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

Answer: The friction factor is directly related to the pressure drop in the flow, meaning higher friction factors result in higher pressure drops. In turn, the pressure drop is directly related to the pumping power requirement for a given mass flow rate; thus, higher pressure drops require more power to move fluid through the tube.

Step-by-step solution

Understand the concept of friction factor

The friction factor (f) is a dimensionless number that represents the resistance of a fluid flowing through a tube due to the wall friction. It is affected by the properties of the fluid, the tube's geometry, and flow characteristics.

Write down Darcy-Weisbach equation

The relationship between the friction factor, pressure drop, and other parameters in the flow can be found using the Darcy-Weisbach equation. The equation is as follows: ΔP = f * (L/D) * (ρ * v^2)/2 where: ΔP is the pressure drop f is the friction factor L is the length of the tube D is the diameter of the tube ρ (rho) is the fluid density v is the fluid velocity

Explain the relationship between friction factor and pressure drop

From the Darcy-Weisbach equation, we can see that the pressure drop (ΔP) is directly proportional to the friction factor (f). A higher friction factor represents greater resistance in the flow, which results in a higher pressure drop. Changing the friction factor will affect the pressure drop accordingly. Now let's move on to relate the pressure drop to the pumping power requirement for a given mass flow rate.

Write the formula for mass flow rate

The mass flow rate (ṁ) is given by the equation: ṁ = ρ * A * v where: ṁ is the mass flow rate ρ (rho) is the fluid density A is the cross-sectional area of the tube v is the fluid velocity

Write the formula for pumping power requirement

The pumping power requirement (P) is given by the equation: P = ΔP * ṁ/η where: P is the pumping power requirement ΔP is the pressure drop ṁ is the mass flow rate η (eta) is the pump efficiency

Explain the relationship between pressure drop and pumping power requirement

From the pumping power requirement equation, we can see that the pumping power requirement (P) is directly proportional to the pressure drop (ΔP). If there is a higher pressure drop, it will require more power to move fluid through the tube. In summary, the friction factor in a tube is directly related to the pressure drop in the flow, and the pressure drop is directly related to the pumping power requirement for a given mass flow rate.