Question

Air at a free stream temperature of T[infinity]=20 ˚C is in parallel flow over a flat plate of length L=2 m and temperature Ts=50 ˚C. However, obstacles placed in the flow intensify mixing with increasing distance x from the leading edge, and the spatial variation of temperatures measured in the boundary layer is correlated by an expression of the form T(˚C)=a+be−ᶜˣʸ, where x and y are in meters. Determine and plot the manner in which the local convection coefficient h varies with x. Evaluate the average convection coefficient hˉ for the plate. Consider a=10 ˚C; b=50˚C; and c=500(1/m²).

Answer 1

The local convection coefficient h varies with x according to the equation h = (k/cb)e^(-cx). The average convection coefficient hˉ can be evaluated by calculating the integral of h with respect to x from x = 0 to x = L.

Step-by-step explanation:

To determine the manner in which the local convection coefficient h varies with x, we can differentiate the given expression for T with respect to x. Taking the derivative of T with respect to x gives us:

dT/dx = -cbe^(-cx)

The local convection coefficient h can be obtained by dividing the negative of the heat flux q by the temperature difference between the plate and the free stream air. Since q = -k(dT/dx), we have:

h = (k/cb)e^(-cx)

To evaluate the average convection coefficient hˉ, we need to calculate the integral of h with respect to x from x = 0 to x = L. The average convection coefficient is then given by:

hˉ = (1/L) * ∫(k/cb)e^(-cx) dx from x = 0 to x = L