8. Natural convection in single-phase fluids and during film condensation
John Richard Thome
25 avril 2008
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 1 / 37 8.1 Scope
The remaining convection mechanisms that we deal with are to a large degree gravity-driven. Unlike forced convection, in which the driving force is external to the fluid, these so-called natural convection processes are driven by body forces exerted directly within the fluid as the result of heating or cooling. Two such mechanisms that are rather alike are :
Natural convection. When we speak of natural convection without any qualifying words, we mean natural convection in a single-phase fluid.
Film condensation. This natural convection process has much in common with single-phase natural convection.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 2 / 37 8.2 The nature of the problems of film condensation and of natural convection
Description
The natural convection problem is sketched in its simplest form on the left-hand side of Fig. 8.1. Here we see a vertical isothermal plate that cools the fluid adjacent to it. The cooled fluid sinks downward to form a b.l. The figure would be inverted if the plate were warmer than the fluid next to it. Then the fluid would buoy upward.
On the right-hand side of Fig. 8.1 is the corresponding film condensation problem in its simplest form. An isothermal vertical plate cools an adjacent vapor, which condenses and forms a liquid film on the wall.1 The film is normally very thin and it flows off, rather like a b.l., as the figure suggests. While natural convection can carry fluid either upward or downward, a condensate film can only move downward. The temperature in the film rises from Tw at the cool wall to Tsat at the outer edge of the film.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 3 / 37 8.2 The nature of the problems of film condensation and of natural convection
Figure 8.1
The convective boundary layers for natural convection and film condensation. In both sketches, but particularly in that for film condensation, the y-coordinate has been stretched.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 4 / 37 8.2 The nature of the problems of film condensation and of natural convection
In both problems, but particularly in film condensation, the b.l. and the film are normally thin enough to accommodate the b.l. assumptions.
A second idiosyncrasy of both problems is that δ and δt are closely related. In the condensing film they are equal, since the edge of the condensate film forms the edge of both b.l.s. In natural convection, δ and δt are approximately equal when Pr is on the order of unity or less, because all cooled (or heated) fluid must buoy downward (or upward). When Pr is large, the cooled (or heated) fluid will fall (or rise) and, although it is all very close to the wall, this fluid, with its high viscosity, will also drag unheated liquid with it.
∼ In this case, δ can exceed δt . We deal with cases for which δ =δt in the subsequent analysis.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 5 / 37 8.2 The nature of the problems of film condensation and of natural convection
Governing equations
To describe laminar film condensation and laminar natural convection, we must add a gravity term to the momentum equation. Equation (6.13) can be written as
∂u ∂u −1 dp ∂2u u + v = + v ∂x ∂y ρ dx ∂y 2
∂p ∼ ∼ where ∂x = dp/dx in the b.l. and where µ = constant. The component of gravity in the x-direction therefore enters the momentum balance as (+g).
∂u ∂u −1 dp ∂2u u + v = + g + v (8.1) ∂x ∂y ρ dx ∂y 2
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 6 / 37 8.2 The nature of the problems of film condensation and of natural convection
In the two problems at hand, the pressure gradient is the hydrostatic gradient outside the b.l. Thus,
dp = ρ g natural convection (8.2) dx ∞
dp = ρ g film condensation (8.2) dx g
where ρ∞ is the density of the undisturbed fluid and ρg (and ρf below) are the saturated vapor and liquid densities.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 7 / 37 8.2 The nature of the problems of film condensation and of natural convection
Equation (8.1) then becomes
∂u ∂u ρ ∂2u u + v = 1 − ∞ g + v for natural convection (8.3) ∂x ∂y ρ ∂y 2
∂u ∂u ρ ∂2u u + v = 1 − g g + v for film condensation (8.4) ∂x ∂y ρ ∂y 2
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 8 / 37 8.2 The nature of the problems of film condensation and of natural convection
Two boundary conditions, which apply to both problems, are
u(y = 0) = 0 the no − slip condition (8.5a)
v(y = 0) = 0 no flow into the wall (8.5a) The third b.c. is different for the film condensation and natural convection problems : ∂u | = 0 condensation : no shear at the edge of the film (8.5b) ∂y y=δ
u(y = δ) = 0 natural convection : undisturbed fluid outside the b.l. (8.5b) The energy equation for either of the two cases is eqn.
