Fluids and Solids Handling Eliminate Iteration from Flow Problems John D. Barry Middough, In. This artile introdues a novel approah to solving flow and pipe-sizing problems based on two new dimensionless quantities that are independent of line size and flow. Proess engineers are often faed with the task of sizing a pipe for a speifi flow. A ommon approah to this problem starts with a typial fluid veloity (e.g., ft/s for a liquid with properties roughly similar to water). Calulating the pressure drop aross a line sized in this way is a straightforward matter, involving the frition fator and either the equivalent length or the sum of resistanes to flow (the K values). This alulation is detailed in many standard referenes. However, that is only one of three possible pipe sizing/flow problems an engineer is likely to enounter. The other two, whih are enountered less ommonly, are: given a required flow and pressure drop riterion, what line size is required? given a line size and a pressure drop, what flow may be expeted? This is the oneptual inverse of the previous problem. The solution of either of these problems typially requires an iterative approah. This artile introdues a novel approah for the diret (non-iterative) solution of suh problems. Frition in pipe flow the lassi approah The Bernoulli equation, also known as the mehanial energy balane, is the basis for understanding flow in pipes: v g g P + z + + lw f + W 0 ( ) g ρ The first term, aounting for kineti energy hanges, is usually small ompared to the other four terms, and thus an be assumed to be negligible. Note that for line sizing purposes, the envelope for a mehanial energy balane typially exludes rotating equipment (e.g., a pump or a turbine) and fouses on the terminal pressure onditions, elevation hanges, and frition losses; the rotating equipment is then sized to meet the onditions derived from this mehanial energy balane. The seond term in the energy balane refers to elevation hanges; those are typially defined as part of the oneptual layout of the proposed line. The third term, referring to the differene in terminal pressures, is also typially defined as part of the piping layout. This approah fouses on the fourth term, lw f, the work lost due to frition: lw f flv gd ( ) In this equation, f is the Dary frition fator: g P D f v The Dary frition fator equals four times the Fanning frition fator, f Fanning. To use the Fanning frition fator, substitute 4f Fanning for f wherever the latter appears. The frition fator is a funtion of both the fluid Reynolds number, Re Dvρ/µ, and the relative roughness, ε/d. The relationship between these quantities and the frition fator is expressed graphially in a Moody plot, or mathematially in various empirial relations (e.g., the Colebrook or Churhill equations). To solve the types of 6 www.aihe.org/ep Marh 008 CEP
problems onsidered here, it is neessary to introdue dimensionless quantities that do not depend on the line size, D, or the fluid veloity, v. Eliminating the diameter Bennett and Myers () suggest that a plot or orrelation of the frition fator, f, as a funtion of Ref / would be useful in solving for line size if the flowrate and pressure drop are known. This dimensionless quantity (Ref / ) is drawn from two quantities already established, namely, the Reynolds number and the Dary frition fator. The veloity, v, appears in both Re and f. It an be eliminated by using the definition of the veloity: Q Q 4Q v A D D 4 v Thus, Re 4Qρ/Dµ and f [g ( P) D ] / [6Q ] [ g ( P)D ] / [8Q ], whih leads to: Re f 6Q ; D v D 6Q ( 4) 4Q g P D ρ D µ 8Q L ρ ( ) 4Qρ D D g P µ 8Q 4 Qρ g P µ 8Q The diameter has now been eliminated. But f, and thus Ref /, depends on the relative roughness, whih presupposes knowledge of the pipe diameter. To