Chapter 6, Problem 6.1P

The temperature distribution within a laminar thermal boundary layer associated with flow over an isothermal flat plate is shown in the sketch. The temperature distribution shown is located at $x=x_2.$

1. Is the plate being heated or cooled by the fluid?
2. Carefully sketch the temperature distributions at $x=x_1$ and $x=x_3.$ Based on your sketch, at which of the three x-locations is the local heat flux largest? At which location is the local heat flux smallest?
3. As the free stream velocity increases, the velocity and thermal boundary layers both become thinner. Carefully sketch the temperature distributions at $x=x_2$ for (i) a low free stream velocity and (ii) a high free stream velocity. Based on your sketch, which velocity condition will induce the larger local convective heat flux?

To determine

Whether the plate being heated or cooled by the fluid.

Answer to Problem 6.1P

The plate is heated by the fluid.

Explanation of Solution

Given information:

  

In the sketch given, it is indicated that the surface temperature is less than the free stream temperature. This concludes that the surface of heat flux is in positive y-direction and can be written by using the equation of Fourier law:

  $q_s^{\text{'}\text{'}}=-k_f{\frac{\partial T}{\partial y}|}_{y=0}$

Thus, the temperature gradient is positive, hence the heat flux is negative. Thus, the fluid heats the plate as the heat is transferred in the negative y-direction.

To determine

Sketch the distribution of temperature at different x locations and locate the places of largest and smallest heat flux.

Answer to Problem 6.1P

The magnitude of heat flux is largest at x₁ and smallest at x₃.

Explanation of Solution

The temperature must vary from surface temperature T_s to free stream temperature T₈ at every location in the boundary layer. With laminar flows, heat transfer to or from the wall varies with distance from the leading edge of a boundary layer. This change is occurred within the thermal boundary layer thickness, as shown in the figure below:

  

The temperature gradient at the surface is proportional to the magnitude of the heat flux and is shown by the dashed line. The temperature gradient is larger at x₁ where the boundary layer is thinner and is smaller at x₃ where the layer is thicker. Therefore, the magnitude of heat flux is largest at x₁ and smallest at x₃.

To determine

Sketch the distribution of temperature at x = x₂for both lower and higher stream velocity and to check the condition at which the larger local convection heat flux will develop.

Answer to Problem 6.1P

The higher convective heat flux is occurred in the case of higher free stream velocity.

Explanation of Solution

Case (i):

Diagram for low free stream velocity is shown below:

  

As the stream velocity increases the boundary layer becomes thinner.

Case (ii):

Diagram for high free stream velocity is shown below:

  

The temperature gradient is thinner for higher free stream velocity.

The larger local convective heat flux:

From the relation of Nusselt number:

  $N_u=\frac{h.L}{K}$

But in case of forced convection the Nusselt number is a function of Reynold’s and Prandtl number. Thus,

  $\begin{array}{l}h=\frac{N_u.K}{L}\\ h=\frac{f\left({\mathrm{R}}_e,P_r\right)K}{L}\end{array}$

As Reynold’s number is a function of velocity of fluid. Hence,

  $h=f\left(V\right)$

Thus, as the velocity increase the heat flux also increases linearly. Hence, the higher convective heat flux is occurred in the case of higher free stream velocity case.