85a422f4-c903-4181-ad26-5f9cba410aca 54 Javier Ochoa 4 5 false false 0 0 decimal

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Estimate friction factor accurately. Author: T. K. Serghides, Kerr-McGee Chemical Corp. Chemical Engineering, March 5, 1984. Here is a very accurate equation for calculating the friction factor directly, without trial-and-error . The Colebrook equation for calculating the friction factor is compared with others that I considered for examination and for a set of ε / D , Reynolds numbers. The x1 parameter is the Darcy friction factor, the x2 parameter is the ε / D ratio and the x3 parameter is the Reynolds number.

x1 x2 x3 g 1 x1 sqrt / 2.0 x2 3.7 / 2.51 x3 x1 sqrt * / + log10 * + :

This is the Colebrook equation arranged to find the values of x1 which makes zero the function g(x1,x2,x3).

x2 x3 h 0.11 x2 68 x3 / + (14 /^* :

Altshul equation. "Hydraulics Aerodynamics. 2nd. Ed. pp. 176 - 196, Strojizdat , Moscow 1975. Published by B. B. Gulyani in Chemical Engineering, February 2001.

a 2.0 x2 3.7 / 12 x3 / + log10 * - :

The Colebrook equation applying Steffenson's accelerated convergence technique, valid for any x2 and x3 > 2100

b 2.0 x2 3.7 / 2.51 a * x3 / + log10 * - :

c 2.0 x2 3.7 / 2.51 b * x3 / + log10 * - :

x2 x3 f a b a - (2^c 2 b * - a + / - (2 -^:

A simpler version of the las equation could be the following, which is valid for any x2 and x3 > 2100.

x2 x3 f1 4.781 a 4.781 - (2^b 2 a * - 4.781 + (/ - (2 -^:

I first try to find a solver which can be used for a wide range of x2, x3 numbers

epsilon 105 -^:

x2 0.015 :

x3 100000 :

iterations 200 :

Bound_P2 1 2.0 x2 3.7 / log10 * - (2^/ : 0.0437

Bound_P1 0.5 :

res x1 x2 x3 g Bound_P1 Bound_P2 BDQRF :

Checked with the Altshul equation and the Steffenson's accelerated convergence technique

res 0.0442

x2 x3 h 0.0389

x2 x3 f 0.0442

x2 x3 f1 0.0442

Let's try to define a set of x2 and x3 values and find the friction factor for each equation. I will begin with the Colebrook equation.

x2 0.00004 0.00005 0.0002 0.0006 0.0015 0.004 0.008 0.015 0.03 0.05 101 mat :

x3 2500400030000100000106^107^108^71 mat :

n x2 length : 10

m x3 length : 7

i 1 n range j 1 m range Bound_P2 i el 1 2.0 x2 i el 3.7 / log10 * - (2^/ : res1 x1 x2 i el x3 j el g Bound_P1 Bound_P2 i el BDQRF : M i j el res1 : 31 line for for

The resulting set of 70 values for the friction factor calculated for the Colebrook equiation are the following:

M 0.0461 0.0399 0.0236 0.0182 0.0125 0.0105 0.0102 0.0461 0.04 0.0236 0.0183 0.0126 0.0109 0.0106 0.0462 0.0401 0.024 0.019 0.0147 0.0138 0.0137 0.0466 0.0405 0.025 0.0207 0.0178 0.0175 0.0174 0.0473 0.0414 0.0271 0.0237 0.022 0.0218 0.0217 0.0493 0.0438 0.0317 0.0295 0.0285 0.0284 0.0284 0.0524 0.0474 0.0374 0.0359 0.0353 0.0352 0.0352 0.0576 0.0531 0.0452 0.0442 0.0437 0.0437 0.0437 0.0677 0.0641 0.0582 0.0575 0.0572 0.0572 0.0572 0.08 0.077 0.0723 0.0718 0.0716 0.0716 0.0716 107 mat

