is usually considered in detail in fluid mechanics.
4.3 Applications of the Mass Rate Balance
4.3.1 Steady-State Application
For a control volume at steady state, the conditions of the mass within the control volume and at the boundary do not vary with time. The mass flow rates also are constant with time.
Example 4.1 illustrates an application of the steady-state form of the mass rate bal- ance to a control volume enclosing a mixing chamber called a feedwater heater. Feed- water heaters are components of the vapor power systems considered in Chap. 8.
Applying the Mass Rate Balance to a Feedwater Heater at Steady State
A feedwater heater operating at steady state has two inlets and one exit. At inlet 1, water vapor enters at p1 5 7 bar, T1 5 2008C with a mass flow rate of 40 kg/s. At inlet 2, liquid water at p2 5 7 bar, T2 5 408C enters through an area A2 5 25 cm2. Saturated liquid at 7 bar exits at 3 with a volumetric flow rate of 0.06 m3/s. Deter- mine the mass flow rates at inlet 2 and at the exit, in kg/s, and the velocity at inlet 2, in m/s.
SOLUTION
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 V2. Schematic and Given Data:
Engineering Model: The control volume shown on the accompanying figure is at steady state.
Fig. E4.1
2 1
3 Control volume boundary A2 = 25 cm2
T2 = 40°C p2 = 7 bar
T1 = 200°C p1 = 7 bar m1 = 40 kg/s
Saturated liquid p3 = 7 bar (AV)3 = 0.06 m3/s
Analysis: The principal relations to be employed are the mass rate balance (Eq. 4.2) and the expression m# 5AV/𝜐 (Eq. 4.4b). At steady state the mass rate balance becomes
➊ dmcv0
dt 5m#
11m#
22m#
3
Solving for m#
2,
m#
25m#
32m#
1
c c c c EXAMPLE 4.1 c
148 Chapter 4 Control Volume Analysis Using Energy
Evaluate the volumetric flow rate, in m3/s, at each inlet.
Ans. (AV)1 5 12 m3/s, (AV)2 5 0.01 m3/s The mass flow rate m#
1 is given. The mass flow rate at the exit can be evaluated from the given volumetric flow rate m#
35 (AV)3
𝜐3
where 𝜐3 is the specific volume at the exit. In writing this expression, one-dimensional flow is assumed. From Table A-3, 𝜐3 5 1.108 3 1023 m3/kg. Hence,
m#
35 0.06 m3/s
(1.10831023 m3/kg) 554.15 kg/s The mass flow rate at inlet 2 is then
m#
25m#
32m#
1554.15240514.15 kg/s For one-dimensional flow at 2, m#
25A2V2y𝜐2, so
V25m#
2𝜐2yA2
State 2 is a compressed liquid. The specific volume at this state can be approxi- mated by 𝜐2<𝜐f(T2) (Eq. 3.11). From Table A-2 at 408C, 𝜐2 5 1.0078 3 1023 m3/kg.
So,
V25 (14.15 kg/s)(1.007831023 m3/kg)
25 cm2 `104 cm2
1 m2 `5 5.7 m/s
➊ 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 volu- metric flow rate at the exit does not equal the sum of the volumetric flow rates at the inlets.
Ability to…
❑ apply the steady-state mass rate balance.
❑apply the mass flow rate expression, Eq. 4.4b.
❑retrieve property data for water.
✓Skills Developed
4.3.2 Time-Dependent (Transient) Application
Many devices undergo periods of operation during which the state changes with time—for example, the startup and shutdown of motors. Other examples include containers being filled or emptied and applications to biological systems. The steady- state model is not appropriate when analyzing time-dependent (transient) cases.
Example 4.2 illustrates a time-dependent, or transient, application of the mass rate balance. In this case, a barrel is filled with water.
