## 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).

##
__
__**Hint**

**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**

**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.

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