C HAPTER 8
Natural Convection·
8.1 INTRODUCTION
As explained i n Chapter 1, natural or free convective heat transfer is heat transfer between a surface and a fluid moving over it with the fluid motion caused entirely by the buoyancy forces that arise due to the density changes that result from the temperature variations in the flow, [1] to [5]. Natural convective flows, like all viscous flows, c aRbeeither laminar or turbulent as indicated in Fig. 8.1. However, because of the low velocities that usually exist in natural convective flows, laminar natural corivective flows occur more frequently i n practice than laminar forced convective flows. In this chapter attention will therefore be initially focused on laminar natural convective flows. The majority of natural convective flows arise due to density changes in the presence o f the gravitational force field. However, similar flows can arise in other force fields, e.g., very large buoyancy type forces can arise in a centrifugal force field as indicated iJ,1 Fig. 8.2. However, in such cases, the flow is frequently not purely natural convective and such flows will not be considered here. A distinction is sometimes made between natural and free convection, the term natural convection then being applied to flows resulting due to the gravitational force field and the term free convectipn being applied to flows arising due to the presence o f any force field. However, t he terms are today both usually used to describe any flow arising due to temperatureinduced density changes in a force field.{md they will be used interchangeably here. ..
8.2 BOUSSINESQ APPROXIMATION
Natural convective flows arise because the density of the fluid involved changes . with temperature i.e., the flow arises because a property of the fluid, i.e., the density
342
CHAPIER
8: Natural Convection 343
Turbulent "Flow
Surface
F IGURES.l Laminar and 'turbulent free" convective boundary layers on a vertical surface.
changes with temperature. The q~estion that immediately t hen arises i s that i f a change i n a fluid property causes t he flow, m ust account be taken o f t he variations o f all o f t he fluid properties with temperature i n t he analysis o f such flows? Now. the changes o f density i n most free "convective flows are relatively small i.e., i f Po is the initial density and Ap is the overall change i n density. then: Ap ~~ 1 (8.1) " Po Consider a twodimensional free convective flow. T he momentum equation for the xdirection, which is assumed to be i n t he vertical direction, is:
pu 
2 au a a a2 + pv  u =   p + f.L (a u +  + g(Po  p) ax ay ax. iJx2 iJy2 " " "
I
u)
."
(8.2)
" the pressure being, of course. measured relative to the local hydrostatic pressure.
Fluid Motion
Arising
Because of Rotation
Cold
Adiabatic
FIGURES.2
Rotating
Free convective flow i n a rotating enclosure.
344 Introduction to Convective Heat Transfer Analysis
The flow is induced by the buoyancy force, i.e., the terms i n this equation must all be o f the order o f Po  P, i.e., o f the order o f ap. Now, Eq. (8.2) can be written as:
2 2 a u) au a u) ap Po (uau +v +ap (u +v =  +P,o (a u a2 + u) ax ay ax ay ax ax ay2
J2u a2u) + aM (ax2 + ay2 + g(ap)
where:
(8.3)
(8.4) .
ap, will have the same order o f magnitude as flp. Hence, Eq. (8.3) has the following form:
Po (u au + v ay + term o f ~rder a p2 . ax a u)
=
2 ap + Mo (a 2u + ay2 + term o f order ap2 + g(flp) ax ax2 a u)
(8.5)
But, since a p/ Po is small, terms o f the order o f ap2 can b e neglected. Eq. (8.5) therefore becomes:
au a Po (u  + v  u) = ax ay
a  p
2 + P,o (a u + 2 ax <Jx
a2  u) + g(po  p) ay2
(8.6)
i.e., the only effect o f the change in density is the generation o f t he buoyancy term. In all o f the other terms, the effect o f the changes in density and other properties o f the fluid due to the temperature changes can be neglected. Now, by definition: (8.7) Po  P = f3Po(T  To) where f3 is the bulk coefficient. I t is a property o f the fluid involved. Substituting Eq. (8.7) into Eq. (~,6) and rearranging gives:
u au ua+vax ay
2 1 ap a2u) =  +v (a u  +  +f3g(TTo) p ax ax2 ay2
.
\
(8.8)
The above discussion indicates that, in the analysis.'of free convective flows, the fluid properties can b e assumed constant except for the density change with temperature which gives rise to the buoyancy force. This is, basically, the "Boussinesq approximation" [6]. This approximation will be adopted in all the analyses given in this chapter.
8.3
GOVERNING EQUATIONS
When the Boussinesq approximation discussed in the previ<?us section is adopted, the equations governiq$ steady laminar twodimensional natural convective flow are:
/
CHAP'IER.
8: Natural Convection 345
Y
~ .I
I
Vertical
I~
x
y component of Buoyancy Force /3gp(T  T1) sintp
I
Buoyancy
Force
/ 3gp(TT1)
x component
of Buoyancy Force /3gp(T  T1) costp
F IGURE 8.3 Buoyancy force components.
au + av = 0 ax ay 22 a a. 1a u  u +v _u =  '  p + v (a u + a u) + {3g(T  Tl) cos.c/J 2 ax ay p ax ax ay2 . v) . . .ua+v =  +v (a 2+a2 +{3g(TTt)smc/J v av 1 ap v 2 ay2 ax ay p ay ax
u
(8.9) (8.10)
(8.11)
aT + v aT ax ay
=
2 (~)(a2T + aT) p Cp ax2 ay2
(8.12)
In these equations, T l is the temperature in the undisturbed fluid far from the surface a ndtlie pressure is, then, measured relative to the local hydrostatic pressure at this temperature o f the undisturbed fluid. The dissipation term has been neglected in the energy equation (8.12) due to the low velocities involved~ The continuity equation (8.9) and the energy equation (8.12) are identical to . those for forced convective flow. T he x  and ymomentum equations, i.e., Eqs. (8.10) and (8: 11), differ, 4owever, from those for forced convective flow due to the presence o f the buoyancy U;rms. The way i n which these terms are derived was discussed i n Chapter 1 when considering the application o f dimensional analysis to convective heat transfer. I n these buoyancy terms, cf> is the angle ~at t he xaxis makes to the vertical as shown i n Fig. 8.3. I t'should b e noted that, i n contrast to forced convective flows, i n natural convective flows, due to the temperaturedependent buoyancy forces in the momentum equations, the velocity and temperature fields are interrelated even though the fluid properties a re assumed to b e constant except for the density change with temperature.
8.4
~JMILARITY
I N F REE CONVECTIVE FLOWS
Consideration was given in Chapter 2 t o t he problem o f determining. the conditions lll1der w hich the flow and temperature. distributions about geometrically similar
346 Introduction to Convective Heat Transfer Analysis
bodies are " similar", Le., the conditions under which the velocity and temperature fields about the bodies will be "similar". B y similar velocity and temperature fields, i t is here meant that the velocity components and temperature when expressed i n dimensionless form are related to the dimensionless coordinate system by the same functions i n all o f t he flows considered. The same problem will now b e briefly considered for free convective flows, i.e., the conditions under which the flow and temperature distributions about geometrically similar bodies are " similar" i n natural convective flows will b e investigated. Attention will be restricted to twodimensional flow i n the present discussion. The governing equations are first expressed in terms o f suitable dimensionless variables. I n order to do this it is necessary to introduce a suitable characteristic velocity. Now, a measure of the buoyancy forces p er u nit volume existing in the flow will be [3 g peTwr  Td where Twr is a measure o f t he wall temperature. The buoyancy forces do work o n t he fluid as it flows over the surface, a measure o f this being the product o f the measure o f the buoyancy forces and a measure o f t he distance over which these forces act. This di~tance will b e t he characteristic size o f the body, D. Hence, a measure o f t he magnitude o f the work being done b y t he buoyancy forces on the fluid per unit volume o f the fluid is [3 g peTw r  T l)D. A s a result o f this work, the fluid gains kinetic energy, a measure o f this kinetic energy per unit volume o f t he fluid is 1I2pu~, Ur being the characteristic velocity i n the flow. Equating the measures of the work done and o f the gain in kinetic energy i n t he fluid then gives:
f3gp(Twr  Tl)D =
i.e.:
'1 PUr 22
U = j2[3g(Twr  Tl)D r
Therefore, the characteristic velocity will b e taken as j[3g(Twr  Tl)D. The following dimensionless variables are then introduced:
X = x lD,
U = u ljf3g(Twr  T1)D, P
Y = ylD, V = vIJrf3g(TwrTl)D~
(J =
(8.13)
= plf3gp(Twr 
T1)D,
(T  Tl)l(Twr  Tl)
Substituting these variables into Eqs. (8.9) to (8.12) then gives:
au av ( ax + ay
=0
(8.14)
u ax + v ay
av
av
=
ay + [/lg(Tw,  T1
ap
v2
)D3]
0.5 ( 2
aX2 + a f2 + 9smq,
(8.16)
.. a v a2 v )
.
CHAPI'BR
8: NaturalConvection 347
u ae Vao ax + ay
=
+ pCpD Jf3g(Twr  Tl)D aJtl + aYz
[k
1
] (aZo
aZo )
(8.17)
From these equations it follows that the relations between U, V, 0, and P and X and Y will be the same in two flows if .the following two parameters have the same values in the two flows:
The first of t~ese is, of course, the Grashof number, Gr, while the second can be written as:
(I':p )[/38(Tw,,;; TI)D'f
= P rGrlfl
(8.19)
where Pr is the Prandtl number. In view of these results and by considering the boundary conditions, it follows .,',' .that two free convective flows over geometrically similar bodies will be similar i f . t he following conditions exist:
.J. The temperature distributions at the smface are similar, i.e., ( Tw  Tl)/(Twr  Tl) is the same function of X and Y in the two flows.
ii. The Grashof number is the same in the two flows. 'ill. The Prandtl number is the same in the two flows.
Froplthe above discussion, it can be concluded that the dimensionless temperature field 8(X, Y) depends only on the dimensionless numbers G r ~d Pro In order to de~ne the implications this has for the local heat transfer rate at any point on the surface it is noted that this heat transfer rate is given by Fouriefs Law as:
qw
,
=
 k an w
aTI
(8.20)
where aTlanl w is the local temperature gradient at the surface measured normal to the surface. Now this derivative can be written as: '
~~Iw ~~L ~~L + ~~Iw ~=L
=
. (8.21)
The derivatives aylanl w and axlanl w will depend only on the shape of the surface . at the point considered. , Now the righthand side ofEq. (8.21) is conveniently rewritten i n dimensionless form to give: (8.22) .
348 Introduction to Convective Heat Transfer Analysis
where
N =
n D
(8.23)
Since aXlaN!w and aYlaNlw will depend only on the shape of the surface and since the dimensionless temperature gradients aOlaxlw and aOlaYlw must depend onthe same variables as the dimensionless temperature field, i.e., on G r and P r, i t follows that at any position on the surface:
an w = (Twr L Tl) function(Gr, P r) . aTI .
Substituting this result into Eq. (8.20) then gives:
(8.24)
(Tw  T1)k =
qwD
(Twr  Tl ction(Gr, Pr) w Tl)fun . T
.
(8.25)
Now the lefthand side· of this equation is the local Nusselt number, Nu, Le.:
qw D N u :;::: (Tw  T1)k
iltfollows, therefore that for a given surface temperature distribution, i.e., for a given ,distribution o f (Tw  Tl)/(Twr  Tl):
N u = function( Gr, P r)
.~This
(8.26)
result was, of course, previously derived by dimensional analysis .
. E X A MP L E 8 .1.
Consider two vertical plane surfaces, both of which have a uniform surface temperature. These surfaces are exposed to stagnant air at a temperature of 15°C. One of surfaces has a height of 0.1 m and a surface temperature of 65°C. The other surface has a height 0[,.0.2 m. Under what conditions will the natural convective flows over the two surfaces be similar and what will be the ratio of the mean heat transfer rates from the two surfaces?
the
Solution. Because both surfaces are exposed to air, the Prandtl numbers will be essentially the same in the two flows. Hence, similar flows will exist when the Grashof numbers i n the two flows are the same, Le., when:
[ f3g(Tw  T 1)D3]
V2
A
= [f3g(TW  ~1)D3]
V2 .
B
subscripts A and B denoting the two flows and D being· the height of the surface. Because the mean fluid temperatures in the two flows will be different, the fluid properties in the two flows will be somewhat different. This will, however, be neglected / here, Le., i t will be assumed that:
f3A = f3B, VA
= VB
/
In this case, similar flows will exist when:
[(Tw

