高等传热学复习守恒方程

1、高等传热学对流部分主讲：朱华zhuzju浙江大学能源工程系及参考书等，高等教育：高等传热学参考书：，20081.高温传热学屠传经等编著，浙江大学，19972. Heat and Mass TransferE.R.G. Eckert & R.M. Drake中译本传热与传质分析3. Convective Heat TransferA. Bejan, 19854.对流传热传质分析杰，西安交大，19915.对流换热理论，19836. Convective Heat TransferL.C.Burmeister, 19837. Convective Heat and Mass Transfer

2、W.M.Kays, 19808.高等传热学杨等编著, 上海交通大学, 2004.第一讲守恒方程Conservation EquationsI. Introduction(1) Convection 对流bulk motion of fluidConvehere 拉丁文，搬东西(流体)流体(2) Continuum 连续体（稀薄气体除外）(3) 两种处理Langrangefixed mass system(取一体0,1atm,air 106molecularsC.V.3.3×10-5cmd)B.L.Eularfixed volume system(open)(4) Convective

3、 heat transfer coefficienta(苏),h(英美)Solution for hl¶th =f×求解贴壁处的温度梯度t¥ - tw¶yy =0+hDt = -l ¶t= -l ¶tfs ¶y¶yy =0+y =0-Newtons lawFourierss lawyttwxh 的影响因素：h =f (u, l,r,m, c,l)¶t¶y流体t 分布贴壁处t 梯度分析法求解+y=0：PDEs 对流换热PDEs Complete egs.(仅方程和结果）B.L.PDEs(exact

4、)IEs以前(approximate)比拟详细，不再论述比拟，比拟等热-动实验大学重点不再重复(experimental )相似理论 PDEs量纲分析数值解(Numerical ) PDEs (专门课)本课重点之一求PDEsII.Complete Eqs.完整的对流换热PDEs(1) Mass conservation Eg.(Continuity eg.)质量守恒原理Rate of creation of mass = 0(a) 3-D,us/s,Compressible ( r¹ Const , r=f (P)¶r¶(ru)¶(rv)¶(r

5、w)¶t+¶x+= 0¶y¶z或Ñ(rrV ) = 0divergence 散度或div(rrV ) = 0V-velocity vector 速度向量C.V.dr另一形式：+ rdivV= 0dt¶¶¶+ w ¶d全导数=+ u+ vdt¶t¶x¶y¶z¶(ru) + ¶(rv) + ¶(rw) = 03-D, s/s, incompressible¶x¶y¶z(b) s/s,incompressible

6、 =const.=Const.incompressible f(P)constant property f(T)divV=0¶u + ¶v = 02-D¶x¶yd r= 1 ( ¶r+ 1 ( ¶rr=f(T,P)d T = Kd P - bd Tr ¶P )Td Pr ¶T )PrT1 ( ¶rK=r ¶P )T为等温压缩系数Tb= 1 ( ¶v )= - 1 ( ¶rr ¶T )P为容积膨胀系数Pv¶Td r= 0IfKT00(液体)ordP0dT0r

7、（气体，当Mach0.3）KT=2×10-5/bar,=2×10-4/Kwaterideal gasb= 1 » 10-3/KT(c) In tensor notation x,y,zx1,x2,x3xi u,v,wu1,u2,u3uiaibi=a1b1+a2b2+a3b3表示三条规则¶r¶(rui )(i)¶t+= 0¶xi¶r¶(ui )¶t+ r ¶x= 0(ii)(iii)i¶(rui ) = 0¶xi¶(ui ) = 0(iV)¶xi(

8、2) Momentum Conservation Egs.Change of Mom. storageMom.C.V.Mom. inflowexternal forces on C.V.outflowRate of creation of Mom. in i-direction=Fisurface forces body forcesNewtonian fluid (牛顿流体)Navier-Stokes Egs.(a)一个éæ ¶u¶v öù¶ éæ ¶w¶P¶¶u

9、¶¶u öùd u2r= -+(2m- mdivV ) +êmç+÷ú +êmç+÷ú + rfxdt¶x¶x¶x¶y ëè ¶y¶x øû¶z ëè ¶x¶z øû3m=Const. 2-D,S/S, 牛r=Const. 3-D,S/S, 牛incomp.Cperties(b)五个3-Drd

10、 u = F - ÑP + mÑudtæ ¶2u¶2u ö¶u¶u¶P+ mçè÷ør(u ¶x + v ¶y ) = rfx-+ ¶y2¶¶2xx2-Dæ ¶2v¶2v ö¶v¶v¶Pr(u ¶x + v ¶y ) = rfy-+ mçè+ ¶y2÷ø¶¶2

11、yxincomp.,const. Properties, 牛顿流体(c)三个用æö¶P¶¶duiuç÷ + rgir= -+ mi¶x ç ¶x÷dt¶xèøijjm( ¶¶u +¶¶u +¶¶u )¶x ¶x¶y ¶y¶z ¶zN = kg × NFx = rfx体积体积力m3m3kgfx质量体积力(3) Energy Con

