Dimensional analysis in heat transfer

As seen in many of the correlations for fluid flow and heat transfer, many dimensionless groups, such as the Reynolds number and Prandtl number, occur in these correlations. Dimensional analysis is often used to group the variables in a given physical situation into dimensionless parameters or numbers which can be useful in experimentation and correlating data. 

An important way of obtaining these dimensionless groups is to use dimensional analysis of differential equations described in Section 3.11. Another useful method is the Buckingham method, in which the listing of the significant variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined. 

Heat transfer inside a pipe. The Buckingham theorem, given in Section 3.11 states that the function relationship among q quantities or variables whose units may be given in terms of u fundamental units or dimensions may be written as q — u) dimensionless groups. 

As an additional example to illustrate the use of this method, let us consider a fluid flowing in turbulent flow at velocity v inside a pipe of diameter D and undergoing heat transfer to the wall. We wish to predict the dimensionless groups relating the heat-transfer coefficient h to the variables D, p, k, and v. The total number of variables is 

The fundamental units or dimensions are = 4 and are mass M, length L, time f, and temperature T. The units of the variables in terms of these fundamental units are as follows  

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