Natural Convection CHAPTER INTRODUCTION

Transcription

1 0 0 CHAPTER Natural Convection YOGESH JALURIA Mechanical and Aerospace Engineering Department Rutgers University New Brunswick, New Jersey. Introduction. Basic mechanisms and governing equations.. Governing equations.. Common approximations.. Dimensionless parameters. Laminar natural convection flow over flat surfaces.. Vertical surfaces.. Inclined and horizontal surfaces. External laminar natural convection flow in other circumstances.. Horizontal cylinder and sphere.. Vertical cylinder.. Transients.. Plumes, wakes, and other free boundary flows. Internal natural convection.. Rectangular enclosures.. Other configurations. Turbulent flow.. Transition from laminar flow to turbulent flow.. Turbulence. Empirical correlations.. Vertical flat surfaces.. Inclined and horizontal flat surfaces.. Cylinders and spheres.. Enclosures. Summary Nomenclature References. INTRODUCTION The convective mode of heat transfer involves fluid flow along with conduction, or diffusion, and is generally divided into two basic processes. If the motion of the [Fi [ -0 No [

2 NATURAL CONVECTION 0 0 fluid arises from an external agent, for instance, a fan, a blower, the wind, or the motion of the heated object itself, which imparts the pressure to drive the flow, the process is termed forced convection. If, on the other hand, no such externally induced flow exists and the flow arises naturally from the effect of a density difference, resulting from a temperature or concentration difference in a body force field such as gravity, the process is termed natural convection. The density difference gives rise to buoyancy forces due to which the flow is generated. A heated body cooling in ambient air generates such a flow in the region surrounding it. The buoyant flow arising from heat or material rejection to the atmosphere, heating and cooling of rooms and buildings, recirculating flow driven by temperature and salinity differences in oceans, and flows generated by fires are other examples of natural convection. There has been growing interest in buoyancy-induced flows and the associated heat and mass transfer over the past three decades, because of the importance of these flows in many different areas, such as cooling of electronic equipment, pollution, materials processing, energy systems, and safety in thermal processes. Several books, reviews, and conference proceedings may be consulted for detailed presentations on this subject. See, for instance, the books by Turner (), Jaluria (0), Kakaç et al. (), and Gebhart et al. (). The main difference between natural and forced convection lies in the mechanism by which flow is generated. In forced convection, externally imposed flow is generally known, whereas in natural convection it results from an interaction of the density difference with the gravitational (or some other body force) field and is therefore inevitably linked with and dependent on the temperature and/or concentration fields. Thus, the motion that arises is not known at the onset and has to be determined from a consideration of the heat and mass transfer process which are coupled with fluid flow mechanisms. Also, velocities and the pressure differences in natural convection are usually much smaller than those in forced convection. The preceding differences between natural and forced convection make the analytical and experimental study of processes involving natural convection much more complicated than those involving forced convection. Special techniques and methods have therefore been devised to study the former, with a view to providing information on the flow and on the heat and mass transfer rates. To understand the physical nature of natural convection transport, let us consider the heat transfer from a heated vertical surface placed in an extensive quiescent medium at a uniform temperature, as shown in Fig... If the plate surface temperature T w is greater than the ambient temperature T, the fluid adjacent to the vertical surface gets heated, becomes lighter (assuming that it expands on heating), and rises. Fluid from the neighboring areas moves in, due to the generated pressure differences, to take the place of this rising fluid. Most fluids expand on heating, resulting in a decrease in density as the temperature increases, a notable exception being water between 0 and C. If the vertical surface is initially at temperature T, and then, at a given instant, heat is turned on, say through an electric current, the flow undergoes a transient before the flow shown is achieved. It is the analysis and study of this time-dependent as well as steady flow that yields the desired information on the heat transfer rates, flow and temperature fields, and other relevant process variables. [ 0.0 Nor [

3 INTRODUCTION 0 0 Figure. system. Flow T w x T w > T Velocity boundary layer Entrainment y T, Body force (gravity field) Tw x Flow () a () b T, Entrainment Natural convection flow over a vertical surface, together with the coordinate The flow adjacent to a cooled surface is downward, as shown in Fig..b, provided that the fluid density decreases with an increase in temperature. Heat transfer from the vertical surface may be expressed in terms of the commonly used Newton s law of cooling, which gives the relationship between the heat transfer rate q and the temperature difference between the surface and the ambient as y T w < q = ha(t w T ) (.) where h is the average convective heat transfer coefficient and A is the total area of the vertical surface. The coefficient h depends on the flow configuration, fluid properties, dimensions of the heated surface, and generally also on the temperature difference, because of which the dependence of q on T w T is not linear. Since the fluid motion becomes zero at the surface due to the no-slip condition, which is generally assumed to apply, the heat transfer from the heated surface to the fluid in its immediate vicinity is by conduction. It is therefore given by Fourier s law as T [ * No * [

