计算传热学第6节-第2章 一维导热3

1、 Sun Jining 2008 BUAA1T=3上上节回顾p一维导热基本问题p一维稳态导热xzyI=1A/m2 U=10V/mT=5=10W/mk， c=0.5*103J/kgk，=7800kg/m33m Sun Jining 2008 BUAA2上上节回顾p一维导热基本问题p一维稳态导热(cT)Pt2-(0T)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面平均量） 源项（体平均量） TxPEWx(e(TE-TP)/xe-w(TP-TW)/xw) yz + SPxyz = 0(e(TE-TP)/xe-w(TP-TW)/xw)+SPx= 0eT

2、E/xe-eTP/xe-wTP/xw+wTW/xw + SPx=0(e/xe)TE-(e/xe+w/xw)TP +(wTW/xw)TW + SPx=0(e/xe)TE +(w/xw)TW + SPx =(e/xe+w/xw)TPxexwewaPTP =aETE +aWTW + baE =e/xeaW =w/xwaP = aE + aW b = SPx Sun Jining 2008 BUAA3x=0.6m上上节回顾p一维导热基本问题p一维稳态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5P233.3TP2=16.7TP3+16.7TP1+650TP

3、1 =16.7TP2+33.3TPa+6P3P550TP5=33.3TPb+16.7TP4+633.3TP3=16.7TP4+16.7TP2+6P433.3TP4=16.7TP5+16.7TP3+6PaPbTPa =3TPb =5TP1 =0.334*4.64+2.118=3.67TP2 =0.602*5.24+1.49=4.64TP3 =0.718*5.45+1.328=5.24TP4 =0.784*5.27+1.322=5.45TP5 =5.27P1TDMA算法（三对角矩阵算法、追赶法）先消元，再回代TP1 =0.334TP2+2.118 Sun Jining 2008 BUAA4=50W

4、/mk=10W/mkx=0.6m上节回顾p一维导热讨论问题p导热系数为x的函数(x)T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP3P4P5P2aP2TP2 =aE2TP3 +aW2TP1 + b2aE2 =e/xe=(1050)/2)/0.6P1aP1TP1 =aE1TP2 +aW1TPa + b1aE1 =e/xe =10/0.6aW1 =w/xw =10/0.3aP1 = aE1 + aW1 =30/0.6b1 = SP1x =1*10*0.6=6P2e/xe=?真的吗真的吗? ? Sun Jining 2008 BUAA5=10W/mk=50W/mkP2x=

5、0.6m上节回顾p一维导热讨论问题p导热系数为x的函数(x)T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP3P4P5P2aP2TP2 =aE2TP3 +aW2TP1 + b2aE2 =e/xe = 1/(xeP/P+ xeE/E) aE2 =1/(0.3/10+0.3/50)=27.8 P1aP1TP1 =aE1TP2 +aW1TPa + b1aE1 =e/xe =10/0.6aW1 =w/xw =10/0.3aP1 = aE1 + aW1 =30/0.6b1 = SP1x =1*10*0.6=6aE2 =e/xe Sun Jining 2008 BUAA6P2x=

6、0.6m上节回顾p一维导热讨论问题p导热系数为T的函数(T)T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP3P4P5=(10+0.5T)W/mkP2aP2(TP1,TP2,TP3)TP2 = aE2(TP2,TP3)TP3+aW2(TP2,TP1)TP1+6P1aP1(TP1,TP2) TP1 = aE1(TP1,TP2)TP2+aW1(TP1)Tpa +6P3P5aP5(TP4,TP5) TP5 = aE5(TP5) TPb+aW5(TP5,TP4)TP4+6aP3(TP2,TP3,TP4)TP3 = aE3(TP3,TP4)TP4+aW3(TP3,TP2)TP2

