OPTIMIZE PIPELINE DESIGN

Fluids/Solids Handling OPTIMIZE PIPELINE DESIGN FOR NON-NEWTONIAN FLUIDS Alejandro Anaya Durand, Cinthya Alejandra Aguilar Guerrero and Edgar Amaro Ronces, National Autonomous University of Mexico Here is a graphical method for determining pipe diameter, fluid temperature and pressure drop for laminar and turbulent flow. Engineers often encounter non- fluids as suspensions, viscous fluids or polymer solutions, among others. Little information exists on how to optimize the design variables when handling these fluids. This article offers a graphical procedure, given a set flowrate, for determining the most economical diameter of a pipeline D opt, as well as its optimum temperature t opt ; included is a method for calculating the pressure drop P. This procedure is based on the effect a given design variable has on costs. Therefore, an optimum value can be established for this variable at which the total costs will be at a minimum. For fluids, the shear stress τ w is directly proportional to the shear rate. The proportionality constant is simply the viscosity. For non- fluids, the shear stress is proportional to the shear rate raised to a power n called the flow behavior index, and its value depends upon whether a fluid is pseudoplastic, Bingham plastic (for both n < ) or dilatant (n >). K is no longer the viscosity, and now is called the fluid consistency index. For non- behavior, we can relate the Fanning friction factor f to the Reynolds number N Re by (): 62 www.cepmagazine.org March 2002 CEP

f = a n /N Re b n () where a n and b n are dimensionless numbers that are functions of n. By knowing n, K, a n and b n, a non- can be characterized. Optimization scheme The costs of a process can be classified in two types: capital and operating costs, f (x) and f 2 (x), respectively, both a function of a design variable x. The total costs C T are the sum of both: C T = f (x) + f 2 (x) (2) To obtain the minimum value of x, the first derivative of Eq. is set = 0; and to ensure that the optimized variable is a minimum, the second derivative must be positive. dc T (3) dx = df (x) + df 2(x) =0 dx dx The fluid properties are set for each application, and we will show that x is a function of the flowrate Q, i.e., x = f(q). Q is set by the particular process application. We will show that this will allow us to solve for D opt and t opt. The graphical method is based on the relationship x = f(q). Plots are made with several values vs. Q using reference values for rheological parameters; this allows for specifying the rest of the variables. Correction factors are then applied to account for fluids with properties that differ from the reference values used; these factors are also found from a series of plots. Optimum pipe diameter The capital costs are assumed to be only those of the piping, and the operating costs are assumed to be the energy of pumping and, if needed, of heating the fluid. Obviously, larger-diameter pipes require higher investments, but Nomenclature a = annual fixed costs as a fraction of installed costs a n = function of n (turbulent regime) A, B = empirical constants in Eq. 2 b = annual maintenance costs as a fraction of total installed costs b n = function of n (turbulent regime) C e = cost of electrical energy, $/kwh C p = specific heat, Btu/lb m F C s = heating (steam) cost, $/million Btu C T = total annual cost/ft of pipe or total annual cost over whole pipe length, $ D = pipe dia., ft D opt = optimum pipe dia., ft E = efficiency of pump and motor F Di = correction factors to D opt, dimensionless (i = 6; see Figures 2 4, 6 8) F r = ratio of total costs for fittings and erection to total purchase cost of pipe F Ti = correction factors to t opt, dimensionless (i = 7; see Figures 0 5, 7) g c = gravitational constant, 32.74 lb m -ft/lb f s 2 h = hours of operation/year K = consistency index, lb f -s n /ft2 L = pipe length, ft n = flow behavior index, dimensionless N Re = Reynolds number, dimensionless p = constant for each pipe material = slope of logarithmic plot of purchase cost of pipe/ft vs. 2D P = pressure drop, psi P 00 = pressure drop per 00 ft, psi/ft Q = volumetric flowrate, ft 3 /s Q c = volumetric flowrate at the end of laminar flow region, ft 3 /s S g = specific gravity, dimensionless t opt = optimum pumping temperature, F T = fluid temperature, F V = average linear flow velocity, ft/s = cost of pipe/ft when 2D = in., $/ft X p Greek letters γ = g c K 8 (n ) µ = viscosity, lb m /ft s µ e = effective viscosity, lb m /ft s ρ = density, lb m /ft 3 τ w = shear stress, lb f /in. 2 D opt = 4.84 0 0 +3n C e hk p a + b F r + X p 2 p E Q 4.05 0 5 π n p ++3n (4) p +5+b n (3n 4) D opt = 2 463 0 0 5 4b n +3b n n g c K 8 n b nan C e Qh g c p a + b F r + X p 2 p ρ b n E Q 3.24 0 6 π 2 b n (2 n ) (5) CEP March 2002 www.cepmagazine.org 63

Fluids/Solids Handling 0.0 0.0 Optimum Dia., in. Optimum Dia., in. 0. 0 00,000 0. 0 00,000 Figure. Laminar flow optimum pipe diameter. Figure 2. Turbulent flow optimum diameter. Correction Factor for Specific Gravity, F D.7.5.3. 0.7 0.3 0.5 0.7..3.5.7.9 2. Specific Gravity Correction Factor for K', F D2 2.0.5 0.5 0.0 0-5 0-4 0-3 0-2 0 - K', lb f -sn' /ft 2 Figure 3. Laminar flow F D, correction factor for specific gravity S g. Figure 4. Laminar flow F D2, correction factor for consistency index K. Table. Reference values used to construct the figures. Variable Reference Value Variable Reference Value ρ 62.37 lb/ft 3 b 0.05 C e 0.076 /kwh Fr. X p $6/ft E 0.7 p'.35 h 7,920 h/yr a 0.5 Variable Bingham n' 0.85.5 0.67 K 0.05 0.05 0.05 0.05 a n 0.077 0.08 0.07 0.078 b n 0.258 0.228 0.27 0.25 Note: a n and b n are necessary in turbulent flow. require less power to pump, so optimizing will determine the pipe diameter that is the minimum of the fixed and variable costs. Thus, f represents the cost per feet of pipeline for a given diameter, and f 2 the energy needed for a section of pipeline. C T takes two forms depending on the flow regime; therefore there are separate equations for the optimum diameter, Eq. 4 for laminar flow and Eq. 5 for turbulent flow (2) (see the box on the previous page). Plotting D opt vs. Q, and keeping the rest of the variables at the reference values generates Figure for laminar flow and Figure 2 for turbulent flow. Each curve represents the most typical non- time-independent fluids with typical rheological properties (K, n, a n and b n ). The reference 64 www.cepmagazine.org March 2002 CEP

Correction Factor for (a + b), F D3.3.2. 0.8 0.05 0.5 0.25 0.35 0.45 (a + b) Correction Factor for K', F D5. 0.7 0-5 0-4 0-3 0-2 0 - K', lb f -sn' /ft 2 Figure 5. Laminar flow F D3, correction factor for (a + b). Figure 7. Turbulent flow F D5, correction factor for consistency index K. Correction Factor for Specific Gravity, F D4.5.3. 0.7 0.25 0.55 0.85.5.45.75 2.05 Specific Gravity Correction Factor for (a + b), F D6.3.2. 0.8 0.05 0.5 0.25 0.35 0.45 (a + b) Figure 6. Turbulent flow F D4, correction factor for specific gravity S g. Figure 8. Turbulent flow F D6, correction factor for (a + b). values used to draw up those graphs are presented in the table below. Correction factors, F Di, are used to account for properties that differ from those used as reference values. In such cases, the optimum diameter can be found by applying the factors in Figures 3 5 for laminar flow and Figures 6 8 for turbulent flow. Thus, the optimum diameter D opt is found by correcting the optimum reference diameter by multiplying by the appropriate factors: D opt = D opt, ref n Π i = F Di Example I A pseudoplastic fluid in a storage vessel will be transported at a flowrate of 50,000 lb/h to a production process 500 ft away. Determine the optimum diameter of the (6) pipe such that the total annual cost of transporting the fluid is