Thermodynamics - closed systems

[MegaDeth]

Homework Statement

A cylinder fitted with a piston contains air at 1.0 Bar and 17°C. The gas is compressed according to the law PVⁿ = const., until the pressure is 4 bar when the specific volume is found to be 28% of the initial value. Heat is then added to the air at constant pressure until the volume is doubled. The same amount of heat is now removed from the air at constant volume.

Determine the value of the index n in the compression process.
Find also
(a) the overall change in internal energy/kg of air and
(b) the final pressure of the air.


R = 0.287 kJ/kgK, Cp = 1.005718kJ/kgK

Homework Equations

(p_s/p₁) = (T₂/T₁)^{(n/n - 1)}

The Attempt at a Solution

4 = (T₂/290)^{(n/n - 1)}

I'm not sure how to find n seeing as I haven't got T₂, any ideas?

 

[Mark44]

Homework Statement

A cylinder fitted with a piston contains air at 1.0 Bar and 17°C. The gas is compressed according to the law PVⁿ = const., until the pressure is 4 bar when the specific volume is found to be 28% of the initial value. Heat is then added to the air at constant pressure until the volume is doubled. The same amount of heat is now removed from the air at constant volume.

Determine the value of the index n in the compression process.
Find also
(a) the overall change in internal energy/kg of air and
(b) the final pressure of the air.


R = 0.287 kJ/kgK, Cp = 1.005718kJ/kgK

Homework Equations

(p_s/p₁) = (T₂/T₁)^{(n/n - 1)}

The Attempt at a Solution

4 = (T₂/290)^{(n/n - 1)}

I'm not sure how to find n seeing as I haven't got T₂, any ideas?

Isn't T₂ = T₁?

The same amount of heat is now removed from the air at constant volume.

BTW, your relevant equation is incorrectly written. In some places you have more parentheses than you need (e.g., (p_s/p₁) is the same as p_s/p₁ ) , and in the exponent on the right side, there are not enough of them.

Your exponent, as written, is equal to zero. n/n - 1 is the same as (n/n) - 1 = 1 - 1 = 0. If you mean $\frac{n}{n - 1}$ rather than $\frac n n - 1$, write it as (n/(n - 1)).

 

[MegaDeth]

How will that work then?

 

[Chestermiller]

You know the ratio of the specific volumes and the ratio of the pressures. This gives you enough information to get n.

 

[rezeew]

I would approach this problem by first understanding the concept of thermodynamics in closed systems. A closed system is one that does not exchange matter with its surroundings, but can exchange energy in the form of heat or work. In this scenario, the cylinder and piston containing air can be considered a closed system.

The first step would be to determine the value of n in the compression process. According to the given information, the law PVn = const. is followed during the compression process. This can be rewritten as P1V1n = P2V2n, where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume. This equation can be further simplified as (P1/P2) = (V2/V1)n. Substituting the given values, we get (1/4) = (2/7)n. Solving for n, we get n = 1.5.

Next, we need to find the overall change in internal energy per kg of air. The change in internal energy can be calculated using the formula ΔU = Q - W, where Q is the heat added and W is the work done. In this case, the work done is zero as the process is carried out at constant pressure. Therefore, the change in internal energy is equal to the heat added. From the given information, we know that the heat added during the constant pressure process is equal to the heat removed during the constant volume process. Thus, the overall change in internal energy is zero.

Finally, we need to determine the final pressure of the air. We can use the ideal gas law, PV = mRT, to calculate the final pressure. We know that the volume is doubled, and the temperature remains constant during the constant pressure process. Therefore, the final pressure can be calculated as P2 = (2V1/V1)P1 = 2P1 = 2(1 bar) = 2 bar.

In conclusion, the value of n in the compression process is 1.5, the overall change in internal energy per kg of air is zero, and the final pressure of the air is 2 bar. These calculations demonstrate the principles of thermodynamics in closed systems and how they can be applied to solve real-world problems.