3) Air at standard conditions enters the compressor shown in Fig. Q3 at a rate of 0.3m3/s. It leaves the tank through a 3 cm diameter pipe with a density of 1.8 kg/m3 and the uniform speed of 210 m/s. (a) Determine the rate (kg/s) at which the mass of air in the tank is increasing or decreasing; (b) Determine the average time rate of change of air density within the tank. Tank Volume is 0.6m^3
To solve this problem, we need to use the principles of conservation of mass and the equation of continuity.
(a) To determine the rate at which the mass of air in the tank is increasing or decreasing, we need to calculate the mass flow rate of air entering and leaving the tank.
The mass flow rate entering the tank can be determined using the equation:
Mass flow rate (kg/s) = Density (kg/m^3) * Volume flow rate (m^3/s)
Given that the air enters the compressor at a rate of 0.3 m^3/s and the density of air is 1.8 kg/m^3, the mass flow rate entering the tank is:
Mass flow rate = 1.8 kg/m^3 * 0.3 m^3/s = 0.54 kg/s
The mass flow rate leaving the tank through the 3 cm diameter pipe can also be calculated using the equation of continuity:
Flow rate (m^3/s) = Area (m^2) * Velocity (m/s)
The area of a pipe is given by the equation:
Area (m^2) = π * (diameter/2)^2
Given that the diameter of the pipe is 3 cm (or 0.03 m) and the velocity of air is 210 m/s, we can calculate the area:
Area = π * (0.03/2)^2 = 0.000706858 m^2
Now we can find the flow rate:
Flow rate = 0.000706858 m^2 * 210 m/s = 0.148
Since the density of air leaving the tank is 1.8 kg/m^3, the mass flow rate leaving the tank is:
Mass flow rate = 1.8 kg/m^3 * 0.148 m^3/s = 0.2664 kg/s
Therefore, the rate at which the mass of air in the tank is changing is:
Rate of change = Mass flow rate entering - Mass flow rate leaving
= 0.54 kg/s - 0.2664 kg/s
= 0.2736 kg/s
So, the mass of air in the tank is increasing at a rate of 0.2736 kg/s.
(b) The average time rate of change of air density within the tank can be calculated using the equation:
Average time rate of change of density (kg/m^3s) = (Final density - Initial density) / Δt
In this case, we don't have enough information to calculate the density at the final time or the time interval (Δt). Therefore, we cannot determine the average time rate of change of air density within the tank without additional information.