7、+6P4aP4(TP3,TP4,TP5)TP4 = aE4(TP4,TP5)TP5+aW4(TP4,TP3)TP3+6PaPbTPa =3TPb =5假设TP1=TP10TP2=TP20TP3=TP30TP4=TP40TP5=TP50代入可得aP1=aP10aP2=aP20aP3=aP30aP4=aP40aP5=aP50解方程可得TP1=TP11TP2=TP21TP3=TP31TP4=TP41TP5=TP51代入可得aP1=aP11aP2=aP21aP3=aP31aP4=aP41aP5=aP51解方程可得TP1=TP12TP2=TP22TP3=TP32TP4=TP42TP5=TP52解方程可得

8、TP1=TP1nTP2=TP2nTP3=TP3nTP4=TP4nTP5=TP5n Sun Jining 2008 BUAA7P2x=0.6m上节回顾p一维导热讨论问题p导热系数为T的函数(T)T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP3P4P5=(10+0.5T)W/mk解方程可得TP1=TP1nTP2=TP2nTP3=TP3nTP4=TP4nTP5=TP5n如果abs(TP1n -TP1n-1)10-xabs(TP2n -TP2n-1)10-x abs(TP3n -TP3n-1)10-xabs(TP4n -TP4n-1)10-x abs(TP5n -TP5n-

9、1)10-xTP1=TP1nTP2=TP2nTP3=TP3nTP4=TP4nTP5=TP5n为方程组的迭代近似解迭代方法 Sun Jining 2008 BUAA82 一维导热p一维导热基本问题p一维稳态导热p一维瞬态导热p一维导热讨论问题p导热系数为x的函数(x)p导热系数为T的函数(T)p源项为x的函数S(x)p源项为T的函数S(T)p小结 Sun Jining 2008 BUAA9T=32 一维导热p一维导热基本问题p一维瞬态导热xzyI=1A/m2 U=10V/mT=5=10W/mk， c=0.5*103J/kgk，=7800kg/m3 Sun Jining 2008 BUAA102

10、一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻 Sun Jining 2008 BUAA112 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） 非稳态项UP(cT)Pt2-(cT)Pt1)xyz假设空间阶梯分布UP=(cT)Pt2-(cT)Pt1)xyz PcP(TPt2-TPt1)xy

11、zxzyt2时刻t1时刻TxPEWewPEWxPew Sun Jining 2008 BUAA12扩散项QT(T/x)e-(T/x)w)yzt假设空间分段线性分布，时间阶梯分布2 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻PEWxQT=(T/x)e-(T/x)w)yzt(e(T/x)et1-(w(T/x)wt1)yzt(e(TEt1-TPt1)/xe-w(TPt1-TWt1)/xw) yztTxPEWewTtt2t1t3xe

12、xwew Sun Jining 2008 BUAA13源项SPxyzt空间一般为已知函数，假设时间阶梯分布2 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwST=SPxyzt SPt1xyztewTtt2t1t3 Sun Jining 2008 BUAA142 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散

13、项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwPcP(TPt2-TPt1)xyz= (e(TEt1-TPt1)/xe-w(TPt1-TWt1)/xw)yzt+ SPt1xyztPcP(TPt2-TPt1)x= (e(TEt1-TPt1)/xe-w(TPt1-TWt1)/xw)t+ SPt1xtPcPxTPt2-PcPxTPt1= (e/xe)tTEt1 -(e/xe)tTPt1-(w/xw)tTPt1+ (w/xw)tTWt1+ SPt1xtPcPxTPt2=PcPxTPt1+ (e/xe)tTEt1 -(e/xe)tTPt1-(w/xw)tTPt1+ (w/xw

14、)tTWt1+ SPt1xtew Sun Jining 2008 BUAA152 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwPcPxTPt2=PcPxTPt1+ (e/xe)tTEt1 -(e/xe)tTPt1-(w/xw)tTPt1+ (w/xw)tTWt1+ SPt1xtaEt1=(e/xe)taWt1=(w/xw)taPt1=PcPx-(e/xe)t-(w/xw)t =PcPx- aEt1 -aWt1a

15、Pt2= aPt1 + aEt1 + aWt1bt1 = SPt1xtaPt2TPt2= aEt1TEt1 + aWt1TWt1 + aPt1TPt1 + bt1ew Sun Jining 2008 BUAA16x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1x=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3xTPaPbP1P2P3P4P5PaPb Sun Jining 2008 BUAA17x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P