a minimum. Note: yr = 7,920 h. Operating data: Specific heat = Btu/ lb F; density = 6 lb/ft 3 ; overall efficiency of pump and motor = 70%; n = 0.85; and K = 0.07 lb f -s n /ft2. Cost data: Steam heating C s = $/million Btu; electrical energy C e = $0.076/kWh; purchase cost of new steel pipe per foot of pipe length X p when 2D = in. is $6/ft; p (a constant for each pipe material) is assumed to be.35 for new carbon steel pipe and 2D opt = in. p is the slope of a logarithmic plot of purchase cost of pipe/ft vs. 2D (2). The average annual interest rate a = 9% of installed costs and annual maintenance charges b = % of installed costs (thus, a + b = 0.09 + 0.0 = 0.); the ratio of total cost for fittings, insulation and installation to the total purchase cost of new pipe F r =. (this is a typical value used in many cost-estimation texts); the anticipated useful life = 0 yr. Straightline depreciation is assumed. CEP March 2002 www.cepmagazine.org 65

Fluids/Solids Handling Optimum Pumping Temperature, F 220 200 80 60 40 20 00 80 60 40 0 00,000 Correction Factor for B, F T3 2.0 0.0 0.05 0.035 0.055 0.075 0.095 B,/ F Figure 9. Optimum pumping temperature t opt. Figure 2. Optimum pumping temperature F T3, correction factor for B. Correction Factor for Cp, F T.2 0.8 0.6 0.4 0 0.5.5 2 Correction Factor for SpecificGravity, F T4.2 0.8 0.6 0.4 0. 0.6..6 Specific Heat, Btu/ F lb m Specific Gravity Figure 0. Optimum pumping temperature F T, correction factor for specific heat C p. Figure 3. Optimum pumping temperature F T4, correction factor for specific gravity S g. Correction Factor for A, F T2 3.0 2.0 0.0.25.75 2.25 2.75 3.25 3.75 4.25 4.75 Correction Factor for Pipe Dia., F T5 3.0 2.5 2.0.5 0.5 0.0 0 2 4 6 8 0 2 4 6 8 A Pipe Dia., in. Figure. Optimum pumping temperature F T2, correction factor for A. Figure 4. Optimum pumping temperature F T5, correction factor for pipe diameter D. 66 www.cepmagazine.org March 2002 CEP

Correction Factor for Pipe Length, F T6.5 0.5 0 00,000 Pipe Length, ft P/ 00, psi/00 ft 0.0 0. 0 00,000 Figure 5. Optimum pumping temperature F T6, correction factor for pipe length L. Figure 7. Laminar flow pressure drop for optimum pipe diameter. Correction Factor for n', F T7 2.25.75.25 0.75 0.25 0.2 0.4 0.6 0.8.2.4.6.8 2 P 00, psi/00 ft 00 0 0 00,000 n' Figure 6. Optimum pumping temperature F T7, correction factor for pipe length flow behavior index n. Figure 8. Turbulent flow pressure drop for optimum pipe diameter. Note that piping length only affects the optimum temperature, not the optimum diameter. The variable p has an exponential effect on the diameter, and it is difficult to correct the diameter and temperature for values that differ from the refence value used. This method is limited to installations using carbon steel or other pipe materials with the same or close value of p. Procedure The volumetric flowrate is 50,000 lb/h ( ft 3 / 6 lb) ( gal/0.337 ft 3 ) ( h/60 min) 00 gpm. From Figure, D opt, ref = 3.82 in. With the known values of specific gravity = 6/62.37 = 78; K = 0.07 lb f -s n /ft 2 and a + b = 0.0; we can obtain from Figures 3, 4 and 5, respectively: F D = 08; F D2 = 8; F D3 =.52. Therefore: D opt = 3.82 in. 08 8.52 = 4.79 in., or D opt 5 in.; the commercial diameter is 6 in. For turbulent flow From Figure 2 for the curve for pseudoplastic fluids (n = 0.85) with a volumetric flowrate of 00 gpm, a reference diameter in turbulent flow is obtained as: D opt, ref = 2.76 in. With a specific gravity = 78; K = 0.07 lb f -s n /ft 2 and a + b = 0.0; we obtain from Figures 6, 7 and 8, respectively: F D4 = 07; F D5 = 2; and F D6 =.23. Therefore: D opt = (2.76 in. 07 2.23) = 3.8 in. 3 in. In the optimization of the diameter, the critical Reynolds number, N Re, crit, determines whether optimum conditions occur in laminar or turbulent flow (2). If N Re, crit is above 2,00, the transition from laminar to turbulent flow, the optimum diameter is for a pipe in turbulent flow. If it is below 2,00, the optimum diameter occurs in laminar flow. CEP March 2002 www.cepmagazine.org 67

