enegy eqn derivation

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Introduction 1

Derivation of energy equation 2references 5


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1

Introduction

Conservation of energy implies that changes in energy, heat

transferred and work done by a system in steady operation are in balance.

The transfer of heat between solid body and a liquid or gaseous flow is a problem whose consideration involves the science of fluid motion. Onthe physical motion of the fluid there is super imposed a flow of heatand generally speaking, the two fields interact, in order to determine thetemperature distribution it is necessary to combine the equations of

motion with those of heat conduction.In this report we establish the energy balance for a fluid element inmotion and to consider it in addition to the equations of motion.


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Considering an elementary volume v=dxdydz of mass m= * v

As its flows along its path the quantity of heat dQ added to the volume in time dt tends to increase itsinternal energy by dE T and performing work by an amount dW, according to the first law of

thermodynamics:

Heat dQ

B y neglecting the transferring of heat by radiation, then it can transfer only by conduction.

According to the fourier s law the heat flux q per unit area A and time is proportional to the temperaturegradient:

K: thermal conductivity of the fluid

The negative sign represent that the heat flux is considered as positive in the direction of the

temperature gradient

Hence the amount of heat transferred into the volume V through the surface elements which arenormal to the x-direction is:

Then the amount of heat leaving is:

Then the net in heat added by conduction to the element during interval to the volume V is:

Ch ange in dE T

dET consists of:

1- Change in internal energy denoted by 2- Change in kinetic energy by the amount

Figure 1: frictional stress on fluid element


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3

B y neglecting the change in potential energy due to displacement in gravitational field then:

Work

According to fig. 1 we consider first the contribution from the component x then the work per unit time

is:

The negative sign is due to we considered that the work added to the fluid is negative.

Then the total work performed by the normal and shearing stresses per unit time is:

(4)

Substituting equation 2, 3 , 4 in equation 1 we get:

Where is dissipation function given by:


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F or a pe rf ect gas where and Employing the equation of continuity, we obtain in the case of a perfect gas:

And with:

We obtain:

In the case of constant thermal conductivity

In case of incompressible fluid we have div w = 0 and equation 5 with yields:


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R e f e r ences

Schlichting, H. - Boundary Layer Theory

enegy eqn derivation

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