get around this, a new dimensionless quantity, the flow funtion, is introdued: Θ ( ) Re f ε 4Qρ g P D D D µ 8Q 4 Qρ ( ) Dµ D g P ε 8Q D ( ) 4Qρε g P µ 8Q ( ) ε g P Θ Re f D µ g P W ρ ( ) µ L ( ) ( Qρ ) ε D ( 6) Nomenlature a, b,, oeffiients in orrelating polynomial (Eq. 9) (represented by y in the general ase of Eq. 0), dimensionless A, B, C, oeffiients in generating polynomial (Eq. 0), dimensionless A pipe ross-setional area normal to flow, ft D pipe (inside) diameter, ft f Dary frition fator, dimensionless g aeleration of gravity,.74 ft/s g onversion fator,.74 ft-lb/lb fore -s Ka Kármán number (Eq. 7), dimensionless lw f lost work due to frition, ft-lb fore /lb L line equivalent length, ft P pressure, psi or lb fore /ft Q volumetri flowrate, ft /s Re Reynolds number, dimensionless v fluid veloity, ft/s W shaft work, ft-lb fore /lb W mass flowrate, lb/s y generi oeffiient in orrelating polynomial (Eq. 9), generated by Eq. 0, dimensionless z elevation, ft Greek Letters ε absolute roughness of pipe, ft Φ frition/roughness funtion (Eq. 8), dimensionless µ dynami visosity, lb/ft-s Θ flow funtion (Eq. 6), dimensionless Θ* modified flow funtion Ref /, dimensionless ρ density, lb/ft whih an be alulated from known quantities. The absolute roughness, ε, is a funtion of the nature of the pipe, whih is known, and is independent of the line diameter. A omplementary dimensionless quantity that eliminates dependene on the flow is also needed. This has already been established in the form of the Kármán number, Ka: D v g P D Ka f ρ Λ Re µ v Dρ lw fgd g D ρ g D ρ ( 7) Many texts provide plots of the frition fator as a funtion of the Kármán number with the relative roughness as a variable. This may not be partiularly helpful, however, sine it presumes knowledge of the pipe diame- CEP Marh 008 www.aihe.org/ep 7
Fluids and Solids Handling ter, whih may not be the ase. To work around this, another new dimensionless quantity, the frition/roughness funtion, is defined: ( ) Φ f ε g P D D 8Q ε D g ( ) P ε D 8Q D g ε 8Q Guidelines for sizing pipe (e.g., Peters and Timmerhaus ()) inlude typial veloities in the range of 0 ft/s for liquids and 0 0 ft/s for gases. A quik hek of Reynolds numbers alulated using veloities within these ranges and properties of ommon industrial liquids (e.g., organi liquids with visosities of approximately P and densities on the same order of magnitude as that of water) shows that flows meeting these guidelines are indeed turbulent; similar omments apply for industrial gases. Most design ourses guide the student to design for turbulent flow in pipelines, sine this is perhaps the most ommon situation in industry. Thus, it is reasonable to assume full turbulene and, as a orollary, define a slightly modified version of Φ to use the fully turbulent Dary frition fator, f T. This modified quantity Φ T, is the produt of f T / and the relative roughness ε/d. This quantity will prove useful in one of the variations of the lassi flow problem presented later. For the alulation of flow based on pipe size and pressure drop, values of f T as a funtion of line size or relative roughness are tabulated in the literature (), and are listed in Tables and. Now we have a dimensionless quantity, Θ, that is independent of line size that an be used to orrelate another dimensionless quantity, Ka. This latter dimensionless quantity is independent of flow. Table. Fully turbulent Dary frition fator as a funtion of line size. Line Size Line Size (nominal), in. f T (nominal), in. 0. 0.07 0.7 0.0 0.0. 