Heat transfer from an isothermal flat plate
Consider the development of the thermal boundary layer due to the heat transfer from a heated wall (compressibility and viscous dissipation are neglected). For a flat–plate flow (i.e. constant velocity U in the external flow) over an isothermal surface (i.e. constant wall temperature Tw) a self–similar thermal boundary layer results, where the shape of the temperature profile still depends on the value of the Prandtl number Pr.
The integral temperature equation, which reads as follows for the problem under consideration: 0
( ) Pr
w
p w
qd T u T T dydx c y
can be used for an approximate estimation of the thermal boundary layer development, when assumed shapes of the velocity and temperature profiles are prescribed. Let now both profiles be approximated by a linear relation, i.e.:
(0 )u y yU
and 1 ( )u yU 1 (0 )T
w T
T T y yT T
and 0 ( )T
w
T T yT T
For the case that δT δ it has been shown during the lectures that: 1/ 3 2 / 3/ ( ) 1Pr Pr/ Pr
w p wT
w
q c T Ts U
where s is the Reynolds–analogy factor. The first expression directly reveals that this situation corresponds to the case that Pr 1.
a. Derive the corresponding expressions for ζ and s for the case that δT δ. Show that this corresponds to Pr 1.
Hints: split the integral in two parts for 0 y δ and for δ y δT. Eliminate δT by introducing ζ (= constant!); δ follows from the integral momentum equation.
b. Investigate the behaviour of both expressions for the limit when Pr goes to zero. N.B.: by “behaviour” it is meant not just the limiting values, but in particular the way in which ζ and s are related to Pr.
c. Give the results of the complete approximation, i.e. for Pr 1 and Pr 1 together, in the form of a graph where 1/ζ and 1/s have been plotted against Pr, on the interval 0 Pr 5. Show that for the limit of Pr going to 1 the results of both approximations match smoothly (continuous in both value and first derivative). N.B.: for this it is sufficient to prove that for ζ and dζ/dPr both limits (i.e. for Pr coming from either side of 1) are equal.
d. For Pr << 1 the exact solution for the temperature profile can be obtained analytically. As in this case δ << δT, it can be assumed in approximation that the development of the thermal boundary layer takes place completely in the inviscid outer flow. This allows the temperature equation to be simplified to: 2
2Pr
T T U x y
– Derive, by means of a similarity transformation, that the above equation can be written as follows: ” 2 ‘ 0
with Θ the transformed temperature profile and η the transformed y–coordinate. Give the expression for η and determine the boundary conditions which θ has to satisfy. N.B.: this η–scaling is not the usual Blasius scaling, which applies to the velocity profile!
– Show that the solution of this equation can be written in terms of the Gaussian error–function (see remark below), and derive from this the heat transfer at the wall, expressed by the local Nusselt number: Nu ( )
wx
w
q x
k T T
as function of Rex and Pr.
– Calculate from this the expression for the Reynolds analogy factor s, if given that the wall shear stress according to the Blasius solution is: 0.332w
UU x
Compare this result to the approximation obtained in part (b).
Remark: the error function is: 2
0
2( )
x terf x e dt
, with ( ) 1erf .
