Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 8
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pV)V -VPrate of increaseof momentumper unitvolumerate ofmomentum gainby convectionper unit volumepressure forceon element perunit volume+V. x+rate ofmomentum gainby viscoustransfer per unitvolumepg(1.43)gravitationalforce on elementper unit volumeEquation 1.43 may be rearranged, with the help of the equation of continuity, to giveDVp - - ~ = - V P + V . x + pg(1.44)The last equation is a statement of Newton's second law of motion in the form mass x acceleration = s u m o f forces.These two forms of the equation of motion (Eqs. 1.43 and 1.44), correspond to the two formsof the equation of continuity (Eqs.
1.40 and 1.41). As indicated, the only body force included inEqs. 1.43 and 1.44 is gravity. In general, electromagnetic forces may also act on a fluid.The scalar components of Eq. 1.44 are listed in Table 1.3 and the components of the stresstensor x are given in Table 1.4.For the flow of a newtonian fluid with varying density but constant viscosity/.t, Eq.
1.44becomesDV1p - ~ = - V P + ff ktV(V • V) +/.tV2V + pg(1.45)If p and ILt are constant, Eq. 1.44 may be simplified by means of the equation of continuity(V • V = 0) for a newtonian fluid to giveDVp~= - V P + ~.I,V2'V -]- pg(1.46)This is the famous Navier-Stokes equation in vector form. The scalar components of Eq. 1.46are given in Table 1.5.For V • x = 0, Eq. 1.44 reduces to Euler's equation:DVp - - ~ = - V P + pg(1.47)which is applicable for describing flow systems in which viscous effects are relatively unimportant.As mentioned before, there is a subset of flow problems, called natural convection, wherethe flow pattern is due to buoyant forces caused by temperature differences. Such buoyantforces are proportional to the coefficient of thermal expansion 13, defined as:13=-p~,(1.48)where T is absolute temperature.
Using an approximation that applies to low fluid velocitiesand small temperature variations, it can be shown [9-11] thatV P - pg = p[Sg(T- Too)(1.49)BASIC CONCEPTS OF HEAT TRANSFER1.15Then Eq. 1.44 becomesDVp - - ~ = V . x - p~g(T- Too)(1.50)buoyant forceon element perunit volumeThe above equation of motion is used for setting up problems in natural convection when theambient temperature T= may be defined.The Energy EquationFor a stationary volume element through which a pure fluid is flowing, the energy equationreads3~)-"~"p ( u + 1//2V2) = - V rate of gain ofenergy per unitvolumep V ( u + 1/2V2) -rate of energy inputper unit volume byconvectionV"q"+rate of energy inputper unit volume byconductionV.
PV+rate of work doneon fluid per unitvolume bypressure forcesp(V • g)rate of work done onfluid per unit volumeby gravitational forcesV. (x. V)rate of work doneon fluid per unitvolume byviscous forces+q"(1.51)rate of heatgeneration perunit volume("source term")where u is the internal energy. The left side of this equation, which represents the rate ofaccumulation of internal and kinetic energy, does not include the potential energy of thefluid, since this form of energy is included in the work term on the right side. Equation1.51 may be rearranged, with the aid of the equations of continuity and motion, to give[10, 191Dup -~= - V • q ' - P ( V . V) + VV:'I: + q'"(1.52)A summary of VV:'~ in different coordinate systems is given in Table 1.6.
For a newtonianfluid,VV:x = BO(1.53)and values of dissipation function • in different coordinate systems are given in Table 1.7.Components of the heat flux vector q " - - k V T are given in Table 1.8 for different coordinatesystems.Often it is more convenient to work with enthalpy rather than internal energy. Using thedefinition of enthalpy, i - u + P/p, and the mass conservation equation, Eq. 1.41, Eq. 1.52 canbe rearranged to giveDiDPp - ~ = V . k V T + - - ~ + . ~ + q"(1.54)1.16CHAPTER ONETABLE 1.3Equation of Motion in Terms of Viscous Stresses (Eq. 1.44)*Rectangular coordinates (x, y, z)x direction(xx= - ~ + --ffx+-~-y+ az ]+pgxp ¥+.~+v~+w~y direction(3V3V312312)3P ~3"[,xy 3T,yy 3"r,zy Ip --~..t-u--~xq-V--~y-t-w--~Z : - - - ~ y - b \ 3x q---~y-b 3Z / qrDgyz directionP ~+~-ffx +~-ffy +WTz =-Tz + ~ x +-~y + az ]+pgzCylindrical coordinates (r, 0, z)r direction3Vr3V r 1)0 3V rp -~+Vr--~-r + r 30])2r + Vz -~Z= -- ~ r +~1 3'l;0r "g00 3Zzr1(rT'rr) + --r 30 _ mr + 3ZJ + Pgr0 direction/ 3Vo3V 0V0 3V 0VrVOP ~ - ~ + V r - ~ r + - -r - - ~ + ~ r+ Vz --~Z = - - -r - ~ +-~-r (r21:rO)+-r - - ~+ 3Z ] + PgoZ direction[3Vz3vz vo 3Vz3v~3P [ 1 313Xoz 3Xzz]P l--~- "~"Vr --~-r -t- --r - ~ + v z "3 Z ] = - "~Z -t- Lr -~r ( r T,rz) + - - r - - ~ + 3 Z J + Pg zSpherical coordinates (r, 0, ~)r directionV, 3V r3Vr3V r VO 3V rP - ~ + V r - - ~ r + - -r - ~ - + - -r sin 0 30+VO2 + V~ ~3PJ3rr[~_ 3~r (r2~rr) +13r s i n 0 30(T,Orsin 0) +1(~l:~rrsin030Zoo +rZ~ ] + Pgr0 direction/ 3V 03V 0]20 3V 0p ~ ' - ~ "~- I/r --~-r -~- - -I-I]~av 0VrV0r -frO-- r sin 0 3~ - I - - -r+5-; (r~/+~rv~ cot0)r= - l_r 3P301313%0 Xr0 X.
