Mindyard | A rigid tank of volume of 8 m 3 is being filled up with air from a pipeline connected through a valve. Initially the valve is closed and the tank is assumed to be completely evacuated. The air pressure and temperature inside the pipeline are maintained at 600 kPa and 306 K, respectively. The filling of the tank begins by opening the valve and the process ends when the tank pressure is equal to the pipeline pressure. During the filling process, heat loss to the surrounding is 1000 kJ. The specific heats of air at constant pressure and at constant volume are 1.005 kJ/kg.K and 0.718 kJ/kg.K, respectively. Neglect changes in kinetic energy and potential energy. The final temperature of the tank after the completion of the filling process is ________

A rigid tank of volume of 8 m³ is being filled up with air from a pipeline connected through a valve. Initially the valve is closed and the tank is assumed to be completely evacuated. The air pressure and temperature inside the pipeline are maintained at 600 kPa and 306 K, respectively. The filling of the tank begins by opening the valve and the process ends when the tank pressure is equal to the pipeline pressure. During the filling process, heat loss to the surrounding is 1000 kJ. The specific heats of air at constant pressure and at constant volume are 1.005 kJ/kg.K and 0.718 kJ/kg.K, respectively. Neglect changes in kinetic energy and potential energy.

The final temperature of the tank after the completion of the filling process is _________ K (round off to the nearest integer)

Correct Answer :

Correct answer is : 395.39

Solution :

The correct answer is 395.39.

1. Analysis of System Parameters from the Image and Problem Description:
Based on the provided schematic diagram and problem details:
- The rigid tank has a constant volume:
$V = 8 m^{3}$
- The tank is initially completely evacuated:
$m_{1} = 0 kg$
- The pipeline air is maintained at pressure:
$P_{i} = 600 kPa$
- The pipeline air is maintained at temperature:
$T_{i} = 306 K$
- The final pressure inside the tank is equal to the pipeline pressure:
$P_{2} = P_{i} = 600 kPa$
- During the filling process, heat is lost to the surroundings:
$Q = - 1000 kJ$
- Specific heat capacities of air are:
$C_{p} = 1.005 kJ / kg \cdot K$ and $C_{v} = 0.718 kJ / kg \cdot K$

2. Gas Constant of Air:
The characteristic gas constant $R$ is calculated as the difference between the constant-pressure and constant-volume specific heats:

$R = C_{p} - C_{v} = 1.005 - 0.718 = 0.287 kJ / kg \cdot K$

3. Unsteady Flow Energy Equation:
For an unsteady flow control volume, the conservation of energy is expressed as:

$\frac{d U}{d t} = {\overset{\cdot}{m}}_{i} h_{i} - {\overset{\cdot}{m}}_{e} h_{e} + \overset{\cdot}{Q} - \overset{\cdot}{W}$
Since there is no exit stream ( ${\overset{\cdot}{m}}_{e} = 0$ ) and the rigid tank performs no boundary work ( $\overset{\cdot}{W} = 0$ ), the mass balance and integrated energy balance reduce to:

$m_{2} - m_{1} = m_{i}$

$m_{2} u_{2} - m_{1} u_{1} = m_{i} h_{i} + Q$
Using the initial condition of the evacuated tank ( $m_{1} = 0$ , meaning $m_{i} = m_{2}$ ):

$m_{2} u_{2} = m_{2} h_{i} + Q$
Substituting internal energy $u_{2} = C_{v} T_{F}$ and enthalpy $h_{i} = C_{p} T_{i}$ (where $T_{F}$ is the final tank temperature):

$C_{v} T_{F} = C_{p} T_{i} + \frac{Q}{m_{2}}$

4. Final Mass Calculation using Ideal Gas Law:
The final mass $m_{2}$ of air inside the tank is given by:

$m_{2} = \frac{P_{2} V}{R T_{F}}$
Substituting this mass expression back into the energy equation:

$C_{v} T_{F} = C_{p} T_{i} + \frac{Q \cdot R \cdot T_{F}}{P_{2} V}$

5. Substituting Numerical Values and Solving for Final Temperature:
Substituting the known values (with $Q = - 1000 kJ$ as heat loss):

$0.718 \cdot T_{F} = 1.005 \cdot 306 - \frac{1000 \cdot 0.287 \cdot T_{F}}{600 \cdot 8}$

$0.718 \cdot T_{F} = 307.53 - \frac{287}{4800} \cdot T_{F}$

$0.718 \cdot T_{F} = 307.53 - 0.059792 \cdot T_{F}$
Grouping the terms containing $T_{F}$ :

$(0.718 + 0.059792) \cdot T_{F} = 307.53$

$0.777792 \cdot T_{F} = 307.53$

$T_{F} = \frac{307.53}{0.777792} \approx 395.39 K$
Hence, the final temperature of the tank after the completion of the filling process is approximately 395.39 K (or 395 K when rounded to the nearest integer).

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