Energy Equation

The conservation of energy can be formulated with applying the first law of thermodynamics on systems. The system here is in the flow field and receives the heat rate of Q. If the work done by the system to the surroundings is W then the change of total energy in the system becomes

ил, • .

— = Q — W (2.44)

At a given time, let the system under consideration coincide with the control volume we choose. If we let Ej denote the internal energy and Ek = V2MV2 the kinetic energy of the total mass in the system, then as the mass independent transferable quantities the specific internal energy becomes e = EJM and the specific kinetic energy reads as Ek/M = VV2. Which means the total specific energy in the control volume is g = e + VV2. Now, we can relate the energy changes of the system and the control volume using Eq. 2.44 in Eq. 2.27 to obtain the integral form of the energy equation

During the flow if we do not provide heat from outside, the system will heat the surroundings by the flux of internal heat from the control surface as follows Q = —&q • dA. On the other hand, the work of the stress tensor throughout the

~ • • dA. Now, if we substitute

the integral forms of the heat flux to the surroundings and the work done by the system on the surrounding, Eq. 2.45 becomes

— йq • dA + & (V •Tj • dA = O-JJJ(e + V2/2)p dV + &(e + V2/2)p ~ • dA

(2.46)

In Eq. 2.46 we have three surface integral terms. If all three area integrals are changed to volume integrals using the divergence theorem, and all the all volume integrals are collected together over the same control volume, we can write the differential form of the energy equation as follows

Here, s = e + VV2 denotes the specific total energy and Eq. 2.39 defines the stress tensor. The heat flux from a unit surface area reads as

q = — kVT + piUihi + qR + <s’v > (2.48)

Wherein, k denotes the heat conduction coefficient, the second term indicates the heat of diffusion, the third term represents radiative heat flux and the last term shows the turbulence heating. In summary, the global continuity is given by Eq. 2.31, continuity of species by 2.35, global momentum by 2.38 and the energy Equation by 2.47. Let us express these equations in Cartesian coordinates in conservative forms.