Haaland Equation, find the Reynolds Number

[Ben Wood]

Hi Guys,

I've just started a HND Building Services Engineering course (for Work) and I have been given my first assignment for hand in on the 11th Nov 2014.

I've not completed any Maths or Physics since Secondary School - 18 years ago (and didn't really pay attention then!) and I'd appreciate a bit of help understanding the equation given to me rather than just getting the answer.

It is as follows: -

Using the Haaland Equation below, determine the Reynolds Number R_e and state whether the flow through the pipe is laminar or turbulent given -

Friction Factor λ = 0.032mm
Absolute Roughness K_s = 1.5 x10^-6m
Internal Bore Diameter d = 0.04m

1/√λ = -1.8log [ (6.9/R_e) + ( (K_s/d) / 3.7)^1.11)

I have the following so far: -

1/√(0.032) = -1.8log [ (6.9/R_e) + ( 1.5 x10^-6 / 3.7)^1.11 ]

1/0.1788854382 = -1.8log [ (6.9/R_e) + ( 3.75 x10^-4 / 3.7)^1.11 ]

5.59 = -1.8log [ (6.9/R_e) + ( 1.013513514 x10^-4 )^1.11 ]

It's at this point I get stuck I'm afraid...

I have been given the answer by another student as R_e = 1275.51 which determines that I have a Streamlined Laminar Flow but I want to be able to solve this myself and check his answer is indeed correct.

So any help in helping me understand how to proceed is much appreciated. I understand once I have worked out as much as I can, it's time to transpose the equation to find R_e which also seems a little tricky to me

Thank you

Ben

 

[SteamKing]

Hi Guys,

I've just started a HND Building Services Engineering course (for Work) and I have been given my first assignment for hand in on the 11th Nov 2014.

I've not completed any Maths or Physics since Secondary School - 18 years ago (and didn't really pay attention then!) and I'd appreciate a bit of help understanding the equation given to me rather than just getting the answer.

It is as follows: -

Using the Haaland Equation below, determine the Reynolds Number R_e and state whether the flow through the pipe is laminar or turbulent given -

Friction Factor λ = 0.032mm
Absolute Roughness K_s = 1.5 x10^-6m
Internal Bore Diameter d = 0.04m

1/√λ = -1.8log [ (6.9/R_e) + ( (K_s/d) / 3.7)^1.11)

I have the following so far: -

1/√(0.032) = -1.8log [ (6.9/R_e) + ( 1.5 x10^-6 / 3.7)^1.11 ]

1/0.1788854382 = -1.8log [ (6.9/R_e) + ( 3.75 x10^-4 / 3.7)^1.11 ]

5.59 = -1.8log [ (6.9/R_e) + ( 1.013513514 x10^-4 )^1.11 ]

It's at this point I get stuck I'm afraid...

I have been given the answer by another student as R_e = 1275.51 which determines that I have a Streamlined Laminar Flow but I want to be able to solve this myself and check his answer is indeed correct.

So any help in helping me understand how to proceed is much appreciated. I understand once I have worked out as much as I can, it's time to transpose the equation to find R_e which also seems a little tricky to me

Thank you

Ben

A couple of comments first:
1. The Darcy friction factor is unitless, so Friction Factor λ = 0.032
2. I calculate the relative roughness K_s/D = 3.75*10^-5, instead of 3.75*10^-4

That said, what you must do is find the Reynold's Number Re such that the Haaland Equation is true, i.e., you know what the LHS is equal to (5.59), so you must find the value of Re which makes the RHS of the equation also equal 5.59 after doing all the calculations.

The log in the Equation is a common, or base 10, log, so you can get rid of it by doing a little algebra:

5.59 = -1.8log [ (6.9/R_e) + ( 1.013513514 x10^-5 )^1.11 ]

divide both sides by -1.8:

5.59/-1.8 = -3.1056 = log [ (6.9/R_e) + ( 1.013513514 x10^-5 )^1.11 ]

by raising both sides as powers of 10:

10^-3.1056 = 10^{log [ (6.9/R_e) + ( 1.013513514 x10-5 )1.11 ]}

the log goes away on the RHS of the equation, leaving:

7.842*10^-4 = [ (6.9/R_e) + ( 1.013513514 x10^-5 )^1.11 ]

Now, by doing some numerical clean up on both sides of the equation and some more algebra, you should be able to calculate Re directly.

I will leave that chore to you.

However, I checked the solution given by the other student for Re = 1275 and found that it did not satisfy this equation.

 

[Ben Wood]

Apologies, my top line should have been as below. The internal bore diameter has been added.

1/√(0.032) = -1.8log [ (6.9/Re) + ( (1.5x10-6 / 0.04) / 3.7)1.11 ]

You were correct with the answer at the top being 3.75 x10-5 (My Bad)

I had a telephone conversation with a classmate and he told me to use the ENG button on the calculator to get 37.5x10-6 so we could carry on using the meters SI Measurement.

Now, the way that he told me that our Lecturer has proposed was to type into the calculator 10^(5.59 / -1.8) = 784.2317924x10-6

So, 784.2317924x10-6 = (6.9/Re) + ( 1.013513514 x10-6 )1.11

Does this ring true with you? Thanks for the help by the way, it's most appreciated!

 

[Ben Wood]

I had a go, see what you think: -

 

[SteamKing]

I had a go, see what you think: -

View attachment 75029

For the term (1.013513514*10^-6)^1.11, I don't get 2.217326039*10^-6. There's some calculation mistake here, as I checked the result also using logs.

Remember, the Re you calculate must also work when you substitute this value into the original equation. This is a way to check your calculations.

 

[Kinggy]

For the term (1.013513514*10^-6)^1.11, I don't get 2.217326039*10^-6. There's some calculation mistake here, as I checked the result also using logs.

Remember, the Re you calculate must also work when you substitute this value into the original equation. This is a way to check your calculations.

I had a go and did it this way, is that right?

Breakdown the brackets, starting with the brackets with the brackets ( ks / d ) / 3.7 )
We know that Ks = 1.5 x 10-6m and d= 0.04m so;

1.5 x 10-6m / 0.04 = 3.75 x10-5
= 3.75 x10-5 / 3.7 = 1.01 x10-3
= 1.01 x10-3 ^1.11 = 2.860x10-6

We now have 7.84x10-4 = 6.9 / Re + 2.860x10-6

 

[Ben Wood]

Yeah, I calculated it wrong by the looks of things. I've tidied up again and it's more in line with Kinggy above. Also, I have been using the ENG button to keep it uniformed to x10^-6

 

[SteamKing]

The previous calculation is fine until you reach the last couple of lines, and then your algebra breaks down.

What you have written here is fine:

784.23179*10^-6 = (6.9/Re) + 2.86069*10^-6

which should then be simplified as:

784.23179*10^-6 - 2.86069*10^-6 = 6.9/Re

781.37110*10^-6 = 6.9/Re

Now, what should be done here is to multiply both sides of the expression above by Re to give:

781.37110*10^-6 Re = 6.9

and solving for Re then gives:

Re = 6.9 / 781.37110*10^-6 = 8830.6, say Re = 8831.

This value of Re, when plugged back into the original Haaland equation in the OP, should then satisfy both sides.

 

[Ben Wood]

Thanks very much, I understand it now. thanks for teaching the thought process.

Ben