## Tuesday, September 1, 2015

### Easy and True solutions of Colebrook-White

This will be easy and true Colebrook-White solution in Excel.

There are about six different equations for Colebrook-White. The six equations are different for different types of designs.  The first one is the maid type of Colebrook-White, but some folks have asked for different equations.  Her are the six different equations. The mode shows some numbers.

1  mode 2.51...   1/sqrt(f)=-2*Log(Rr/3.7+2.51/(Re*sqrt(f)))
2  mode 1.74...   1/sqrt(f)=1.74-2*Log(2*Rr+18.7/(Re*sqrt(f)))
3  mode 1.14...  1/sqrt(f)=1.14+2*Log(1/Rr)-2*Log(1+(9.3/(Re*Rr*sqrt(f)))
4  mode 9.35...   1/sqrt(f)=1.14-2*Log(Rr+9.35/(Re*sqrt(f)))
5  mode 3.71...   1/sqrt(f)=-2*Log(Rr/3.71+2.51/(Re*sqrt(f)))
6  mode 3.72...   1/sqrt(f)=-2*Log(Rr/3.72+2.51/(Re*sqrt(f)))

Now to compute f  not  the "1/sqrt(f)"  just add  "f=1/(" at the first and ")^2 and the end for all of the six equations.  Example of the main mode, I call  "mode 2.51" the first one. It will be...

f= 1/(-2*Log(Rr/3.7+2.51/(Re*sqrt(f))))^2  (this mode 2.51, but you can use all 6 modes)

In Excel the first Guess f number can be 1 or up to 1000, and the Guess can be changed, but the final f solution will not change.  For this test I am entering the guess in Cell B3  as 100, but 1 might save a loop. But you can change the Guess to see the changes.

The in Cell B1 enter the Rr and Cell B2 enter the Re and in Cell B3 there the Guess f.

And in Cell C4 there   =B4=B3   and it will enter True or False to the B4=B3

But enter the number of Rr and Re and for the f, enter the Cell number that is above, I said that my Guess was in Cell B3.    For this example, I am using  Rr= 0.01954 and the Re = 611,040 and the guess as 100 in cell B3

Her is the Excel picture for the first cells from A1 to C4.
-----A------|----------B-----------------|-----C------|
1          Rr =           0.01954
2          Re =          611,040
3  Guess f =             100
4           f1 =  0.0482082215337859    FALSE
===============================================
Then below the guess enter of 100 is this, in Cell B4 ...            =1/(-2*LOG(\$B\$1/3.7+2.51/(\$B\$2*SQRT(B3))))^2
And C4 is....       =B4=B3
(Note: the two \$'s will not let the cope's change,  but with out the \$, like the B3, it will change the f per cells copied.

Then copy Cells B4 and C4 down to about 20 or more cells. And in Cells A4 to A23 I enter "f1=" to "f20=", but when the C cells say to TRUE then the f is right, and does not change.

Rr =           0.01954
Re =           611,040
Guess f =            100
f1=       0.0482082215337859 FALSE
f2= 0.0482718790546499 FALSE
f3= 0.0482718361564094 FALSE
f4= 0.0482718361852896 FALSE
f5= 0.0482718361852702 FALSE
f6= 0.0482718361852702 TRUE
f7= 0.0482718361852702 TRUE
f8= 0.0482718361852702 TRUE
f9= 0.0482718361852702 TRUE
f10= 0.0482718361852702 TRUE
f11= 0.0482718361852702 TRUE
f12= 0.0482718361852702 TRUE
f13= 0.0482718361852702 TRUE
f14= 0.0482718361852702 TRUE
f15= 0.0482718361852702 TRUE
f16= 0.0482718361852702 TRUE
f17= 0.0482718361852702 TRUE
f18= 0.0482718361852702 TRUE
f19= 0.0482718361852702 TRUE
f20= 0.0482718361852702 TRUE

Now you can change the Rr and Re to compute the mode 2.51
But the cell B4 will change for each modes.

This will be the equations in B4 for different modes...
1 'mode 2.51   =1/(-2*Log(\$B\$1/3.7+2.51/(\$B\$2*sqrt(B3))))^2
2 'mode 1.74   =1/(1.74-2*Log(2*\$B\$1+18.7/(\$B\$2*sqrt(B3))))^2
3 'mode 1.14 =1/(1.14+2*Log(1/\$B\$1)-2*Log(1+(9.3/(\$B\$2*\$B\$1)*1/sqrt(B3))))^2
4 'mode 9.35   =1/(1.14-2*Log(\$B\$1+9.35/(\$B\$2*sqrt(B3))))^2
5 'mode 3.71   =1/(-2*Log(\$B\$1/3.71+2.51/(\$B\$2*sqrt(B3))))^2
6 'mode 3.72   =1/(-2*LOG(\$B\$1/3.72+2.51/(\$B\$2*SQRT(B3))))^2

After you enter B4 cells, then copy the B4 down to the 20 cells.

Then to test it, just compute the 1/sqrt(f) and then compute the right side of this equation using Rr and Re and f  to see if they are te same.
1  mode 2.51...   1/sqrt(f)=-2*Log(Rr/3.7+2.51/(Re*sqrt(f)))
2  mode 1.74...   1/sqrt(f)=1.74-2*Log(2*Rr+18.7/(Re*sqrt(f)))
3  mode 1.14...  1/sqrt(f)=1.14+2*Log(1/Rr)-2*Log(1+(9.3/(Re*Rr*sqrt(f)))
4  mode 9.35...   1/sqrt(f)=1.14-2*Log(Rr+9.35/(Re*sqrt(f)))
5  mode 3.71...   1/sqrt(f)=-2*Log(Rr/3.71+2.51/(Re*sqrt(f)))
6  mode 3.72...   1/sqrt(f)=-2*Log(Rr/3.72+2.51/(Re*sqrt(f)))