Module 3. Dimensional analysis and heat transfer

Lesson 13

THERMAL SIMILARITY: DIMENSIONLESS NUMBERS

13.1  Conditions of Thermal Similarity

Thermal similarity determines the condition in which geometrically and hydromechanically similar system are similar thermal as well.

Thermal similarity implies similarity of temperature fields and heat fluxes.

Let us consider two similar systems and express it by equation:

13.2  Equation of Heat Transfer

This is applicable to boundary of substance for heating & cooling process.

Writing these equation for IInd phenomenon,

Let us transform the equation of second phenomenon into first phenomenon with help of similarity constant.

This is under unsteady state condition,

Thus thermal similarity of two or several system requires the Fourier, Peclet, Nusseltt number to be numerically equal at any corresponding point of system.

Further, various modification are carried out for practical application.

Where Pr = Prandtle Number

Such replacement is very convenient for practical application. Reynold Number characterizes hydromechanical similarity and Prandtle number is made up of only physical parameter of fluid.

Prandtle Number characterizes the mechanism of heat exchanged and also made of heat propagation in liquid medium.

The goal of any experimental study of heat flow is usually to find out heat transfer coefficient α. Therefore, the criterial equation of heat transfer by convection is presented in the form that Nusselt Number = f(Fo, Pe) or f(Fo, Re_, Pr), since thermal similarity is impossible without hydro mechanical similarity. Therefore, the dimensionless group Reynold and Grashoff must be introduced into the criterial equation as independent variable. Therefore final criterial equation of heat transfer can have following form

Nu = f (Fo, Re, Pe, Gr)

The general criterial equation can be simplified when applied to any individual problem for example- Fourier’s number is omitted in the case of steady state flow, in forced turbulent flow the influence of free convection can be ignored, therefore, Grasshoff number is eliminated. Hence in the case of steady state forced flow the criterial equation is of the form-

Nu = f (Re, Pr)

Conversely in case of free flow of fluid or free convection, Reynolds’s number can be eliminated and criterial equation will be-

Nu = f (Gr, Pr)

Finally for gases of equal valency, whose Prandle number is equal and constant, the criterial equation will be

Nu = f(Re)…..for forced flow

Nu = f(Gr)…..for free flow