16、3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE1t0=(e/xe)t=(10/0.6)*10=167aW1t0=(w/xw)t=(10/0.3)*10=333aP1t0=PcPx- aE1t0 aW1t0=7800*0.5*0.6-167-333=1840aP1t1=aP1t0+aE1t0+aW1t0=2340b1t0= SP1t0 xt=1*10*0.6*10=602340TP1t1=167TP2t0 + 333 TPa+1840TP1t0+ 60TP1t1=(167*3+333*3+1840*3+60)/2340=3.0256P1 Sun Jining 20

17、08 BUAA18x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P2PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE2t0=(e/xe)t=(10/0.6)*10=167aW2t0=(w/xw)t=(10/0.6)*10=167aP2t0=PcPx- aE2t0 aW2t0=7800*0.5*0.6-167-167=2006aP2t1=aP2t0+aE2t0+aW2t0=2340b2t0= SP2t0 xt=1*10*0.6*10=602340TP2t1=167TP3t0 + 16

18、7 TP1t0+2006TP2t0+ 60TP2t1=(167*3+167*3+2006*3+60)/2340=3.0256P1 Sun Jining 2008 BUAA19x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：2340TP3t1=167TP4t0 + 167 TP2t0+2006TP3t0+ 60TP3t1=(167*3+167*3+2006*3+60)/2340=3.0256P1P42340TP4t1=167TP5t0 +

19、 167 TP3t0+2006TP4t0+ 60TP3t1=(167*3+167*3+2006*3+60)/2340=3.0256 Sun Jining 2008 BUAA20 x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P5PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE5t0=(e/xe)t=(10/0.3)*10=333aW5t0=(w/xw)t=(10/0.6)*10=167aP5t0=PcPx- aE5t0 aW5t0=7800*0.5*0.6-333-167=1840

20、aP5t1=aP5t0+aE5t0+aW5t0=2340b5t0= SP5t0 xt=1*10*0.6*10=602340TP5t1= 333 TPb + 167TP4t0+1840TP5t0+ 60TP5t1=(333*5+167*3+1840*3+60)/2340=3.31P1 Sun Jining 2008 BUAA21x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5xTPaPbP1P2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻： TP1=3.0256 TP2=

21、3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA22x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：设t=10s,t2=20s时刻：aE1t1=(e/xe)t=(10/0.6)*100=167aW1t1=(w/xw)t=(10/0.3)*100=333aP1t1=PcPx- aE1t1 aW1t1=7800*0.5*0.6-167-333=1840aP1t2=aP1t1

22、+aE1t1+aW1t1=2340b1t1= SP1t1xt=1*10*0.6*10=602340TP1t2=167TP2t1 + 333 TPa+1840TP1t1+ 60TP1t2=(167*3.0256+333*3+1840*3.0256+60)/2340=3.0476P1TP1=3.0256 TP2=3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA23x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P2PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均

23、匀，3设t=10s,t1=10s时刻：设t=10s,t2=20s时刻：aE2t1=(e/xe)t=(10/0.6)*100=167aW2t1=(w/xw)t=(10/0.6)*100=167aP2t1=PcPx- aE2t1 aW2t1=7800*0.5*0.6-167-167=2006aP2t2=aP2t1+aE2t1+aW2t1=2340b2t1= SP2t1xt=1*10*0.6*10=602340TP2t2=167TP3t1 + 167 TP1t1+2006TP2t1+ 60TP2t2=(167*3.0256+167*3.0256+2006*3.0256+60)/2340=3.0512

24、P1TP1=3.0256 TP2=3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA24x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：设t=10s,t2=20s时刻：2340TP3t2=167TP4t1 + 167 TP2t1+2006TP3t1+ 60TP3t2=(167*3.0256+167*3.0256+2006*3.0256+60)/2340=3.0512P1P

25、42340TP4t2=167TP5t1 + 167 TP3t1+2006TP4t1+ 60TP2t2=(167*3.31+167*3.0256+2006*3.0256+600)/2340=3.0716P52340TP5t2=333TPb + 167 TP4t1+1840TP5t1+ 60TP5t2=(333*5+167*3.0256+1840*3.31+60)/2340=3.556TP1=3.0256 TP2=3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA25x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=

