An adiabatic 0.2-m^3 storage tank that is initially...

An adiabatic $0.2-\mathrm{m}^{3}$ storage tank that is initially evacuated is connected to a supply line that carries nitrogen ane at $225 \mathrm{K}$ and $10 \mathrm{MPa}$. A valve is opened, and nitrogen flows depe into the tank from the supply line. The valve is closed when the pressure in the tank reaches 10 MPa. Determine the final temperature in the tank ( $a$ ) treating nitrogen as an ideal gas, and $(b)$ using generalized charts. Compare your results to the actual value of $293 \mathrm{K}$.

Key Concepts

Real Gas Behavior and Throttling Effects

In processes such as flow through a valve, the gas may experience throttling, which is an irreversible expansion that can cause temperature changes not predicted by ideal gas analysis. Generalized charts incorporate real gas properties and the Joule?Thomson effect to yield more accurate predictions of final temperatures when intermolecular forces and non-ideal behaviors are significant.

Adiabatic Process

An adiabatic process involves no heat transfer between the system and its surroundings. In analyses where the storage tank is insulated, energy changes occur solely due to work or mass flow, which simplifies the energy balance by eliminating heat interactions.

Ideal Gas Behavior

Assuming ideal gas behavior uses the ideal gas law (PV = nRT) to relate pressure, volume, and temperature. This simplification ignores molecular interactions and the volume of the gas particles, allowing for straightforward prediction of final states under adiabatic conditions.

First Law of Thermodynamics for Open Systems

This concept extends energy conservation to flowing systems by accounting for energy carried by the mass entering or leaving the system. When a gas flows into a closed, adiabatic tank, the energy balance informs how internal energy and hence temperature evolve during the filling process.

Transcript

00:01 For this problem on the topic of thermodynamic property relations, we are told that an adiabatic storage tank is initially evacuated and then connected to a supply line that carries nitrogen at a given state.

00:13 The valve is open and nitrogen flows into the tank from the supply line until the pressure in the tank reaches a given value.

00:20 We want to determine the final temperature in the tank if we first assume that the nitrogen is an ideal gas and secondly by using the generalized charts.

00:29 So firstly, we'll write the mass balance equation.

00:36 So the mass in minus the mass out must equal to the change in mass of the system, delta m.

00:48 Now, here we assume there is uniform flow conditions and the kinetic and potential energy changes are negligible.

00:57 So we can see from this that the mass in must obviously equal the final mass of the system since the tank is initially evacuated.

01:10 We can also write an energy balance equation which tells us that the energy in minus the energy out must equal we change the energy of the system delta e.

01:24 We have a closed system.

01:28 And from here, we can see that the energy, the inlet enthalpy, must equal to the final internal energy of the system, m2, u2.

01:47 So the enthalpy of the supply line must equal the final internal energy of tank.

01:53 So if we combine the two balances, we get that the final internal energy, the final specific internal energy u2 must equal the inlet enthalpy h .i.

02:10 So now we can go about finding the final temperature of the system using an ideal gas relation.

02:18 So if we go to the ideal gas property table of nitrogen at 225 kelvin, we get the specific internal energy on the molar basis u2, which will be equal to hi is equal to the enthalpy at a temperature of 225 kelvin.

02:45 And from our tables, this value is 6 ,000, 537 kilojoules per kilo -mole.

03:02 So that's on a molar basis.

03:05 Now we can find a temperature that corresponds to this internal energy.

03:11 And this temperature, t2, is equal to 300 and 14 .8 kelvin.

03:23 So using our ideal gas relation for nitrogen, we find a 7 .4 % error from the theoretical value of 293 kelvin.

03:36 So that's our first answer.

03:39 Now secondly we want to assume, we want to use the generalize charts and find this final temperature again.

03:48 So if we use the generalize the end -dalpi departure chart, we can first find the reduced temperature of the inlet state, tri, which is simply the temperature of the inlet line ti over the critical temperature for nitrogen t -c, and this is 225 over 126 .2, which gives us a reduced temperature of 1 .78.

04:17 The reduced pressure, the r .i, of 4 .7...

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