TdD3]A == [(Tw  Tl)D 3]B
Le.: (65  15) X 0.1 3
= (TwB 
15) X 0.23
C HAPlER _8:
Natural Convection
~49
i.e.:
TwB
= 21.25°C
_Therefore. the flows over the two plates will be similar when the larger plate has a surface temperature o f 21.25°C. I f t he flows are similar, the Nusselt numbers in the two flows w ill b e the same, Le.:
[
qwD
(Tw  T})k
]
A
[ qw D ) ] == (Tw  Tl)k B
Hence, because it is being assumed that:
kA = kB
i t follows that:
;::,,~
qwAO.l
(65  15)
==
qwBO.2
(21.25  15)
i.e.:
qwB = 0.0625 qwA
Therefore, the mean heat transfer rate p er unit surface area for the large plate will be 0.0625 times that for the smaller plate.
, 8.5 , 'BOUNDARY LAYER EQUATIONS FOR NATURAL CONVECTIVE FLOWS
As in the case~with forced' convective flows, there are many free convective flows that can be analyzed with suffipient accuracy by adopting the boundary layer assumption. Essentially this boundary layer assumption is that the flow consists of two regions:
i. A thin region adjacent to the surface in which the effects of viscosity and heat transfer are important. ii. An outer regioIl/in which the effects of viscosity and heat transferare negligible.
Since the effects of heat transfer are negligible i n the outer region, the temperature in this region will be effectively constant and, therefore, equal to the ambient temperature, T l. There are, therefore, no buoyancy forees in this outer region. The boundary layer equations for free convective flow will be deduced using essentially the same approach as was adopted in forced convective flow. Attention will, as discussed above, be restricted to the case of twodimensional laminar boundary layer flow. Attention will initially be focused on a plane surface that is at 8Jl angle, q" the vertical as shown in Fig. 8.4. The xaxis is chosen to be parallel to this surface as shown in Fig. 8.4. Inorder to derive the boundary layer equations, it is necessary to introduce a velocity that is characteristic of that existing in the boundary layer. For the reasons given in the previous section thischaracteristic velocity will be taken as: (8.27)
/
350
Introduction to Convective Heat Transfer Analysis
Vertical
y
FIGURES.4
Free convective flow considered.
Here, L i s a reference length that characterizes the size o f t he surface, e.g., its length. T wr is again some reference wall temperature. I f some measure o f t he boundary layer thickness, S , is also introduced, then the governing equations c an b e written in t enns o f the following dimensionless variables:
X
=
xlL,
Y
P
=
ylS, . U = uIJ{3g(Twr  Tt)L,
(J =
V = v IJ{3g(Twr  Tl)L
(8.28)
= pl{3gp(Twr  TdL,
( T  Tt)/(Twr  T 1)
No distinction is made at this stage between the velocity a nd t emperature boundary layer thicknesses, both being assumed to b e o f t he same order o f magnitude. S is, therefore, a measure o f the order o f m agnitude o f b oth boundary layer thicknesses. I t will b e r ecalled that, by basic assumption, s /L is small. ' Because o f the way in which they are defined, all o f t he above dimensionless variables except V have the order o f m agnitude Of I , Le.:
I
x = 0(1),
Y = 0(1),
U
= 0(1),
P = 0(1),
(J
=
0 (1)
T he o rder o f magnitude o f V is detemrined from the continuity equation, i.e., Eq. (8.9), which can be written as:
au
F rom t his i t follows that:
ax + ay (SIL)
o (U)
av
1
=
0
(8.29)
o (V) = o (X) X oCY) X 0 L
(S)
(8.30) .
Therefore, since, as discussed above, o (U), o(Y), a nd o (X) a re all 1, t his equation indicates that: (8.31) S ince t he basic assumption o f b oundary layer theory is that (SIL) is small, i t follows from this that V is also small. Thus, i n a free convective boundary layer, as i n t he forced convective boundary layer, the lateral velocity component, v, is very much smaller than the longitudinal component, u, w hich i s, o f course, what i s p hysically to b e e xpected.
CHAPlER
8:' Natural Convection 351
Now the xmomentum equation (8.10) can b e written in terms o f the' dimension;. less variables introduced above as:
au
U ax
+ v a y ( 5IL)
au 1
[{3g(T v2Tl)L3 ]0.5 =  ax +
ap
wr 
x
(aaXZ + au(8IL)2 ) + 1 a y2
2U 2
(8.32)
(J
coscf>
i.e., introducing the orders o f magnitude o f t he various terms i n this equation gives the following: ~0(1) 0(1) 1 0(1) 0 (1) + 0 (8IL) 0(1) (8IL)
= 0(1) + Gr~5
.
1
[ 0(1) 0(1)
+ 0(1) (atL)2 + 0(1)
(8.33)
0(1)
1
]
Le.:
0(1)
+
0(1)  0(1)
+ G~5 )+ o~~;]/)] + H
=
0(1)
0(1)
T he second te~ on the righthand side o f this equation shows that i f ( 8/L) is' sniall, as it m ust b e i f the boundary layer approximations are to. b e applicable, then GrL nlUst b e large, its order o f magnitude being such that:
o (l/G r °.5) 0 [(5IL)2]
o (GrL)
=
0
[(8/~)4 ]
(8.34)
From this i t also follows that 51L = 0[lIGr~25J. As a consequence o f the above result, i t c an b e seen that the first term within the square bracket on the righthand side o fEq. (8.33), i.e., the term originating from the term (f.La2ulax2), will have 0(5IL)2 Whereas, the other terms in the equation have 0(1). For this reason;/when dealing with boundary layer flows, this term in the xwise momentum equation is negligible. For such flows, therefore, the xmomentum equation is:
au U ,+ ax a u" v = ay
1 ap  . p ax
+ ( f.L)a u + {3g(T  ' . .
p ay2
2
 Tl)COS.p .
(8.35)
To find the order o f magnitude o f t he pressure term, i.e., o f t he difference between the dimensionless pressure i n the boundary layer ·and the pressure that would exist in the absence o f the flow, attention i s turned to the ymomentum equation. I n terms o f the dimensionless variables introduced above this equation becomes:
av
U ax + v
av 1 ay ( 8/L)
ap =  at +
[{3g(T~rv2T )L3 ]0.5
1
x (aXZ +
a
2V
a2v
a y2(8/L)2
1)+ .
(8.36)
(J SlUcf>
352 Introduction to Convective Heat Transfer Analysis
i.e., introducing the orders o f magnitude o f t he various terms in this equation gives the following: .
o (8IL) 0(1) 0(1)
+ 0 (8IL)
0 (8IL) 1 0(1) ( 8IL) 0 (8P) = 0 (8/L)
+ Gr 2· 5
1 [ 0(8IL)
0(1)
+
0 (6/L) 1 ] 0(1) (8IL)2 + 0(1)
H ere, /:lP i s a measure o f the dimensionless pressure change across the boundary layer. B ut i t w as shown that o (Grd = 0 (8/L)4. Hence, the above equation shows that: (I1P) . 0 (8IL) + 0 (8IL) = ; (8IL) + [o(8IL~3 + 0 (8IL)] + 0(1) F rom this i t follows that:
o(/:lP) = 0 (8IL)
(8.37)
S ince t erms o f the order 81L a nd less are being assumed to b e negligible, this .. equation indicates that the changes i n pressure in the ydirection acrosS t he boundary l ayer c an b e ignored, i.e., that the pressure in the boundary l ayer i s e qual t<) the pressure a t t he outer edge o f the. boundary layer. But i n a f ree convective flow, '. the pressure outside the boundary layer is equal to the hydrostatic pressure i n the ,stagnant surrounding fluid. Hence, since the pressure, p, b eing u sed is measured relative to the local hydrostatic pressure this means that p is equal t o 0 every.. '. , where i n the boundary layer. I t follows that the xdirection momentum equation( for twodimensional free convective boundary layer flows, i.e., Eq. (8.35), can be written as: u ax'+ v ay =
aui
au
2 (M)iJy2 + f3g(T Ii a U
Tt)cos<p
(8.38)
I t should b e noted that i t w as assumed i n the above analysis that () cos <p was o f the order o f 1. This will not b e t rue when cos (4)) is near 0, i.e., w hen <p is n ear 90°, i.e., the above equation will not apply to a nearhorizontal surface. This is ,because in such a case there is essentially no component o f t he buoyancy force i n t he direction o f flow, t he flow actually being caused b y t he pressure differences i n t he flow that are induced b y t he buoyancy forces acting across the flow. T hese pressure differences were, o f course, ignored in the above analysis. Next, consider the energy equation. I n t enns o f the dimensionless variables introduced above this becomes:
a() iJO 1 ( 1 ) [ u ax + v a f (81L) = P r f3g(T
0(1)
p2
wr
~ T dL3
]0.5 (a 20
'.
ao) aX2 + a y2
2
(8:39)
U sing t he results concerning the orders o f magnitude o f V a nd GrL d erived above gives the orders o f magnitude o f t he various terms in Eq. (8.39) as:
/
+ 0(1)
= o (Pr) [0(8/L)2
1
+ 0(1)]
(8.40)
, CHAPTER
8: Natural Convecti9n 353
I f t ennsof t he order ( 8/L) and less are again neglected a nd i f i t i s assumed t hatthe Prandtl number, Pr; has order 1 or greater, i.e., is not small, i t follows that 2 the xwise diffusion term, i.e., a T/ail, is negligible compared to the other terms. Hence, the energy equation for free convective laminar twodimensional boundary layer flow becomes:
u aT + v aT , 'ax ay
=
(~)a2T p Cp ay2
(8.41)
To summarize, therefore, the equations for twodimensional, constant fluid property boundary l ayer flow over a plane surface are:
au + av ax ay au au lA,ax + v ay
~
2
=0
(IL)ay2 +,""  'Tl)coS4> u " pa ,{3g(T
aT aT u ax + v ay
(pk )aa2T = Cp y2
.I
These equations were derived for flow over a plane surface. T hey m ay b e applied to flow over a curved surface provided that the boundary layer thickness remains small compared to the radius o f curvature o f t he surface. When applied to ~ow over a curved sUrface, x is measured along the surface and y is measured :nonnal to i t a t all points, i.e., body fitted coordinates are used. The value o f cfJ i s t hen dependent on x. T hisis jllustrated in Fig. 8.5. I n order to solve the above set o f boundary layer equations, the b oundary conditions on the variables invo~ved, i.e., u, v, a nd T, must b e known. A t t he wall, the noslip condition requires that the velocity b e t he same as that o f the wall, Le., usually zero, so that one such boundary condition is: At Y
= 0:
u= v=0
(8.42)
I f there is blowing suction at t he sUrface, t he rate o f blowing or sucking will dete~e v a t t he walL
Vertical
0;
8 1R«1
F IGURE 8.5
Free convective flow over' a curved surface.
354 Introduction to Convective Heat Transfer Analysis
The' boundary conditions on temperature at the wall depend on the thermal conditions speCified at the wall. I f the distribution of the temperature o f the waIl is specified then since the fluid in contact with the wall must be at the same temperature as the waIl, the boundary condition on temperature is:
A ty
== 0: T = Tw
(8.43)
where T w is the temperature of the wall at the particular value of x being considered. I t is also possible for the distribution of the heat transfer rate at the wall to be specifi~d ,and in this case the boundary condition become~:
' aT qw At y = 0: a y =  k
(8.44)
where qw is the specified local heat transfer rate per unit area at the wall at the particular value of x being considered. " Because there are assumed to be no changes in temperature outside of the bound ' ary layer, there is no flow in the xdirection outside of the boundary layer. Therefore, since the temperature is constant outside the boundary layer, the following boundary conditions apply:
·J· 1
. For large y :
u
~
0,
T
~
Tl
(8.45) •
. It sho~ld again be noted that the velocity and temperature field's are interrelated by the buoyancy term in the momentum equations. Therefore, it is not possible, as 'in forced convecti.on, to solve for the' velocitY components independently of the tem. perature field. '
8.6. S IMILARrfY SOLUTIONS F OR F REE C ONVECTIVE LAMINAR BOUNDARY LAYER F LOWS
'I
I n order to illustrate how. similarity solutions are obtained for free convective flows.. see [7] ·to [23], conSideration will initially be given to tWbdimensional flow over a vertical fiat plate with a uniform surface temperature. The situation being considered is thus, as shown in Fig. 8.6.
Vertical Surface
Held at a Unifonn
Surface Temperature
.
y
v
.'
./
F IGURE 8.6 Flow situation considered.
CHAPTER
8:, Natural C onvootion3SS
Because the boundary layer assumptions are being adopted~ the equations' governing the flow are, as discussed above:
ax au u ax a u,
au+ av = 0 '
ay
(8A6)
+ v a y='
,,(f.L) ay2 '+ (3g(T u , pa
2
Tl)
(8A7)
aT . . a T '( k , )a2 T u+v=  ax ay p Cp ay2
(8.48)
. cos cp for a vertical surface being equal to 1. The boundary conditions on the, solution 'are:' At y
= 0:
"
U~ V =
0,
T
= Tw
Tl
(8.49)
For large y:
u ~ 0, T
~
As was done in dealing with forced convective flow over a uniform temperature plate, it is assumed that the velocity and temperature profiles are similar at all values x, i.e., that:
of
'U
u ' fun' ::;::: c tion5, r"
(y)
and that:
T  Tl = funCtiOn('?) Tw  Tl U
where Ur is again a~reference velocity and 5 is a measure of both the local velocity and thermal boundary layer thiq1messes. . The profiles are thus assumed to be similar in ,the sense that although 8 varies with x and although the velocity and:temperature' at a fixed distance, y, from the wall vary with x, the velocity relative to the local reference value and temperature difference at fixed values 9 fyl5 remain constant The proof that such similar profiles do in fact exist follows ,from the' analysis given below. ' , ' , , ' ,Now, i f the distance, x, from the leading edge o f the plate to the point under consideration is taken as the reference l~ngth, L, it follows from the derivation of the boundary layer equations presented earlier that:
81x ,':;:: o[1/Gr~·2S]
where G rx is the Grashofnumber based/on~, i.e.:'
, G rx ' 'f3g(Tw  Tl)X3
(8.50)
=
v2
Therefore, a measure o f the local value of yl5 is:
, ~ = '~dr~·25 x
(8.51)
This variable, 'YJ, is termed the similarity variable.
356 Introd,uction to Convective Heat Transfer Analysis
t herefore, i t w ill b e assumed that:
~
Ur
= function( 'TJ)
a nd that:
T Tl Tw  Tl = function(l1)
F or the reasons given previously i n this chapter, the reference velocity is taken as:
Ur
=
J f3g(Tw  Tl)X
Therefore, i t is a ssumed that the velocity and temperature profiles have the form: (8.52) and: (8.53) The prime, o f course, again denotes differentiation with respect to 'TJ. T he differentiated function p I is, as i n forced flow, used for convenience i n describing the velocity profile. . T he following dimensionless variables are now introduced:
.. U
=
U
.
J f3g(Tw  T1)x
=
=
v:±:::
v J f3g(T
, (J
w  Tt)x ..
(ux)arv (vx)arx
O .5
~". x
'
O5 •
(8.54)
= ( T  T1)/(Tw 
Tt)
Us~g t hese t he continuity equation can b e written. as:
i.e.:
au " u o V  ++=0
o x' 2x
r Jy"
(8.55)
I n t erms o f t he similarity variable, 'TJ" t his becomes:
OU 011 + 011 a x . 2 x
!!. + o V 0'TJ
a'TJ a y
=0
(8.56)
/
B ut u sing t he definition of'TJ g iven i n Eq. (8.51):
a'TJ = ' LGrO.25 = a x . 4 x2 x ."
 !L
4x
(8.57)
CHAPTER
8: Natural Convection 357
01'/ = Gr~·25 IJy X
(8.58)
Substituting these two results into Eq. (8.56) then gives:
d F'
d1'/
(.!L) + F' + oV Gr~·25
4x 2x
=0
01'/
x
i.e.:
oV _
1 01'/  2G~·25
(1'/ dF' _ FI]
'2 d1'/
:
(8.59)
The boundary conditions on the solution g.ive: .
'.
At Y = 0: v = 0
But when y plies:
= 0,
1'/ = 0, and wh~n v
At 1'/
= 0" V
= 0 so this boUQ.dary condition im=0
= 0:
V
Eq. (8.59) can be integrated using this boundary condition to give:
V=
2Gr~.25
1
(J' 12'/ ddF' dTJ 1'/
F
].
i.e.:
i.e.: (8.60) The first tenn in the bracket having been integrated by parts which gives:
I
i.e.:
71 dTJ d1'/ = 2 dF'
J' dd (1'2/ F ')dTJ  J'rdTJ F' 71
:(J
Consideration can now be given to the momentum ~quation which c~n be.written in terms ,of the dimensionless variables intrOduced'8;bove as: , ." ',
oU U2 o U' 02U 1 U++V,=viJx ' 2x o y, uy2 Jf3g(TVY'.,Tt)x
+x
358 Introduction 'to Convective Heat Transfer Analysis
i.e.:
F 'F" (  !L ) + F,2
4x

