Next: 2.3.5 Ideal gas with Up: 2.3 A collection of Previous: 2.3.3 CO5BOLD equation of   Contents   Index

2.3.4 Derived thermodynamic coefficients

First, the missing derivative can be found from the relation:
 (38)

which is obtained from the equality of the mixed derivatives in Eq.(20), written as:
 (39)

Then
 (40)

 (41)

This relation is obtained by combining Eq.(20) with the identity
 (42)

The adiabatic sound speed is then obtained as
 (43)

 (44)

This relation is obtained by combining Eq.(20) with the identity
 (45)

 (46)

since
 (47)

 (48)

or
 (49)

We define the coefficients and through the relation
 (50)

Entropy change at constant density:
 (51)

This relation is obtained from the equality of the mixed derivatives in Eq.(20) together with Eq.(44). Entropy change at constant pressure:
 (52)

This relation is obtained from the equality of the mixed derivatives in Eq.(21) together with Eq.(46). Specific heat at constant density:
 (53)

To derive the specific heat at constant pressure, we start from the relation
 (54)

from which we get
 (55)

Using Eqs.(44) and (53), we obtain
 (56)

Now
 (57)

or
 (58)

hence
 (59)

and finally
 (60)

or
 (61)

Using Eqs.(27), (44), (52), (56), we finally obtain the relation for the specific heat at constant pressure:
 (62)

Alternatively, can be obtained from Eq.(32)
 (63)

or from
 (64)

once and are known (see below).
We can now express the thermodynamic coefficients provided by CO5BOLD in terms of , , , and :
 (65)

 (66)

 (67)

 (68)

 (69)

 (70)

We consider again Eq.(39), replacing by
 (71)

so
 (72)

The requirement that the mixed derivatives must be equal then yields
 (73)

or
 (74)

Finally,
 (75)

Comparison with Eq.(69) implies
 (76)

Similarly, replacing by
 (77)

in Eq.(39), we get
 (78)

and the requirement that the mixed derivatives must be equal then yields
 (79)

or
 (80)

or
 (81)

Since
 (82)

we finally obtain, using Eqs.(29), (65) and (70),
 (83)

and
 (84)

The isothermal sound speed is then obtained as
 (85)

Next: 2.3.5 Ideal gas with Up: 2.3 A collection of Previous: 2.3.3 CO5BOLD equation of   Contents   Index