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Explicit representation of the implicit Colebrook–White equation
Case Studies in Thermal Engineering 5 (2015) 41–47
Contents lists available at ScienceDirect
Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite
Explicit representation of the implicit Colebrook–White equation P. Rollmann n, K. Spindler Institute of Thermodynamics and Thermal Engineering (ITW), University of Stuttgart, Pfaffenwaldring 6, 70550 Stuttgart, Germany
a r t i c l e i n f o
abstract It is shown that the Colebrook–White equation 1/ λ = − 2lg⎡⎣2.51/Re λ + ϵ/3.71D⎤⎦ can be solved analytically for the friction factor λ. The solution contains two infinite sums. For given Reynolds numbers Re and relative roughnesses ϵ/D , one can create an own approximation with the required accuracy by adding a finite number of summands. The computing time of both the iterative calculation and several approximations is being compared. In all cases, the approximation is much faster than the iteration. Two examples for practical applications are given. & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Article history: Received 9 October 2014 Received in revised form 30 November 2014 Accepted 7 December 2014 Available online 10 December 2014 Keywords: Friction factor Drag coefficient Analytical solution Explicit Colebrook White
1. Introduction The friction factor for turbulent flows in rough pipes is needed to calculate the pressure drop. It is also needed to estimate the Nusselt number with the Gnielinski equation:
Nu = 1+
λ RePr 8 λ 12.7 8 (Pr 2/3
⎡ ⎛ ⎞2/3⎤ ⎢1 + ⎜ D ⎟ ⎥ ⎝ L ⎠ ⎥⎦ − 1) ⎢⎣
(1)
It can be calculated with the Colebrook–White equation [1]:
⎡ 2.51 1 ϵ ⎤ = − 2lg⎢ + ⎥ ⎣ Re λ 3.71D ⎦ λ
(2)
Here, ϵ is the average roughness height. For ϵ = 0, the Colebrook–White equation becomes the Prandtl equation for the friction factor in smooth tubes:
⎡ 2.51 ⎤ 1 = − 2lg⎢ ⎥ = 2lg(Re λ ) − 0.8 ⎣ Re λ ⎦ λ For Eq. (3), Goudar and Sonnad [2] already presented an explicit solution. n
Corresponding author. E-mail address: [email protected] (P. Rollmann).
http://dx.doi.org/10.1016/j.csite.2014.12.001 2214-157X/& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
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P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
Nomenclature D L lg ln Nu Pr Re
diameter (m) length (m) logarithm to the base 10 natural logarithm Nusselt number (dimensionless) Prandtl number (dimensionless) Reynolds number (dimensionless)
ta ti W ⎡x⎤ ⎣ y⎦
ϵ λ τ
duration of approximation (s) duration of iteration (s) Lambert W function unsigned Stirling numbers of the first kind (dimensionless) average roughness height (m) Darcy–Weisbach friction factor (dimensionless) dimensionless duration τ = ti/ta (dimensionless)
Calculating the total pressure drop by integrating the local pressure drops and during numerical simulations the iterations for λ can require long computing times due to high spatial and temporal resolutions. Therefore, numerous approximations have been developed. Brkić [3] gives an overview of 20 approximations from the literature and compares them with his own equation. Brkić [4] also presented the three approximations of Boyd [5], Barry et al. [6] and Winitzki [7] that are actually approximations of the Lambert W function. In 2013, Ćojbašić and Brkić [8] developed a very accurate approximation. With this, the friction factor can be calculated with an accuracy of less than 0.01%.
