5.3-Continuity Equation, fluid mech

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Continuity Equation
5.3 Continuity Equation
The continuity equation derives from the conservation of mass, which, in Lagrangian form, simply states that the
mass of the system is constant.
The Eulerian form is derived by applying the Reynolds transport theorem. In this case the extensive property of
the system is its mass,
B
cv
=
m
sys
. The corresponding value for
b
is the mass per unit mass, or simply, unity.
General Form of the Continuity Equation
The general form of the continuity equation is obtained by substituting the properties for mass into the Reynolds
transport theorem, Eq. (5.21), resulting in
However,
dm
sys
/
dt
= 0, so the general, or integral, form of the
continuity equation
is
(5.24)
This equation can be expressed in words as
If the mass crosses the control surface through a number of inlet and exit ports, the continuity equation
simplifies to
(5.25)
where
m
cv
is the mass of fluid in the control volume. Note that the two summation terms represent the net mass
outflow through the control surface.
Example 5.4 shows an application of the continuity equation to calculating the mass accumulation rate in a tank.
EXAMPLE 5.4 MASS ACCUMULATIO I A TAK
A jet of water discharges into an open tank, and water leaves the tank through an orifice in the
bottom at a rate of 0.003 m
3
/s. If the crosssectional area of the jet is 0.0025 m
2
where the velocity of
water is 7 m/s, at what rate is water accumulating in (or evacuating from) the tank?
PROBLEM DEFINITION
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Continuity Equation
Situation:
Jet of water (7 m/s at 0.0025 m
2
) entering tank and water leaving at 0.003 m
3
/s through
orifice.
Find:
Rate of accumulation (or evacuation) in tank (kg/s).
Sketch:
Assumptions:
Water density is 1000 kg/m
3
.
PLAN
A control volume is drawn around the tank as shown. There is one inlet and one outlet.
1. Develop equation for accumulation rate by applying the continuity equation, Eq. (5.25).
2. Analyze equation term by term.
3. Calculate the accumulation rate.
SOLUTION
1. Continuity equation
Because there is only one inlet and outlet, the equation reduces to
2. Termbyterm analysis
·
The inlet mass flow rate is calculated using Eq. (5.5)
·
Outlet flow rate is
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Continuity Equation
3. Accumulation rate:
REVIEW
Note that the result is positive so water is accumulating in the tank.
The rate of water level rise in a reservoir is an oftenused application of the continuity equation. Example 5.5
illustrates this application.
EXAMPLE 5.5 RATE OF WATER RISE I RESERVOIR
A river discharges into a reservoir at a rate of 400,000 ft
3
/s (cfs), and the outflow rate from the
reservoir through the flow passages in the dam is 250,000 cfs. If the reservoir surface area is 40 mi
2
,
what is the rate of rise of water in the reservoir?
PROBLEM DEFINITION
Situation:
Reservoir with 400,000 cfs entering and 250,000 cfs leaving. Area is 40 mi
2
.
Find:
Rate of water rise (ft/hr) in reservoir.
Sketch:
Assumptions:
Water density is constant.
PLAN
The control volume selected is shown in the sketch. There is an inlet from the river at location 1 and
an outlet at location 2. The control surface 3 is just below the water surface and is stationary. Mass
passes through control surface 3 as the water level in the reservoir rises (or falls). The mass in the
control volume is constant.
1. Apply the continuity equation, Eq. (5.25).
2. Analyze term by term.
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Continuity Equation
3. Evaluate rise rate using Eq. (5.1).
SOLUTION
Continuity equation:
2 Termbyterm analysis:
·
Mass in the control volume is constant, so
dm
cv
/
dt
= 0.
·
Inlet port 1 is river flow rate,
.
·
Outlets are reservoir surface and dam outlet,
Substitution of terms back into continuity equation:
3 Rise rate calculation using Eq. (5.1):
Example 5.6 illustrates how to predict the emptying rate of a tank. In this example, a control volume of varying
size is chosen.
EXAMPLE 5.6 WATER LEVEL DROP RATE I DRAIIG
TAK
A 10 cm jet of water issues from a 1 m diameter tank. Assume that the velocity in the jet is m/s
where h is the elevation of the water surface above the outlet jet. How long will it take for the water
surface in the tank to drop from
h
0
= 2 m to
h
f
= 0.50 m?
PROBLEM DEFINITION
Situation:
Water draining by a 10 cm jet from 1 m diameter tank.
Find:
Time (in seconds) to drain from depth of 2 m to 0.5 m.
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Continuity Equation
Sketch:
PLAN
The control selected is shown in the sketch. The control surface is located at and moves with the
water surface. Water crosses control surface at location 1.
1. Apply the continuity equation, Eq. (5.25).
2. Analyze term by term.
3. Solve the equation for elapsed time.
4. Calculate time to change levels.
SOLUTION
1. Continuity equation
2. Termbyterm analysis
·
Accumulation rate term
where
A
T
is crosssectional area of tank.
·
Inlet mass flow rate with no inflow is
·
Outlet mass flow rate
Substitution of terms in continuity equation:
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