2x
+
1
4Gr~·25
G 0.25 ('TIF'  3F)F" rx
= x + x· x
/ If
F ill
6
I.e.:
'TI F'F "
4
+2 +
F,2
'TIFfF"
4
 4 = F
3FF "
+G
i.e.:
F lit
+    + G = 0 4 2
3 FF"
Ff2
.
(8.61)
Lastly,·consider the energy equation. Because Tw a nd T l are constants, the energy equation can b e written as:
,
;
i.e.:
F' Gf
i.e.:
(!L) +
4x
1 ('TIFf _ 3F)G' G~25 4G1'2· 25 x x
=
G" Pr x
Gil
.4
+ ~PrFGf
=0
(8.62)
Eqs. (8.61) a nd (8.62) constitute a pair o f simultaneous ordinary differential equations for the velocity and1temperature functions, F a nd G. They must be solved subject to the following boundary conditions:
A t y = 0: u = 0 i.e., at 'TI = 0: F ' = 0 A t); = 0: v = 0 i.!!., a t 11 = 0: F = 0
T· = Tw i.e;, a t 11 = 0: G '= 1 F or large y: u ~ 0 i.e .• for large 'TI: 1;' ~ 0
A t Y == 0:
F or large y :
(8.63)
T ~ T l i.e., forlarge 11:· G
+
0
The fact t hat the original partial differential equations have been reduced to a pair o f ordinary differential equations confirms the assumption that similarity solutions do i n fact exist. Eq. (8.62) h as t he Prandtl number, Pr, as a parameter and a separate solution for the variation o f F and G with 'TI is, thus, obtained for each value o f t he Prandtl number, Pro T he h eat transfer rate at the wall is then obtained b y noting that:
a qw =  k  T
oy
I
dG Grf!.·25 =  k(Tw  Tl) .x y=o d'TI 'I}=o x
I
CHAPTER
8: Natural Convection 359
i.e.:
GrQ.25 = x
Nux
G'I71=0
(8.64)
where Nux is, as before, the local Nusselt number. I f the boundary conditions listed i n Eq. (8.63) are considered, it will be seen that there are conditions at both 'YJ = 0 and at large TJ. Eqs. (8.61) and (8.62) cannot therefore, be directly integrated. One way of simultaneously solving these'equations is to guess the values of F " and G' at TJ = 0 and to then simultaneously :Q.umerically integrate Eqs. (8.61) and (8.62) to give the variations of F and G with 7]. The solutions so obtained will not generally satisfy the boundary conditions on F and G at large TJ. The solution is then obtained for other guessed values of F " and G ' a t TJ = oand these results are used to deduce the values o f these quantiti~ that give solutions that Satisfy the boundary conditions F ' = 0 and G = 0 at large 11; While there are some very elegant ways of obtaining the solution in this general manner, ,the simplest procedure is to guess F " and G' at TJ = 0 and obtain the solution andthen increase, first, the guessed value of Fit at TJ = 0 by a small amount and obtain the solution and, second, the guessed value of G ' at 7] = 0 by a' small amount and obtain the solution. From these solutions the effect Of changes in the guessed values of F" and G ' at TJ ,: O'on the differences between the values of F,' and G at large TJ given by the solution and the required values can be determined. Improved guesses for the values of F "and G ' at 7] = 0 can then be deduced and the process repeated until converged values of F " and G' at TJ = 0 are obtained. A computer program, SIMPLNAT, written in FORTRAN, that implements this procedure is available as discussed, i n the Preface. As set up, this program will give results for Prandtl number values up to about 30. The initial guessed values must be altered to obtain solutions at higher values o f Prandtl number. Some typical veloCity and t~mperatUre profiles obtained using this procedure are shown in Figs. 8.7,and 8.8 while some typical values of G'I'I1=o are shown i n Table 8.1.
,
'
0.6
.......,..~..
0.4
F'
0.2
2
4
6
8
TI
F IGURES.' Dimensionless velocity profiles i n natural convective boundary layer on a vertical plate for various values of Prandtl number.
/
360 Introduction to Convective Heat Transfer Analysis
1.0
r..,..,.~_...
0.8
0.6 G 0.4
0.2 0.0 L~~l:....."":::::::_...L::::::::::;;=_ _ _ ' _0 2 4 6 8
1'J
F IGURE 8.8 Dimensionless temperature profiles in natural convective boundary layer on a vertical plat~ for various values of Prandtl number.
T ABLE 8 .1
Values of G'I,p=o for various values of P r
Pr
0.01 0.03 0.09 0.72 1 2 5 10 100 1000
G'ltJ~o
G'ltJ",oIP,.o.zs
0.1802 0.2312 0.2828 0.3873 0.4010 0.4260 0.4511 0.4644 0.4898 0.4992
.~
0.0570 0.0962 0.1549 0.3568 0.4010 0.5066 0.6746 0.8259 1.549 2.807
I t has been fou~d.that the values of G'I'1=o can be approximately fitted by the
following empirical equation:
:~~ = G·I.~o = [2.44 + 4~:S~~~: 4:,!SPr
1 (L qw = 1,Jo qw dx
r
(8.65)
(8.66)
The mean heat transfer rate for a plate of length. L (see Fig. 8.9) can be obtained by noting that:
Hence, using Eq. (8.64):
1 (L[ G rO .2S _] qw = 1, Jo  k(Tw  Tl)G'I.,,=o ; d x
/
CHAPTER
8: Natural Convection 361
L
FIGURES.9 Flat plate being considered.
. i.e.:
where GrL is the Grashof number based on 'the plate length, L. The mean Nusselt number for the whole plate N UL ( = / iu k) is, therefore, given by:
GI}.25 =
L E XAMPLE 8 .1.
N UL
3'G
4
II
' 1]=0
(8.67)
A 30em high vertical isothennal :8at plate with a sutface temperature of 500C is exposed to stagnant air at 10°C and ambient pressure. Plot the velocity and temperature profiles i n the boundary layer at the top of the plate and the variation of the localheat transfer rate along the surface. Assume the flow remains laminar.
Solution. The similarity solution given above indicates that the temperature and velocity distributions are given by:
(a)
and:
(b)
where:
1j
= !x :;c 2S ..GrO.
The functions G and F' depend on the value of the Prandtl number, Pro The mean air temperature is:
Tav
= (50 + 10)
2
p
= 300C
At this temperature air, at standard ambient pressure, has the following properties:
fJ : : : 0.0033 K  1,
k=
0.026~8
= 16 X 106 m2ls.
W/mK, Pr == 0.7
Therefore, atthe top o f the plate where x
= 0.3 m:
= !.. [fJg(Tw ~ T..,)x.3 f2S = L
.,., X p2 .
[ 0.0033 X 9.81 X (50  10)0.3 0.3 (16.X 10:6)2. .
. .
3f2S
.
3 62 Introduction to Convective Heat Transfer Analysis This equation gives on rearrangement:
. y = 366.4 (c)
Also:
u J2{3g(Tw  Tt)x
and:
u
J 2 X 0.0033 x 9.81 x (50  lO) x 0.3
T  Tl Tw  Tt
T lO
u
0.8815
(d)
=
5 0lO
=
T lO 40
(e)
T he similarity solution gives the variations o f the functions G and F ' with 11 for P r = 0.7. F or any value o f 11, Eq. (c) allows the corresponding value o f y to b e found while Eqs. (d) and (a) together allow the corresponding value o f u to b e found and Eqs. (e) a nd (b) together allow the corresponding value o f T to b e found. T he variations o f u a nd T with yobtained using this procedure are shown i n Fig. E8;2a. For a Prandtl number of 0.7, t he similarity solution ~so gives:
G rO•25 = 0.353
x
Nux
.
i.e.:
, qw x " k(Tw  Tt)
~ 0.353[{3g(Tw TaJ )xlf25
v2
,
In t he situation here being considered it therefore follows 'that:
. qwx = 0 353[0.0033 x X (50  'lO) X3 0.02638 X (50  lO), . (16 X 106 )2
9.8~
f
25
Le.:
0.4
j .l mls
0.2
0.0
o
' _....I..._'_'=:::..L_..I
5
10
15
20
25
s
10:
15
2S
y mm
, ymm '
/
FIGURE ES.2a
CHAPTER
Mm~~~~~
8: Natural Convection 363
300
200
100L~~~~~
o
10
20
30
x em
F IGURE E8.2b
The v~ation o f the local heat transfer rate qw with x as given b y this equation is shown in Fig. E8.2b.
E XAMPLE·8.3.
Plot the variation of the boundary layer thicknesses with distance along a lm·high vertical pla,e which is maintained at a uniform surface temperature of 50°C and exposed to stagnariI air'at ambient pressure and a temperature of H)oC. Assume the flow remains laminar. ..
Solution. As indicated schematically in Fig. ES.3a, it can. be deduced from Figs. 8.7 ·and 8".8 that the velocity decre~es to 0.01 of its maximum value and the temperature difference decreases to 0.01 of its overill value for a Prandtl n.umber of 0.7 (the assumed value for air) ap~r()xima~ely when:
11 ~ 7 and 11
= 5.5
respectively.
 1
F'max
F'
and G
F IGURE ES.3a
364 Introduction to Convective Heat Transfer Analysis
Hence:
8 Gr?:25 x"
and:
/)
=7
.2:Gr2·25 = 5.5 x where /) and /)T" are the effective thicknesses of the velocity "and temperature boundary layers. respectively. The average air temperature is given by:
2 At this temperature, air at standard ambient pressure, h~s the following properties:
Tav
= (50 + 10) = 300C
' 11
f3
Hence:
G
1'"
= 0.0033 K  1,
= 16 X 1 06 m2/s
=
f3g(Tw  T1)x3 = 0.0033
112
X 9.81 X 4 0 3 = 5 058 280xl 0.0000162 X • ,
where x is is the distance up the plate in m. Using this expression for Gr" then gives:
/) = (5,058,~;Ox3)o.2S
:and:
8
T=
= 0.1476i>·25
S.5x (5,058,280x3)o.2S
= 0.1160xo.2S
The variations of 8 jUld 8T with x as given by these two equations are shoWn in Fig. E8.3b.
1.0
. :"!r,."T1
0.8
0.6
X m
0.4
0.2
0.0 __
~
~d:::::::==_
_ _. ..L__ _ __ '
0.00
0.05
0.10
O.IS
FIGURE ES.3b
CHAPTER
8: Natural Convection
365
Vertical
Plate
Tw =function (x)
L
F IGURES.tO
Stagnant Fluid at Temp. Too
Flow situation considered in the discussion o f the numerical solution o f l aminar natural convection.
8.7
NUMERICAL S OLUTION O F T HE NATURAL CONVECTIVE BOUNDARY LAYER EQUATIONS
A numerical solution to the laminar 90undary: layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. A computer program, LAMBNAT, written in FORTRAN that is based on this procedure is available as discussed in the Preface. This program applies to twodimensional flow over a vertical flat plate with a surface temperature that in general is a fu~ction of the distance along the plate, x, i.e., to the flow situation shown in Fig. 8.10.
U se the computer program, LAMBNAT, for twodimensional laminar boundary layer flow over a vertical plate to find the dimensionless heat transfer rate variation along a plate whose surface temperature varies linearly in such a way that T w is equal to T a t t he leading edge and equal to T + 40°C at the trailing edge.
E XAM'PLE 8 .4.
r:t:J
f»
Solution. T he dimensionless temperature used i n the numerical solution is given by:
8
= (T 
Tl)/(Twr  Tl)
I n the present situation T w  Tl varies from 0 at X = 0 to 40°C at X wall temperature will therefore be taken to b e s uch that:
//
=
1. T he reference
The dimensionless wall temperature will then vary from 8w
X
= 1, Le.:
8w
= 0 at X = 0 to 8w = 2 at
= 2X
The only change required to the program as available is in the subroutine TEMP which becomes: .
* *
SUBROtJ'l'INB TEMP (X, TW)
*********** THIS DBTBRMINBS THB WALL TBMPBRATURB *******************
*
*
TW
= 2.0*X
RBTURN
366 Introduction to Convective Heat Transfer Analysis
1.0
r ...............
0.5
0.0 '      1   _    1 ._ _ _. .1.._ _ 1 0.01 0.1 1 10 100
Pr
F IGURE ES.4
T he program when run with this change gives the variation of NU:JfJGr~·25 with X where here: . .
G rx == [3g(Twr
i
.
V2
Tl)X
3
T he variation obtained in this way is shown i n Fig. E8.4.
8.8 . . F REE C ONVECTIVE F LOW T HROUGH A V ERTICAL C HANNEL
Attention wit! b e given in this~ection to buoyancyinduced flows through a vertical channel i n a large surroupdiIig environment. The flow arises because all or part o f the walls o f t he channel are at a different temperature from that o f the fluid surrounding the channel, see [24] to [33]. It will be assumed in the present discussion that the walls are heated and that the flow is consequently upward through the channel. The same analysi~ c an be used, o f course, when the walls are cooled, the flow then being downward through the channel. The flow situation here being considered is, therefore, as shown in Fig. 8.11. Flows o f this general type occur, for example, i n a ~umber o f situations involving the cooling o f electrical and electronic equipment and in the flow through certain types o f fin ·aiTangement. .. . Here i t w ill b e assumed, as shown in Fig. 8.11, that the flow enters the channel through a smooth, convergent inlet i n which the viscous losses are negligible. O n the inlet plane o f t he channel, the velocity will consequently be uniform and equal to the mean velocity a t a ny section o f the channel, U m. This is illustrated in Fig. 8.12. Because viscous effects are· assumed to be negligible i n the inlet section, Bernoulli's equation gives across this inlet section:
poo = Pi
+
p u2
/
2m
cllAPrER
8~
Natural Convection 367
t! t tit
tt
Discharge to
.Environment
Adiabatic . Inlet Section Flow Through Channel From Environment
FIGURE. 8.11 Free convective flow through a vertical channel.
Velocity Distribution on InI~t Plane
J .._..:Pr.:.='P.li:li:nll'P.
=PI
on this plane
."Velocity =0
F IGURE 8.12 . Assumed velocity distribution on inlet pl~e. . '. . .
Le.: (8.68) The pressure therefore falls across the inlet section from poo to:.poo  pu~!2, Le., the pressure on the inlet plane is below ambient. As the fluid flows up the channel, viscous forces tend to cause the pressure to drop while the buoyancy forces tend to cause the pressure to rise. Because the flow leaving the channel is essentiaJ1y parmlel to the walls of the channel, the pressure on the exit plane is essentially uniform and equal to the ambient pressure. In the present analysis, it will therefore be assumed that across ·the exit plane the pressure is equal to. the· ambient pressure, poo. This is illustrated in Fig. 8.13.
/
" ''I
36S Introduction to'Convective Heat Transfer Analysis
Nearly Parallel Flow on Exit Plane Exit Pressure on this =Poo
FIGURES.13 Assumed conditions on channel exit plane.
z
P
FIGURES.14 Pressure variation along channel.
T he pressure variation up the channel therefore resembles that shown in Fig. 8.14. Near the inlet the viscous forces are high and the pressure falls. However, further up the channel, the buoyancy forces become dominant and the pressure rises, reaching the ambient value on the exit plane. In order to illustrate how natural convection in a vertical channel can be analyzed, attention will be given to flow through a· wide rectangular channel, i.e., to laminar, twodimensional flow i n a plane channel as shown in Fig. 8.15. This type o f flow is a good model o f a number o f flows o f practical importance. In the present analysis i t will be assumed that both walls o f the channel· are heated to the same uniform temperature and that the flow is therefore symmetrical about the channercenter line. The coordinate system shown in Fig. 8.16 will therefore b e used i n the analysis and, because o f the assumed symmetry, the solution will only b e obtained for y values between 0 and W12, W b~ing, as indicated i n Fig. 8.16, the full width o f the channel.
FIG~S.IS
/
'JYpe of flow considered.
CHAPTER
8: Natural Convection 369
z
y • v i ..
i
Flow
t tt !
F IGURE 8.16 Coordinate system used.
I f it is assumed that W is small compared to the height of the channel, H , which w ill mean that: '
i Jp« 1 iJy
,i.e., i f the "parabolic" flow assumptions are adopted, the momentum equation in the zdirection gives:
"
2 iJu iJu dp p u+pv =  +JL (iJ u) pg iJz iJy dz iJy2
(8.69) "
where p is the absolute pressure. The "parabolized" forms of the governing equation are thus being used. It is convenient to work with the pressure relative to ambient pressure, i.e., with the "gauge" pressure. I t i s therefore noted that:
p
=
(p  pcx» + pcx>
where pcx> is the local ambient pressure at the value o f z being considered. The above equation gives:
d p =,~ ( p,_ P<x» + dp<x> dz dz dz
(8.70)
But:
Hence, Eq. (8.70) gives: (8.71) , Substituting Eq. (8.71) into Eq. (8.69) and dividing through by p then gives: ,,'
2 1d ui JuiJu =  (pp<x»+v (iJ u) + (P<x> ~ p)g , , + viJz i Jy, P d z iJy2 P
/
(8.72)
370 Introduction to Convective Heat Transfer Analysis Hence, using:
poo  p = f3(T  Too) P
Eq. (8.72), i.e., the zmomentum equation, becomes:
uau + v au az ay
=
1d  p dz(P 
poo)
+v
2 (ay2 + f3g(T...;.. Too) a u)
(8.73)
The Boussinesq approximation has, of course, been used in deriving the above eqUation. The continuity equation, as before, gives:
au + av az ay
I f it is assumed that:
=0
(8.74)
and i f the other assumptions used in deriving the zmomentum equation are adopted? the energy equation gives for the flow here being considered:
u aT + v aT az ay
=
(k
v
p Cp
T) )(a2 ay2.
(8.7,5)
The boundary conditions on·the above set of equations are as follows, it being . recalled that it is being assumed that the flow is symmetrical about the center line:
At y
= 0: ~; = 0,
u
= 0,
aT ay
=0
(8.76)
and:
A t Y = W/2:
= 0,
v
= 0, T = T w
(8.77)
the ycoordina~;as discussed before and as shown in Fig. 8.17, being measured from the center line towards the wall. Eq. (8.76) follows from the assumed symmetry of the flow about the center line. I n addition, as discussed before, on the inlet ~d exit planes of the channel, i t is assumed that: . .
At z
= 0: u = Urn,
At Z
pu~ p poo =  2
(8.78)
and:
=.e:
P = poo
(8.79)
where .e is the vertical height of the c h a n n e l . . . I t is also assumed that there is no heat transfer to the fluid the inlet section so that on the inlet plane the fluid is at the ambient temperature, Le.: /
m
A t z = 0:
T = Too ,
CHAPI'ER
8: Natural Convection 371
t,
W/2
W/2
I Flow i
t
<?+ i
i . ! i i
!
I
y
Wall: u = 0, v = 0,
T=Tw
i
Flow Symmetrical About Center L ineSov
J
=°
FIGURES.17 Boundary conditions on solution.
1\vo limiting types o f flow can exist. I f the channel is short and the Rayleigh number is. high (see later), the flow will essentially consist o f boundary layers on each wall with a uniform flow at temperature T<Xl between the boundary layers as shown in Fig. 8.18. Under these circumstances it is to be expected that a boundarytype relation for the heat transfer rate will apply. T he other limit would occur i f the channel was long and the Rayleigh number was low. I n this case, a type o f fully developed flow would b e expected to exist. Approximate solutions for the two limiting cases discussed above can be obtained (see below). However, most real flows are not well described by either o f th~se two limiting solutions. For this reason, a numerical solution o f the governing equations must usually be pbtained. To illustrate how such solutions can be obtained, a simple forwardmarching, explicit finitedifference solution will be discussed here. Although i t i s not a necessary step, the governing equations and the boundary conditions will b e written in dimensionless form before obtaining the numerical
~
Boundary Layers
':\
Temperature Variation Across Channel
F IGURES.1S . Boun~ layer ftow reglme . .
~;,
'
372 Introduction to Convective Heat Transfer Analysis
solution. For this purpose, the following is defined:
{3.g(Tw  Too) W4 = Grw W (8.80) v2 f f where G rw is the Grashof number based on the channel width, W. The following dimensionless variables are then defined:
G
Z
=
=~
f G,
y
=L
W;
0
= ( T  Too)/(Tw  Too)
(8.81)
u=
(u:)(;;;).
av + au = 0 ' ay' az au au d P (a 2 U az + v ay ==  dZ + ay2
U az
In terms o f these dimensionless variables, the governing equations become: . (8.82)
u).+
ao ay
. .
,
~
0
(8.83) (8.84)
*
ao
+ v ay
ao
=
2 (P r )(aYZ 1 a ())
0,
=0
I n terms o f these dimensionless variables, the boundary c~mditions are:
At and:
y = 0: au = 0, v = ay
U = 0,
(8.85)
A t Y = 0.5,
V = 0,
() = 1
(8.86)
while the conditions on the inlet and exit planes are:
A tZ = 0:
and:
U = Urn,
P =  U;
2,
() = 0
(8.87)
A tZ where:
= L:
P=0
(8.88)
(8.89) and:
f 1 (8.90) L= =fG G are the dimensionless mean velocity and dimensionless channel length respectively. Eqs. (8.82), (8.83), and (8.84) are the equations that mustbe solved. A t first viewing, they may appear to be a set o f three equations in four variables, i.e.,U, V, P, and (). However, i f the boundary conditions given i n Eqs. (8.85) and (8.86) are
CHAPI'ER
8: Natural Convection 373
examined, it will be seen that there are more boundary conditions than are required to solve Eqs; (8.82) to (8.83). In fact, the requirement that V be equal to zeto at both Y = 0 and at Y = 0.5, which by virtue of Eq. (8.82) is equivalent to:
a
az
1°·5 UdY = 0
°
(8.91)
(this is equivalent to:
1 °
0.5
U d Y = constant
.
i.e., is a statement of the fact that mass neither leaves or enters the duct) gives the extra relationship required to give four equatipns in four variables. Essentially, this equation is used to determine the pressure gr~dient which must be such that continuity of mass flow is ensured. As previously mentioned, the solution to the above set of equations will here ~ obtained using a forwardmarching, explicit finitedifference procedure. The solution starts with the known conditions on the inlet .plane and marches forward in the zdiiection from grid line to grid line as indicated in Fig. 8.19. Consider the nodal ',points shown in Fig. 8.20.
eo
,.....
i,
I
0
•
0
0
, . Zf
1
I
.
,
,,..
t
.
J ,
ForwardMarcbing SolutionSolution Advances from one Yli ne to the next, linebyllne .
_J_,!r"
~nlet PlaneI I
t Y
,Conditions Known
F IGURE 8.19
Forwardmarching solution procedure used.
AY
AY
  ...~.:......_.
I
i ,j
i l,jl i l,j
i l,j + 1
Equation Applied Here
FIGURE 8.20 Nodal point considered.
374 Introduction to Convective Heat Transfer Analysis
In terms of the values of the variables at these points~ the momentum equation, i.e., Eq. (8.83), gives the following, it being recalled that; since an explicit solution procedure is being used, all the Y derivatives· are evaluated on the line from which the solution is advancing, i.e., on the i  1 line:
U.
, I,}
. ( Ui,i  UiI,i) + v .
az
, 1.}
. ( UiI,j+1  UiI,i I ) 2 ay
P i  PiI
az
+
( UiI,i+I
+ U iI,jI
a y2
 2Ui I,j)
+
(J
i I,j
which gives:
' TT
V i,j 
_
U'
Pi iI,j ,,   . " U il,j ( UiI,j+I
P iI' V iI,} +      ( UiI,j+I'U iI,j U iI,j ,  2UiI,j)
UiI,jI .u. , 2aY ,
)'AZ
+.
i.e.:,
+ U iI,jI
a f2 '
A Z·
U i'I,j
+
8.
. dZ
, I,}
Ui~I,j
U'·· ,}
=
U·,  I,} '
,
p'
_ __ Ui
i I,j
+ A· }
(8.92)
where:
P i l _ V iI,j ( Uil,i+ l  Uil,jI ) az U il,j U il,j 2 ay
+
( Uil,j+l
+ U iI,jI
af2
 2Ui l ,j)
az + (J ;  1 ,az 'U i  l , i ' ) U iI,j
(8.93)
This equation applies from j = 2 to j = N  1. A j is a known quantity at any stage of the calculati9n because it only involves the values of the variables on.the ( i  1) line which are known. Attention will next be given to the continuity equation which can be wri~en as: , d Y =  az
av
au
(8.94)
This equation is again applied to the nodal points shown in Fig. 8.21. The equation is actually applied to the point 0 , shown in Fig. 8.21 which lies halfway between nodal points i, j and i, j  : 1. The value of a UlaZ at point 0 is, as before, taken as the average of the values' of a Ulaz at these two points. Hence, the continuity equation gives:'
V i,j
~;i,jl
=
_~ [ Ui,j ~~il,j +
U i,jI
~~iI,jl]
i.e.:
ay V i,j  Vi,jI =  2az[Ui,f  UiI,j
+ Ui,j~
i
/
 UiJ,jtJ
(&.95)
. CHAPl'BR
8: Natural Convection 375
LlY
I
ll.Z
II
i ,jl
"'~j Continuity I
Equation Applied
_.
F IGURE 8.21 Nodal point used with continuity equation.
___. •!_l~=~____..
I
Here
i l,j
r)
Substituting Eq. (8.92) into this equation then gives:
Yi,j  Vi,jl aY =," 2aZ [.PiI ,}. + A j , Pi 1. + A jI ].  V' U''., , I ,}
/
(8.96)
which can be written as:
. Vi,j  Yi,jl =  Pi 2 aZ U iI,j
[ aY (
1
+
U ;I,jI
1)~
~  2az(Aj + A jI)
aY
"
(8.97)
i.e., as: y '. .  y;. I ,} ' ,}where:
Bj =
,
=  B} . p.
"
 C· }
a y.
2 aZ V il,j
[
1
+
1
,
U il,jI
]
(8.98)
and: (8;99)
I t should be noted that Eq. (8.92) only applies for j = 2 to N  1. The boundary ' conditions, however~ give Uj,N = U iI,N = 0 so:
(8.100) and:
CN = a Y 1 2 aZ U il,Nl
(8.101)
It will be noted that since at any stage of the solution the conditions on the ( i ...:.. 1) line are known, the coefficients A j' A jl, B j. a ndCj are known qllantities. Now on the center line, i.e., at j = 1, Vi,j is zero. Hence i f Eq. (8.97) is applied' sequentially outward from point j = 2 to the point j = N which lies on the ,Wall,
/
376 Introduction to Convective Heat Transfer Analysis
the following is obtained:
Y i,2  0 =  B2 P i

