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Unformatted text preview: EFM (IF—Tm): / I 41 UNSTEADYSTATE
CONDUCTION INTRODUCTION If a solid body is suddenly subjected to a change in environment, some time
must elapse before an equilibrium temperature condition will prevail in the
body. We refer to the equilibrium condition as the steady state and calculate
the temperature distribution and heat transfer by methods described in Chaps.
2 and 3. In the transient heating or cooling process which takes place in the
interim period before equilibrium is established, the analysis must be modiﬁed
to take into account the change in internal energy of the body with time, and
the boundary conditions must be adjusted to match the physical situation which
is apparent in the unsteadystate heattransfer problem. Unsteadystate heat
transfer analysis is obviously of signiﬁcant practical interest because of the
large number of heating and cooling processes which must be calculated in industrial applications. The differential equation is GIT
ﬁx: in
(tar (41) I 42 LUMPEDHEATCAPACITY SYSTEM _. t1?“
,Cp}, 4—! We continue our discussion of transient heat conduction by analyzing systems
which may be considered uniform in temperature. This type of analysis is called
the lumped~heat—capacity method. Such systems are obviously idealized be
cause a temperature gradient must exist in a material if heat is to be conducted
into or out of the material. In general, the smaller the physical size ofthe body,
the more realistic the assumption of a uniform temperature throughout; in the
limit a differential volume could be employed as in the derivation of the general
heat—conduction equation. If a hot steel ball were immersed in a cool pan of water. the lumped—heat
capacity method of analysis might be used if We could justify an assumption
of uniform ball temperature during the cooling process. Clearly, the temperature
distribution in the ball would depend on the thermal conductivity of the ball
material and the heattransfer conditions from the surface of the ball to the
surrounding fluid, i.e., the surfaceconvection heattransfer coefﬁcient. We
should obtain a reasonably uniform temperature distribution in the ball if the
resistance to heat transfer by conduction were small compared with the con
vection resistance at the surface, so that the major temperature gradient would
occur through the fluid layer at the surface. The lumpedheatcapacity analysis,
then, is one which assumes that the internal resistance of the body is negligible
in comparison with the external resistance. The convection heat loss from the body is evidenced as a decrease in the
internal energy of the body, as shown in Fig. 42. Thus dT cool/— 5:}1A(T* Ta.) :
(1'7 (44)
where A is the surface area for convection and V is the volume. The initial
condition is written at 'T I 0 so that the solution to Eq. (44.) is T _ Tx _ ekuxA/pr'v]? —1ﬁ 45
TO_Tx ( ) I 46 TRANSIENT NUMERICAL METHOD The charts described above are very useful for calculating temperatures in
certain regularshaped solids under transient heatflow conditions. Unfortu nately, many geometric shapes of practical interest do not fall into these cat~
egories; in addition, one is frequently faced with problems in which the bound BET BET 5T
k + 6y2) = pC E (424} assuming constant properties. We recall from Chap. 3 that the second partial
derivatives may be approximated by 621‘ I , ‘ ,
6x3 2 (Ar)2 (fr7i+ I.” + 3"m 71.1: k 21mm) (4'23)
BZT l a 7: (Tmm + l + Th1.”  l i 2Tfn'Jl) The time derivative in Eq. (4—24) is approximated by if“ Tﬁiinl _ (4—27)
(31' A7
L
t ‘i ‘1 T
m. n + I _L
‘I T T— —' T
or
L ’m—im mn ﬁtlJil—t
A}!
