FLUID-FLOW DESIGN METHODS R. P. GENEREAUX E. I. du Pont de Nemours & Company, Inc., Wilmington, Del.
to 4000) in which flow changes from viscous to turbulent or vice versa, and above Re = about 4000 lies the turbulent region. Most plant flows are in the turbulent region, for which the theoretical relations are not so well known. Pipe wall roughness does affect the friction factor. The plot of f vs. Re data forms a relatively narrow band indicating a curve of negative slope, the slope decreasing as the Reynolds number increases. No such simple and accurate formula as that for viscous flow has been obtained. However, two methods are used which give adequate accuracy for design purposes.
INCE most chemical engineering plant designs require consideration of fluid transportation, a familiarity with the fundamental principles involved is of considerable value in obtaining suitable and economic results. In the following text certain fundamental principles are adopted for use in solving fluid-flow problems.
Calculation of Flow in Pipes The most common problem is the determination of pipe size and pressure drop. Many of the publications cited in a bulletin of the National Research Council (2) describe adequately the use of the Reynolds number and the Fanning equation which indicate that all Newtonian fluids1 follow the same laws of flow. Thus, a single type of relationship may be used for solving problems of flow in pipes. The Fanning equation may be expressed as : A P = 4 fLp V 2 / 2gD
(1)
and the Reynolds number as: Re = DVp/p = D G / p
(2)
Any consistent set of units, such as feet, pounds, seconds, may be used in these equations. By transposing Equation l to obtain f, we find by substituting units that f is dimensionless. This is also true of Re. A plot on double logarithmic paper of f and Re values calculated from the available pressure drop data exhibits the following characteristics, Below Re = 2000, the data fall on a line expressed by the formula
f
=
(3)
16/Re
which, when substituted in Equation 1, resul s in the Poiseuille equation for viscous flow:
/
which has been proved theoretically (4). This equation holds for all circular pipe, whether smooth oq the so-called commercial pipe. It is simple and accurate, Above Re = 2000 lies the “critical” region (Re = 2000 1 Newtonian fluids are those whose values of shearing stress us. rate of shear, when plotted on arithmetic paper, follow a straight line running through the origin; i. e., the viscosity does not vary with the rate of flow. This class includes all the gases and most of the liquids. A further desoription is given on page 1271 of Perry’s handbook (4).
385
T
INDUSTRIAL AND ENGINEERING CHEMISTRY
386
TURBULENT REGION
DIAMETER
WEIGHT FLOW
MASS VELOCITY
PIPE FLOW CHART
VOL. 29, NO. 4
BASED ON CLEAN STEEL PIPE
TEMPERATURE:
PRESSURE DROP
DECREES CENTleRADE
(CENT IPOISES)OJ~ LBS/CU.FT. AT I ATM. 70.16
ACTUAL INSIDE THOUSANDS OF TWUSANDS OF IIAMETER I N INCHES POUNDS PER HR POUNDS PER HR.
100
5 0 3
0.5
05
0.1
0.001
ao I
0.0051
0 . mI
0.001-J
FIGUR 1~ To determine the pipe size for calcium chloride brine LIQUIDEXAMPLE. To obtain pressure drop per foot of pipe for air a t 100' C. G a s EXAMPLHI. (25 per cent solution) a t 20' C. flowing a t 40,000 pounds per hour with an and 10 atmospheres absolute pressure flowing a t 250 pounds per hour i n a allowable pressure drop per foot of pipe length A p / L = 0.01 pound per 1.05 to m = 0.25, crossing reference 1-inch standard pipe: Conneot D square inch per foot: Connect 20' C. t o point 20, reading 0.016 on the 29 t o read Zo.*@/p = line a t 7.1. Connect l o O D C. t o molecular weight ZO.l@/pline, Connect this value t o A p / L = 0.01, crossing the reference line 6.2. Connect thie value to the 7.1 on the referenoe line and read 0.05 on et 11.25. Connect this value t o m = 40, orossing D 2.8, which indicates the A p P / L line. A p / L = 0.05/10 = 0.005 pound per square inch per a 3-inch pipe. foot of pipe length. K H ~ TO Y LIQUIDS 14 Hydrochloric acid, 31.5% 23 Sulfur dioxide 2 Butanol 15 Acetic acid 100% 17 Sulfuric acid, 111% 11 Acetic acid: 70% 20 Calcium chloride brine, 25% 24 Mercury 5 Methyl alcohol 100% 18 Sulfuric acid 9 8 7 8 Ether 7 Acetone 22 Nitric aoid 9 5 9 19 Sulfuric acid: 7 8 d 10 Ammonia, anhydrous 13 Ethyl acetate 21 Nitric acid: 6 0 8 6 Turpentine 4 Aniline 3 Ethyl alcohol, 95% 16 Water 1 Octane 9 Benzene 12 Glycerol, 50%
-
-
I n the first method the tools are the Fanning equation and the friction factor plot (f vs. Re). I n calculating pressure drop, Re is calculated according to Equation 2 and f determined from the plot. Pressure drop is then calculated by means of Equation 1. The equation and plot may be expressed in any convenient set of units. When calculating pipe diameter or pipe capacity (rate of flow), this method necessitates trial and error, since these variables are included in Reynolds number. In the second method the data are expressed by an equation for f which is then substituted in the Fanning equation. Numerous expressions for this relation have been offered in the forms: f = a b R P (a curve) (5) f = aRen (a straight line) (6)
+
Although Equation 5 fits the data curve more closely than does Equation 6, a cumbersome equation of additive terms results on substitution. Approximations of the type of Equation 6 when substituted give simple equations sufficiently accurate for plant design, provided the error is on the side of safety-i. e., that a higher pressure drop, larger pipe size, or smaller flow is indicated. The derivation of such approximation and its application to Equation 1 is described below. In discussing a recent paper by Miller (3) Drew and Genereaux (1) submitted a plot of the majority of the available data for the turbulent region. The plot was given in a different dimensionless form, f-o.6 vs. (Re)(fo.7, in keeping with the substance of the paper which concerned the use of the von Karman number, (Re)Cf0.6). To avoid recalculation and plotting of all the data on the f vs. Re basis to obtain an approximation according to Equation 6, the following method was used: A line was drawn on the "safe" side of the data on the von Karman plot; the equation of the line is f - 0 . 5 = 2.7 loglo(Re)(f0.6) 2.2. Values o f f and Re were calculated from this equation, as given in Table I.