4 0 0 NATURAL CONVECTION ( ) T q = ka y 0 (.) Here the temperature gradient is evaluated at the surface, y = 0, in the fluid and k is the thermal conductivity of the fluid. From this equation it is obvious that the natural convection flow largely affects the temperature gradient at the surface, since the remaining parameters remain essentially unaltered. The analysis is therefore directed at determining this gradient, which in turn depends on the nature and characteristics of the flow, temperature field, and fluid properties. The heat transfer coefficient h represents an integrated value for the heat transfer rate over the entire surface, since, in general, the local value h x would vary with the vertical distance from the leading edge, x = 0, of the vertical surface. The local heat transfer coefficient h x is defined by the equation q = h x (T w T ) (.) where q is the rate of heat transfer per unit area per unit time at a location x, where the surface temperature difference is T w T, which may itself be a function of x. The average heat transfer coefficient h is obtained from eq. (.) through integration over the entire surface area. Both h and h x are generally given in terms of a nondimensional parameter called the Nusselt number Nu. Again, an overall (or average) value Nu, and a local value Nu x, may be defined as Nu = hl k Nu x = h xx k (.) where L is the height of the vertical surface and thus represents a characteristic dimension. The fluid far from the vertical surface is stationary, since an extensive medium is considered. The fluid next to the surface is also stationary, due to the no-slip condition. Therefore, flow exists in a layer adjacent to the surface, with zero vertical velocity on either side, as shown in Fig... A small normal velocity component does exist at the edge of this layer, due to entrainment into the flow. The temperature varies from T w to T. Therefore, the maximum vertical velocity occurs at some distance away from the surface. Its exact location and magnitude have to be determined through analysis or experimentation. The flow near the bottom or leading edge of the surface is laminar, as indicated by a well-ordered and well-layered flow, with no significant disturbance. However, as the flow proceeds vertically upward or downstream, the flow gets more and more disorderly and disturbed, because of flow instability, eventually becoming chaotic and random, a condition termed turbulent flow. The region between the laminar and turbulent flow regimes is termed the transition region. Its location and extent depend on several variables, such as the temperature of the surface, the fluid, and the nature and magnitude of external disturbances in the vicinity of the flow. Most of the processes encountered in nature are generally turbulent. However, flows in many [. Nor [

5 BASIC MECHANISMS AND GOVERNING EQUATIONS 0 0 Figure. surface. Velocity and temperature distributions in natural convection flow over a vertical industrial applications, such as those in electronic systems, are often in the laminar or transition regime. A determination of the regime of the flow and its effect on the flow parameters and heat transfer rates is therefore important. Natural convection flow may also arise in enclosed regions. This flow, which is generally termed internal natural convection, is different in many ways from the external natural convection considered in the preceding discussion on a vertical heated surface immersed in an extensive, quiescent, isothermal medium. Buoyancy-induced flows in rooms, transport in complete or partial enclosures containing electronic equipment, flows in enclosed water bodies, and flows in the liquid melts of solidifying materials are examples of internal natural convection. In this chapter we discuss both external and internal natural convection for a variety of flow configurations and circumstances.. BASIC MECHANISMS AND GOVERNING EQUATIONS.. Governing Equations The governing equations for a convective heat transfer process are obtained by considerations of mass and energy conservation and of the balance between the rate of momentum change and applied forces. These equations may be written, for constant viscosity µ and zero bulk viscosity, as (Gebhart et al., ) Dρ Dt = ρ t + V ρ = ρ V (.) [ 0. No [

6 0 0 NATURAL CONVECTION ρ DV Dt ρc p DT Dt ( ) V = ρ + V V = F p + µ V + µ ( V) (.) t = ρc p ( T t ) + V T = (k T)+ q + βt Dp Dt + µφ v (.) where V is the velocity vector, T the local temperature, t the time, F the body force per unit volume, c p the specific heat at constant pressure, p the static pressure, ρ the fluid density, β the coefficient of thermal expansion of the fluid, Φ v the viscous dissipation (which is the irreversible part of the energy transfer due to viscous forces), and q the energy generation per unit volume. The coefficient of thermal expansion β = (/ρ)( ρ/ T ) p, where the subscript p denotes constant pressure. For a perfect gas, β = /T, where T is the absolute temperature. The total, or particle, derivative D/Dt may be expressed in terms of local derivative as / t + V. As mentioned earlier, in natural convection flows, the basic driving force arises from the temperature (or concentration) field. The temperature variation causes a difference in density, which then results in a buoyancy force due to the presence of the body force field. For a gravitational field, the body force F = ρg, where g is the gravitational acceleration. Therefore, it is the variation of ρ with temperature that gives rise to the flow. The temperature field is linked with the flow, and all the preceding conservation equations are coupled through variation in the density ρ. Therefore, these equations have to be solved simultaneously to determine the velocity, pressure, and temperature distributions in space and in time. Due to this complexity in the analysis of the flow, several simplifying assumptions and approximations are generally made to solve natural convection flows. In the momentum equation, the local static pressure p may be broken down into two terms: one, p a, due to the hydrostatic pressure, and other other, p d, the dynamic pressure due to the motion of the fluid (i.e., p = p a + p d ). The former pressure component, coupled with the body force acting on the fluid, constitutes the buoyancy force that is driving mechanism for the flow. If ρ is the density of the fluid in the ambient medium, we may write the buoyancy term as F p = (ρg p a ) p d = (ρg ρ g) p d = (ρ ρ )g p d (.) If g is downward and the x direction is upward (i.e., g = ig, where i is the unit vector in the x direction and g is the magnitude of the gravitational acceleration, as is generally the case for vertical buoyant flows), then F p = (ρ ρ)gi p d (.) and the buoyancy term appears only in the x-direction momentum equation. Therefore, the resulting governing equations for natural convection are the continuity equation, eq. (.), the energy equation, eq. (.), and the momentum equation, which becomes ρ DV Dt = (ρ ρ )g p d + µ V + µ ( V) (.) [.0 Lon * [

