UnsteadyCond - EFM (IF—Tm): / I 4-1 UNSTEADY-STATE...

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Unformatted text preview: EFM (IF—Tm): / I 4-1 UNSTEADY-STATE 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 unsteady-state heat-transfer problem. Unsteady-state 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 (ta-r (4-1) I 4-2 LUMPED-HEAT-CAPACITY 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 heat-transfer conditions from the surface of the ball to the surrounding fluid, i.e., the surface-convection heat-transfer 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 lumped-heat-capacity 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. 4-2. Thus dT cool/— 5:}1A(T* Ta.) : (1'7 (4-4) 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. (4-4.) is T _ Tx _ ekuxA/pr'v]? —1fi 4-5 TO_Tx ( ) I 4-6 TRANSIENT NUMERICAL METHOD The charts described above are very useful for calculating temperatures in certain regular-shaped solids under transient heat-flow 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 (4-24} 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 fi-tlJil—t A}! m,nii ._. I— Fig. 4-19 Nomenclature for oumerical solu- ra.~c:‘l;ex{ tion of two~dimensi0flal unsteady-state 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 (4-28} 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 + n-i— + Tglrr— + l ‘ Tfiin (4-29) 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 two-dimensional system or a value of 2 for a one-dimensional 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 finite-difference equations limit the values of A:- which may be used once AI is chosen. Note that if l/Q < 2 in Eq. (4-37), 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 finite-difference 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 99-103 2 one-dimensional systems 4 two—dimensional systems lV IV (1.57 Forward and Backward Differences The equations above have been developed on the basis ofaforwm-d~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 backward-difference formulation because the time de- rivative moves backward from the times for heat conduction into the node. The equation equivalent to Eq. (4-28) would then be Tart—Slut + Tfiftll n —' 2131+): ijjH-i—Jll’lrl + Tgifiil—l _ 2725:} (Ax)? (Ar)2 1 T”+1 — T9 = _ UL” HIJI at AT ( ) The equivalence to Eq. (4-29) is — Ar 13-7,”; = (3)02 (Tfilt'llJl + Thrill.” + Tfizjhl-t—l + Tfilfill-l) 40: Ar + 1 + T fifn' 4'37 i: (Axlli ’ ( ) We may now note that this backward-difference 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 backward-difference 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 forward-difference 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 x-direction 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 left-hand 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,M-3 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.

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UnsteadyCond - EFM (IF—Tm): / I 4-1 UNSTEADY-STATE...

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