Question

How is the friction factor for flow in a tube related to the pressure drop? How is the pressure drop related to the pumping power requirement for a given mass flow rate?

Short answer

Question: Explain the relationship between the friction factor for flow in a tube, the pressure drop, and the pumping power requirement for a given mass flow rate. Answer: The friction factor is a dimensionless number that represents the resistance to fluid flow due to friction between fluid particles and the tube walls. The pressure drop refers to the difference in pressure between the entrance and exit of the tube. These two quantities are directly related through the Darcy-Weisbach equation. The pumping power requirement is the mechanical work needed to move a given mass flow rate of fluid through a tube. It is directly proportional to the pressure drop, and therefore, also related to the friction factor. A higher friction factor results in a higher pressure drop, which in turn requires a greater amount of mechanical power to move the fluid through the tube.

Step-by-step solution

Definition of friction factor and pressure drop

The friction factor, denoted as 'f', is a dimensionless number that characterizes the resistance of fluid flow due to the friction between fluid particles and the walls of the tube. The pressure drop, denoted as 'ΔP', is the difference in pressure between the entrance and exit of the tube.

Relationship between friction factor and pressure drop

The friction factor is related to the pressure drop through the Darcy-Weisbach equation, which is given as follows: ΔP = f * (ρ * L * v² / (2 * D)) where ΔP = pressure drop (Pa) f = friction factor (dimensionless) ρ = fluid density (kg/m³) L = length of the tube (m) v = flow velocity (m/s) D = diameter of the tube (m). This equation shows that the pressure drop is directly proportional to the friction factor. The higher the friction factor, the greater the resistance to fluid flow, and thus, the greater the pressure drop.

Definition of pumping power requirement

Pumping power requirement, denoted as 'P', is the mechanical work required by a pump to move a given mass flow rate (ṁ) of fluid through a tube. It is usually expressed in units of Watts (W) or horsepower (hp).

Relationship between pressure drop and pumping power requirement

The pumping power requirement for a given mass flow rate can be calculated using the following equation: P = ΔP * ṁ / η where P = pumping power requirement (W) ΔP = pressure drop (Pa) ṁ = mass flow rate (kg/s) η = pump efficiency (dimensionless, usually in decimal form) This equation shows that the pumping power requirement is directly proportional to the pressure drop. The higher the pressure drop, the greater the amount of mechanical power needed to move the fluid through the tube.

Key concepts

Pressure Drop

Pressure drop is a crucial concept in fluid dynamics and refers to the reduction in pressure as fluid flows through a tube or pipe. It is directly influenced by the friction between the fluid particles and the walls of the tube. This drop in pressure is important because it tells us how much energy is lost in the process of moving fluid between two points.
The calculation of pressure drop uses the Darcy-Weisbach Equation which incorporates several factors: fluid density, length of the pipe, fluid velocity, and the pipe's diameter. The pressure drop is calculated using the formula:\[\Delta P = f \times \frac{\rho \times L \times v^2}{2 \times D}\]where:

• \(\Delta P\) is the pressure drop in Pascals (Pa)
• \(f\) is the friction factor
• \(\rho\) is the fluid density in kilograms per cubic meter (kg/m³)
• \(L\) is the length of the tube in meters (m)
• \(v\) is the flow velocity in meters per second (m/s)
• \(D\) is the diameter of the tube in meters (m)

This equation highlights that as the friction factor increases, so does the pressure drop, indicating more resistance to fluid flow.

Pumping Power Requirement

The pumping power requirement is a measure of the mechanical energy needed to transport a fluid through a pipe. This is an essential calculation in engineering as it determines how powerful a pump must be to achieve the desired flow rate.
The power needed is affected by the pressure drop across the system. An increase in pressure drop demands more power from the pump, and vice versa. The pumping power can be found using the equation:\[P = \frac{\Delta P \times \dot{m}}{\eta}\]Here:

• \(P\) is the pumping power requirement in Watts (W)
• \(\Delta P\) is the pressure drop in Pascals (Pa)
• \(\dot{m}\) is the mass flow rate in kilograms per second (kg/s)
• \(\eta\) is the pump efficiency (a dimensionless value usually shown as a decimal)

The equation shows that more energy (or power) is needed to overcome a greater pressure drop. Efficient pumps use less power to move the same amount of fluid, which can significantly decrease operational costs.

Darcy-Weisbach Equation

The Darcy-Weisbach Equation is a fundamental formula in fluid mechanics used to calculate the pressure drop due to friction along a given length of pipe. This equation provides a relationship between several factors affecting fluid flow, allowing engineers to predict how these variables impact the system.
The formula is:\[\Delta P = f \times \frac{\rho \times L \times v^2}{2 \times D}\]This equation depends on:

• The friction factor \(f\), which is crucial for understanding resistance within the pipe
• The fluid density \(\rho\), which affects how much pressure is lost due to the weight of the fluid
• The length \(L\) and diameter \(D\) of the pipe, which determine the physical space the fluid must navigate
• The velocity of the fluid \(v\), which indicates the speed with which the fluid is moving

By using the Darcy-Weisbach equation, engineers can optimize the design of piping systems to reduce energy loss and required pumping power, saving costs and improving efficiency.