FreeConvection

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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

Case I: Unstable temperature gradient

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

Case II: Stable temperature gradient

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 → ∞.

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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

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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 coeďŹƒcient, local N ux can be written as

−k ∂T | x qx /(Ts − T∞ ) x hx ∂y y=0 N ux = = = k k (Ts − T∞ )k 1 1 Grx 4 dT ∗ Grx 4 |Ρ=0 = g(P r) (10.17) = 4 dΡ 4 where, Grx =

βg [Ts − T∞ ] x3 ν2

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 ≤ P r ≤ ∞. Average N u is given by N uL = ÂŻ= where h

10.2.3

1 L

L

hdx =

0

k L

gβ(Ts −T∞ ) 4ν 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 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 ≈ 1, then both free and forced convection mechanisms are important. Re2 L

(Note that only when free convection.)

GrL Re2L

→ ∞, 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.

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CHAPTER 10. NATURAL OR FREE CONVECTION

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