0.0 0.09 0.08 Soure: (). f T 4 0.07 6 0.0 8 0 0.04 6 0.0 8 4 0.0 ( 8) Plotting log Ka as a funtion of log Θ and Θ T yields a family of parallel urves. Multiple regression analysis shows that these urves an be represented by polynomials with log Θ as the independent variable and the form: + + + + ( 9) log Ka a b log Θ log Θ d log Θ We ll all this relationship the orrelating polynomial. A ubi polynomial is generally suffiient to orrelate log Ka as a funtion of log Θ (for reasons that will be disussed later). Multiple regression analysis shows that a, b,, d are well-orrelated by polynomials in log Φ T. Thus: y A B log ΦT C log ΦT D log Φ E log Φ where y is a generi oeffiient for Eq. 9. We ll all this relationship the generating polynomial to distinguish it from the orrelating polynomial that relates log Ka to log Θ. A quarti (fourth-degree) polynomial is suffiiently aurate for most work. The oeffiients A, B, C, D and E for Eq. 0 are given in Table. Knowing the physial definition of the system (line [equivalent] length, pressure drop, nature of the pipe), Table. Fully turbulent Dary frition fator as a funtion of relative roughness. ε/d f T ε/d f T 0.07 0.0840 0.06 0.0780 0.0 0.07 0.04 0.06467 0.0 0.0774 0.0 0.04867 0.0 0.0469 0.0 0.07904 0.009 0.0688 0.008 0.097 0.007 0.074 0.006 0.06 0.00 0.0067 0.004 0.0846 0.00 0.066 0.00 0.04 0.00 0.077 0.00 0.096 0.0009 0.094 0.0008 0.086 0.0007 0.0806 0.0006 0.07404 0.000 0.06699 0.0004 0.089 T 4 T + + + + 0.000 0.0497 0.000 0.07 0.000 0.096 0.000 0.098 0.00009 0.074 0.00008 0.0487 0.00007 0.007 0.00006 0.00896 0.0000 0.0044 0.00004 0.007 0.0000 0.00964 0.0000 0.0090 0.0000 0.00898 0.0000 0.00806 0.000009 0.0079 0.000008 0.00779 0.000007 0.0076 0.000006 0.00747 0.00000 0.0077 0.000004 0.0070 0.00000 0.00678 0.00000 0.0066 0.00000 0.0069 0.00000 0.0079 ( 0) Note: These values are alulated from the Churhill relationship for the frition fator using a Reynolds number ontribution of zero. 8 www.aihe.org/ep Marh 008 CEP
Table. Generating polynomial oeffiients (for use in alulating orrelating polynomial oeffiients). A B C D E Quadrati a 0.769 0.864047 0.0040069 0.00664 0.0000 b.09400 0.074000 0.0080 0.0000060 0.0000084 0.0090 0.000846 0.00007 000000668 0.00000046 Cubi a 0.09777.040908 0.0089 0.00976 0.00048466 b.090664 0.76886 0.048046 0.0048687 0.000808 0.08 0.067707 0.009 0.0000476 0000089 d 0.00044 0.0009 0.0000880 0.00000 0.000004 Table 4. Data for Example. Given Data Density, ρ 4.7 lb/ft Visosity, µ.8 P Line Length, L,000 ft Flowrate, Q,70 gal/min Absolute Roughness, ε 0.000 ft () Derived Data Pressure Drop Between Tie-In Point and Maximum Downstream Terminal Pressure, P 7 psi Correlating polynomial oeffiients, smooth pipe Quadrati a b 0.09009 0.88869 0.004669 the flowrate, and the fluid s physial properties (density, visosity) allows Θ, log Θ, Φ T and log Φ T to be determined. Log Φ T an be used to alulate the oeffiients a, b, and d (via Eq. 0) and the tabulated values for A, B, C, D and E, whih are used with Eq. 9 to yield log Ka and therefore Ka. Finally, knowing Ka leads diretly to a theoretial line size: Ka Ka D Cubi a b d 0.09040 0.79788 0.09470 0.000706 g g P D ρ P D ρ µ Ka ρ g µ Ka D ρ g L L It is unlikely that the diameter alulated in this way will be equal to a ommerial pipe size. In most ases, the pressure drop speifiation is a maximum allowable pressure drop, so the next larger pipe size should be hosen. (Conversely, if a minimum pressure drop were speified, the next smaller line size would be hosen.) Example A refinery needs to move 60,000 bbl/d (,70 gal/min) of a produt with an API speifi gravity of 0 (approximately 4.7 lb/ft ) and a visosity of.8 P. The pressure at the new tie-in point (i.e., the soure) for this new line is 90 psig. Due to the design of existing equipment, the disharge pressure downstream annot exeed psig. The proposed line routing (approximately,000 equivalent ft, aounting for fittings) is essentially flat. Refinery speifiations all for arbon steel pipe to be used. What size pipe is needed? Solution. The data for the problem are given in Table 4. The alulations are detailed in the box on the next page.. Calulate Θ using Eq. 6, and then log Θ. Θ 9.6, log Θ 0.980.. Calulate Φ using Eq. 8, and then log Φ. Φ,, log Φ 4.097.. Use the generating polynomial, Eq. 0, to get oeffiients for the Ka-Θ relationship (for simpliity, use the quadrati form). These oeffiients are a.707, b.06, and 0.0008. 