Physical properties of fluid and fluid transport in piping system
Laminar and turbulent flow in pipe
Pressure drop equation in isothermal flow
Flow in pipes and valves theory - full content
The flow in long pipelines closely approximates isothermal conditions. The pressure drop in such lines is often large compared
to inlet pressure and solution of this problem is outside the limitations of the Darcy equation. An accurate solution is made by
isothermal equation where weight flow w in [kg/s] is:
Adiabatic flow is the flow where no heat is transferred to or from pipe line in short, perfectly insulated pipe.
The heat which is created due to flow friction is added to the flow and that amount of energy is actually acceptable.
For adiabatic flow is:
where is:
- p1,2 - pressure on the begging and on the end of pipe line
- w - mass flow rate
- v1 - specific weight
- f - friction factor
- L - pipe length
- D - internal pipe diameter
- A - pipe cross section area
Substituting specific weight using equation of state pressure drop due to friction can be written as:
where is:
- p1,2 - pressure on the begging and on the end of pipe line
- w - mass flow rate
- Zm - mean compressibility factor
- R - gas constant
- T - temperature
- f - friction factor
- L - pipe length
- D - internal pipe diameter
- A - pipe cross section area
Both two above equations are developed with following assumptions:
- flow is isothermal
- no mechanical work is added or subtracted
- flow is steady in time
- gas is perfect
- velocity is represented as the average velocity in cross section
- the friction factor is constant
- pipe line is straight and horizontal
In the practice of gas pipe line engineering, another assumption is added:
Acceleration can be neglected because the pipe line is long.
With that assumption, equation for isothermal flow in horizontal pipe line is:
or:
where is:
- p1,2 - pressure on the begging and on the end of pipe line
- w - mass flow rate
- v1 - specific weight
- f - friction factor
- L - pipe length
- D - internal pipe diameter
- A - pipe cross section area
- Zm - mean compressibility factor
- R - gas constant
- T - temperature
Mean compressibility factor is calculated as:
where is:
- Zm - mean compressibility factor
- Z1 - compressibility on the begging of pipeline
- Z2 - compressibility on the end of pipeline
Volumetric flow rate can be calculated for defined conditions, like normal conditions, standard conditions or based on
the actual flow condition used appropriate density in following equation:
where is:
- q - volumetric flow rate
- w - mass flow rate
- &rho - density
Normal conditions: p=101325 Pa, T=273,15 K
Standard conditions: p=101325 Pa, T=288,15 K (15OC)
When volumetric flow at specified condition is known, using continuity equation volumetric flow rate on some other condition
can be calculated using:
where is:
- q - volumetric flow rate
- p - pressure
- T - temperature
As it is usually more common to express flow rates in terms of cubic meter per hour at standard conditions equation
for isothermal flow can be written:
where is:
- qh - volumetric flow rate [m3/h]
- p - pressure [Pa]
- T - temperature [K]
- Sg - relative density [ - ]
- Lm - pipe length [km]
- d - internal pipe diameter [mm]
- f - friction factor [ - ]
Other equations are used for compressible flow in long pipe lines like Weymouth and Panhandle formula:
Weymouth formula is:
where is:
- qh - volumetric flow rate [m3/h]
- p - pressure [Pa]
- T - temperature [K]
- Sg - relative density [ - ]
- Lm - pipe length [km]
- d - internal pipe diameter [mm]
Friction factor used in Weymouth formula is: f=0.094/d 1/3. This friction factor is identical with one obtained from Moody
diagram for fully turbulent flow for 20 inch inside diameter. For pipe diameters that are smaller than 20 inch, Weymouth
friction factors are larger and for pipes bigger than 20 inch, Weymouth friction factor is smaller than in friction factors
obtained from Moody diagram for same pipe sizes.
Panhandle formula is:
where is:
- qh - volumetric flow rate [m3/h]
- p - pressure [Pa]
- Lm - pipe length [km]
- d - internal pipe diameter [mm]
- E - flow efficiency factor E=0.92
Panhandle formula is for natural gas pipe sizes from 6" to 24", and for Reynolds numbers between Re = 5x106 and
Re = 14x106, with specific gravity for natural gas Sg=0,6. The flow efficiency factor E is defined as an experience
factor and is usually assumed to be 0.92 for average operating conditions.
The Panhandle friction factor is defined as: f = 0.0454 (d/qhSg)0.1461. In the range where Panhandle formula is applicable,
the friction factors are smaller than one from Moody diagram as because of that the flow rate are usually greater than those
calculated using equation for isothermal flow.
Calculation of natural gas flow and pressure drop through natural gas pipe line can be made using Renouard equation:
where is:
- p1 - absolute pressure on the start of pipe line [bar]
- p2 - absolute pressure on the end of pipe line [bar]
- Sg - relative density [ - ]
- L - pipe length [km]
- qh - volumetric flow rate [m3/h]
- D - internal pipe diameter [mm]
Volumetric flow rate qh in Renouard equation is to be used at standard conditions (p=101325 Pa, T=288,15 K (15OC)).
Relative density in Renouard equation is calculated as follows:
where is:
- Sg - relative density (around 0.64 kg/m3)
- &rhoNG - natural gas density on standard conditions, which depends on the natural gas mixture (around 0.78 kg/m3)
- &rhoAIR - air density at standard conditions, which is 1.226 kg/m3
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