Fluids/Solids Handling N Re, crit = Q C = π 4 D n V 2 n ρ n g c K 8 crit 2,00γ ρ 2 n D 4 3n 2 n = D n 4Q C πd 2 2 n ρ γ (7) Eq. 4, corresponding to laminar flow, may be substituted for D in Eq. 8. Substituting (M/ρ) +n for (3,600Qc) +n and rearranging yields Eq. 9 (see the box). This equation depicts conditions just at the end of the laminar flow regime. To solve the problem:. Evaluate the right-hand side (RHS) of Eq. 9. This yields 6.2. 2. Substitute the proposed volumetric flowrate Q into the left-hand side (LHS) of Eq. 9. The conversion is: (00 gpm)( min/60 s)( ft 3 /7.48 gal) = 0.223 ft 3 /s. Using this and the values for n and p yields 8.463 0 2. 3. If the LHS is < the RHS, the flow will be laminar in the pipe of optimum diameter, which is then evaluated from Eq. 4. This is the case here, and laminar flow should be used. 4. If the LHS is > the RHS, the flow may be regarded as turbulent, which is evaluated from Eq. 5. 5. When n is <, substitute the corresponding value of N Re, crit from Eq. 7. Optimum pumping temperature If the temperature decreases, it will be more difficult to pump the fluid and the electric costs will increase. On the other hand, steam or another heating medium will be needed when the temperature must be increased, along with the associated costs. (8) Following the scheme described by Eqs. 2 and 3, the optimum temperature in the pipeline can be obtained. When a heater must be installed at the pipeline inlet, it will be necessary to consider the fixed costs of the heater in the optimization analysis. If however, an adequate heater is already available, then there is no fixed cost, i.e., f (x) = 0. Thus, f 2 (x) is relevant, and consideration must be given to the costs of the steam and of pumping the fluid. We will make the latter assumption here. Two equations are used: Eq. 0 for laminar flow and Eq. for turbulent flow (2) (see the box). K and t are related by: K = (/g c ) 0 (A Bt) (2) Taking common logarithms: log(k g c ) = A Bt (3) In industry, optimizing the temperature at turbulent flow is infrequent, therefore, the graphical analysis will be limited to laminar flow. (Our own calculations for t opt in turbulent flow indicated numerous times that such a temperature did not have a significant effect vs. flowrate; e.g., at gpm, a typical value was 00 F, and only 05 F for 00 gpm.) In this case, t opt is plotted vs. Q, with the rest of the variables at reference values. Thus, the optimum temperature of pumping will be: Where the F Ti are correction factors, and the optit opt = t optref Σ n Π i = F Ti (4) Q C 3,600Q C +n 4 3n 2 n p ++3n = π 4 2,00 g c K 8 ρ n 2 n 5.064 0 7 +3n C e hk p a + b F r + X p 2 p E 2.5π n 4 3n 2 n p ++3n (9) t opt = B A log C pc s g c ρed +3n 3.4625BC e L 4.05 0 5 π Q n (0) t opt = B A b n log C pc s g c ρ b ned 5+ b n 3n 4.7343a n b n C e L8 b n n B 3.24 0 6 π Q 2 b n 2 n () 68 www.cepmagazine.org March 2002 CEP

mum temperature at reference conditions t opt, ref is given in Figure 9 as a function of Q. The additional reference values that were used to draw up the graphs are: C p = Btu/lb m F; D = in.; L = 00 ft; C s = $/0 6 Btu; A = 2.5; and B = 0.05/ F. Note that if D is not = in., then it should be corrected by the factor F T5. This optimization is independent of the diameter. Other references values (ρ, C e and E) are the same as in the first example. Example II We will refer to the same example as before, and find t opt. A pseudoplastic fluid in a storage vessel is pumped at 00 gpm to a production unit 500 ft away. The vessel is equipped with adequate heating facilities in the form of a steam jacket and heating coils. The vessel contents are at 60 F before heating. Determine the optimum temperature of this system, so that the total