Imagine a room filled with air. The air is heated and it is allowed to escape the through cracks in the walls.
a. Take the heating to occur at constant power (P_heating). There is no conductive entropy transfer through the walls of the room. The pressure p in the room is assumed to stay constant. The molar temperature coefficient of enthalpy (molar heat at constant pressure) is taken to be 7/2R (R = 8.314 J/(K·mole) is the universal gas constant). Show that T(t) = T0·exp(P_heating·t/(7/2·p·V)) where V is the volume of the room and T0 is the initial temperature of the air in the room. b. Show that the pressure of the air in the room can be expressed in terms of the amount of air and the molar entropy according to p =
((T_ref*n*8.314/Volume)^(1.4*8.314/(1.4-1))
*p_ref^(-8.314)
*EXP(molar_entropy))^((1.4-1)/8.314). | | c. Create a dynamical model for the air in the room of the following type. Express the laws of balance of entropy and the amount of substance (of the air in the room). Allow for addition of entropy due to heating (dissipation at constant rate) and convective entropy transfer. There is a flow of amount of substance of air out of the room because the pressure of the air rises above the ambient (equal to the initial) pressure. Use a laminar flow relation for the air escaping from the room. Also program the solution from problem a. Does the simulated temperature follow the theoretical one from problem a if the flow resistance is small and there is no entropy loss due to conduction. d. Include conductive loss of entropy in the dynamical model. Perform parameter studies by varying the flow resistance and the entropy conductance for conductive loss. |
