A fan drawing electricity at a rate of 1.5kw is located within rigid enclosure of 3m*4m*5m. Enclosure is filled with air at 27oC, 0.1 MPa. The fan operates for 30 minutes. Assume the ideal gas model, determine
a) mass (in kg) b) final T (in oC) and c) final pressure (in MPa). There is no heat transfer between the enclosure and surroundings. Ignore the volume occupied by the fan and assume the fan stores no energy.
Find the mass using density.
Find final temp using the specific heat capacity of air
for final pressure, assume ideal gas law
To solve the problem, we need to perform the following steps:
Step 1: Determine the initial conditions
Given:
- The dimensions of the enclosure: 3m x 4m x 5m
- The initial temperature: 27°C
- The initial pressure: 0.1 MPa
Step 2: Calculate the volume of the enclosure
The volume of the enclosure is 3m x 4m x 5m = 60 cubic meters.
Step 3: Calculate the initial mass of the air inside the enclosure
We can use the ideal gas equation to calculate the initial mass of the air inside the enclosure.
The ideal gas equation is given by PV = mRT, where:
- P is the pressure
- V is the volume
- m is the mass
- R is the specific gas constant for air, which is 0.287 kJ/kg·K
- T is the temperature converted to Kelvin
Using the given values, we have:
0.1 MPa x 60 m^3 = m x 0.287 kJ/kg·K x (27 + 273) K
Simplifying the equation, we get:
6 MPa = m x 0.287 x 300
Solving for m, we find:
m = 6 MPa / (0.287 x 300) = 0.693 kg
So, the initial mass of the air inside the enclosure is 0.693 kg.
Step 4: Calculate the work done by the fan
The work done by the fan can be calculated using the formula:
Work = Power x Time
Given:
- Power = 1.5 kW
- Time = 30 minutes = 30/60 hours = 0.5 hours
Using the given values, we have:
Work = 1.5 kW x 0.5 hours = 0.75 kWh
Step 5: Calculate the final temperature and pressure
Since no heat transfer occurs between the enclosure and surroundings, the process is adiabatic. In an adiabatic process, we can use the equation:
T2 / T1 = (P2 / P1)^((γ-1)/γ)
where:
- T2 is the final temperature
- T1 is the initial temperature
- P2 is the final pressure
- P1 is the initial pressure
- γ is the heat capacity ratio (for air, γ ≈ 1.4)
We can rearrange the equation to solve for T2:
T2 = T1 x (P2 / P1)^((γ-1)/γ)
Using the given values, we have:
T1 = 27°C = 27 + 273 = 300 K
P1 = 0.1 MPa
We need to determine the final pressure P2. Since the fan has done work on the system, the pressure will increase. However, without more information, we cannot determine the exact final pressure.
Let's assume a final pressure P2 = 2 P1 = 0.2 MPa.
Using the equation, we can calculate T2:
T2 = 300 K x (0.2 MPa / 0.1 MPa)^((1.4-1)/1.4)
Calculating the value, we find:
T2 ≈ 300 K x 1.122 ≈ 336.7 K
To convert the final temperature back to °C, we subtract 273:
Final T ≈ 336.7 - 273 ≈ 63.7°C
So, the final temperature of the air inside the enclosure is approximately 63.7°C.
Step 6: Calculate the final mass
Since the mass of air inside the enclosure remains constant and no additional air is introduced or removed, the final mass will be the same as the initial mass.
Therefore, the final mass of the air inside the enclosure is 0.693 kg.
Step 7: Summarize the results
a) The mass of the air inside the enclosure is 0.693 kg.
b) The final temperature of the air inside the enclosure is approximately 63.7°C.
c) The final pressure of the air inside the enclosure is assumed to be 0.2 MPa.
To solve this problem, we need to apply the laws of thermodynamics to determine the mass, final temperature, and final pressure of the air inside the enclosure after the fan has operated for 30 minutes.
Let's break down the problem step by step:
Step 1: Calculate the total energy supplied by the fan.
- We are given that the fan is drawing electricity at a rate of 1.5 kW for 30 minutes.
- Convert 30 minutes to seconds: 1 minute = 60 seconds, so 30 minutes = 30 * 60 = 1800 seconds.
- Calculate the total energy supplied: Energy (in Joules) = Power (in Watts) × Time (in seconds).
- Energy = 1.5 kW × 1800 s = 2700 kJ = 2700 × 10^3 J.
Step 2: Calculate the initial volume of the enclosure.
- The initial volume of the enclosure is given as 3 m × 4 m × 5 m = 60 m^3.
Step 3: Calculate the initial mass of the air inside the enclosure.
- We can use the ideal gas law: PV = mRT, where P is the pressure, V is the volume, m is the mass, R is the gas constant, and T is the temperature in Kelvin.
- Rearrange the equation to solve for the mass: m = PV/RT.
- The given conditions inside the enclosure are: P = 0.1 MPa = 0.1 × 10^6 Pa, V = 60 m^3, T = 27°C = 27 + 273 = 300 K.
- The specific gas constant for air is R = 287 J/(kg·K).
- Calculate the initial mass: m = (0.1 × 10^6 Pa) × (60 m^3) / (287 J/(kg·K) × 300 K).
Step 4: Calculate the final temperature.
- When the energy is supplied by the fan, it causes a change in temperature inside the enclosure.
- We need to know how much the temperature increased in order to calculate the final temperature.
- We can use the first law of thermodynamics: ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat transfer, and W is the work done by the system.
- Since there is no heat transfer, Q = 0.
- The work done by the fan is given by W = Energy supplied = 2700 × 10^3 J.
- ΔU = Q - W = 0 - 2700 × 10^3 J.
- ΔU can also be expressed as ΔU = m × Cv × ΔT, where Cv is the specific heat capacity at constant volume and ΔT is the change in temperature.
- Rearranging the equation to solve for ΔT: ΔT = -W / (m × Cv).
- The specific heat capacity at constant volume for air is approximately Cv = 20.8 J/(kg·K).
- Calculate the change in temperature: ΔT = -2700 × 10^3 J / (m × 20.8 J/(kg·K)).
Step 5: Calculate the final pressure.
- Since there is no heat transfer and no work done by or on the system, the process is adiabatic and isentropic.
- For an adiabatic process, we can use the ideal gas law with the relationship P × V^γ = constant, where γ is the heat capacity ratio for air (approximately 1.4).
- Rearranging the equation to solve for the final pressure: P2 = P1 × (V1 / V2)^(γ).
- We know P1 = 0.1 MPa = 0.1 × 10^6 Pa and V1 = 60 m^3. We need to find V2.
- The change in volume is given by: ΔV = V2 - V1 = m × R / P2 × (ΔT).
- Rearranging the equation to solve for V2: V2 = V1 + (m × R / P2 × (ΔT)).
By following these steps and substituting the relevant values into the equations, you should be able to find the mass, final temperature, and final pressure of the air inside the enclosure after the fan has operated for 30 minutes.