12、servation Eg.Rate of creation of energy = 0C.V.中能量增加internal heat sourceEuler point of view:E=W+ Fconv+ Fcond+ Frad+ Fs (固) Langrange :E- Fconv= W+ Fcond+ Frad+Fs (移)C.M.中能量增加0rde(a)formdt¶(re0 d x d y d z)+ v2 + w22+ uE =(i)e= e0总能¶t2动能(w)(J/g)内能C.V.= -é ¶ (re0u)+)ù¶&#

13、182;y(re0v)+¶¶z(ii)Fr0净e w ú d x d y d zê¶xconvëûx-directionnet.(energy in-energy out)rud yd z ×e0 - érud yd z ×e0 + ¶ (rue0 d yd z ×)d xù = - ¶ (rue0 d yd z d x)êëúû¶x¶xy-directionz-directioné &

14、#182; æ¶t ö¶ æ¶t ö¶ æ¶t öù¶(iii)F=l+çl÷ +ld xd y d z = -ç÷ç÷qi d xd y d zêúcond¶¶¶¶¶¶¶xèxøyyzèzøûxèøëi- l¶t d y d z

15、x-directionenergy in¶xenergy out - l¶t¶(-l¶t dy d z d x)d y d z +¶x¶x¶xFRad.= qRad. d xd y d z(iV)(V)Fs = sd xd yd z分别代入E=W+ Fconv+ Fcond+ Frad+ Fs得：éù¶0rd edt d x d y d z = ê- ¶x qi+ qRad.+ S ú d x d y d z + wëûird e(b)formd

16、t+ v2 + w2u 2internal energy forme0= e +2Work done at surface & body (Wsurf.+Wbody=W)éù+ v2 + w2u2d= êr+ gy) - P divV + mfú d x d y d zWsurf .(dt2动能ëû位能Compressive workViscous dissipation粘性耗散函数C.V.膨胀或压缩所需的功Mechanical energy(reversible 可逆 )(可逆 )(irreversible 不可逆 )= -

17、rgv d x d y d zWbody= Wsurf . + WbodyWde0W及e0代入rform得dt¶rd e = -q + q+ S - P divV + mfiRdt¶xi¶(l ¶t ), or div(lgrad t)¶xi¶xi热éæ ¶u ö2æ ¶wö2 ùéæ ¶v¶u ö2 ùæ ¶v ö2¶u ö2¶v &#

18、246;2æ ¶wæ ¶w23m(divV )2mf= 2mêç+ ç÷+ çú + mêç+÷+ ç+÷+ ç+ú -úû÷÷÷êëè ¶x øè ¶y øè ¶z øêëè ¶x¶y øè

19、82;y¶z øè ¶x¶z øúûViscous normal stressViscous shear stressi = e + Prd i(c)formEnthalpy 焓rdt连续 =rdivVd i = d e + 1 d P - P d rdtdtr dtr2dt代入得¶¶xrd iq + q+ S + mf+ d P= -iRdtdtid Trc(d)formi=f(T,P)pdtb= - 1 ( ¶r¶i¶id id Td Pr ¶T )P

20、= (+ ()PTdt¶Tcpdt¶Pdt= v - T ¶v= 1 + T ( 1 ¶r= 1- bTrr r¶T )P¶TrPd T + 1- bT d Pd i= cPdtdtrdt代入式(6-3-13)得¶d Tdtq + q+ S + mf+ bT d Prc= -piR¶xdtiIf incompressible =0,s=0,qR=0,=0，S/S，3-D，const. property,，则：æ ¶2t¶2t ö¶t¶t¶t

21、82;2tç÷ør+ v+ w) = l+ ¶y2 + ¶z2c (up¶¶¶¶2xyzxèd T(e)formrcdr1vv =, dv = -dtrr2e=f(T,v)dr1d v = - 1=Ñ×Vd e = ¶ed T + ¶ed vdtr2dtrdt¶Tcvdt¶vdt= -rÑ ×V (or - rdivV )vT¶p= - p + T ¶Trrd e = rcd T + T d P&

22、#209; ×V - PÑ ×Vr代入式(6-3-10)得vdtdtd T¶q - T ( ¶P )d Trc= -Ñ×V + mfrvidt¶x¶Ti. Boundary Layer Eqs.Complete egs.B.L.egs数量级比较法estimating order of magnitude(1) Assumptions(8个)(a)(b)(c)(d)(e)(f)(g)(h)incompressible Constant properties No S. qr2-Ds/sno viscous

23、dissipation Newtonian fluidForced convection ( no body force or neglected )(2) Complete egs.¶u + ¶v = 0(a)¶x¶yæ ¶2u¶2u ö¶u¶u1 ¶Pç÷ø+ v= -+n+x¶y2u(b)¶¶r¶¶2xyxèæ ¶2v¶2v ö¶v¶

24、;v1 ¶P+ v= -+nçè+ ¶y2÷øu(c)¶¶r ¶¶2xyyxæ ¶2t¶2t ö¶t¶t= aç÷+ v+uç÷(d)¶x¶y¶¶y22xèø(3) 用数量级比较法B.L.egs应用边界层性质d << 1, d » 1,d d, x LuudtdLt¶Tw 1Ly d,d, u ud¥t¶yv ud(eq.(a),¶ 1 , t t L¥w¶xLp r