Filling a Barrel with Water
Water flows into the top of an open barrel at a constant mass flow rate of 7 kg/s. Water exits through a pipe near the base with a mass flow rate proportional to the height of liquid inside: m#
e 51.4 L, where L is the instantaneous liquid height, in m. The area of the base is 0.2 m2, and the density of water is 1000 kg/m3. If the barrel is initially empty, plot the variation of liquid height with time and comment on the result.
SOLUTION
Known: Water enters and exits an initially empty barrel. The mass flow rate at the inlet is constant. At the exit, the mass flow rate is proportional to the height of the liquid in the barrel.
Find: Plot the variation of liquid height with time and comment.
c c c c EXAMPLE 4.2 c
4.3 Applications of the Mass Rate Balance 149 Schematic and Given Data:
Analysis: For the one-inlet, one-exit control volume, Eq. 4.2 reduces to dmcv
dt 5m#
i 2m#
e
The mass of water contained within the barrel at time t is given by mcv (t)5𝜌AL(t)
where 𝜌 is density, A is the area of the base, and L(t) is the instantaneous liquid height. Substituting this into the mass rate balance together with the given mass flow rates
d (𝜌AL)
dt 5721.4 L Since density and area are constant, this equation can be written as
dL
dt 1a1.4
𝜌AbL5 7 𝜌A
which is a first-order, ordinary differential equation with constant coefficients. The solution is L551C expa21.4t
𝜌Ab
where C is a constant of integration. The solution can be verified by substitution into the differential equation.
To evaluate C, use the initial condition: at t50, L50. Thus, C5 25.0, and the solution can be written as L55.0[12exp(21.4ty𝜌A)]
Substituting 𝜌51000 kg/m3 and A50.2 m2 results in
L55[12exp(20.007t)]
This relation can be plotted by hand or using appropriate software. The result is Engineering Model:
1. The control volume is defined by the dashed line on the accompanying diagram.
2. The water density is constant.
mi = 7 kg/s
Boundary of control volume
A = 0.2 m2 L (m)
me = 1.4 L kg/s
Fig. E4.2a
➊
150 Chapter 4 Control Volume Analysis Using Energy
If the mass flow rate of the water flowing into the barrel were 12.25 kg/s while all other data remained the same, what would be the limiting value of the liquid height, L, in m? Ans. 0.9 m
From the graph, we see that initially the liquid height increases rapidly and then levels out as steady-state opera- tion is approached. After about 100 s, the height stays constant with time. At this point, the rate of water flow into the barrel equals the rate of flow out of the barrel. From the graph, the limiting value of L is 5 m, which also can be verified by taking the limit of the analytical solution as tS`.
➊ Alternatively, this differential equation can be solved using Interactive Thermodynamics: IT. The differential equation can be expressed as
der(L, t) 1 (1.4 * L)/(rho * A) 5 7/(rho * A) rho 5 1000 // kg/m3
A 5 0.2 // m2
where der(L,t) is dL/dt, rho is density 𝜌, and A is area. Using the Explore button, set the initial condition at L50, and sweep t from 0 to 200 in steps of 0.5. Then, the plot can be constructed using the Graph button.
Ability to…
❑ apply the time-dependent mass rate balance.
❑solve an ordinary differen- tial equation and plot the solution.
✓Skills Developed
BIOCONNECTIONS The human heart provides a good example of how bio- logical systems can be modeled as control volumes. Figure 4.4 shows the cross section of a human heart.
The flow is controlled by valves that intermittently allow blood to enter from veins and exit through arteries as the heart muscles pump. Work is done to increase the pressure of the blood leaving the heart to a level that will propel it through the cardiovascular system of the body.
Observe that the boundary of the control volume enclosing the heart is not fixed but moves with time as the heart pulses.
Aorta Artery
Left ventricle Left atrium
Valve
Cardiac muscle Inferior
vena cava
Right ventricle Valve
Right atrium Veins
Superior vena cava
Boundary Valve
Valve
Fig. 4.4 Control volume enclosing the heart.
Fig. E4.2b
1 0 2 3 4 5 6
200 400 600 800
Time t, s
Height L, m