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.