Module 2. Convection heat transfer

Lesson 9

DIFFERENTIAL EQUATIONS OF HEAT TRANSFER: DISCUSSION ON VARIOUS FORMS

9.1  Heat Transfer Equation

As discussed earlier heat transfer calculations are based on Newtonian formula. However, in using this formula to determine Q one must know the value of Heat Transfer Coefficient ‘α’

Q = α F (∆t)

The relationship between the heat transfer coefficient and the conditions of the process may be established by analyzing the boundary conditions. Since heat is transferred through the laminar boundary layer of the fluid only by conduction. Therefore, according to Fourier’s law we have the differential amount of heat transfer, given by

On the other hand according to Newtonian law, the amount of heat transferred dQ shall also be equal to:

On equating the two equations, we get

In this form it is known as differential equation of heat transfer which describes the heating and cooling process at the boundaries of a substance. (Fig. 9.1 )

9.1.1  Differential equation of conduction

To find the heat transfer coefficient one must know the temperature gradient and consequently temperature distribution in the fluid. The temperature distribution in the fluid can be obtained from the differential equation of conduction which is based on the energy conservation law.

Now let us consider in the flow of fluid an elementary parallelopiped limited by dx, dy and dz and we can write the heat balance equation for this.

Let us assume that the physical properties of the parallelopiped

1. i.e. λ, C_p and r are constant

2. Ignoring the variation in pressure, the amount of imparted heat will be equal to the change in heat content or enthalpy of the substance in accordance with fundamental laws of thermodynamics.(Fig. 9.2)

Let us first calculate the amount of heat imparted to the body by conduction through the sides of element. Therefore, according to Fourier’s law is the amount of heat flowing per unit time dτ in direction of X-axis through the side ABCD

Through the face EFGH whose temperature is:

t + temperature gradient x d_x

Therefore amount of heat flowing through the face EFGH

Now by subtracting we get:

 Similarly,

Total amount of heat remaining in the element of volume d_x d_y d_z per unit time d_τ

This amount of heat imparted to the element will raise its temperature by-

And, we know that its enthalpy will be

Equating this equation and rewriting it per unit volume per unit time

So,

This is Fourier-Kirchhoff’s Differential Equation of Conduction.

It establishes the relation between time and space variation in temperature at any point of a moving medium.

In this form the equation is employed in investigating heat conduction in moving fluids. If it is applied to the solids then the equation becomes:

In this form it is known as the Fourier’s differential equation. It assumes the simplest form in case of steady state one-dimensional conduction, namely, On solving this equation, we get the calculation formula for a plane well.