Heat Transfer Handbook part 55 ppt

7.3 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 7.3.1 Vertical Surfaces The classical problem of natural-convection heat transfer from an isothermal heated vertical surface, shown

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δT

δ = O

1

Pr1/2



(7.13)

where Gr is the Grashof number based on a characteristic length L and Pr is the

Prandtl number These are defined as

Gr=gβL3(T w − T∞)

k =

ν

whereν is the kinematic viscosity and α the thermal diffusivity of the fluid These

dimensionless parameters are important in characterizing the flow, as discussed in the next section

The resulting boundary layer equations for a two-dimensional vertical flow, with variable fluid properties except density, for which the Boussinesq approximations are used, are then written as (Jaluria, 1980; Gebhart et al., 1988)

∂u

∂x +

∂v

u ∂u ∂x + v ∂u ∂y = gβ(T − T∞) +1ρ∂y ∂

µ∂u ∂y



(7.16)

ρc p



u ∂T

∂x + v

∂T

∂y



= ∂

∂y

k ∂T

∂y



+ q+ βT u ∂p a

∂x + µ

∂u

∂y

2

(7.17)

where the last two terms in the energy equation are the dominant terms from pressure work and viscous dissipation effects Hereu and v are the velocity components in the

x and y directions, respectively Although these equations are written for a vertical,

two-dimensional flow, similar approximations can be employed for many other flow circumstances, such as axisymmetric flow over a vertical cylinder and the wake above

a concentrated heat source

There are several other approximations that are commonly employed in the anal-ysis of natural convection flows The fluid properties, except density, for which the Boussinesq approximations are generally employed, are often taken as constant The viscous dissipation and pressure work terms are generally small and can be neglected

However, the importance of various terms can be best considered by nondimension-alizing the governing equations and the boundary conditions, as outlined next

7.2.3 Dimensionless Parameters

To generalize the natural convection transport processes, a study of the basic nondi-mensional parameters must be carried out These parameters are important not only

in simplifying the governing equations and the analysis, but also in guiding experi-ments that may be carried out to obtain desired information on the process and in the presentation of the data for use in simulation, modeling, and design

In natural convection, there is no free stream velocity, and a convection velocity

V cis employed for the nondimensionalization of the velocity V, whereV cis given by

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V c = [gβ(T w − T∞)]1/2 (7.18)

The governing equations may be nondimensionalized by employing the following dimensionless variables (indicated by primes):

V= V

V c p=

p

ρV2

c θ= T − T∞

T w − T∞

Φ

v = ΦvL

2

V2

c t= t t

c ∇= L∇ (∇)2 = L2∇2 (7.19) wheret cis a characteristic time scale The dimensionless equations are obtained as

Sr

∂v

∂t + V· ∇v

= − eθ− ∇p

d+√1

Sr

∂θ

∂t + V· ∇θ

Pr√

Gr(∇)2θ+ (q)+ βT gβL

c p

Sr∂p

∂t + V· ∇p +gβL

c p

1

√

GrΦ

where e is the unit vector in the direction of the gravitational force.

Here Sr = L/V c t c is the Strouhal number andq is nondimensionalized with

ρc p (T w − T∞)V c /L to yield the dimensionless value (q) It is clear from the equa-tions above that√

Gr replaces Re, which arises as the main dimensionless parameter

in forced convection Similarly, the Eckert number is replaced bygβL/c p, which

now determines the importance of the pressure and viscous dissipation terms The Grashof number indicates the relative importance of the buoyancy term compared to the viscous term A large value of Gr, therefore, indicates small viscous effects in the momentum equation, similar to the physical significance of Re in forced flow

The Prandtl number Pr represents a comparison between momentum and thermal diffusion Thus, the Nusselt number may be expressed as a function of the Grashof and Prandtl numbers for steady flows if pressure work and viscous dissipation are neglected The primes used for denoting dimensionless variables are dropped for convenience in the following sections

7.3 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 7.3.1 Vertical Surfaces