2. Analytical solution In the following, it is shown how Eq. (2) can be rearranged analytically for the friction factor λ. The result contains two infinite sums. By truncating after a finite number, approximations for the friction factor with arbitrary precision can be created. The analytical solution can be done as follows:
⎡ 2.51 ⎡ 2.51 ϵ ⎤ ϵ ⎤ 1 2 = − 2lg⎢ + + ln⎢ ⎥=− ⎥ ⎣ Re λ 3.71D ⎦ ln(10) ⎣ Re λ 3.71D ⎦ λ ⇒−
(4)
⎡ 2.51 ln(10) ϵ ⎤ = ln⎢ + ⎥ ⎣ 3.71D ⎦ 2 λ Re λ
(5)
ϵ 2.51 + = Z* 3.71D Re λ
(6)
Substitution:
⇒ λ =
2.51 ⎛ ϵ ⎞⎟ * ⎜ Re Z − ⎝ 3.71D ⎠
(7)
Inserting Eqs. (6) and (7) into Eq. (5) yields
ln(10) ln(Z *) = − · 2 ⇒ln(Z *) =
⎛ ϵ ⎞⎟ Re⎜Z * − ⎝ 3.71D ⎠ 2.51
Re ·ϵln(10) ReZ *ln(10) − 2·2.51·3.71D 2·2.51
Substitution:
(10)
Re ·ϵln(10) = Re2* 2·2.51·3.71D
(11)
Re1* = exp(Re2* − Re1*Z *) Re* 1
Substitution:
(9)
Reln(10) = Re1* 2·2.51
⇒ln(Z *) = Re2* − Re1*Z * ⇒Z *
(8)
Re1*Z * = Z
(12)
(13) (14)
P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
* 1 ⇒Z = Re1*e Re2 Z e
Substitution:
43
(15) * ˜ Re1*e Re2 = Re
(16)
˜ ⇒Ze Z = Re
(17)
Eq. (17) can be solved with a Lambert W function:
˜) Z = W (Re
(18)
Therefore Corless et al. [9] presented a function: ^ ^ =∞ k =∞ m
˜ ) = ln(Re ˜ ) − ln[ln(Re ˜ )] + W (Re
∑ ∑ k=0 m=1
⎡k + m⎤ 1 ⎥ ⎢ ˜ )⎤⎦ (−1)k ⎢ k + 1 ⎥ ln⎡⎣ln(Re m! ⎦ ⎣
{
m
}
⎡⎣ln(Re ˜ )⎤⎦−k − m
k+m
The coefficients [ k + 1 ] are unsigned Stirling numbers of the first kind (see Section 5).
^ ^ = 1 (top) and k^ = 1, m ^ = 2 (bottom). Fig. 1. Percentage deviation for k = 0 , m
(19)
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P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
The back substitution yields
˜ ) = Re*Z * = ln(10) + Reln(10)ϵ W (Re 1 2·2.51·3.71D 2 λ
(20)
Finally, one obtains
⎡ 2 ⎤−2 Re ·ϵ ˜)− λ=⎢ W (Re ⎥ ⎣ ln(10) 2.51· 3.71D ⎦
(21)
3. Discussion Good approximations are being obtained by not adding an infinite number of summands in Eq. (19), but a finite number. ^ ^ = 1 is shown. In this case only one summand On top of Fig. 1 the percentage deviation of the approximation for k = 0 and m is calculated each. The relative roughness is being plotted over the Reynolds number. The dimensionless time τ = ti/ta indicates how many times longer the iterative determination of the friction factor has taken than the calculation with the approximation. Therefore, the friction factor was being calculated for 367 Reynolds numbers and 401 relative roughnesses ^ ^ =1 (i.e. 147 167 data points) and the ratio of the measured times was formed. For the first approximation with k = 0 and m the ratio is τ = 11 735.
^ ^ = 3 (top) and k^ = 3, m ^ = 4 (bottom). Fig. 2. Percentage deviation for k = 2 , m
P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
45
^ ^ = 5 (top) and k^ = 10 , m ^ = 11 (bottom). Fig. 3. Percentage deviation for k = 4 , m
^ ^ = 2); (k^ = 2, m ^ = 3); (k^ = 3, m ^ = 1); (k^ = 1, m ^ = 4 ) and (k^ = 4 , Figs. 1–3 show the percentage deviation for (k = 0, m ^ m = 5). While the computation time ta increases and thus the dimensionless time τ decreases, the accuracy is steadily ^ ^ = 11 is shown at the bottom of Fig. 3. The calculation of the friction factor with the increasing. Finally, the case for k = 10, m approximation is still faster by a factor of 164 than the iterative calculation. The maximum percentage deviation is being reduced to 3.68 × 10−6% .