C2
Vi,3  Yi,2 =  B3 P;  C3 Y '4  Y'3 = 1,  C4 4 4 B4 P V '5  V'4 = 1,  C5 4 4 B5 P
Yi,N  Yi,NI =  BNPi  CN
Adding this set of equations together then gives:
N N

Yi,N =  PiLBk
k =2
LCk
k='2
(8.102)
But the boundary conditions give Vi,N = 0 so Eq.(8.102) gives:
N
L Ck
=2 PI. = . kN
(8.103)
L Bk
k =2
Eq. (8.103) allows the dimensionless pressure, Pi, to be detentrlned. Once this ihas been found, Eq. (8.92) can be used to find the U i,j values and Eq. (8.97) applied sequentially outward from j = 2 as discussed above allows the V i,j values to 'be found. The boundary conditions give Ui,N = 0 and since to first otder of accuracy the zero gradient condition on the center line gives:
U i,2 U;,l =
0
AY
it follows that Ui,l = V i,2' Lastly, consider the finitedifference form of the dimensionless energy equation, i.e., Eq. (8.84), which is used to determine the distribution of 8 i,j' The finitedifference form of Eq. (8.84) is obtained using the same nodal points as used in dealing with the momentum equation (see Fig. 8.20).\ The finitedifference form of Eq. (8.84) is:
U. . (8 i ,j  8i 1 + Y; . (8 i  1 ;j) ,j+1 8i  I,jI) I I,}!:l.Z II,} 2 AY
=
~ (8i1,j+l + 8i l,jI """ 28 i  I,j)
Pr A y2
which gives:
,j+I .:.. 8il,jI)~ 8 . . = 8. . + ( !:l.Z ) [(8;I,j+l + 8 i l,j1  28iI,j )"_ y . . ( 8 i 1 I,} . II,} U '· . P A y2 II,} 2 AyJ .
I
I,}r~
;
(8.104)
CHAPTER
8: Natural Convection 377
This equation allows the 8 i,j values for j = 2 to N to b e found. The boundary conditions give 8 i,N = 1 and, to first order accuracy, the symmetry condition gives
8 '1 I,
==
8I'2' ,
TJle above set of equations allows the values of the variables on the iline to b e detenirlned from the known values o f the variables on the ( i  1) line. Therefore, startiIi.g with the known conditions on the inlet, the solution can b e marched up the chamtel from iline to iline. h1 general, the solution procedure adopted should allow the value o f the dimensionless duct length, L, to be selected and the solution should give the value of the dimensionless mean velocity, Um . I t is, however, more convenient to select a value' of U111, a nd use the solution procedure to calculate the value o f L corresponding to this value o f Um. I f the result for a particular value of L i s required i t can be obtained by interpolating between results obtained for several different values o f Um. T he actual solution procedure therefore involves the following steps:
1. ' A value o f Um is selected. . 2.' The values o f the vatiables on the first iline which lies on the inlet plane are
specified, i.e.:
i
= 1,
i
= 1 to N = 1, j = N:
j
i
1:
U
= Um,
V
V
= 0,
8
8
=0
U
= 0,
= 0,
=1
Also on the inlet plane:
= 1: P =  U;,/2
3. 4. 5. '6. 7. 8.
Using the values o f U l,j, V l,j, and P I find the values o f A j, B j, and C j. Use Eq. (8.103) to find the value o f P2. U seEq. (8.92) to find the values'of U 2;j. Use Eq. (8.97) to find the values o f V 2,j' Use Eq. (8.104) to find the values o f 8 2,j. Having found the values o f the variables in this way on the second iline, the saine procedure can be used to advance the solution to the next iline and so on up the chaI)llel. 9. The process is continued until the value of the dimensionless pressure, Pi, becomes positive. The actual dimensionless duct length that corresponds to the chosen value o f Um, i.e., the Z value that gives P 'T, 0 is then given b y interpolation, i.e., by: L i.e.:
1 0= Z il + ( Pi _ Piil ) (Zi P
Zil)'
L = Z,, 1 ,
(
P Pi l p
i
iI
) (Z' ,
z,
I 1)
Having discussed how the values o f variables can be found, attentionwill b e turned to the determination o f the heat transfer rate. ~e mean heat transfer'rate up
378 Introduction to Convective Heat Transfer Analysis
to any value of Z, i.e., qw, can be obtained by noting that: Total heat transfer" _ Rate enthalpy crosses _ Rate enthalpy from inlet to z  channel section at ~ enters the chann~l i.e., considering unit width of the duct and recalling that the flow is being assumed to be symmetrical about the center line: 2qwz = 2 fOW/2pUCpTdY  pumcpTfYOW
=2
JI
{~
o
pucp(T  TfYO)dy  2
,0
J, '
{~""
'
pucpTfYO d y  pumcpT'rA W
""
But conservation of mass requires that:
2 Jo
{WI2
p udy
= pumW
so Eq. (S.105) becomes: 2qwz i.e.: qw = =
2 Jo
fWI2
pucp(T  TfYO)dy
11 z
W2 ' pucp(T
 TaJ)dy
, (S.106)
0
In terms of the dimensionless variables defined in Eq. (S.SI) this equation be~ comes:
i.e.:
w qW
k(Tw  Tt:/J)Pr
=_
1 Z
10.5 U 8dY. '
0 "
\
(S.107)
i.e., defining: (S.108) Eq. (8.107) can be written:
Qa = 1 2 '0
1°.5 U8dY .
/
(8.109)
"
.'
CHAPIER
8: Natural Convection 379
U sing the values o f U and 8 given b y the numerical solution at a ny value o f Z, t he value' o f the righthand side o f this equation can b e obtained by numerical integration so allowing Qa to b e found. For example, to the same order o f accuracy as used in obtaining the solution for the flow, Eq. (8.109) gives:
. Qa =
~i [Ui,~8i.l + Ui,2 8 i,2 + Ui,3~t,3 + ... + Ui'N18i'Nl].~Y
(8.110)
i t h aving been noted that the boundary conditions give Ui,N = O. T he m ean dimensionless heat transfer rate for the entire channel will be give~ by: .
QL = 
. lio.
L
0
5
U8dY
the integral being evaluated on the e xit plan~ o f the channel. A computer program, NATCHAN, written in FORTRAN that is based o n'the procedure that is outlined above is available as discussed in the Preface. Some results . obtained using this program for P r = 0.7 are shown in Figs. 8.22 and 8.23. B ecause an explicit finitedifference procedure i s being used tQ solve the: momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds i t can diverge increasingly from the actual solution. The analysis o f t he conditions under which such an instability will develop,fliat was given in Chapter 4 for the case o f forced convection in a duct essentially applies here and shows that in order t o avoid instability, az should b e selected so that: .
. dZ
< O.5AY Ui l,Nl
.
2
Becaljse this criterion was obtained using an approximate analysis, i t i s usual to assume that for stability:
·0.04 ,      r    . . ,      r  r       ,
0.03
0.01
.
0.00 ' _ _. ..._ _. ....I_ _ .L_.._' 0.000 0.003 0.001 0.002
L
FIGURES.22 Variation o f Um with L for Pr =
0.7.
380 Introduction to Convective Heat Transfer Analysis
0.Q15
rr~r__~r_____...,
0.010
0.005
P r= 0.7
0.000 1 _1.._ _.....1._ _. ...1..._1 0.000 0.001 0.002 0.003 0.004
L
F IGURE 8.23 Variation of Qa with L for
Pr
=
0.7.
where K is less than 0.5. This stability criterion is incorporated into the progiam
:NATC~.
;
As previously discussed, there are two limiting cases for natural convective flow :through a vertical ,channel. One o f these occurs when elw is large and the Rayleigh number is low. Under these circumstances all the fluid will be heated to very near ., :, the wall temperature within a relatively short distance up the channel and a tYPe of , fully developed flow will exist in which the velocity pro~iIe is not changing with Z 'and in which the dimensionless crossstream velocity component, V, is essentially zero, i.e., i n this limiting solution:
.~
U = F (y),
V = 0,
(J
=1
the dimensionless wall temperature being 1. The velocity function F is independent .ofZ. In this limiting <;ase, then, Eq. (8.83) reduces to:
, ,
dP "  dZ
+
(tPu) + 1 d y2
=K
=0
i,
(8.111)
The pressure gradient will be constant i n fully developed flow so this equation shows that i n this limiting case: '
d2U d y2
where K is a constant whose value has to be determined. Integrating this equation once gives: 
dU
d Y = K Y+Cl 
/
CHAPTER
8: Natural Convection 381
where Cl is a constant of integration. Integrating this equation in turn th~n gives:
K y2 U =  2
+ C IY + C2 y2
(8.112)
C2 being another constant of integration.
The two constants of integration are detenmned by applying the bOundary conditions, i.e., by using:
Y = 0: dU d Y = 0, Y = 0.5: U=0
Applying ~ese to Eq. (8.112) gives:
C 1 = 0,
"8 + O.SC} + C2
K C2 =  S
K
=
0
These equationS give:
Cl = ' 0,
Substituting these values back into Eq. (8.112) then gives the velocity profile
as:
U = _K .24
(! _
y 2)
(S.113}
. The dimensionless mean velocity is such that:
.' 1 Um = 0.5
1°·5 UdY °
=' _ K
Substituting Eq. (8.\113) into this equation then gives:
Um
=  K (0.5
10
(! _y2)dY
4.
12
i.e.:
/
K
=  12Um
.....
. ,.(8.114)
Substituting ~s result back into Eq. (s.ll~) then gives:
. :U== 6Um(! ...y2 ) . ' 4
It follows that: .
,
(~:;)=K =12Um .
(S.116)
Because the effects of the developing region are being neglected i n the present limiting case analysis and because the pressure gradient is constant in the flow,being ,
382 Introduction to Convective Heat Transfer Analysis
considered, it follows that:
_ ~: = _ (Pent; Pin )
where P in and Pexit are the pressures on the inlet and outlet planes of the duct. But, as discussed before: '
0
Pin therefore:
=
2 Um
2
and
P exit
=0
~: = (~)
Substituting Eqs. (8.116) and (8.117) into Eq. (8.111) then gives: '
.!!!: =
(8.117)
U2 2L
1  12U
m
i.e.:
It should be noted that according to this equation the highest possible value of Um is equal to 1112 and this is approached as L ~ 0 0. '; Because in the limiting flow situation here being considered (J = 1, Eq. (8.109) ,
gives at the exit of the pipe: '
QL ::\:: L
11°·5
°
U(JdY
=
11°·5
°
=
UdY
=
U O.5~
L
,
L
Substituting Eq. (8.118) into this equation then gives:
QL = 1 
Um
l~Um
1 (0_ 12) U
m
(8.119)
Eqs. (8.118) and (8.119) allow the variations of Um and QL with L to be determined for the limiting case of large t'lW and low Rayleigh number to ,be found. The other limiting solution is that in which the flow essentially consists of bound , a t}' layers on each wall of the duct, these boundary layers being so thin compared to W that there is no interaction between the flows in the two boundary layers, i.e., the boundary layer on each wall o (the'duct behaves as a boundary layer op. a vertical plate in a large environment. Now, for free convective boundary layer flow over a vertical plate of height e, it was shown earlierin this chapter that:
/
A be~g a function of P r alone.
CHAP'I'ER
8: Natural Convection 383
I n terms o f the dimensionless variables being used here i n t he analysis o f duct flow, this equation can be written as:
q ;w
k (Tw
i.e.:

