Principles of Fluid Flows in Pipes- Pressure Drop & Flow Rate

November 8, 2015 | By | Reply More

A practical application for the concepts of pressure, viscosity, and the Reynolds number is the flow of fluids through pipes, hoses, and ducts. In addition to distributing water, gasoline, natural gas, air, and other fluids, pipe flow is also an important topic for biomedical studies of the human circulatory system (Figure). Blood flows through the arteries and veins in your body in order to transport oxygen and nutrients to tissue and to remove carbon dioxide and other waste products. The vascular system comprises relatively large arteries and veins that branch out into many, much smaller capillaries extending throughout the body. In some respects, the flow of blood through those vessels is similar to that encountered in such engineering applications as hydraulics and pneumatics.

fluid flow in human arteries

Figure: The flow of blood in the human circulatory system is similar in many respects to the flow of fluids through pipes in other engineering applications. Images such as this of the human pulmonary system are obtained through magnetic resonance imaging, and they provide physicians and surgeons with the information they need to make accurate diagnoses and devise treatment plans.

The flow of Fluid in Pipes

Fluids tend to flow from a location of high pressure to one of lower pressure. As the fluid moves in response, it develops viscous shear stresses that balance the pressure differential and produce steady flow. In the human circulatory system, with all other factors being equal, the greater the difference in pressure between the heart and femoral artery, the faster the blood will flow.

Pressure Drop fluid in Pipes

The change in pressure along the length of a pipe, hose, or duct is called the pressure drop, denoted by Δp. The more viscous a fluid is the greater the pressure differential that is necessary to produce motion. The figure below depicts a free body diagram of a volume of fluid that has been conceptually removed from a pipe. Since the pressure drop is related to the shear stress, we expect that Δp will increase with the fluid’s viscosity and speed.

flow in pipes

Figure: Free body diagram of a volume of fluid within a pipe. The pressure difference between two locations balances the viscous shear stresses between the fluid and the pipe’s inner surface. The fluid is in equilibrium, and it moves with a constant speed.

 

In a section of the pipe that is away from disturbances (such as an inlet, pump, valve, or corner) and for low enough values of the Reynolds number, the flow in the pipe is laminar. Experimental evidence shows that laminar flow occurs in pipes for Re < 2000. Recalling the no-slip condition, the velocity of the fluid is precisely zero on the inner surface of the pipe. By the principle of symmetry, the fluid will move fastest along the pipe’s centerline and decrease to zero velocity at the pipe’s radius R (Figure below ). In fact, the velocity distribution in laminar flow is a parabolic function of radius, as given by the equation

fluid flow in pipe equations                                                                          (Special case: Re < 2000)  ————  1

 

Maximum velocity of fluid in pipes

Figure: Steady laminar flow of a fluid in a pipe. The fluid’s velocity is the greatest along the pipe’s centerline, changes parabolically over the cross-section, and falls to zero on the surface of the pipe.

where r is measured outward from the pipe’s centerline. The maximum velocity of the fluid

fluid flow equation                                                                                                                      —————————– 2
occurs at the pipe’s centerline, and it depends on the pressure drop, the pipe’s diameter d = 2R, the fluid’s viscosity , and the pipe’s length L. Engineers will typically specify the diameter of a pipe, rather than its radius, because the diameter is easier to measure. The term Δp/L in Equation (2) is interpreted as the drop in pressure that occurs per unit length of the pipe.

Volumetric Flow rate

Aside from the fluid’s speed, we are often more interested in knowing the volume ΔV of fluid that flows through the pipe during a certain time interval Δt. In that respect, the quantity is called the volumetric flow rate, ΔV/Δt and it has the dimensions of m^3/s in the SI and ft^3/s or gal/s in the USCS. Conversion factors between those dimensions are given in Table. We can read off the conversion factors for

flow rate conversion table

Table: Conversion Factors Between USCS and SI Units for Volumetric Flow Rate

 

volumetric flow rate in a pipe

figure: volumetric flow rate in a pipe

The volumetric flow rate is related to the pipe’s diameter and to the velocity of the fluid flowing through it. The figure above depicts a cylindrical element of fluid having cross-sectional area A and length Δx flowing through a pipe. In the time interval Δt, the volume of fluid that flows past any cross section of the pipe is given by ΔV = A Δx Since the average speed of the fluid in the pipe is  vavg=Δx/Δt, The Volumetric flow rate is given by

q = Avavg ——————————- 3 

 

When the flow is laminar, the fluid’s average velocity and the maximum velocity in Equation (2) are related by (Special case: Re < 2000) as shown in Figure (b).

vavg = 0.5 vmax

In calculating the Reynolds number for fluid flow in pipes, the average velocity v and the pipe’s diameter d should be used in Equation of Reynolds Number.

Poiseuille’s law of fluid flow in the pipe 

Combining Equations (1), (2), and (3), the volumetric flow rate in a pipe for steady, incompressible, laminar flow is

Poiseuille’s law of fluid flow in pipes

This is called Poiseuille’s law and it is limited to laminar flow conditions. As measured by volume, the rate of fluid flowing in a pipe grows with the fourth power of its diameter, is directly proportional to the pressure drop, and is inversely proportional to the pipe’s length. Poiseuille’s law can be used to determine the volumetric flow rate when the pipe’s length, diameter, and pressure drop are known; to find the pressure drop; or to determine the necessary diameter for a pipe when q, L, and D p are given.

When a fluid’s compressibility is insignificant, the volumetric flow rate will remain constant even when there are changes in the pipe’s diameter, as depicted in Figure below.

Fluid fl ow in a pipe having a constriction.

Fig: Fluid fl ow in a pipe having a constriction.

In essence, because fluid can’t build up and become concentrated at some point in the pipe, the amount of fluid that flows into the pipe must also flow out of it. In Figure, the cross-sectional area of the pipe decreases between sections 1 and 2. For the same volume of fluid to flow out of the constriction per unit time as flows into it, the fluid’s velocity in section 2 must be higher. By applying Equation (3), the average speed of the flowing fluid changes according to the equation

A1v1 = A2v2

If the cross-sectional area of a pipe, hose, or duct becomes smaller, the fluid flows faster, and vice versa. You probably have experimented with volumetric flow rate without realizing it when you place your finger over the end of a garden hose to cause the water to spray farther.

Further Readings

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Category: Fluidics, Mechanical

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