4. Use these oeffiients in Eq. 9 and the value of Θ alulated in Step to generate Ka: log Ka.707 + (.06)(0.980) + ( 0.0008)(0.980) 4.86. So, Ka 8,69.8904.. Solve for D using Eq.. D 0.88 ft 0.89 in. Sine a minimum pressure drop was speified, the next-smaller ommerially available line size would be hosen to ensure that this minimum drop riterion is met. Thus, a 0-in. line (atual ID 0.0 in. for Sh. 40 pipe) should be seleted. Estimating flow This approah an also be used to alulate the expeted flow through a pipe of a speified size with a known pressure drop. The terms used to alulate Ka (and thus log Ka) are known. Again fully turbulent flow is assumed, sine that is ommon in industry. Then ε/d an be alulated based on the known line size and the nature of the pipe, and f T an be obtained from the tabulated values as mentioned previously. That allows Φ T to be alulated, as well as the oeffiients a, b, and d for the orrelating polynomial relating log Ka to log Θ. Artile ontinues on next page CEP Marh 008 www.aihe.org/ep 9
Fluids and Solids Handling Calulations for Example Step :.74 ft-lb 4 ( 0.000 ft) Θ lbf -s ( ) 44 in 7 psi ft 67. 0 ft (.8 P) 4 -lb/s 8(,000 ft) P,70 gal min 0.00 ft /s lb gal/min 4. 7 ft 9. 6 Step : Φ 8,70 gal min Step : ft-lb/s (.8 P) 67 0 4. P D lb 4. 7 ft.74 ft-lb 7 psi lbf -s ( 44 in ) ft 0.00 ft /s 4.7lb gal/min ft (,000 ft) ( 8, 69. 89) 0.000 ft,.4 - (,000 ft) ft-lb. 74 lb -s 7 psi 44 in ( ) f ft 0.88 ft 0.89 in. This new approah uses the fully turbulent frition fator for a given line size as a ontributing parameter to the generating equation (Eq. 0). The results of the generating equation are ultimately employed to get a value of Θ and thus the flow. One ould then re-evaluate the frition fator based on this flow and the known line size and use the generating equation again, but it s up to the user to evaluate whether this would have pratial value. In the traditional approah (speifying physial properties and terminal onditions), one may alulate flow diretly by postulating omplete turbulene in addition to the known onstraints (pressure drop, line length, physial properties). In that ase, it is neessary to verify that the alulated flow does indeed yield a Reynolds number that qualifies as fully turbulent. Again, further refinement by iteration would be at the user s disretion. It was suggested previously that a ubi polynomial should suffie to relate log Ka to log Θ, based on the observation that a ubi polynomial is the highest order equation that an relatively easily be solved analytially. (It is possible to solve a quarti polynomial analytially, but the solution is onsiderably more involved than solving a ubi polynomial. There is no general analyti solution for higher-order polynomials and it s unlikely that there would be muh, if any, benefit to using one.) Methods of solving ubi and quadrati equations are desribed in detail at http://mathworld.wolfram.om (4). Of the three roots arising from the solution of a ubi equation, only one is of interest. Values of log? within the sope of this orrelation fall within the approximate range of. log Θ 7.. With log Θ and thus Θ known, a flowrate an be found by solving Eq. 6: µ Qρ Θ g Q ρ µ Θ g µ W ρ Θ g W ρ µ Θ g P ( P ) P Example Chilled water at 4 F flows from a onstant-level reservoir through a -in. Sh. 40 steel pipe, the end of whih is open to the atmosphere. The pipe has an equivalent length of 7 ft, and the outlet is ft below the liquid level in ( P ) ( ) ( ) 40 www.aihe.org/ep Marh 008 CEP
