Known A stream of water vapor mixes with a liquid water stream to produce a saturated liquid stream at the exit. The states at the inlets and exit are specified. Mass flow rate and volumetric flow rate data are given at one inlet and at the exit, respectively.
Find Determine the mass flow rates at inlet 2 and at the exit, and the velocity V _{2}.
Schematic and Given Data:
Engineering Model The control volume shown on the accompanying figure is at steady state.
Analysis The principal relations to be employed are the mass rate balance (Eq. 4.2) and the expression \dot{m}= AV / v (Eq. 4.4b). At steady state the mass rate balance becomes
\frac{d m_{ cv }}{d t}=\sum_{i} \dot{m}_{i}-\sum_{e} \dot{m}_{e} (4.2)
\dot{m}=\frac{ AV }{v} (one-dimensional flow) (4.4b)
1 \frac{dm _{cv}^{\nearrow0}}{d t}=\dot{m}_{1}+\dot{m}_{2}-\dot{m}_{3}
Solving for \dot{m}_{2},
\dot{m}_{2}=\dot{m}_{3}-\dot{m}_{1}
The mass flow rate \dot{m}_{1} is given. The mass flow rate at the exit can be evaluated from the given volumetric flow rate
\dot{m}_{3}=\frac{( AV )_{3}}{v_{3}}
where v_{3} is the specific volume at the exit. In writing this expression, one-dimensional flow is assumed. From Table A-3, v_{3}=1.108 \times 10^{-3} m ^{3} / kg. Hence,
\dot{m}_{3}=\frac{0.06 m ^{3} / s }{\left(1.108 \times 10^{-3} m ^{3} / kg \right)}=54.15 kg / s
The mass flow rate at inlet 2 is then
\dot{m}_{2}=\dot{m}_{3}-\dot{m}_{1}=54.15-40=14.15 kg / s
For one-dimensional flow at 2, \dot{m}_{2}= A _{2} V _{2} / v_{2}, so
V _{2}=\dot{m}_{2} v_{2} / A _{2}
State 2 is a compressed liquid. The specific volume at this state can be approximated by v_{2} \approx v_{ f }\left(T_{2}\right) (Eq. 3.11). From Table A-2 at 40°C, v_{2}=1.0078 \times 10^{-3} m ^{3} / kg. So,
v(T, p) \approx v_{ f }(T) (3.11)
V _{2}=\frac{(14.15 kg / s )\left(1.0078 \times 10^{-3} m ^{3} / kg \right)}{25 cm ^{2}}\left|\frac{10^{4} cm ^{2}}{1 m ^{2}}\right|=5.7 m / s
1 In accord with Eq. 4.6, the mass flow rate at the exit equals the sum of the mass flow rates at the inlets. It is left as an exercise to show that the volumetric flow rate at the exit does not equal the sum of the volumetric flow rates at the inlets.
\begin{aligned}\sum_{i} \dot{m}_{i} & = \sum_{e} \dot{m}_{e}\\\text { (mass rate in) } &\text { (mass rate out) }\end{aligned} (4.6)
Skills Developed
Ability to…
• apply the steady-state mass rate balance.
• apply the mass flow rate expression, Eq. 4.4b.
• retrieve property data for water.
Quick Quiz
Evaluate the volumetric flow rate, in m ^{3} / s, at each inlet. Ans. ( AV )_{1}=12 m ^{3} / s ,( AV )_{2}=0.01 m ^{3} / s
