Modern Engineering Thermodynamics

6.10 Open System Unsteady State Processes 191
Also, the system has a single-inlet flow stream but no outlet flow stream. To carry out the analysis, we note that
the system does not contain a work mode, so _W = 0: Also, the system is not moving, so
½dðKE + PEÞ/dtŠ system
= 0
Finally, we assume that the tank is filled slowly enough that we can ignore the inlet flow stream’s specific kinetic and
potential energy terms. Under these conditions, the generalized energy rate balance equation becomes
or
_Q − 0 + _m in ðh in + 0 + 0Þ− 0 = ðdU/dt + 0 + 0
_Q + _m in h in = ½dmu
ð Þ/dt
Then, multiplying through by dt and integrating over the filling process gives
1Q 2 +
Z 2
1
Š tank
h in dm in = ðm 2 u 2 − m 1 u 1 Þ tank
where state 1 is the initial state inside the tank and state 2 is the state inside the tank after it has been filled. When
the tank is filled from a pipe of unlimited supply, as shown in Figure 6.14, h in = constant and this equation becomes
1Q 2 + h in m in = ðm 2 u 2 − m 1 u 1 Þ tank
where the conservation of mass law for the filling process gives
Combining the previous two equations, we obtain
m in = ðm 2 − m 1 Þ tank
u 2 = u 1 ðm 1 /m 2 Þ+ 1 Q 2 /m 2 + h in ð1 − m 1 /m 2 Þ (6.33)
where the numerical subscripts refer exclusively to states inside the tank from this point on.
By knowing p 2 from the filling process and calculating u 2 from Eq. (6.33), we have fixed the final thermodynamic
state of the filled tank and can then find the value of any other property we desire, say its final temperature T 2 .To
illustrate this, let us assume the tank is insulated ( 1 Q 2 = 0) and initially evacuated (m 1 = 0). Then, Eq. (6.33)
becomes:
Þ tank
u 2 = h in (6.34)
If we now assume that the tank is filled with an incompressible fluid, then we can utilize Eq. (6.19) and write
u 2 = cT 2 = h in = cT in + vp in
or
T 2
filling
incomp:
fluid
= T in + ðvp in Þ/c (6.35)
and the compression process of pressurizing the tank would cause T 2 to be greater than T in by an amount ðvp in Þ/c:
EXAMPLE 6.7
A high-pressure water storage system is used to fill initially empty, rigid, insulated tanks with liquid water. The temperature
of the water entering the tank is 20.0°C and the final pressure of the water in the tank is 50.0 MPa. Determine the final temperature
of the water in the tank immediately after it has been filled.
Solution
First, draw a sketch of the system (Figure 6.15).
(Continued )