Transcription
Chapter 9Natural Convection
Forced convection--with external forcing condition Natural (or free) convection--driven by buoyancy force, whichis induced by body force with the presence of density gradient9.1 Physical Considerations
9.2 The Governing Equationsx-mom. eq.:2 u u p u1u v ρ g ν 2 x y x y(9.1)With p / y 0 from the y-mom. eq., the x-pressure gradient in theb.l. must equal to that in the quiescent region outside the b.l., pi.e., ρ g(9.2) xSo, u u g 2uu v ( ρ ρ ) ν 2(9.3) x y ρ yIntroducing the volumetric thermal expansion coefficientρ ρ ρ 11 β ρ T pρ T T ρ 1 p 1) (Note: For ideal gases, β 1 ρ T p ρ RT 2 T u u 2u u v g β (T T ) ν x y y2(9.4)(9.9)(9.5)
The set of governing equations for laminar freeconvection associated with a vertical heated plate are:continuity eq.: u v 0(9.6) x yx-mom. eq.: u u 2uuenergy eq.: x v y gβ (T T ) ν T T 2Tu v α 2 x y y y2(9.7)(9.8)Note the dissipation is neglected in (9.8) and Eqs. (9.6)(9.8) are strongly coupled and must be solvedsimultaneously.
9.3 Similarity ConsiderationsDefining x* x and y* y , L is the characteristic lengthLLu* u and v* v , u0 is an arbitrary reference velocityu0u0T* T T T s T 1 u0 [gβ(Ts-T )L]1/2Eqs. (9.7) and (9.8) reduce to u* u * g β (Ts T ) L * 1 2u *u* v* T 2ReL y *2u0 x* y* T * T *1 2T *u* v* x* y * ReL Pr y *22 u* u* u**1(9.10) u * v* T (GrL )1/2 y *2 x* y*whereg β (Ts T ) L3GrL 2ν(9.10)(9.11)(9.10a)(9.12) GrL plays the same role in free convection that ReL plays in forcedconvection.
If there is a non-zero free stream velocity, u , we may use u0 u .Then g β (T T ) L g β (T T ) L3 u L 2 Grs Ls 2ν2u 2 ν ReL u* u * GrL * 1 2u * Eq. 9.10 becomes u * v* 2T ReL y *2 x* y * ReL(9.10b)Generally,(GrL / ReL2 ) 1 both free & forced convection to be considered NuL f ( ReL , GrL , Pr )(GrL / ReL2 ) 1 forced convection NuL f ( ReL ,Pr )(GrL / ReL2 ) 1 free convection NuL f (GrL , Pr )
Alternative derivation of Gr under purely natural convectionEqs. (9.10) can be also written as u* u * g β (Ts T ) L * ν 2u *u* v* T 2u0 L y *2u0 x* y*If u0 is set to make u0L/ν 1, or u0 ν/L u* u * gβ(Ts T )L3 * 2u * u * v* T 2 x* y*ν y *2 T * T * 1 2T *u* v* x* y * Pr y *2where GrL g β (Ts T ) L3ν2(9.10’)(9.11’)(9.12)
9.4 Laminar Free Convection on a Vertical SurfaceIntroducing the similarity parameter Grx 1/ 4 and ψ ( x, y ) f (η ) 4ν 4 Eqs. (9.6 to 9.8) can be reduced to1/ 4y Grx η x 4 f' ' ' 3 ff " 2( f ' ) 2 T * 0T *" 3 Pr f T *n 0(9.17)(9.18)The numerical results are shown in Fig. 9.4.1/ 41/ 4*GrGr Nu x hx x dT x g ( Pr )k 4 dη η 0 4 where g(Pr) is determined numerically determined as (9.20).1/ 4And Grx hL4 Nu L g ( Pr ) 3 4 k(9.19)(9.21)
9.5 The Effects of TurbulenceFor vertical plates the transition occurs atRax ,c Grx ,c Pr g β (Ts T ) x 3να 109(9.23)EX 9.19.6 Empirical Correlations: External Free Convection FlowsGenerally,Nu L h L CRa nL , n 1/4 for laminar, n 1/3 for turbulent flowkTable 9.2 (p. 583) summarizes the empirical correlations for differentimmersed geometries.EX 9.2
F Kreith & MS Bohn,Principles of HeatTransfer, 2001
Some other flowconditions in 9.6
Flow Pattern
9.7 Natural Heat Transfer Between Parallel PlatesVertical Parallel Plates: C1C2Nu S 2RaS S / L ( RaS S / L)where RaS g β (Ts T ) S 3 / αν 1/ 2,(9.45)(or qs” c)Cengel, Heat TransferReference: A. Bar-Cohen and W.M.Rohsenow, Thermally optimum spacing ofvertical, natural convection cooled, parallelplates, ASME J. Heat Transfer, 106 (1984)116-123.
Eq. (9.45) is suitable for different thermal conditions ofthe plates, isothermal or isoflux plates, symmetric orwith one plate adiabatic. The different values of C1 andC2 for each condition are given in Table 9.3.
Eq. (9.45) is commonly used for vertical plate heat sinks,although this can be inaccurate for short fins (H/S 5) due toadditional boundary layers near the base plate corners.For inclined parallel plates, for 0 θ 45 and within the isolateplate limit, RaS(S/L) 200,Nu S 0.645( RaS S / L)1/ 4(9.47)Cengel, Heat Transfer
F Kreith & MS Bohn, Principles of Heat Transfer, 2001. Some other flow conditions in 9.6. Flow Pattern. Reference: A. Bar-Cohen and W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, ASME J. Heat Transfer, 106 (1984) 116-123. 9.7 Natural Heat Transfer Between Parallel Plates 1/2 12 2 3 , (/)/ where ( ) / S S S Ss CC Nu Ra S L Ra S L Ra g T T .
