ExACT Exact Analytical Conduction Toolbox

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1 ExACT Exact Analytical Conduction Toolbox X23B60T0 case Kevin D. Cole and Lukas G. Brettmann April 28, 2014 "Copyright (c) by Kevin D. Cole and Lukas Brettmann" 1 Description Slab body initially at ambient temperature with cosine-periodic heat flux variation at x=0 and dissipating heat by convection into a zero temperature environment at x=l 2 Schematic 3 Governing Equations The boundary value problem defining the temperature is given by (1) ( ) ( ) ( ) "Cite this article as follows: Kevin D. Cole and Lukas Brettmann, Slab body initially at ambient temperature with cosine-periodic heat flux variation at x=0 and dissipating heat by convection into a zero temperature environment at x=l, Exact Analytical Conduction Toolbox, exact.unl.edu, 2014."

2 Where is the ambient temperature. To normalize the temperature the following dimensionless variables are employed: (2) Then the dimensionless boundary value problem is given by ( ) ( ) (3) ( ) 4 Dimensionless Temperature The temperature solution is given in series form by ( ) { ( ), ( ) ( ) -} (4) ( ) ( ) The solution method is similar to that given in [1, p. 204] using the large-cotime Green s function, but with cosine heating instead of steady heating. Note that the temperature has two series added together. The first part is the steady-periodic part, and the second series is the decaying-transient part. The steady periodic part shown above is a slowly converging series; this term can be replaced by a non-series form [1, chap. 9] to give the recommended form of the solution: ( ) * ( ( ) ( ) ) + ( ) (5) Here the notation Real refers to the real part of a complex expression for which 2

3 5 Intrinsic Verification The solution given in Eq. (5) has been verified in two ways. First the steady-periodic portion of the solution has been computed from the series form given in Eq. (4) and in the non-series form given in Eq. (5). These two forms were shown to agree within 8 decimal places, however higher precision is limited only by the machine precision. Second, at early time (t<0.02) the temperature is zero in the center of the body because heat from the boundary has not yet arrived. The zero temperature is produced mathematically by cancellation of the two terms in Eq. (5), which provides an opportunity for intrinsic verification. That is, Eq. (5) is the sum of a steady-periodic (sp) term and a complementary transient term (ct) of the form ( ) ( ) ( ) (6) At and at early time ( ) the temperature is zero: ( ) ( ) or ( ) ( ) (7) Table 1 shows the temperature at for several small times ( ) along with the steady-periodic and the complementary transient portions of the solution. The number of terms used to compute the series was chosen by examining the exponential [1, p. 153]. Table 1 demonstrates agreement to ten decimal places at early time, thus providing verification of the temperature expression and the computer program. The Matlab code used to produce this table is given in the Appendix. 3

4 Table 1: Dimensionless temperature at early time demonstrating intrinsic verification between steady-periodic (sp) and Complimentary transient (ct) portions of the solution. # terms E E E E E E E E E References 1. Cole, K.D., Beck, J. V., Haji-Sheikh, A., and Litkouhi, B., Heat Conduction Using Green s Functions, Second Edition, Taylor and Francis,

5 Figure 1: Temperature versus position for case X23B60T0 at several dimensionless times and at dimensionless frequency at Bi =1 Figure 2: Temperature versus time for case X23B60T0 at location x/l =0.5 for three dimensionless frequencies at Bi=1. 5

6 7 Appendix: Matlab Code % MATLAB function fdx23b60to.m % Slab X23 heated by cosine periodic heat transfer. % Provides for intrinsic verification at small time. % Lukas Brettmann, Kevin D. Cole, 28 May 2014 % input values % x location, normalized as x/l % t time, normalized as alpha*t/l/l % w frequency, normalized as w*l*l/alpha % output values % td temperature, normalized as (T - T0)/(q L/K) % td1 complementary transient portion of temperature % td2 steady-periodic portion of temperature % n number of series terms for comp. trans. portion % other quantities % bm eigenvalue for x21 case, pi*(m - 1/2) % beta complex quantity from steady-periodic solution % comments % slab of thickness L, conductivity k, diffusivity alpha % heat flux at x=0 has form q = q0 *cos(w*t) = real[ exp(i*w*t) ] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [td,td1,td2,n] = fdx23b60t0(x,t,w,b2,n) % compute Fourier series portion of solution % Use exponential to choose the number of terms in series. % Last term in series will be of order exp(-23) = 1.E-10 kmax = 1 + (4./pi/pi*23./t)^(1/2); mmax=82; %if ( sizet > 1 ) dt = td(2) - td(1); % first entry might be zero %else %dt = td(1); %end %mmax=3*floor(1/2+1/pi*sqrt(a*log(10)/dt)); betar=feigxij(mmax,1.0e-15,b2); betav=betar(:,2); b2=b2; % initialize sum sum = 0.; for m=1:mmax; bm=betav(m); %Num=(bm^2+B2^2); RR=2*Num/(Num+B2)*exp(-bm^2*td1); % begin loop on k %for m=1:mmax a1 = (((bm*bm)/w)*exp(-bm*bm*t))/(((bm*bm)/w)^2+1); a2 = ((bm*bm+b2*b2)/(bm*bm+b2*b2+b2))*cos(bm*x); sum = sum - (1/w)*2.*a1*a2; % end of k loop td1 = sum; n = round(kmax); % save nearest integer, number of terms end % compute non-series form of steady-periodic portion sigma = sqrt(w)*sqrt(i); % note i is sqrt(-1), complex R=(sigma-b2)/(sigma+b2); 6