26、5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：设t=10s,t2=20s时刻：xTPaPbP1P2P3P4P5TP1=3.0476 TP2=3.0512 TP3=3.0512 TP4=3.0716 TP5=3.556TP1=3.0256 TP2=3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA26x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P

27、4P5 Sun Jining 2008 BUAA27x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：设t=10s,t2=20s时刻：t300=3000s时刻：xTPaPbP1P2P3P4P5TP1 =3.65 TP2 =4.59 TP3 =5.17 TP4 =5.39 TP5 =5.25TP1=3.0476 TP2=3.0512 TP3=3.0512 TP4=3.0716 TP5=3.556TP1=3.0256 TP2=3.0256 T

28、P3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA28x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5xTPaPbP1P2P3P4P5TP1 =3.67 TP2 =4.64 TP3 =5.24 TP4 =5.45 TP5 =5.27稳态解：理论解 Tmax=5.348 x=2.166稳态理论解：TP1 =3.65 TP2 =4.59 TP3 =5.17 TP4 =5.39 TP5 =5.25瞬态稳定解： Sun Jining 2008 BUAA292

29、 一维导热p一维导热基本问题p一维瞬态导热xtPaPbP1P2P3P4P5t0PaPbP1P2P3P4P5t1PaPbP1P2P3P4P5t2PaPbP1P2P3P4P5t3PaPbP1P2P3P4P5t4P1P2P3P4P5P1P2? Sun Jining 2008 BUAA302 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） 非稳态项UP(cT)Pt2-(cT)Pt1)xyz假设空间阶梯分布UP=(cT)Pt2-(cT)Pt1)xyz PcP(T

30、Pt2-TPt1)xyzxzyt2时刻t1时刻TxPEWewPEWxPew Sun Jining 2008 BUAA31扩散项QT(T/x)e-(T/x)w)yzt假设空间分段线性分布，时间阶梯分布2 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻PEWxQT=(T/x)e-(T/x)w)yzt(e(T/x)et1-(w(T/x)wt1)yzt(e(TEt1-TPt1)/xe-w(TPt1-TWt1)/xw) yztTxPEWe

31、wTtt2t1t3xexwew Sun Jining 2008 BUAA32源项SPxyzt空间一般为已知函数，假设时间阶梯分布2 一维导热p一维导热基本问题p一维瞬态导热(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwST=SPxyzt SPt1xyztewTtt2t1t3 Sun Jining 2008 BUAA332 一维导热p一维导热基本问题p一维瞬态导热xtPaPbP1P2P3P4P5t0PaPbP1P2P3P4P5t1PaPbP1P2P3P4

32、P5t2PaPbP1P2P3P4P5t3PaPbP1P2P3P4P5t4P1P2P3P4P5P1P2显式时间格式 Sun Jining 2008 BUAA34x=0.6m2 一维导热p一维导热基本问题p一维瞬态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5xTPaPbP1P2P3P4P5TP1 =3.67 TP2 =4.64 TP3 =5.24 TP4 =5.45 TP5 =5.27稳态解：理论解 Tmax=5.348 x=2.166稳态理论解：TP1 =3.65 TP2 =4.59 TP3 =5.17 TP4 =5.39 TP5 =5.25瞬态稳

33、定解：计算传热学作业2一维瞬态导热程序V1.0 Sun Jining 2008 BUAA352 一维导热p计算传热学作业2：一维瞬态导热程序V1.0p功能要求：给定x方向长度，给定常数源项和导热系数、密度、比热，给定网格数，给定常数时间步长，给定总计算时间，给定两端点温度，可自动划分均匀网格，并计算网格中心点温度。p输入输出要求：计算条件由文本文件读入，计算结果以（t，x，T）格式输出到文本文件中。 Sun Jining 2008 BUAA362 一维导热p计算传热学作业2：一维瞬态导热程序V1.0pPPT需包括：程序结构图；程序正确性验证算例；讨论显式时间步长对瞬态解的影响；瞬态解和稳态解的