7 BASIC MECHANISMS AND GOVERNING EQUATIONS Common Approximations The governing equations for natural convection flow are coupled, elliptic, partial differential equations and are therefore of considerable complexity. Another problem in obtaining a solution to these equations lies in the inevitable variation of the density ρ with temperature or concentration. Several approximations are generally made to simplify these equations. Two of the most important among these are the Boussinesq and the boundary layer approximations. The Boussinesq approximations involve two aspects. First, the density variation in the continuity equation is neglected. Thus, the continuity equation, eq. (.), becomes V = 0. Second, the density difference, which causes the flow, is approximated as a pure temperature or concentration effect (i.e., the effect of pressure on the density is neglected). In fact, the density difference is estimated for thermal buoyancy as ρ ρ = ρβ(t T ) (.) These approximations are employed very extensively for natural convection. An important condition for the validity of these approximations is that β(t T ) (Jaluria, 0). Therefore, the approximations are valid for small temperature differences if β is essentially unchanged. However, they are not valid near the density maximum of water at C, where β is zero and changes sign as the temperature varies across this value (Gebhart, ). Similarly, for large temperature differences encountered in fire and combustion systems, these approximations are generally not applicable. Another approximation made in the governing equations is the extensively employed boundary layer assumption. The basic concepts involved in using the boundary layer approximation in natural convection flows are very similar to those in forced flow. The main difference lies in the fact that the pressure in the region outside the boundary layer is hydrostatic instead of being the externally imposed pressure, as is the case in forced convection. The velocity outside the layer is only the entrainment velocity due to the motion pressure and is not an imposed free stream velocity. However, the basic treatment and analysis are quite similar. It is assumed that the flow and the energy, or mass, transfer, from which it arises, are restricted predominantly to a thin region close to the surface. Several experimental studies have corroborated this assumption. As a consequence, the gradients along the surface are assumed to be much smaller than those normal to it. The main consequences of the boundary layer approximations are that the downstream diffusion terms in the momentum and energy equations are neglected in comparison with the normal diffusion terms. The normal momentum balance is neglected since it is found to be of negligible importance compared to the downstream balance. Also, the velocity and thermal boundary layer thicknesses, δ and δ T, respectively, are given by the order-of-magnitude expressions ( ) δ L = O Gr / (.) [.0 Lon [

8 0 0 NATURAL CONVECTION δ T δ = O ( Pr / ) (.) where Gr is the Grashof number based on a characteristic length L and Pr is the Prandtl number. These are defined as Gr = gβl (T w T ) ν Pr = µc p k = ν α (.) where ν 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, 0; Gebhart et al., ) u x + v y = 0 (.) u u x + v u ρc p [ u T x + v T y y = gβ(t T ) + ρ y ] = ( k T y y ( µ u y ) ) + q + βtu p a x + µ (.) ( ) u (.) y where the last two terms in the energy equation are the dominant terms from pressure work and viscous dissipation effects. Here u 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 analysis 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 nondimensionalizing the governing equations and the boundary conditions, as outlined next... Dimensionless Parameters To generalize the natural convection transport processes, a study of the basic nondimensional parameters must be carried out. These parameters are important not only in simplifying the governing equations and the analysis, but also in guiding experiments 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 c is employed for the nondimensionalization of the velocity V, where V c is given by [. Lon * [

