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Unit 42: Heat Transfer and Combustion Lesson 3: The Thermal Boundary Layer Aim • LO2: Understanding Heat Transfer Mechanisms and Coefficients NDGTA The Thermal Boundary Layer NDGTA • In hydrodynamics, the boundary layer is defined as that region of the flow where viscous forces are felt. • A thermal boundary layer may be defined as that region where temperature gradients are present in the flow. • These thermal gradients would result from heat-exchange process between the fluid and the wall Temperature Profile in the Thermal Boundary Layer y x T∞ δt Tw qw = -k ∂T ∂y w NDGTA The Thermal Boundary Layer NDGTA • The temperature of the wall is Tw, the temperature of the fluid outside the thermal boundary is T∞ and the thickness of the thermal boundary layer is designated as δt. • At the wall the velocity is zero and the heat transfer takes place by conduction. • Thus the heat flux per unit area q” is… The Thermal Boundary Layer q = q” = - k ∂T A ∂y NDGTA w From newton’s law of cooling, q” = h(Tw – T∞) Where h is the convection heat-transfer coefficient. Thus combining the equations gives… The Thermal Boundary Layer NDGTA h = - k(∂T/∂y)wall (Tw – T∞) • So we need only find the temperature gradient at the wall in order to evaluate the heat-transfer coefficient. • This means that we need to obtain an expression for the temperature distribution The Thermal Boundary Layer NDGTA • Thus consider the following 4-conditions… T = Tw at y = 0 ∂T/∂y = 0 at y = δt T = T∞ at y = δt ∂2T = 0 at y = 0, since the velocities at the wall is zero ∂y2 Temperature Profile in the Thermal Boundary Layer y NDGTA A A x T∞ u∞ H δ u δt dx Tw 1 2 dqw = -k ∂T ∂y w The Thermal Boundary Layer NDGTA • The four conditions may be fitted into a cubic polynomial so that… θ = T – Tw = 3 y – 1 y 3 θ∞ T∞ - Tw 2 δt 2 δt • We now need to determine δt, the thermalboundary-layer-thickness. • Consider the volume bounded by AA12 • Thus the heat given up to the fluid over the length dx is dqw The Thermal Boundary Layer NDGTA • Thus… Energy convected in + viscous work within element + heat transfer at wall = energy convected out. Energy convected in through plane A1 is… H ρcp uTdy 0 Energy convected out of plane A2 is… H ρcp uTdy 0 + d ρcp uTdy dx dx The Thermal Boundary Layer The mass flow through the plane AA is… H d dx ρudy dx 0 And this carries with it an Energy equal to… H cpT∞ d dx ρudy dx 0 NDGTA The Thermal Boundary Layer NDGTA The net viscous work done within this element is… μ H du 2dy dx dy 0 And the heat transfer at the wall is… dqw = -k dx ∂T ∂y w Combining these energy equations gives the integral energy equation of the boundary layer for constant properties and constant free-stream temperature T∞. Handout 1 – p233 & 234 Heat Transfer 10th Edition Holtzman The Prandtl Number NDGTA • Note Pr = ν/α is called the Prandtl number after Ludwig Prandtl – he introduced the concept of boundary-layer theory • Thus when the plate is heated over its entire length, x0 = 0 and… δt = ζ = 1 Pr-1/3 δ 1.026 The Prandtl Number NDGTA • In the handout 1 given the analysis was based fundamentally on ζ < 1. This assumption is satisfactory for fluids having Prandtl numbers greater than 0.7 • Fortunately most gases and fluids fall within this category. • Liquid metals are a notable exception since they have Prandtl numbers of the order 0.01 • The Prandtl number ν/α relates the relative thickness of the hydrodynamic and thermal boundary layers. The Prandtl Number NDGTA • The kinematic viscosity of a fluid conveys information about the rate at which momentum may diffuse through the fluid because of molecular motion. • The thermal diffusivity tells us the same thing in regard to the diffusion of heat in the fluid • Thus the ratio of these two quantities should express the relative magnitudes of diffusion of momentum and heat in a fluid. The Prandtl Number NDGTA • But these diffusion rates are precisely the quantities that determine how thick the boundary layers will be for a given external flow field; large diffusivities mean that the viscous or temperature influence is felt further out in the flow field. • The Prandtl number is thus the connecting link between the velocity field and the temperature field. Pr = ν = μ/ρ = cpμ α k/Ρcp k The Prandtl Number NDGTA • In SI units typically μ is in kg/s/m; cp in kJ/kg/oC and k in kW/m/oC. h = -k(∂T/∂y)w = 3 k = 3 k Tw – T∞ 2 δt 2 ζδ Substituting for the hydrodynamic-layer boundary thickness gives… The Nusselt Number NDGTA hx = 0.332kPr1/3 (u∞/νx)1/2 (1 – (x0/x)3/4)-1/3 This equation may be non-dimensionalised by multiplying both sides by x/k producing the dimensionless group on the LHS… Nux = hxx/k Which is called the Nusselt number after William Nusselt The Nusselt Number NDGTA Hence Nux = 0.332Pr1/3 Rex1/2 (1 – (x0/x)3/4)-1/3 Or for a heated plate over its entire length, xo = 0 thus… Nux = 0.332Pr1/3 Rex1/2 Constant Heat Flux NDGTA • Most of the preceding analysis has considered the laminar heat transfer from an isothermal surface. • In many practical problems the surface heat is essentially constant and the objective is to find the distribution of the plate surface temperature for given fluid-flow conditions. • For constant heat flux it can be shown that the Nusselt number is given by… Nux = hx = 0.453Pr1/2 Rex1/3 Constant Heat Flux NDGTA • This may be expressed in terms of the wall heat flux and temperature difference… Nux = qwx /[k(Tw – T∞)] The average temperature difference along the plate for constant heat flux condition may be obtained by performing integration. The result being… qw = (3/2)hx=L(Tw – T∞)