
#19
Jan1113, 02:05 AM

Sci Advisor
P: 3,369

The forces and fluxes of linear thermodynamics are identified using the fundamental law
##dS=dU/T+p/TdV\sum_i\mu_i/T dN_i ##. So entropy production due to a flux of energy U is due to the gradient of 1/T, due to volume change to grad p/T and due to particle number flux due to grad μ/T. See, e.g., http://en.wikipedia.org/wiki/Onsager...ocal_relations 



#20
Jan1113, 09:50 AM

Sci Advisor
HW Helper
Thanks
PF Gold
P: 4,437

In any event, I didn't mean to imply that the piston is moving at a speed exceeding the speed of sound, but only at the speed necessary to drop the pressure at the piston face to a somewhat lower value (say 50% of the original pressure) and hold it there. I also didn't mean to imply that I am creating anything close to a pure vacuum in the right hand portion of the cylinder. In both regions of the gas, the pressure and density are high enough for the continuum gas dynamics equations to apply. The mean free path of the molecules will still be very short, as will the time between collisions. The MaxwellBoltzmann distribution should not be seriously disturbed. Gas dynamics problems of this type can nowadays be routinely solved using computational fluid dynamics (CFD) software. Although I have not followed the literature in great detail, I am confident that the results of such calculations have been extensively verified experimentally over the years. If you are interested in seeing how simple problems are analyzed using gas dynamics, I refer you to our old friend BSL, Transport Phenomena. A worked example, presenting all the assumptions used, is presented in section 11.4, p. 350, Example 11.47 OneDimensional Compressible Flow: Velocity, Temperature, and Pressure Profiles in a Stationary Shock Wave (If I remember correctly, you were interested in learning how all this applies to shock waves). The system they are analyzing is an open system (flow system) rather than a closed (fixed mass) system that I discussed in my post, but I'm sure you will get the connection. Once the gas dynamics equations have been solved (say for the closed system situation that I discussed), you can take the results and calculate the local transient rate of entropy generation from temperature gradients and velocity gradients, and plug these rates into the entropy change equation I gave (which applies to both the reversible and irreversible process in which diffusional mass transport is not occurring). You can thereby calculate the change in entropy for the system, between the two equilibrium states as well as at all points along the reversible path. All this is valid as long as the assumption of continuum behavior for all portions of the system is valid (short mean free path and short time between collisions). Another interesting feature that the equation I provided for the entropy change tells us is that, while the dQ/T at the boundary is less for an irreversible path than for a reversible path, the entropy generation from temperature and velocity gradients for the irreversible process compensates for this so that the total entropy change between the starting and final equilibrium states is the same for both paths. 



#21
Jan1213, 04:25 AM

P: 5,462

First let me observe that this thread has veered way off topic but I assume the mods are allowing it to continue for interest's sake.
Dr Du has posted the link between Rap's analysis and that of ChesterMiller. For those (like me) who are not deeply into this part of subject I found pages 307 to 320 of Callen (Thermodynamics and an Introduction to Thermostatistics) highly illuminating, offering a tight but readable introduction to Onsager. In particular, Rap, you will find the answer to your question in post#18 there. One caution, however, like so many workers in This subject Callen has some differences in terminology and sign conventions. In relation to this thread his use of the words closed or closure should be checked. 



#22
Jan1213, 11:11 AM

P: 789





#23
Jan2313, 11:05 PM

P: 789

I feel sure that a gradient in fugacity will produce entropy in general, but there appear to be cases where it does not, depending on what is held constant. I think I have learned that the statement that differences (or gradients) in X cause changes (or fluxes) in Y means nothing until you have specified which other state variables have been held constant. 


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