Non equilibrium thermodynamics: T as a function of time

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When two separate gas systems with different temperatures, pressures, and volumes are combined, their initial inhomogeneous distribution requires careful analysis. The Navier–Stokes equations are suggested as a foundational approach for simulating the system's behavior over time. Additionally, incorporating the differential energy balance equation is essential for accounting for heat conduction within the combined system. The discussion highlights the complexity of deriving an average temperature equation as a function of time. Overall, a combination of simulation and analytical methods will be necessary for accurate modeling.
pterodox
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Hi.
Let's say we have two separate systems (containing gases), each with their own temperature, pressure and volume. Then these systems are brought together, with no walls in between. Of course, one might consider they are now technically the same system, but the distribution of pressure and temperature is inhomogeneous for a while.
Is there an equation that I could use to calculate the average temperature of any volume inside this space as a function of time?
Thanks.
 
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You can simulate the system, sure. The Navier–Stokes equations are probably a good start.
Analytic equations might give reasonable approximations for some setups.
 
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In addition to what mfb said, you would also have to include the differential energy balance equation to include heat conduction within the system.
 
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Thank you both! I'll look into those. This might be a bit tougher than I initially expected.
 

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