34、对比p作业目的：p通过编写计算程序掌握有限体积方法的瞬态离散方程，并掌握时间显式推进算法p通过对比稳态解和瞬态解，理解稳态解的时间推进算法p通过改变时间步长大小理解显式时间步长的局限性 Sun Jining 2008 BUAA37T=32 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式xzyI=1A/m2 U=10V/mT=5=10W/mk， c=0.5*103J/kgk，=7800kg/m3 Sun Jining 2008 BUAA382 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项

35、（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻 Sun Jining 2008 BUAA392 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） 非稳态项UP(cT)Pt2-(cT)Pt1)xyz假设空间阶梯分布UP=(cT)Pt2-(cT)Pt1)xyz PcP(TPt2-TPt1)xyzxzyt2时刻t1时刻TxPEWewPEWxPew Sun Jining 2008 BUAA40扩散项QT(T/x)e-(

36、T/x)w)yzt假设空间分段线性分布，时间阶梯分布2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻PEWxQT=(T/x)e-(T/x)w)yzt(e(T/x)et2-(w(T/x)wt2)yzt(e(TEt2-TPt2)/xe-w(TPt2-TWt2)/xw) yztTxPEWewTtt2t1t3xexwew Sun Jining 2008 BUAA41源项SPxyzt空间一般为已知函数，假设时间阶梯分布2 一

37、维导热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwST=SPxyzt SPt2xyztewTtt2t1t3 Sun Jining 2008 BUAA422 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwPcP(TP

38、t2-TPt1)xyz= (e(TEt2-TPt2)/xe-w(TPt2-TWt2)/xw)yzt+ SPt2xyztPcP(TPt2-TPt1)x= (e(TEt2-TPt2)/xe-w(TPt2-TWt2)/xw)t+ SPt2xtPcPxTPt2-PcPxTPt1= (e/xe)tTEt2 -(e/xe)tTPt2-(w/xw)tTPt2+ (w/xw)tTWt2+ SPt2xtPcPxTPt2=PcPxTPt1+ (e/xe)tTEt2 -(e/xe)tTPt2-(w/xw)tTPt2+ (w/xw)tTWt2+ SPt2xtew Sun Jining 2008 BUAA432 一维导

39、热p一维导热讨论问题p一维瞬态导热全隐时间格式(cT)Pt2-(cT)Pt1)xyz=(T/x)e-(T/x)w)yzt+SPxyzt非稳态项（体平均量） 扩散项（面时平均量） 源项（体时平均量） xzyt2时刻t1时刻xePEWxxwPcPxTPt2=PcPxTPt1+ (e/xe)tTEt2 -(e/xe)tTPt2-(w/xw)tTPt2+ (w/xw)tTWt2+ SPt2xtaEt2=(e/xe)taWt2=(w/xw)taPt1=PcPxaPt2= aPt1 + aEt2 + aWt2bt2 = SPt2xtaPt2TPt2= aEt2TEt2 + aWt2TWt2 + aPt1T

40、Pt1 + bt2ew Sun Jining 2008 BUAA44x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1x=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3xTPaPbP1P2P3P4P5PaPb Sun Jining 2008 BUAA45x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE1t1=(e/xe)t=(10/0.6)*

41、10=167aW1t1=(w/xw)t=(10/0.3)*10=333aP1t0=PcPx=7800*0.5*0.6=2340aP1t1=aP1t0+aE1t0+aW1t0=2840b1t1= SP1t1xt=1*10*0.6*10=602840TP1t1=167TP2t1 + 333 TPa+2340TP1t0+ 60P1 Sun Jining 2008 BUAA46x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P2PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE2t

42、1=(e/xe)t=(10/0.6)*10=167aW2t1=(w/xw)t=(10/0.6)*10=167aP2t0=PcPx=7800*0.5*0.6=2340aP2t1=aP2t0+aE2t1+aW2t1=2674b2t1= SP2t1xt=1*10*0.6*10=602674TP2t1=167TP3t0 + 167 TP1t0+2340TP2t0+ 60P1 Sun Jining 2008 BUAA47x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，

