Differential Equations for a Gas-Filled Chamber System

• Xtof
In summary, the general system described by a certain molar mass of gas occupies a volume with pressure, and changes in volume, mass, and heat are simultaneously varied.
Xtof
I'm trying to figure out which set of differential equations describe the following quite general but relatively simple system:

A chamber which volume can be changed by moving a piston filled with a gas. Additionally, gas can be released from the chamber through a relief valve and heat can be added/extracted through the chamber walls.

State of the system described by a certain molar mass of a gas n occupying a volume V at pressure P (assume ideal gas).

The question thus is: what are the differential changes in pressure dP for given differential changes in volume dV, molar mass dn and heat dQ.

In other words: how does this system evolve when volume and mass are simultaneously varied while heat is supplied/extracted?

Additionally, which equations describe the evolution of this system for a non-ideal gas which heat capacities are general functions of temperature (Cp(T), Cv(T)).

I have a basic knowledge of thermodynamics and assume you need to use concepts of internal energy and enthalpy but I cannot figure out how to apply them since neither volume nor pressure are constant.

Thanks for your assistance or discussion!

Christophe

You could try googling '1st law of thermodynamics open system'. (I found this: http://www.tech.plym.ac.uk/sme/ther103/ther103-first law open systems.pdf.)

The equation in the center of page 5 pretty much summarizes it all. Should be pretty easy to understand if you're familiar with closed systems already.

I'd tell you more, but it's been a long while since I had to think of similar problems. Also, the inlet velocity would be zero. And you should be able to substitute the work and mass rates and changes in enthalpy with expressions that have dP, dV, dT, and dn in them.

I'm not too sure about the pressure relief valve (and maybe a few other things as well). Could be I'm seriously oversimplifying, in fact, but this should get you going somewhat, at least.

1. What is a gas-filled chamber system?

A gas-filled chamber system is a closed container that contains a gas at a certain pressure and temperature. It is often used in scientific experiments to study the behavior of gases under different conditions.

2. What are differential equations and how do they relate to gas-filled chamber systems?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. In the context of gas-filled chamber systems, they are used to model the changes in pressure, volume, and temperature of the gas inside the chamber over time.

3. What factors affect the behavior of a gas-filled chamber system?

The behavior of a gas-filled chamber system is affected by several factors, including the type of gas, the initial pressure and temperature, the volume of the chamber, and the presence of any external forces or interactions.

4. How are differential equations solved for a gas-filled chamber system?

Differential equations for a gas-filled chamber system can be solved using various mathematical techniques, such as separation of variables, substitution, and integrating factors. These solutions can then be used to predict the behavior of the gas inside the chamber.

5. What are some real-world applications of differential equations for gas-filled chamber systems?

Differential equations for gas-filled chamber systems have many practical applications, including in the fields of chemistry, physics, and engineering. They can be used to study the behavior of gases in industrial processes, such as in chemical reactors or combustion engines, and in medical devices, such as ventilators and oxygen tanks.

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