4. Practical application
^ ^ can be For known relative roughnesses, Reynolds numbers and a required accuracy, the upper summation limits k and m selected from pictures 1–3. 4.1. Example 1
^ ^ = 1 can For a relative roughness ϵ/D = 10−3 and Reynolds numbers Re ≥ 106 the very fast approximation with k = 0 and m be chosen. The maximum percentage deviation is smaller than 0.02%. Even when there is fouling and the relative roughness increases, the maximum percentage deviation is still smaller than 0.02%. The approximation is 11 735 times as fast as the iteration.
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P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
4.2. Example 2
^ ^ = 5 could be chosen. The For applications with very high required accuracies the upper summation limits k = 4 and m range for the Reynolds number as well as the relative roughness where there is the smallest percentage deviation (<0.0005% ) is wide. The maximum percentage deviation for all Reynolds numbers and all relative roughnesses is <0.00405% . The approximation is 767 times as fast as the iteration. ˜ = Re*e Re2* is sometimes not possible. Since Re ˜ is not The exponent Re2* can assume large values and the calculation of Re 1 ˜ ), one can rearrange the equation as follows: needed but only ln(Re
⎛˜⎞ ⎛ ⎛ ⎞ *⎞ ⎟ = ln⎜Re1*e Re2 ⎟ = ln⎜Re1*⎟ + Re2* ln⎜Re ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
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This step avoids the limitations described by Sonnad and Goudar [10], Clammond [11] and Brkić [12].
5. Unsigned Stirling numbers of the first kind In the following matrix 21 11 unsigned Stirling numbers of the first kind are presented. Thus, the friction factor until ^ ^ = 11 can be determined. In the matrix, the first element at the top left has the coordinate (0,0): k = 10 and m ⎡1 0 0 0 0 0 ⎢ 1 0 0 0 0 ⎢0 ⎢0 1 1 0 0 0 ⎢0 2 3 1 0 0 ⎢ 0 6 11 6 1 0 ⎢ ⎢0 24 50 35 10 1 ⎢ 120 274 225 85 15 ⎢0 720 1764 1624 735 175 ⎢0 ⎢0 5040 13 068 13 132 6769 1960 ⎢ 40 320 109 584 118 124 67 284 22 449 ⎢0 ⎢0 362 880 1 026 576 1 172 700 723 680 269 325 ⎢0 3 628 800 10 628 640 12 753 576 8 409 500 3 416 930 ⎢ 39 916 800 120 543 840 150 917 976 105 258 076 45 995 730 ⎢0 ⎢0 479 001 600 1 486 442 880 1 931 559 552 1 414 014 888 657 206 836 ⎢ 6 227 020 800 19 802 759 040 26 596 717 056 20 313 753 096 9 957 703 756 ⎢0 ⎢0 87 178 291 200 283 465 647 360 392 156 797 824 310 989 260 400 159 721 605 680 ⎢0 1 307 674 368 000 339 163 001 600 6 165 817 614 720 5 056 995 703 824 2 706 813 345 600 ⎢ 20 922 789 888 000 70 734 282 393 600 102 992 244 837 100 87 077 748 875 900 48 366 009 233 420 ⎢0 ⎢0 355 687 428 096 000 1 223 405 590 579 000 821 602 444 625 000 158 331 397 572 7000 909 299 905 844 100 ⎢ 22 376 988 058 520 000 34 012 249 593 820 000 30 321 254 007 720 000 17 950 712 280 920 000 ⎢ 0 6 402 373 705 728 000 ⎣ 0 121 645 100 408 800 000 431 565 146 817 600 000 668 609 730 341 200 000 610 116 075 740 500 000 371 384 787 345 200 000 ⎤ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ 1 0 0 0 0 ⎥ 21 1 0 0 0 ⎥ ⎥ 322 28 1 0 0 ⎥ 4536 546 36 1 0 ⎥ ⎥ 63 273 9450 870 45 1 ⎥ 902 055 157 773 18 150 1320 55 ⎥ 13 339 535 2 637 558 357 423 32 670 1925 ⎥ ⎥ 206 070 150 44 990 231 6 926 634 749 463 55 770 ⎥ 3 336 118 786 790 943 153 135 036 473 16 669 653 1 474 473 ⎥ ⎥ 56 663 366 760 14 409 322 928 2 681 453 775 368 411 615 37 312 275 ⎥ 1 009 672 107 080 272 803 210 680 5 4631 129 553 8 207 628 000 928 095 740 ⎥ 18 861 567 058 880 5 374 523 477 960 1 146 901 283 528 185 953 177 553 23 057 159 840 ⎥ 369 012 649 234 400 110 228 466 184 200 2 4871 845 297 940 4 308 105 301 929 577 924 894 833 ⎥ ⎥ 7 551 527 592 063 000 2 353 125 040 550 000 557 921 681 547 000 102 417 740 732 700 14 710 753 408 920 ⎥ 161 429 736 530 100 000 52 260 903 362 510 000 12 953 636 989 940 000 2 503 858 755 468 000 381 922 055 502 200 ⎦
References [1] C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes, Proc. R. Soc. A: Math. Phys. Eng. Sci. 161 (906) (1937) 367–381, http: //dx.doi.org/10.1098/rspa.1937.0150. [2] C.T. Goudar, J.R. Sonnad, Explicit friction factor correlation for turbulent flow in smooth pipes, Ind. Eng. Chem. Res. 42 (12) (2003) 2878–2880, http: //dx.doi.org/10.1021/ie0300676. [3] D. Brkić, Review of explicit approximations to the Colebrook relation for flow friction, J. Pet. Sci. Eng. 77 (1) (2011) 34–48, http://dx.doi.org/10.1016/j. petrol.2011.02.006. [4] D. Brkić, W solutions of the CW equation for flow friction, Appl. Math. Lett. 24 (8) (2011) 1379–1383, http://dx.doi.org/10.1016/j.aml.2011.03.014. [5] J.P. Boyd, Global approximations to the principal real-valued branch of the lambert W-function, Appl. Math. Lett. 11 (6) (1998) 27–31, http://dx.doi.org/ 10.1016/S0893-9659(98)00097-4. [6] D.A. Barry, J.-Y. Parlange, L. Li, H. Prommer, C.J. Cunningham, F. Stagnitti, Analytical approximations for real values of the Lambert W-function, Math. Comput. Simul. 53 (1-2) (2000) 95–103, http://dx.doi.org/10.1016/S0378-4754(00)00172-5. [7] S. Winitzki, Uniform approximations for transcendental functions, in: V. Kumar, M.L. Gavrilova, C.J.K. Tan, P. L'Ecuyer (Eds.), Computational Science and its Applications - ICCSA 2003, Lecture Notes in Computer Science, vol. 2667, Springer, Berlin and New York and Paris, 2003, pp. 780–789. [8] Žarko Ćojbašic, Dejan Brkić, Very accurate explicit approximations for calculation of the colebrook friction factor, Int. J. Mech. Sci. 67 (2013) 10–13, http://dx.doi.org/10.1016/j.ijmecsci.2012.11.017.
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[9] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1) (1996) 329–359, http://dx.doi.org/ 10.1007/BF02124750. [10] J.R. Sonnad, C.T. Goudar, Constraints for using lambert function-based explicit colebrook–white equation, J. Hydraul. Eng. 130 (9) (2004) 929–931, http://dx.doi.org/10.1061/(ASCE)0733-9429(2004)130:9(929). [11] D. Clamond, Efficient resolution of the Colebrook equation, Ind. Eng. Chem. Res. 48 (7) (2009) 3665–3671, http://dx.doi.org/10.1021/ie801626g. [12] D. Brkić, Comparison of the Lambert W–function based solutions to the Colebrook equation, Eng. Comput. 29 (6) (2012) 617–630, http://dx.doi.org/ 10.1108/02644401211246337.
