Thermodynamics Piston Cylinder Problem

[thorwynn]

Homework Statement

We were given a problem and asked to write a c++ program about this problem:

A sample of gas is inside a piston cylinder arrangement initially at a temperature of 303K. The gas is leaked to a final temperature of 323K. According to the process PV^n=constant. The cylinder is made of aluminum with dimensions of L=2D and with a uniform thickness T.

Input data (given):
1. User chooses the kind of gas (nitrogen,etc.)
2. mass of the gas, in kg.
3. polytropic index, n (1 to 3 only)
4. initial pressure, P1 (value from 1 to 3 bar only)

Find the following:
1. Final Pressure
2. Final Volume
3. Work Transfer
4. Heat Transfer
5. Thickness of cylinder
6. Mass of the container including the gas.

Can someone help me out on this, just giving me some points or equations on where to start will do. I am completely blank on this problem. Give me some tips on how to derive the right equation to be used in the problem. Thanks in advanced.

Homework Equations

The Attempt at a Solution

This is a closed polytropic system, so

P1(V1)ⁿ = P2(V2)ⁿ,

Using the ideal gas law: V2 = mRT2/P2, thus

P1(V1)ⁿ = P2(mRT2/P2)ⁿ = (P2)^(1-n) (mRT2)ⁿ,

(P2)^(1-n) = P1(V1/mRT2)ⁿ,

P2 = P1(V1/mRT2)^(n/1-n)

Where V1 = π(D²/4)2D,

V2 = mRT2/P2

W = (P2V2 – P1V1)/(n-1)

Q – W = ∆U

m(gas) is given.

m(cyl) = ρV= (2.6x10^-3)( π(D²/4 – (D-t)²/4)2D)

 

[KataKoniK]

= 2.6x10^-3( π(D²/4 – (D-t)²/4)2D)

These are just some initial equations and steps that can be used to solve this problem. The first step is to identify the given information and variables, and then use the appropriate equations to solve for the unknowns. In this case, the ideal gas law and the polytropic process equation are the main equations that will be used. It is important to also consider the units of the given variables and ensure they are consistent throughout the calculations. Additionally, it may be helpful to sketch a diagram of the system to better visualize the problem and identify any other potential variables that may be needed. Good luck!

 

[Mene]

= 3.3x10^-2 kg

T1 = 303K, T2 = 323K

To start, we need to understand the given conditions and variables, as well as the equations and principles of thermodynamics that apply to this problem. The first principle of thermodynamics states that energy cannot be created or destroyed, only transferred or converted from one form to another. This is important to keep in mind when considering the work and heat transfer in this problem.

Next, we need to consider the ideal gas law, which relates the pressure, volume, and temperature of a gas. This law can be rearranged to solve for any of these variables, depending on what information is given.

In this problem, we are given the initial and final temperatures, as well as the initial pressure. We also know that the gas follows a polytropic process, meaning that the relationship between pressure and volume is described by the equation PV^n=constant. This equation can be used to find the final pressure and volume, as shown in the Attempt at a Solution above.

To find the work transfer, we can use the equation W = (P2V2 – P1V1)/(n-1), which is derived from the first law of thermodynamics and the ideal gas law. This equation takes into account the change in volume and the polytropic index.

To find the heat transfer, we can use the equation Q – W = ∆U, where Q is the heat transfer, W is the work transfer, and ∆U is the change in internal energy. In this problem, we are assuming that the system is adiabatic (no heat transfer), so the heat transfer would be equal to zero.

To find the thickness of the cylinder, we can use the equation for volume of a cylinder (V=πr²h) and solve for the thickness (h), using the known volume and diameter of the cylinder.

Lastly, to find the mass of the container including the gas, we can use the density of aluminum (ρ) and the volume of the cylinder to calculate the mass (m=ρV).

Overall, the key to solving this problem is understanding the principles of thermodynamics and how they apply to this specific situation. By carefully considering the given information and using the appropriate equations, we can solve for the desired variables and provide a complete response to the problem.