Austin, P.P. (1975) How to simplify fluid flow calculations

Prévia do material em texto

liow to simplify fluid
flow calculations
Solving f/uid f/ow problems has a/ways
been done by s/ow, inefficient trial and
error methods. Here is a way to solve
these prob/ems direct/y, in two steps
Paul Page Austin, Arthur G. MeKee & Co., San Mateo,
Calif.
THE USUAL METHOD of solving a fluid flow problem,
using rational formulas, is by the use of a chart showing
the relationship between the Reynolds number R and the
Fanning Coeffieient or frietion eoeffieient f in the well
known Fanning equation:
f v2 Lh=--D2 g
In most fluid flow problems either the Aow rate or the
hydraulie gradient is unknown. If the flow rate is known,
a two-step solution is made by ealeulating the Revnolds
number, determining the frietion faetor from the R vs f
ehart (Fig. 1) and then ealculating the hydraulic gradient
by placing the value of f in the Fanning formula.
If the hydraulie gradient is fixed by physical eonditions,
and the flow rate is unknown, a trial and errar solution is
usually made by assuming a flow rate, ealculating the
Flg. 1-Friction factor vs. Reynolds number
Reynolds number, determining the friction faetor and
ealculating the hydraulic gradient, whieh must then be
compared with the required gradient.
Realizing that both the Reynolds Number R and
Fanning friction faetor f are really dimensionless parame-
ters, the late Sidney P. Johnson1 reasoned that there must
be three other dimensionless pararneters, eaeh with one or
two variables missing from its equation, the same as in the
R and I e q u a t i o n s. By the applieation of dimensional
TABLE l-Dimensionless parameters pertinent to the problem of resistance to fluid flow in pipes
QD MD QD I u,D uD
!System of units used f!. PIl f!. PIl (hg) Il
\
(hg) Il f!. PIl
Name Variable
Symbols used of not (I) (2) (3) I (4) (5)
formula present
«') D'h IDarey number ( 7r2)~ (7r2 )~ (2) g~h (2)~Fanning IlIII j Coeffieient 8 pQ2 8 M2P 8 g Q2 ; U pu
II2
Poiseuille (t28) !Q:. See II3 See III See III (32)~number P 7r D4f!. D2f!.
II3 R Reynolds p (±) Qp (~)~ (±) Qp Dup Dupnumber 7r J.l.D 11" J.l.D J.I. J.I.
II4 (Q) 1'5 D
Q3p p4 M3pp Q3h/
5 s s=:r:IJ. Il J.I.
3 3 2
II4 (V) T5 D ~ ~
hg Il I!.IJ.
Q
pD
3
P pD
3
P ghD3 p2 ghD3 p2 pD3 PIIó S2 M 2 2 2J.I. J.I. IJ. IJ. IJ.
u
Ali numbers in any horizontal row are the sarne. irrespective 01 the svstern 01 units used. The numerical coefficients in parenthesis are those required to give the para-
rneters the numerical values most frequentlv used in practice. All quarrrit ies are consistently from lhe same evstem. either English or rnetric, in which the force unit
gives acceleration to a unit of mass.
HYDROCARBONPROCESSING September 1975 197
HOW TO SIMPLIFY FLUID FLOW CALCULATIONS
analysisv" he derived the formulas for the three other
dirnensionless parameters.
Two of these parameters plotted versus t numbers, to-
gether with the Reynolds number vs. f chart, makes it
possible to solve any flow problem directly, without trial
and error.
ln 1883, Osborn Reynolds' published his paper in which
he showed the difference between viscous or nonturbulent
fiow and turbulent flow and gave his formula which deter-
mined which of the two conditions existed in a pipe, for
any set of given fíow conditions. Since then many papers
have been published giving curves showing the relation-
ship between the Reynolds number (R) and the friction
factor f. These curves, alI obtained by experiment, were
for various sizes of pipe and were often difficult to corre-
late over the entire range of pipe sizes. Some of them also
had the fault that the lines continued to droop as the
Reynolds numbers became larger, whereas modern curves
of this kind flatten out to constant values of f =. high
Reynolds numbers. ln 1944, Lewis Moody' published his
paper in which he introduced a new variable, inside pipe
walI roughness. This was done by plotting a series of
curves showing the R vs. f relationship, each line for a
fixed ratio of roughness (expressed in thousands of an
inch) divided by inside pipe radius in inches. These curves
flatten out to constant values of f at high Reynolds
numbers.