9 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 0 0 V c = [gβ(t w T )] / (.) The governing equations may be nondimensionalized by employing the following dimensionless variables (indicated by primes): Φ v = Φ v V = V V c p = p ρv c L Vc θ = T T T w T t = t t c = L ( ) = L (.) where t c is a characteristic time scale. The dimensionless equations are obtained as V = 0 (.0) ( v ) Sr + V v = eθ p t d + ( ) V (.) Gr ( θ ) Sr + V θ = t Pr Gr ( ) θ + (q ) + βt gβl ) (Sr p + V p c p t + gβl c p Gr Φ v (.) where e is the unit vector in the direction of the gravitational force. Here Sr = L/V c t c is the Strouhal number and q is nondimensionalized with ρc p (T w T )V c /L to yield the dimensionless value (q ). It is clear from the equations above that Gr replaces Re, which arises as the main dimensionless parameter in forced convection. Similarly, the Eckert number is replaced by gβl/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.. LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES.. Vertical Surfaces The classical problem of natural-convection heat transfer from an isothermal heated vertical surface, shown in Fig.., 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 [. Lon [

10 NATURAL CONVECTION 0 0 heat source is considered within the flow. Therefore, the problem is considerably simplified, although the complications due to the coupled partial differential equations remain. The governing differential equations may be obtained from eqs. (.) (.) 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 function ψ(x,y) is first defined so that it satisfies the continuity equation. Thus, we define ψ by the equations u = ψ y v = ψ x (.) Then the similarity variable η, the dimensionless stream function f, and the temperature θ are defined so as to convert the governing partial differential equations into ordinary differential equations. Gebhart et al. () have presented a general approach 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 ( ) / ( ) / Grx Grx ψ = νf(η) θ = T T (.) x T w T where The boundary conditions are: Gr x = gβx (T w T ) ν (.) at y = 0: u = v = 0,T = T w ; as y : u 0,T T (.) These must also be written in terms of the similarity variables in order to obtain the solution. Note that the velocity component v 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 transformations as f + ff (f ) + θ = 0 (.) θ Pr + f θ = 0 (.) where the primes here indicate differentiation of f(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 at η = 0: f = f = θ = 0; as η : f 0, θ 0 [ 0.0 Cus [

11 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 0 0 which may be written more concisely as f(0) = f (0) = θ(0) = f ( ) = θ( ) = 0 (.) 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 () gave results for various values of the Prandtl number Pr, employing approximate methods. Ostrach () numerically obtained the solution for the Pr range 0.0 to 00. The velocity and temperature profiles thus obtained are shown in Figs.. and.. An increase in Pr is found to cause a decrease in the thermal boundary 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 velocity is also found to decrease and the velocity 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 higher η 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 (), Bejan (), and Oosthuizen and Naylor (). The heat transfer from the heated surface may be obtained as q x = k ( T y ) = k(t w T ) ( ) / ( ) Grx θ 0 x η 0 ( ) / Grx (.) = [ θ (0) ] k(t w T ) x The local Nusselt number Nu x is given by Nu x = h xx q x x = k T w T k We have for an isothermal surface Nu x = [ θ (0) ] ( Gr x ) / = θ (0) Gr / x = φ(pr)gr / x (.) where φ(pr) = [ θ (0) ] /. Therefore, the local surface heat transfer coefficient h x varies as [ ] h x = Bx / where B = k[ θ (0)] gβ(tw T ) / ν [.0 Cu * [

12 NATURAL CONVECTION 0 0 ν x ( ) = ux Gr f Pr = f ( ) y Gr / = x ( x ( Pr = Figure. Calculated velocity distributions in the boundary layer for flow over an isothermal vertical surface. (From Ostrach,.) [ * Nor * [

13 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 0 0 w ( ) = T T T T ( ) Pr = 0.0 y Gr / = x ( x ( Pr = Figure. Calculated temperature distributions in the boundary layer for flow over an isothermal vertical surface. (From Ostrach,.) 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 Therefore, h = L L 0 h x dx = B L L/ [ 0. No * [

14 0 0 NATURAL CONVECTION Nu = [ θ (0) ] Gr / = φ(pr) Gr/ = Nu L (.) 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.. The significance of n and the uniform heat flux data in the table is discussed later. An approximate curve fit to the numerical results for φ(pr) has been given by Oosthuizen and Naylor () as ( 0.Pr / ) / φ(pr) =. +.Pr / (.) +.Pr 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 positive q value 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 () to be the power law and exponential distributions, given as T w T = Nx n and T w T = Me mx (.) TABLE. Computed Values of the Parameter φ(pr)for a Vertical Heated Surface φ(pr) φ ( Pr, ) (Isothermal), (Uniform Heat Flux), Pr n = 0 n = Pr / 0.Pr / Pr / 0.Pr / Source: Gebhart (). [. Nor [

