6.5 Natural Cnvectin in Enclsures Enclsures are finite spaces bunded by walls and filled with fluid. Natural cnvectin in enclsures, als knwn as internal cnvectin, takes place in rms and buildings, furnaces, cling twers, as well as electrnic cling systems. Internal natural cnvectin is different frm the cases f external cnvectin, where a heated r cled wall is in cntact with the quiescent fluid and the bundary layer can be develped withut any restrictin. Internal cnvectin usually cannt be treated using simple bundary layer thery because the entire fluid in the enclsure engages t the cnvectin. 1
L L L T C T T C T T C T γ (a) shallw enclsure (b) tall enclsure (c) inclined enclsure L D D i T T C (d) enclsure with vertical partitins (a) (e) cncentric annulus L D D i T γ T C D (f) bx enclsure (g) truncated annular enclsure (b) Figure 6.17 Different cnfiguratin f natural cnvectin in enclsures 2
Tw-dimensinal natural cnvectin in a rectangular enclsure with tw differentially heated sides and insulated tp and bttm surfaces (Fig. 6.18) will be cnsidered. Assumed t be Newtnian and incmpressible. Initially at a unifrm temperature f zer. At time zer the tw sides are instantaneusly heated and cled t T / 2 and T / 2, respectively. The transient behavir f the system during the establishment f the natural cnvectin is the subject f analysis 6.5.1 Scale Analysis 3
y, υ g T=+ΔT/2 Thermal bundary layer Flw circulatin directin δ T T=-ΔT/2 T= ΔT 0 0 L x, u Figure 6.18 Tw-dimensinal natural cnvectin in rectangular enclsure. 4
It is assumed that the fluid is single-cmpnent and that there is n internal heat generatin in the fluid. Therefre, the gverning equatin fr this internal cnvectin prblem can be btained by simplifying eqs. (6.8), (6.13) and (6.14): u v + = 0 (6.205) x y 2 2 u u u 1 p u u + u + v = + ν + 2 2 t x y ρ x x y 2 2 v v v 1 p v v + u + v = + ν + g[1 β ( T T 2 2 0 )] t x y ρ y x y 2 2 T T T T T u v α + + = + 2 2 t x y x y (6.206) (6.207) (6.208) 5
Immediately after impsing f the temperature difference, the fluid is still mtinless, hence the energy equatin (6.208) reflects the balance between the thermal inertia and the cnductin in the fluid. The scales f the tw terms enclsed in the parentheses n the right-hand side 2 2 f eq. (6.208) are T / δ t and T /, respectively. Since, 2 2 2 2 δ t = ne can cnclude that T / y = T / x. The balance f scales fr eq. (6.208) then becmes: T t : α T δ Thus, the scale f the thermal bundary layer thickness becmes: 1/ 2 δ T ~ ( α t) (6.209) 2 t 6
T estimate the scale f the velcity, ne can cmbine eqs. (6.206) and (6.207) by eliminating the pressure t btain: v v v u u u + u + v + u + v x t x y y t x y 2 2 2 2 v v u u T = ν + g β 2 2 + 2 2 + x x y y x y x (6.210) where the left-hand side represents the inertia terms, and the right-hand side represents the viscsity and buyancy terms. The scales f these three effects are shwn belw Inertia Viscsity Buyancy v v g β T ν δ t δ δ 3 T T T (6.211) 7
T examine the relative strength f each effect, ne can divide the abve expressin by the scale f viscsity effect t btain Inertia Viscsity Buyancy 1 Pr 1 g β Tδ where eq. (6.209) was used t simplify the inertia term. Fr the fluid with Pr>1, the mmentum balance at requires a balance between the viscsity and buyancy terms: 2 g β Tδ T 1 ~ ν v ν v 2 T 8
Substituting eq. (6.209) int the abve expressin and rearranging the resultant expressin, the scale f vertical velcity at the initiatin f the natural cnvectin is btained as fllwing: ~ g β T α v t ν (6.212) As time increases, the effect f the inertia term in eq. (7.208) weakens, hence the effect f advectin becmes strnger. This trend cntinues until a final time, t f, when the energy balance requires balance between the advectin and cnductin terms, i.e., T T T v ~ α ~ δ t 2 T, f f 9