Let's evaluate the other correlations:

i 1 n range j 1 m range res1 x2 i el x3 j el h : N i j el res1 : 21 line for for

N 0.0447 0.0397 0.0241 0.018 0.0112 0.0091 0.0088 0.0447 0.0397 0.0241 0.0181 0.0115 0.0095 0.0093 0.0448 0.0398 0.0245 0.0189 0.0141 0.0132 0.0131 0.0449 0.0401 0.0255 0.0208 0.0177 0.0173 0.0172 0.0453 0.0406 0.0273 0.0238 0.0219 0.0217 0.0217 0.0462 0.0419 0.0309 0.0288 0.0278 0.0277 0.0277 0.0476 0.0437 0.035 0.0336 0.033 0.0329 0.0329 0.0499 0.0465 0.0399 0.0389 0.0385 0.0385 0.0385 0.0538 0.0512 0.0466 0.046 0.0458 0.0458 0.0458 0.058 0.056 0.0526 0.0522 0.052 0.052 0.052 107 mat

i 1 n range j 1 m range a 2.0 x2 i el 3.7 / 12 x3 j el / + log10 * - : b 2.0 x2 i el 3.7 / 2.51 a * x3 j el / + log10 * - : c 2.0 x2 i el 3.7 / 2.51 b * x3 j el / + log10 * - : res1 a b a - (2^c 2 b * - a + / - (2 -^: L i j el res1 : 51 line for for

L 0.0461 0.0399 0.0236 0.0182 0.0125 0.0105 0.0102 0.0461 0.04 0.0236 0.0183 0.0126 0.0109 0.0106 0.0462 0.0401 0.024 0.019 0.0147 0.0138 0.0137 0.0466 0.0405 0.025 0.0207 0.0178 0.0175 0.0174 0.0473 0.0414 0.0271 0.0237 0.022 0.0218 0.0217 0.0493 0.0438 0.0317 0.0295 0.0285 0.0284 0.0284 0.0524 0.0474 0.0374 0.0359 0.0353 0.0352 0.0352 0.0576 0.0531 0.0452 0.0442 0.0437 0.0437 0.0437 0.0677 0.0641 0.0582 0.0575 0.0572 0.0572 0.0572 0.08 0.077 0.0723 0.0718 0.0716 0.0716 0.0716 107 mat

i 1 n range j 1 m range a 2.0 x2 i el 3.7 / 12 x3 j el / + log10 * - : b 2.0 x2 i el 3.7 / 2.51 a * x3 j el / + log10 * - : res1 4.781 a 4.781 - (2^b 2 a * - 4.781 + (/ - (2 -^: H i j el res1 : 41 line for for

H 0.0461 0.0399 0.0236 0.0182 0.0125 0.0105 0.0102 0.0461 0.04 0.0236 0.0182 0.0126 0.0109 0.0106 0.0462 0.0401 0.024 0.019 0.0147 0.0138 0.0137 0.0466 0.0405 0.025 0.0207 0.0178 0.0175 0.0174 0.0473 0.0414 0.0271 0.0237 0.022 0.0218 0.0217 0.0493 0.0438 0.0317 0.0295 0.0285 0.0284 0.0284 0.0524 0.0474 0.0374 0.0359 0.0353 0.0352 0.0352 0.0576 0.0531 0.0452 0.0442 0.0437 0.0437 0.0437 0.0677 0.0641 0.0582 0.0575 0.0572 0.0572 0.0572 0.08 0.077 0.0723 0.0718 0.0716 0.0716 0.0716 107 mat

Now, we cand find the deviation ( E ) as the fractional difference between the Colebrook friction factor value and the numerical solutions proposed, without the iteration process.

i 1 n range j 1 m range E1 i j el M i j el N i j el - (M i j el / : for for

E1 0.0304 0.0051 0.0218 - 0.0102 0.1007 0.1342 0.1368 0.0304 0.0052 0.0218 - 0.0096 0.0936 0.1207 0.1226 0.0317 0.0068 0.0207 - 0.0031 0.0415 0.0468 0.0472 0.0352 0.011 0.017 - 0.0038 - 0.0093 0.0107 0.0108 0.0427 0.0202 0.0068 - 0.0014 - 0.0031 0.0036 0.0036 0.0625 0.0437 0.0229 0.0247 0.0263 0.0264 0.0265 0.091 0.0768 0.0635 0.0645 0.0652 0.0653 0.0653 0.1338 0.1245 0.1178 0.1184 0.1188 0.1189 0.1189 0.2051 0.2007 0.1987 0.1991 0.1993 0.1993 0.1993 0.2751 0.2731 0.2727 0.2729 0.273 0.273 0.273 107 mat

