Chapter 10 Natural or free convection (Material presented in this chapter are based on those in Chapter 9, ”Fundamentals of Heat and Mass Transfer”, Fifth Edition by Incropera and DeWitt) In the previous two chapters, analytical expressions and correlations to estimate the heat transport coeﬃcients from a ﬂuid to a surface of an object during (a) forced external ﬂow, that is, the ﬂuid is forced to ﬂow past the object and (b) force internal ﬂow, that is, the ﬂuid is forced to ﬂow through the object and as a result the internal surface is exposed the the ﬂuid for heat transport. In this chapter, the heat transport due to natural convection. Natural convection occurs when the the body force acts on a ﬂuid where there are density gradients. In contrast to the forced convection case, the ﬂuid is not forced to ﬂow in fully free convection case. When the ﬂuid, at rest, experiences a density diﬀerence due to certain disturbances, then the resulting buoyancy force induces free convection currents under certain conditions. The density gradient could be due to the temperature gradient and the situations where convection currents are introduced, the net motion could be due to the gravitational ﬁeld or centrifugal force or Coriolis force. The free convection ﬂow rates are usually much smaller than the forced convection ﬂow rates. Therefore, the heat transfer coeﬃcients in free or natural convection is smaller than in the case of forced convection. Free convection strongly inﬂuences heat transfer from pipes, transmission lines, stream radiation to room air etc. Typically for gases, the density gradient with respect to temperature ∂ρ < 0, that is, the density decreases with increase in the temperature. ∂T 147

148

10.1

CHAPTER 10. NATURAL OR FREE CONVECTION

Fluid in a box

Consider the case of a ﬂuid at a quiescent state, that is, at rest between two large parallel plates (Fig. 10.1). Free convection can be induced in this system depending the temperatures of the plates, which will be discussed in the following sub-sections.

Figure 10.1: Fluid at rest between two parallel plates. (a) Free convection due to positive temperature gradient. (b) No free convection.

10.1.1

Suppose that the upper plate is maintained at temperature T1 (Fig. 10.1a), and the lower plate at T2 > T1 . The density ρ2 of the ﬂuid near the lower plate will be lower than the density ρ1 of the ﬂuid near the top plate. But, denser ﬂuid being on top of a lighter ﬂuid is an unstable situation resulting in the re-circulation of the ﬂuid. Gravity will pull the denser ﬂuid down resulting in a net ﬂow of the ﬂuid carrying heat, that is, free convection. Heat transfer from the lower plate to the upper plate occurs via free convection.

10.1.2

Suppose that the upper plate is maintained at temperature T1 > T2 (Fig. 10.1b), then the ﬂuid near the lower plate will have a higher density than that near the upper plate. This is a stable situation and the heat transfer in this case is only by conduction.

10.2. HEATED PLATE IN A QUIESCENT FLUID

10.2

149

Heated plate in a quiescent ﬂuid

Consider the case of an immersed heated plate in a quiescent medium (Fig. 10.2). Assume the temperature of the plate Ts is greater than the temperature of the quiescent medium T∞ . Assume that the density of the quiescent medium is ρ∞ and that the gravity acts in the negative x − dir (see Fig. 10.2). The density of the ﬂuid near the plate will be lesser than that far away from the plate. As a result, the buoyancy forces will induce a free convection and the ﬂuid will rise vertically along the plate, entraining ﬂuid from the quiescent media. Free convection will lead to boundary layer formation.

Figure 10.2: Natural convection past a heated plate placed in a quiescent ﬂuid. Assume laminar conditions, no viscous dissipation and steady 2-D ﬂow. Assume the ﬂuid to be incompressible and the boundary layer approximations to be valid. The velocity of the ﬂuid is zero at y → ∞.