∂T ∂T ∂2T u + v = α ∂x ∂y ∂y 2
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 9 / 37 8.3 Laminar natural convection on a vertical isothermal surface
Figure 8.2 Natural convection from a vertical heated plate.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 10 / 37 8.3 Laminar natural convection on a vertical isothermal surface
Prediction of h in natural convection on a vertical surface
The analysis of natural convection using an integral method was done independently by Squire and by Eckert. We shall refer to this important development as the Squire-Eckert formulation. The analysis begins with the integrated momentum and energy equations. We assume δ = δt and integrate both equations to the same value of δ :
Z δ Z δ d 2 ∂u (u − uu∞)dy = −v + gβ (T − T∞)dy (8.6) dx 0 ∂y 0 and equation 6.47
Z δ d qw ∂T u(T − T∞)dy = = −α |y=0 dx 0 ρcp ∂y
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 11 / 37 8.3 Laminar natural convection on a vertical isothermal surface
The integrated momentum equation is the same as eqn. (6.24) except that it includes the buoyancy term, which was added to the differential momentum equation using the following equation
∂u ∂u ∂2u u + v = −gβ(T − T ) + v (8.7) ∂x ∂y ∞ ∂y 2
Since the temperature profile has a fairly simple shape, a simple quadratic expression can be used :
T − T y y 2 ∞ = a + b + c (8.8) Tw − T∞ δ δ
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 12 / 37 8.3 Laminar natural convection on a vertical isothermal surface
Notice that the thermal boundary layer thickness, δt , is assumed equal to δ in eqn. (8.7). This would seemingly limit the results to Prandtl numbers not too much larger than unity. Actually, the analysis will also prove useful for large Prandtl numbers because the velocity profile exerts diminishing influence on the temperature profile as Pr increases. We require the following things to be true of this profile :
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 13 / 37 8.3 Laminar natural convection on a vertical isothermal surface
So a = 1, b = -2, and c = 1. This gives the following dimensionless temperature profile : T − T y y 2 y 2 ∞ = 1 − 2 + = 1 − (8.9) Tw − T∞ δ δ δ
We anticipate a somewhat complicated velocity profile (recall Fig. 8.1) and seek to represent it with a cubic function :
y y 2 y 3 u = u (x) + c + d (8.10) c δ δ δ
where, since there is no obvious characteristic velocity in the problem, we write uc as an as-yet-unknown function. (uc will have to increase with x, since u must increase with x.) We know three things about u :
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 14 / 37 8.3 Laminar natural convection on a vertical isothermal surface
u(y=0) = 0
u(y = δ) = 0 or u = 0 = (1 + c + d) uc
∂u ∂y |y=δ = 0
These give c = -2 and d = 1, so
u y y 2 = 1 − (8.11) uc (x) δ δ
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 15 / 37 8.3 Laminar natural convection on a vertical isothermal surface
We could also have written the momentum equation (8.7) at the wall, where u = v = 0, and created a fourth condition :
∂2u gβ(T − T ) | = − w ∞ ∂y 2 y=0 v
δ2 and then we could have evaluated uc (x) as βg|Tw − T∞| 4v . A correct expression for uc will eventually depend upon these variables, but we will not attempt to make uc fit this particular condition. Doing so would yield two equations, (8.6) and (6.47), in a single unknown, δ(x). It would be impossible to satisfy both of them. Instead, we shall allow the velocity profile to violate this condition slightly and write βg|T − T | u (x) = C w ∞ δ2(x)(8.12) c 1 v Then we shall solve the two integrated conservation equations for the two unknowns, C1 (which should =∼ 1/4) and δ(x).
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 16 / 37 8.3 Laminar natural convection on a vertical isothermal surface
The dimensionless temperature and velocity profiles are plotted in Fig. 8.3.
With them are included Schmidt and Beckmanns exact calculation for air (Pr = 0.7). Notice that the integral approximation to the temperature profile is better than the approximation to the velocity profile.
That is fortunate, since the temperature profile exerts the major influence in the heat transfer solution.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 17 / 37 8.3 Laminar natural convection on a vertical isothermal surface
Figure 8.3 The temperature and velocity profiles for air (Pr = 0.7) in a laminar convection b.l.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 18 / 37 8.3 Laminar natural convection on a vertical isothermal surface
This is the Squire-Eckert result for the local heat transfer from a vertical isothermal wall during laminar natural convection :
Pr 1/4 Nu = 0.508Ra (8.13) x x 0.952 + Pr
with
gβ∆TL3 Ra = Gr Pr = Rayleigh number x x αv
where Grx is the the Grashof number and β equals 1/T∞.