The classical problem of natural-convection heat transfer from an isothermal heated vertical surface, shown in Fig 7.1, with the flow assumed to be steady and laminar and the fluid properties (except density) taken as constant, has been of interest to investigators for a very long time Viscous dissipation effects are neglected, and no

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heat source is considered within the flow Therefore, the problem is considerably sim-plified, although the complications due to the coupled partial differential equations remain The governing differential equations may be obtained from eqs (7.15)–(7.17)

by using these simplifications

An important method for solving the boundary layer flow over a heated vertical surface is the similarity variable method A stream functionψ(x,y) is first defined so

that it satisfies the continuity equation Thus, we defineψ by the equations

u = ∂ψ

∂y v = −

∂ψ

Then the similarity variableη, the dimensionless stream function f , and the

temper-atureθ are defined so as to convert the governing partial differential equations into

ordinary differential equations Gebhart et al (1988) have presented a general ap-proach to determine the conditions for similarity in a variety of flow circumstances

For flow over a vertical isothermal surface, the similarity variables which have been used in the literature and which may also be derived from this general approach may

be written as

η = y

x

Grx 4

1/4

ψ = 4νf (η)

Grx 4

1/4

θ = T − T∞

T w − T∞ (7.24) where

Grx =gβx3(T w − T∞)

The boundary conditions are:

aty = 0: u = v = 0, T = T w; asy → ∞: u → 0, T → T∞ (7.26) These must also be written in terms of the similarity variables in order to obtain the solution Note that the velocity componentv for y → ∞ is not specified as zero in

order to account for the ambient fluid entrainment into the boundary layer

The governing equations are obtained from the preceding similarity transforma-tions as

f+ 3ff− 2(f)2+ θ = 0 (7.27)

θ

where the primes here indicate differentiation off (η) and θ(η) with respect to the

similarity variable η, one prime representing the first derivative, two primes the

second derivatives, and three primes the third derivative The corresponding boundary conditions are

atη = 0: f = f= 1 − θ = 0; asη → ∞: f→ 0, θ → 0

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which may be written more concisely as

f (0) = f(0) = 1 − θ(0) = f(∞) = θ(∞) = 0 (7.29) where the quantity in parentheses indicates the location where the condition is applied

The solution of these equations has been considered by several investigators

Schuh (1948) gave results for various values of the Prandtl number Pr, employing approximate methods Ostrach (1953) numerically obtained the solution for the Pr range 0.01 to 1000 The velocity and temperature profiles thus obtained are shown in Figs 7.3 and 7.4 An increase in Pr is found to cause a decrease in the thermal bound-ary layer thickness and an increase in the absolute value of the temperature gradient

at the surface This is expected from the physical nature of the Prandtl number, which represents the comparison between momentum and thermal diffusion An increasing value of Pr indicates increasing viscous effects The dimensionless maximum ve-locity is also found to decrease and the veve-locity gradient at the surface to decrease with increasing Pr, indicating the effect of greater viscous forces The location of this maximum value is found to shift to higherη as Pr is decreased The velocity boundary

layer thickness is also found to increase as Pr is decreased to low values These trends are expected from the physical mechanisms that govern this boundary layer flow, as dicussed earlier It is also worth noting that the results indicate the coupling between the velocity and temperature fields, as evidenced by the presence of flow wherever

a temperature difference exists, such as the profiles at low Pr Additional results and discussion on the flow are given in several books; see, for instance, the books by Kaviany (1994), Bejan (1995), and Oosthuizen and Naylor (1999)

The heat transfer from the heated surface may be obtained as

q

x = −k

∂T

∂y

 0

= −k(T w − T∞) x1

Grx 4

1/4
∂θ

∂η

 0

=−θ(0)  k(T w − T∞)

x

Grx 4

1/4

(7.30) The local Nusselt number Nuxis given by

Nux =h k x x = q x

T w − T∞

x k

We have for an isothermal surface

Nux =−θ(0)
Grx

4

1/4

=−θ√(0)

