CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran
CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW
LAMINAR BOUNDARY LAYER FLOW Boundary layer theory was proposed by Prandtl shortly after the completion of his doctoral dissertation in 1904
Introduction TWO FUNDAMENTAL PROBLEMS IN CONVECTIVE HEAT TRANSFER 1. The net force exerted by the stream on a plate (to calculate the pressure drop and consequently the pumping power ) 2. The resistance to the transfer of heat from the plate to the stream (to calculate the heat transfer rate) Key Question: How can we calculate F and q? We must first determine the flow and temperature fields in the vicinity of the solid wall by solving the continuity, Navier-Stokes and energy equations
Introduction The conservation equations for an incompressible flow with constant property boundary conditions + Solve four equations with the boundary conditions to obtain four unknowns (u, v, P, T) But!!!! The mathematical complexity of convection heat transfer is traced to the non-inearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields. The boundary layer concept provides major simplifications.
The Boundary Layer Concept This concept is based on the notion that under special conditions certain terms in the governing equations are much smaller than others and therefore can be neglected without significantly affecting the accuracy of the solution. Two questions are raised: (1) What are the conditions under which terms in the governing equations can be dropped? (2) What terms can be dropped? velocity or viscous boundary layer: Under certain conditions the effect of viscosity is confined to a thin region near the surface. Thermal boundary layer: under certain conditions the effect of thermal interaction between the surface and the moving fluid is confined to a thin region near the surface.
The Boundary Layer Concept The velocity boundary layer (1)slender body without flow separation (2) high Reynolds number (Re >100). The conditions for the formation The thermal boundary layer : of the two boundary layers. (1) Slender body without flow separation and (2) high product of Reynolds and Prandtl numbers (Pe=Re Pr > 100) Important observations (1) Zero fluid velocity or so called no slip condition at the surface. (2) Fluid velocity and temperature change rapidly in the boundary layer. (Free stream velocity and temperature at the edge of the boundary layer) (3) At high Re and Pr both velocity and thermal boundary layers are thin. (4) Viscosity plays no role outside the viscous boundary layer. Thus, the flow field is devided into a viscosity dominated region (boundary layer), and an inviscid region (outside the boundary layer). (5) Boundary layers can exist in both forced and free convection flows.
Boundary Layer Equations- Scale analysis
Boundary layer equations- Scale analysis The scale of each of the five terms in the x-momentum equation is determined to see which term can be neglected Free stream characteristics scales for changes in x, y, and u From mass continuity equation The order of both inertia terms is The inertia terms can not be neglected (slenderness of the boundary layer) 2u/ x2 term can be neglected at the expense of the 2u/ y2 term The next step is to simplify these two terms
Boundary layer equations- Scale analysis pressure friction balance Moreover, we know that : (mass continuity) inside the boundary layer, the pressure varies chiefly in the x direction; at any x, the pressure inside the boundary layer region is practically the same as the pressure immediately outside it,
Boundary layer equations- Scale analysis Now, we can write the boundary layer equations. boundary layer equation for momentum The boundary layer equation for energy Performing a scale analysis to the energy equation we can simply show that the thermal diffusion in the x direction can be neglected. x L, Three unknowns (u,v,t) are obtained from these equations. Compare this with the four equations and four unknowns problem contemplated originally. The disappearance of the 2/ x2 diffusion terms from the momentum and energy equations makes this new problem solvable in a variety of ways.
SCALE ANALYSIS- boundary layer thickness and wall friction What is the order of magnitude of the boundary layer thickness and wall friction? The order of magnitude of the boundary layer thickness is obtained from momentum equation Assume a free stream with uniform pressure (simplest free stream possible) Consider the inertia friction balance in the boundary layer momentum equation From mass continuity equation Reynolds number based on the longitudinal dimension of the boundary layer region slenderness postulate on which the boundary layer theory is based (δ <<L) is valid provided that ReL>>1 dimensionless skin friction coefficient The real (measured or calculated) value of τ will differ from factor of order unity by only a
SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number Scaling will now be used to find the order of magnitude of h and δt The δt scale needed for estimating h k/δt can be determined from the energy equation x L, there is always a balance between conduction from the wall into the stream and convection Scales for u and v depend on whether δt is larger or smaller than δ
For this case the axial velocity u within the thermal boundary layer is of the order of the free stream velocity: Theaxial velocity u within the thermal boundary layer is smaller than the free stream velocity. Pretending that the velocity profile is linear, similarity of triangles gives a scale for u as
SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number Case1- Thick thermal boundary layer, δt >>δ (mass continuity ) The 2 nd term can be neglected Peclet number the convection conduction balance becomes: δt >>δ, the range occupied by liquid metals
SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number Case2- Thin thermal boundary layer, δt <<δ fluids with Prandtl numbers of order 1 (e.g., air) or greater than 1 (e.g., water or oils). These scaling results agree within a factor of order unity with the classical analytical results discussed next
Comments
Comments The meaning of Reynolds number.: In most treatments of fluid mechanics, the Reynolds number is described as the order of magnitude of the inertia/friction ratio in a particular flow. This interpretation is not correct because in the boundary layer region, there is always a balance between inertia and friction, whereas Re L can reach as high as 10 5 before the transition to turbulent flow. The only physical interpretation of the Reynolds number in boundary layer flow is geometric δ is proportional to L 1/2. Moreover, along the wall (0 < x < L), the boundary layer thickness increases as x 1/2. Now, one particular property of the x1/2 function is that its slope is infinite at x = 0. This geometric feature of the boundary layer is inexplicably absent from the graphics employed by most texts.
Summary of Boundary Layer Equations for Steady Laminar Flow Simplifying assumptions Continuity: Continuum Newtonian fluid two-dimensional process negligible changes in kinetic and potential energy constant properties Slender surface high Reynolds number (Re > 100) high Peclet number (Pe > 100) steady state laminar flow no dissipation no gravity no energy generation x-momentum: Energy The pressure term is obtained from the solution to inviscid flow outside the boundary layer. Thus, the momentum equation has two unknowns: u and v. To include the effect of buoyancy, the following term should be added to the right-hand-side of the momentum equation