annual cost of transporting the fluid is a minimum. Operating data: C p = 0.85 Btu/lb m F; ρ = 6 lb/ft 3 ; L = 500 ft; E = 0.70; K at 60 F = 0.063 lb f -s n /ft 2 ; at 70 F = 0.03; and at 75 F = 0.02; n = 0.85. Cost data: C s = $/million Btu; C e = $0.076/kWh; anticipated useful life = 0 yr. Solution Using Eq. 3 and fitting a linear relationship to the data yields A = 2.2079 and B = 3.644 0-2 / F. D = 6 in. from the first example. From Figure 9 for a pseudoplastic fluid, t opt, ref = 3 F. Figures 0 6 are used to determine F T F T7, respectively: F T = 68; F T2 = 0.88; F T3 = 38; F T4 = 96; F T5 = 2.24; F T6 = 0.69; and F T7 = 2. Therefore: t opt =3 F 68 0.88 38 96 2.24 0.69 2 = 8.8 F. Pressure drop This method derives from Refs. 3 6. First, from a force balance, the relationship between shear stress τ w and P is: τ w = D P/4L (5) The friction factor is defined as: f = [τ w g c /(ρv 2 /2)] (6) Substituting Eq. 5 into Eq. 6 and rearranging yields: P/L = 2fρ 2 /g c D (7) In the case of the turbulent regime, the expressions developed by Dodge and Metzner () can be used: f = 6/R e (8) We have already shown that: f = a n /N Re b n (9) The effective viscosity of a non- fluid is: µ e = τ w /[8(V/D)] = K g c 8 (n ) V (n ) D ( n ) (20) Writing N Re for a non- fluid as: N Re = DVρ/µ e = D n V (2 n ) ρ/g c K 8 (n ) (2) And using Eqs. 7 2, Figure 7 (laminar flow) and Figure 8 (turbulent flow) were created, which are used to determine the pressure drop for the D opt for a given volumetric flowrate. CEP Literature Cited. Dodge, D. W., and A. B. Metzner, Turbulent Flow of Non- Systems, AIChE J., 5 (2), pp. 89 204 (June, 959). Errata: AIChE J., 8 (), p. 43 (Mar. 962). 2. Skelland, D. P., Non- Flow and Heat Transfer, John Willey, New York, pp. 24 269 (967). 3. Anaya Durand, A., et al., Optimización de Sistemas de Manejo de Fluídos No-os en Tuberías, Memories of the XL National Convention of IMIQ [Mexican Institute of Chemical Engineers], México City (Oct. 2000). 4. Metzner, A. B., Non- Fluid Flow: Relationships between Recent Pressure-Drop Correlations, Ind. & Eng. Chem., 49 (9), pp. 429 432 (Sept. 957). 5. Metzner, A. B., and J. C. Reed, Flow of Non- Fluids: Correlation of the Laminar, Transition, and Turbulent Flow Regions, AIChE J., (4), pp. 434 440 (Dec. 955). 6. Sultán, A. A., Sizing Pipe for Non- Flow, Chem. Eng., 95, pp. 40 46 (Dec. 9, 988). ALEJANDRO ANAYA DURAND (Parque España, St 5B Col. Condesa, México, D.F:, México, 0640; Phone and Fax: 5255-52-0385; E-mail: aanayadurand@hotmail.com) is a professor of chemical engineering at the National Autonomous University of Mexico (UNAM). He has been working as a process advisor to Bufete Industrial, S.A. (from 998 until now) in Grupo Industrial Resistol S.A. de C.V. (from 999 until now) and in Consultoría Empresarial Ejecutiva, S.A. de C.V, He also advises TECHINT, S.A. He has 40 years of experience in process engineering, project engineering and equipment design. He retired from the Mexican Petroleum Institute after 30 years of holding several top-level positions. Anaya Durand has 37 years of experience as a professor of chemical engineering. He holds a master s in project enginering from UNAM. He is a Fellow of AIChE and, in 997, he won the National Award in Chemistry. CINTHYA ALEJANDRA AGUILAR GUERRERO (Miguel Alemán 2a secc. 3-202 Lomas la Trinidad, Texcoco, Estado de México, México; Phone: 5259-5954- 6722; E-mail: cinthyaaag@yahoo.com.mx or ephf@hotmail.com) is a ninthsemester chemical engineering honor student at UNAM. She is an active member of the Mexican Institute of Chemical Engineers (IMIQ). EDGAR AMARO RONCES (Isla Sn. Diego 3 Col. Jardines de Morelos Ecatepec, Estado de México, México; E-mail: edgaramaro@hotmail.com) is a ninthsemester chemical engineering honor student at UNAM. He is an active member of the Mexican Institute of Chemical Engineers (IMIQ). CEP March 2002 www.cepmagazine.org 69