A tank with an internal volume of 1 m3 contains air at 800 kPa and 25°C. A valve on the tank is opened, allowing air to escape, and the pressure inside quickly drops to 150 kPa, at which point the valve is closed. Assume there is negligible heat transfer from the tank to the air left in the tank.
- (a) Using the approximation he ≈ constant = he,avg = 0.5 (h1 + h2), calculate the mass withdrawn during the process.
- (b) Consider the same process but broken into two parts. That is, consider an intermediate state at P2 = 400 kPa, calculate the mass removed during the process from P1 = 800 kPa to P2 and then the mass removed during the process from P2 to P3 = 150 kPa, using the type of approximation used in part (a), and add the two to get the total mass removed.
- (c) Calculate the mass removed if the variation of he is accounted for.
FIGURE P5–185
(a)
The mass withdrawn during the process.
Answer to Problem 185RP
The mass withdrawn during the process is
Explanation of Solution
Write the equation of mass balance.
Here, the inlet mass is
The change in mass of the system for the control volume is expressed as,
Here, the suffixes 1 and 2 indicates the initial and final states of the system.
Consider the tank as the control volume. Initially the tank is filled with air and the valve is in closed position, further no other mass is allowed to enter the tank. Hence, the inlet mass is neglected i.e.
Rewrite the Equation (I) as follows.
Write the formula for initial and final mass of air present in the tank.
Here, the mass of air is
Write the energy balance equation.
Here, the heat transfer is
When the valve is opened and air starts escape from the tank. Neglect the heat transfer and work done i.e.
The Equation (V) reduced as follows.
The enthalpy and internal energy in terms of temperature and specific heats are expressed as follows.
Rewrite the Equation (VI) as follows.
The temperature of the air while exiting the tank is considered as the average temperature of initial and final temperatures.
Refer Table A-1, “Molar mass, gas constant, and critical-point properties”.
The gas constant
Refer Table A-2b, “Ideal-gas specific heats of various common gases”.
The specific heat at constant pressure
Conclusion:
Substitute
Substitute
Substitute
Use Engineering Equation Solver (EES) or online calculator to solve the Equation (VIII) and obtain the value of
Substitute
Substitute
Thus, the mass withdrawn during the process is
(b)
The mass withdrawn during the pressure reduced from
Answer to Problem 185RP
The total mass withdrawn during the process 1-3 is
Explanation of Solution
Consider Process 1-2:
The pressure drop from
Substitute
Substitute
Substitute
Use Engineering Equation Solver (EES) or online calculator to solve the Equation (IX) and obtain the value of
Substitute
Substitute
Thus, the mass withdrawn during the process 1-2 is
Consider Process 2-3:
The pressure drop from
Here,
Substitute
Substitute
Substitute
Use Engineering Equation Solver (EES) or online calculator to solve the Equation (X) and obtain the value of
Substitute
Substitute
Thus, the mass withdrawn during the process 2-3 is
The total mass withdrawn during the process 1-3 is as follows.
Thus, the total mass withdrawn during the process 1-3 is
(c)
The mass withdrawn during the process if there is variation in
Answer to Problem 185RP
The mass withdrawn during the process is
Explanation of Solution
Write the general mass balance equation.
Here, the inlet mass flow rate is
Refer Equation (XI).
Write the mass balance equation for the given system.
Rewrite the Equation (XII) as follows.
Write the general energy rate balance equation.
Here, the rate of total energy in is
The system is at steady state. Hence, the rate of change in net energy of the system becomes zero.
Refer Equation (XIII).
Write the energy balance equation for the given system.
Here, the mass is
Substitute
The enthalpy and internal energy is expressed as follows.
Substitute
The mass of air in terms ideal gas is expressed as follows.
Rewrite the Equation (XVI) as follows.
Using
Substitute
Here,
Integrate the Equation (XVIII) at the initial-1 and final-2 states.
Refer Table A-2(a), “Ideal-gas specific heats of various common gases”.
The specific heat ratio
Conclusion:
Substitute
Substitute
Substitute
Thus, the mass withdrawn during the process is
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