Entropy generation (1 Viewer)

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Consider a rod which is insulated on its lateral surface, now this rod is brought in contact with a source at temperature T1 and sink at temperature T2 now a temperature gradient sets up in the rod after steady state is reached temperature at some distance X from the source end is given as
now this rod is removed and ends are insulated as well
then my question is as heat flows from high temp. region to low temp. region is any entropy generated within the rod...?
what i feel is it should not because there are no irreversibility in the rod during heat transfer, even the heat transfer across any rod element takes place at temp. difference tending to zero, hence in eqn.
ds=δq/T+Δsgen, Δsgen should be zero so that eqn. becomes
check me if i am correct and if i have missed something notify me.
Your argument is that if an isolated system is not at equilibrium, it will not move toward equilibrium with time (since that obviously would increase entropy)??? Seriously?!! S = ∫dQ/T and you think the heat flow magically stops once the ends are insulated? I draw your attention to the second law of thermodynamics. (as well as the zeroth law).
Entropy will increase but that increase will be due to heat transfer from hot end to cold end and not because of some generations due to irreversibility and thats what i have said above in last equation i have taken entropy generatio(Sgen=0) and not change in total entropy equal to zero(dS!=0)

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