## Nozzles

In an experiment, a machine uses several nozzles that can only individually produce 20 kg/s of an ideal gas mixture with an average molecular weight of 15. How many nozzles are required to produce a volumetric flow rate of 30 m^3/s at 300 K and 500 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 ( $$\rho$$ ): $$\dot{m}=\frac{\dot{V}}{v}=\frac{\dot{V}MP}{\bar{R}T}$$$
$$=\frac{(15kg/kmol)(500kPa)(30m^3/s)}{(300K)(8.314kPa\cdot m^3/(kmol\cdot K))}=\frac{225,000}{2,494.2}=90.2\:kg/s$$$Since each nozzle produces 20 kg/s: $$\frac{90.2kg/s}{20kg/s}=4.5\:nozzles$$$
The machine will need 5 nozzles.
5
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