Page 37 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: JYD/GKJ
CB908/Date
0521853265c01
16
INTRODUCTION
P 0 521 85326 5 May 20, 2005 12:20

PISTON
Figure 1.6. Equilibrium of an isothermal gas.

P T INSULATED
CYLINDER

8. Consider a constant-pressure and constant-mass reactor so that volume change
is permitted. Assume Q w = 0. Hence, show that

dM cv ω k dH cv ˙
= R k V cv and = Q chem V cv ,
dt dt
where V cv = M cv R u T /(pM mix ), R u is the universal gas constant, the mixture
−1
molecular weight M mix = ( ω k /M k ) , T = H cv /(M cv C p mix ), and H cv =
k
ρ m V cv h.
9. Consider a 2D natural convection problem in which the direction of gravity is
aligned with the negative x 2 direction. Use the definition of the coefficient of
cubical expansion β =−ρ −1 ∂ρ/∂T and express the B 2 term in Equation 1.3
ref
in terms of β. Now, examine whether it is possible to redefine pressure as,
∗
say, p = p + ρ ref gx 2 in Equations 1.3 for i = 1 and 2. If so, recognise that
ρ ref gx 2 is nothing but a hydrostatic variation of pressure.
10. Consider a frictionless piston–cylinder assembly containing isothermal gas
as shown in Figure 1.6. The assembly is perfectly insulated. Now, consider
the unlikely circumstance in which the external pressure p is not equal to
internal pressure p. Discuss the consequences if the gas temperature is to remain
constant.