7. HEAT TRANSFER SPRING 2009 7.1 Definitions 7.2 Modes of heat transfer 7.3 The Prandtl number 7.4 Dimensionless numbers in free and forced convection 7.5 The energy equation 7.6 Laminar boundary layers with isothermal walls 7.7 Temperature profile in a turbulent boundary layer 7.8 Heat-transfer coefficients 7.9 Temperature integral results 7.10 Engineering heat-transfer calculations 7.11 References Examples 7.1 Definitions Heat transfer is energy in transit due to a temperature difference. The heat flux qh is the rate of heat transfer per unit area. 7.2 Modes of Heat Transfer 7.2.1 Conduction Heat transfer due to molecular activity in the absence of a bulk motion. Fourier ’s Law = − ∇ qh kh T (1) kh is the conductivity of heat . –1 –1 –1 –1 At 20 ºC, kh(water) = 0.604 W m K ; kh(air) = 0.0257 W m K . 7.2.2 Advection Heat transfer due to bulk motion of fluid. = − qh c p (T Te )u (2) cp is the specific heat capacity (at constant pressure). Te is a reference (usually the external or free-stream) temperature. –1 –1 –1 –1 At 20 ºC, cp(water) = 4182 J kg K ; cp(air) = 1007 J kg K . Turbulent Boundary Layers 7 - 1 David Apsley 7.2.3 Radiation Heat transfer by emission of electromagnetic radiation. Stefan-Boltzmann Law = 4 qh Ts (3) (= 5.670 ×10 -8 W m–2 K–4) is the Stefan-Boltzmann constant . is the emissivity ( = 1 for a perfect black body) 7.2.4 Free and Forced Convection For fluids, conduction and advection are usually combined as convection , which may be either: forced convection – flow driven by external means; free (or natural ) convection – flow driven by buoyancy. At low speeds, both are present and we refer to mixed convection . Regardless of the convective heat-transfer mechanism it is common to write Newton ’s law of cooling = − qh h(Ts Te ) (4) h is the convective heat-transfer coefficient ; Ts is the surface temperature; Te is the free- stream temperature (external flows) or bulk temperature (internal flows). 7.3 The Prandtl Number The Prandtl number c p Pr = = (5) h kh is the ratio of momentum diffusivity ( ) to heat diffusivity ( h = kh/ cp). It is a measure of the relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers. For air at normal temperatures, Pr = 0.71. For a laminar boundary layer: gases: Pr ≈ 1 ≈ T liquid metals: Pr « 1 » T oils: Pr » 1 « T where and T are, repectively, the thicknesses of momentum and thermal boundary layers. In general, / T = f(Pr). Note that many fluid properties, notably viscosity, vary substantially with temperature. In boundary-layer calculations it is common to evaluate them at the film temperature Tf, the average of the free-stream and surface temperatures: = 1 + T f 2 (Ts Te ) (6) Turbulent Boundary Layers 7 - 2 David Apsley 7.4 Dimensionless Numbers in Free and Forced Convection 7.4.1 Free Convection 1 ∂ Thermal expansion coefficient = − . The buoyancy force per unit mass is g T. ∂T 3 = g T L Grashof number : Gr 2 (7) Ratio of buoyancy to viscous forces. Its role in free convection is much the same as that of the Reynolds number in forced convection. g T L3 Rayleigh number : Ra = = Gr Pr. (8) h 7.4.2 Forced Convection qh L Nusselt number : Nu = (9) kh T Ratio of actual to conductive heat flux q Nu Stanton number : St = h = (10) Uc p T Re.Pr Ratio of actual to advective heat flux. In all cases it is necessary to specify what length, velocity and temperature scales are being used. For calculations, engineers require cf and Nu or St. By dimensional analysis, Nu = Nu (Re, Pr) 7.5 The Energy Equation The energy equation for fluid flow can be written in any of the convenient forms: De Internal energy : = − p∇ •u − ∇ • q + (11a) Dt h h p Dh Dp Enthalpy ( h = e + ): = − ∇ • q + (11b) ρ Dt Dt h h p Dh ∂p ∂ Total enthalpy ( h = e + + 1 u 2 ): 0 = − ∇ • q + (u ) (11c) 0 ρ 2 ∂ h ∂ i ij Dt t x j The rate of transfer of mechanical to heat energy (i.e. the rate of working of viscous forces) is given by ∂u = i = 2 (S − 1 S )2 (12) h ∂ ij ij 3 kk ij x j The LHS of any of the energy equations can also be written in conservative form ; e.g. Turbulent Boundary Layers 7 - 3 David Apsley De ∂( e) → + ∇ • ( eu) Dt ∂t For ideal gases, = = e cvT , h c pT (13) where cv and cp are the specific heat capacities at constant volume and constant pressure, respectively. The Eckert number U 2 Ec = (14) c p T is the ratio of kinetic energy to enthalpy change across a layer and is usually assumed small when heat transfer is of interest. At low speeds a change in energy or enthalpy is proportional to a change in temperature: = (enthalpy ) cP T (per unit mass) (15) and hence the energy equation can be written as a temperature equation. The incompressible boundary-layer equations for momentum and energy may be written: ∂u ∂u dP ∂ ∂u (u + v ) = − e + ( − u v′′ ) ∂x ∂y dx ∂y ∂y (16) ∂T ∂T ∂ ∂T c (u + v ) = (k − c v′T ′) p ∂x ∂y ∂y h ∂y p The mean vertical heat flux is: ∂T q = −k + c v′T ′ (17) h h ∂y p The effect of turbulence is an additional net heat flux due to vertical migration of fluid elements carrying their own temperature or energy content. 