It applies for either Tw >T∞ or Tw John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 19 / 37 8.3 Laminar natural convection on a vertical isothermal surface The average heat transfer coefficient can be obtained from R L q(x)dx R L h(x)dx h = 0 = 0 L∆T L Thus, Pr 1/4 Nu = 0.678Ra1/4 (8.14) L L 0.952 + Pr All properties in eqn. (8.14) and the preceding equations should be evaluated at T = (Tw + T∞)/2 except in gases, where β should be evaluated at T∞. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 20 / 37 8.3 Laminar natural convection on a vertical isothermal surface Variable-properties problem Sparrow and Gregg provide an extended discussion of the influence of physical property variations on predicted values of Nu. They found that while β for gases should be evaluated at T∞, all other properties should be evaluated at Tr , where Tr = Tw − C(Tw − T∞)(8.15) and where C=0.38 for gases. It has also been shown by Barrow and Sitharamarao that when β∆T is no longer «1, the Squire-Eckert formula should be corrected as follows : 3 1/4 Nu = Nu 1 + β∆T + o(β∆T )2 (8.16) sq−Ek 5 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 21 / 37 8.3 Laminar natural convection on a vertical isothermal surface Note on the validity of the boundary layer approximations The boundary layer approximations are sometimes put to a rather severe test in natural convection problems. Thermal b.l. thicknesses are often fairly large, and the usual analyses that take the b.l. to be thin can be significantly in error. This is particularly true as Gr becomes small. Figure 8.5 includes three pictures that illustrate this. These pictures are interferograms (or in the case of Fig. 8.5c, data deduced from interferograms). An interferogram is a photograph made in a kind of lighting that causes regions of uniform density to appear as alternating light and dark bands. Figure 8.5a was made at the University of Kentucky by G.S. Wang and R. Eichhorn. The Grashof number based on the radius of the leading edge is 2250 in this case. This is low enough to result in a b.l. that is larger than the radius near the leading edge. Figure 8.5b and c are from Krauss classic study of natural convection visualization methods. Figure 8.5c shows that, at Gr = 585, the b.l. assumptions are quite unreasonable since the cylinder is small in comparison with the large region of thermal disturbance. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 22 / 37 8.3 Laminar natural convection on a vertical isothermal surface Figure 8.4 The thickening of the b.l. during natural convection at low Gr, as illustrated by interferograms made on two-dimensional bodies. (The dark lines in the pictures are isotherms.) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 23 / 37 8.4 Natural convection in other situations Natural convection from horizontal isothermal cylinders Churchill and Chu provide yet another comprehensive correlation of existing data. Horizontal cylinder data from a variety of sources, over about 24 orders of magnitude of the Rayleigh number based on the diameter, RaD, are shown in Fig. 8.5. The equation that correlates them is 0.518Ra1/4 Nu = 0.36 + D (8.17) D [1 + (0.559/Pr)9/16]4/9 9 When RaD is greater than 10 , the flow becomes turbulent. The following equation is a little more complex, but it gives comparable accuracy over a larger range : " #2 Ra 1/6 Nu = 0.6 + 0.387 D (8.18) D [1 + (0.559/Pr)9/16]16/9 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 24 / 37 8.4 Natural convection in other situations Figure 8.5 The data of many investigators for heat transfer from isothermal horizontal cylinders during natural convection, as correlated by Churchill and Chu. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 25 / 37 8.4 Natural convection in other situations Natural convection from vertical cylinders The heat transfer from the wall of a cylinder with its axis running vertically is the same as that from a vertical plate, so long as the thermal b.l. is thin. However, if the b.l. is thick, as is indicated in Fig. 8.6, heat transfer will be enhanced by the curvature of the thermal b.l. This correction was first considered some years ago by Sparrow and Gregg, and the analysis was subsequently extended with the help of more powerful numerical methods by Cebeci. Figure 8.6 includes the corrections to the vertical plate results that were calculated for many Prandtl numbers by Cebeci. The left-hand graph gives a correction that must be multiplied by the local flat-plate Nusselt number to get the vertical cylinder result. Notice that the correction increases when the Grashof number decreases. The right-hand curve gives a similar correction for the overall Nusselt number on a cylinder of height L. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 26 / 37 8.4 Natural convection in other situations Figure 8.6 Corrections for h and h on vertical isothermal plates to make them apply to vertical isothermal cylinders. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 27 / 37 8.4 Natural convection in other situations Heat transfer from general submerged bodies Spheres. The sphere is an interesting case because it has a clearly specifiable value of NuD as RaD → 0. We look first at this limit. When the buoyancy forces approach zero by virtue of : low gravity small diameter very high viscosity a very small value of β then heated fluid will no longer be buoyed away convectively. In that case, only conduction will serve to remove heat. Example : Yuge’s correlation for spheres immersed in gases : 1/4 NuD = 2 + 0.43RaD (8.19) 5 where RaD is smaller than 10 . John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 28 / 37 8.4 Natural convection in other situations Laminar heat transfer from inclined and horizontal plates In 1953, Rich showed that heat transfer from inclined plates could be predicted by vertical plate formulas if the component of the gravity vector along the surface of the plate was used in the calculation of the Grashof number. Thus, g is replaced by gcosθ, where θ is the angle of inclination measured from the vertical, as shown in Fig.8.7. The heat transfer rate decreases as (cosθ)1/4. Subsequent studies have shown that Richs result is substantially correct for the lower surface of a heated plate or the upper surface of a cooled plate. For the upper surface of a heated plate or the lower surface of a cooled plate, the boundary layer becomes unstable and separates at a relatively low value of Gr. