Untitled - Aerobib - Universidad Politécnica de Madrid

66 CHAPTER 3. GENERAL EQUATIONS
and the internal energy of the mass contained in V is ρV u. Similarly, the kinetic
energy of the mass contained in V is ρV 1 2 v2 . Therefore, the total energy 8 of the
mass contained in V is ρV ( u + 1 2 v2) and its time variation when following the
motion of V is
(
D
ρV (u + 1 )
Dt 2 v2 ) = ρV D Dt (u + 1 2 v2 ). (3.20)
Consequently, the time variation of the energy of the mass contained in the unit
volume is
b) Work.
The work done by ¯f per unit time is
ρ D Dt (u + 1 2 v2 ). (3.21)
¯v · ¯f = ¯v · (¯n · τ e ) = ¯n · (¯v · τ e ) , (3.22)
and the work per unit time done by the surface forces that act upon the unit
volume is ∇ · (¯v · τ e ). 9 The work per unit time done by the mass forces that act
upon the unit volume is ρ ¯F · ¯v. Consequently, the work per unit time done by all
the forces that act upon the unit volume is
∇ · (¯v · τ e ) + ρ ¯F · ¯v. (3.23)
c) Heat.
The heat received by the element V through dσ per unit time is −¯q · ¯n dσ, where
¯q is the vector of the heat flux defined in chapter 2. Therefore, the heat per unit
time received by the unit volume is
−∇ · ¯q. (3.24)
In the phenomena normally studied by the Aerothermochemistry the radiation
effects are of secondary importance. Furthermore, great difficulty is generally
encountered for the study of the influence of these effects. Nevertheless, if they
are to be taken into consideration, the vector ¯q r of the radiation flux must be
added to the vector ¯q of the heat flux. Information regarding the radiation flux
can be found in the work of Hirschfelder, Curtiss and Bird, mentioned in Ref.
[4], pp. 720 and following.
8 Other forms of energy different from the internal and the kinetic energies, as for example the radiation
energy, are not considered in the present study.
9 It can be proved in the same manner as for the calculation of the resultant of the surface forces.