Final1y in 1943, Hunter Rouse" published his paper in
which he had derived an empirical formula containing
the four variables, R, f, r (pipe radius) and k (roughness).
This formula gives relationships between f and R which
reflect correct values from the lowest values of R in the
beginning of the turbulent region, out to the higher values,
where f becomes constant. This is the set of curves in
Fig. 1. The roughness is that of new clean steel pipe,
k = 0.0018 incho
The advantage of using the Hunter Rouse formula to
produce a set of R vs. f curves (with an assumed fixed
value for k) is that the curves will always be practically
identical, irrespective of who ca1culates and plots them.
Using Buckingham's I1 designation, Table 1 shows the
five dimensionless parametric equations written in various
forms and the variable missing from each.
The first (I1l) and third (I1a) are the f and R parame-
ters, respectively, and the other three are those derived
by Johnson.
Thus, it becomes evident that any two parameters can
be plotted against each other, but it is also clear that most
of the 10 charts that could thus be obtainable would be of
little value.
In any chart of two parameters plotted against each
other, one must be an independent variable, and the other
thereby automatically becomes the dependent variable.
For example, in the R vs. f chart, the Reynolds number
is the one that is always calculated and is thus the inde-
pendent variable and f becomes the dependent variable,
the value of which is required.
The parameter I1~ excludes density. As density does not
influence the head loss in either turbulent or nonturbulent
flow, and since density is never an unknown variable in a
fluid flow problem, no further consideration will be given
to I12•
TABLE2-Rational flow formulas
F1uld Reynolds number Head or preseure lose
Liquide
7742 Dv Aj~fR= h =
11
Q QS BjQ2sLR = C-- = C- P=DI' DIJ. D6
I
Values of A Values of B
Q - Rate 01 Values 01 C Q ~ Rate of
flow in flow in L in miles L in M leet L in miles L in M feet
gpm 3162 gpm 164.3 31.1 71.1 13.47
bph I 2213 bph 80.5 15.25 34.8 6.60bpd 92.2 bpd 0.1398 0.0265 0.0605 0.0115I
Gaeea-
BjZT~~2L*Volume p/ _ P
2
2Baele
C
QG =
R = IJ.D
P = Bj TGQ2L**
2P1D5
Values of B
Q = Rate of fíow Values of C Q == Volume flow rate at standard conditions***
expressed in*** L in miles L in M leet
scfm 29.0 scfm 0.2767 0.0524
Mscfh 483.6 Mscfh 76.56 14.5
MMscf per 24 hrs. 20,150 MMscf per 24 hrs, 133.580 25,300
Wel~h
M2Rate
Any
M Pt = 0.00336 j pD6Fluld
R = 6.32 IJ.D
2
Valid for ali fluids P = 1.294 j V ~L ****
For nomenclature-See Table 4
·See note I, Table 4 ·.See note 2. Table 4 • •• See note 3. Table 4 ·*·*See note 4. TabIe 4 tPressure drop per 1.000 ft.
198 September 1975 HYDROCARBON PROCESSING
With the above facts in mind, it now becomes apparent
"that the only two additional charts really necessary to
make a direct solution of a fiow problem are the inde-
pendent parameters Il, (with diameters unknown) and
Iló (with flow rate unknown) each plotted versus Il1 (I)
as the dependent parameter.
Only occasionally is the unknown variable the pipe
diameter. In the smaller sizes, only commercial sizes are
available so a very rough estimate will give the range of
size within one, or at the most, two commercial sizes. This
parameter will perhaps be most usefui for determining
larger sizes of pipe, where commercial sizes above 42
inches are not available.