43、3设t=10s,t1=10s时刻：2674TP3t1=167TP4t0 + 167 TP2t0+2340TP3t0+ 60P1P42674TP4t1=167TP5t0 + 167 TP3t0+2340TP4t0+ 60 Sun Jining 2008 BUAA48x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P5PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：aE5t1=(e/xe)t=(10/0.3)*10=333aW5t1=(w/xw)t=(10/0.6)*10=167

44、aP5t0=PcPx=7800*0.5*0.6=2340aP5t1=aP5t0+aE5t0+aW5t0=2840b5t1= SP5t1xt=1*10*0.6*10=602840TP5t1=167Tbt1 + 333 T4t1+2340TP5t0+ 60P1 Sun Jining 2008 BUAA49x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：2674TP3t1=167TP4t0 + 167 TP2t0+2340TP3t

45、0+ 60P1P42674TP4t1=167TP5t0 + 167 TP3t0+2340TP4t0+ 60P22674TP2t1=167TP3t0 + 167 TP1t0+2340TP2t0+ 60P12840TP1t1=167TP2t1 + 333 TPa+2340TP1t0+ 60P52840TP5t1=167Tbt1 + 333 T4t1+2340TP5t0+ 60 Sun Jining 2008 BUAA50 x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5xTPaPbP1P2P

46、3P4P5t0=0s时刻：全场温度均匀，3设t=10s,t1=10s时刻：隐式时间格式解显式时间格式解TP1=3.0226 TP2=3.0255 TP3=3.0265 TP4=3.0402 TP5=3.258TP1=3.0256 TP2=3.0256 TP3=3.0256 TP4=3.0256 TP5=3.31 Sun Jining 2008 BUAA51x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=1000s,t1=1000s时刻：aE1t1=(

47、e/xe)t=(10/0.6)*1000=16667aW1t1=(w/xw)t=(10/0.3)*1000=33333aP1t0=PcPx=7800*0.5*0.6=2340aP1t1=aP1t0+aE1t0+aW1t0=52340b1t1= SP1t1xt=1*10*0.6*1000=600052340TP1t1=16667TP2t1 + 33333 TPa+2340TP1t0+ 6000P1 Sun Jining 2008 BUAA52x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P2PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5

48、t0=0s时刻：全场温度均匀，3设t=1000s,t1=1000s时刻：aE2t1=(e/xe)t=(10/0.6)*1000=16667aW2t1=(w/xw)t=(10/0.6)*1000=16667aP2t0=PcPx=7800*0.5*0.6=2340aP2t1=aP2t0+aE2t1+aW2t1=35674b2t1= SP2t1xt=1*10*0.6*1000=600035674TP2t1=16667TP3t0 + 16667 TP1t0+2340TP2t0+ 6000P1 Sun Jining 2008 BUAA53x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格

49、式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=1000s,t1=1000s时刻：35674TP3t1=16667TP4t0 + 16667 TP2t0+2340TP3t0+ 6000P1P435674TP4t1=16667TP5t0 + 16667 TP3t0+2340TP4t0+ 6000 Sun Jining 2008 BUAA54x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P5PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度

50、均匀，3设t=1000s,t1=1000s时刻：aE5t1=(e/xe)t=(10/0.3)*1000=33333aW5t1=(w/xw)t=(10/0.6)*1000=16667aP5t0=PcPx=7800*0.5*0.6=2340aP5t1=aP5t0+aE5t0+aW5t0=52340b5t1= SP5t1xt=1*10*0.6*1000=600052340TP5t1=16667Tbt1 + 33333 T4t1+2340TP5t0+ 6000P1 Sun Jining 2008 BUAA55x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx

51、=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=1000s,t1=1000s时刻：35674TP3t1=16667TP4t0 + 16667 TP2t0+2340TP3t0+ 6000P1P435674TP4t1=16667TP5t0 + 16667 TP3t0+2340TP4t0+ 6000P235674TP2t1=16667TP3t0 + 16667 TP1t0+2340TP2t0+ 6000P152340TP1t1=16667TP2t1 + 33333 TPa+2340TP1t0+ 6000P552340TP5t1=16667Tbt1