15 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 0 0 where N,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 for n = 0. From the expression for q x, eq. (.), it can be shown that q x varies with x as x (n )/. Therefore, a uniform heat flux condition, q x = constant, arises for n =. It can also be shown that physically realistic solutions are obtained for n< (Sparrow and Gregg, ; Jaluria, 0). The governing equations are obtained for the power law case as f + (n + )ff (n + )(f ) + θ = 0 (.) θ Pr + (n + )f θ nf θ = 0 (.) The local Nusselt number Nu x is obtained as Nu x Gr / = θ (0) = φ(pr,n) (.) x The function Nu x /Grx / is plotted against n in Fig... For n<, the function is found to be negative, indicating the physically unrealistic circumstance of heat transfer to the surface for T w >T. The surface is adiabatic for n =, 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 for x>0. For the case of uniform heat flux, n = and q x = q, a constant. Therefore, from eq. (.), q = k [ θ (0) ] ( ) gβn / N ν which gives { q } / ( ν ) / N = k[ θ (.) (0)] gβ Therefore, for a given heat flux q, which may be known, for example, from the electrical input into the surface, the temperature of the surface varies as x / and its magnitude may be determined as a function of the heat flux and fluid properties from eq. (.). The parameter θ (0) is obtained from a numerical solution of the governing equations for n = at the given value of Pr. Some results obtained from Gebhart () are shown in Table. as φ ( Pr, )... 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 inclination γ from the vertical. Boundary layer approximations, [ 0. No [

16 0 NATURAL CONVECTION 0 0 / Nu /(Gr /) x x n Pr =.0 Pr = 0. Figure. Dependence of the local Nusselt number on the value of n for a power law surface temperature distribution. (From Sparrow and Gregg,.) similar to those for a vertical surface, may be made for this flow. It can be shown that if x is taken along the surface and y normal to it, the continuity and energy equations, eqs. (.) and (.), respectively, remain unchanged and the x-direction momentum equation becomes u u x + v u y = gβ(t T ) cos γ + ρ y ( µ u ) y (.) [0 0.0 Nor * [0

17 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 0 0 Therefore, the problem is identical to that for flow over a vertical surface except that g is replaced by g 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 corresponding results for an inclined surface. This implies using Gr x cos γ for Gr x and 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 (), and his data are in good agreement with the values predicted. The data obtained by Vliet () for a uniformflux heated surface in air and in water indicate the validity of this procedure up to inclination angles as large as 0. 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 and, to a close approximation, up to a maximum angle of 0. Detailed experimental results on this problem were obtained by Fujii and Imura (). 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 () 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 () 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.., the dynamic or motion pressure p d drives the flow. Physically, the upper side of a heated surface heats up the fluid adjacent to it. This fluid becomes lighter than the ambient, 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 considerations apply for the lower side of a cooled surface. The governing equations are the continuity and energy equations (.) and (.) and the momentum equations u u x + v u y = ρ y gβ(t T ) = p d ρ y ( µ u ) p d y ρ x (.0) (.) This problem may be solved by similarity analysis, as discussed earlier for vertical surfaces. The similarity variables, given by Pera and Gebhart (), are [. No [

18 NATURAL CONVECTION 0 0 Figure. Natural convection boundary layer flow over a semi-infinite horizontal surface, with the heated surface facing upward. η = y ( ) / ( ) / Grx Grx ψ = νf(η) (.) x Figure. shows the computed velocity and temperature profiles for flow over a heated horizontal surface facing upward or a cooled surface facing downward. For a heated surface facing downward or a cooled surface facing upward, a boundary layer type of flow is not obtained for a fluid that expands on heating. This is because the fluid does not flow away from the surface due to buoyancy. The local Nusselt number for horizontal surfaces is given by Pera and Gebhart () for both the isothermal and the uniform-heat-flux surface conditions. The Nusselt number was found to be approximately proportional to Pr / over the Pr range 0. to 0. The expression given for an isothermal surface is Nu x = h xx k and that for a uniform-flux surface, Nu x,q,is Nu x,q = h xx k = 0.Gr / x Pr / (.) = 0.0Gr / x Pr / (.) It can be shown by integrating over the surface that for the isothermal surface, the average Nusselt number is times the value of the local Nusselt number at x = L. Therefore, the natural convection heat transfer from inclined surfaces can be treated in terms of small inclinations from the vertical and horizontal positions, detailed results on which are available. For intermediate values of γ, an interpolation between these two regimes may be used to determine the resulting heat transfer rate. Numerical methods, such as the finite difference method, can also be used to solve the governing equations to obtain the flow and temperature distributions and the heat transfer rate. This regime has not received as much attention as the horizontal and [ 0. Nor [

