1. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 서울대학교 기계항공공학부 담당 : 김 찬 중 열전달 강의 4 2013. 03. 26

2. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 2 스펙트럼 흑체방사도 ■ 플랑크(Planck)의 식 ■ 스테판-볼츠만 법칙 (Stefan-Boltzmann law)

3. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 3 예제 2 #> c1 = 3.7420e8; c2 = 14388; lam = 3; c1 = 3.742e+008 c2 = 14388 lam = 3 #> solve.T {1000} ( 800 = c1 / ( lam^5*(exp(c2/(lam*T))-1) ) ); ans = 634.12755

4. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 4 흑체복사함수 #> double fb(z) = int.x(600,z) ( 3.742e8/x^5/(exp(14388/x)-1))/_sigma; #> fb(3000); ans = 0.27321438 #> fb(4000); ans = 0.48084737

5. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 5 흑체복사함수 double fb(z) = int.x(600,z) ( 3.742e8/x^5/(exp(14388/x)-1))/_sigma; double fbinv(gsum) { dz = 100; z = 600; sum = fb(z); if( sum > gsum ) return z; while( 1 ) { z += dz; sum = fb(z); if( sum > gsum ) { z -= dz; dz /= 10; continue; } if( |gsum-sum| < 0.0001 ) break; } return z; } double fb12(z1,z2) = int.x(z1,z2) ( 3.742e8/x^5/(exp(14388/x)-1))/_sigma; // example 5-4 #> T = 5800; fb12(0.4*T,0.76*T); T = 5800 ans = 0.42604032

6. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 6 흑체복사함수, 예제 3,4 // example 5-3 #> T = 1500; T = 1500 #> fbinv(0.1); ans = 2195 #> lam1 = ans/T; lam1 = 1.4633333 #> fbinv(0.9); ans = 9373 #> lam2 = ans/T; lam2 = 6.2486667 // example 5-4 #> T = 5800; fb12(0.4*T,0.76*T); T = 5800 ans = 0.42604032

7. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 7 ans = [ 700 1.74515e-006 4100 0.498713 8000 0.856239 ] [ 800 1.63379e-005 4200 0.515984 8500 0.874557 ] [ 900 8.69151e-005 4300 0.532666 9000 0.889978 ] [ 1000 0.000320615 4400 0.548764 9500 0.903033 ] [ 1100 0.000910982 4500 0.564288 10000 0.914146 ] [ 1200 0.00213378 4600 0.579247 10500 0.923656 ] [ 1300 0.00431575 4700 0.593655 11000 0.931836 ] [ 1400 0.00778929 4800 0.607524 11500 0.938905 ] [ 1500 0.0128485 4900 0.620871 12000 0.945044 ] [ 1600 0.0197169 5000 0.633711 12500 0.950396 ] [ 1700 0.0285312 5100 0.646059 13000 0.955084 ] [ 1800 0.0393383 5200 0.657932 13500 0.959205 ] [ 1900 0.0521036 5300 0.669347 14000 0.962842 ] [ 2000 0.0667243 5400 0.68032 14500 0.966063 ] [ 2100 0.0830469 5500 0.690868 15000 0.968925 ] [ 2200 0.100882 5600 0.701006 15500 0.971477 ] [ 2300 0.120022 5700 0.710751 16000 0.973758 ] [ 2400 0.140248 5800 0.720117 16500 0.975805 ] [ 2500 0.161346 5900 0.72912 17000 0.977645 ] [ 2600 0.18311 6000 0.737775 17500 0.979304 ] [ 2700 0.205346 6100 0.746096 18000 0.980804 ] [ 2800 0.227878 6200 0.754097 18500 0.982163 ] [ 2900 0.250547 6300 0.76179 19000 0.983397 ] [ 3000 0.273216 6400 0.769189 19500 0.984521 ] [ 3100 0.295762 6500 0.776307 20000 0.985546 ] [ 3200 0.318083 6600 0.783154 25000 0.992162 ] [ 3300 0.340091 6700 0.789742 30000 0.995295 ] [ 3400 0.361714 6800 0.796083 35000 0.996963 ] [ 3500 0.382894 6900 0.802186 40000 0.997904 ] [ 3600 0.403583 7000 0.808062 45000 0.998423 ] [ 3700 0.423747 7100 0.81372 50000 0.998662 ] [ 3800 0.443357 7200 0.81917 55000 0.998712 ] [ 3900 0.462395 7300 0.82442 0 0 ] [ 4000 0.480849 7400 0.829479 0 0 ] [ 0 0 7500 0.834354 0 0 ] #> fbm++( ( 700:100:4000)' ) | fbm++( (4100:100:7500)' ) | (fbm++( (8000:500:19500)' ) _ fbm++( (20000:5000:55000)' )); matrix fbm(double z) = [ z, int.x(600,z) ( 3.742e8/x^5/(exp(14388/x)-1))/_sigma ];

8. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 8 double paracir(ri,rj,L=1) { double fij ; ri /= L ; rj /= L ; x = 1+(1+rj^2)/ri^2 ; fij = (x-sqrt(x*x-4*(rj/ri)^2))/2; return fij; } double pararec(x,y,H=1) { double fij; x /= H; y /= H; x2 = sqrt(1+x*x); y2 = sqrt(1+y*y); fij =0.5*log((1+x^2)*(1+y^2)/(1+x^2+y^2)) -x*atan(x)-y*atan(y) +x*y2*atan(x/y2)+y*x2*atan(y/x2); fij *= 2/pi/x/y; return fij; } double edgerec(x,y,L=1) { double fij; x = L/x; y = L/y; z = x*y/sqrt(x^2+y^2) ; fij = atan(x)/x+atan(y)/y-atan(z)/z +0.25*log( (1+x^(-2))*(1+y^(-2))/(1+z^(-2)) *((1+z^2)/(1+x^2))^(1/x^2)* ((1+z^2)/(1+y^2))^(1/y^2) ); fij *= x/pi; return fij; } 형상계수

9. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 9 #> paracir(12,6,15) ; ans = 0.091891197 예제 10

10. Computer Aided Thermal Design Laboratory www.msharpmath.com / 00 10 #> j1 = 23224; r2 = (1-0.8)/(pi*0.06^2*0.8); e2 = _sigma*300^4; #> r12 = 240.3; r13 = 24.3; r23 = 140; #> solve.q1.j2.j3 ( q1 = (j1-j2)/r12+(j1-j3)/r13, (e2-j2)/r2 = (j2-j1)/r12+(j2-j3)/r23, 0 = (j3-j1)/r13+(j3-j2)/r23 ); j1 = 23224 r2 = 22.104853 e2 = 459.3024 r12 = 240.3 r13 = 24.3 r23 = 140 ans = [ 190.204 ] [ 4663.73 ] [ 20478.9 ] #> solve [ q1=1, j2=2, j3=3 ] ( q1 = (j1-j2)/r12+(j1-j3)/r13, (e2-j2)/r2 = (j2-j1)/r12+(j2-j3)/r23, 0 = (j3-j1)/r13+(j3-j2)/r23 ); #> q1;j2;j3; q1 = 190.20368 j2 = 4663.7268 j3 = 20478.93 예제 11