TUTORIAL ON EXTERNAL FORCED CONVECTION A. THERMAL BOUNDARY LAYER - ORDER OF MAGNITUDE ANALYSIS AND REYNOLDS ANALOGY 1. Consider laminar flow over a cooled flat plate with µ C p = k . If the wall temperature is Tw , the free stream conditions are U ∞ , T∞ , and the wall shear stress is τ w , then determine the wall heat flux. Is it possible to use the same result for flow over a curved plate? Give reasons.

2. Engine oil at 60°C flows over a 5-m long horizontal flat plate whose wall temperature is kept at 30°C. The incipient flow is parallel to the length of the plate. Various properties of the fluid are as follows: density ρ= 870 kg/m3, kinematic viscosity ν=250 ×10-6 m2/s, thermal conductivity k=0.145 W/(m.K), specific heat at constant pressure Cp1= 1900 J/(kg.K). If the specific heat at constant pressure changes to Cp2=3800 J/(kg.K), with all other conditions and properties remaining unaltered, determine the relative change in the ratio of thermal boundary layer thickness (δT) to hydrodynamic boundary layer thickness (δ) by executing order of magnitude analysis.

3. Atmospheric air is in parallel flow = m/s, T∞ 15 o C ) over a flat heater ( u∞ 15= surface that is to be maintained at a temperature of 140 o C . The heater surface area is 0.25 m2, and the airflow is known to induce a drag force of 0.25 N on the heater. What is the electrical power needed to maintain the prescribed surface temperature? Air properties may be approximated as ρ = 0.995 kg/m3 , c p = 1009 J/kg·K and Pr = 0.7 .

B. MOMENTUM AND ENERGY INTEGRAL METHOD 4. Consider a heated flat plate of axial length L and width b. Fluid is forced to flow over the plate with a free stream velocity of U ∞ and temperature of T∞ . In an effort to have an enhanced thermal performance, small holes are drilled in the plate and fluid is injected into the boundary layer through those holes with a uniform velocity Vw . The plate boundary is subjected to a uniform temperature, Tw . All properties of the fluid are taken as constants. (a) Derive the momentum integral equation and energy integral equation corresponding to the above situation.

(b) Assuming a non-dimensional velocity profile of the form:

u y = , derive an U∞ δ

ordinary differential equation depicting the growth of the hydrodynamic boundary layer ( δ ) as a function of x . Solve that equation in δ for the special case in the limit Vw → 0 . (c) Assuming a non-dimensional temperature profile in the thermal boundary layer of the same form as that of the non-dimensional velocity profile in the hydrodynamic boundary layer, derive an expression for the growth of the thermal boundary layer as a function of x , assuming the momentum diffusivity of fluid to be negligible as compared to its thermal diffusivity. Hence, derive an expression for the local Nusselt number, for the special case in the limit Vw → 0 .

5. Consider the laminar flow of a two-dimensional thin liquid film (water) on a flat wall that is inclined at an angle θ relative to the horizontal direction. The film flow is essentially driven by the gravitational acceleration component ( g sin θ ) acting parallel to the wall. (a) Considering that viscosity of air is significantly less than that of water, derive the fully developed velocity profile in the water film. (b) Consider heat transfer from the wall to the liquid film of thickness H in the case where the film temperature is T0 everywhere upstream of x = 0 , whereas the wall temperature is raised to T0 + ∆T downstream of x = 0 . Considering that the thermal boundary layer thickness ( δ T ) is much less than the hydrodynamic boundary layer thickness ( δ ) at immediate downstream from x = 0 , obtain an order of magnitude of δ T as a function of α , ν , x, H , g sin θ . (c) Derive the energy integral equation for the thin film, considering that the fluid above y = δ T is isothermal (at T0 ). (d) Is the following temperature profile an appropriate choice for substituting in the energy integral equation? T − T0 y  y  =1 − 2 +   δT  δT  ∆T T T0 =

2

for 0 ≤ y ≤ δ T for δ T ≤ y ≤ δ