150

CHAPTER 10. NATURAL OR FREE CONVECTION

The governing equations can be obtained using Eqs (7.19) - (7.23). As the free convection velocities are small, the viscous dissipation can be assumed negligible. If there is no body force in y−dir, then the y−momentum balance ∂p will lead to ∂y = 0, that is, pressure gradient in the y − dir is zero. ∂u ∂v + =0 ∂x ∂y 

 ∂u ∂u 1 ∂P ∂ 2 u X˙ u +v =− +ν 2 + ∂x ∂y ρ ∂x ∂y ρ    2  ∂T ∂T ∂ T u +v =α ∂x ∂y ∂y 2

(10.1)

(10.2) (10.3)

As the y − dir pressure at any x position in the boundary layer must be equal to the pressure in that x position in the quiescent ﬂuid, the pressure gradient in x − dir, that is, ∂P in the boundary layer will be equal to ∂P in ∂x ∂x the quiescent stream. As u = v = 0 in the quiescent stream, using x − dir momentum balance in the quiescent region, ∂P ∂P |BL = |Quiescent = −ρ∞ g ∂x ∂x

(10.4)

Using Eq. (10.4) and setting X˙ = ρg, Eq. (10.2) will become 

∂u ∂u u +v ∂x ∂y

 =g

∂2u Δρ +ν 2 ρ ∂y

(10.5)

is the buoyancy force that is exerted on the ﬂuid where, Δρ = ρ∞ − ρ. g Δρ ρ due to density variations. Note that Eq. (10.5) is valid in the whole boundary layer.

10.2.1

Volumetric thermal expansion coeﬃcient

If the density diﬀerences are only due to temperature variations, then the volumetric thermal expansion coeﬃcient β is given by     1 ∂ρ 1 ρ∞ − ρ β=− ⇒ (ρ∞ − ρ) ≈ βρ(T − T∞ ) ≈− (10.6) ρ ∂T P ρ T∞ − T

10.2. HEATED PLATE IN A QUIESCENT FLUID

151

The approximation made in Eq. (10.6) is called the Bousinessq approximation, substituting which Eq. (10.5) becomes   ∂u ∂u ∂2u u +v = βg [T − T∞ ] + ν 2 (10.7) ∂x ∂y ∂y If the ﬂuid were to be an ideal gas, then   1 ∂ρ 1 β=− = ρ ∂T P T

10.2.2

(10.8)

Heat transfer coeﬃcient

Introducing the non-dimensional quantities x∗ = Lx , y ∗ = Ly , u∗ = ∞ T ∗ = TTs−T into Eqs (10.1), (10.7), (10.3) leads to −T∞

 u

∗ ∂u

∂x∗



βg [Ts − T∞ ] L ∗ 1 ∂ 2 u∗ T + u20 ReL ∂y ∗2     ∗ ∗ ∂2T ∗ 1 ∗ ∂T ∗ ∂T u = +v ∂x∗ ∂y ∗ ReL P r ∂y ∗2

+

∂u∗ v∗ ∗ ∂y

∂u∗ ∂v ∗ + =0 ∂x∗ ∂y ∗ =

u , u0

v∗ =

v , u0

(10.9) (10.10) (10.11)

The governing equations are subject to the boundary conditions u(y = 0) = v(y = 0) = 0, T (y = 0) = Ts , u(y → ∞) → 0, T (y → ∞) → T∞ . As there is no free stream velocity, u0 is an unknown quantity, and hence ReL can be used in deﬁning a non-dimensional quantity for the coeﬃcient L of T ∗ term in Eq. (10.10). This coeﬃcient can be identiﬁed as Gr where Re2L Grasshof number Gr is given by  2 βg [Ts − T∞ ] L u0 L βg [Ts − T∞ ] L 2 βg [Ts − T∞ ] L3 GrL = = Re = L u20 ν u20 ν2 (10.12) Grasshof number is essentially the ratio of the body to viscous forces. The governing equations can be solved using stream function formula 1 tions. Setting similarity variable η = xy Gr4 x 4 and deﬁning stream function as   1 Grx 4 ψ(x, y) = f (η) 4ν (10.13) 4

152

CHAPTER 10. NATURAL OR FREE CONVECTION

Continuity equation (Eq. 10.9) is naturally satisﬁed. Equations (10.10) and (10.11) will become    f + 3f f − 2f 2 + T ∗ = 0 (10.14) and





