Viscous Flow In Pipes

Reynolds Number:

Reynolds Number is the ratio of Inertial forces to the Viscous forces. By knowing this number, one can describe the flow whether the flow is laminar, turbulent or transitional flow.

There are three possibilities of the range of Reynolds Number:

• If Reynolds number < 2100 or 2300  ⟺  Laminar Flow
• If Reynolds number > 4000  ⟺  Turbulent Flow
• If 2300 < Reynolds number < 4000  ⟺  Transitional Flow

In case of different shapes other than circle, the formula for Reynolds Number remain the same however diameter involves in the formula become Hydraulic Diameter which is given by:

For pipe:

• Critical Reynolds Number  ⟾  The Reynolds number at which the flow becomes turbulent Re_cr

For different shapes, the hydraulic diameters are given by:

Completely Filled Flow In Closed Channel:

If pipe is is considered to filled completely with closed channel, then following are the conditions which are generated to flow liquid.

• If Pressure at 1 < Pressure at 2  ⟼  Flow will be in the direction 1➙2.
• If Pressure at 1 = Pressure at 2  ⟼  No flow of liquid, requires external pressure (pump) to flow liquid.

Partially Filled Flow In Open Channel:

If pipe is is considered to filled completely with closed channel, then following are the conditions which are generated to flow liquid.

There is only one condition arises which is:

• To flow liquid, pump is required because pressures are equal at any two states because it is open to atmospheric pressure.

Types Of Flow:

There are three types of flow depending upon the behavior of fluid particles which are described as follows:

1. Laminar Flow:

It is characterized by the by smooth streamlines and highly ordered motion. The Reynolds number for Laminar flow is lesser than 2300. 

2. Turbulent Flow:

It is characterized by fluctuation in velocity and highly disordered motion. The Reynolds number for Turbulent flow is greater than 4000.

3. Transitional Flow:

It is the transition from laminar to turbulent flow does not occur suddenly.

Boundary Layer Theory:

When liquid want to flow through a closed or open channel, the layer closes to wall or surface resists due to friction and viscosity which increases with time. The velocity of boundary layer is 0.99 times the free velocity or middle layer velocity.

The Entrance Region:

The length from entrance to the uniform velocity profile is called Hydrodynamic Entry Length. And this region is called Hydrodynamic Entrance Region.

• The region after the attainment of uniform velocity profile is called Fully Developed Region.

The hydrodynamic entry length for laminar and turbulent flow are:

 Laminar Flow In Pipes:

Consider a ring-shaped differential volume element of radius 'r', thickness 'dr' and length 'dx' oriented co-axially with the pipe. A force balance on the volume element in the flow direction gives:

Pressure & Head Loss For Inclined Pipes:

For inclined pipes, the pressure and head loss is given by introducing sine of θ.

Energy Equation In Terms Of Heads:

Energy equation for calculating various parameters in terms of head is given by:

Where;

• Head for Turbine  ↔  output energy which we are extracted.
• Head Loss  ↔  due to decrease in pressure.
• Correction Factor, α  ↔  it determines the flow is ideal or actual. Its value is in between 0-1.

Turbulent Flow In Pipes:

There are four types of layers which are described as follows:

1. Viscous (or Laminar or Linear or Wall) Sub-Layer  ↣  layer close to wall in which viscous effects are dominant, velocity profile is linear.
2. Buffer Layer  ↣  viscous effects are still dominant but turbulent effects are becoming significant.
3. Overlap (or Transition) Layer or Inertial Sub-Layer  ↣  layer in which turbulent effects are much more significant.
4. Outer (or Turbulent) Layer  ↣  turbulent effects are dominant.

Calculation Of Darcy Friction Factor:

We calculate Darcy Friction Factor by two following ways:

1. By Moody Chart:

To calculate friction factor from moody chart, you should have Reynolds number 'Re' and relative roughness ε/D.

2. By Colebrook Equation:

To calculate Darcy friction factor from Colebrook method, following equation is used. 

Minor Losses:

Losses due to elbows or fittings are called Minor Losses. In case of minor losses, a minor loss coefficient or resistance coefficient will appear. Minor losses are given by:

The total head loss is given by:

For number of fittings or elbows, the above equation after summation becomes:

Piping Network:

There are two types of piping network which are discussed ahead:

1. Series Piping System:

The total head loss in the series piping system is equal to the sum of individual head losses and the flow rate remains same.

2. Parallel Piping System:

The total head loss in the series piping system is equal to the sum of flow rate. The pressure drop in each individual pipe connected in parallel are same.

Note: If you want to learn piping system in advance, go and watch it:

References:

• Materials From Class Lectures + Own Knowledge + Book named Fundamentals of Fluid Mechanics by Munson, Young and Okiishi's (8th Edition).
• Photos from Google Images.
• Videos from YouTube + Google.