A piston/cylinder arrangement, shown in Fig. P3.111,...

A piston/cylinder arrangement, shown in Fig. P3.111, contains air at 250 $\mathrm{kPa}$ and $300^{\circ} \mathrm{C}$. The 50 -kg piston has a diameter of $0.1 \mathrm{m}$ and initially pushes against the stops. The atmosphere is at 100 $\mathrm{kPa}$ and $20^{\circ} \mathrm{C}$. The cylinder now cools as heat is transferred to the ambient surroundings. a. At what temperature does the piston begin to move down? b. How far has the piston dropped when the temperature reaches ambient?

A piston/cylinder arrangement, shown in Fig. P3.111, contains air at 250 $\mathrm{kPa}$ and $300^{\circ} \mathrm{C}$. The 50 -kg piston has a diameter of $0.1 \mathrm{m}$ and initially pushes against the stops. The atmosphere is at 100 $\mathrm{kPa}$ and $20^{\circ} \mathrm{C}$. The cylinder now cools as heat is transferred to the ambient surroundings.
a. At what temperature does the piston begin to move down?
b. How far has the piston dropped when the temperature reaches ambient?

Key Concepts

Conservation of Mass

In closed systems like a piston-cylinder assembly, the mass of the gas remains constant even as its state changes. This principle is fundamental when applying the ideal gas law during thermodynamic processes, ensuring that the analysis of temperature, pressure, and volume relationships is based on a fixed amount of gas.

Thermodynamic Process Analysis

Understanding thermodynamic processes, particularly cooling and heat transfer, is crucial for analyzing how the state of the gas changes over time. In processes where the gas is cooled, energy is transferred to the surroundings, which impacts the pressure and volume of the gas and dictates the timing and extent of the piston's movement.

Mechanical Equilibrium of Pistons

Mechanical equilibrium in a piston-cylinder arrangement is achieved when the force generated by the internal gas pressure acting on the piston surface equals the total external forces, such as atmospheric pressure and the weight of the piston. This balance condition is essential for determining at what point the piston will start to move as the gas conditions change due to cooling.

Ideal Gas Law

The ideal gas law establishes a relationship between pressure, volume, and temperature (PV = mRT) for a given mass of gas. In a piston-cylinder system, as the gas cools, this relationship is used to determine how the internal pressure changes with temperature and volume, which in turn affects the movement of the piston.

Transcript

00:01 For this problem on the topic of properties of pure substances, we are given a piston cylinder arrangement as shown, which contains air at a given state.

00:10 We have a piston with a given mass and diameter, which initially pushes against the stops.

00:16 If we know the atmospheric state and the cylinder it then cools as heat is transferred to the ambient, we want to know the temperature at which the piston begins to move down, as well as the distance that the piston has dropped when the temperature reaches ambient.

00:35 So let's first look at our piston, and our piston has surface area ap, which is pi over four times the square of its diameter, 0 .1 squared.

00:51 And so the surface area of the piston is 0 .0075 square meter.

01:01 So we can balance the forces when the piston begins to float.

01:08 So the pressure at which this happens, we'll call it p float is equal to the atmospheric pressure p -0 plus the weight of the piston mp times g over its area ap.

01:28 So this is the ambient pressure 100 plus the mass of the piston is 50 times g 9 .807 over the surface area of the piston, which is 0 .00785, and we'll multiply this by a thousand to convert our answer to kilo -pascals.

01:53 And hence, we get the float pressure for the piston to be 162 .5 kilo -pascals.

02:07 And this is equal to p2.

02:13 And p3.

02:15 So after this the pressure is then constant.

02:19 Now to find the temperature at state 2 will assume an ideal gas behavior.

02:25 So the temperature t2 using the ideal gas relation is equal to t1 times the ratio of pressures p2 over p1 and so the final temperature is the initial one 573 .15 times the ratio of pressures 162 .5 over the initial pressure 250.

02:51 This gives a final temperature.

02:55 So the temperature at which the piston begins to move down to be 372 .5 kelvin.

03:07 So now we know the temperature at which the piston begins to move down initially...

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