T ∗ + 3P rf T ∗ = 0

(10.15)

and the the boundary conditions will become 



At η = 0, f = f = 0, T ∗ = 1; At η → ∞, f → 0, T ∗ → 0

(10.16)

(Note that Eq. (10.14) is a function of temperature and hence the velocity boundary layer is also a function of temperature.) The governing equations in terms of stream functions cannot be solved analytically. Numerically simulated velocity proﬁle (in terms of the similarity variable) and the temperature proﬁles for various P r are presented in Fig. (10.3). Figure (10.3a) and (10.3b) can be used to obtain the velocity and

Figure 10.3: (A) Velocity proﬁle (in terms of the similarity variable) and (B) Temperature proﬁle during natural convection for various P r under laminar conditions during free convection along a heated ﬂat plate placed in a quiescent ﬂuid. temperature at any position in the boundary layer and the appropriate heat transfer coeﬃcient.

10.2. HEATED PLATE IN A QUIESCENT FLUID

153

In order to estimate the heat transfer coeďŹ&#x192;cient, local N ux can be written as

 â&#x2C6;&#x2019;k â&#x2C6;&#x201A;T | x qx /(Ts â&#x2C6;&#x2019; Tâ&#x2C6;&#x17E; ) x hx â&#x2C6;&#x201A;y y=0 N ux = = = k k (Ts â&#x2C6;&#x2019; Tâ&#x2C6;&#x17E; )k  1 1  Grx 4 dT â&#x2C6;&#x2014; Grx 4 |Îˇ=0 = g(P r) (10.17) = 4 dÎˇ 4 where, Grx =

Î˛g [Ts â&#x2C6;&#x2019; Tâ&#x2C6;&#x17E; ] x3 Î˝2

and

(10.18)

1

0.75P r 2

g(P r) =

1 2

0.609 + 1.221P r + 1.238P r

14

(10.19)

is valid 0 â&#x2030;¤ P r â&#x2030;¤ â&#x2C6;&#x17E;. Average N u is given by N uL = ÂŻ= where h

10.2.3

1 L

L

hdx =

0

k L

gÎ˛(Ts â&#x2C6;&#x2019;Tâ&#x2C6;&#x17E; ) 4Î˝ 2

ÂŻ 4 hL = N ux k 3

14

g(P r)

L

(10.20)

dx 1

4 0 x

Forced and free convection

In principle, both forced and free convection can co-exist when a hot plate looses heat to the surrounding ďŹ&#x201A;uid. It is useful to estimate the conditions under which each of these will dominate and under which both are important. Grasshof number can be used to identify these conditions. L When Gr  1 then, free convection is negligible and forced convection Re2L is the dominant convection mechanism that governs heat transport in this L problem. In this case N uL = f (ReL , P r). Similarly when, Gr  1 then, free Re2L convection dominates over forced convection and N uL = f (P r, GrL ). When GrL â&#x2030;&#x2C6; 1, then both free and forced convection mechanisms are important. Re2 L

(Note that only when free convection.)

GrL Re2L

â&#x2020;&#x2019; â&#x2C6;&#x17E;, then the heat transport occurs purely by

154

10.2.4

CHAPTER 10. NATURAL OR FREE CONVECTION

Turbulent conditions

Thermally turbulent boundary layer occurs in natural convection at the crit3 ∞ )x ≈ 109 . Governing equaical Rayleigh number RaL,c = Grx P r = gβ(Ts −T νx tions under these conditions cannot be solved analytically and the heat transport coeﬃcient can be estimated by using correlations.

