Volumetric Flow Rate In Pipes
The volumetric flow rate of a fluid through a pipe is not constant across the cross-section. In fact, the flow rate will exhibit a trend similar to that shown in the figure below:
As seen in the figure, the average velocity, Vavg, is used to calculate the volumetric flow rate Q, which is accomplished with the following formula:
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where D is the pipe diameter, with SI units of m, and Vavg has SI units of m/s.
Friction Loss
The fluid flow at the pipe boundary is subject to friction loss due to the interaction with the pipe. The fluid particles in contact with the surface of the pipe will stop moving. This will impart friction to adjacent particles, and through a chain reaction the flow will be diminished due to the friction from the pipe boundary to the middle of the flow. It is this changing friction that results in the velocity gradient as the fluid at the center of the pipe must flow faster than closer to the pipe boundary to maintain a constant mass flow rate.
The first step in calculating the friction loss is to determine if the flow is laminar or turbulent, which is accomplished via the Reynold’s number.
Reynold’s Number
The Reynold’s number Re of the flow is calculated using the following equation:
where ρ is the fluid density, with SI units of kg/m3, and μ is the dynamic viscosity of the fluid, with SI units of kg/m∙s. The density and dynamic viscosity of the fluid will vary with the fluid and other parameters of the specific situation, but these values can usually be found in a look-up table.
In a circular pipe, a Reynold’s number of less than approximately 2300 is generally considered laminar flow, between 2300 and 4000 flow is considered transitional, and anything over approximately 4000 is considered turbulent flow. The difference between laminar and turbulent fluid flow can be seen in the following image:
Laminar Flow Friction Factor
To calculate the friction factor f under laminar flow conditions, the following equation is used:
Turbulent Flow Friction Factor
Calculation of the friction factor under turbulent flow conditions is more involved but can be accomplished via either the Colebrook or the Zigrang-Sylvester Equation. However, both equations are based on extensive experimentation, so the calculated friction factor results are only valid within the experimental uncertainty.
The Colebrook Equation is:
where ε is the average roughness of the pipe interior, a value that can be found using a look-up table, with SI units of mm. Some examples of roughness are shown in the following table:
Using the Colebrook Equation involves iteration. But the Zigrang-Sylvester Equation is one equation that enables the friction factor to be calculated without iteration. This equation is shown below:
Other equations used to estimate the friction factor without iteration include the Swamee-Jain equation and Haaland equation.
Head Loss/Pressure Drop
Using the friction factor, the head loss due to friction hf, which is the pressure lost as the fluid flows through the pipe, can be calculated using the following equation:
where L is the length of the pipe, with SI units of m, and g is gravitational acceleration, equal to 9.81 m/s2.
Additionally, the pressure drop due to friction ΔP can be calculated with the following formula:
To overcome the head loss, a certain amount of power is required. This power can be calculated as follows:
where γ is the specific weight of the fluid, with SI units of N/m3, and can be calculated as follows:
Critical Velocity
The velocity at which the flow transitions from laminar to turbulent is called the critical velocity. Below the critical velocity, streamlines in the flow will be straight and parallel. As the velocity increases and the flow approaches a Reynold’s number that is in the transition region, the streamlines will become less steady. Above the critical velocity, when the flow is turbulent, the streamlines will be dispersed and random.
Boundary Layer
The viscous flow through the pipe will create a boundary layer. The thickness of this region δ will vary with the pipe diameter and is a function of the Reynold’s number. It can be calculated using the following formula:
The thickness of the boundary layer will increase as the flow length increases, and will ultimately reach the center of the pipe. This leads to the development of the flow.
Flow Development
The flow of a fluid through a pipe will undergo something called development. This can be described as the difference between the initial flow at the pipe inlet and the flow further along the pipe that has reached a steady state. The flow will undergo several changes, which can be seen in the following figure:
Application Of Fluid Flow In A Pipe
The calculation of the flow of a fluid in a pipe can be used when developing the design of systems to carry gasses or liquids. Based on the design needs for how quickly the fluid most get from one point to another and allowable parameters such as the change in pressure, the pipe can be designed. Pipe parameters that need to be considered in the design will include the diameter and, if turbulent flow is expected, the pipe material.
Regardless of whether the flow will be laminar or turbulent, there will be friction acting on the fluid. This friction will cause the pressure of the fluid to decrease. Due to the nature of this friction, laminar flow will have less friction, and that friction will not be dependent on the pipe material. The pressure loss will also be a function of the pipe length, so designing a system with the shortest pipe length allowable will be a key engineering requirement.