m,nii
._. I— Fig. 419 Nomenclature for oumerical solu
ra.~c:‘l;ex{ tion of two~dimensi0ﬂal unsteadystate cone dLiCilOﬂ problem T£1+i,rl+ TﬁlilJr “ 2Tp “1,1: + Tgtm+l + T‘fim. —l _ 2733:.” I Tfhtll ﬂ TEFL”
(A10 (Ayf‘ 0: M
(428} 11 at any particular
may be calculated by writing ining the values of T1” 1. The In," ion after any desired number coordinates are chosen such
that
Ax = Ay
the resulting equation for Tﬁll'j becomes
at AT 4a A":
jail:1] : ; T{;!+ n + T527 n + ni— + Tglrr— + l ‘ Tﬁin
(429)
If the time and dIGIdUCE increments are conveniently chosen so that
Ax 2
( ) = 4 (4—30)
0: A7 A 2
T‘ZiH = (2x): (T3141 + Tfjpl) + [J * 0A7] Tﬁi governs the ease with which We may proceed to effect the numerical solution;
the choice of a value of 4 for a twodimensional system or a value of 2 for a
onedimensional system makes the calculation particularly easy. Once the distance increments and the value of Q are established, the time
increment is ﬁxed; and We may not alter it without changing the value of either
Ax or Q . or both. Clearly. the larger the values of A): and AT, the more rapidly
our solution will proceed. On the other hand, the smaller the value of these
increments in the independent variables. the more aCCuracy will be obtained.
At ﬁrst glance one might assume that small distance increments could be used
for greater accuracy in combination with large time increments to speed the
solution. This is not the case, however, because the ﬁnitedifference equations
limit the values of A: which may be used once AI is chosen. Note that if l/Q
< 2 in Eq. (437), the coefﬁcient of Tﬁ, becomes negative, and we generate a
condition which will violate the second law of thermodynamics. Suppose, for
example, that the adjoining nodes are equal in temperature but less than 1%,.
After the time increment Ar, Til, may not be IOWer than these adjoining tem
peratures; otherwise heat would have to ﬂow uphill on the temperature scale,
and this is impossible. A value l/Q < 2 would produce just such an effect;
so we must restrict the values ot'Q LO : (Ax): :{ This restriction automatically limits our choice of Ar, once Ax is established. it so happens that the above restrictions, which are imposed in a physical
sense, may also be derived on mathematical grounds. It may be shown that
the ﬁnitedifference solutions will not converge unless these conditions are
fulﬁlled. The problems of stability and convergence of numerical solutions are d. . _
15mm m SEE SCHETZ (1993) PAGES 99103 2 onedimensional systems
4 two—dimensional systems lV IV (1.57 Forward and Backward Differences The equations above have been developed on the basis ofaforwmd~dfj‘ference
technique in that the temperature of a node at a future time increment is
expressed in terms of the surrounding nodal temperatures at the beginning of
the time increment. The expressions are called explicit formulations because
it is possible to write the nodal temperatures TENT; explicitly in terms of the
previous nodal temperatures TﬁLH. In this formulation. the calculation proceeds
directly from one time increment to the next until the temperature distribution
is calculated at the desired ﬁnal state. The difference equation may also be formulated by CDmputing the space
derivatives in terms of the temperatures at the p + 1 time increment. Such an arrangement is called a backwarddifference formulation because the time de
rivative moves backward from the times for heat conduction into the node.
The equation equivalent to Eq. (428) would then be Tart—Slut + Tﬁftll n —' 2131+): ijjHi—Jll’lrl + Tgiﬁil—l _ 2725:}
(Ax)? (Ar)2
1 T”+1 — T9
= _ UL” HIJI at AT ( )
The equivalence to Eq. (429) is
— Ar
137,”; = (3)02 (Tﬁlt'llJl + Thrill.” + Tﬁzjhlt—l + Tﬁlﬁlll)
40: Ar
+ 1 + T ﬁfn' 4'37
i: (Axlli ’ ( ) We may now note that this backwarddifference formulation does not permit
the explicit calculation of the T”+1 in terms of T”. Rather, a whole set of
equations must be written for the entire nodal system and solved simultaneously
to determine the temperatures T9 H. Thus we say that the backwarddifference
method produces an implicit formulation for the future temperatures in the
transient analysis. The solution to the set of equations can be performed with
the methods discussed in SEE SCHETZ (1993) PAGES 102—103 The advantage of an explicit forwarddifference procedure is the direct cal
culation of future nodal temperatures; however, the stability of this calculation
is governed by the selection of the values of Ax and AT. A selection of a small
value of Ax automatically forces the selection of some maximum value of Ar.
