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Chapter 8: Problem 90

The vented storage tank shown in Fig. \(\mathrm{P} 8.90\) is used to refuel race cars at a race track. A total of \(40 \mathrm{ft}\) of steel pipe \((\mathrm{I.D},=0.957 \text { in. })\) two \(90^{\circ}\) regular elbows, and a globe valve make up the system. Calculate the time needed to put 20 gal of fuel in a car tank. The pressures, \(p_{2}\) and \(p_{1},\) are aqual, the connections are threaded, and the fuel has the properties of \(68^{\circ} \mathrm{F}\) normal octane \(\left(\nu=8.31 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\right)\)

Short Answer

Expert verified
The calculation involves multiple steps including finding the Darcy friction factor, calculating the Reynolds number, determining the head loss, applying Bernoulli's equation, calculating the flow rate, and then determining the time needed for refuelling. Each step builds on the previous one to eventually reach the solution.

Step by step solution

01

Calculation of Darcy friction factor

First, the Darcy friction factor (f) for the steel pipe needs to be calculated using the Colebrook equation in fluid dynamics: \(1/ \sqrt{f} = -2.0 log_{10}(k/3.7D + 2.51/Re \sqrt{f})\), where k is the roughness, D is the internal diameter of the pipe and Re is the Reynolds number.
02

Calculation of Reynolds number

Reynolds number (Re) can be calculated using the equation: \(Re=VD/v\), where V is the velocity of the fluid, D is the internal diameter of the pipe, and v is the kinematic viscosity of the fluid. Note: the Reynolds number must be an assumption initially for the Colebrook equation.
03

Calculation of head loss

Calculate the head loss (h_f) using the Darcy-Weisbach equation: \(h_f = f L/D V^2/2g\), where f is the friction factor, L is the length of the pipe, D is the diameter of the pipe, V is the velocity of the fluid and g is the acceleration due to gravity. Also, we need to consider the head loss due to the presence of valves and bends in the system.
04

Application of Bernoulli's theorem

The Bernoulli's equation which is \(p1/ρ + V1^2 /2g + z1 = p2/ρ + V2^2/2g + z2 + h_f\) can now be applied, assuming that initial and final velocities and elevations for the fluid are negligible and the pressures at the two ends are equal.
05

Calculation of flow rate

The equation can be rearranged to solve for V (flow velocity), then the flow rate (Q) can be calculated as \(Q=V π (D^2)/4\).
06

Calculation of refuel time

Finally, the time needed to refuel can be found by dividing the volume of the fuel by the flow rate.

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Most popular questions from this chapter

A motor-driven centrifugal pump delivers \(15^{\circ} \mathrm{C}\) water at the rate of \(10 \mathrm{m}^{3} / \mathrm{min}\) from a reservoir, through a 2500 -m-long, \(30-\mathrm{cm} 1 . \mathrm{D} .\) plastic pipe, to a second reservoir. The water level in the second reservoir is \(40 \mathrm{m}\) above the water level in the first reservoir. The pump efficiency is \(75 \% .\) Find the motor output power. The pipe entrance is square edged.

A 10 -m-long \(, 5.042\) -cm I.D. copper pipe has two fully open gate valves, a swing check valve, and a sudden enlargement to a \(9.919-\mathrm{cm}\) I.D. copper pipe. The \(9.919 \mathrm{cm}\) copper pipe is \(5.0 \mathrm{m}\) Iong and then has a sudden contraction to another 5.042 -cm copper pipe. Find the head loss for a \(20^{\circ} \mathrm{C}\) water flow rate of \(0.05 \mathrm{m}^{3} / \mathrm{s}\)

A fluid flows through a horizontal 0.1-in.-diameter pipe. When the Reynolds number is \(1500,\) the head loss over a \(20-f t\) lenzth of the pipe is \(6.4 \mathrm{ft}\). Determine the fluid velocity.

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Water is pumped between two large open reservoirs through \(1.5 \mathrm{km}\) of smooth pipe. The water surfaces in the two reservoirs are at the same elevation. When the pump adds \(20 \mathrm{kW}\) to the water, the flowrate is \(1 \mathrm{m}^{3}\) is. If minor losses are negligible, determine the pipe diameter.
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