Thus, the scale f t f becmes ν t f ~ g β Tα The thermal bundary layer thickness at time t f is: δ (6.213) (6.214) At time t f, natural cnvectin in the rectangular enclsure reaches steady-state and the thickness f the thermal bundary layer n lnger increases with time. 1/ 2, ~ ( α t ) ~ Ra 1/ 2 1/ 4 T f f 10
The wall jet thickness increases with time until t = t f, when the maximum wall jet thickness, δ v,f, is reached (see Fig. 6.19). Outside the thermal bundary layer, the buyancy frce is absent and the thickness f the wall jet can be determined by balancing the inertia and viscsity terms in eq. (6.210): v v ~ ν which can be rearranged t btain: 1/ 2 1/ 2 δ ~ ( ν t) ~ Pr δ v δ v t (6.215) Fr t > t f, steady-state has been reached, and the wall jet thickness is related t 1/ 2the thermal bundary layer δ v, f ~ Pr δ T, f thickness by. δ 3 v T 11
T v α t δ, 0 x t f ν t δ, 0 x Figure 6.19 Tw-layer structure near the heated wall. v f 12
Similarly, the cnditin t have distinct vertical wall jets r mmentum bundary layers is δ < L, r equivalently: L <Ra 1/ 4 1/ 2 (6.216) When the vertical wall jet encunters the hrizntal wall, it will turn t the hrizntal directin and becme a hrizntal jet. This hrizntal jet will cntribute t the cnvective heat transfer frm the heated wall t the cled wall: q ~ ρ v δ c T Cnsidering eqs. (6.212) and (6.214), the abve scale f cnvective heat transfer becmes: 1/ 4 q ~ k TRa Pr cnv f T, f p cnv v, f 13
When a warm jet is frmed at the tp and a cld jet is frmed at the bttm, there will be a temperature gradient alng the vertical directin. The heat cnductin due t this temperature gradient is: T q cnd ~ kl The cnditin under which that the hrizntal wall jets can maintain their temperature identity is that the heat cnductin alng the vertical directin is negligible cmpared t the energy carried by the hrizntal jets: T kl < k TRa 1/ 4 14
r equivalently L > Ra 1/ 4 (6.217) The characteristics f varius heat transfer regimes are summarized in Table 6.2. Table 6.2 Characteristics f natural cnvectin in a rectangular enclsure heated frm the side Regimes I: Cnductin II: Tall Systems III: Bundary layer IV: Shallw systems Cnditin f ccurrence < 1/ 4 1/ 4 1/ 4 / L>Ra Ra 1 Ra < / L<Ra / L<Ra 1/ 4 Flw pattern Clckwise circulatin Distinct bundary layer n tp and bttm walls Bundary layer n all fur walls. Cre remains stagnant Tw hrizntal wall jets flw in ppsite directins. Effect f flw n heat transfer Insignificant Insignificant Significant Significant eat transfer mechanism Cnductin in hrizntal directin Cnductin in hrizntal directin Bundary layer cnvectin Cnductin in vertical directin eat transfer q : k T / L q k T / L q : ( k / δ ) T : q : ( k / δ T, f ) T T, f 15
6.5.2 Rectangular Enclsures eated frm the Side eat transfer in regimes I and II, shwn in Table 6.2, is dminated by cnductin and the fluid circulatin plays an insignificant rle. Therefre, heat transfer in these tw regimes is practically equal t the heat transfer rate estimated using Furier s law. The flw pattern in regime III is f the bundary layer type and the cre f the fluid is stagnant and stratified. The fluid flw and heat transfer in regime III can be btained via bundary layer analysis. 16
(a) (b) (c) (d) Figure 6.20 Natural cnvectin in a square enclsure (Ra =10 6, Pr=0.71). Steadystate distributins f dimensinless (a) temperature, (b) pressure, (c) hrizntal velcity, and (d) vertical velcity (Wang et al., 2010) 17