2 Gr

1/4

x = φ(Pr)Gr1/4

whereφ(Pr) =−θ(0)/√2 Therefore, the local surface heat transfer coefficient

h xvaries as

h x = Bx −1/4 whereB = k[−θ√(0)]

2

gβ(T

w − T∞)

ν2

1/4

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

( (

= 2G

0.01

Pr = 0.01

1 0.72

2 10 100 1000

0 0.1 0.2 0.3

Figure 7.3 Calculated velocity distributions in the boundary layer for flow over an isothermal vertical surface (From Ostrach, 1953.)

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0 0.2

0.8

0.4

1.0

0.6

( (

TT w

Pr = 0.01

Pr = 0.01

1 0.72

2 10

100 1000

0 0.2 0.4 0.6

Figure 7.4 Calculated temperature distributions in the boundary layer for flow over an isothermal vertical surface (From Ostrach, 1953.)

The average value of the heat transfer coefficient ¯h may be obtained by averaging

the heat transfer over the entire length of the vertical surface, to yield

¯h = L1  L

0

h x dx = 4

3

B

L L3/4

Therefore,

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Nu=4

3



−θ(0)

√

1/4= 4

3φ(Pr) Gr1/4= 4

The values of φ(Pr) can be obtained from a numerical solution of the governing

differential equations Values obtained at various Pr are listed in Table 7.1 The significance ofn and the uniform heat flux data in the table is discussed later An

approximate curve fit to the numerical results forφ(Pr) has been given by Oosthuizen

and Naylor (1999) as

φ(Pr) =

0.316Pr5/4

2.44 + 4.88Pr1/2 + 4.95Pr

1/4

(7.33)

It must be mentioned that these results can be used for both heated and cooled surfaces (i.e.,T w > or < T∞), yielding respectively a positiveqvalue for heat transfer from the surface and a negative value for heat transfer to the surface

In several problems of practical interest, the surface from which heat transfer occurs is nonisothermal The two families of surface temperature variation that give rise to similarity in the governing laminar boundary layer equations have been shown

by Sparrow and Gregg (1958) to be the power law and exponential distributions, given as

T w − T∞= Nx n and T w − T∞= Me mx (7.34)

TABLE 7.1 Computed Values of the Parameter φ(Pr)for a Vertical

Heated Surface

5

5

Source: Gebhart (1973).

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whereN , M, n, and m are constants The power law distribution is of particular

interest, since it represents many practical circumstances The isothermal surface is obtained forn = 0 From the expression for q

x, eq (7.30), it can be shown thatq

x

varies withx as x (5n−1)/4 Therefore, a uniform heat flux condition,q

x = constant,

arises forn =1

5 It can also be shown that physically realistic solutions are obtained for−3

5 ≤ n < 1 (Sparrow and Gregg, 1958; Jaluria, 1980) The governing equations

are obtained for the power law case as

f+ (n + 3)ff− 2(n + 1)(f)2+ θ = 0 (7.35)

θ

Pr + (n + 3)f θ− 4nfθ = 0 (7.36) The local Nusselt number Nuxis obtained as

Nux

Gr1x /4 =−θ√(0)

The function Nux /Gr1/4

x is plotted againstn in Fig 7.5 For n < −3

5, the function

is found to be negative, indicating the physically unrealistic circumstance of heat transfer to the surface forT w > T∞ The surface is adiabatic forn = −3

5, which thus represents the case of a line source at the leading edge of a vertical adiabatic surface,

so that no energy transfer occurs at the surface forx > 0.