Contents lists available at ScienceDirect
Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite
Explicit representation of the implicit Colebrook–White equation P. Rollmann n, K. Spindler Institute of Thermodynamics and Thermal Engineering (ITW), University of Stuttgart, Pfaffenwaldring 6, 70550 Stuttgart, Germany
a r t i c l e i n f o
abstract It is shown that the Colebrook–White equation 1/ λ = − 2lg⎡⎣2.51/Re λ + ϵ/3.71D⎤⎦ can be solved analytically for the friction factor λ. The solution contains two infinite sums. For given Reynolds numbers Re and relative roughnesses ϵ/D , one can create an own approximation with the required accuracy by adding a finite number of summands. The computing time of both the iterative calculation and several approximations is being compared. In all cases, the approximation is much faster than the iteration. Two examples for practical applications are given. & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Article history: Received 9 October 2014 Received in revised form 30 November 2014 Accepted 7 December 2014 Available online 10 December 2014 Keywords: Friction factor Drag coefficient Analytical solution Explicit Colebrook White
1. Introduction The friction factor for turbulent flows in rough pipes is needed to calculate the pressure drop. It is also needed to estimate the Nusselt number with the Gnielinski equation:
Nu = 1+
λ RePr 8 λ 12.7 8 (Pr 2/3
⎡ ⎛ ⎞2/3⎤ ⎢1 + ⎜ D ⎟ ⎥ ⎝ L ⎠ ⎥⎦ − 1) ⎢⎣
(1)
It can be calculated with the Colebrook–White equation [1]:
⎡ 2.51 1 ϵ ⎤ = − 2lg⎢ + ⎥ ⎣ Re λ 3.71D ⎦ λ
(2)
Here, ϵ is the average roughness height. For ϵ = 0, the Colebrook–White equation becomes the Prandtl equation for the friction factor in smooth tubes:
⎡ 2.51 ⎤ 1 = − 2lg⎢ ⎥ = 2lg(Re λ ) − 0.8 ⎣ Re λ ⎦ λ For Eq. (3), Goudar and Sonnad [2] already presented an explicit solution. n
Corresponding author. E-mail address: [email protected] (P. Rollmann).
http://dx.doi.org/10.1016/j.csite.2014.12.001 2214-157X/& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
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P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
Nomenclature D L lg ln Nu Pr Re
diameter (m) length (m) logarithm to the base 10 natural logarithm Nusselt number (dimensionless) Prandtl number (dimensionless) Reynolds number (dimensionless)
ta ti W ⎡x⎤ ⎣ y⎦
ϵ λ τ
duration of approximation (s) duration of iteration (s) Lambert W function unsigned Stirling numbers of the first kind (dimensionless) average roughness height (m) Darcy–Weisbach friction factor (dimensionless) dimensionless duration τ = ti/ta (dimensionless)
Calculating the total pressure drop by integrating the local pressure drops and during numerical simulations the iterations for λ can require long computing times due to high spatial and temporal resolutions. Therefore, numerous approximations have been developed. Brkić [3] gives an overview of 20 approximations from the literature and compares them with his own equation. Brkić [4] also presented the three approximations of Boyd [5], Barry et al. [6] and Winitzki [7] that are actually approximations of the Lambert W function. In 2013, Ćojbašić and Brkić [8] developed a very accurate approximation. With this, the friction factor can be calculated with an accuracy of less than 0.01%.
2. Analytical solution In the following, it is shown how Eq. (2) can be rearranged analytically for the friction factor λ. The result contains two infinite sums. By truncating after a finite number, approximations for the friction factor with arbitrary precision can be created. The analytical solution can be done as follows:
⎡ 2.51 ⎡ 2.51 ϵ ⎤ ϵ ⎤ 1 2 = − 2lg⎢ + + ln⎢ ⎥=− ⎥ ⎣ Re λ 3.71D ⎦ ln(10) ⎣ Re λ 3.71D ⎦ λ ⇒−
(4)
⎡ 2.51 ln(10) ϵ ⎤ = ln⎢ + ⎥ ⎣ 3.71D ⎦ 2 λ Re λ
(5)
ϵ 2.51 + = Z* 3.71D Re λ
(6)
Substitution:
⇒ λ =
2.51 ⎛ ϵ ⎞⎟ * ⎜ Re Z − ⎝ 3.71D ⎠
(7)
Inserting Eqs. (6) and (7) into Eq. (5) yields
ln(10) ln(Z *) = − · 2 ⇒ln(Z *) =
⎛ ϵ ⎞⎟ Re⎜Z * − ⎝ 3.71D ⎠ 2.51
Re ·ϵln(10) ReZ *ln(10) − 2·2.51·3.71D 2·2.51
Substitution:
(10)
Re ·ϵln(10) = Re2* 2·2.51·3.71D
(11)
Re1* = exp(Re2* − Re1*Z *) Re* 1
Substitution:
(9)
Reln(10) = Re1* 2·2.51
⇒ln(Z *) = Re2* − Re1*Z * ⇒Z *
(8)
Re1*Z * = Z
(12)
(13) (14)
P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
* 1 ⇒Z = Re1*e Re2 Z e
Substitution:
43
(15) * ˜ Re1*e Re2 = Re
(16)
˜ ⇒Ze Z = Re
(17)
Eq. (17) can be solved with a Lambert W function:
˜) Z = W (Re
(18)
Therefore Corless et al. [9] presented a function: ^ ^ =∞ k =∞ m
˜ ) = ln(Re ˜ ) − ln[ln(Re ˜ )] + W (Re
∑ ∑ k=0 m=1
⎡k + m⎤ 1 ⎥ ⎢ ˜ )⎤⎦ (−1)k ⎢ k + 1 ⎥ ln⎡⎣ln(Re m! ⎦ ⎣
{
m
}
⎡⎣ln(Re ˜ )⎤⎦−k − m
k+m
The coefficients [ k + 1 ] are unsigned Stirling numbers of the first kind (see Section 5).