10.3

External ﬂows free convection

External free convection can occur over diﬀerent geometries such as plate, cylindrical surface depending on the application. No analytical expressions exist for most cases under all conditions. However, several experiments based correlations are available. Correlations for obtaining heat transport coeﬃcients for ﬂow over various geometries will be developed in this section. Vertical ﬂat plate The general form of the Nusselt number is given by N uL =

¯ hL = CRanL k

(10.21)

3

∞ )L , n = 14 for laminar ﬂow and n = 13 for turbulent where RaL = gβ(Ts −T να ﬂows. General correlation for Nusselt number that is valid for entire range of RaL is ⎡ 8 ⎤2 1 27 6 0.387RaL N uL = ⎣0.825 +  (10.22)  0.492  169  ⎦ 1 + Pr

Inclined heated plate: Lower surface exposed to ﬂuid Correlations that can be used to compute the heat transfer coeﬃcient from an inclined ﬂat plate to ﬂuid below it is 1

N uL = 0.54RaL4 (104 ≤ RaL ≤ 107 ) 1 3

N uL = 0.15RaL (107 ≤ RaL ≤ 101 1)

(10.23) (10.24)

10.4. LONG HORIZONTAL CYLINDER

155

Inclined heated plate: Upper surface exposed to ﬂuid Correlation that can be used to compute the heat transfer coeﬃcient from an inclined ﬂat plate to ﬂuid above it is 1

N uL = 0.27RaL4 (105 ≤ RaL ≤ 1010 )

10.4

(10.25)

Long horizontal cylinder

Consider a cylinder under isothermal conditions. Free convection boundary layer will be formed as shown in Fig. (10.4). The average Nu number will be inﬂuenced by the boundary layer formation. The correlation that can be used for estimating heat transport coeﬃcient for the case in Fig. (10.4) is given by ⎧ ⎫2 1 ⎪ ⎪ ⎨ ⎬ 0.387RaD6 N uD = 0.6 +

RaL ≤ 1012 8 9   ⎪ ⎪ 16 ⎩ 27 ⎭ 1 + 0.559 Pr

Figure 10.4: Boundary layer development around a cylinder. N u decreases from θ = 0 to θ = π as shown in Fig. (10.4).

(10.26)

156

10.5

CHAPTER 10. NATURAL OR FREE CONVECTION

Sphere

Correlation for estimating heat transfer coeﬃcient from the sphere to the ﬂuid due to free convection is 1

0.589RaD4 N uD = 2 +

 169 49  1 + 0.469 Pr

10.6

(10.27)

Free convection within parallel plates

Consider a quiescent ﬂuid at T∞ between two parallel plates maintained at a certain temperatures (see Fig. 10.5). Assume that the parallel plates are oriented at a certain angle θ with the vertical. Free convection ﬂow can be 3-D. Therefore, there will be a buoyancy component in the direction perpendicular to the parallel plates.

Figure 10.5: Free convection within parallel plates

10.6. FREE CONVECTION WITHIN PARALLEL PLATES

10.6.1

157

Vertical channels

For isothermal, symmetric temperature system, the correlation to estimate heat transfer coeﬃcient under laminar conditions for average Nu is   34   35 S 1   1 − exp − N uS = Ras 24 L Ras LS where, Nusselt and Rayleigh numbers are N uS = gβ(Ts −T∞ αν

)S 3

q/A Ts −T∞

!

(10.28) S k

and RaS =

. For symmetric isoﬂux conditions, under fully developed conditions, the correlation is  12  ∗S N uS,L(f d) = 0.144 Ras (10.29) L and for asymmetric isoﬂux conditions, under fully developed conditions, the correlation is  12  ∗S N uS,L(f d) = 0.204 RaS (10.30) L    qS S where, N uS = Ts −T k ∞ Alternative general formulation of the above correlations for isothermal surface is − 12  C1 C2 N uL = + (10.31) (RaS S/L)3 (RaS S/L)2 where, C1 = 576; C2 = 2.87 for symmetric isothermal plates and for isoﬂux conditions − 12  C2 C3 N uS,L = + (10.32) Ras S/L (Ras S/L) 25 where, C1 = 48; C2 = 2.51 for symmetric isoﬂux conditions.

158

CHAPTER 10. NATURAL OR FREE CONVECTION

FreeConvection
FreeConvection

&lt;0,thatis,thedensitydecreaseswithincreaseinthetemperature. 147 ∂ρ ∂T