgn+(Z+l)-sgn_(Z-1),
X=X--,
Y = y,
a
a
z))]},
Z = z__ a
z?
R~(X, Y, Z) = sgn•
+ 1)/_
Ro(X,
Y, () dr,
Ro(X, Y, r = CI(X, tl, r - C2(X, Y, (),
oc
Cl(X, Y, r
= lnv+v~-')'+V2-,
C2(X, Y, r = lnT+Tf,
~'~ = Y ? + x/(1 + Z)2 + (1 - X) 2 + r
7~ = ( 1 + X ) 2 + r 2,
~'~ = Y? + V/(1 +
Z)2 + (1 + X) 2 + r
Ri2(X, Y, Z) = s g n •
fo
C, (X, Y, r de,
z+
R~3(X, Y, Z) : sgn• R~(X, II, Z) = sgn• X+ D•162
(1+X)2+r
+ 1)
[
+ 1)/-~o [D+(X'
[//
D+(X, II, r de + Y~
V/(I+X)2+(I+y)2+r
x,~=ll+xl,
Y' r + D-(X, Y, r de,
Y~=II:i:Y l,
f+
D•
]
Y, r de , Y~-
2 + v/(l:t:X)2+(l_y)2+r
] '
z.2=lZ+ll.
On the basis of (i6), using (17), we determine the integral characteristic To. To study the temperature field in this system it is necessary to use some specific law to prescribe the coefficient of thermal conductivity as a function of temperature. In [5, 7], for example, the following dependence was assumed: A(T) = AoX.(T), (18) where A.(T) = 1 - kT, k = 0.132 and T is the dimensionless temperature referred to the base to, which in the present case assumes the value to = 333 ~ C. Using (18) and relation (4), we find
T=~1 (1 -
~/1 -
2kKiF(x, y, z)),
(19)
where Ki = 9_e is the Kirpichev criterion [4], and the function F(x, y, z) is defined either by the expression to (14) in the case of cavities of arbitrary dimensions or by (17) for small values of the linear dimension of a cubic cavity. The expression (19) is a real solution of the steady-state problem of heat conduction for the heatsensitive space with cubic cavity under consideration. Since the temperature distribution is symmetric in the spatial coordinates x, y, z, we study the dependence of the temperature on the dimensionless variable 1475
.4 2
X referred to the linear dimension of the cubic cavity with fixed values of Y and Z and different values of the criterion Ki. To study the temperature field we wrote a software package in FORTRAN-77. In computing the values of the integrals that occur in the expressions that determine the unknowns we used Simpson's rule, whose error bound was estimated using the method of double computation. The values of the coefficients a,~m of the infinite system of linear algebraic equations were found using the standard subroutine GELG [8]. In doing this, at most 5 • 5 terms of the expansion (10) were required to obtain a 1% relative error in the solution of the system (15). Increasing the number of terms further did not have any significant influence on the results of the computations. The solution of the system (15) is given in the table. The results of the numerical study of the temperature field are given in the figure, where the dashed line shows the temperature of the heat-sensitive space and the dash-dotted line the approximate solution for a heat-sensitive medium obtained using the integral characteristic. The solid line corresponds to the temperature in a heat-sensitive space with a cubic cavity obtained using the expansions of the unknown values over the surfaces of the cubic cavity into double functional series. The behavior of the temperature T as a function of the coordinate X for different values of Ki is shown in the figure (the curves are l - - K i = 0.01, 2 - - K i = 0.1, 3--Ki = 1, ~{--Ki = 10, 5 - - K i = 100, Y = 1, Z = 0). These studies showed that taking account of the dependence of the coefficient of thermal conductivity on the temperature leads to a decrease in temperature in comparison with a non-heat-sensitive body, the maximal discrepancy between them being about 9%. The difference between the temperature defined by the solution obtained using expansions of Fourier series type (10) and the temperature obtained using the integral characteristic (16) in a heat-sensitive body is of the order of 3~. Increasing the Kirpichev criterion leads, as one should expect, to an increase in the temperature. As the coordinate X increases the temperature decreases, and a drop by more than two orders of magnitude in comparison with the value on the boundary Y = 1 along the X-axis occurs at a distance commensurate with one linear dimension of the cubic cavity. Literature Cited 1. V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1976). 2. Yu. M. Kolyano and I. I. Verba, "The Dirichlet problem for regions with rectangular grooves," Differents. Uravn., 21, No. 19, 1624-1626 (1985). 3. H. Korn and T. Korn, Handbook of Mathematics for Scientists and Engineers [in Russian], Nauka, Moscow (1975).
1476
4. A. V. Lykov. Theory of Heat Conduction [in Russian], Vysshaya Shkola, Moscow (1967). 5. J. Nowinski 'The transient thermoelasticity problem in an infinite medium with a spherical cavity and exhibiting temperature-dependent properties," Applied Mechanics: Proc. Amer. Soc. Mech. Eng., 29, No. 2, 197-205 (1962). 6. Ya. S. Podstrigach. V. A. Lomakhin, and Yu. M. Kolyano, Thermoelasticity of Bodies of Inhomogeneous Structure [in Russian], Nauka, Moscow (1984). 7. Ya. S. Podstrigach and Yu. M. Kolyano, Nonstationary Temperature Fields and Stresses in Thin Plates [in Russian], Naukova Dumka, Kiev (1972). 8. A Collection of Scientific Programs in FORTRAN: Programmer's Handbook. Matrix Algebra and Linear Algebra [in Russian], Statistika, Moscow (1979), No. 2.
1477