(a) Extend the solution found in example 11.4-1 to determine the total amount of heat (Q) transferred to the fluid from the wall per foot of distance
in the z direction as shown in Fig. 11.4-1 and over the length L in the x direction from the leading edge of the plate.
 

(b) Transform your answer to (a) to determine the average heat transfer coefficient (h) such that Q=h*L*(T₀-T_inf); Then show that you can find a relation that gives the Nusselt Number:  (h*L)/k as a function of the Reynolds Number:  (L*vinf*rho)/mu and the Prandtl number of the fluid.
 
 

Note: The solution shown for Example 11.4-1 must be executed before using the Maple commands shown below.

(a) Now, that phi(etaT) has been chosen and delta(x) is known, the temperature distribution T as a function of x and y is determined from the assumed form of the temperature distribution.

> T(x,y);

The heat flux in the y-direction can be determined using Fourier's Law of Heat Conduction, where k is the thermal conductivity of the fluid.

> qy:=-k*diff(T(x,y),y);

> qy:=unapply(qy,x,y);

The heat flux from the heated flat plate to the fluid can be found by evaluating qy at the surface of the heated flat plate (y=0).

> qy(x,0);

The total amount of heat transferred to the fluid from the wall per foot of distance in the z-direction as shown in Fig. 11.4-1 is found by integrating qy(x,0) from x=0 to x=L, the edge of the plate.

> Q:=int(qy(x,0),x=0...L);

Q simplifies to the following:

> evalf(Q);

The heat flux at the plate-fluid interface is related to the difference between the temperature at the plate surface and the temperature outside the thermal boundary layer by Newton's Law of Cooling. Newton's Law of Cooling gives the relation qy=h*(T-Tinf). An average heat transfer coefficient h is calculated by taking the heat flux qy to be an average over the surface of the plate. This average qy is calculated by Q/L.

> h:=Q/(L*(T0-Tinf));

(b) We must now find the Nusselt number in terms of the Reynolds number and Prandtl number. The Nusselt number is defined below.

> Nu:=h*L/k;

nu( v ) is equal to mu/rho. The Reynolds number Ren is equal to (L*vinf*mu)/rho. Thus, v is given by (L*vinf)/Ren.

> nu:=L*vinf/Ren;

Substituting for v , we find the Nusselt number to be:

> Nu:= simplify(Nu,assume=positive);

The Nusselt number simplifies to the following answer:

> Nu:=evalf(Nu);