Fluid Mechanics for MAP/Energy Considerations

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Losses and Energy consideration[edit | edit source]

Remember the conservation of energy equation:


The velocity does not have necessarily a plug profile at the inlet and outlet of a control volume and the cross sectional areas might also differ at the inlet and outlet of a CV. The variations in the cross sections of flow conduits and the velocity profiles have to be accounted to calculate the correct pressure drop due to the viscous losses. Therefore, the surface integrals has to be calculated by considering these variations:

For the calculation of the profile effects one can use the kinetic energy coefficient as follows:

For fully developed laminar velocity profile, where

For fully developed turbulent flow, due to relatively flat velocity profile,

Losses energy.png
Velocity profiles for laminar (upper) and turbulent (lower) states at the same mass flow rate

Rearranging the energy equation:

The conversion of mechanical energy (A) and the irreversible conversion and loss of mechanical energy with viscous dissipation (B) can be better distinguished.

Total pressure drop[edit | edit source]

Finally the total pressure drop due to viscous effects can be written as

Major and minor pressure losses[edit | edit source]

Piping system on a chemical tanker.jpg

The total pressure drop consist of major losses, which are due to frictional effects in fully developed flow in constant area tubes, and minor losses, which occur at entrances, sudden area changes, bends, elbows, fittings, valves, contraction and diffusers, etc..

Major losses in pipe flows[edit | edit source]

Moody diagram

The pressure drop along a pipe, which has no elevation change, can be written as:

Nondimensionalizing with the kinetic energy per unit mass of the flow:

where is the loss coefficient and f is the friction factor.

Non-circular ducts[edit | edit source]

For non circular ducts, which appears in air conditioning, heating and ventilating applications, an equivalent diameter is calculated so that correlations for circular pipes can be utilized.

This equivalent diameter is called as hydraulic diameter:

Thus for a rectangular duct with a width of b and height of h:

And for a square duct

Minor losses[edit | edit source]

For the other components which might cause pressure drop, the loss coefficient (resistance coefficient) can similarly be defined:

However, for the vast variety of geometries, it is not possible to obtain relations as for the pipe flows. Thus, for each component is measured and documented.


Loss coefficient of standard components[edit | edit source]

Elbow geometry
Range of loss coefficients for different valves and flaps
Range of loss coefficients for circular bellmouth inlet(collector)
The loss coeffecicent data for various elbow dimensions
Range of loss coefficients for the entrance into a straight tube of constant cross section
Range of loss coefficients for sudden expansion of a flow having a uniform velocity distribution

Calculation of total loss in a single pipe system[edit | edit source]

A single pipe system may have many minor losses. Since all are correlated with , they can be summed into a single total system loss if the pipe has constant diameter:

However, we must sum the losses separately, if the pipe size changes, i.e. mean velocity changes.

Multiple path system[edit | edit source]

Losses energie11 2.png

The solution for pipe network problems is often carried out by use of node and loop equations similar in many ways to that done in electrical circuits.

  1. The net flow into any junction must be zero.
  2. The net head loss around any closed loop must be zero. At each junction there is one single pressure.
  3. All head losses must satisfy the major and minor-loss friction correlations.

In such problems the pipes and components might have different areas, thus pressure loss at each pipe and component should be calculated separately. Moreover, there can be components with unknown properties.

For such cases it is found useful to formulate the pressure loss as a function of the flow rate:

Resultant loss coefficients, Kr, for serial and parallel connections[edit | edit source]

Serial connection: Flow rate is the same for all components

Losses energie12.png

Parallel connection: Pressure drop is the same inlet and outlet junctions

Losses energie13.png

Design of a dry powder inhaler[edit | edit source]

The utilization of the explained concepts will be demonstrated in the class by developing an inhaler from scratch. The basic steps of the design are as follows:

Schematics for Inhaler flow
  1. Make a conceptual design
  2. Decide pressure drop-flow rate characteristic of the inhaler (Inhaler patient matching)
  3. Dimension each component of the inhaler to ensure
    • Pressure drop-flow rate characteristic
    • Full emptying of the blister
    • Sufficient dispersion of the powder composition

An overview of dry powder inhaler design could be found in in the following publication [1]

Reference[edit | edit source]

  1. Int J Pharm. 2011 Sep 15;416(1):25-34. doi: 10.1016/j.ijpharm.2011.05.045. Epub 2011 Jun 28. A method for the aerodynamic design of dry powder inhalers. Ertunç O, Köksoy C, Wachtel H, Delgado A.