Page 43 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

28 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

In fact the first law of thermodynamics postulates the possibility to

transform the heat (thermal energy) into mechanical energy. More pre-
cisely within a thermodynamical process (when the deformable material
subsystem passes from a thermodynamical state to a “neighboring” one),
E
the rate of change of the total energy re is equal to the elemental power

6L , ;
Tt of the direct forces exerted on the system plus the quantity of heat

oO
added to or taken out per unit time 2. so we have

DE 6L 6Q
Di dt’ dt

If 6Q = 0, ie., there is not a heat change with the surrounding, the
process (and the motion) are called adiabatic. Generally 6Q = 6Q.+6Qa,
where 6Q, and 6Qgq are, respectively, “contact actions” (the conduction
heat) and “distance actions” (the radiation heat). By accepting (to

introduce the corresponding densities) that Se and Sea are absolutely
continuous functions of surface and, respectively, mass, we will have that

OQ _ 6Qaq _
we / g(v,n,t)da, “4 = / pra(t,t)dv,

S D
D being, at the respective moment, the configuration of the subsystem

P and § its boundary.
Under these circumstances, for any deformable continuum subsystem
P, the first law of thermodynamics can be written

D 1
iy f oles ge?) do= fw Bt ardat f pe-v+ra)av

D S D

On the other side the energy variation principle, stated in the previous
section, is

D 1 9» _
=f 5° dw [ 2wdv = fv Tda+ f pt-vav

D D S D

such that, using also the transport formula and the continuity equation,
the first law of thermodynamics could be written

| vedo = faten,s) da+ [ (2w+ pra) dv.

D S D