max1 E1 max : 0.2751

avg1 E1 i j el j 1 m sum i 1 n sum n m * (/ (: 0.0846

i 1 n range j 1 m range E2 i j el M i j el L i j el - (M i j el / : for for

E2 1.77 106 -^* - 1.87 109 -^* 2.31 105 -^* 2.28 105 -^* 3.76 106 -^* 1.29 106 -^* - 3.08 107 -^* 2.25 105 -^* 2.12 105 -^* 2.41 106 -^* 4.4 107 -^* 1.19 106 -^* 1.53 105 -^* 9.2 106 -^* 2.88 106 -^* - 8.32 107 -^* 9.04 107 -^* 7.09 106 -^* 9.84 107 -^* 1.95 108 -^* - 1.46 105 -^* 7.69 106 -^* 1.06 106 -^* 4.23 106 -^* - 4.91 106 -^* 2.15 106 -^* 2.03 107 -^* 6.03 106 -^* - 4.37 108 -^* - 1.49 107 -^* 1.6 106 -^* 1.18 107 -^* 2.44 107 -^* - 8.94 106 -^* 1.08 108 -^* 1.17 106 -^* 2.33 105 -^* - 2.3 105 -^* 7.9 107 -^* 8.72 1010 -^* 3.38 106 -^* - 7.23 107 -^* - 1.24 106 -^* 3.81 108 -^* 41011 -^* 1.53 105 -^* 1.02 105 -^* 6.89 108 -^* 2.06 109 -^* 2.14 1012 -^* 105 mat

max2 E2 max : 3.7896 105 -^*

avg2 E2 i j el j 1 m sum i 1 n sum n m * (/ (: 3.7347 106 -^*

i 1 x2 length range j 1 x3 length range E3 i j el M i j el H i j el - (M i j el / : for for

E3 1.8537 105 -^* 5.2089 105 -^* 0.0014 0.002 1.9123 105 -^* 5.1998 105 -^* 0.0014 0.0019 2.2434 105 -^* 4.722 105 -^* 0.0011 0.001 2.7106 105 -^* 3.4249 105 -^* 0.0006 0.0002 5.0568 105 -^* 2.2415 105 -^* 0.0002 2.6552 105 -^* 6.2837 105 -^* 8.9248 107 -^* 8.9049 106 -^* 1.1797 106 -^* 8.8752 105 -^* 1.0199 105 -^* 5.2838 107 -^* 8.9863 106 -^* 0.0001 8.7299 107 -^* 2.3055 105 -^* 7.9179 107 -^* 8.3704 105 -^* 2.4074 105 -^* 1.3192 106 -^* 4.038 108 -^* 7.4493 105 -^* 2.7019 105 -^* 1.2076 107 -^* 3.6191 109 -^* 104 mat

max3 E3 max : 0.002

avg3 E3 i j el j 1 m sum i 1 n sum n m * (/ (: 0.0002

We can make a Table to show the values for de absolute deviation and de maximum absolute deviation for each method, when compared to the Colebrook equation, in the range of x2 and x3 parameters proposed.

U Equation Average Maximum Althsul avg1 max1 Steffenson method avg2 max2 Steffenson simplified avg3 max3 43 mat :

U avg3 max3 avg2 max2 avg1 max1 res1 res Bound_P1 epsilon f1 f c b a h g U avg3 max3 avg2 max2 avg1 max1 res1 res Bound_P1 Bound_P2 x3 x2 epsilon f1 f c b a h g avg3 max3 avg2 max2 avg1 max1 res1 res Bound_P1 epsilon f1 f c b a h g

U
Equation	Average	Maximum
Althsul	0.0846206	0.2750947
Steffenson method	0.0000037	0.0000379
Steffenson simplified	0.0001752	0.0019812
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The equation of Althsul has better results than others presented by the author in the article, like Chen and Churchil equations. For Reynolds numbers greater than 4000 or 5000 it is also better than the Moody, Jain and Zigrang and Sylvester equations, when compared for the points within equation ' s limits.

The Colebrook equation solved by direct substitution, after finite number of iterations, called the Steffenson method is more accurate than the others proposed for Reynolds numbers greater than 2500, also the Steffenson simplified is still better than the others looked at here, and easier to use.

These equations could help to speed up the design stage when we want to evaluate the pressure drop, the flow rate and the diameter of pipes. I will make some evaluations in other worksheet