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Whence o - C,, - + C: or - Do

1) The document discusses analysis of flow through cylindrical tubes, including annular flow with an inner cylinder moving axially and analysis of a capillary flowmeter. 2) Equations are derived for the mass flow rate and force on a cylinder in annular flow, and the mass flow rate is related to fluid properties, gravity, and height for a capillary flowmeter. 3) Analysis of low-density phenomena in compressible tube flow relates the velocity distribution to pressure change along the tube length via an ideal gas law relationship between pressure, density, and temperature.

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Arsal Maqbool
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61 views4 pages

Whence o - C,, - + C: or - Do

1) The document discusses analysis of flow through cylindrical tubes, including annular flow with an inner cylinder moving axially and analysis of a capillary flowmeter. 2) Equations are derived for the mass flow rate and force on a cylinder in annular flow, and the mass flow rate is related to fluid properties, gravity, and height for a capillary flowmeter. 3) Analysis of low-density phenomena in compressible tube flow relates the velocity distribution to pressure change along the tube length via an ideal gas law relationship between pressure, density, and temperature.

Uploaded by

Arsal Maqbool
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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2B.

7 Annular flow with inner cylinder moving axially


o. The momentum balance is the same as that in Eph. 2.3-11
or Eq. 2.4-2, but with the pressure-difference term omitted. We can
substitute Newton's law of viscosity into this equation to get

whence
dr r _ _ c, „, or — = —Do + D2
’0
oz . + C2

That is, we select new integration constants, so that they are


dimensionless. These integration constants are determined from the
no-slip conditions at the cylindrical surfaces: t›, (cR) = t›0 £tnd
r, (R) = 0. The constants of integration are D2 = 0 and D1 = —1/ln c.
This leads then directly to the result given in the book.
â. The mass rate of flow is

oR2
rdrd8 —— 2ap

— { 2 )|t = 2c
2
ln c —4 (1—r 2 ))
4
2:r:p
2ap

which is equivalent to the answer in the text.


c. The force on a length L of the rod

L 2* (/&A)
0 0 0
In›r

which gives the expression in the book.


d. When we replacer by l— e and expand in a Taylor series,
we get

+2 1 _ 2 nLy)v0
(i— 21
—(• 2 3*3 + 4
To get this last result one has to do a long division involving the
polynomial in the next-to-last step.
2B.8 Analysis of a capillary flowmeter
Designate the water by fluid "I" and the carbon
tetrachloride Dy ii". Later the distance irom B to C as "J". One mass
rate or ñow in the tube section "AB" ñ given by

8yL 8y£

Since the fluid in the manometer is not moving, the pressures at D


and E must be equal; hence

from which we get

EA GB + PI8 h —— (PII PI) gH

Insertion of this into the first equation above gives the expression
for the mass rate of flow in terms of the difference in the densities
of the two fluids, the acceleration of gravity, and the height H.
2B.9 Low-density phenomena in compressible tube flow
When we replace no-slip boundary condition of Eq. 2.3-17
by ñq. 2ñ.9-i, we get

“ 4yL 2yfi

so that the velocity distribution in the tube is

2
T (/0 "PL)•r
2pL

2
w —— * p(z)n (r, z)rdrd8 —— 2 z1
o RET

where we have introduced the ideal gas law, with g being the gas
constant (we use a subscript g here to distinguish the gas constant
from the tube radius). We have also introduced a dimensionless
radial coordinate. When we introduce the velocity distribution
above, we get

• 2 2zE
fo 1...
4

4p (,,Tj‹,\—°
../\ —*

This is now integrated over the length of the tube, keeping mind that
the mass flow rate in is constant over the entire length

4(o
dp

2— 14
2
R4 M — p2 + 4 \}
#0 0
(P0 " /L)

This gives

8L RT 2 R

which leads then to Eq. 2B.9-2.

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