Question 7.2: Dry air at a constant temperature of 20°C (293 K) passes thr...
Dry air at a constant temperature of 20°C (293 K) passes through a valve into a rigid cubic container of 1 m on each side (see Figure 7.2). The pressure pi at the inlet of the valve is constant, and it is greater than p. The valve resistance is approximately linear, and R = 1000 Pa·s/kg. Assume the filling process is isothermal. Develop a mathematical model of the pressure p in the container.
Solution
Applying the law of conservation of mass gives
Note that
\frac{dm}{dt}=\frac{dm}{dp} \frac{dp}{dt}= C \frac{dp}{dt}.Air at room temperature and low pressure can be approximated as an ideal gas. For an isothermal process, the pneumatic capacitance of the container is
C=\frac{V}{R_{air}T}.
The linear valve resistance R is defined as
R=\frac{p_i-p}{p_iq_i}.Thus, the differential equation of the system is
\frac{V}{R_{air}T} \frac{dp}{dt}=\frac{p_i-p}{R}or
\frac{RV}{R_{air}T} \frac{dp}{dt} +p=p_iwhere RV/( R_{air} T)= 1000 \times 1^3/(287.06 \times 293) = 1.19 \times 10^{−2} s.