ECONOMIC PIPE DIAMETER NEIGHT FLOW THOUSANDS OF POUNDS PER HOUR
DIAMETER INSIDE DIAMETER Of PIPE I N INCHES
D
+
TABLEI. VALUES CALCULATED FROM f - o . s p . 5 )
10.3 13.0 15.7 18.4
1,000 10,000 100,000 1,000,000
f 0.01324 0.009164 0.006340 0.004386 0.003035
0.0094 0.0059 0.00405 0.00295
10,300 130,000 1 570 000 18:400:000
P.6
0.1151 0.09574 0.07964 0.06624 0.05509
c 1-20
TABLE11. VALUES CALCULATED FROM f Re 1,000 10,000 100,000 1,000,000 10,000,000
= 2.7 loglo(Re)
+ 2.2
=
f -0.6
8.688 10.44 12.56 15.10 18.15
0.04 Re-o.16 (Re)(fW 115.1 957.4 7,964 66,240 550,900
When plotted on double logarithmic paper, these values gave a curve to which was fitted a straight line, the formula for which is:
E
f = 0.04 Re-O.16 (7) Values of and (Re)(fo.6) were calculated from this expression (Table 11) and, when plotted on the von Karman plot, gave a curve which represented the data equally as well as the curve originally drawn. Equation 7 was then adopted as being amply correct for engineering computations. It holds for the Reynolds number range of 4,000 to 20,000,000 (the entire range of available data). Substitution of Equation 7 in Equation 1 gives, in consistent units (such as feet, pounds, seconds), f-Oa6
A P = 0.00249
LpO.16 p0.84 Vl.84/D1.16
L Z0.l6rn1.84/pD,4.84
0.1
o.mi
FIQURE 2 EXAMPLE.T o find the economio pipe diameter for handling 1500 pounds per hour of air at 60 pounds per square inch gage pressure and 20' C.: Calculate the density = 0.34 pound per oubic foot and connect this value to m = 1.5, reading diameter D = 3.8,and use a 4-inoh pipe.
(8)
which can be rearranged to determine any one of the variables when all the others are known. The exponents serve to indicate the relative effect that changes in a given variable will have on the others. The terms in Equation 8 can be transformed to any units desired, for example: Ap = 0.1325
tl-
matic viscosity, Z o J 6 / p . A scale of mass velocity, M , has been added to permit its determination and provide for using it with Di or M when M or Di, respectively, is unknown. Temperature and molecular weight scales for gases are given in the form of a line-coordinate chart by which values of Z O . l 6 / p are determined directly. For liquids a temperature scale and points for various liquids are given. Points for liquids not listed may be located by determining the intersection of lines connecting corresponding values of Z o . l 6 / p and temperature. The values of Z0.I6/p obtained from Figure 1 may be in error by not more than 2 per cent. Examples of calculations are given underneath the chart. Calculations in the viscous region may be made directly from Equation 4 converted to any convenient units. The critical region is generally treated as a part of the turbulent
(9)
Pipe Flow Chart The alignment chart shown in Figure 1 was constructed from Equation 9. It affords rapid calculations, permits determination of the effect of altering the value of any variable, and avoids the lengthy solution of fractional exponents. The chart consists essentially of scales for pipe diameter Di, weight rate of flow m, pressure drop Ap/L, and a kine387
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INDUSTRIAL AND ENGINEERING CHEMISTRY
388
region, and Equation 7 can safely be used down to Re = 2000.
