This is an unsteady state flow problem where we have steam flowing at high temperature
and high pressure through a pipeline and we have a valve separating the flow stream from
an empty tank, so this tank is going to be filled to ten megapascals pressure, so this
is megapascals, by opening the valve and the valve will be closed when the pressures are
equalized, and the question is what is the temperature in the tank when the pressure
is equalized. So the approach is to do an energy balance for a flow system where its
unsteady state, so we're going to write down the change in internal energy and this is
the total change, let me put a "t" here to indicate "total" account for what ever number
of moles or kilograms, total internal energy of the tank, so we're doing a balance with
this as our system, okay? And the reason the energy changes is if we look at the general
balance there'd be some mass flow into the tank and the enthalpy per unit mass coming
in, well a general would be mass flow out, enthalpy out, that term is zero, heat transfer,
but this is an adiabatic system and we should specify that, so that term is zero, and then
there is shaft work, and there is no shaft work, so that's zero. So we have a simplified
equation that we can integrate because what we're interested in is the initial and final
conditions, so zero to whatever the time is, and that means we integrate this where initially
the tank is empty so internal energy is zero, until the total final internal energy, so
the left side is just the internal energy final in the system, the right side, well
H in is a constant, it's the enthalpy at 450 degrees and ten megapascals, and so we have
the integral of Mdot in dt so this is a flow rate, the dot means a flow rate mass in, and
so the total mass that's added, I'll write it like this, so this is mass without a dot,
so this is the total mass that's in the tank when we're finished, well the left side is
the total internal energy which is going to be the mass in the tank, which is the amount
that we added, right, because this is at the final time, times internal energy final per
mass equals H in Mass in, so this is independent of the size of the tank, we end up with the
same equation, internal energy per kilogram equals enthalpy in per kilogram, we can look
this value up in the steam tables, so that means that of course U final is equal to this
value and so we have the final internal energy and we have the final pressure, the pressure
is 10 megapascals, single component system, two variables completely specify the system,
so we should be able to go to the steam tables at ten megapascals pressure and find the conditions
with this internal energy and at the steam tables, the temperature that corresponds to
this internal energy at ten megapascals is around 600 degrees centigrade so this shows
that the temperature increases rather significantly from 450 centigrade to 600 degrees centigrade
when we fill this tank and this happens because we're doing work, that flow work, on the gas
and that work goes into the internal energy.