^ ^ = 1 (top) and k^ = 1, m ^ = 2 (bottom). Fig. 1. Percentage deviation for k = 0 , m
(19)
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P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
The back substitution yields
˜ ) = Re*Z * = ln(10) + Reln(10)ϵ W (Re 1 2·2.51·3.71D 2 λ
(20)
Finally, one obtains
⎡ 2 ⎤−2 Re ·ϵ ˜)− λ=⎢ W (Re ⎥ ⎣ ln(10) 2.51· 3.71D ⎦
(21)
3. Discussion Good approximations are being obtained by not adding an infinite number of summands in Eq. (19), but a finite number. ^ ^ = 1 is shown. In this case only one summand On top of Fig. 1 the percentage deviation of the approximation for k = 0 and m is calculated each. The relative roughness is being plotted over the Reynolds number. The dimensionless time τ = ti/ta indicates how many times longer the iterative determination of the friction factor has taken than the calculation with the approximation. Therefore, the friction factor was being calculated for 367 Reynolds numbers and 401 relative roughnesses ^ ^ =1 (i.e. 147 167 data points) and the ratio of the measured times was formed. For the first approximation with k = 0 and m the ratio is τ = 11 735.
^ ^ = 3 (top) and k^ = 3, m ^ = 4 (bottom). Fig. 2. Percentage deviation for k = 2 , m
P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
45
^ ^ = 5 (top) and k^ = 10 , m ^ = 11 (bottom). Fig. 3. Percentage deviation for k = 4 , m
^ ^ = 2); (k^ = 2, m ^ = 3); (k^ = 3, m ^ = 1); (k^ = 1, m ^ = 4 ) and (k^ = 4 , Figs. 1–3 show the percentage deviation for (k = 0, m ^ m = 5). While the computation time ta increases and thus the dimensionless time τ decreases, the accuracy is steadily ^ ^ = 11 is shown at the bottom of Fig. 3. The calculation of the friction factor with the increasing. Finally, the case for k = 10, m approximation is still faster by a factor of 164 than the iterative calculation. The maximum percentage deviation is being reduced to 3.68 × 10−6% .
4. Practical application
^ ^ can be For known relative roughnesses, Reynolds numbers and a required accuracy, the upper summation limits k and m selected from pictures 1–3. 4.1. Example 1
^ ^ = 1 can For a relative roughness ϵ/D = 10−3 and Reynolds numbers Re ≥ 106 the very fast approximation with k = 0 and m be chosen. The maximum percentage deviation is smaller than 0.02%. Even when there is fouling and the relative roughness increases, the maximum percentage deviation is still smaller than 0.02%. The approximation is 11 735 times as fast as the iteration.