19 EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 0 0 Figure. Calculated (a) velocity and (b) temperature distribution in natural convection boundary layer flow over a horizontal surface with a uniform heat flux. (From Pera and Gebhart,.) vertical surfaces, although some numerical and experimental results are available such as those of Fujii and Imura ().. EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES.. Horizontal Cylinder and Sphere Much of the information on laminar natural convection over heated surfaces, discussed in Section., has been obtained through similarity analysis. However, neither the horizontal cylindrical nor the spherical configuration gives rise to similarity, and therefore several other methods have been employed for obtaining a solution to the governing equations. Among the earliest detailed studies was that by Merk and Prins ( ), who employed integral methods with the velocity and thermal boundary layer thicknesses assumed to be equal. The variation of the local Nusselt number with φ, the angular position from the lower stagnation point φ = 0, is shown in Fig.. for a horizontal cylinder and also for a sphere. The local Nusselt number Nu φ decreases downstream due to the increase in the boundary layer thickness, which is predicted to be infinite at φ = 0, resulting in a zero value for Nu φ there. However, Merk and Prins ( ) indicated the inapplicability of the analysis for φ due to boundary layer separation and realignment into a plume flow near the top. [ 0. No [

20 NATURAL CONVECTION 0 0 / Nu /(Gr Pr) / Nu /(Gr Pr) (deg) () a 0 (deg) () b Pr = Pr = Figure. Variation of the local Nusselt number with downstream angular position φ for (a) a horizontal cylinder and (b) a sphere. (From Merk and Prins,.) The mean value of the Nusselt number Nu is given by Merk and Prins ( ) for a horizontal isothermal cylinder as 0 0 Nu = hd k = C(Pr)(Gr Pr)/ (.) where Nu and Gr are based on the diameter D. The constant C(Pr) was calculated as 0., 0., 0.0, 0., and 0. for Pr values of 0.,.0,.0, 0.0, and, respectively. [ 0. Lon [

21 EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 0 0 The preceding expression is also suggested for spheres by Merk and Prins ( ), with C(Pr) given for the Pr values of 0.,.0,.0, 0.0, and as 0., 0., 0., 0., and 0., respectively. There are many other analytical and experimental studies of the natural convection flow over spheres. Since this configuration is of particular interest in chemical processes, it has also been studied in detail for mass transfer. Chiang et al. () solved the governing equations, using a series method, and presented heat transfer results. Trends similar to those discussed earlier were obtained. A considerable amount of experimental work has been done on the heat transfer from spheres. Amato and Tien () have discussed such studies and have given the heat transfer correlation as Nu = + 0.(Gr Pr) / (.) where the constant in the expression can be shown analytically to apply for pure conduction. Additional correlations for transport from horizontal cylinders and spheres are given later. Work has also been done on the separation of the flow to form a wake near the top of the body. This realignment of the flow can significantly affect the heat transfer rate in the vicinity of the top of a cylinder or a sphere (Jaluria and Gebhart, )... Vertical Cylinder Natural convection flow over vertical cylinders is another important problem, being relevant to many practical applications, such as flow over tubes and rods (as in nuclear reactors), over cylindrical heating elements, and over various closed bodies (including the human body) that can be approximated as a vertical cylinder. For large values of D/L, where D is the diameter of the cylinder and L its height, the flow can be approximated as that over a flat plate, since the boundary layer thickness is small compared to the diameter of the cylinder. As a result, the governing equations are the same as those for a flat plate. However, since this result is based on the boundary layer thickness, which in turn depends on the Grashof number, the deviation of the results obtained for a vertical cylinder from those for a flat plate must be given in terms of D/L and the Grashof number. Sparrow and Gregg () obtained the following criterion for a difference in heat transfer from a vertical cylinder of less than % from the flat plate solution, for Pr values of 0. and.0: D L Gr / (.) where Gr is the Grashof number based on L. When D/L is not large enough to ignore the effects of curvature, the relevant governing equations must be solved. Sparrow and Gregg () employed similarity methods for obtaining a solution to these equations. Minkowycz and Sparrow () obtained the solution using the local nonsimilarity method. Cebeci () gave results on vertical slender cylinders. LeFevre and Ede () employed an integral method to solve the governing equations and gave the following expression for the Nusselt number Nu: [ 0. Lon * [