7 ex = exp( - sigma*x); e2b = exp(-2.*sigma); e2mx = exp(-sigma*(2. - x)); dum1 = exp(i*w*t) * (ex + R*e2mx)/(sigma*(1.-R*e2b)); td2 = real(dum1); td = td1 + td2; % total temp. is sum of two parts % end file % function callx23b60t0.m % Calling routine to print a table of temperature values % L. Brettmann, K. Cole 1/20/14 % input values % xmax max location, actually max x/l % t time value, normalized as alpha*t/l/l % w frequency, normalized as w*l*l/alpha % n number of times to be computed % fname 'name' of output file must be in single quotes as shown % output values % t time, subdivided as tmax/n, normalized as alpha*t/l/l % tmp1 steady-periodic part, non-series form % tmp2 decaying transient, Fourier series % tmp temperature normalized as (T - T0)/(q0 *L/k) % other comments % slab of thickness L, conductivity k, diffusivity alpha % heat flux at x=0 has form q = q0 *cos(w*t) = real[ exp(i*w*t) ] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function eff = xcallx23b60t0(x,tmax,w,n,b2,fname) % open output file fid = fopen(char(fname),'w'); % print out table column headings head = ' x time freq T_ct T_sp T_total #terms'; fprintf(fid,'%s\n',[head]); for k = 1:n % loop over times % compute temperatures at each time t = tmax*k/n; [tmp,tmp1,tmp2,num] = fdx23b60t0(x,t,w,b2,n); % print out one row of table fprintf(fid,'%7.3f %7.3f %7.3f %14.10f %14.10f %14.10f %4d \n',[x,t,w,tmp1,tmp2,tmp,num]); end; fclose(fid); % close output file % end file %feigxij.m 6/20/07 Matlab subroutine written by James V. Beck based on programming and method of Prof. A. Haji-Sheikh, %Haji-Sheikh, A. and Beck, J.V., "An Efficient Method of Computing Eigenvalues in Heat Conduction", %Numerical Heat Transfer Part B: Fundamentals, Vol 38 No. 2, pp , %It gives the eigenvalues for the X11, X12, X13,..., X33 cases. X33 is the general case and includes the others. 7

8 %However, the X11, X12, X21, X22 eigenvalues are well-known, for X11 and X22, the eigenvalues are n *pi, for n = 1, 2, 3,.... %(Also n = 0 for X22.) For X12 and X21, the eigenvalues are (2*n-1)*pi/2 for n = 1, 2, 3, function XIJ=feigXIJ(m,B1,B2) %Input quantities %B1 is the one of the Biot numbers. For X31, it is for Bi_1. %B2 is the other Biot number. For X13, it is for Bi_2. Actually the eigenvalues for X13 and X31 are the same. %For the boundary condition of the first kind (given temperature), Bi value goes to infinity. Use BI1=1.0e+15 or BI2 = 1.0e+15. %For the case of bc of the seoond kind, the Bi value goes to zero. Use BI1=0 or BI2 = 0 %m is the number of eigenvalues %Accuracy: Machine accuracy is expected, about 14 decimal place accuracy BI1=B1; BI2=B2; AN=1:1:m; %Immediately below generates the X33 eigenvalues using the Haji subroutine CN =(AN-0.75)*pi; DN=(AN-0.5)*pi; D=1.22*(AN-1)+0.76; P23 = 1./AN; BX = BI1*BI2; BS = BI1 + BI2; GAM = *(sqrt((BS + BX+CN-pi/4.0)./(BS +BX+D))- (BS+BX+sqrt(D.*(CN-pi/4.0)))./(BS+BX+D)); X23 = (BI1+BI2-CN)./(BI1+BI2+CN); X13 = BX./(BX+0.2+BS*(pi*pi*(AN-0.5)/2.0)); G13=1+X13.*(1-X13).*(1-(0.85./AN)-( /AN).*(X13+1).*(X /AN)); E23 = GAM.*(P23.*X23 + (1.0 - P23).*tanh(X23)/tanh(1)) ; E13 = 2.0*G13.*X13; Y23=CN+pi*E23/4.0; Y13=DN+pi*E13/4.0; ZN =(Y13*BX./(BX+DN.^2)+Y23.*(1.0-BX./(BX+DN.^2))); for i=1:3 A0=(( *BS+BX-ZN.*ZN).*cos(ZN)-(6.0+BS) *ZN.*sin(ZN))/6.0; B0=(ZN.^2-BS-BX).*cos(ZN)+(2.0+BS)*ZN.*sin(ZN); C0=(ZN.^2-BX).*sin(ZN)-BS*ZN.*cos(ZN); A0=(1.0+BS)*sin(ZN)+2.0*ZN.*cos(ZN)-C0/2.0; E0=-C0./B0-(-A0.*C0.^3+A0.*C0.^2.*B0)./(3.0*A0.*C0.*C0.*B0-2.0*A0.*C0.*B0.^2+B0.^4); ZN = ZN+E0; end if BI1 == 0 & BI2 == 0 ZN(1)=0; end num=1:1:m; XIJ=[num' ZN']; 8

Handbook of Computational Analytical Heat Conduction

Handbook of Computational Analytical Heat Conduction Filippo de Monte, James V. Beck, et al. January 7, 2013 1 Contents 1. Problem description 2 2. Dimensional governing equations. 2 3. Dimensionless variables. 3 4. Dimensionless governing equations.. 3