Question 7.5.2: Pressurizing an Air Cylinder Air at temperature T passes thr...
Pressurizing an Air Cylinder
Air at temperature T passes through a valve into a rigid cylinder of volume V, as shown in Figure 7.5.1. The mass flow rate through the valve depends on the pressure difference Δp = p_{i} − p, and is given by an experimentally determined function:
q_{mi} = f (Δp) (1)
Develop a dynamic model of the gage pressure p in the container as a function of the input pressure p_{i} . Assume the filling process is isothermal.
From conservation of mass, the rate of mass increase in the container equals the mass flow rate through the valve. Thus, if p_{i} − p > 0, from equation (1)
\frac{d m}{d t} = q_{mi} = f (\Delta p)
But
\frac{d m}{d t} = \frac{d m}{d p} \frac{d p}{d t} = C \frac{d p}{d t}
and thus,
C \frac{d p}{d t} = f (\Delta p) = f (p_{i} − p) (2)
where the capacitance C is given by (7.5.6) with n = 1.
C = \frac{m V}{np V} = \frac{V}{n R_{g} T} (7.5.6)
C = \frac{V}{R_{g} T}
If the function f is nonlinear, then the dynamic model is nonlinear.