# Calculation Using the Correlation Formulas

First the dimensionless characteristics such as Re and Pr in forced convection, or Gr and Pr in free convection, have to be determined. De­pending on the range of validity of the equations, an appropriate correla­tion is chosen and the Nu value calculated. The equation defining the Nusselt number is

Nu = ^ (4.191)

A

Given the heat transfer factor a. Using the equation

Q = | = «0, (4.192)

The heat flow density is determined, which represents a certain temperature difference or the temperature difference between the wall and the fluid for a certain heat flow rate.

As an example, for free convective heat transfer from a vertical wall,

Nu = 0.13- (GrPr)1/3 = 0.13-

Equations (4.191), (4.192), and (4.193) give

This equation does not incorporate the characteristic length L; hence the wall height has no influence on heat transfer.

In problems of forced convection, it is usually the cooling mass flow that has to be found to determine the temperature difference between the cooling substance and the wall for a given heat flow. In turbulent pipe flow, the fol­lowing equation is valid:

Nu = 0.0395 Re3/4Pr1/3 = (4.195)

A vi

The mass flow is found using the continuity equation m= pwrrd2/4 and the Reynolds number formula Re = 4m/{<rrp dv):

 4/3
 Qd
 O. O395Pr1/30A
 . 77 f) d V
 M = (4.196)

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final ve­locity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully devel­oped flow.

The reason the heat transfer is improved can be seen from the equations Nu = as/A (where s is the thickness of the boundary layer) and q = aO, giving

NuAQ (4.197!

5 ‘

Thin boundary layers provide the highest values of heat flow density. Be­cause the boundary layer gradually develops upstream from the inlet

Point, the heat flow density is highest at the inlet point. Heat flow density

Decreases and achieves its final value in the region of fully developed flow. The correction is noted in the equations by means of the quotients d/L and d/x.

For some fluids, such as oils, the viscosity is temperature dependent. Here the correction factor (rjp/’qw)°’i4 is used, where rip is the viscosity at the mean fluid temperature and rfw is the viscosity at the wall temperature.