A [(3g(Tw  Too)W 4 ]O.25
v2
To)Pr = P r;
e
A
= LO.25 P r
(8.120)
I n the limiting boundary layer situation here being considered, the total flow through the channel will be small, i.e., effectively: Um = 0
. . .
.
(8.121)
Eqs. (8.120) and (8.121) represent the limiting boundary layer solution for natural convective flow through a vertical plahe duct. For the particular case o f P r :::; 0.7, . the similarity solution for natural convective boundary layer flow on a vertical plate was shown to give A = 0.477. Hence, for P r = 0.7, Eq. (8:120) gives:
Q _ 0.681
L
LO.25
T he two limiting solutions, as defined by Eqs. (8.l1S) and (S.119) and b y Eqs. (S.120) and (S.121) can be compared with the full numerical solutions shown in Figs. S.22 and S.23 and it will be seen that the numerical results do tend to these limiting solutions at high and at low values of L. . The analysis o f natural convective flow through a vertical plane duct that was described abo.Ye is easily extended to deal with other geometrical situation~, such as natural convective flow through a vertical pipe, and to deal w ith other thermal boundary conditions at the wall.
Consider the natural convective flow of air though a plane vertical channel with isothermal walls whose temperature and height'ate such that the Grashof number based on we height of these walls is lOS. Determine how the dimensionless heat transfer rate basedoti thlhelght of these heated walls varies with the gap between the walls.
E XAMPLE 8 .5.
,
.
Solution. Here the height, .e, of the channel and the wall temperature an.d air temperature ar~ fixed. Now:
G··~
But:
f3$(Tw  T=)
v2
W4
.e
= f3g(T w  T=).e3 = Gr . · ,;2 '.e. e ..e
(W)4. .
·(W)4
so:
Hence:
.L
/
= ..!.. =
G
0.00001
(W/.e)4
(a)
384 Introduction to Convective Heat Transfer Analysis
15
r r....
13L~~~~
0.2
0.4 Wil
0.6
0.8
F IGUREE8.5
It is also noted that: Q _ q;;;w
a
=
q;;;.e
k (Tw

k(Tw  To)Pr
W To)Pr .e
i.e.
where:
q;;;e Ql = k(Tw  To)Pr
(b)
TJte numerical solution discussed above gives the variation of Qa with L. For any selected· value of WI.e, Eq. (a) allows the value of L to be found. The numerical results· then allow the value of Qa corresponding to this value of L to be found. Eq. (b) then allows the value of Qt10 be found. The variation of Qe with WI.e found using this procedure is showri in Fig; E8.5. . Because .e is constant in the situation being considered, Qe will be a measure of the heat transfer rate from the walls to the flow. I t will be s~n from Fig. E8.5 that there is therefore a value of WI.e that gives the highest heat transfer rate.
E XAMPLE 8 .6.
A ir flows by natural convection through the channel formed between between 2 lOcm high plates which are kept at a temperature of 50°C. I f the distance between the 2 plates is 2 cm and i f the ambient air temperature is lOoC, find the rate of heat transfer from the 2 plates to the air. Also find the mean velocity of the air through the channel. .
Solution. The mean air temperature will be taken as:
Tav
= (50 ;
lO)
= 300C
CHAPl'ER,
8: NatUral Convection 385
At this temperature, air at standard ambient pressure, has the following properties: .
f3
= 0.0033 K  1 , v = 16 X 106 m'iIs , k = 0.02638 W/mK

Hence:
G = f3g(Tw
T..,) W4
v '1.
e
= 0.0033 X9.81 x (50 (16 x 1 06)2
10) 0.024 = 8093 0 .1 . . .
From this it follows that:
L
= 1 G
= .  = 00001236
1
8 093'
.
From the numerically detennined variations of Um and Qa with L for P r that for this value of L: Hence:
= .0. 7 it follows
:
Um = 0.00481 and Qa = 7.44
qw W
44
Qa == k (Tw  To)Pr = 7.
and:
U. (umvW)(~) = 0.00481 m
=
Hence:
l'
.
qw
= 7.44 x 0.02638 x (50 .0.02
10) x 0.7
= 2 748 WI
..
m
2
Therefore, considering unit width of the channel, the total heat transfer ~ate·from thetwo walls w ill be: .
Q = qwA
= 274.8X2XO.1 x 1 = 54.96W
X
Also:
un. =
0.00481 x 0;1 ~.~~3
16 X 1 06 == 0.003114 mls
Therefore the heat transfer rate from the walls per uiri.t channel width is 54..96 W . and the mean velocity through the channel is 0.003114 mls.
8.9 NATURAL CONVECTIVE HEAT TRANSFER ACROSS A R ECTANGULAR ENCLOSURE
Heat transfer ~y natural convection across an enclosed space, called an enclosure or, sometimes, a cavity, occurs in many real situations, see [34] to [67]. For example, the heat transfer between the panes of glass in a double pane window, the heat trans· . fer between the collector plate and the glass cover i n .a .so~ collector and in· numy electroJ;1ic and electrical systems basically involves. natural convective flow across. an enclosure. . . .
386 Introduction to: Convective Heat Transfer Analysis
,
,
,
Horizonta1~'··~~.J:
FIGURE8~
1YPe of enclosure considered.
Attention will here ,be restricted here to flow in a rectangular enclosure as shown in Fig. 8.24. I n general, the enclosure is inclined to the vertical as illustrated in Fig. 8.24. For simplicity, in order to illustrate how enclosure flows can be analyzed, it , , will be assumed that one wallofthe enclosure (AB i n Fig. 8.24) is at a unifonn high temperature, TH, and that the opposite wall (CD in Fig. 8.24) is at a unifonn low temperature, Te . 1\vo boundary conditions on temperature are usually considered .. on the two remaining "end" walls ( Be and DA i n Fig. ,8.24). I f these walls are made , from a material that has a low thermal conductivity. it is usual to assume that these walls are adiabatic, i.e., that there is no net heat transfer to or from the waIl at any poirii on 'the 'waIl~ Alternatively,' if these walis' are made from a ma~rial that has a relatively high thermal conductivity, it is usual to assume that these 'walls are "perfectly conducting" and that the temperature on these "end" walls varies linearly with distance from the hot wall from TH to Te. It will be assumed here that the flow i n the enclosure is steady and remains laminar. I t will ,also ,be assumed that 'the flow is twodimensional. :With these assumptions and using the Boussinesq approximation that was discussed earlier. the eq~tions gqverning the flow in the enclosur,e are: : .
au + av = 0 ax ay au au 1 ap,' (q2 U a2 u)" , u  + v  = ...;.. + 'V 2 + .. :2' +.f3g(T""'7 T I) cos </>' ax ay pax " ' ax, . a.r '' ,, 22 av ,av 1 ap . u  +V' = :  + v (a v+ J v) + f3g(T  Tt)slUcP 2 i ,a~ ~y. p ay" ax ay2. ,: ' ..
.
,
".
(8.122) (8.123) (8.124)
".
.
.
~
. '.
:.'
.
(8.125)
/
CHAPTER
8: NaturalConvection 387
In writing these equations. the density at the cold wall temperature, T c, has. been used as the reference relative to which the density changes are specified and'p, is then ,the gauge, pressure measured relative to the ambient pressure that w()uld exist at the point considered in the end()sure i f all ,the fluid in the enclosure, was at rest and at temperature, T c . The boundary conditions on the solution are:
•
. Ori: aii walls,'i.e., ~n AB, BC, CD, 'and DA: u ~. v' = 0' , On wall AB:,T =."TH " On wa:llCD:·T =, Tc '
•
~.
~.
_.
'.
'."
:.
•4
, ,(S~126)
.. '. the'bound~ ·conditions on temp,~ni~e on','Y~~ BC ~d 'DA' depend ~n 'whether these walls are assumed to be adiabatic or perfectly ,coiiduc~ng.· I f ·th~y, a,re as~umed to be adiaQatic" the bo~ndary conditio~ o~ these' walls· are, ',using FQiuier~~ law:·o~ ,th~se waIls, i.e., 'usirig: . ' '. ' . ',~ .. .
;'
'r,
q'  k , . ....,. ' dy
,ar
0:
'
and noting that i f the wall is adiabatic q
On w'alls BC and DA:
."
=
.
aT
ay = 0
(S.127a)
On· the other hand, i f the end waJIs are'· perfectly conducting, the temperature is, '" as discussed above, assumed to vary linearly along these walls, i.e., the boundary conditions .on these end walls thenis:
. ...•.. . O n walls B Cand D AfT ' " T " ,  (: )<TH '  ·Tc )
(&.127b)
I t should be noted that when enclosure contairis a gas; :the convective heat transfer rates ~an b e ~ov( and rad.iant he~t transfer ,can bfj, significant. Some $~e~,
the
such as carlJon dioxide.and water ,vapor, ~bsorb and.·emit radi~on and ~nsuch ~es the energy equation bas to be modified to 'account for this. However: even when the 'gas in the en~los!ire is tr~parent to radiation. there can be an interaction between ,the radiant arid ·convective 'heat transfer. For exampie,.for t:J:J,e;case where the end walls can be assumed to be adiabatic, i f qrab and qrem are the'rates at which radiant at any point on these ,end energy is being absorbed and emitted per unit wall walls then the actual thermal boundary conditions on these, walls are:
area
"
"0n wallBC" qrab :

k··ay = qrem ,aT'
and:
.. On wall DA: qrab
= qrem ~:~::qy.
.
aT
/
The difference between these two expression~ is due t o the fact that the ydire¢on is from fluidtowall on BC and from wall.;.tofiwd on DA.. .. '
388 Introduction to Convective Heat Transfer Analysis
I t will b e assumed here that the convective heat transfer rates are high enough to allow the effects of radiation on the convective motion t o b e ignored, i.e., to assume that the convection· and the radiation can be considered separately and that the total heat transfer rate will be the sum of the separately evaluated convective and radiant heat transfer rates. Various approximate solutions to the governing equations for natural convection i n an enclosure have been obt8ined. For example, the flow at high Rayleigh numbers . . can be assumed to consist of boundary layer flow up the hot wall and a boundary . layer flow down the cold wall, these two boundary layers being t hin enough to ensure '. that there is no significant inieraction between the two boundary layer flows. More consideration will be given to these approximate solutions later in this chapter. In genei'al, however, the full governing equati<)ns must be numerically solved and this will be discussed here. . . The govermng equations given above, i.e., Eqs., (8.122) to' (8.125), are given in terms of the socalled primitive variables, i.e., u; v,p, and"T. The solution procedure discussed here is based on equations involving the stream function, !/I, the vorticity; w, and .the temperature, T, as variables. The stream function and vorticity are as befote defined by:
a
u= 
at/J at/J , ay' v = .~ ax
(8.128) (8.129) ".
OJ
' ..
=
(!> ~~)
As shown in Chapter 2, the stream function as so defined satisfies the continuity .
equation. The vorticity equation ~s obtained by eliminating the pressure between the two momentum equations, i e., by taking the yderivative ofEq. (8.123) and subtracting from it the xderivative ofEq. (8.124). This gives,: .
. '  +u +  =+v .,  u     ,  v. 2
au au ayax
a2 u av au a2 u au av a2v' av av a2 v ayax ayay oy2 ax ax , ax ax ay axay 3 3u' .' (' a3 u. +'a,  a3v'..  v)'+f3gcosCPSlDcP) (aT .' aT,. =v 3 , dXa '  a y2 ayax2 ay3 ax ·ay , ax·
.. .
i.e.:
OJ
=
(av au) \ ax 8y
/
CHAPIBR.8: Natural Convection 389
and using the continuity equation, this equation can be written' as:
a aw u w +vax
2 2 aT. , aT =; (a x{J) +  y2  J3g (Stne/>  cose/>) ' _. a {J)) oy a2 a ax ay (a 2 + a2  J3g (aT.tne/>  a T,) {J) w) ax2 ay2 ax S ay coscf>
I n tenns of the stream function, this equation becomes:
af/! aw  af/! aw ayax ax ay
=
P
(8.130)
This equation, which basically expresses conservation of angular momentum, will b e one of the equations on which the present numerical analysis ,of flow i p an enclosure is based.' . ,' , I n'tenns o,f the stre~ function, the fiquation defining the vorticity" i.e'::
{~
becomes:

: ; ) , . b)
(8.131) while in tenns o f the stream function the energy equation becomes:
at/! aT _ af/! aT = (~)(a2T + ayax ax ay , pCp ax2 , ay2
conditions on these equations are, as <ij.scussed before:
;PT)
af/!
uy
(6.132),
Eqs. (&d30), (8.131), and (8.132) which involve the three variables'"" (J), and T are the set of equations that are used in the present numerical solution. The boundary
On all walls, t.e., on AB, Be, CD, and DA:  ; =  ; = 0
, ' oX
.
a",
On wall AB: T = TH On wall CD: T = Tc
On w alls B e and DA: cither~r = 0 or T
(8.133)
TH  ( ; ) (TH  T c l
=
The actual value of the stream function is quite arbitrary because only the deriva.. ' tives of f/! appearin the set o f equations and'in the boundary conditio~s discussed above. The value of ' " at point A in Fig. 8.24 will therefore arbitrarily ~ t~en as , O. Because the boundary C9nditions give a",tay' = 0 along AB which indicates that '" is not changing along AB, if! will be oeverywhere along AB. This means that f/i Will be 0 at point B in Fig. 8.24. Because the boundary conditions give af/!Iox = 0 along BC which indicates that I/J is not changing'along Be, I/J be 0 everyWhere along Be. This means that f/! will be 0 at point .c tn Fig. 8.24. I n the same way i t follows because af/ilay = 0 on CD and al/Jlax = 0 on DA t hat", will be 0 on CD and on DA. Therefore I/J will beO everywhere on'ABCDA.
will
/
390
Introduction to Convective Heat Transfer Analysis
Hence the boundary conditions on str~am functi(;m are: , On walls on AB and CD: ,l/I = 0, () x = 0
(8.134) On walls on BC and DA:
.
,01/1

.

l/I :::;: 0, :l/I vy
= ,0
This can be written as: On walls on AB, BC, CD, and DA: l/I = 0 , :l/I = 0 vn '
(8.135)
where n is the coordinate measured normal to the surface being considered. Before considering the numerical solution to the above set of equations, it is convenient t o rewrite the equations in terms of dimensionless variables. This i s not, i t must be stressed, a necessary step i n the solution' process. The numerical ~oluti.on can be obtained directly in. terms of dimensional variables. The following dimensionless variables will be used here:
' l' = l/IPr,o,
p
=
(r)W 2 Pr
p
x'y X= y=
W'
W
(8.136)
:' ' !f
() = ' ( T  Tc) (TH  Tc)
where Pr is the Prandtl number of the fluid. In ~rms o~ these v~ables, ,the yortic~ty eq~ation becomes:
, a'\ft an. ' a'\ft an _ '(0 20,' a2n.) ' oyax  ax ay  pr,aX2 ,+ 'aYZ
B tlt:
' f3g(TH ~ T c)W 3 PrZ  SInCP  p2 ' ,' , oX
(ao . ' ;, ao cos</>) . ay ,
Ra : :;: ,
JI
r.lg(TH  T c )W3
p2'
',
Pr =, Grw Pr
is the Rayle~gh number based on ,the e~closure width W. The dimensionless, vorticity .transport equatio~ given above can therefore be'written as: '
an + af2 := P1r' (a'l' an ~'ax.ay'+Ra(ao . . aY~os</> ' (8.1"37) o" ow. an')" ,~lr.2 a a,'yax aX,smf/J.o2 2
" 00'
')
',In te~s'ofthe <;fuiien~io~~ss'vanables, Eq. (8.131) becomes: . .. ... .
,
.
~,:
" '.
'a2'1':' ()2qr
ax2 + aYZ' =  n,
,
"
(8.138)
/
\J
CHAPmR.
8: NaturalConvection 391
Similarly, the energy equation, i .e.,Eq. (8.132), becomes i n t erms o f t he dimensionless variables:
a2() a'lt a() aqr a() aX2 + ay2 = ay ax  ax ay
O n walls o n AB. B C, C D a nd DA: qr
a2()
(8.139)
I n t erms o f t hese dimensionless variables, the boundary conditions ~: _
= 0,
aqr
aN
=0
w here N = n lW i s the dimensionless coordinate measured n ormal t o t he surface being considered, e.g., i t i s X for AB a nd Y for BC. O n w all AB, i.e., a t X = 0: () = 1 O n w all CD, i.e:, a t X = ' () 1: .. .
=0
.
O n walls B C a nd DA, i.e., at Y = 0 and Y = e ither 8() = 0 o r () = 1  X
A:
(8.140)
, ay

whereA = H IW i s t he socalled aspect ratio o f the enclosure~ A n i terative finitedifference solution procedure will b e d iscussed here. Many other more efficient solution procedures a re available b ut t he s imple p rocedure considered here should serve to indicate t he m ain features o f s uch m ethods. ' A s eries o f e venly spaced nodal lines as shown i n Fig. 8.25 will b e u sed. The" a nd grid spacings i n b oth the X and Ydirections are thus constant a nd ~qual to A Y r espectively as indicated i n Fig. 8.25: C onsider the n odal points shown in Fig. 8.26. I n t erms o f t he v alues a t these nodal p oints t he finitedifference f orm o f t he dimensionless energy equation, i.e., Eq. (8.139),becom~s: 
ax
()'+1 . ( ',J
+ ( J'lJ. I.
A.XJ 
26.}) +' .+ ().'.J (6"+1,
:" I ,) _
,
' 1  28.}) I,
A y2
()il,j ) __
:
= (Wi,j+l~ W'i,jl ) (()i+l,j .' 2AY _ '.."
2 AX.
.
(qri+1,j  Wil,i ) ((J;,}+l  ()i,jl)
2 aX
2 AY
• } .:\Y
/
F IGURE 8.25 Grid system used.
392 Introduction to Convective Heat Transfer Analysis
F IGURE 8.26 Nodal points used in obtaining finitedifference equations.
which can be rearranged to give: ,.
64i = {
(~4Y )('P4i +1 : 'P'il)(6H~j [
 ('I';+l,i  ':t'il,j)(Oi,j+l  (Ji,jl)
..T r ]
6.1,i)
+ + (fh+1,jf 1X26i I,i )
(8.141)
<'
+ (6,j+~;:"il ) }
I (~ + 4;')
i)
Similarly, the finitedifference form o f the vorticity transport equation. i.e.. .Eq. (8.137). gives: .
(Oi+LJ
+ 'i ~

