Group 2 Volumetric Flowrate and Average Velocity Sage Ross Randy Goll Lisa Gourley Marci Wyatt...
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Transcript of Group 2 Volumetric Flowrate and Average Velocity Sage Ross Randy Goll Lisa Gourley Marci Wyatt...
Group 2Volumetric Flowrate and
Average Velocity
Sage Ross
Randy Goll
Lisa Gourley
Marci Wyatt
Jennifer Oberlag
April 21, 2001
Problem Statement
Fluid flows through a conduit with a quarter circle cross-section. The profile of the velocity in the z-direction can beapproximated by:
])/4(1[ ])/(1[ )/( 22 RrRrvv Az
With the coordinate system as shown and where vA is a constant equal to 20 cm/s and R=30 cm. Find the Volumetric flowrate Q and the average velocity.
The circular shape of the conduit suggests that we use cylindricalcoordinates to solve the problem.
vA=20 cm/s R=30 cm
3 dimensional view of conduit
Quarter circle cross section
])/4(1[ ])/(1[ )/( 22 RrRrvv Az
We need to use the following equations to solve the problem:
A
nvQ dA )(
A
Qvave
A
dAnvQ )(
Solution
zvnv drrddA
R
zvQ0
4
4
dr dr
R
Rr
Rr
A rvQ0
4
43
162 dr - ]1[ 2
3
Volumetric flowrate across element dA: dAnv )(
To find the volumetric flow we must integrate:
Such that: and
The limits for integration can be found from geometry.
Substitute the given value for vz:
R
AvQ0
4
4
242
Rr
Rr dr d]r -][1-[1
Integrate with respect to
1 0
2
3
16
3
16
44 2
64
3
2
64
3
R
Rr
Rr
A rdrvQ
R R
RrrA
RrA
R
vdrr
R
vQ
0 0532
2
53
2
4
312122
22
53 cm 30cm/s 2045
2
15
2
332
53
Rv
R
vQ AR
RRA
sQ /cm 2500 3
Evaluate the integral at the limits:
Simplify and integrate with respect to r:
Simplify and evaluate:
2
3
24
1 cm) 30(
)/cm 2500(4
s
R
Q
A
Qvave
cm/s 5.3avev
The average velocity is found by diving volumetric flowrate by area.
vz
Vave represents the average of all of the velocities over the velocity profile.