Solution Found!
Answer: Applying the Entropy Balance: Closed SystemsAt
Chapter 6, Problem 76P(choose chapter or problem)
At steady state, an insulated mixing chamber receives two liquid streams of the same substance at temperatures \(T_{1}\) and \(T_{2}\) and mass flow rates \(\dot{m}_{1}\) and \(\dot{m}_{2}\), respectively. A single stream exits at \(T_{3}\) and \(\dot{m}_{3}\). Using the incompressible substance model with constant specific heat c, obtain an expression for
(a) \(T_{3}\) in terms of \(T_{1}, T_{2}\), and the ratio of mass flow rates \(\dot{m}_{1} / \dot{m}_{3}\).
(b) the rate of entropy production per unit of mass exiting the chamber in terms of c, \(T_{1} / T_{2}\) and \(\dot{m}_{1} / \dot{m}_{3}\).
(c) For fixed values of c and \(T_{1} / T_{2}\), determine the value of \(\dot{m}_{1} / \dot{m}_{3}\) for which the rate of entropy production is a maximum.
Questions & Answers
QUESTION:
At steady state, an insulated mixing chamber receives two liquid streams of the same substance at temperatures \(T_{1}\) and \(T_{2}\) and mass flow rates \(\dot{m}_{1}\) and \(\dot{m}_{2}\), respectively. A single stream exits at \(T_{3}\) and \(\dot{m}_{3}\). Using the incompressible substance model with constant specific heat c, obtain an expression for
(a) \(T_{3}\) in terms of \(T_{1}, T_{2}\), and the ratio of mass flow rates \(\dot{m}_{1} / \dot{m}_{3}\).
(b) the rate of entropy production per unit of mass exiting the chamber in terms of c, \(T_{1} / T_{2}\) and \(\dot{m}_{1} / \dot{m}_{3}\).
(c) For fixed values of c and \(T_{1} / T_{2}\), determine the value of \(\dot{m}_{1} / \dot{m}_{3}\) for which the rate of entropy production is a maximum.
ANSWER:a.)
Step 1 of 3
We have to obtain an expression for in terms of
,
, and the ratio of mass flow rates
using the incompressible substance model with constant specific heat
.
At steady state, the mass balance equation is given by
=
+
The energy balance equation with the assumption that for control volume =
= 0 and all kinetic and potential effects are negligible becomes,
+
-
= 0
Combining the above two equations
+
-
=0
(
)= 0
Now, by modeling the liquid stream as incompressible with specific heat and using the equation
The above equation can be rewritten as
[
]+
= 0 (effect of pressure is neglected)
(
)
Therefore, the required equation is