cot0- (x00 sin 0) + r sin 0 3¢ + --r - ~ rsin 0 -30+ Pgo# direction{ 3V¢~3V,V0 3V¢~=V,3V¢~r sin 0 3~VC~Vrrvov, cot 0r)13P [ 1 31 31:o~13x.Xr# 2X0~cot0]r sin 0 3~ + [ 7 ~ r (r2"l:r~)+ - - r - ~ + -r sin- 0 3~ + --r +-r+ pg** Components of the stress tensor (x) for newtonian fluids are given in Table 1.4.
This equation may also be used fordescribing nonnewtonian flow. However, we need relations between the components of x and the various velocitygradients; in other words, we have to replace the expressions given in Table 1.4 with other relations appropriate for thenonnewtonian fluid of interest. The expressions for x for some nonnewtonian fluid models are given in Ref.
10. See alsoChap. 10.BASIC CONCEPTS OF HEAT TRANSFERTABLE 1.41.17Components of the Stress Tensor x for Newtonian Fluids*Rectangular coordinates (x, y, z)bu23v2Xxx=. 2 -ffX--X--~- (V • V)]Xzz=. 2--~z - ~- (V • V)bu3vT,xy "- T,yx "-- ~J,Jr "~X"Cyz = "Czy= ~t+"Czx= "Gz= kt+bw~v.
v):-G-x +-c:oy +bzCylindrical coordinates (r, 0, z)I~V r]2[(l vo~ - g 2 7--~-+Xzz = g/2--~(v.v)]2OVz-bTz--~ (v. v)]~r~=~Or=~t r ~1+--ro,,or-~10vz 1r~z~,]"~zr-- T'rz = ~'l'[--~-T "~- 3Z J~(V.V)=I1 ~vo--r --~r ( r v r ) + - r - ~3Vz+ ~zSpherical coordinates (r, 0, ~)3Vr2]T,rr=~l, 2 "-~F --'~- (V • V )~-g 2 7-~+[ (x~=~t 21--~(v.v)3v,r sin 0 /)~[ ~ (~-)"Or0-- "l~0r-- ~LI,r-fir-rvrVoCOtO) 2+ ~ + ~r(V'V)]l Ovr1+ - -r - - ~rsino~( v° )x0~ = %0 = l a [ ~-r~1~vo]+ r sin 0 /)~1()Vr~ (~)1%r='r'r~=l't[ rsinO3~ + r - ~ r1 ~(V. V) = ~- ~ r1~1~v,(r2vr) + rsinO 30 (vo sin O) + rsinO ~* It should be noted that the sign convention adopted here for components of the stress tensor is consistent with thatfound in many fluid mechanics and heat transfer books; however, it is opposite to that found in some books on transportphenomena, e.g., Refs.
10, 11, and 14.1.18CHAPTER ONETABLE 1.5(Eq. 1.46)Equation of Motion in Terms of Velocity Gradients for a Newtonian Fluid with Constant p and laRectangular coordinates (x, y, z)x direction(~u ~u ~u ~.) ~P l~u ~,, ~u~p¥ + . ~ + ~ y + W ~ --~+,~-~-Vx~+~y---7+ ~z~i+pgxy direction(~~v ~~)~Pl~P ~ + . ~ x + ~ y +WTz : - ~ + , [ ~ +~~~y~+~z~I+Pgrz direction(~w ~w ~w ~w~ ~P /~w ~w ~w~p --~+u--~x+V--~y+W--~z ,]: - --~-z + ~t[--~x2+~+3y2 3z 2 ) + pgzCylindrical coordinates (r, 0, z)r direction/aVraVrvo aVrPk--~- + Vr-~r + . r .
