An oxygen cylinder of volume 7 cubic metre has oxygen pressure 150 kg/cm^2. In OPEN state, the nozzle has diameter 1 cm. How much time is required for pressure to come down to 4 kg/cm^2 if nozzle is opened?
To find the time required for the pressure to come down to 4 kg/cm^2, we need to use the principles of gas flow through a nozzle and apply the ideal gas law.
The first step is to convert the pressure from kg/cm^2 to pascals (Pa), as the ideal gas law uses SI units.
Given:
Volume of oxygen cylinder (V) = 7 cubic meters
Initial pressure (P1) = 150 kg/cm^2
Final pressure (P2) = 4 kg/cm^2
Nozzle diameter (d) = 1 cm
We can calculate the initial pressure in pascals (Pa) using the conversion factor:
1 kg/cm^2 = 98,066.5 pascal
Therefore, P1 = 150 kg/cm^2 * 98,066.5 Pa/kg.cm^2 = 14,709,975 Pa
Similarly, we can calculate the final pressure (P2) in pascals:
P2 = 4 kg/cm^2 * 98,066.5 Pa/kg.cm^2 = 392,266 Pa
Next, we need to determine the flow rate through the nozzle. The flow rate (Q) can be calculated using the ideal gas law equation:
Q = (A * v)
where,
Q = Flow rate (in cubic meters per second)
A = Area of the nozzle (in square meters)
v = Velocity of the gas (in meters per second)
The area of the nozzle (A) can be calculated using the formula for the area of a circle:
A = π * (d/2)^2
where,
π (pi) is a constant approximately equal to 3.14159
Substituting the given diameter, we have:
A = 3.14159 * (1 cm/2)^2
Next, we need to determine the velocity of the gas (v) at the nozzle. This can be calculated using Bernoulli's equation, assuming steady flow of the gas:
P1 + 0.5 * ρ * v1^2 = P2 + 0.5 * ρ * v2^2
where,
ρ (rho) = Density of the gas (assumed constant)
v1 = Initial velocity of the gas (at the cylinder)
v2 = Final velocity of the gas (at the nozzle)
In this case, since the cylinder and nozzle are both open to the atmosphere, we can assume that both v1 and v2 are approximately equal to the velocity of the gas in the room, which is negligible. Therefore, we can assume that v1^2 and v2^2 are both approximately equal to zero.
Using Bernoulli's equation, we can simplify it to:
P1 = P2, since ρ * v1^2 = ρ * v2^2 ≈ 0
Therefore, P1 = 14,709,975 Pa = P2 = 392,266 Pa
Since the pressures are equal, the flow rate (Q) through the nozzle is also equal to zero.
Hence, there is no flow through the nozzle and the time required for the pressure to come down to 4 kg/cm^2 will be infinite or extremely large.