·杆上一小段dx的熵变为： dS-JCpdT/T=JCp"pAdxdlnT Adk=dVC,:单位质量的热容 -CppAdx-In[((T+T2)/2)/(T+(T2-T)x/Lo)] 。 整个杆的熵变为dS的积分： AS=JdS=JCppA[ln((T+T2)/2)-In(T+(T2-T)x/Lo)]dx =CppAln((T1+T2)/2)Lo-CppAJIn(Tr+(T2-T1x/Lo)(Lo/(T2-T))d(T+(T2-T)x/Lo) =Cpln((T1+T2)/2)-Cp/(T2-Tlyry2Inydy y=Ti+(T2-Tx/Lo y1-T1y2=T2 JInydy=y2(Iny2-1)-yi(Iny1-1)=T2(InT2-1)-Ti(nTi-1) AS=Cp[In((T+T2)/2)-1/(T2-T(T2(InT2-1)-T(InT-1)) ● =CplIn((T+T2)/2)-((T2InT2-TInTD)-(T2-TD)/(T2-TDl =CplIn((T+T2)/2)-(T2InT2-TInT/(T2-T+1] (2) (2)式即为长杆熵变的数学表达式 ·证明熵变大于零： 令：A=ln(T1+T2)/2)-(T2lnT2TlnT/(T2-T)+1 若能证明A>0,则必有△S>0.• 杆上一小段dx的熵变为: • dS=∫CpdT/T=∫Cp 0AdxdlnT Adx=dV Cp 0 :单位质量的热容 • =Cp 0Adx·ln[((T1+T2 )/2)/(T1+(T2 -T1 )x/L0 )] • 整个杆的熵变为dS的积分: • S=∫dS=∫Cp 0A[ln((T1+T2 )/2)-ln(T1+(T2 -T1 )x/L0 )]dx • =Cp 0Aln((T1+T2 )/2)L0 -Cp 0A∫ln(T1+(T2 -T1 )x/L0 )·(L0 /(T2 -T1 ))d(T1+(T2 -T1 )x/L0 ) • =Cp ln((T1+T2 )/2)-Cp /(T2 -T1 )∫y1 y2lnydy • y=T1+(T2 -T1 )x/L0 y1=T1 y2=T2 • ∫lnydy=y2 (lny2 -1)-y1 (lny1 -1)=T2 (lnT2 -1)-T1 (lnT1 -1) • S=Cp [ln((T1+T2 )/2)-1/(T2 -T1 )(T2 (lnT2 -1)-T1 (lnT1 -1)) • =Cp [ln((T1+T2 )/2)-((T2 lnT2 -T1 lnT1 )-(T2 -T1 ))/(T2 -T1 )] • =Cp [ln((T1+T2 )/2) － (T2 lnT2 -T1 lnT1 )/(T2 -T1 ) + 1] (2) • (2)式即为长杆熵变的数学表达式. • 证明熵变大于零: • 令: A= ln((T1+T2 )/2) － (T2 lnT2 -T1 lnT1 )/(T2 -T1 ) + 1 • 若能证明A>0,则必有S>0