30. .]202aVr~aPr + Vz 3z ] : - ~+[a (1 a) 1 a2Vr 2 av o a2Vr]. . .g ~r-~r (rVr) + .r 2 . 302r 2 30 + 3z2J + Pgr0 direction{3vo3vo vo 3voP ~ - - ~ + V r - ~ r + - -r - ~ +v,vo3vo \1 3P~r + ~ -~z) : - r - ~ + ~' TrbTr (rv0) + 7 ~+7 ~+ az ~ J + pg0z directionPl--ff't'- + Vr"-~r + - -r - - ~ + Vz" 3Z) =-- ~+ ~tLr-ffr-r \ r 3 r ) + ~ - - - ~+ 3Z2j + PgzSpherical coordinates (r, 0, ¢)*r direction0Vrp \| ~dt+ V r ~ + -ar-r --frO-+ r sin 0 30--r) =-- ~"~-~l,~VZvrr2r 2 a0 --0cot0r22av, )r 2sin0 3¢ +Pgr0 direction/aVeaveve aVev,avevrvov~cot0~1 aP[2 3Vra0) = .
r 30. +kt. V2VO+rE.P~--~ + Vr--~r + - -r --0-0-+ r sin 0 3¢ + . r . . . rdirection{av~av~ ve av,v~ av, V$Vr vev$p~-~- + Vr-~-r + - -r - - ~ + r s i n 0 a¢ + r + r cot 0Vor 2sin 202 cos 0 3v,r 2 sin 2 0 3¢ ) + Pg0)13P(v_______L__~2aVrr sin 0 0 ~ + g V2v* - F sin 2 0 + r 2 sin 0 3¢* For spherical coordinates the laplacian isV2=~--~--r-r r2-~r + r 2 s i n e - ~ sine+ r2sin2 e2 cos 0 av0r 2sin 20 3 ¢ ) + pg¢BASIC CONCEPTS OF HEAT TRANSFER1.19TABLE 1.8 Summary of Dissipation Term VV:x in Different Coordinate SystemsRectangular coordinates (x, y, z):VV:~, -- T,xx(~) (~~)(o~)(~~ ~:)(~: ~~)(~: ~:)d- T,yy"]-"CZZ"]-"Cxy-I-at- T,yzd-d- "Czx"t-Cylindrical coordinates (r, O, z):fDVr~(1DV 0 _.~_) [DVz~[D (_~_)1DVrl(1DV z DVO~ (Dr z DVr~VV:~-~'~rrt--~r ) "l-'r,oo --~-Jt+ Xzzt-~z j + Xro r-~r+r DOJ4"'~Oz --~ dr" DZ / -I-'r'rz -~-r "Jr"DZ /Spherical coordinates (r, O, ~):(~)(1Dvo_~ _~)( l rsin 0 Dv,D~ v, vocotO)[DvolDvr_~)"at"%O++ T,~+ --r + ~ r+ T'r°t--~-r + --r ~DO -VV:'~ "- T,rr[DvO +\or+ ~I-mT_.1 D V r V - - ~ 7 ) (+ x0, 1Dr° rsinl DVOoD~r sin 0 Dt~ ~ ~ +v,C2tr0 )TABLE 1.7 The Viscous Dissipation Function •Rectangular coordinates (x, y, z):~rr~u~ ;~v~ ~w~l (~v ~uv (~w ~v~ (~ ~w~ ~(~u ~v ~w~+tOy/ + \ O z / J + -~x + Dy J + --~y +--~z] + -~z + Dx ) - 7 -~x +-~y +--~z )¢ = Lt Ox )Cylindrical coordinates (r, 0, z):2r(~vrVO= l\Dr] +(2;~vo ~r)2+\DZ]J(DVZ~21 [ D (_~)1 Dv,12+ r-~r+r-~+--DOJi1~ ~vol~ roar ~Vz~ ~ra~l Ovo ~Vzl~Dz J +\Dz +--~-r]--3 [r-o-r-r (rVr) + r - - ~ +--~-zJ+ Lr-~+Spherical coordinates (r, O, ~):2[(DVr~2(1Dv 0¢II = L\ Dr ] +-~~r)2 ( 1 DV$ Vr vo cot O) 2]+ r sin 0 D4~ rr+ --[ D (_~) 1 Dvr]2 [~_0__~(+ r-~-r+--r - ~ J +v, )~+1Dvo]2 [ 1 DvrD (_~)] 2+ r sin 0 Dt~ +r -~rr sin 0 D~2 1 1~/ )~r (r2v') + 1 D (vo sin O) + 1 Dk'$]2r sin 0 DOr sin 0 Dt~'TABLE 1.8 Scalar Components of the Heat Flux Vector q"Rectangular (x, y, z)DTq'; = - k -~xCylindrical (r, 0, z)~Tq~=-k OrOTq'y"=-k Dyq~ = -k -- ~r DO1 DTq~ = -k -r DODTq'z'=-k D--z~Tqz' =-k Dz1 ~Tq~ = - k ~r sin 0 D~1 DTSpherical (r, 0, 0)OTq~ =-k D--~1.20CHAPTER ONEFor most engineering applications it is convenient to have the equation of thermal energy interms of the fluid temperature and heat capacity rather than the internal energy or enthalpy.In general, for pure substances [11],DiDt -()()o,OiDPOiDT1 (1-~T)--~+ce-~ r--~ + ~e Dt - pDtwhere 13is defined by Eq.
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