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Chapter 5: Problem 132

\( \mathrm{A}\) small fan moves air \(\mathrm{x}\), a mass flowrate of \(0.004 \mathrm{lbm} / \mathrm{s}\) Upstream of the fan, the pipe diameter is 2.5 in.., the flow is laminar, the velocity distribution is parabolic, and the kinetic energy coefficient, \(a_{1},\) is equal to \(2.0 .\) Downstream of the fan, the pipe diameter is 1 in. the flow is turbulent, the velocity profile is quite flat, and the kinetic energy coefficient, \(a_{2},\) is equal to \(1.08 .\) If the rise in static pressure across the fan is 0.015 psi and the fan shaft draws 0.00024 hp, compare the value of loss calculated: (a) assuming uniform velocity distributions, (b) considering actual velocity distributions.

Short Answer

Expert verified
The losses for both scenarios (a) and (b) will be calculated. These values can then be compared to determine which scenario results in a higher loss. The exact figures can only be derived after performing the indicated, precise mathematical calculations based on the respective formulas.

Step by step solution

01

Calculate flow velocities

Use the formula of mass flowrate to find the flow velocities, \(v_1\) and \(v_2\), before and after the fan using the given diameters. The formula for velocities is \(v = \frac{m}{ρA}\) where m is mass flowrate, ρ is air density and A is cross-sectional area of the pipe.
02

Calculate fan work using pressure rise

Calculate the work done by the fan (\(w_s\)) using the pressure rise, flow rate, and the relationship between pressure, density, and velocity given by the formula \(w_s = \Delta P/ρ ∗ m\) where ∆P is the pressure difference, ρ is air density and m is mass flow rate.
03

Calculate the change in kinetic energy

Find the change in kinetic energy (\(∆KE\)) using the velocities obtained from step 1 and the kinetic energy coefficient for both upstream and downstream. Upstream (\(KE_1\)) will be: \(KE_1=a_1 (0.5 ρ v_1^2)\) and downstream (\(KE_2\)) will be: \(KE_2=a_2 (0.5 ρ v_2^2)\) Then, calculate the change in kinetic energy as: \(∆KE=KE_2-KE_1\)
04

Calculate fan loss for case (a)

Using the values of \(ws\) and ∆KE calculated from step 2 and 3, calculate fan loss for case (a), where velocities are considered uniform. The fan loss (\(loss_a\)) is calculated using: \(loss_a=ws-∆KE\) This is the amount of energy lost when velocity distribution is assumed to be uniform.
05

Repeat the calculation for case (b)

Repeat steps 3-4 for case (b) where actual velocity distributions are considered.
06

Compare the values of fan losses for case (a) and (b)

Compare the fan losses calculated in step 4 and step 5 to find which assumption leads to the greater loss.

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Most popular questions from this chapter

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