Slope n of the cost curves for steel pipe is essentially 1.5, giving upon substitution and solution for D: 20.025 Di = m0.448 P ~
Economic Pipe Diameter The selection of the diameter of pipe for greater economy in meeting the conditions of flow is often made on the basis of a narrow range of velocity which is considered as the optimum. Although this method is generally correct for the common fluids, water, air, and steam, experience with other fluids which occur in chemical processes is not so general, and application of data used in the velocity method for common fluids might result in error. The factors to be considered are cost of pipe (which varies directly with diameter) and cost of pressure drop (pumping or blowing costs) which varies inversely with pipe diameter. A general expression for the economic diameter can be determined by balancing these costs. The annual cost of pipe may be expressed by the following equation: Cpipe
(a
+ b) ( F + 1 ) XDin
(10)
The annual cost of pressure drop is evaluated by determining the cost of compressing gases or pumping liquids to overcome the pressure drop. This cost is eliminated when not charged to the operation, as in drawing water from a main. Steam is a separate case not to be covered here, since the pressure drop involves a loss in available energy the value of which varies according to the pressure and temperature level of the steam. The annual cost of compressing or pumping may be expressed as follows: For pumping liquids, the product of flow and pressure drop is an exact expression for the work done. For gases this is not exact. The percentage error in assuming this to be true for compressing gases is approximately half the percentage pressure drop. Thus, if pressure drop is 10 per cent of initial pressure, the error in assuming Q A p equal to the work of adiabatic compression is approximately 5 per cent. For the sake of simplicity, however, this approximation will be made:
ft*lb* = 1000 Q 144ApY -
year kw-hr. -= year dollars -= year
=
1000 9 144ApY P
144,000 mApY 2,654,200 Ep 0.0542 mApYK EP
Substituting for Ap pressure drop Equation 9, the annual cost of pressure drop per foot of pipe is:
The total annual cost in dollars per foot of pipe for both pipe and pressure drop is the sum of Equations 10 and 11:
Differentiating total annual cost with respect to the diameter:
doi cT = n(a + b) (F + l)XDin-l+
(-4.84) 0.0072m2.B42 0 . 1 6 Y K Di5.84 p2E
Equating dCT/dDi to zero to obtain minimum cost:
0.0232 Y K
*
[ ( a~
+ b) (F + 1 ) XE] ~
0.168
~
(14)
The terms outside the brackets determine the flow and those inside determine the fixed cost. The cost expression can be evaluated according to actual conditions, but for simplification extreme and normal values have been taken to determine an average value to the 0.158 power. The values obtained were 1.7 and 2.6 for the extremes and an average of 2.2 is used below. Further simplification of Equation 14 is made by neglecting Zo.02s which is approximately equal to unity for most values of Z. The simplified equation thus becomes: = 2.2mo.44a/pa.ai6
(15)
This equation is adequate for ordinary plant conditions and for fluids whose viscosities lie between 0.02 and 30 centipoises. An alignment chart constructed from Equation 15 is given in Figure 2. It is economical to select the standard pipe size above the actual diameter obtained on the chart.
Nomenclature The terms for which a consistent set of units, such as feet, pounds, seconds, should be used are: AP = pressure drop, Ib./sq. ft. f = Fanning friction factor, dimensionless L = length, ft. p = density, lb./cu. f t . V = velocity, ft./sec. g = acceleration of gravity, ft./sec.a D = pipe diameter, ft. Re = Reynolds No. p = viscosity, Ib./ft. sec. G = mass velocity, lb./(hr.) (sq. ft.) = Vp Terms used in the text with units other than those given above are: A p = pressure drop, lb./sq. in.
weight rate of flow, thousands of lb./hr. pipe diameter, in. = viscosity, centipoises M = mass velocity, thousands of lb./(hr.) (Hq. ft.) a, b, n = constants m Di 2
= =
Additional terms used in evaluating economic pipe diameter: a = amortization (expressed as a fraction, i. e., 0.10 for 10 per cent) b = maintenance (expressed as a fraction, i. e., 0.01 for 1 per cent) C = annual cost Cpip,= annual cost of pipe Cp,d.= annual cost of pressure drop CT = total annual cost F = factor for fittings and erection (i. e., if these items cost the same as the pipe, F = 1 ) X = cost of 1 in. diameter pipe/ft. length Di = nominal diameter of pipe, in. = slope of plot of diameter us. cost of pipe (1.5 for steel n pipe) Q = thousands of cu. ft./hr. Y = hours of o eration per year E = over-all efKciency of motor and pump or blower, fractional K = cost of electrical energy, dollars per kw-hr.
Literature Cited (1) Drew,
T. B., and Genereaux, R. P., Trans. Am. Inst. Chem.
Engrs., 32,17-19 (1936).
Equation 4 is the general form for economic pipe diameter, including all the terms considered to be important. Certain assumptions are made in the following simplification
(2) Dryden, Murnaghan, and Bateman, Natl. Research Council, BUZZ.84 (1932j. (3) Miller, B., Trans. Am. Inst. Chem. Engrs., 32, 1-14 (1936).
(4)Perry, J. H., Chemical Engineers' Handbook, New York, McGraw-Hill Book Co., 1934, RECEIVED February 24, 1937.