Figure 6.20 shws cnturs f dimensinless temperature, pressure, and hrizntal and vertical velcities fr natural cnvectin in a square enclsure ( / L = 1) heated frm its left side and cled frm its right side while insulated frm the tp and bttm. Fr a high Rayleigh number, the average Nusselt number fr regime III, as btained frm bundary layer analysis, is L 1/ 4 Nu = 0.364 Ra (6.218) which indicates that the heat transfer in the natural cnvectin in a rectangular enclsure is dminated by bth the Rayleigh number and aspect rati. Fr laminar natural cnvectin in a rectangular enclsure with aspect rati /L between 1 and 10, the fllwing crrelatins were suggested by Berkvsky and Plevikv: 18
which is valid fr 0.28 0.09 Pr L Nu = 0.22 Ra 0.2 + Pr 2 < / L < 10, Pr < 10, Ra < 10 5 13 0.29 0.13 Pr L Nu = 0.18 Ra 0.2 + Pr, and (6.219) (6.220) 1 < / L < 2, 10 < Pr < 10, 10 < [Pr/(0.2 + Pr)]Ra ( L / ) which is valid fr. -3 5 3 3 Fr a rectangular enclsure with a larger aspect rati, MacGregr and Emery (1969) recmmended the 0.3 fllwing crrelatins: 1/ 4 0.012 Nu L = 0.42Ra L Pr L (6.221) 19
4 4 7 which is valid fr 10 < / L < 40, 1< Pr < 2 10, 10 <Ra L < 10, and 1/ 3 Nu (6.222) L = 0.046Ra L which is valid fr 6 9 1 < / L < 40, 1< Pr < 20, 10 < Ra. It L < 10 shuld be pinted ut that the characteristic length fr the Rayleigh and Nusselt numbers used in eqs. (6.221) and (6.222) is the width f the enclsure, L. Fr the shallw enclsure represented by regime IV in Table 6.2, the hrizntal wall jet flws frm the left t right n the tp wall and frm right t the left n the bttm wall. Figure 6.21 shws streamlines and istherms fr natural cnvectin in a shallw enclsure ( / L = 0.1) heated frm the right side and cled frm the left side. 20
Figure 6.21 Natural cnvectin in a shallw enclsure: (a) Ra = 10 6, (b) Ra = 10 8 (/L = 0.1, Pr = 1.0; Tichy and Gadgil, 1982). 21
Eq. Figure 6.22 Average Nusselt number fr natural cnvectin in a shallw enclsure (Bejan and Tien, 1978). Bejan and Tien perfrmed a scale analysis and btained an analytical slutin via analysis f fluid flw and heat transfer in the cre regin. As indicated by Fig. 6.22, their results cmpared favrably with experimental and numerical results. Fr a limited case, the asympttic heat transfer results can be expressed as: 22
which is valid when line in Fig. 6.22. 1 Nu = 1 + Ra 362880 L 2 ( / L) Ra 0 2 (6.223), and is shwn as dashed 23
eated frm the Bttm Fr a rectangular enclsure filled with fluid and heated frm the side, natural cnvectin will be initiated as sn as the temperature difference between the tw vertical walls is established. Fr a rectangular enclsure heated frm belw, natural cnvectin may r may nt ccur depending n whether the temperature difference between the tp and bttm walls exceeds a critical value. The cnditin fr the nset f natural cnvectin can be expressed in terms f a critical Rayleigh number. 24
Fr the case that the hrizntal dimensin is much larger than the height f the enclsure, the criterin fr the nset f natural cnvectin is: Ra 3 g β T = > ν α 1708 (6.224) Figure 6.23 Rlls and hexagnal cells in natural cnvectin in enclsure heated frm belw (Osthuizen and Naylr, 1999). 25
When the Rayleigh number just exceeds the abve critical Rayleigh number, the flw pattern is tw-dimensinal cunter rtating rlls referred t as Bénard cells [see Fig. 6.23(a)]. As the Rayleigh number further increases t ne r tw rders f magnitude higher than the abve critical Rayleigh number, the tw-dimensinal cells breakup t three dimensinal cells whse tp view is hexagns [see Fig. 6.23(b)]. The functin f the twdimensinal rlls and three-dimensinal hexagnal cells is t prmte heat transfer frm the heated bttm wall t the cled tp wall. Glbe and Drpkin suggested the fllwing empirical crrelatin 1/ 3 0.074 Nu = 0.069Ra Pr (6.225) where all thermphysical prperties are evaluated at ( T + Tc ) / 2 5 9 Equatin (6.225) is valid fr 3 10 < Ra < 7 10. In additin, /L must be sufficiently large s that the effect f the sidewalls can be negligible. 26
