Continuity Equation Tutorial

Dr. Eyad Abushandi

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Continuity Equation

• Continuity equation represents the law mass conservation.

• Unsteady flow the continuity equation

• For steady state condition

(Mass flow rate into the system)

(Mass flow rate out of the system)- =

Rate of change of storage.

(Mass flow rate into the system)

(Mass flow rate out of the system)- =

0

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Example No. 1

• 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 m3/s. If the cross-sectional area of the jet is 0.0025 m2 where the velocity of water is 7 m/s, at what rate is water accumulating in (or evacuating from) the tank?

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Example No. 1

• Situation: Jet of water (7 m/s at 0.0025 m2) entering tank and water leaving at 0.003 m3/s through orifice.

• Find: Rate of accumulation (or evacuation) in tank (kg/s).

Water density is 1000 kg/m3.

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Example No. 1

PlanThere is one inlet and one outlet.

1. Develop an equation for accumulation rate by applying the continuity equation.

2. Analyze equation term by term.

3. Calculate the accumulation rate.

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Example No. 1

1. Continuity equation

cs cs

iocv mmmdt

d0

The accumulation rateof mass in the control

volume

The net outflow rate ofmass through the control

Surface + = 0

0mmmdt

dicv

Because there is only one inlet and outlet, the equation reduces to:

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Example No. 1

2. Term-by-term analysis: The inlet mass flow rate is calculated using:

VAmi

23 0025.0/7/1000 msmmkgmi

skgmi /5.17

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Example No. 1

2. Term-by-term analysis: The outlet mass flow rate is calculated using:

Qmo

skgsmmkgmo /3/003.0/1000 33

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Example No. 1

3. Accumulation rate:

oicv mmmdt

d

skgskgmdt

dcv /3/5.17

skgmdt

dcv /5.14

Note that the result is positive so water is accumulating in the tank.

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Example No. 2

• A river discharges into a reservoir at a rate of 400,000 ft3/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 mi2, what is the rate of rise of water in the reservoir?

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Example No. 2

Situation: Reservoir with 400,000 cfs entering and 250,000 cfs leaving. Area is 40 mi2.

Find: Rate of water rise (ft/hr) in reservoir

Assumptions: • Water density is constant• The mass in the control volume is constant.

Plan

1. Apply the continuity equation.

2. Analyze term by term.

3. Evaluate rise rate.

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Example No. 2

Solution

1.Apply Continuity equation:

cs cs

iocv mmmdt

d0

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Example No. 2

Solution

2. Term-by-term analysis:

Mass in the control volume is constant, so dmcv/dt = 0.

Inlet port 1 is river flow rate 11 Qmcs

Outlets are reservoir surface and dam outlet,

risecs

o QQm 2

Substitution of terms back into continuity equation:

12 QQQ rise 12 QQQ rise

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Example No. 2

Solution

3. Rise rate calculation

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3 A

QQ

A

QV riserise

22 )/5280(40

000,250000,400

miftmi

cfscfsVrise

hrftsftVrise /482.0/1034.1 4

A tank has a hole in the bottom with a cross-sectional area of 0.0025 m2. The cross-sectional area of the tank is 0.1 m2. The velocity of the liquid flowing out the bottom hole is V=(2gh)0.5, where h is the height of the water surface in the tank above the outlet. At a certain time the surface level in the tank is 1 m and rising at the rate of 0.1 cm/s. The liquid is incompressible. Find the velocity of the liquid through the inlet.

Example No. 3

Solution: cm/s.dt

dh10

outincv mm

dt

dM

hAVM cv dt

dhA

dt

dMcv

outoutout

ininin

AghAVm

AVm

2

outinin AghAVdt

dhA 2

).(.).(V)/.(. in 0025018192002501001010 m/s.Vin 4684

Example No. 3

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Example No. 4

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 h0 = 2 m to hf = 0.50 m?

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Example No. 4

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.

Plan

1. Apply the continuity equation.

2. Analyze term by term.3. Solve the equation for elapsed time.4. Calculate time to change levels.