Throughout the entire range of turbulent flow, Il1 (I)
varies only from 0.04 down to 0.006 in numerical value.
Its use as the dependent parameter in ali problems of
turbulent How, therefore, besides being consistent with
current engineeringpractice, permits an accurate graph-
ical presentation readable to three significant places if ali
data to be displayed are less than two cycles of logrithmetic
paper. Therefore the use of III (I) as the dependent
parameter in turbulent flow problems is the most con-
venient, and it has been so plotted in Figs. 1, 2 and 3.
These are necessary to maintain equality. Also g, the
acceleration of gravity appears in some equations. Here it
is a dimensionless constant, having the property of force
per unit of mass, which is necessary if h is regarded as a
slope. If h is considered in the nature of energy loss per
unit mass of fluid and length of pipe, which is equally
permissible, g becomes merely a co n s t a n t of propor-
tionality.
There are several interesting singularities in Table 1
where some of the Ils contain only three variables. For
example III can be expressed in three variables, D, Q and
h.•but this is no particular advantage, as III is never cal-
culated as an independent parameter. The parameter Ila
Fig. 2-Friction facto r VS. S number.
(R) can be expressed in terms of mass rate offiow, diame-
ter and viscosity, without p.
This permits the general method of fiow calculation to
be applied to almost any case of gas flow, irrespective of
the pressure drop. When any one value of Il is determined
on a curve, all the others become fixed. In the case of
gases f.J. is a function of temperature but substantially inde-
pendent of pressure or density, and therefore IlJ is con-
stant from one end of a line of uniform diameter to the
other so long as the temperature in the line is fairly con-
stant. All the Ils are constant if the expansion of gas as it
fiows through a line is isothermal.
The scale of f has been made rather large so it can be
easily read to three significant figures. The horizontal
length of 4 cycles has been compressed into about 1.5
times the one cycle vertical scale. This makes for the
smooth line slope downward roughly 30 degrees and gives
the greatest accuracy of representation possible in a given
amount of chart space.
Figs. 2 and 3 do not show TI" and TI., respectively, directly
in the form given in Table 1, but instead show the square
TABLE3-Dimensionless parameter formulas with coefficients for use with commercial (engineering) units
Fluld Parameter
T number T = (Ilj/5 S number S= (Il,/12
Liquida (Q3p 4) 1/5 (Q3 hs5) 1/5 (pD3)1/2 (hD3)1/2T = F = N-- S = X IIS1/2 = y--IJ. IJ. 11
Values of F Va!ues of M Values of X Va!ues of Y
Q - Rate of
L in rniles L in M feetflow expressed in L in miles L in M feet L in H rniles L in M feet L in rniles L in M feet
gprn 10.1 14.1 8.58 12.0 2.65 6.09 1.74 4.01
bph 8.19 11.4 6.93 9.66
bpd 1.22 1.70 1.03 1.44
Gases
[ ( P/
T
-: :22) Q3G4J 1/5 S ( p/ _ p/ yl2 D3/2Gl/2T= = X TOL IJ.F
IJ.
Values or F Va!ues of X
Q = Ra te of flow expressed in L in rniles L in M feet L in miles L in M feet
scfrn 0.282 0.394 0.390 0.895
MMscf per 24 hours 14.3 20.0
Vapors ( 3 ) 1/5 ( 3) 1/2Steam T = 0.149 0!J!pJ... S 0.771 P pD=IJ. IJ.