20 41
)+ (°. + tl  2O. ~
. /+ 1
=
Ur )r"i+~~:"il t'+l,~O'I'J)
_
('P'+1.~;.;::'PI ~J t 4i+ ';;.:"1 )]
2DJ(
(Jil,i)Sill cP
+ Ra((Ji+l,l ~
_
(8 ,J+l2dY(Ji,jI)coScP· .,
i
.
which can be rearranged to give:
0" =
{(4~;ypr )[('P',J+l 
'P',JI)(O.+I,i  O.Li)
. A  ('I'i+l,j  'l'iI,j)(Oi,i+ 1  Hi,iI) ]
+ + (ni+l'l.11)(2o.i~'J)
+ (Oi'i+1 + n i,i 1).,;. R a [(fJ i + 1,J ' fJil,i ~f2 + . 2AX
)Sin cP _ (8t.1+1 aY8i,jl )coscP]} 2
(8~142)
CHAPTER
8: Natural Convection 393
Lastly, the finite. .difference form of the equation relating the dimensionless vorticity to the dimensionless stream function, i.e., Eq. (8.138), is:
' lIi+ l ,j
+ '\fI'il,j
Al{2
 2"l!i,j
+ 'lIi,j+ 1 + " I"t,jl
Ay2
 2'\f1'i,j
=  Ui,j
n
which can be rearranged to give:
(8.143)
Bqs. (8.141), (8.142), and (8.143) apply to all ''internal'' points, i.e., to all points . from i = 2 to M  1 and for j = 2 to N  1 where M and N are the number of nodal points in the Xand Ydirections, respectively. The boundary conditions give the values of the variables on the boundaries. These boundary conditions direCtly give: (8.144) ' \fI'l,j= 0 , "l!M,j = 0, 'lIj,1 = 0 , '\fI'j,N = 0 and:
Ol,j = 1 ,
O M,j
=0
(8.145)
Consider the boundary conditions on the end walls. When the temperature gradient is zero on the end walls, i.e., when the end walls are adiabatic, since to the same order of accuracy as u sed in deriving the finitedifference equations:
. dY .
dOl
=
1.1
[(0,1,2 _(J i,l ) _
(Oi,3  (Ji,l)]
4
(2) aY
and:
dJy 'N =.[(0.I,N  o.I,Nl ). _ (Oi,N  4Oi,N2)] (2). Ol i AY
I,
. the boundary conditions give:
Oi,l =
4 f. 3,8;'2  0'3) .4,
(8.146)
and: (8.147) Alternatively, i f the end walls are perfectly conducting; the boundary conditions on these walls are: (8.148) 0'1 = I X'I I, and:
~
OI , ·= I X' 'N I
(8.149)
394 Introduction to Convective Heat Transfer Analysis
Now, .a consideration of the boundary conditions indicates· that· there' are no boundary conditions on dimensionless vorticity on the walls.· However, it will be noted that the boundary condition:
aw
aN
=0
where N is the dimensionless coordinate measured normal to the surface at the point considered, has nqt been used. To illustrate how this can be u s¢ to obtain a boundary condition on f i, consider a point on wall surface AB such as that sho~n i n Fig. 8.27. Because ' I' = 0 at all points on AB, it follows that a2w/aYZ = 0 at all points on AB. Hence when Eq. (8.138) is applied to any point on AB it gives:
. ..
fix=o
=
a2'1'
aX2
X=O'
(8.l~O')
cOnsider the two'poi~ts~adjacimt to the surface AB that are shown in Fig. 8.27. . A Taylor expansion gives to the same order o f accuracy as u sed in obtaining the finitedifference approximations used in the rest of the analysis.
' l'2,j
= ' l'l.j
+
(~~)
.
. AX +
LJ
(~~ \ . ~
~'J
.=
0
(8.151)
But tlte:bot;lndary c~nditions give: .
..
" I'l,j =
.
0 and
(a'l') ax
LJ
B
F IGURE 8.27 Wall points considered in' obtaining boundary con~tion qn vorticity.
/
CHAPTER
8: Natural Convection 395
so Eq. (8.151) gives:
*2.'} = (aa2'l'). X2
i.~:
IlX2
2
1 ,i
.
(~~) . = 2!i~
l ,J
Substituting this into Eq. (8.150) then gives for any point on AB:
qr2 ; . f ll , J. =  2'} /lX.2
(8.1.52)
.E q.. (8.152) supplies a boundary condition on the dimensionless vorticity at the .walL Applying the· same· procedur~ to points on surfaces BC, CD, and DA gives:
{ l.
I,N
=
2 qri,N~l l ly2
.
(8.153)
and·:
o M,}, and:
O i,l =
2  qrMI,i 1lX1
(8.154)
 2 IlY2
qr12
(8.155)
The set of nonlinear dimensionless finitedifference equations with their associated boundary conditions that have been presented above are'solve&iteratively starting with guessed values of the variables at all points. The procedure, therefore, involves the following steps:: (1) Guess the values 'of 'l'i,j, f li,j and (h,i at all points. Typically, it could be assumed that imtial1y:
.. • . .. I ,)
'l" . = 0' Il··} .0 f. = '
(J I. . .}
== .1 .:x·I ,J:, . ..
the assumed dimensionless temperature variation stemming from the fact that i f there is nQ .fluid 'motion the temperature· distribution will be that. existing with pure conduction across the enclosure, i.e., in view of the boundary conditions, it will vary linearly with X. ' .': (2) .Eq.. (8.142) ·is applied sequentially at all "internal" nodal points·, i.e., all points. that are not on a wall, to find· new values of fli,j. The righthand side of this equation is found using the assumed values of the variables. Underrelaxation. is actually used so the ''updated''.values of f li,j are actually taken as:
o ~ . == 0 Q. + r(O ~~culated ~ f l 9 . ) '
I ,} I ,} I.}
I ,}
where n~1cu1ated is· the value given directly by Eq. '(8.142) and r is the relaXation factor « l)~ The superscripts 0 and 1 refer to conditions at the beginning and the
396 Introduction to Convective Heat Transfer Analysis
end of the iteration step. The solution procedure, as mentioned ab,ove, is applied to all internal points, i.e., for i = 2 to M  1 and for j = 2 to N  1. The whole step is repeated before proceeding to step (3) of the solution procedure, this having been found to accelerate the convergence. (3) Eq. (8.143) is applied sequentially at all internal nodal points to get updated values of W i.} at these points, the righthand side of this equation again being found using the assumed values of the variables. Underrelaxation is used so the "updated" values of Wi.} are actually taken as:
'\{I}. = "VP. L} L}
+ r(W~culated  "V~0}.) L}
where Wi~fCUlated is the value given directly by Eq. (8.143) and r is the relaxation factor « 1). The solution procedure, as mentioned above. is applied to all internal points, i.e., for i = 2 to M  1 and for j = 2 to N  1. The boundary conditions give the values of W at the nodal points on the boundaries and these do not change. This . whole step is also repeated before proceceding to step (4) o f the solution procedure, this having been found to accelerate the convergence. (4) Eqs. (8.152) to (8.155) are then used to get updated values of,{} on the boundary points. Underrelaxation is also used for these points so the "updated" values of f ),i;j at the wall points are actually taken as: .
f ),!. = , aQ.
t .}
t .}
+ r b (,{}~culated l ,}
nQ.)
t .}
.
where f),~'CUlated is the value given directly by Eqs. (8.152) to (8.155) and Tb is the relaxation factor « 1). The value of the relaxation factor for the wall points, as indicated by the subscript on the r, is often taken to be larger than that used for the internal points, this being found to accelerate convergence. (5) Eq. (8.141) is applied sequentially at all internal nodal points to get updated values of (Ji,jafthese points, the righthand side of this equation again being found . using the assumed values o f the variables. Underrelaxation is also used s o the "updated" values of (Ji.j are actually taken as:
( Jl. = (JQ. t,} l ,}
+ r«(J<:a!culated l ,}
(J9 )
I ,j
where (J~rated is ,the ,value given directly by Eq. (8.141) and r is the relaxation factor « 1). The solution procedure, as mentioned above, is applied to all internal points, i.e., for i = 2 to M  1 and for j = 2 to N  1. The boundary conditions 'give the values of (J at the nodal points on the boundaries!AB and CD and on Be and DA i f perfectly conducting end walls are assumed and these values do not change. I f . adiabatic end walls are assumed the updated values of (Ii, 1 and fh,N are found using Eqs. (8.146) and (8.147). This whole step is also repeated before proceeding to step (6) of the solution procedure, this having been found to accelerate the convergence. (6) Steps 2 to 5 are repeated over and over until the values o f the variables cease to change from one iteration to the next by less than some prescribed value. A minimum number of iterations is usually prescribed in order to avoid false convergence and a maximum number of iterations also has to be prescribed because convergenCe will not always occur. . (7) Once the converged solution is obtained, the values of the heat transfer rate at the nodal points on the hot and cold walls can be found by applying Fourier's law
/
CHAPTER.
8: Natural Convection 397
at these wall points. For example, consider a point on wall surface AB such as that shown in'Fig. S.27. Now:
( hi =
( h,)
+
(;~)
AX +
L)
(~; \~,}. ~
(8.156)
But the velocity components are both 0 at the wall and O i,} is cOnstant ( = 1 ) everywhere on AB with the result t hata 2 0lay2 is 0 everywhere on AB. Using these ' results, it follows from the energy equation that on AB:
Hence. Eq. (S.156) gives:
_(ao)
a x. 1,}
= 8t,} 
02.1 = 1  02,}
' 1lX
AX
. (S.157~
But Fourier's law gives at the nodal points on AB:
qw
= ' k aTI
ax
x =O
which becomes in tenns of the dimensionless variables:
qwW
k(TH  Tc) = 
aOI ax 1,}
(S.158)
i.e., using Sq. (S.157):
i qw 1  lh N u ' ). I ,} = k(TH  Tc) =  ax I .} = /),X
w aol
qwW _
For points on the wall surface CD, i.e., on the cold wall, the equivalent equation which can be derived using the same procedure is:
N U M.} 
O Mt,}
k(TH  Tc) 
llX
(S159) .
The mean heat transfer rates on AB and CD are then found by integrating the ' local distributions, i.e., by using:
\
qwll
1 =H
Jo
fH
qw dx
Numerical integration using the values of the local dimensionless heat transfer rate at the points considered therefore gives:
N UH= . k(TH _ T c) 1
qwW
I = AX [NUll + A
~
N Ul,2
' NUIN] + N Ul,3 + ... + N Ul,Nl +
T
/
(S.160)
398 Introduction to Convective Heat Transfer Analysis
:
Similarly~
for the cold wall:
qN uc = k(TH ;wTc)
I
M
=A
!:,.X [NUMI
2'
+ NUM.2 + NUM.3 + ... + NUM.Nl +
.
NUM.N] 2
(8.161)
N UH and N Uc being the mean Nusselt numbers, based on W, for the hot and the cold
walls, respectively. With adiabatic end walls, because a steady state situation is being considered, these two values should have the 'same numerical value, i.e., the rate at f which heat is· transferred _rom the hot wall to the fluid should be equal to ~e rate at which heat is transferred from the fluid to the cold wall. Small differences usually exist between the values of N UH and N Uc given by the numerical solution due to the small numerical roundoff errors and due to the finite convergence criterion used in the numerical solution. Ail examination of the governing equations aIid the boundary conditions on these equations indicates that the solution to the equations contains four par~eters, i e., the Rayleigh number, Ra: thePrandtl number, Pr, the angle of inclination, cfJ, and the aspect ratio A = H IW. _ _. A computer program, ENCLREC, that is based on the procedure outlined above which is written in FORTRAN is available as discussed in the Preface. Besides the . solution parameters, the inputs to the program are the number of nodal points i n the X and Y directions M and. N , the relaxation factors r and rh, and the IIlllft.imum and minimum number of iteration steps. T he prograni and the numericatprocedufe 'discussed above are very simplistic, intended· only to illustrate the main features of . more advanced solution procedures. . Some results obtained using the program are presented in Figs. 8.2a, 8.29, and 8.30. Tl1~_~e results are all for the adiabatic end wall boundary condition.' Figures
CONTOUR VALT,JES
1;000  2.000 3.000 4.000 5.000 6.000  7.000 .
STREAMLINES
FIGURE 8.28 Streamlines for Ra = 3 X 104, P r and end walls adiabatic.
/
= 0.7, cp = 90°, A = I,
i
CHAPfER:
8: Natural Convection 399
CONTOUR
VALUES 0.100 0.200 0.300
0.400
0.500 0.600 0.700 . 0.800 0.9.00
ISOTHERMS
FIGURE 8.29 Isotherm for R a = 3 X 1Q4. P r end walls adiabatic.
= 0.7, c/J
= 90°, A = 1, and
5~~r~
4
3
Approxima~
Nu
2
Boundary Uayer s o!uti,on\ .
/ j,A
./
.,./
1 I".'~~.~::; . ................. , . ........... . Pure Conduction (Nu 1)
=
l e+2
o~~~~~
F IGURE 8.30 .
l e+3
l e+4
l e+S
Ra
Variation of N u with R a for P r = 0.7, q, = 90°, A = 1. and end walls adiabatic.
8.28 and 8.29 show typical streamline and isothenn (lines of constant temperature) patterns in the enclosure while Fig. 8.30 shows the variation of the mean Nusselt / number, N u, with Rayleigh number, Ra, these results being for the particular case of P r = 0.7 (air), cp = 900 (hot and cold walls vertical) and A = 1 (a square enclosure).
Use the computer program discussed· above for natural convective flow in an enclosure to derive a relation between Nusselt number and Rayleigh number for a square enclosure with isothennal vertical walls and perf~t1y conducting horizontal
E XAMPLE 8 .7.
400 Introduction to Convective Heat li"ansfer Analysis
" walls. To do this, find the Nusselt numbers for Rayleigh numbers of between 10 and about 100,000 and plot the variation of N u with Ra. Thy to obtain a correlation of the form:
Nu
= aRa
R
a and n being constants, for the results at the larger Rayleigh numbers.
Solution,. The program has been run with the following input values:
IBND.2 ; PR = 0.7 1 RAl0, 3 0, 1 00, • •••••••••• , 3 0000, 100000, 300000
The values of N u obtained i n this way are plotted against R a in Fig. E8.7. The values of N u obtained at three of the larger values of R a are listed in the following table:
Ra
10,000 30,000 "100,009
Nu L751 2.395 3.400
Now if:
N u = a Ran
then:
logN~ =
l oga + n logRa
I f the results at any two values in the above table are con"sidered, these being indicated by subscripts 1 and 2, then the above equation gives:
n ==
==:...=~=
logNu2  logNu] Iog.Ra2  log Ral
AppI.ymg this equation to the results in the table gives:
n = 0.29
4
3
Nu
2
Nu=O.12I~
/0.29
_ _...J
Ie + 6
/
O'~...I..1'
Ie + I
Ie + 2
Ie t 3
Ie + 4 Ra
Ie + 5
"FIGUREES.7 "
CHAPTER
8: Natural Convection 401
It then follows that:
Nu a =  29 
RaO.
Applying this to the results in the table gives:
a
= o .lil
Hence, the results indicate that at the higher Rayleigh numbers.:
N u = O.12IRao. 29
. The values given by this equation are also shown in Fig. ES.7 and it will be seen that it describes the results quite well for Rayleigh numbers greater than about 3000.
As mentioned before, there are analytical 'solutions for the heat transfer that apply under certain limiting conditions. For example, i f the Rayleigh number is very low, the convective motion will be so weak that it has no influence on the heat transfer rate and the heat transfer will effectively be by pure conduction. I n this case the heat . transfer rate is uniform on the hot and cold walls and is given by:
q w=
k(TH  Tc) .
W
. 1.e.,Nu=1
(8.162)
It will be seen from Fig. 8.30 that the numerically calculated values of N u do indeed "' approach a value of 1 at low values of R a . · Another approximate limiting solution for a vertical enclosure (i.e., <p < 90°) is .obtained, as mentioned before, by assuming that the flow consists of boundary layers on the hot and cold walls with an effectively stagnant layer between them and that; the presence of these end walls has a negligible effect on the boundary layer flows. The assu.l:Iled flow is therefore as shown i n Fig. 8.31. . The boundary flow up the hot wall will be identical to the flow down the cold wall and the temperature of the fluid between the two layers will therefore be ( TH +Tc)/2. The ~mperature difference across both boundary layers will therefore be ( TH Because th~/hot and cold walls are' vertical and at a uniform temperature and because the fluid between the two boundary layers is ass~med to b e stagnant and at