22 NATURAL CONVECTION 0 0 Nu = hl k = [ Gr Pr ] / ( + Pr)L + (0 + Pr) ( + Pr)D (.) where both Nu and Gr are based on the height L of the cylinder. Other studies on vertical axisymmetric bodies are reviewed by Gebhart et al. ()... Transients We have so far considered steady natural convection flows in which the velocity and temperature fields do not vary with time. However, time dependence is important in many practical circumstances (Jaluria, ). For instance, the change in the thermal condition that generates the natural convection flow could be a sudden or a periodic one, leading to a time-dependent variation in the flow. The startup and shutdown of thermal systems, such as furnaces, ovens, and nuclear reactors, involves a consideration of time-dependent or unsteady natural convection if buoyancy effects are significant. If the heat input at a surface is suddenly changed from zero to a specific value, the steady natural convection flow is eventually obtained following a transient process. As soon as the heat is turned on, the surface starts heating up, this change being essentially a step variation if the thermal capacity of the body is very small. In response to this sudden change, the fluid adjacent to the surface gets heated, becomes buoyant, and rises, if the fluid expands on heating. However, the flow at a given location is initially unaffected by flow at other portions of the surface. This implies that the fluid element behaves as isolated, and the heat transfer mechanisms are initially not influenced by the fluid motion. Consequently, the initial transport mechanism is predominantly conduction and can be approximated as a one-dimensional conduction problem up to the leading edge effect, which results from flow originating at the leading edge and which propagates downstream along the flow, is felt at a given location x. The heat transfer rates due to pure conduction being much smaller than those due to convection, it is to be expected that for a step change in the heat flux input, there may initially be an overshoot in the temperature above the steady-state value. Similarly, for a step change in temperature, a lower heat flux is expected initially, ultimately approaching the steady-state value, as the flow itself progresses through a transient regime to steady-state conditions. The preceding discussion implies that at the initial stages of the transient, the solution for a step change in the surface temperature, or in the heat flux, is independent of the vertical location and is of the form obtained for semi-infinite conduction solutions. Employing Laplace transforms for a step change in the heat flux, the solution is obtained as (Ozisik, ) θ = q [ αt exp( η ] ) η erfc(η) (.) k π where η = y/ αt,α being the thermal diffusivity of the fluid. Here, erfc(η) is the conjugate of the error function and q is the constant heat flux input imposed at [ -0. Nor [

23 EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 0 0 time t = 0, starting from a no-flow, zero-heat-input condition. The temperature θ is simply the physical temperature excess over the initial temperature T. The heat transfer coefficient h is obtained from the preceding temperature expression, by using Fourier s law, as h = q = k π (.0) [ θ] 0 αt Similarly, for a step change in the surface temperature, the solution is (Ozisik, ) T T = θ = erf(η) (.) T w T The velocity profile is obtained by substituting the preceding temperature solution into the momentum equation and solving the resulting equation by Laplace transforms to obtain u(y). Numerical solutions to the governing time-dependent boundary layer equations have been obtained by Hellums and Churchill () for a vertical surface subjected to a step change in the surface temperature. The results converge to the steady-state solution at large time and show a minimum in the local Nusselt number during the transient, as shown in Fig... An integral method for analyzing unsteady natural convection has also been developed for time-dependent heat input and for finite thermal capacity of the surface element. This work has been summarized by Gebhart (, ) and is based on the analytical and experimental work of Gebhart and co-workers. This analysis is particularly suited to practical problems, since it considers the thermal capacity of the bounding material and determines the temperature variation with time over the entire transient regime. x x / Nu /Gr Steady state Conduction solution 0 0 νt x Gr x Figure. Variation of the heat transfer rate with time for a step change in the surface temperature of a vertical plate. (From Hellums and Churchill,.) [ 0.0 No [

24 NATURAL CONVECTION 0 0 Churchill () has given a correlation for transient natural convection from a heated vertical plate, subjected to a step change in the heat flux. The thermal capacity of the plate is taken as negligible, and the local Nusselt number is given as ( [ ] n πx ) n/ { } n/ Ra x / Nux,q = + [ αt ] (.) + (0./Pr) / / where Ra x = gβ(t w T )x να (.) Employing the available experimental information, the appropriate value of n is given as. With this value of n, the preceding correlation was found to give Nusselt number values quite close to the experimental results. A temperature overshoot was not considered, since the experimental studies of Gebhart () showed no significant overshoot. For a step change in surface temperature, Churchill and Usagi () have obtained an empirical correlation approximating the entire transient domain... Plumes, Wakes, and Other Free Boundary Flows In the preceding sections we have considered external natural convection adjacent to heated or cooled surfaces. However, there are many important natural convection flows that, although generated by a heated or cooled surface, move beyond the buoyancy input so that they occur without the presence of a solid boundary. Figure. shows the sketches of a few common flows, which are often termed free boundary flows. Many of these flows are of interest in nature and in pollution and are usually turbulent (Gebhart et al., ). Thermal plumes which are assumed to arise from heat input at point or horizontal line sources represent the wakes above heated bodies. The former circumstance is an axisymmetric flow and is generated by a heated body such as a sphere, whereas the latter case is a two-dimensional flow generated by a long, thin, heat source such as an electric heater. Figure. shows a sketch of the flow in a two-dimensional plume, ( a ) Plume ( b ) Thermal Figure. ( c ) Starting plume Common free boundary flows. ( d) Jet Nozzle [ 0. Cus [

