Question

A commercial steel annulus 40 ft long, with a = 1 in and b = ½ in, connects two reservoirs that differ in surface height by 20 ft. Compute the flow rate in ft³/s through the annulus if the fluid is water at 20°C.

Solution

Verified
Answered 2 years ago
Answered 2 years ago
Step 1
1 of 5

The flow rate can be easily solved using Bernoulli's Energy Equation. Solve first the hydraulic diameter which will be used as a diameter in solving other values. Hydraulic diameter has a formula Dh=4APD_h=\dfrac{4A}{\mathcal{P}} where AA is the cross sectional area of the outlet and P\mathcal{P} is the wetted perimeter of the outlet.

The area for annulus is the area of the larger circle minus the area of the smaller one A=πa2πb2A=\pi{a^2}-\pi{b^2}. The wetted perimeter is the perimeter of the smaller and larger circles P=2πa+2πb\mathcal{P}=2\pi a+2 \pi b. Solve now for the hydraulic diameter.

Dh=4AP=4(πa2πb2)2πa+2πb=4(π(12)π(0.52))2π(1)+2π(0.5)=1 in=112 ft\begin{align*} D_h&=\dfrac{4A}{\mathcal{P}}\\ &=\dfrac{4(\pi{a^2}-\pi{b^2})}{2\pi a+2 \pi b}\\ &=\dfrac{4(\pi(1^2)-\pi(0.5^2))}{2\pi(1)+2 \pi (0.5)}\\ &=1~\text{in}=\frac{1}{12}~\text{ft} \end{align*}

Solve the friction factor which is needed for the head loss in the Bernoulli's Energy Equation.

The friction factor can be solved by solving first the Reynold's number and the ratio ϵDh\dfrac{\epsilon}{D_h} where ϵ\epsilon is the roughness coefficient from Table 6.1 and DhD_h is the hydraulic diameter. ϵ\epsilon of a commercial steel is equal to 0.0015 ft0.0015~\text{ft}.

ϵDh=0.00015112ϵDh=0.0018\begin{align*} &\frac{\epsilon}{D_h}=\frac{0.00015}{\frac{1}{12}}\\ &\frac{\epsilon}{D_h}=0.0018 \end{align*}

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Fluid Mechanics 8th Edition by Frank M. White

Fluid Mechanics

8th EditionISBN: 9780073398273 (1 more)Frank M. White
1,641 solutions
Fundamentals of Electric Circuits 6th Edition by Charles Alexander, Matthew Sadiku

Fundamentals of Electric Circuits

6th EditionISBN: 9780078028229 (9 more)Charles Alexander, Matthew Sadiku
2,120 solutions
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics 4th Edition by Randall D. Knight

Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

4th EditionISBN: 9780133942651 (8 more)Randall D. Knight
3,508 solutions
Advanced Engineering Mathematics 10th Edition by Erwin Kreyszig

Advanced Engineering Mathematics

10th EditionISBN: 9780470458365 (8 more)Erwin Kreyszig
4,134 solutions