Boundary . Layers
Tc)/2.
Stagnant Core Region
Tc
/
F IGURE 8.31 Assumed limiting form of flow.
401 Introduction to Convective Heat Transfer Analysis
a uniform temperature, the similarity solution for the boundary layer on a vertical plate derived earlier in this chapter will describe the flow in the boundary layers. This solution gives:
NUB _ [ p,.s/4 . ] Gr£?·25  2.44 + 4.88Prl12 + 4.95Pr
w
.
where N uH is the mean Nusselt number based on the height, H , o f the plate and the temperature difference across the boundary layer and GrH is the Grashof number based on the height, H , of the plate and the temperature difference across' the boundary layer, the temperature difference across the boundary layers being. half the overall temperature difference. I n terms of the variables being used here to d~cribe the flow in the enclosure, this equation gives:
G,.Q~2S
.Nu· AO•25 '  [
 . 2.44
. 0.()313P,.s/4 . ]1/4 + 4.88Pr l12 + 4.95Pr
where Gr is the Grashof number based on the width of the enclosure and the overall te~perature difference. This equation can be written as:
_[ . 0.0313Prl/4 ]114. 0.25 N u  A(2.44 + 4.88Prl12 + 4.95Pr) R a.
(~.163)
The variation of of N u with Ra given by this equa~on for P r = 0.7 and A = 1 is compared with the numerical results in Fig. 8.30 and will be seen t o compare quite well with the numerical results at higher values of Ra.
E XAMPLE 8.S.
Consider the vorticity equation for flow in an enclosure, i.e.:
= Pr
2 a2n aa . a p + aY2
1
(O'Jyax I' aD, a
a't' an) ( ao. ao ) ax ay  Ra ax sm<fJ  aY cos<fJ
.
I f the Prandtl number is large, the first term on the righthand side o f this equation. i.e., the/inertia tenn, will be negligible, i.e., the equation will effectively be:
a2D, a2 n aXl + aY2
= Ra
(afJ. afJ ) ax sm<fJ  ay cos<fJ
\
With this approximation, the equations governing the flow i n the enclosure are: /
a2't' ala alD, axz + aY2
a2'l' aXZ + af2
=  Ra
=  a.
(afJ. a8 ) ax sm<fJ  ay cos<p .
a2 fJ· + a28 = a'lt ao _ aw aO a p aY2 ay ax ax ay
/
The only parameter in these equations is the Rayleigh number, Ra, i.e.~ the solution depends only on the Rayleigh number and no~ on the Prandtl number. From this it follows
CHAPTER
8: Natural Convection 403
3.6
r .,...,,
3.5
Nu 3.4
3.3
R a = 10 5
3.2
L   _ _ _ J.._ _~_
_ '___ _ _ l
0.1
1
10
100
Pr
FIGUREE8.8
that: N u= function (Ra) Verify this conclusion by calculating the Nusselt number for a fixed Rayleigh number o f 105 for the perfectly conducting wall case for Prandtl numbers o f between 0.5 and 30 and verify that the Nusselt number tends to a constant value at the larger values o f Pro
Solution. The program has been run with the following input values:
IEND=2 ; RA=100000.0 ; PR=0.5, 0 .7, 1 , 2 , 3 , 1 0, 30
T he variation o f N u with Pr obtained in this way is shown in Fig. E8.8. I t will be seen from the results given in Fig. E8.8 that N u does appear to be essentially independent o f P r when P r > 2. I t will also be seen that the change in N u over the entire Pr range considered is small. This indicates that the inertia term in the vorticity equation, i.e.:
J... (a'l' an _ a'l'. an ) P r ayax ax. aY
is small compared to the viscosity and buoyancy terins because the inertia term is the only term containing Pr as a separate parameter.
8.10 HORIZONTAL ENCLOSURES HEATED FROM BELOW
The above discussion was concerned with enclosures in which the heated and cooled walls were at an angle to the horizontal. In the present section, the concern is with rectangular enclosures in which the hot and cold walls are horizontal and in which the hot wall is atthe bottom, see [68] to [831. The flow ~ituation being considered is therefore as shown in Fig. 8.32.
404
Introduction to Convective Heat Transfer Analysis
FIGURE8.3~
Horizontal enclosure heated from below.
Because the hot wall is at the bottom, the buoyancy forces will tend to cause a fluid motion, hot fluid flowing upwards from the lower surface and cold fluid flowing downward from the upper surface. This motion will only develop however. when the Rayleigh number based on the difference between the temperatures of ~e upper~ andlower surfaces and on the size of the gap between the lower and upper surfaces exceeds a certain critical value. The value of this critical Rayleigh number can be', . detennmed by imposing a small disturbance on the initially stationary fluid betw~n tli.e two w~1s. At low Rayleigh numbers this disturbance will decay wi~ ti~e. However, at high Rayleigh numbers, the disturbance will grow with time. The Rayleigh number at which the disturbances neither grow nor decay is the required critical Rayleigh number. This process is illustrated in Chapter 10 where it is applied to the case where the gap between the two surfaces is 'filled, with a porous medium. In the present case, an atteJ:fipt will be made to find the' critical Rayleigh number by using a numerical approach. The computer program, ENCLREC, discussed above is easily applied to a horizontal enclosure, i.e., an enclosUre with horizontal hot and cold walls as shown in Fig. 8.32, and with perfectly conducting end walls, by setting 4> equal to 0, This modified program has been used to determine, the variation of mean Nusselt with Rayleigh'number for a horizontal enclosure for various enclosure aspect ratios for a Prandtl number of 0.7. A typical such variatign, this being for an aspect ratio of 5, is shown in Fig. 8.33. I t will be seen that at low Rayleigh numbers the Nusselt number has a value of 1. the pure conduction value. At these Rayleigh numbers there is no fluid/motion in the enclosure. However it will be seen that at a Rayleigh number of approximately between 1700 and 1750, the Nusselt number starts to rise above the pure conduction value of 1. i.e., fluid motion develops in the enclosure. This is further illustrated by the results given in Fig~ 8.34. These results are for an enclosure with an aspect ratio of 10. Figure 8.34 shows the variation of the maximum positive value of the stream function with Rayleigh number for Rayleigh numbers!'near 1700. Because of the nature of the flow in the enclosure (see below) there are both positive and negative
/
CHAPTER
8: Natural Convection 405
1.03
. ........wr,
Aspect Ratio =5
1.02
• •
~u
1.01
Rayleigh 1.00
S~ .LUmI
• ••
  ~.~ ~Pure Conduction (Nu =1)
..
..
0.99 ' _ _1 . .  . _   1_ _. ......_ _. ......_ _ ' 1800 1900 2000 1500 1600 1700 Ra
F IGURE 8.33 Variation of mean Nusselt number with Rayleigh number for an enclosure with an aspect ratio of 5 for P r = 0.7.
.0•.8.... ..'..r.,.;,
Aspect Ratio =10
0.6
' I'I/IQX 0.4
0.2
0.0 L...._ _ _ _ 1600 1700
..:::::==_.....L_ ___....I
1800 1900
Ra
F IGURE 8.34 Variation of maximum value of the stream function with Rayleigh number for an enclosure with an aspect ratio of 10 for P r = 0.7.
values of the stream function in the enclosure flow. The maximum negative value of the stream function varies with Rayleigh number in basically the same way as the maximum positive value. The maximum value.ofthe stream function is indication / of the intensity of the fluid motion in the enclpsure. The results given in Fig. 8.34 therefore also indicate that a significant fluid motion develops in the enclosure when the Rayleigh number reaches a value of approXimately 1700 to 1720. . In running the program discussed above, the sm8ll errorsintroduCed by numeri/ cal roundoff are sufficient to trigger the instability. In some related situations, however, it may be necessary to introduce very small rand,om disturbances in order.to trigger the instability. .
an
406 Introduction to Convective Heat Transfer Analysis
(a)
Threedimensional Hexagonal Cells
(b)
F IGURE 8.35
F1o~;pattems
in a horizontal enclosure.
Using the small disturbance approach gives a value of 1708 for the Rayleigh nlimber at which disturbances start to grow, i.e., at which fluid motion will develop i n the enclosure. This value will be independent of Prandtl number because the fluid motion for Rayleigh numbers near the critical value is very weak and the effect of the convective terms in the momentum equation is then negligible and the governing equations, i.e., Eqs. (8.137) to (8.139) then are:
a2 .o, a2.o, ax2 + ay2
=
( a(). a() )  Ra aX sme(>  ay cose/>
.
a2'1t
ax2 + ay2
=
a2'1t
=  .0,
a2 () a2 () aX2 + ay2
a'lt a() a'lt a() ay ax  ax ay
The only. parameter in these equations is the Rayleigh number. Therefore, the Rayleigh number a t which the flow starts to develop in a horizontal eR.closure will be independent of the Prandtl number. When the Rayleigh number exceeds the cnti.cal value, fluid motion develops. Initially, this consists of a series o fparallel twodimensional vortices as indicated in Fig; 8.35a. However a t higher Rayleigh numbers 8\lthree.,dimension3I cellular flow of the type indicated in Fig. 8.35b develops. These. threedimensional cells have a hexagonal'shape as indicated in the figure. This type of flow is termed Benard cells or Benard convection.
Two large horizontal plane surfaces are separa~by aJlayerof air. The . lowest surface is at a wriform temperatUre of 40°C and the upper surface is at a unifoim temperature of 20°C. Findthe gap between the two plates at which significant conveCtive . motion i n the air layer can be expected to first occur: .
E XAMPLE 8 .9.
Solutio,n. Convective motion will be assumed to begin when:
/
CHAPTER
8: Natural Convection 407
The mean air temperature is:
Tav
= (40 + 20) = 300C
2
At this temperature, air at standard ambient pressure, has the following properties:
f3
=
0.0033K 1, V
=
16 X 106 m2/s, P r
. .3
=
0.7
Hence, convective motion will be assumed to begin when: 0.0033
x 9.81 X (40 (16 X 1 06 )2
20)W
07
X.
= 1708
From which it follows that:
W
= 0.00988 m =
9.88 mm
Therefore, cO,llvective motion'will first occur when the gap between the plates.is 9.88mm. .
8.11 . T URBULENT NATURAL CONVECTIVE F LOWS
I n the discussions of naruriu conwective flows presented so f ar in'this chapter it haS b een assumed that the flow is laminar. Turbulent flow can, ,however, as discussed before, occur i n natural convective flows, s ee [84]' to f~5], this being illustrated schematically in Fig. 8.36. .' " F or flow over a vertical surface, i t has often been assumed that transition occurs when t helocal R ayleighnumber is equal to 108 o r when the'local Grashof number is equal to 1 08 • For fluids\ with Prandtl numbers near one, the difference between these two criteria is small. For fluid~ with Prandtl numbers that are very different from one, available analyses suggest that the Grashof numberbased criterion should be used.
E XAMPLE a.to. A vertical isothermal flat plate with a surface temperature of 50°C is exposed to stagnant air at 10°C and ambient pressure. Find the distance from the bottom of the plate at which transition to turbulence can be ~xpected to occur.
Vertical Flat Plate Turbulent
/
F IGURE 8.36 ." . . Transition to turbulence in .the natural convective flow over . . I a vertical plate.
408 Introduction to Convective Heat Transfer Analysis
,Solution. Transition will be assumed to occur when:
O rx
=
f3g(TH
Jl2
Tc):i3
= 108
The mean air temperature is:
Tav
= ( 50+ 10) = 300C
2
A t this temperature air, at standard ambient pressure"has the following properties:
f3
=
0.0033K 1,
JI
=
16 X 1O6m2/s
Hence, transition is assumed to occur when: 0.0033
x 9.81 X (50  1O):i3 = 108
(16 X 106)2
'
,
From which it fOllows. that transition occurs ~hen:
x
= 0.2704m
Therefore, transition will occur at a distance of 0.2704 m from the bottom of thp plate.
Available analyses of turbulent natural convection mostly rely in some way on dhe:assumption that the turbulence structure is similar to that which exists in turbulent forced convection, see [96] to [105]. In fact, the buoyancy forces influence the turbulence and the direct use of empirical information obtained from studies c;>f forced convection to the analysis of natural convection is not always appropriate. This will be discussed further in Chapter 9. Here, however, a discussion of one of the earliest analyses of turbulent natural convective boundary layer flow on a flat plate will be presented. This analysis involves assumptions that are typical of those used in the majority of available analyses of turbulent natural convection. This analysis is based on the use of the momentum and energy iritegral equations which for natural convective flow over a vertical plate are: (8.164) and:
~ [(8 u(T _ T l)d Y] d x Jo
=
.!l!!...
p Cp
(8.165)
These equations are the same as for forced convection except for th~ presence of the integrated buoyancy force term on the righthand side ofEq. (8.164). An overbar here will not be used to denote timeaveraged values of the variables, all variables used in the present section being timeaveraged. In using these equations, the forms of the veiocity and temperature profiles in the boundary layer are assumed. Now, in turbulent forced convective boundaty layer flows It has been found that the velocity profile is well described by:
/
CHAPrER
8: Natural Convection 409
U 1  = 1J"
Ul
(8.166)
where: (8.167) 8 being t he local boundary layer thickness and U l being the local freestream velocity; n i s a n integer with a value o f n ear 7. Equation (8.166) cannot be directly applied to natural convective boundary layer flows because in such flows the velocity is zero a t the outer edge o f the boundary layer. However, Eq. (8.166) should give a good description o f t he velocity dislributionnear the wall. I t is therefore assumed that in a turbulent natural convective boundary layer:
 ' = 1J"(1  1J) .
Ul
\
U
1
2
. (8.168)
where 1.41 is a characteristic velocity for the near wall flow. I t will have a value that is close to the maximum velocity in the boundary layer. Eq. (8.168) satisfies the boundary conditions:
A ty A ty
= 0: = 8:
U
U
=0 =0
A ty = 8: a y = 0
and ensures that the velocity distribution near the wall (i.e., for small values o f TJ) is effectively described by Eq. (8.166). The fonn o f the temperature profile must also be assumed. I t will b e assumed that the velocity and temperature·boundary layers have the same thickness, 8 . This i s ~ relatively good assumption in turbulent boundary layer flows. Now, i n turbulent forced convective boundary layer flows i t has been found that the temperature profile is well described by: . .
T T i l l =  TJn Tw  Tl .
au
.(8.169)
where T 1 i s the freestream temperature and T w is the wall temperature. This equa / tion satisfies the boundary conditions:
A ty = 0: T = Tw A ty = 8: T = Tl
these boundary conditions applying in both forced and natural convective boundary layer flows. Eq. (8.169) is really only expected to describe the temperature distribution n ear t he wall in natural convective flows b ut because i t does satisfy the boundary conditions i t will be assuined here to apply over the w~ole boundary layer.
/
410 Introduction to Convective Heat Transfer Analysis
Now, Eqs. (8.164) and (8.165) can be written as:
:x (U~8 [f.' (:.)' d,,])
and:
=
fJ
g(Tw  T,)8 f.' (~ __ i }" ~
(8.170)
(8.171)
I f the following are then defined:
It =
f.' (:. J = f.' ,,!(l d" Tw  Tl
Tl
,,)4 d "
·(8.172) (8.113) (8.174)
h = {I (T  "Tl ) dY= { \1 
. Jo
Jo
TJ~)dTJ
h
=
Jo
(I (.!!.)( T  Tl .)
Ul
Tw