7.6 Laminar Boundary Layers With Isothermal Walls Substituting the Blasius solution into the temperature equation gives d 2 Pr d + f = 0 (18) d 2 2 d where the boundary-layer coordinate η and dimensionless stream function f(η) are: U = e = x y U e f ( ) (19) x U e and θ is the dimensionless temperature defined by T − T = e , )0( = ,1 (∞) = 0 (20) − Ts Te The temperature equation in boundary-layer coordinates is readily solved in terms of the Blasius function f (see Examples). The solution, which is clearly a function of Pr, can be obtained numerically. Turbulent Boundary Layers 7 - 4 David Apsley To a good approximation, d ≈ − 3/1 ≤ ≤ .0 332 Pr 1.0( Pr 10000 ) d =0 and hence q x ≡ w = − 2/1 d ≈ 2/1 3/1 Nu x Re x .0 332 Re x Pr (21) kh T d =0 = − 2/1 Comparing with the skin-friction result c f .0 664 Re x , ≈ 1 3/1 Nu x 2 c f Re x Pr. or 1 c St ≈ f (22) Pr 3/2 2 This is sometimes called the Chilton-Colburn formula . –1/2 Since qw ∝ x , integrating the heat flux over a finite length L and averaging gives = = ≈ 2/1 3/1 Nu 2Nu x (x L) .0 664 Re L Pr (23) = = − 2/1 (Again, compare the momentum result: cD (L) 2c f (L) .1 328 Re L .) 7.7 Temperature Profile in a Turbulent Boundary Layer ′ ′ The effect of turbulence is an additional vertical heat flux c p v T due to the wall-normal fluctuations of particles carrying their temperature with them. In turbulent forced convection one would expect the same eddying motions to be responsible for diffusing both heat and momentum so that ≈ T A Reynolds analogy leads to an eddy-diffusivity model: ∂u momentum : − u v′′ = t ∂y ∂T heat : − v′T ′ = (24) t ∂y where t = Pr t (sometimes t) (25) t is the turbulent Prandtl number . Pr t is a function of the molecular Prandtl number Pr, but usually (except in liquid metals) Pr t ≈ 0.9 – 1.0. ≡ ¡ For momentum we have already defined a velocity scale u* ( u ) from the wall shear stress: = u 2 (26) w * In like manner, for heat transfer we can define a temperature scale T* from the wall heat flux: q = c u T (27) w p * * Turbulent Boundary Layers 7 - 5 David Apsley (Note that T* can have either sign.) In wall units, + u + T − T U = , T = w (28) u T * * In the overlap layer dimensional analysis gives + y ∂T + ∂T = y = constant T ∂y ∂y + * from which one may recover a logarithmic profile. The gradient-transfer hypothesis gives ∂T q = − c ( + ) h p h t ∂y and, writing q = q = c u T , w p * * ∂T u T = −( + t ) * * ∂ Pr Pr t y whence we have the temperature law of the wall + + y + ∂T 1 + ⌠ dy = or T = (29) ∂ + −1 + −1 −1 + −1 y Pr Pr t ( t / ) ⌡0 Pr Pr t ( t / ) In the near-wall limit ( t / → 0), T + = Pry + (30) In the overlap region ( = u y ⇒ / = y + >> 1), t * t + Pr + T = ln y + A(Pr) (31) The function A(Pr) is derived by integration of (29). An accepted curve fit for Pr 0.7 is A(Pr) = 13 Pr 3/2 − 7 (32) but more complicated expressions have been given (see Kader, 1981). 7.8 Heat-Transfer Coefficients We obtained skin-friction coefficients (defining momentum transfer) by assuming the logarithmic profile to apply across the shear layer. An exactly analogous technique can be used for heat transfer. Note that the results for flat plate and pipe flows below are strictly for constant-temperature (isothermal) walls; varying wall temperatures may be treated by an integral analysis – see Section 7.8. 7.8.1 External Flow – Flat-Plate Boundary Layer Applying temperature and velocity laws at the boundary layer depth: Turbulent Boundary Layers 7 - 6 David Apsley T − T Pr T u∗ w e = t ln + A ( A = 13 Pr 3/2 − 7 ) T∗ U 1 u∗ 2 e = ln + B + ( B = 5.0, = 0.45) u∗ Subtracting, assuming Pr t = 1 and / T = 1, T − T U 2 w e − e = A − B − T∗ u∗ Substituting for T* in terms of the heat flux gives − u cPU e (Tw Te ) ∗ 2 U e = A − B − + qw U e u∗ which can be rearranged to give c 2/ St = f + − − 1 (A B 2 / ) c f /2 q where St = w is the Stanton number.