Untitled - Aerobib - Universidad Politécnica de Madrid

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 79
By virtue of the second assumption, the diffusion equation (3.80) reduces to
which is the expression of Fick’s Law.
dY
dx = m (Y − ε), (3.84)
ρD 12
If, furthermore, the heat capacity c p is independent from temperature, then
equation (3.83), considering (3.60), takes the form
m
(c p T − qε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e, (3.85)
3 dx
where e must now include the constant terms that come from (3.60).
3.10 Stationary, one-dimensional motion of ideal gases
with heat addition
Let us assume in the above case that the action of viscosity and thermal conductivity
can be neglected in the equation of motion (3.72) and energy (3.85). This is justified if
the gradients of velocity and temperature are not very ”large”. 21 Then (3.72) reduces
to
p + mv = i. (3.86)
Likewise (3.85) reduces to
m
(c p T − qε + 1 )
2 v2 = e. (3.87)
These two equations together with (3.65)
ρv = m (3.88)
and the equation of state (3.39)
p
ρ = R mT, (3.89)
determine the values for p, ρ, T and v corresponding to each value of ε, if the values of
such variables corresponding to a given value of ε are known, for example, to ε = 0.
This enables the determination of the constants of Eqs. (3.86), (3.87) and (3.88). In
such case, in Eq. (3.87) the term Q = qε represents the heat added to the gas per unit
mass from the initial state ε = 0. The result is independent from the way in which the
heat is added; be it by chemical reactions or transmitted from external sources.
21 See chapter 5 for the exact meaning of this expression.