10.1.1Analysis for energy equationA correct energy equation is:aaa(pcpT)a(pcpUT)aTa(ocpuT)OT+ST.axatayayaxaxdy(2)Rewrite into the previous format of G.E.:aaSTa(pT)a(puT)a(poT)(AT)(AOT)axataxayaxayCp ayCpCpacpaTacpOTOCnuTpCpOTataxaxayay(3)This term was neglected in the previous G.EWhen it can be neglected?(1) Cp is constant;6/68
6/68 A correct energy equation is: Rewrite into the previous format of G.E.: This term was neglected in the previous G.E. （3） （2） 10.1.1 Analysis for energy equation When it can be neglected? p （1）c is constant；
(2) Or following three terms simultaneously equal zero)lJCOCpuT-a=0, pCpoT-入=0and(4)1oCp.ataxay(3) Or the sum of the three terms equal zero!Obviously,suchchances are quitelimited!Hence,simulationresults based on previous G.E. inherently include some errors!10.1.2 Improved format of the general G.E.The framework(框架)of the previous general G.E.isretained（保留）butthediffusioncoefficientisresumedto恢复到）its original value（a)by introducinga nominaldensity (pc,) as follows:(5)a(p*d)+ div(p*pU) = div(Tsgradp) + Sat7/68
7/68 Obviously, such chances are quite limited! Hence, simulation results based on previous G.E. inherently include some errors! 10.1.2 Improved format of the general G.E. The framework (框架) of the previous general G.E. is retained (保留), but the diffusion coefficient is resumed to （恢复到） its original value () by introducing a nominal density (cp ) as follows: * * * ( ) div U div grad S ( ) ( ) t + = + （5） （2）Or following three terms simultaneously equal zero: （3）Or the sum of the three terms equal zero! （4）
The new form of G.E. for different variables are:SpEquationdToO010ContinuityequationapMomentumegn.(xdirection)uofu6.Momentum eqn. (y direction)uUpfyayTST2Energy equationocpThat is, now we regard pcp in the energy equationas ageneral density:aa(pcpT)(pcpuT)a(pcpoT)OTdSTataxaxdyaudrSuch a treatment is much better than taking a / cas a nominal diffusion coefficient and Sr / cpas anominalsourcetermLi W, Yu B, Wang Y, et al. Communications in Computational Physics, 2012, 12(5):1482-14948/68
8/68 That is, now we regard in the energy equation as a general density： p c Such a treatment is much better than taking as a nominal diffusion coefficient and as a nominal source term . / p c / T p S c The new form of G.E. for different variables are: Li W, Yu B, Wang Y，et al. Communications in Computational Physics, 2012, 12(5): 1482-1494
10.1.3 Comparisons of energy G.ESTaaa(pT)a(puT)a(pT)AaTa)(Old)(1)axataxayCpaxCpQyCpa(pcpuT)a(pcpoT)aa(pcpT)aTaaT(New)+ST.(5)ataxaxayaxdydy1) Example 1 ---Natural convection in enclosure witha moving lid and variable fluid heat capacityUT,=-10CT =-10°C3500((O.p/r)3000E-02500Cp=(0.001174-0.009T3+2000cs0.09T?+111T+1993)J(kg·K)150(Th=50°C100to560304010T (c)9/68
9/68 10.1.3 Comparisons of energy G.E. 1）Example 1 -Natural convection in enclosure with a moving lid and variable fluid heat capacity 4 3 2 -1 (0.0011 0.009 0.09 111 1993)J(kg K) p c T T T T = − + + + • T c = −10 C Th = 50 C (Old) (New) （5） （1）
For the new G.E. of Eg.5, fluid heat capacity can bedeterminedbythelocaltemperature(correctone)For original G.E. of Eq.1, five average c, are usedwhich lead to large deviations with the correct one.3060Eq (1) OldEq.(1)-o--C,C1C24-48--- C2C+-C360C4三E1224Eq (5) New12ResultsfromEq.(5)00.20.40.60.80.20.01.00.00.80.40.6XX(b)Relativedeviation(a)Nusseltnumber10/68
10/68 For original G.E. of Eq.1, five average cp are used, which lead to large deviations with the correct one. Eq（1）Old Eq（5）New Eq.(1) Results from Eq.(5) For the new G.E. of Eq.5, fluid heat capacity can be determined by the local temperature (correct one)