1、传热学大作业传热学第四章大作业二维稳态导热问题的数值解法姓名:班级:学号:第一题:如图所示,一个无限长矩形柱体,其横截面的边长分别为和,常物性。该问题可视为二维稳态导热问题,边界条件如图中所示,其中=0.6m, =0.4m, =60,=20,。1) 编写程序求解二维导热方程。2) 绘制x=/2和y=/2处的温度场,并与解析解进行比较。已知矩形内的温度场的解析解为: 解:(1)建立控制方程及定解条件控制方程: 定解条件:(2)区域离散化(确立节点)将矩形区域分为M*N个网格,其中x方向上的步长;y方向的步长。设节点为(m,n)。(3)建立节点离散方程对节点(m,n)有:内节点:化简得边界节点:(
2、4)编程求解,程序见附录取M=N=50得到矩形区域各节点温度(见附件一),为方便在这里仅给出M=N=10时温度分布数据,如下表:606060606060606060606060.4486560.8287961.0919261.213461.1883861.0274160.7531460.39773606060.9277461.7095862.2468162.491962.4371862.1051761.5423660.81427606061.4716862.6990863.5315463.9036663.8092363.285562.4050261.26912606062.1241863.86
3、41265.0208665.5227665.3720864.6243963.381261.78318606062.9474565.2883666.8012267.4319167.2002766.1835164.5150962.37953606064.040367.0806368.9736469.7243169.377468.0322465.856663.08431606065.5775969.3828171.6537872.5045271.998270.2500867.4632863.92776606067.9025872.3668974.9651375.8878875.1709172.929
4、5569.4027464.945536066.1803471.7557176.1803479.021138079.0211376.1803471.7557166.1803460M=N=50时矩形区域节点温度分布图如图:M=N=50时x=/2处数值解与解析解温度场分布如下:M=N=10时将数值解与理论解进行比较,如下表所示:y00.4/90.4/9*20.4/9*30.4/9*40.4/9*50.4/9*60.4/9*70.4/9*80.4数值解6061.213462.491963.9036665.5227667.4319169.7243172.5045275.8878880理论解6061.17
5、44762.4128263.7824465.3578167.2246469.4844972.2602975.7030380相对误差00.064%0.127%0.190%0.252%0.308%0.345%0.338%0.244%0M=N=50时y=/2处数值解与解析解温度场分布如下图:M=N=10时将数值解与理论解进行比较,如下表所示:x00.6/90.6/9*20.6/9*30.6/9*40.6/9*50.6/9*60.6/9*70.6/9*80.6数值解6062.1241863.8641265.0208665.5227665.3720864.6243963.381261.7831860理论
6、解6062.1372464.016765.4116966.1539566.1539565.4116964.016762.1372460相对误差00.021%0.238%0.598%0.954%1.182%1.204%0.993%0.569%0第二题:将第一题中处的边界条件变为,其他条件不变。1) 编写程序求解二维导热方程并计算从y=0处导入的热量。2) 当时,该二维导热问题可简化为一维导热问题。在一维的近似下,试计算从y=0处导入的热量,并比较不同L2/L1下的比值。由该问题的解析解可知:L2/L10.0070.010.050.080.10.99870.99120.9560.930.912解:
7、编程得到M=N=50温度分布数据见附件二,得到温度分布图如下:这里也仅给出M=N=10时温度分布数据,见下表:606060606060606060606058.9248458.0307557.4169257.1105157.1105157.4169258.0307558.92484606057.7692255.9369354.697254.0848354.0848354.697255.9369357.76922606056.4364853.5797551.6986550.78750.78751.6986553.5797556.43648606054.7896250.7889648.269244
8、7.0839947.0839948.2692450.7889654.78962606052.605147.3399744.2467242.854242.854244.2467247.3399752.6051606049.4740242.925739.4683438.0055138.0055139.4683442.925749.47402606044.5750837.1376833.8034832.5066732.5066733.8034837.1376844.57508606036.1261529.5260127.2331326.4314826.4314827.2331329.5260136.
