# Density and pressure drop relationship memes thèse une fois pour toutes, même si l 'on s'engueulait beaucoup trop à cause de cela .. density Direct mass flow measurement is preferable because it offers a .. because the relationship between flowrate and the pressure drop in the. velocity of the fluids in the pipes and pressure drop. velocity in pipes. The average speed is based on the relationship: volume flow (m3 / s) / section area ( m²). In fluid dynamics, the Darcy–Weisbach equation is an empirical equation, which relates the head loss, or pressure loss, due to friction along a given length of pipe to ρ, the density of the fluid (kg/m3);: D, the hydraulic diameter of the pipe (for a pipe .. While the Colebrook–White relation is, in the general case, an iterative.

Note that this is equivalent to a Multi-Step Cullender and Smith calculation.

Flow Correlations Many single-phase correlations exist that were derived for different operating conditions or from laboratory experiments. Generally speaking, these only account for the friction component, i. For example, if the Gray correlation was selected but there was only gas in the system, the Fanning gas correlation is used. Single-phase correlations can be used for vertical or inclined flow provided that the hydrostatic pressure drop is accounted for in addition to the friction component.

Even though a particular correlation may have been developed for flow in a horizontal pipe, incorporation of the hydrostatic pressure drop allows that correlation to be used for flow in a vertical pipe. Multiphase Flow Multiphase pressure loss calculations parallel single-phase pressure loss calculations. Essentially, each multiphase correlation makes its own particular modifications to the hydrostatic pressure difference and the friction pressure loss calculations, in order to make them applicable to multiphase situations.

The presence of multiple phases greatly complicates pressure drop calculations. This is due to the fact that the properties of each fluid present must be taken into account. Also, the interactions between each phase must be considered. Mixture properties must be used, and therefore the gas and liquid in-situ volume fractions throughout the pipe need to be determined.

In general, multiphase correlations are essentially two-phase and not three-phase. Accordingly, the oil and water phases are combined, and treated as a pseudo single-liquid phase, while gas is considered a separate phase. The hydrostatic pressure difference calculation is modified by defining a mixture density.

This is determined by a calculation of in-situ liquid holdup amount of liquid in the pipe section. Some correlations determine holdup based on defined flow patterns. They can be grouped as follows: Do not account for flow patterns: Gray — Developed using data from gas and condensate wells. Hagedorn and Brown — Derived using a test well running different oils and air Consider flow patterns: Beggs and Brill — Correlation derived from experimental data for vertical, horizontal, inclined uphill and downhill flow of gas-water mixtures Petalas and Aziz — Mechanistic model combined with empirical correlations.

This multi-purpose correlation is applicable for all pipe geometries, inclinations and fluid properties. These models can be used for gas-liquid multiphase flow, single-phase gas or single-phase liquid, because in single-phase mode, they revert back to the Fanning equation, which is equally applicable to either gas or liquid.

Poiseuille's Law - Pressure Difference, Volume Flow Rate, Fluid Power Physics Problems

The Gray and Hagedorn and Brown correlations were derived for vertical wells and may not apply to horizontal pipes. Flow Fluid Properties Superficial Velocities The superficial velocity of each phase is defined as the volumetric flow rate of the phase divided by the cross-sectional area of the pipe as though that phase alone was flowing through the pipe.

Since the liquid phase accounts for both oil and water: The oil, water, and gas formation volume factors BO, BW, and Bg are used to convert the flow rates from standard or stock tank conditions to the prevailing pressure and temperature conditions in the pipe.

Since the actual cross-sectional area occupied by each phase is less than the cross-sectional area of the entire pipe, the superficial velocity is always less than the true in-situ velocity of each phase. Mixture Velocity Mixture velocity is another parameter often used in multiphase flow correlations. The mixture velocity is given by: These in-situ velocities depend on the density and viscosity of each phase. Typically the phase that is less dense flows faster than the other. This causes a "slip" effect between the phases.

As a consequence, the in-situ volume fractions of each phase under flowing conditions will differ from the input volume fractions of the pipe.

### Darcy–Weisbach equation - Wikipedia

If the slip condition is omitted, the in-situ volume fraction of each phase is equal to the input volume fraction. Because of slippage between phases, the liquid holdup EL can be significantly different from the input liquid fraction CL. In other words, the liquid slip holdup EL is the fraction of the pipe that is filled with liquid when the phases are flowing at different velocities. It can be defined as follows: We can also write them in function of the superficial velocities as: QL is the liquid rate at the prevailing pressure and temperature. Similarly, QGBg is the gas rate at the prevailing pressure and temperature. The input volume fractions, CL and EL, are known quantities, and are often used as correlating variables in empirical multiphase correlations. Actual Velocities Once the liquid holdup has been determined, the actual velocities for each phase can be determined as follows: Note that this is in contrast to the way density is calculated for friction pressure loss.

The friction factor represented in these regions is given by the Colebrook formula which is used throughout industry and accurately represents the transition and turbulent flow regions of the Moody diagram. The Colebrook formula for Reynolds number over is given in equation 3.

The roughness factor is defined as the absolute roughness divided by the pipe diameter or. Typical values of absolute roughness are 5. The Colebrook equation has two terms. The second term,is dominate for fluid flow where the relative roughness lines converge smooth pipes. Liquid Incompressible Flow For liquid flow, equation 4 has been used by engineers for over years to calculate the pressure drop in pipe due to friction.

This equation relates the various parameters that contribute to the friction loss. This equation is the modified form of the Darcy-Weisbach formula which was derived by dimensional analysis. The friction factor in this equation is calculated by equation 3 for a specified Reynolds number and roughness factor using an iterative method or a trial and error procedure. Gas Compressible Flow For gas flow, density is a strong function of pressure and temperature, and the gas density may vary considerably along the pipeline.

### Pressure Loss Calculations

Due to the variation of density, equation 5 should be used for calculation of friction pressure drop. Again, the friction factor in this equation is calculated by equation 3 for a specified Reynolds number and roughness factor using a trial and error procedure. Actual volume flow rate is needed to calculate the velocity of gas in the line from which the Reynolds number is calculated.

Equation 6 may be used to convert the volume flow rate at standard condition to the actual volume flow rate. The total length of pipe is 55 km.

## Darcy–Weisbach equation

Total Fluid Energy Daniel Bernoulli, a Swiss mathematician and physicist, theorized that the total energy of a fluid remains constant along a streamline assuming no work is done on or by the fluid and no heat is transferred into or out of the fluid.

The total energy of the fluid is the sum of the energy the fluid possesses due to its elevation elevation headvelocity velocity headand static pressure pressure head. The energy loss, or head loss, is seen as some heat lost from the fluid, vibration of the piping, or noise generated by the fluid flow. Between two points, the Bernoulli Equation can be expressed as: In other words, the upstream location can be at a lower or higher elevation than the downstream location.

If the fluid is flowing up to a higher elevation, this energy conversion will act to decrease the static pressure. If the fluid flows down to a lower elevation, the change in elevation head will act to increase the static pressure. Conversely, if the fluid is flowing down hill from an elevation of 75 ft to 25 ft, the result would be negative and there will be a