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传热大作业
传热学大作业
——二维导热问题的数值解法
学院：
核科学与技术学院
班级：
20121515班
学号：
2012151510
姓名：
JiXiaochao
第一题
图2.3所示是一个长矩形柱体的横截面，边长分别为
和
，材料为常物性。
由于柱体很长，且边界上的换热条件与坐标z几乎无关，因而可看作二维稳态导热，边界条件如图中所示，试求柱体内温度场的表达式。
由条件知，柱体的边界上的换热条件与坐标z几乎无关，因而可看作二维稳态导热所以我们先假设柱体的一些参数：
L1=.6m，L2=.4m，Tw1=60℃，Tw2=20℃，λ=200W/（mK）。
又经过查资料得到矩形内温度场的解析解为：
因此下面首先通过对柱体进行数值计算，求出柱体内部的数值解，然后再根据上述解析解的温度公式求出其解析解，然后将数值解与解析解进行对比，求出其误差。
1、通过对二维稳态导热问题的分析，取X和Y方向上的步长为△x=△y=1cm，把矩形区域划分为40×60的网格。
Cccc,l,cls,l
把矩形区域划分为40×60（共有41×61个节点）的网格后，则由柱体内部的导热微分方程：
建立的柱体二维平面离散方程为：
t（x,y）=.25*（t（x+1,y）+t（x-1,y）+t（x,y+1）+t（x,y-1））
其中，1＜y＜41，1＜x＜61。
又由假设和题意知柱体的边界条件为：
℃
℃
℃
℃
2、由离散方程，根据迭代算法，运用matlab语言编程，求解上述柱体内部温度的离散解，从而求解出其温度场的分布。
由上述分析的出求解柱体内部温度分布的matlab语言程序如下所示：
m=61;
n=41;
globalab;
tw1=60;
tw2=20;
L1=0.6;
L2=0.4;
y=zeros（1,n）;
x=zeros（m,1）;
t=zeros（m,n）;
T=t;
R=t;
er1=t;
Er=t;
t（:
1）=tw1;
t（1,:
）=tw1;
t（61,:
）=tw1;
Er=1;
i=;
while（Er>0.00000001）
fora=2:
m-1
forb=2:
n-1
z=0.01*a;
t（:
41）=60+20*sin（pi*z/.6）;
t（a,b）=（t（a-1,b）+t（a+1,b）+t（a,b-1）+t（a,b+1））/4;
x=0.01*（a-1）;
y=.01*（b-1）;
R（a,b）=tw1+（tw2/sinh（pi*L2/L1））*sin（pi*x/L1）*sinh（pi*y/L1）;
Eps=abs（t./R-1）;
Er=abs（t（a,b）/T（a,b）-1）;
T=t;
end
end
i=i+1
end
i
t
R
Eps
Er
figure
（1）,
mesh（t）;
title（'数值解温度场'）
figure
（2）,
mesh（R）;
title（'解析解温度场'）
figure（3）
mesh（Eps）;
title（'数值解解析解温度场误差分布'）
x=.6:
-（0.6/（m-1））:
;
y=.4:
-（0.4/（n-1））:
;
t1=t（:
（n-1）/2）;
R1=R（:
（n-1）/2）;
Er1=Eps（:
（n-1）/2）;
t2=t（（m-1）/2,:
）;
R2=R（（m-1）/2,:
）;
Er2=Eps（（m-1）/2,:
）;
figure（4）
plot（x,t1','g',x,R1','b',x,Er1','r'）
xlabel（'横坐标x'）
ylabel（'纵坐标t1R1Er1'）
title（'绘制y=L2/2处数值解解析解温度曲线及误差曲线'）
figure（5）
plot（y,t2','g',y,R2','b',y,Er2','r'）
xlabel（'横坐标y'）
ylabel（'纵坐标t2R2Er2'）
title（'绘制x=L1/2处数值解解析解温度曲线及误差曲线'）
Er=
9.9951e-09
数值解及其温度场如下：
t=
Columns1through9
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Columns10through18
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