the reservoir. Negleting any kineti energy ontributions, determine the flow. Solution. The data for this problem appear in Table. In the interest of spae, the alulations are not detailed here.. Using Eq. 7, alulate Ka 6,80.. Then, log Ka 4... Sine the nature of the pipe and the line size are known, ε/d, as well as the fully turbulent frition fator for that relative roughness, are also known. Therefore, Φ is obtained from its definition, Eq. 8: Φ [f / (ε/d)],6.98. So, log Φ.4047.. Use Φ and the generating polynomial (Eq. 0) to alulate the oeffiients a, b and of the orrelating polynomial (Eq. 9). For the purpose of illustration, use a quadrati generating polynomial. Thus: a.96, b 0.970, and 0.00. 4. Solve the quadrati orrelating polynomial for Θ: x β ± β 4 αγ α Table. Data for Example. Given Data Density, ρ 6.4 lb/ft Visosity, µ.47 P Line Length, L 7 ft Pipe ID, D.067 in. Absolute Roughness, ε 0.000 ft () Derived Data Head Change, P ft liquid. psi Relative Roughness, ε/d 0.00087 Fully Turbulent Dary Frition Fator, f T 0.09 () where β b 0.970, γ a log Ka.04, α 0.00. Thus, x log Θ.40 and 47.766. The first root reflets the use of the positive sign of the radial; the seond root, the negative sign. Only the first root has any physial signifiane, so with log Θ.40, Θ.878.. Use Eq. to derive the flowrate: Q 0.9 ft /s 6.4 gal/min. log Ka.0 0.0 8.0 6.0 4.0.0 0 0.0 4.0 6.0 8.0 0.0.0 4.0 log Θ* Figure. Kármán number as a funtion of Q* for smooth pipe (ε 0). The speial ase of smooth pipe By definition, smooth pipe has zero roughness, whih would render the definition of Θ useless. However, Gilmont s work () an be modified to orrelate Ka as a funtion of Θ*, where Θ* Ref /, to obtain a relationship where one variable is independent of the line size and the other is independent of the flowrate. Like the Θ-Ka orrelation, this Θ*-Ka orrelation an be represented well by a ubi polynomial. Therefore, the tehniques disussed above for rough pipe an be applied to smooth pipe using the data at the bottom of Table and the relationship in Figure. A final note Line sizing is not exat or rigorous, sine it involves disrete standard ommerial sizes rather than values of a ontinuous funtion. Calulations may indiate that a pipe diameter of, say,.4 in., is required to aommodate a given flow, but that s simply the solution to an equation. Rather, one hooses the losest ommerial size to suit the appliation based on the theoretial results of alulations, engineering judgment, and experiene. Literature Cited. Bennett, C. O., and J. E. Meyers, Momentum, Heat and Mass Transfer, MGraw-Hill, New York, NY (974).. Peters, M. S., and K. M. Timmerhaus, Plant Design and Eonomis for Chemial Engineers, MGraw-Hill, New York, NY (968).. Flow of Fluids through Valves, Fittings and Pipe, Crane Tehnial Paper No. 40, th printing (988). 4. Weisstein, E. W., Cubi Formula and Quadrati Equation, from MathWorld A Wolfman Web Resoure, http://mathworld.wolfram.om.. Gilmont, R., Pipeline Pressure Drop: A New Design Correlation, Chem. Eng. Progress, 0 (6), pp. 4 4 (June 006). CEP JOHN D. BARRY, P.E., is a senior proess engineer with Middough, In. (Mt. Laurel, NJ; E-mail: john.barry.engr@omast.net). Previously, he worked for several engineering firms in the greater Philadelphia area, as well as performed ontrat engineering work for DuPont (Deepwater, NJ and Wilmington, DE) Premor (now Valero; Delaware City, DE).He holds a BS in hemistry from the Univ. of Delaware, and an MS in hemial engineering from the Univ. of Maryland. He is a professional engineer liensed in New Jersey and Delaware, and is a member of AIChE. CEP Marh 008 www.aihe.org/ep 4