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 29 / 37 8.4 Natural convection in other situations Figure 8.7 Natural convection b.l.s on some inclined and horizontal surfaces. The b.l.separation, shown here for the unstable cases in (a) and (b), occurs only at sufficiently large values of Gr. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 30 / 37 8.4 Natural convection in other situations In the limit θ = 90 - a horizontal plate - the fluid flow above a hot plate or below a cold plate must form one or more plumes, as shown in Fig. 8.7c and d. In such cases, the b.l. is unstable for all but small Rayleigh numbers, and even then a plume must leave the center of the plate. The unstable cases can only be represented with empirical correlations. Theoretical considerations, and experiments, show that the Nusselt number for laminar b.l.s on horizontal and slightly inclined plates varies as Ra1/5. For the unstable cases, when the Rayleigh number exceeds 104 or so, the experimental variation is as Ra1/4, and once the flow is fully turbulent, for Rayleigh numbers above about 107, experiments show a Ra1/3 variation of the Nusselt number. In 1/3 the latter case, both NuL and RaL are proportional to L, so that the heat transfer coefficient is independent of L. Moreover, the flow field in these situations is driven mainly by the component of gravity normal to the plate. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 31 / 37 8.4 Natural convection in other situations Natural convection with uniform heat flux When qw is specified instead of ∆T ≡ (Tw − T∞), ∆T becomes the unknown qw dependent variable. Because h = ∆T , the dependent variable appears in the Nusselt number ; however, for natural convection, it also appears in the Rayleigh number. Thus, the situation is more complicated than in forced convection. Raithby and Hollands give the following correlations for laminar natural convection from vertical plates with a uniform wall heat flux ∗ 1/5 Rax Pr Nux = 0.63 √ (8.20a) 4 + 9 Pr + 10Pr ∗ 1/5 6 RaLPr NuL = √ (8.20b) 5 4 + 9 Pr + 10Pr These equations apply for all Pr and for Nu ≥ 5. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 32 / 37 8.5 Film condensation Laminar film condensation on a vertical plate Consider the following feature of film condensation. The latent heat of a liquid is normally a very large number. Therefore, even a high rate of heat transfer will typically result in only very thin films. These films move relatively slowly, so it is safe to ignore the inertia terms in the momentum equation (8.4) : 2 ∂u ∂u ρg ∂ u u + v = 1 − g + v 2 ∂x ∂y ρf ∂y This result will give u = u(y, δ) (where δ is the local b.l. thickness) when it is integrated. We recognize that δ = δ(x), so that u is not strictly dependent on y alone. However, the y-dependence is predominant, and it is reasonable to use the approximate momentum equation 2 d u ρf − ρg g 2 = − (8.21) dy ρf v John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 33 / 37 8.5 Film condensation This simplification was made by Nusselt in 1916 when he set down the original analysis of film condensation. He also eliminated the convective terms from the energy equation (6.40) : ∂T ∂T ∂2T u + v = α ∂x ∂y ∂y 2 From calculations we get 4k(T − T )µx 1/4 δ = sat w (8.22) ρf (ρf − ρg )ghfg 0 cp(Tsat − Tw ) hfg = hfg 1 + 0.68 (8.23) hfg John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 34 / 37 8.5 Film condensation And hx x Nu = = (8.24) x k δ Thus, with the help of eqn.(8.23), we substitute eqn. (8.22) in eqn.(8.24). And we get " 0 3 #1/4 ρf (ρf − ρg )ghfg x Nux = 0.707 (8.25) µk(Tsat − Tw ) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 35 / 37 8.5 Film condensation Figure 8.8 Heat and mass flow in an element of a condensing film. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 36 / 37 8.5 Film condensation The average heat transfer coefficient is calculated in the usual way for Twall = constant : 1 Z L 4 h = h(x)dx = h(L) L 0 3 so " 0 3 #1/4 ρf (ρf − ρg )ghfg L NuL = 0.9428 (8.26) µk(Tsat − Tw ) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 37 / 37 8.5 Film condensation Two final issues in natural convection film condensation Condensation in tube bundles. Nusselt showed that if n horizontal tubes are arrayed over one another, and if the condensate leaves each one and flows directly onto the one below it without splashing, then Nu Nu = D1tube (8.27) Dforntubes n1/4 This is a fairly optimistic extension of the theory, of course. In addition, the effects of vapor shear stress on the condensate and of pressure losses on the saturation temperature are often important in tube bundles. Condensation in the presence of noncondensable gases. When the condensing vapor is mixed with noncondensable air, uncondensed air must constantly diffuse away from the condensing film and vapor must diffuse inward toward the film. This coupled diffusion process can considerably slow condensation. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 25 avril 2008 38 / 37