Question

Consider fully developed laminar flow in a circular pipe. If the diameter of the pipe is reduced by half while the flow rate and the pipe length are held constant, the pressure drop will \((a)\) double, \((b)\) triple, \((c)\) quadruple, \((d)\) increase by a factor of 8 , or (e) increase by a factor of 16 .

Short answer

Answer: The pressure drop increases by a factor of 16.

Step-by-step solution

1. Understand the Hagen-Poiseuille equation

The Hagen-Poiseuille equation for fully developed laminar flow in a circular pipe is given by: ΔP = \frac{8μQL}{πR^4} Where ΔP is the pressure drop, μ is the dynamic viscosity of the fluid, Q is the volumetric flow rate, L is the length of the pipe, and R is the radius of the pipe (which is half of the diameter).

2. Determine the initial pressure drop

According to the given problem, the diameter of the pipe is reduced by half while the flow rate (Q) and the pipe length (L) are held constant. The initial pressure drop will be: ΔP_initial = \frac{8μQL}{πR_initial^4}

3. Determine the pressure drop when the diameter is reduced by half

When the diameter of the pipe is reduced by half, the radius is also reduced by half: R_final = \frac{R_initial}{2} Now, let's determine the new pressure drop: ΔP_final = \frac{8μQL}{πR_final^4} = \frac{8μQL}{π(\frac{R_initial}{2})^4}

4. Find the change in pressure drop

Now that we have the initial and final pressure drops, we can find the change in pressure drop. To do this, we divide ΔP_final by ΔP_initial: Change in pressure drop = \frac{ΔP_final}{ΔP_initial} = \frac{\frac{8μQL}{π(\frac{R_initial}{2})^4}}{\frac{8μQL}{πR_initial^4}} The 8μQL and π terms in the numerator and denominator cancel out, leaving: Change in pressure drop = \frac{1}{(\frac{R_initial}{2})^4} \cdot \frac{1}{R_initial^4} The expression simplifies to: Change in pressure drop = 2^4 = 16

5. Present the final answer

The pressure drop in the laminar flow in a circular pipe when the diameter is reduced by half, while the flow rate and the pipe length are held constant, increases by a factor of 16 (Option (e)).