Inclined Rectangular Enclsure When the rectangular enclsure heated frm the side is tilted relative t the directin f gravity, additinal unstable stratificatin and thermal instability will affect the fluid flw and heat transfer. The variatin f Nusselt number as functin f tilt angle γ is qualitatively shwn in Fig. 6.24. 0 90 γ c 180 Figure 6.24 Effect f inclinatin angle n natural cnvectin in enclsure 27
The istherms and the streamlines fr Ra=10 5 are shwn in Fig. 6.25. At γ=135, which, accrding t Table 6.3, is less than the critical inclinatin angle, the istherms start t exhibit sme behavirs f thermally unstable cnditins. This is the crrelatin fr natural cnvectin f air in a squared enclsure ( / L = 1) in the regin 0 < γ < 90 K γ Nu (0 ) Nu ( γ ) Nu (0 ) 2 = = γ sin γ Nu (90 ) Nu (0 ) π (6.226) where is fr pure cnductin. While eq. (6.226) is gd fr air in a squared enclsure, the fllwing crrelatin can be applied t ther situatins: L L 1 + Nu (90 ) 1 sin γ 0 < γ < 90 Nu ( γ ) = L Nu (90 )(sin γ ) 90 < γ < γ c (6.227) 28
Table 7.3 Critical inclinatin angle fr different aspect rati (Arnld et al., 1976) Aspect rati, /L 1 3 6 12 >12 Critical tilt angle, γ c 155 127 120 113 110 Isthermals Streamlines Figure 6.25 Natural cnvectin in inclined squared enclsures (Zhng et al, 1983). 29
Example 6.5 A rectangular cavity is frmed by tw parallel plates, each with a dimensin f 0.5 m by 0.5 m, which are separated by a distance f 5 cm. The temperatures f the tw plates are 37 C and 17 C, respectively. Find the heat transfer rate frm ht plate t cld plate fr the inclinatin angles f 0, 45, 90, and 180. L T h Air T c γ Figure 7.27 Natural cnvectin in inclined squared enclsure. 30
Slutin: T = 37 C, T = 17 C, T m = 27 C = 300K. h c = 0.05 m, L = 0.5 m, D = 0.5 m 6 2 Frm Table C.1: ν = 15.89 10 m /s, k = 0.0263 W/ 6 2 m-k, α = 22.5 10, and m /spr = 0.707 At the inclinatin angle 0, heat transfer is slely by cnductin. The rate f heat transfer is Th Tc 37 17 q(0 ) = kd = 0.0263 0.5 0.5 = 2.63W L 0.05 31
The Nusselt number fr the case f pure cnductin is Nu L (0 ) = 1 Nu (0 r ) = ( / L)Nu L (0 ) = 10. eat transfer fr γ the = 45case f can be calculated using eq. (6.227), which requires the Nusselt number γ = 90fr. Thus, the heat transfer γ = rate 90 fr will be calculated first. The Rayleigh number is: 3 3 g β ( Th Tc ) 9.807 1/ 300 (37 17) 0.5 Ra = = = 2.286 10 6 6 ν α 15.89 10 22.5 10 The Nusselt number fr can be btained frm eq. (6.220) : Nu (90 ) 0.18 Ra 0.28 0.13 Pr L = 0.2 + Pr 0.28 0.13 0.707 8 0.05 0.18 2.286 10 49.6 = = 0.2 + 0.707 0.5 8 32
The heat transfer cefficient is: knu (90 ) 0.0263 49.6 2 h(90 ) = = = 2.61 W/m -K 0.5 γ = 90 is: Therefre, the heat transfer rate fr q h D T T (90 ) = (90 ) ( h c ) = 2.61 0.5 0.5 (37 17) = 6.53 W γ = 45 = π / 4 When the inclinatin angle is L L Nu (45 ) = 1 + Nu (90 ) 1 sin 0.05 = + = 0.5 1 49.6 1 sin 45 3.80, eq. (6.227) yields: Thus, the Nusselt number is Nu (45 ) = 38.0 and the crrespnding heat transfer cefficient is: knu (45 ) 0.0263 38.0 2 h(45 ) = = = 2.00 W/m -K 0.5 γ 33
The heat transfer rate fr q h D T T γ = 45 is therefre: (45 ) = (45 ) ( h c ) = 2 0.5 0.5 (37 17) = 5.00W When the inclinatin angle is γ = 180, the prblem becmes natural cnvectin in an enclsure heated frm the belw. The Rayleigh number is: 3 3 g β ( Th Tc ) L 9.807 1/ 300 (37 17) 0.05 Ra L = = = 2.286 10 6 6 ν α 15.89 10 22.5 10 The Nusselt number in this case can be btained frm eq. (6.225): Nu L (180 ) = 0.069Ra Pr 1/ 3 0.074 = = 5 1/ 3 0.074 0.069 (2.286 10 ) 0.707 4.11 5 34
The heat transfer cefficient is knu L (180 ) 0.0263 4.11 2 h(180 ) = = = 2.16 W/m -K L 0.05 The heat transfer rate fr is therefre, q h D T T (180 ) = (180 ) ( h c ) = 2.16 0.5 0.5 (37 17) = 10.80 W 35
