# 2nd Law of Thermodynamics

Please see attachment for original question format.

A combined air filled, spring loaded cylinder has a frictionless piston of area 0.012 m^2 that rests against the spring. The spring loaded end of the cylinder is open to atmosphere. The spring force in terms of piston movement is given by:

Spring Force = k X

Show that i) the change in pressure is proportional to change in volume and spring position X such that:

dP/dV = k/A^2 where A is the piston cross section area

The air temperature is increased by means of an electric heater so that the piston moves by 0.04 m with the pressure in the cylinder rising to 5 bars. The piston. The cylinder is then allowed to cool down to a temperature of 300K at which the pressure in the cylinder has dropped to 1 bar. ii) Calculate the volumetric change in the cylinder due to heating. If the air in the cylinder had an initial volume of 0.0005 m^3, iii) calculate the mass of air.

The cylinder is heated again so the volume becomes 1.3 times the original volume. Calculate iv) the final pressure, v) final temperature, vi) the work done if it is assumed that when the pressure-volume graph is extended, it would pass through origin, vii) the change in internal energy and the heat transfer.

Note: Cp = 1005J/kgK , Cv = 718J/kgK for air.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

Please see attachment.

Problem:

A combined air filled, spring loaded cylinder has a frictionless piston of area 0.012 m2 that rests against the spring. The spring loaded end of the cylinder is open to atmosphere. The spring force in terms of piston movement is given by: âˆ†Spring Force = k âˆ†X Show that:

i) the change in pressure is proportional to change in volume and spring position X such that: âˆ†P/âˆ†V = k/A2 where A is the piston cross section area;

The air temperature is increased by means of an electric heater so that the piston moves by 0.04 m with the pressure in the cylinder rising to 5 bars. The cylinder is then allowed to cool down to a temperature of 300K at which the pressure in the cylinder has dropped to 1 bar.

ii) Calculate the volumetric change in the cylinder due to heating.

If the air in the cylinder had an initial volume of 0.0005 m3,

iii) calculate the mass of air.

The cylinder is heated again so the volume becomes 1.3 times the original volume. Calculate

iv) the final pressure,

v) final temperature,

vi) the work done if it is assumed that when the pressure-volume graph is

extended, it would pass through origin,

vii) the change in internal energy and the heat transfer.

Note: Cp = 1005J/kgK , Cv = 718J/kgK for air

Solution:

i) The model of analysis is shown in the figure below:

If one denotes p = actual pressure in the cylinder and p0 = ambient pressure, the equation of the mechanical equilibrium can be written as:

( 1)

where k = elastic constant of the spring (unknown)

Î”x = displacement from the initial equilibrium position (unloaded spring)

The same equation can be put in another form:

( 2)

Since the variation of the volume is

( 3)

it follows from the previous equation that:

( 4)

Hence:

( 5)

ii) The volumetric change can be computed using equation (3):

( 6)

Since the final pressure after heating becomes 5 bar, the elastic constant of the spring can be determined, using (2):

( 7)

Assuming that p0 = 1 bar = 105 Pa, we will get:

( 8)

iii) The mass of the air in the cylinder will be given by the law of ideal gases, applied to the state of the air after heating and cooling, respectively when the pressure is p0 = 1 bar, temperature is T0 = 300 K and the volume is

V0 = 0.0005 m3 (the spring is unloaded, since p = p0 = 1 bar):

( 9)

where R = gas constant for air, R = 287 J/kg.K. Actually, this can be derived in this problem by using Mayer's formula:

( 10)

Introducing in (9), the mass of the air in the cylinder will be found:

( 11)

iv) The new volume after the second heating is

( 12)

so that

( 13)

Using the result (4), the final pressure will be determined:

( 14)

Numerically:

( 15)

v) The final temperature will be found by using the ideal gases law (9):

( 16)

vi) According to the problem statement, the evolution of the air in a p-V diagram will have a linear form passing through the origin, as in the figure below:

The work done by the piston is the hatched area in the figure of above and it is given by the equation:

( 17)

(the pressure difference shows that the piston needs to "defeat" the ambient pressure on the open side of the cylinder).

Using the formula of the area of a right triangle and the result (13), we will have:

( 18)

Check: the work received by the spring (which increases its potential energy) is given by the formula:

( 19)

Numerically:

( 20)

vii) Assuming constant specific heats, the change in the internal energy is given by formula:

( 21)

Numerically:

( 22)

Using the 1-st law of Thermodynamics, the heat exchange with the surroundings will be found:

( 23)

(the work is considered as positive when it is done by the system and the heat is positive when it is absorbed by the system).

Using the results (20) and (22), we will find:

( 24).

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