Nozzle Count
In an experiment, a machine uses several nozzles that can only individually produce 40 kg/s of an ideal gas mixture with an average molecular weight of 20. How many nozzles are required to produce a volumetric flow rate of 25 m^3/s at 320 K and 420 kPa? Note the universal gas constant is 8.314 kPa∙m^3/(kmol∙K).
Expand Hint
Ideal gas formula:
$$$Pv=RT$$$
where
$$P$$
is pressure,
$$v$$
is the specific volume,
$$R$$
is the gas constant, and
$$T$$
is the absolute temperature.
Hint 2
$$$\dot{m}=\rho \times \dot{V}$$$
where
$$\dot{m}$$
is the mass flow rate,
$$\rho$$
is density of the fluid, and
$$\dot V$$
is volumetric flow rate.
Ideal gas formula:
$$$Pv=RT$$$
where
$$P$$
is pressure,
$$v$$
is the specific volume,
$$R$$
is the gas constant, and
$$T$$
is the absolute temperature. Since
$$R=\bar{R}/M$$
, where
$$\bar{R}$$
is the universal gas constant and
$$M$$
is the molecular weight:
$$$v=\frac{\bar{R}T}{MP}$$$
To solve for mass flow rate of the entire machine:
$$$\dot{m}=\rho \times \dot{V}$$$
where
$$\rho$$
is density of the fluid, and
$$\dot V$$
is volumetric flow rate. Since the specific volume (
$$v$$
) is the reciprocal of the fluid’s density (
$$\rho$$
):
$$$\dot{m}=\frac{\dot{V}}{v}=\frac{\dot{V}MP}{\bar{R}T}$$$
$$$=\frac{(20kg/kmol)(420kPa)(25m^3/s)}{(320K)(8.314kPa\cdot m^3/(kmol\cdot K))}=\frac{210,000}{2,660.5}=78.9\:kg/s$$$
Since each nozzle produces 40 kg/s:
$$$\frac{78.9kg/s}{40kg/s}=1.9\:nozzles$$$
The machine will need 2 nozzles.
2
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Similar Problems from FE Section: PVT Behavior
179. PVT Behavior of an Ideal Gas
183. Specific Weight
253. Ideal Gas
288. New Volume
513. Nozzles
555. Ideal Volume