For nornenclature-See Tab!e 4
HYDROCARBON PROCESSING September 1975 199
OW TO SIMPLlFY FLUID FLOW CALCULATIONS
Fig. 3-Friction factor VS. T number.
root of lI, designated S and the fifth root of TI. designated
T. The reason for this alternation in the first place is to
give S and T ab o u t the same magnitude as R, the
Reynolds number. The S number has roughly one-seventh
the value of R, while the T number has roughly two-fifths
the value of R. In the second place, the numerical values
of both values TI,. and IT. are uncomfortably large to
handle even with the notation of powers of 10. In particu-
lar, the maximum value of IT I looks like a figure of astro-
nomical magnitude. Fractional exponents introduced by
using S instead of TI" are no great objection, as thev can be
read easily from a slide rule. Use of the fifth root of TI. is
not quite so easy to calculate on a slide rule, but actually
T will not be used as often as S,
The Hunter Rouse formula contains the variable k
that takes care of and varies the value of f, according to
the pipe wall roughness. But since nearly ali fluid flow
ca!culations are made for new steel pipe, having a K value
of 0.0018 inches, Fig. 1 was calculated and plotted for this
value of k, When calculations of fiuid flow are made for
pipe either rougher or smoother, a simple ca!culation can
be made to obtain proper value of f·
Table 5 gives the range of pipe roughness for the vari-
ous types of pipe in commercial use.
Riveted pipe has of course not been used for many
years, but is shown here as ao example of high roughness
pIpe.
Suppose the pipe to be used is 12-inch galvanized. The
roughness of galvanized pipe k = 0.006 inches. Then the
relative roughness diameter of 12-inch galvanized pipe is
12 X 0.0018 = 3.6 inches. Calculate the Reynolds num-
0.0060
ber for the fiow conditions on the 12-inch pipe, but read
off the friction factor for a pipe diameter of just under
4 inches.
Fig. + is a chart on which the relative roughness diame-
ter can be graphically obtained for any size pipe.
APPLlCATIONS TO WATER HAMMER CALCULATIONS
Howard Moere" has found a uniqueapplication for the
S number in connection with pipe line water harnmer
calculations, When a centrifugal pump is discharging up-
hill through a pipe line and the pump stops, reversal of
flow and the sudden closing of the check valve at the
pump may cause severe water hammer in the line. Severity
of the stress set up by the water hammer is a function
primarily of water velocity, time the check valve takes to
elose. length of line and severa I other factors. When the
pllmp stops the velocity in the line quickly drops to zero,
and then reverses, increasing until the line friction or head
loss is equal to the hydraulic gradient. At this point equi-
librium is established, and the velocitv can go no higher.
The check valve can, of course, elose and probably will,
before maximum velocity is attained. but there is no way
of determining the velocitv at the time of closure except
by experiment on the completed line. However, maximum
severity of water hammer can be ca!Culated by assuming
that valve elosing takes place when the back flow velocity
reaches its maximurn possible magnitude.
Nomenclature
TABLE4-Nomenclature, abbreviations and notes for Tables 2 and 3
Abbrevations. etc.
D = Internal pipe dia meter. in.
Q = Volume rate of fiow. See Tables 2 anel 3 for units
.\1 =- Mass rale of fíow, lbs. per hour
p -= Total pressure 10505 in psi in lerigt h of Line L
h = Total head loss in it. of liquid in length of Line L
P = Total loss of pressure in psi in length of Line L
(For gas flow only)
L = Length of pipe in thousands of it. or miles
gpm
11 = Velocity of fiow in fps = 0.408-
D2
TO = Absolute temperature of flowing gas. °F + 460 = T
G = Specific gravity of gas referred to air as unity
P, & P2 = Initial and final pressure psi absolute of gas fiow in line of
of Length L
s = Specific gravitv of liquíd referred to water as unit v
PV
Z = Compressibilitv factor for gases = - at average fiow conditions
KT
K = Gas constant
". = (mu) = Absolute viscositv in centipoises
\I = (nu) = Kinematic víscosit y in centistokes
p = (rho) = Densit y flowing Iiquid in lbs. per cu. ft.
gprn =- Gallons per minute
bph = Barrels 142 ~al.) per hour
bpd = Barrels (42 gal.) per 24 hours
M = 1.000
scirn = Standard cubic feet per minute
Mscf h = Thousands ai standard cubic feet per hour
r...1~lsi per 24 hrs. = Millions or standard cubic ieet in 24 hours
CONST.-\:-.ITS USED
G is taken as 62.32 ft./sec.2
Densit y of water = 62.37 lb./cu. ft. @ 60° F
NOTES: See Table 2
O) Compressibility facto r Z wi ll normally be uni t y for normal
pressures, If pressure is hig h, select value of Z from cornpressibilit v
curves for zasbei ng pi ped ,
(2) This formula valid for short tine lengths where pressure drops do
not exceed 0.10 of final pressure.