d~ .= JeTJ~(1  TJ)2(},TJ~)dTJ o
The values of these integrals can be determined for any chosen value of n. With the integrals defined in this way, Eqs. (8.170) and (8.171) can be writtep as: (8. 175} and: (8.176) To proceed further, relationships for the wall shear stress, 'Tw, and the wall heat transfer rate, qw, must be assumed. It is consistent with the ass~ption that the flow near the wall in"a turbulent natural convective boundarY layer is similar to that in a turbulent forced convective boundary layer to assume that the expressions for 'Tw and qw that have been found to apply in forced convection should apply in natural convection. It will therefore be assumed here that the following apply in a natural . convective boundary layer:
'Tw _
0.023
, pu~  (pu15IJL)0.25
i.e.:
pO.75 U 1.75 JLO.25
'Tw
= 0.023
8~.25
(8.177)
/
and that:
CHAPTER
8: Natural Convection 411
Le.:
(8.178) it being assumed here that the characteristic forced velocity in these equations is the same as that used in defining the velocity profile. Using these equations, Eqs. (8.175) and (8.176) become:
d2 I I d x ( uI 8 ) = h f3 g(Tw

Tl)8  0.023
(f.L )0.25 80.25 P ups
25 (Tw 
(8.179)
and:
h d x [Ul (Tw  Td 8] =
,
d
q p; 0.023PrO.67 vO.
P
Tl) 8~.25
~~
(8.180)
These two equatio~s together describe the variations of 8 and Ul. An inspection . of the form of these two equations indicates that the solutions are of the form:
8
= Rx, Ul = S r
j
(8.181)
This form of solution implies that the boundary layer can be assumed to be t urbulent from the leading edge of the surface, i.e., from x = 0, or that x is measured from some artificial origin. ,. Substituting Eq. (8.181) into Eqs. (8.179) and (8.180) and equating the indices of x in the various terms gives:
2 s+rl=r
and:
2s + r  1 = 1. 75s  0.25r
and:
s + r  1 = 0.75s  0.25r
Adding s ~() both sides of the third equation shows that it is the same as the second equati()n. Solving between the first and second equation then gives:
s
= 0.5,
r
= 0.7
. S1.75
(8.182)
Substituting these results back into the governing equations and rearranging gives:
1.7S2Rlt = hf3g(Tw  Tl)R  0.023vo. 25 RO.25
and:
1 .2hSR = 0.023PrO~7 vO. 25 RO.25
Solving between these two equations gives: S
SO.75
=
[
] 2 1.7lt + 1.2hp,o.67
J.
Q5
[
f3g(Tw  Tl)
]0.5
(8.183)
412 Introduction to Convective Heat Transfer Analysis
and:
R=
[
0.023 ]0.8 1. 2 h p,o.67
[1.7h + 1.2hPro.67]0.1 [ .
h
v2 {3g(Tw  Tl)
r·
l
(8.184)
These two equations can be written as: S = Fl[{3g(Tw  TI)]0.5 and:
(8.185)
(8.186) where: . (8.187') and: . F2 = [ 0.023 ]0.8 1.2hp,o·67
[1.7h
. 01 + 1.2hP,.o.67] .
h
(R188)
Combining the results derived above then gives:
UI =
FI[{3g(Tw  Tl)f 5 xO. 5
(8. 189h
and:
8=
F2[,Bg(T~Tll Xo.,
(8.190)
. Therefore, because Eq. (8.178) can be written as: )0.75 ( )0.25 . qw x = 0.023Pr0.33 ( U IX ~ k(Tw  TI) v 5 It follows that:
FO. 75 1  GrOA N u X · 023PrO. 33 _O.25 x =0 p
2
(8.191)
The present analysis therefore predicts that in turbulent boundary .layer flbW over a vertical surface:
:~~4 = function(Pr) =
Nux
A
(~.192)
I f n is assumed to be 7, the above equation gives for P r = 0.7:
= 0.0185Gr~A
. (8.193)
CHAP1'BR 8: Natural Convection 413
T he mean heat transfer rate for a plate o f length, L, c an b e obtained b y noting that:
Hence, using Eq. (8.192):
i.e.:
qw L
( Tw

_A
0.4
Tl)k  1.2 GrL
,where GrL is the Grashof number b~ed on the plate length, L . T he me8:fl NussCflt n umber for the whole plate, N UL( = .h U k) is, therefore, given by:
N UL A GI};25 = 1.2
(8.194)
I f n is assumed to b e?, t he above equation gives for Pi = 0.7:
N UL =
O.0154Gr~4
(8,19~
A comparison o f Eq. (8.195) with some experimental results is shown i n Fig. 8.37. Quite good agreement will b e seen to b e obtained. . 'I'ttfbulent natural convective flows can also be analyzed b y numerically solv': ing the governing equations together with some form o f t urbulence model. This is
P r=0.7
1000
Nu 100
F IGURE 8.37 Comparison of measured and predicted Nusselt number 10 ......._ ........._ ___....I.,I....JI.~ variations for turbulent, l e+ 7 Ie + 8 Ie + 9 l e+ 10 Ie + 11 Ie + 12 . natural convective flow over Ra ( = GrPr) . a vertical plate.
/
414 Introduction to ConveCtive Heat Transfer Analysis
however; hampered by the fact that many turbulence models do not correctly describe the effect o f the buoyancy forces on the turbulence structure. .
 _/
Plot the variations of boundary layer thickness, maximum velocity, and local heat transfer rate along a lOm high vertical plate which is maintained at a uniform surface temperature of 50°C and exposed to stagnant air at ambient pressure and a temperature of lOoC. Assume the flow is turbulent from the leading edge.
E XAMPLES.11.
Solution. The following integrals arise in the a~proximate solution for turbulent natural convective boundary layer flow over a flat plate discussed above:
II
=
L1]~(1
1
.
 1]/ d1]
I'
, lz =
. IJ =e 1]1(1 _1])2(1 17 i )d1] Jo
These integrals are easily evaluated for any chosen value of the index, n. It will here be assumed that:
i°
l
(1  1]ii)d1]
,
n =7
and in this case:
11
.
= 0.10972,12
= 0.125, IJ
= 0.052723
The following functions of these integrals and of the Prandtl number were also de:~ " fined i n the analysis: .,
Fl
= [ 1.7lt + 1.2hp,o.67
lz
]0.5
and:
F2
=[
0.023 JO.8[1.7lt + 1.2IJP,.o.67]0.1 1.2hp,o·67 12
Because air flow is being considered it will be a~sumed that P r = 0.7. I n this case, using the values of the integrals given above, the following are obtained:
Fl
= 0.72856; F2 = 0.57408
= F 1[f3g(Tw Tl)]0.5 XO.5
The analysis then gives:
Ul
qX FO.75 w = 0 023PrO•33 _ 1 _ Gr0.4 k(Tw  T 1 ) ' Fg.25 , x
CHAPI'ER
8: Natural Convection
415
T he m ean air temperature i n t he boundary layer is:
Tav = ( 50 + 10) = 300C 2
A t this temperature, air a t standard ambient pressure has the following properties:
f3
Hence:
= 0.0033 K  1, v = 16 X
0.72856[0.0033
1 0 6 m2/s, k
= 0.02638 W /mK
= 0.8292xoos
U1 =
x 9.81 x (50  1O)]oosxoos
_ ( 16 >< 1 06 ) 2 007 _ 007 ~  0.57408 [ 0.0033 x 9.81 x ( 50 _ 10) . x  0.04882x
rot
3 r,,
2
U t m/s
1
o~~~
o
5
10
FIGURE E8.11a
x m
0.3
r ,'t
0.2
8 m
0.1
0.0
~
_ _________..J....._ _ _ _ _ _ _ _ _ _  ' 5 10
o
x m
FIGI1RE E8.11b
416 Introduction to Convective Heat Transfer Analysis
3oor~~~
200
q W/m2
100
o~~~ 5 10
o
FIGURE ES.llc
x m
and:
. qwx _ 0.33 0.72856°·75 0.0033 X 9.81 X (50  lO)x 0.02638 X (50  10)  0.023 x 0.7 x 0.574080.25 x (16 X 106)2
[
3]0.4
i.e.:
qw = 148.8xo. 2
The variations o f U l, 8 , and qw with x as given by these equations is shown in Figs. E8.11a, E8.11b, and E8.11c respectively.
8.12 CONCLUDING REMARKS
Some of the mQre· commonly used methods of obtaining solutions to problems involving natura1 convective flow have been discussed in this chapter. Attention has . been given to laminar natural convective flows over the outside of bodies, to laminar natural convection through vertical openended ch811Ilels, to laminar natural convec.,. don in a rectangular enclosure, and to turbulent natural convective boundary layer flow. Solutions to the boundary layer forms of the governing equations and to the full governing equations have been discussed.
I
P ROBLEMS
8.1. A v ertical isothermal flat plate with a surface temperature o f 5 0°C is exposed to stagnant air at 20°C and standard ambient pressure. Plot the velocity and temperature profiles in the boundary layer at distances o f IS, 30, and 45 c m from the leading edge.
/
CHAPnm.
8: Natural Convection 417
, 8.2. A 0.3m vertical plate is maintained at a surface temperature of 65°C and is exposed to stagnant air at a temperature of 15°C and standard ambient pressure. Compare the natural convective heat transfer rate from this plate with that which would result from forcing air over the plate at a velocity equal to the maximum velocity that occurs in the natural convective boundary layer. 8.3. Compare the heattransfer coefficients for laminar forced and free convection over vertical flat plates. Develop an approximate relation between the Reynolds and Grashof numbers such that the heattransfer coefficients for pure forced convection and pure free convection are equal. 8.4. A 0.2m square vertical plate is heated to 400°C and placed in room air at 25°C. Calculate the heat loss from one side of the plate. 8.5. A vertical plate 10 cm high is immersed in a stagnant 'fluid. The plate is maintained at a tempe~ature of 50°C and th~ fluid temperature is 10°C. Determine the avera~heat . transfer coefficient for this situation if the fluid is: (i) air at standard atmospheric pressure (ii) ~r at 0.01 X standard atmospheric pressure (iii) water 8.6. Plot the freeconvection boundarylayer thickness along a O.3m high vertical plate which is maintained at a uniform surface temperature of 50°C and exposed to stagnant air at ambient pressure and a temperature of lOoC. Assume the flow remains laminar. 8.7. Using the ~imilarity solution results, derive an expression for the maximum velocity in the natural convective boundary layer on a vertical flat plate. At what position in the boundaryJayer does this maximum velocity occur? 8.8. Two vertical flat plates Jleld at a uniform surface temperature of at 40°C are placed in a tank: of water Which is at a temperature 'Of 20°C. I f the plates are 10 cm high, estimate the minimum spacing between the plates if there is to be no interference between the boundary layers on the two plates. ' 8.9. A vertical flat pl~te is maintained at a uniform surface temperature and is exposed to air at standard ambient pressure. At a distance of 10 cm from the leading edge of the plate the boundary layer thickness is 2 cm. Estimate the thiclrness of the boundary layer at a distance of 25 em from the leading edge. Assume a laminar boundary layer flow. 8.10. Consider laminar freeconvective flow over a vertical flat plate at whose surface the '. heat transfer rate per unit area, qw, is constant. Show t hata similarity solution to the twodimensional laminar boundary layer equations can be derived for this case. 8.11. It will be seen from the results given by the similarity solution that the velocities are very low in natural convective boundary layers in fluids with high Prandtl numbers. I n such circumstances, the inertia terms (Le., the convective terms) in the momentum equation are negligible and the boundary layer momentum equation for a vertical surface effectively is:
418 Introduction to Convective Heat Transfer Analysis
i.e., there is a balance between the buoyancy and the viscous forces. I f the following are assumed:
u xla 1 R ax12
= F '('TI )
and:
where: 71
.
= ~Ra1l4 xx
T1).x3
Ra x being the Rayleigh number based on x, i.~.: Ra x
= 13g(T 
va
= GrxPr
the prime, .of course, again denotes differentiation with respect to 71, show that the boundary layer equations give: /
F ill
+G
4
=0 =0
Gil
+ ~FG'
with boundary conditions:
At 71 At 71 At 71
=
0 : F'
=
0
= 0: F = 0 = 0: G = 1
~
~
For large 71 : F' For large 71 : G
Nux
0 0
Show from these results that this solution indicates that
= function(Ra x )
i.e., show that at large values o fthe Prandtl number the local Nusselt number depends only on the local Rayleigh number and not separately on the Prandtl number and the local Grashof number.
8.12. A 30cm high vertical plate has a surface temperature that varies linearly from 15°C at the lower edge to 45°C a t the upper edge. This plate is exposed to air at 15°C and ambient pressure. Use the computer program for natural convective boundary lay~r flow to determine how the local heat transfer rate varies with distance up the plate. from the lower edge.
8.1~.
Use the computer program for twodimensional laminar free convective flow to find the Nusselt number variation along a vertical plate whose surface temperature varies / .in such a way that Tw  T", is equal to 10°C over the lower half of the plate and equal to 30°C over the upper half of the plate.
CHAPl'ER
8: Natural Convection 419
8.14. When the density variation with temperature of water is considered, a maximum is found to occur near 4°C as indicated schematically in the following figure.
I
Pmax
._=::::::=00'_
Density,
p
FIGURE P8.14
Near this point of maXimum density, the density is approximately given by:
Pmax  P
= A (T 
4)2
where A i sa constant. Modify the program for solving the boundary layer equation for natural convective flow discussed in this text to apply to this situation. I n this case define:
(J
=
T 4
T w 4
Twbeing the uniform wall temperature which is between O°C and 4°C.
8.15. Air flows by natural convection through the channel formed between 2 20cm high plates kept at a temperature of 50°C. I f the distance between the 2 plates is 3 cm and i f the ambient air temperature is 20°C, find the rate of heat transfer from the 2 plates to the air and the mean velocity of the air through the channel.
8.16. Consider the natural convective flow of air at 10°C though a plane vertical channel with isothermal walls whose temperature is 40°C and whose height is 10 cm. Determine how the mean h eattransfer rate from the heated walls varies with the gap between the walls. 8.17. Consider heat transfer across an airfilled inclined square enclosure with one wall heated to a uniforril temperature and the parallel wall cooled to a uniform temperature with the remaining two walls being perfectly cOl').ducting. The wall temperature and enclosure size are such that the Rayleigh number based on the enclosure size and on the difference between the temperatures of the hot and cold walls is lOS. Examine the effect of the angle of the heated wall to the vertical on the heat transfer rate by·nUinerically solving the governing equations. Consider angles of inclination to the vertical of between +60° and  60°. 8.18. When the density variation with temperature of water is considered, a maximum is found to occur near 4°C as discussed in Problem 8 J4. Near this point o f maximum, density i s approximately given by:
Pmax  P :::: A (T  4i
/
420 Introduction to Convective Heat Transfer Analysis where A is a constant. M()djfy the program for solving for the natural convective flow in an enclosure to apply to this situation. Consider the case of a vertical square enclosure with perfectly.conducting top and bottom walls whose cold wall is at O°C and determine how the dimensionless hot wall temperature affects the mean Nusselt number. Use a constant suitably modified Rayleigh number based on the difference between the temperatures of the hot and cold walls of 105 • Define:
8=
T 4 T HTc
8.19. The discussion of free convective flows given in this chapter has implicitly been concerned with flows that arise due to the presence of temperature gradients in a fluid acted' on by gravity. Free convective flows can be caused by the presence of temperature gradients in a fluid acted upon by centrifugal forces. Consider a 5cm high plane fin kept at a uniform temperature of 60°C and exposed to air at 20°C. This fin is part of a rotating electrical machine and the center o f the fin is rotating on a radius of 0.5 m at a speed of 600 rpm. Estimate the mean heat transfer rate from the plate assuming that the air to which the plate is .exposed is rotating with the machine. . 8.20. Plot the variations of boundary layer thickness and local heat transfer rate along a 2m high vertical plate which is maintained at a uniform surface temperature of 600C and exposed to stagnant air at ambient pressure and a temperature of 20°C assuming that (i) the flow is laminar from the leading edge and (ii) the flow is turbulent from the leading edge. 8.21. A vertical isothermal flat plate with a surface temperature of 20°C is exposed to stagnant water at 10°C. Find the distance from the bottom of the plate at which transitiop to turbulence can be expected to opcur. . 8.22. A wide vertical plate 1 m high is immersed in a stagnant air which has a temperature of 30°C.' The plate is maintained at a uniform temperature of 50°C. Plot the variation of average heat transfer rate from plate with air pressure for air pressures between 0.1 and 10 times standard atmospheric pressure. .
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CHAPTER
8: Natural Convection 421
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422 Introduction to Convective Heat Transfer Analysis
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/
1 ",\
I,
CHAPTER
8: Natural Convection 423
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424 Introduction to Convective Heat Transfer Analysis
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CHAPI'ER
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