6.2 Governing Equations for Natural Convection


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1 6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed in Chapter. Consider a multicomponent system with N components, the governing equations for a stationary reference frame can be expressed as Dρ DV + ρ V = (6.4) ρ = ρ gτ p + (6.5) where the viscous stress tensor, τ, can be determined by using Newton s law of viscosity [see eq. (1.53)] 0 1
2 where D is the rate of strain tensor: τ = µ D µ ( V) I 3 1 D = V + V ( ) (6.6) (6.7) For natural convection problems, it is often assumed that the fluid is incompressible, except in the first term on the righthand side of eq. (6.5); this is referred to as the Boussinesq assumption. Under this assumption, the continuity equation (6.4) becomes: V = 0 (6.8) According to eq. (6.8), the second term on the righthand side of eq. will be zero. The momentum equation (6.5) then becomes: DV ρ = ρ g p + ( µ V) T (6.9)
3 where the lefthand side is the inertial term; the three terms on the righthand side represent body force per unit volume, pressure force per unit volume, and viscous force per unit volume, respectively. The density of a mixture is a function of its temperature and the mass fractions of its species. It can be expanded using a Taylor s series near the vicinity of a reference point ( T ω ω Lω ):,,, N 1,,, ρ N ρ ρ = ρ + ( T T ) + ( ω ω ) + i i, T i = 1 ω i L ρ where is density at the reference point. By defining the coefficient of thermal expansion, β, and composition coefficient of volume expansion, β m,i, as follows: 3
4 β 1 ρ = T ρ p (6.10) β m, i 1 ρ = ρ ω i (6.11) and neglecting the higher order term in the Taylor s series expansion, one obtains: ρ ρ ρ β ( T T ) ρ β ( ω ω ) m i i, (6.1) i = 1 which is valid only if β ( T T ) = 1 and β m ( ω i ρ ω i, ) = 1. Substituting eq. (6.1) into eq. (6.9), the momentum equation for natural convection is obtained: DV ρ = + ρ ρ β ( p g ) g ( T T ) N i = 1 p N ρ g β ( ω ω ) + ( µ V) m, i i i, (6.13) 4
5 which is a generalized momentum equation because the effects of buoyancy forces due to both temperature and composition variations are considered. If it is assumed that the Dufour effect is negligible and the fluid is incompressible, the energy equation is: DT ρ cp = ( k T ) + q + Vτ: (6.14) The conservation of species mass in terms of mass fraction for the i th species can be expressed as Dω i ρ = J i + m& i (6.15) For a binary system of A and B, one can apply Fick s law to eq. (6.15) to obtain: Dω A ρ = ρ ( DAB ω A) + m& A (6.16) 5
6 6.. External Natural Convection from Heated Vertical Plate For external natural convection near a vertical flat plate as shown in Fig. 6.1, the boundary layer assumption can be applied to simplify the above generalized governing equations. The boundary layer treatment for the case of natural convection is very similar to that for the case of forced convection that was discussed in Chapter 4. The difference between the natural convection problem shown in Fig. 6.1 and forced convection over a flat plate is that the free stream velocity in the outside of the velocity boundary layer is zero. In addition, the pressure outside the boundary layer is hydrostatic for the case of natural convection, instead of being externally imposed as in the case of forced convection. 6
7 For D external convection of an incompressible fluid as shown in Fig. 6.1, the continuity equation becomes u x v + = y (6.17) If one assumes that the fluid is single component so that the natural convection is driven by the density difference induced by the temperature gradient, eq. (6.13) becomes: DV ρ = + ρ ρ β + µ Applying the boundary layer assumption and assuming constant thermophysical properties, the momentum equation becomes 0 ( p g ) g ( T T ) ( V) u v g g T T u 1 + u = p + β ( ) ν u + x y ρ x y (6.18) (6.19) 7
8 Since the pressure in the boundary layer is independent of y ( p / y = 0 ), the pressure inside the boundary layer, p, is same as the pressure outside the boundary layer at the same longitudinal position, p, i.e., p dp dp = = x dx dx The hydrostatic pressure, p, is dictated by the density and the longitudinal position: dp dx = ρ Substituting the above two equations into eq. (6.19), the momentum equation becomes: u u u + v = ν u + g β ( T T ) x y y g (6.0) 8
9 After applying the boundary layer assumption and assuming the viscous dissipation is negligible, the energy equation becomes: u + v = α x y y T T T (6.1) At the heated wall, the nonslip and impermeable conditions yield the following boundary condition for the momentum equation u = v = 0, at y = 0 The temperature at the heated wall is specified, i.e., T = T, at 0 w y = (6.) (6.3) 9
10 Since the quiescent fluid far away from the heated plate is not disturbed by the existence of the heated plate, the velocity at the locations away from the flat plate should be zero: u = v = 0, y (6.4) Also, the temperature of the fluid outside the thermal boundary layer is not affected by the heated wall: T = T, y (6.5) 10
11 6..3 Dimensionless Parameters While the Reynolds number was used as a dimensionless parameter to characterize the flow for the case of forced convection, it cannot be used to characterize the natural convection because the characteristic velocity is not available. To identify the appropriate dimensionless parameters for description of natural convection, one defines the following dimensionless variables: where u 0 is a reference velocity that is unknown at this point. x y u v T T X =, Y =, U =, V =, θ = L L u u T T 0 0 w (6.6) 11
12 Equations (6.17), (6.0) and (6.1) can be respectively nondimensionalized as follows U X V + = Y 0 (6.7) U g β ( T T ) L θ U U 1 U w + V = + X Y ReL Y u0 U + V = X Y Re Pr Y θ θ 1 θ L (6.8) (6.9) where Re L = u L 0 ν (6.30) 1
13 is the Reynolds number based on the reference velocity. To simplify the dimensionless governing equations, one can choose the reference velocity to be: u (6.31) 0 = g β ( Tw T ) L so that the factor before dimensionless temperature, θ, in eq. (6.8) becomes unity. With this choice of reference velocity, the Reynolds number becomes: Re L = gβ ( T T ) L w (6.3) In natural convection problems, we define Grashof number, Gr L, to be w GrL = ReL = ν g β ( T T ) L ν 3 3 (6.33) 13
14 as the dimensionless number which represents the ratio of the buoyancy force and the viscosity force acting on the fluid. Equations (6.8) and (6.9) then become: U U 1 U + V = U + θ X Y Gr Y 1/ L U + V = X Y Gr Pr Y θ θ 1 θ 1/ L (6.34) (6.35) The role of the Grashof number for a natural convection problem is similar to the role of Reynolds number for a forced convection problem. 14
15 The Prandtl number, which reflects the ratio between momentum and thermal diffusions, is another parameter that affects natural convection. Therefore, it is expected that the average Nusselt number for natural convection, Nu L = hl / k, will be a function of the Grashof number and the Prandtl number: Nu L = f (Gr,Pr) The objective of this chapter is thus to identify the appropriate forms of the above function for various geometric configurations and physical conditions. L (6.36) 15
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