On the other hand, no such restriction is imposed on the solution of the equa
tions which are obtained from the implicit formulation. This means that larger
time increments can be selected to speed the calculation. The obvious disad—
vantage of the implicit method is the larger number of calculations for each
time step. For problems involving a large number of nodes, however, the
implicit method may result in less total computer time expended for the ﬁnal
solution because very small time increments may be imposed in the explicit
method from stability requirements. Much larger increments in Ar can be em
ployed with the implicit method to speed the solution. A Earlier, we stated that an implicit method involves solving a system of equa
tions whose matrix of coefﬁcients is tridiagonal. Here, we shall show how the sys
tem can be solved directly in a simple manner. Referring back to Eq. (4—61) and
Fig. 4—l3tB), one can see that the algebraic problem at each step from n to n + l in
the xdirection will involve solving a system of equations for truth“ from m I 2 to
m = M —1ofthe form Anti: + A33“; + A241“ '5' Azsits + ' ' ' 'i' A2,M*l“M*l : 82
A33“; 'l" A3311; + 1434144 + A35?“ + ' ‘ ‘ + A3.M—l“M—I = BE
(4—63)
Ail/file”? 'l' Air—1.3“: + ' ' ‘ + AM—l.M—iuM—l Z BM—l First, one can ask what happened to m and HM, and what about the boundary condi
tions? On a surface, in = O, and it“ is known as in, = Ue(n + l). The boundary
conditions are implemented by ﬁrst taking Ami. from its apparent place at the ex
treme left of the ﬁrst equation in the matrix and putting it into 82, since that term
involves all known quantities. Next, the term AM_,M11M is taken from its apparent
place at the right end of the lefthand side of the last equation in the matrix and is put
into Bids, on the righthand side. since it is known. Now, the system of interest here is rridiagonal, since Eq. (4—61) has only
three of the unknowns for any given m. Thus, in the ﬁrst equation in the preceding
matrix, A24 to AMH are all identically zero. in the second equation, A35 to Arm,
are zero. And the last equation will have AM” to AM_1,M3 zero. For the model
equation, all the coefﬁcients on the main diagonal are simply (I + 2Q), and all the
coefﬁcients on the diagonals above and below the main diagonal are simply —Q (see
Eq. (4e61). The matrix of coefﬁcients will thus have a lot of zeroes, and one says
that such a matrix is sparse. The system might still be large, but fortunately, there is
a convenient method of direct solution for the tridiagonal case known as the Thomas
algorithm. According to this method, we rewrite the system for a generic interior
point m as Amuﬂ'i—l + Bmum + Cmum+l 2 Der and introduce Xm and Y”, by
him : Xmumf] + Ym where
“'C Dm _ A Y —[
m E m Ym E m m X B," + Ame_1’ Bm + Ameil ( The reader can substitute these two expressions into the original equation and conﬁrm that it is satisﬁed. That is, _Cm“m+l + Dm 7 Am Ym—l um : m
B," + Amei} Hm + Ame71 (4— 67) will satisfy the original equation. Now consider the situation for m = 2. We have
A2111 + Blitz + Can; = D: —> Bzuz : —C2u3 + (D2 — Azui) (4*68)
and
u; = X2u3 + Y2 (4—69)
Comparing these two expressions, we see that the following relationship must hold: E _C2 7 D2 _ A2111
1 X2 — Y1 (4—70) We also can write —C2 D11 _ A: Y1 X:——. r2:
2 B2+A2X, 32+A2X1 (4‘71) Comparing Eqs. (4—70) and (4w71), we see that we must have
XI = 0, Y I H] = 0 This leads to the ﬁnal, direct procedure: We solve for X”, and Ym from m = 2 to m =
M — 1, using Eq. (4—66), starting from X1 = Y1 = 0. We then solve for u,,. from
m = M — 1 down to m I 2 with Eq. (4—65), using the top boundary condition for
to”, i.e., MM = U.(n + l). ...
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This note was uploaded on 01/23/2012 for the course AOE 3044 taught by Professor Schetz,j during the Fall '08 term at Virginia Tech.
 Fall '08
 Schetz,J

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