# WILSON PLOT

The Overall Heat Transfer Coefficient for any heat transfer equipment is obtained from , where is the heat flow per unit time (watts), A is the heated surface area and ΔT the overall temperature difference. It is usual to design equipment using practical values of U rather than from a series of film coefficients. However, for the important case of heat transfer from one fluid to another across a metal surface, Wilson (1915) developed a method for doing so. By way of illustration, consider cold water flowing through a tube with steam condensing on the outside. The overall and individual coefficients are given by

(1)

where Ri is the fouling resistance inside the tube and α0 and αi are the heat transfer coefficients on the outside and inside of the tube and tw is the thickness and λw is the conductivity of the wall. For turbulent flow, the fouling resistance is constant, α0 is approximately independent of the velocity of the water and since for a smooth circular tube

(2)

where Nu is Nusselt Number, Re is Reynolds Number and Pr is Prandtl Number and the values of the constants are a = 0.8 and b = 0.4, the inside coefficient is given by

(3)

where λ is the fluid conductivity, d is the inside tube diameter, ρ is fluid density and η is fluid viscosity, and v is fluid velocity.

Hence, Equation (1) reduces to

(4)

which, for a plot of 1/U against l/v0.8 is a straight line of slope 1/C2 and intercept C3. Note that the value of C2 is the inside coefficient for unit inside velocity.

For a clean tube, the fouling resistance is zero and, knowing the thickness and conductivity of the wall, C3 gives α0. By repeating measurements as fouling develops values of Ri can be determined from the changes in the intercept.

The method has been used extensively and is in current use, sometimes with modifications, examples being for coiled tubes and annuli and for determining α0 for condensing organic vapors.

The method is based on a number of conditions and assumptions, namely that the outside coefficient and the fouling resistance are constant and C1, a and b are known. These last three parameters will vary with tube design and with inner surface roughness. Thus any degree of roughness imposed artificially, for example by machining, or naturally by scaling or fouling, will alter these parameters. Moreover, it is known [Wilkie (1966)] that Equation (1) does not apply in these conditions, and hence a nonlinear version of Equation (3) is needed.

Even if the form of Equation (1) is correct, it is necessary to assume known values of the three parameters and, if it is not correct, further coefficients must be determined. The validity or otherwise of Equation (1) and the values of all coefficients can be obtained from suitably designed experiments, using least squares curvilinear regression [Wilkie (1962 and 1986)].

#### REFERENCES

Wilkie, D. (1962) A method of analysis of mixed level factorial experiments, Appl. Statistics, XI Pt 3, 184-195.

Wilkie, D. (1966) Forced convection heat transfer from surfaces roughened by transverse ribs, 3rd Int. Heat Transfer Conf., Chicago Amer. Inst. Chem. E., paper No. 1, 1-19 New York.

Wilkie, D. (1987) Analysis of factorial experiments and least squares polynomial fitting by the method of orthogonal polynomials for any spacing of the levels of independent variables, J. Appl. Statistics, 14, 1, 83-89.

Wilson, E. E. (1915) A basis for rational design of heat transfer apparatus, Trans. Am. Soc. Mech. Engrs., 37, 47-70.

#### References

1. Wilkie, D. (1962) A method of analysis of mixed level factorial experiments, Appl. Statistics, XI Pt 3, 184-195. DOI: 10.2307/2985364
2. Wilkie, D. (1966) Forced convection heat transfer from surfaces roughened by transverse ribs, 3rd Int. Heat Transfer Conf., Chicago Amer. Inst. Chem. E., paper No. 1, 1-19 New York.
3. Wilkie, D. (1987) Analysis of factorial experiments and least squares polynomial fitting by the method of orthogonal polynomials for any spacing of the levels of independent variables, J. Appl. Statistics, 14, 1, 83-89.
4. Wilson, E. E. (1915) A basis for rational design of heat transfer apparatus, Trans. Am. Soc. Mech. Engrs., 37, 47-70.