9、126156020202020202020202020从y=0处导入的热量可以近似看作从y=0向y=0.4/9处传递的热量,y=0.4/9处的温度分别为:x00.6/90.6/9*20.6/9*30.6/9*40.6/9*50.6/9*60.6/9*70.6/9*80.6T6058.9248458.0307557.4169257.1105157.1105157.4169258.0307558.9248460则整理得当时,该二维导热问题可简化为一维导热问题。当时,令M=N=100,得=1129571.429看作一维导热时不同下比值如下表:L2/L10.0070.010.050.080.11129
10、571.42978954015281292890729081142857.143800000160000100000800000.98840.98690.95510.92890.91140.99870.99120.9560.930.912附录Matlab程序:第一题:function chuanredazuoyeclearclcL1=0.6; %矩形长度L2=0.4; %矩形宽度Tw1=60;Tw2=20;global M N %设置网格数M=input(请输入将区间0,L1等分的个数M:);N=input(请输入将区间0,L2等分的个数N:);XDIF=L1/M; %x方向上的步长YDIF=
11、L2/N; %y方向上的步长axis(0,M,0,N);grid %设置网格U=initial(M,N,Tw1,Tw2,L1,XDIF);AIM=YDIF/XDIF;AIP=YDIF/XDIF;AJM=XDIF/YDIF;AJP=XDIF/YDIF;AP=2*YDIF/XDIF+2*XDIF/YDIF;CON=0; %离散方程系数T=rechuandao(U,M,N,AIM,AIP,AJM,AJP,AP);A=flipud(T);mesh(A);title(温度分布)xlabel(x)ylabel(y)zlabel(T)function U=rechuandao(U,M,N,AIM,AIP,A
12、JM,AJP,AP)while 1 temp=U; for i=2:M-1 for j=2:N-1 U(i,j)=(AIM*U(i,j-1)+AIP*U(i,j+1)+AJM*U(i-1,j)+AJP*U(i+1,j)/AP; %设置迭代条件 end end eps=abs(U-temp); if max(max(eps)1e-8 break; %限制迭代次数 endend function U=initial(M,N,Tw1,Tw2,L1,XDIF)U=zeros(M,N); %赋温度矩阵初值U(:,1)=Tw1;U(:,M)=Tw1;U(N,:)=Tw1;for i=1:MU(1,i)=T
13、w1+Tw2*sin(pi*i*XDIF/L1); end %初始和边界条件的设定第二题:function gaibianjietiaojianclearclcL1=0.6; %矩形长度L2=0.4; %矩形宽度Tw1=60;Tw2=20;global M N %设置网格数M=input(请输入将区间0,L1等分的个数M:);N=input(请输入将区间0,L2等分的个数N:);XDIF=L1/M; %x方向上的步长YDIF=L2/N; %y方向上的步长axis(0,M,0,N);grid %设置网格U=initial(M,N,Tw1,Tw2);AIM=YDIF/XDIF;AIP=YDIF/X
14、DIF;AJM=XDIF/YDIF;AJP=XDIF/YDIF;AP=2*YDIF/XDIF+2*XDIF/YDIF;CON=0; %离散方程系数T=rechuandao(U,M,N,AIM,AIP,AJM,AJP,AP);A=flipud(T);mesh(A);title(改变边界条件后温度分布)xlabel(x)ylabel(y)zlabel(T)function U=rechuandao(U,M,N,AIM,AIP,AJM,AJP,AP)while 1 temp=U; for i=2:M-1 for j=2:N-1 U(i,j)=(AIM*U(i,j-1)+AIP*U(i,j+1)+AJM*U(i-1,j)+AJP*U(i+1,j)/AP; %设置迭代条件 end end eps=abs(U-temp); if max(max(eps)1e-8 break; %限制迭代次数 endend function U=initial(M,N,Tw1,Tw2)U=zeros(M,N); %赋温度矩阵初值U(:,1)=Tw1;U(:,M)=Tw1;U(N,:)=Tw1;U(1,:)=Tw2; %初始和边界条件的设定
copyright@ 2008-2023 冰点文库 网站版权所有
经营许可证编号:鄂ICP备19020893号-2