For the case of uniform heat flux,n = 1

5andq

x = q, a constant Therefore, from

eq (7.30),

q= k−θ(0)N

gβN

4ν2

1/4

which gives

N = k[−θ q(0)]

4/5

4ν2

gβ

1/5

(7.38)

Therefore, for a given heat fluxq, which may be known, for example, from the electrical input into the surface, the temperature of the surface varies asx1/5 and

its magnitude may be determined as a function of the heat flux and fluid properties from eq (7.38) The parameter−θ(0) is obtained from a numerical solution of the

governing equations forn = 1

5 at the given value of Pr Some results obtained from Gebhart (1973) are shown in Table 7.1 asφPr,1

5

7.3.2 Inclined and Horizontal Surfaces

In many natural convection flows, the thermal input occurs at a surface that is itself curved or inclined with respect to the direction of the gravity field Consider, first, a flat surface at a small inclinationγ from the vertical Boundary layer approximations,

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⫺1.0

⫺0.8

⫺0.6

⫺0.4

⫺0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

n

Figure 7.5 Dependence of the local Nusselt number on the value of n for a power law surface

temperature distribution (From Sparrow and Gregg, 1958.)

similar to those for a vertical surface, may be made for this flow It can be shown that

ifx is taken along the surface and y normal to it, the continuity and energy equations,

eqs (7.15) and (7.17), respectively, remain unchanged and thex-direction momentum

equation becomes

u ∂u

∂x + v

∂u

∂y = gβ(T − T∞) cos γ +

1

ρ

∂

∂y

µ∂u

∂y



(7.39)

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Therefore, the problem is identical to that for flow over a vertical surface except thatg

is replaced byg cos γ in the buoyancy term Therefore, a replacement of g by g cos γ

in all the expressions derived earlier for a vertical surface would yield the correspond-ing results for an inclined surface This implies uscorrespond-ing Grxcosγ for Grxand assuming equal rates of heat transfer on the two sides of the surface This is strictly not the case since the buoyancy force is directed away from the surface at the top and toward the surface at the bottom, resulting in differences in boundary layer thicknesses and heat transfer rates However, this difference is neglected in this approximation

The preceding procedure for obtaining the heat transfer rate from an inclined surface was first suggested theoretically by Rich (1953), and his data are in good agreement with the values predicted The data obtained by Vliet (1969) for a uniform-flux heated surface in air and in water indicate the validity of this procedure up to inclination angles as large as 60° Additional experiments have confirmed that the replacement of g by g cos γ in the Grashof number is appropriate for inclination

angles up to around 45° and, to a close approximation, up to a maximum angle of 60° Detailed experimental results on this problem were obtained by Fujii and Imura (1972) They also discuss the separation of the boundary layer for the inclined surface facing upward

The natural convection flow over horizontal surfaces is of considerable importance

in a variety of applications, for instance, in the cooling of electronic systems and

in flows over the ground and water surfaces Rotem and Claassen (1969) obtained solutions to the boundary layer equations for flow over a semi-infinite isothermal horizontal surface Various values of Pr, including the extreme cases of very large and small Pr, were treated Experimental results indicated the existence of a boundary layer near the leading edge on the upper side of a heated horizontal surface These boundary layer flows merge near the middle of the surface to generate a wake or plume that rises above the surface Equations were presented for the power law case,

T w − T∞= Nx n, and solved for the isothermal case,n = 0 Pera and Gebhart (1972)

have considered flow over surfaces slightly inclined from the horizontal

For a semi-infinite horizontal surface with a single leading edge, as shown in Fig

7.6, the dynamic or motion pressurep ddrives the flow Physically, the upper side of a

heated surface heats up the fluid adjacent to it This fluid becomes lighter than the am-bient, if it expands on heating, and rises This results in a pressure difference, which causes a boundary layer flow over the surface near the leading edge Similar consid-erations apply for the lower side of a cooled surface The governing equations are the continuity and energy equations (7.15) and (7.17) and the momentum equations

u ∂u

∂x + v

∂u

∂y =

1

ρ

∂

∂y

µ∂u

∂y



−1 ρ

∂p d

gβ(T − T∞) = 1

ρ

∂p d

This problem may be solved by similarity analysis, as discussed earlier for vertical surfaces The similarity variables, given by Pera and Gebhart (1972), are