46
P. Rollmann, K. Spindler / Case Studies in Thermal Engineering 5 (2015) 41–47
4.2. Example 2
^ ^ = 5 could be chosen. The For applications with very high required accuracies the upper summation limits k = 4 and m range for the Reynolds number as well as the relative roughness where there is the smallest percentage deviation (<0.0005% ) is wide. The maximum percentage deviation for all Reynolds numbers and all relative roughnesses is <0.00405% . The approximation is 767 times as fast as the iteration. ˜ = Re*e Re2* is sometimes not possible. Since Re ˜ is not The exponent Re2* can assume large values and the calculation of Re 1 ˜ ), one can rearrange the equation as follows: needed but only ln(Re
⎛˜⎞ ⎛ ⎛ ⎞ *⎞ ⎟ = ln⎜Re1*e Re2 ⎟ = ln⎜Re1*⎟ + Re2* ln⎜Re ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(22)
This step avoids the limitations described by Sonnad and Goudar [10], Clammond [11] and Brkić [12].
5. Unsigned Stirling numbers of the first kind In the following matrix 21 11 unsigned Stirling numbers of the first kind are presented. Thus, the friction factor until ^ ^ = 11 can be determined. In the matrix, the first element at the top left has the coordinate (0,0): k = 10 and m ⎡1 0 0 0 0 0 ⎢ 1 0 0 0 0 ⎢0 ⎢0 1 1 0 0 0 ⎢0 2 3 1 0 0 ⎢ 0 6 11 6 1 0 ⎢ ⎢0 24 50 35 10 1 ⎢ 120 274 225 85 15 ⎢0 720 1764 1624 735 175 ⎢0 ⎢0 5040 13 068 13 132 6769 1960 ⎢ 40 320 109 584 118 124 67 284 22 449 ⎢0 ⎢0 362 880 1 026 576 1 172 700 723 680 269 325 ⎢0 3 628 800 10 628 640 12 753 576 8 409 500 3 416 930 ⎢ 39 916 800 120 543 840 150 917 976 105 258 076 45 995 730 ⎢0 ⎢0 479 001 600 1 486 442 880 1 931 559 552 1 414 014 888 657 206 836 ⎢ 6 227 020 800 19 802 759 040 26 596 717 056 20 313 753 096 9 957 703 756 ⎢0 ⎢0 87 178 291 200 283 465 647 360 392 156 797 824 310 989 260 400 159 721 605 680 ⎢0 1 307 674 368 000 339 163 001 600 6 165 817 614 720 5 056 995 703 824 2 706 813 345 600 ⎢ 20 922 789 888 000 70 734 282 393 600 102 992 244 837 100 87 077 748 875 900 48 366 009 233 420 ⎢0 ⎢0 355 687 428 096 000 1 223 405 590 579 000 821 602 444 625 000 158 331 397 572 7000 909 299 905 844 100 ⎢ 22 376 988 058 520 000 34 012 249 593 820 000 30 321 254 007 720 000 17 950 712 280 920 000 ⎢ 0 6 402 373 705 728 000 ⎣ 0 121 645 100 408 800 000 431 565 146 817 600 000 668 609 730 341 200 000 610 116 075 740 500 000 371 384 787 345 200 000 ⎤ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ 1 0 0 0 0 ⎥ 21 1 0 0 0 ⎥ ⎥ 322 28 1 0 0 ⎥ 4536 546 36 1 0 ⎥ ⎥ 63 273 9450 870 45 1 ⎥ 902 055 157 773 18 150 1320 55 ⎥ 13 339 535 2 637 558 357 423 32 670 1925 ⎥ ⎥ 206 070 150 44 990 231 6 926 634 749 463 55 770 ⎥ 3 336 118 786 790 943 153 135 036 473 16 669 653 1 474 473 ⎥ ⎥ 56 663 366 760 14 409 322 928 2 681 453 775 368 411 615 37 312 275 ⎥ 1 009 672 107 080 272 803 210 680 5 4631 129 553 8 207 628 000 928 095 740 ⎥ 18 861 567 058 880 5 374 523 477 960 1 146 901 283 528 185 953 177 553 23 057 159 840 ⎥ 369 012 649 234 400 110 228 466 184 200 2 4871 845 297 940 4 308 105 301 929 577 924 894 833 ⎥ ⎥ 7 551 527 592 063 000 2 353 125 040 550 000 557 921 681 547 000 102 417 740 732 700 14 710 753 408 920 ⎥ 161 429 736 530 100 000 52 260 903 362 510 000 12 953 636 989 940 000 2 503 858 755 468 000 381 922 055 502 200 ⎦
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