Radial Flow through Tubes

1
Radial Flow through Tubes

• MSE 4984

2
Flow Through a Tube
R
r
z
3
Flow Through a Tube

• Flow Pattern Flow fastest near middle of pipe,
  symmetric flow pattern which must mean that shear
  stress must be zero in the middle of the tube
• v(z) 0 at r R
• Fully Developed, incompressible flow
• Newtonian Fluid

4
Flow Through a Tube

• Momentum Flux due to fluid flow
• Momentum Flux Across Surface at z z
• 2?r?rvz(?vz), evaluated at z z
• Momentum Flux Across Surface at z z?z
• 2?r?rvz(?vz), evaluated at z z?z
• Momentum due to Shear
• Momentum Flux Across Cylindrical Surface
• ??r?z?rz evaluated at r r
• Momentum Flux Across Cylindrical Surface
• ??r?z?rz evaluated at r r ?r

5
Flow Through a Tube

• Momentum Flux Due to Pressure Loss
• Pressure Force at z z Pz 2?r?r
• Pressure Force at z ?z Pz ?z2?r?r
• Sum of Momentum Flux

6
Flow Through a Tube

• ??r?z?rz evaluated at r r ?r- ??r?z?rz
  evaluated at r r Pz ?z2?r?r - Pz 2?r?r
  0
• d(r?rz )/dr - (dP/dz) r 0
• r?rz (dP/dz) r2/2 C1
• ?rz (dP/dz) r/2 C1/r
• BC ?rz 0 at r0, C1 0
• (dP/dz) r/2 -????dv(z)/dr
• v(z) -(dP/dz) r2 /4? C2
• BC V(z) 0 at r R
• C2 -(dP/dz) R2 /4?

7
Flow Through a Tube

• v(z) -(dP/dz) R2 /4? 1-(r/R)2
• v(max) ?
• Q ?
• v(avg) ?

8
Flow Through an Annulus
R
r
aR
z
9
Flow Through an Annulus

• Flow Pattern
• v(z) 0 at r lt aR, a lt 1, no flow in annular
  center
• z(z) 0 at r R...no slip
• Fully Developed, incompressible flow
• Newtonian Fluid

10
Flow Through an Annulus

• Momentum Flux due to fluid flow
• Momentum Flux Across Surface at z z
• 2?r?rvz(?vz), evaluated at z z
• Momentum Flux Across Surface at z z?z
• 2?r?rvz(?vz), evaluated at z z?z
• Momentum due to Shear
• Momentum Flux Across Cylindrical Surface
• ??r?z?rz evaluated at r r
• Momentum Flux Across Cylindrical Surface
• ??r?z?rz evaluated at r r ?r

11
Flow Through an Annulus

• Momentum Flux Due to Pressure Loss
• Pressure Force at z z Pz 2?r?r
• Pressure Force at z ?z Pz ?z2?r?r
• Sum of Momentum Flux

12
Flow Through an Annulus

• ??r?z?rz evaluated at r r ?r- ??r?z?rz
  evaluated at r r Pz ?z2?r?r - Pz 2?r?r
  0
• d(r?rz )/dr - (dP/dz) r 0
• r?rz (dP/dz) r2/2 C1
• ?rz (dP/dz) r/2 C1/r -????dv(z)/dr
• v(z) -(dP/dz) r2 /4? - C1ln(r)/? C2
• B. C.s V(z) 0 at r aR and R

13
Flow Through an Annulus

• C1 (dP/dz)(R2 /4)(1-a2)/ln a
• C2 (dP/dz)(R2 /4 ?)1 (1- a2)lnR/ ln a
• Substitute in for C1 and C2 to get v(z)
• v(max) ?
• v(avg) ?
• Q ?

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Final Comments

• Integrate the velocity function across the flow
  cross-sectional area to get Q
• Divide Flow rate by integrated cross-sectional
  area to get average flow rate
• Simple geometry...solvable DEQs
• Flat Plate Flow
• Flow on an Incline
• Flow Through a Pipe
• Flow Through and Annulus