52、+ 33333 T4t1+2340TP5t0+ 6000 Sun Jining 2008 BUAA56x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5xTPaPbP1P2P3P4P5t0=0s时刻：全场温度均匀，3设t=1000s,t1=1000s时刻： TP1=3.46 TP2=4.09 TP3=4.52 TP4=4.79 TP5=4.96 Sun Jining 2008 BUAA57x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPbx=0.6

53、mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：aE1t1=(e/xe)t=(10/0.6)*10000=166667aW1t1=(w/xw)t=(10/0.3)*10000=333333aP1t0=PcPx=7800*0.5*0.6=2340aP1t1=aP1t0+aE1t0+aW1t0=502340b1t1= SP1t1xt=1*10*0.6*10000=60000502340TP1t1=166667TP2t1 + 333333 TPa+2340TP1t0+ 60000P1 Sun Jining 2008 B

54、UAA58x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P2PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：aE2t1=(e/xe)t=(10/0.6)*1000=166667aW2t1=(w/xw)t=(10/0.6)*1000=166667aP2t0=PcPx=7800*0.5*0.6=2340aP2t1=aP2t0+aE2t1+aW2t1=335674b2t1= SP2t1xt=1*10*0.6*10000=60000335674TP2t1=166667T

55、P3t0 + 166667 TP1t0+2340TP2t0+ 60000P1 Sun Jining 2008 BUAA59x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：335674TP3t1=166667TP4t0 + 166667 TP2t0+2340TP3t0+ 60000P1P4335674TP4t1=166667TP5t0 + 166667 TP3t0+2340TP4t0+ 60000 Sun Jini

56、ng 2008 BUAA60 x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P5PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：aE5t1=(e/xe)t=(10/0.3)*10000=333333aW5t1=(w/xw)t=(10/0.6)*10000=166667aP5t0=PcPx=7800*0.5*0.6=2340aP5t1=aP5t0+aE5t0+aW5t0=502340b5t1= SP5t1xt=1*10*0.6*10000=60000502340T

57、P5t1=166667Tbt1 + 333333 T4t1+2340TP5t0+ 60000P1 Sun Jining 2008 BUAA61x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：335674TP3t1=166667TP4t0 + 166667 TP2t0+2340TP3t0+ 60000P1P4335674TP4t1=166667TP5t0 + 166667 TP3t0+2340TP4t0+ 6000

58、0P2335674TP2t1=166667TP3t0 + 166667 TP1t0+2340TP2t0+ 60000P1502340TP1t1=166667TP2t1 + 333333 TPa+2340TP1t0+ 60000P5502340TP5t1=166667Tbt1 + 333333 T4t1+2340TP5t0+ 60000 Sun Jining 2008 BUAA62x=0.6m2 一维导热p一维导热基本问题p一维稳态导热T=3T=5P1PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5P233.3TP2=16.7TP3+16.7TP1+6P150TP1 =1

59、6.7TP2+33.3TPa+6P3P550TP5=33.3TPb+16.7TP4+633.3TP3=16.7TP4+16.7TP2+6P433.3TP4=16.7TP5+16.7TP3+6PaPbTPa =3TPb =5 Sun Jining 2008 BUAA63x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P3PaPbx=0.6mx=0.6mx=0.6mx=0.6mP2P3P4P5t0=0s时刻：全场温度均匀，3设t=10000s,t1=10000s时刻：335674TP3t1=166667TP4t0 + 166667 TP2t0+2340TP3t0+

60、60000P1P4335674TP4t1=166667TP5t0 + 166667 TP3t0+2340TP4t0+ 60000P2335674TP2t1=166667TP3t0 + 166667 TP1t0+2340TP2t0+ 60000P1502340TP1t1=166667TP2t1 + 333333 TPa+2340TP1t0+ 60000P5502340TP5t1=166667Tbt1 + 333333 T4t1+2340TP5t0+ 60000 Sun Jining 2008 BUAA64x=0.6m2 一维导热p一维导热讨论问题p一维瞬态导热全隐时间格式T=3T=5P1PaPb