25 EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 0 0 Figure. (a) Sketch and coordinate system for a two-dimensional thermal plume arising from a horizontal line source. Also shown are the calculated (b) velocity and (c) temperature distributions. (From Gebhart et al., 0.) along with a coordinate system which is similar to that for flow over a heated vertical surface. Using the nomenclature and analysis given earlier for a vertical surface, it can easily be seen that n = because the centerline, y = 0, is adiabatic due to symmetry. Also, the vertical velocity is not zero there but a maximum, since a noshear condition applies there rather than the no-slip condition. Then the similarity variables given earlier in eq. (.) may be used with T w T = Nx /, where T w is the centerline temperature. The governing equations are then obtained from eqs. (.) and (.) as f + ff (f ) + θ = 0 (.) θ Pr + (f θ + f θ) = 0 (.) [. Cu * [

26 0 NATURAL CONVECTION 0 0 The boundary conditions for a two-dimensional plume are θ (0) = f(0) = f (0) = θ(0) = f ( ) = 0 (.) The first and third conditions arise from symmetry at y = 0, or η = 0. The overall energy balance can also be written in terms of the total convected energy in the boundary layer q c as q c = ρc p u(t T )dy ( ) gβn / = µc p N x (n+)/ f (η)θ(η)dη (.) ν Therefore, the x dependence drops out for n = and q c represents the total energy input Q per unit length of the line source. Then the constant N in the centerline temperature distribution, T w T = Nx /, is obtained from eq. (.) as ( ) / Q N = gβρ µ cp I (.) where I is the integral I = f (η)θ(η)dη (.) This integral I can be determined numerically by solving the governing similarity equations and then evaluating the integral. Values of I at several Prandtl numbers are given by Gebhart et al. (0). For instance, the values of I calculated at Pr = 0.,.0,., and.0 are given as.,.0, 0.0, and 0., respectively. Figure. also presents some calculated velocity and temperature profiles in a two-dimensional plume from Gebhart et al. (0). These results can be used to calculate the velocity and thermal boundary layer thicknesses, which can be shown to vary as x /, and the centerline velocity, which can be shown to increase with x as x /. The centerline temperature, which decays with x as x /, can be calculated by obtaining the value of N from eq. (.) for a given Pr and heat input Q. Note that this analysis applies for a line source on a vertical adiabatic surface as well, since q c is constant in this case, too, resulting in n = (Jaluria and Gebhart, ). Therefore, the governing equations are eqs. (.) and (.). However, the boundary condition f (0) = 0 is replaced by f (0) = 0 because of the no-slip condition at the wall. Similarly, a laminar axisymmetric plume can be analyzed to yield the temperature and velocity distributions (Jaluria, a). The wake rising above a finite heated body is expected finally to approach the conditions of an axisymmetric plume far downstream of the heat input as the effect of the size of the source diminishes. However, as mentioned earlier, most of these flows are turbulent in nature and in most practical applications. Simple integral analyses have been carried out, along [0. Nor [0

27 INTERNAL NATURAL CONVECTION 0 0 with appropriate experimentation, to understand and characterize these flows (Turner, ). Detailed numerical studies have also been carried out on a variety of free boundary flows to provide results that are of particular interest in pollution, fires, and environmental processes.. INTERNAL NATURAL CONVECTION In the preceding sections we have considered largely external natural convection in which the ambient medium away from the flow is extensive and stationary. However, there are many natural convection flows that occur within enclosed regions, such as flows in rooms and buildings, cooling towers, solar ponds, and furnaces. The flow domain may be completely enclosed by solid boundaries or may be a partial enclosure with openings through which exchange with the ambient occurs. There has been growing interest and research activity in buoyancy-induced flows arising in partial or complete enclosures. Much of this interest has arisen because of applications such as cooling of electronic circuitry (Jaluria, b; Incropera, ), building fires (Emmons,, 0), materials processing (Jaluria, 00), geothermal energy extraction (Torrance, ), and environmental processes. The basic mechanisms and heat transfer results in internal natural convection have been reviewed by several researchers, such as Yang () and Ostrach (). Some of the important basic considerations are presented here... Rectangular Enclosures The two-dimensional natural convection flow in a rectangular enclosure, with the two vertical walls at different temperatures and the horizontal boundaries taken as adiabatic or at a temperature varying linearly between those of the vertical boundaries, has been thoroughly investigated over the past three decades. Figure.a shows a typical vertical enclosure with the two vertical walls at temperatures T h and T c and the horizontal surfaces being taken as insulated. The dimensionless governing equations may be written as Pr V ω = ω Ra θ Y (.0) V θ = θ (.) where the vorticity ω = ψ, θ = (T T c )/(T h T c ), Y = y/d, and the Rayleigh number Ra = Gr Pr. The width d of the enclosure and the temperature difference T h T c are taken as characteristic quantities for nondimensionalization of the variables. The velocity is nondimensionalized by α/d here. This problem has been investigated numerically and experimentally for a wide range of Rayleigh and Prandtl numbers and of the aspect ratio A = H/d. Figure.b shows the calculated isotherms at a moderate value of Pr. A recirculating flow arises and distorts the temperature field resulting from pure conduction. [ - No [