Two-dimensional flow analysis over a flat plate serves well to illustrate several key concepts in forced convection heat transfer.
The viscosity of the fluid requires that the fluid have zero velocity at the plate's surface. As a result a boundary layer exists where the fluid velocity changes from in the free stream (far from the plate) to zero at the plate. Within this boundary layer, the flow is initially laminar but can proceed to turbulence once the Reynolds Number Re of the flow is sufficiently high. The transition from laminar to turbulent for flow over a flat plate occurs in the range, |
3 x 10^{5} < Re_{x} < 3 x 10^{6} , | ||
For the current problem, we consider that the plate is heated starting at a point x = x_{0} and continuing downstream. Furthermore we assume that the plate is maintained at constant temperature T_{w}, making this problem isothermal. We are interested only in laminar flow, so it is assumed that the plate length L is sufficiently short such that turbulent flow is never triggered (i.e. Re_{x} < 3 x 10^{5}). |
An analysis of the fluid flow over the plate that considers conservation of momentum and energy, including the effects of viscosity, temperature, and heat entering the fluid from the plate, results in the following equation for the Nusselt Number Nu as a function of x, The Nusselt Number is a non-dimensional ratio of the heat entering the fluid from the surface to the heat conducted away by the fluid. It is defined by the equation, The Prandtl Number Pr is a non-dimensional ratio of the viscous boundary layer thickness to the thermal boundary layer thickness. It is defined by the equation, If the plate (of length L) is uniformly heated over its entirety (x_{0} = 0), then the average Nusselt Number is found to be, |
The Nusselt Number equation can be used to calculate the heat transfer coefficient h via, |
which can then be used to calculate the heat convected away by the fluid via Newton's Law of cooling, |
Since fluid properties (such as viscosity, diffusivity, etc.) can vary significantly with temperature, there can be some ambiguity as to which temperature one should use to select property values. The recommended approach is the use the average of the wall and free-stream temperatures, defined as the film temperature T_{f}, |
This problem is executed in a calculator. |