6.5.3 Annular Space between Cncentric Cylinders and Spheres Natural cnvectin in spaces between lng hrizntal cncentric cylinders r between spheres is very cmplicated. The nly practical apprach t analyze the prblem is via numerical slutin. The physical mdel f natural cnvectin in annular space between cncentric cylinders is shw in Fig. 6.27. r δ r i T i θ, u r, v g T Figure 6.27 Natural cnvectin in hrizntal annular space between cncentric cylinders 36
Since the prblem is axisymmetric, ne nly needs t study the right half f the dmain. In the crdinate system shwn in Fig. 6.27, the gverning equatins are 1 u v v (6.228) + + = 0 r θ r r 2 2 u u u uv 1 p 1 u 1 u u u 2 v + v + = + ν + + + 2 2 2 2 2 r θ r r ρ r θ r θ r r r r r θ g β ( T T ref )sinθ (6.229) 2 2 2 u v v u 1 p 1 v 1 v v v 2 u + v = + ν + + 2 2 2 2 2 r θ r r ρ r r θ r r r r r θ + g β ( T T ref )sinθ (6.230) 37
2 2 u T T 1 T T 1 T + v = α + + 2 2 2 r θ r r θ r r r (6.231) where T ref is a reference temperature and: peff = p + ρ g sinθ ρ g csθ (6.232) is the effective pressure. Equatins (6.228) (6.231) are subject t the fllwing bundary cnditins: v T (6.233) u = 0, = = 0, at θ = 0 r π θ θ u = v = 0, T = T, at r = r u = v = 0, T = T, at r = r i i (6.235) (6.236) 38
Date (1986) slved this prblem numerically using a mdified SIMPLE algrithm fr δ / D i = 0.8 and 0.15. Figure 6.28 shws the cnturs f the stream functins and istherms at Ra = 4.7 x 10 4. The dimensinless stream functin, dimensinless temperature and Rayleigh number are defined as the fllwing ψ T T gβ Tδ Ψ =, Θ =, Ra = α T T ν α i 3 (6.236) Where the stream functin, ψ, is defined as u ψ =, v = r 1 r ψ θ (6.237) 39
It can be seen that istherms cncentrate near the lwer prtin f the surface f the inner cylinder and the upper prtin f the surface f the uter cylinder, which are indicatins f the develpment f thermal bundaries near the heated and cled surfaces. While the cnturs f the streamlines have kidney-like shapes with the center f the flw rtatin mves upward due t effect f natural cnvectin. The rate f heat transfer per unit length f the annulus can be calculated by the fllwing crrelatin 1/4 2.425 k( T Pr Ra i T ) D i q 3/5 5/ 4 [1 + ( Di / D ) ] 0.861 + Pr (6.238) Where the Rayleigh number is defined as 3 gβ ( Ti T ) Di Ra D i = ν α (6.239) 40
Equatin (6.238) is valid fr 0.7 < Pr < 6000 and Ra < 10 7. The thermphysical prperties f the fluid shuld be evaluated at the mean temperature ( Ti + T ) / 2. The scales f the thermal bundary layer n the inner surface f the uter cylinder and n the uter surface f the inner cylinder are: δ ~ D Ra, δ ~ 1/ 4 1/4 D i i D It is bvius that δ > δ i since D > D i. Equatin will be valid nly if the bundary layer thickness is less than the gap between the tw cylinders, i.e. nly if δ < D D i. Under lwer Rayleigh numbers, n the ther hand, we have D Ra > D D 1/ 4 D i D Ra (6.240) and the heat transfer mechanism between tw cylinders will apprach pure cnductin. i 41
Instead f using eq. (6.240) t check the validity f eq. (6.238), anther methd is t calculate the heat transfer rate via eq. (6.238) and pure cnductin mdel, and the larger f the tw heat transfer rate shuld be used. Fr natural cnvectin in the annulus between tw cncentric spheres, the trends fr the evlutin f the flw pattern and istherms are similar t the cncentric cylinder except the circulatin between cncentric spheres has the shape f a dughnut. The empirical crrelatin fr the heat transfer rate is 7 / 5 5/ 4 [1 + ( Di / D ) ] 0.861 + Pr (6.241) where the definitin f Rayleigh number is same as fr eq. (6.239). Equatin (6.241) is valid fr 0.7 < Pr < 4000 and Ra < 10 4. 2.325 π kd ( ) Pr Ra i Ti T D q i = 1/ 4 42
Ψ Θ Figure 6.28 Natural cnvectin in a hrizntal annulus (Pr =0.7, δ/d i =0.8, Ra =4.7 10 4 ; Date, 1986) 43