Internal Flow

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[1]Fluid flows can be either external, where the fluid is made to flow over or around a body, or internal, where a fluid is made to flow through a system. This article discusses internal flow, introduces the entrance and fully developed regions, discusses the link between flow and the Reynolds Number and discusses the various calculations associated with internal flows.

Contents

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 337 - 360 (Chapter 8).

Introductory Comments

[2]Some important notes to keep in mind in this chapter are:

  • The terms pipe, duct and conduit can be used interchangeably for any circular pipe to minimise confusion, even though they each have their own specific meaning as:
  • Pipes have circular cross sections
  • Ducts have non circular sections
  • Circular pipes are the most popular as they minimise deformations when pressure differences between the inside and outside of the pipe exist
  • However rectangular ducts are used in buildings as they are cheaper and pressure differences are usually very low
  • The theory of fluid flow is mostly obtained from experiments and empirical evidence, rather than pure mathematical solutions
  • Due to the no slip condition, the velocity in a pipe changes from zero at the walls to a maximum at the center of the pipe cross section
  • It is therefore more convenient to work with the average velocity, Vavg, even though the density might change due to heat transfer, as the losses are minor
  • Since friction plays a minor role in internal flows, we usually ignore it and care more about its overall result, which is a pressure drop (or head loss)

Laminar and Turbulent Flow

[3]It is noticed that fluid flow in low velocity is highly ordered and streamlined, whereas higher velocities yield chaotic, zig-zagging motion. The ordered flow is known as laminar while the unordered flow is known as turbulent. Turbulent flow causes larger momentum transfer between fluid particles and hence increases friction. Whether a flow is turbulent or laminar can be determined by Reynolds number.

Reynolds Number

[4]The Reynolds Number is determined by the ratio of the inertial forces of a fluid to the viscous forces. Reynolds number can be determined by:

ReynoldsNumber.png

Where:

Vavg = Average Velocity
D = Characteristic length of the pipe (in internal flows this is generally the diameter) (m)
v = μ/ρ = kinematic viscosity of the fluid (m²/s)
μ = viscosity of fluid (kg/m.s)

A high Reynolds Number suggests that the inertial forces are greater than the viscous forces, and hence the fluid cannot prevent the random motion of the fluid (and hence seen as turbulent). It is generally accepted that for internal flow, a reynolds number of:

Re ≤ 2300 is laminar
2300 ≤ Re ≤ 4000 is transitional
Re ≥ 4000 is turbulent

Non Circular Pipe Sections

[5]For non-circular pipe sections, the characteristic length (or diameter) is determined by the hydraulic diameter, Dh, defined as:

HydraulicDiameter.png

Where:

Ac = the pipe cross section
p = wetted perimeter

The Entrance Region

[6]Fluid flow in pipes is constant only after a certain length of pipe, known as the hydrodynamic entrance length. This is due to the no slip condition, which creates a boundary layer in which friction and viscous effects take place. Eventually the boundary layer reaches the center of the flow, at which point the fluid flow begins to develop properly, as the friction forces slow down fluid flows and the center flow has to increase in velocity in order to maintain a constant mass flow rate. The result is a parabolic velocity profile in laminar flow, and flatter velocity profile for turbulent flows.

[7]For turbulent flow, the developed velocity profile is flatter, and it has four layers:

  1. Viscous Sublayer: where the viscosity effects are not negligible. The velocity profile is nearly linear and its flow almost fully streamlined
  2. Buffer Layer: flow is still dominated by viscous effects but turbulent effect begin to become important
  3. Overlap Layer: turbulent effects are very important, but the flow is still dominated by viscous effects
  4. Turbulent Layer: where turbulent effects dominate the flow

Pressure and Head Losses

As mentioned above, friction effects are small relative to pressure (or head) losses. Recall that the general energy equation (which can be expressed in terms of heads) involves an item for mechanical and head losses. The calculations for these terms are discussed here.

Laminar Flow Calculations

[8]For laminar flow, the pressure drop can be calculated by :

LaminarPressureDrop.png

and the pressure drop as a head loss can be evaluated by:

LaminarHeadLoss.png

Where:

f = Darcy Friction Factor
L = Pipe Length
D = Pipe Diameter
Vavg = Average Velocity

For circular pipes in laminar flow, the Darcy friction factor is given by:

LaminarDarcyFactor.png

From these, the required pumping power to overcome the losses is given by:

LaminarRequiredPumpingPower.png

Note that the required volume flow rate and average velocity can be found by:

LaminarAverageVelocity.png
LaminarFlowRate.png

Turbulent Flow Calculations

[9]Since turbulent flow is chaotic, mathematical solutions only help with the analysis, but the relationships are determined by experiments and empirical evidence. The pressure loss is given by the same equation as for laminar flow, that is:

LaminarPressureDrop.png

but the Darcy Friction factor is not determined by the equation for laminar flow. It is determined from the Moody Chart by calculating the Reynolds Number first, then the relative roughness, ε/D (usually given for materials) and then by comparing the two on the Moody chart to get the Darcy Friction Factor.

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).

  1. Textbook p. 337
  2. Textbook p. 338
  3. Textbook p. 339
  4. Textbook p. 340
  5. Textbook p. 340
  6. Textbook p. 341
  7. Textbook p. 354
  8. Textbook p. 345
  9. Textbook p. 357
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