Principles of Fluid Flows in Pipes- Pressure Drop & Flow Rate
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.
- Properties of fluids and Their Applications
- Fundamentals of Fluid mechanics
- Dimensionless Numbers used in Fluid Mechanics
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.
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
(Special case: Re < 2000) ———— 1
where r is measured outward from the pipe’s centerline. The maximum velocity of the fluid
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
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
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.
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.