(3) Standard conditions for fiowing gas are 14.72 psi absolute and 60° F.
(4) Valid for incompressible fíuids.
For steam or other vapor. accu ra te only if PI - P2 is less than 0.1
of P2. For other conditions use formula for gases. volume basis.
200 September 1975 HVDROCARBON PROCESSING
Absence of flow rate or velocity from the formula for S
makes it possible to calculate the maximum velocity with-
out trial and error calculations.
(hgD3p2)Y, (hD3)Y,S = = 4.01 "'-----'--p. v
In the above equations, the first having no numerical
coefficient is in consistent units, and the second is in the
usual engineering units defined in Table 4. Here h is the
average rate of head loss over the entire length of the
line. Having calculated the value of S, the friction factor
f can be read off on Fig. 2. Then the Reynolds number
for the same value of f can be read off of Fig. 1 or calcu-
lated by the formula:
Having the Reynolds number the maximum flow rate can
be calculated from the formula Q = :1~;where Q is in
gallons per minute and from Q the maximum velocity
can be calculated.
With the maximum possible velocity in hand, the pres-
sure rise at the instant of closing can be calculated from
any one of six or more empirical Iorrnulas.!" or if it is de-
sired to assume instantaneous closing of the check valve,
the head rise is:
where:
h= av
g
h = Head rise in feet
a = Surge wave velocity, fps
v = Maximum possible velocity in fps
g = Acceleration of gravity.
From the rise in head, the resultant rise in pipe stress can
be calculated.
RELATIONSHIPS OF PARAMEnRS
between theThere are three interesting relationships
parameters which may at times be useful.
(1) S= R~~
(2) T = .li. (8 7T3 f) 1/54
(3) _ 7T
2 (T )10/3f-- -8 S
Formula 1 was used for calculating the S numbers for
Fig. 2 and Formula 2 was used for calculating the T num-
bers for Fig. 3.
About the author
PAUL P. AUSTIN is senior engineer with
Arthur G. McKee & c«, Weste'/'n
Knapp Engineering Division, San
Mateo, Calif. His duties include specifi-
cation writing, engineering calculations
and evaluation of competitive bids. Mr.
Austin holds an AB deçree in mechani-
cal engineering [rom M.I.T. Past projee-
sional experience includes over 30
. yearsin mechanical engineering activi-
ties including both domestic and foreign assignments. He is a
member of ASME and is a registered professional engineer
in California.
HYDROCARBON PROCESSING September 1975
Fig. 4-Correction for pipe roughness other than new steel pipe.
TABLE5-Typical interna I roughness of commercial
pipe
Klnd o( pípe Internal rougbnese K Ia.
Riveted pipe .
Concretc pipe .
Wood 8tave ..........•............... · ·
Cast íron .
Gal vanized .
Asphalted cast iron .
Smooth rubher hose ' ............•
New steel pipe (base) . . . . .
Drawn tubing (copper or steel) .
0.36 to 0.036
0.12 to 0.012
0.036 to 0.0072omo
0.006
0.0049
0.0023
0.0018
0.()()()()8
NOMENCLATURE
For Table I, units must be consistent, i.e., metric or all
English. For engineering units nonnally used, see Nomenclature
for Table .J..
D = Inside diameter of pipe
Q = Flow rate. volume per unit of time
M = Mass rate of flow, per unit of time
P = Pressure. absolute (for gases)
p = Pressure, gage
v = Mean velocity of fluid over cross-section of pipe
L = Length of pipe line
x = Variable length along line
h = - :~ Head 1055 rate or hydraulic gradient
dp 1 diP = -"'dx' Pressure oss rate, or pressure gra ient
S = Specific gravity of a liquid referred to water as unity
g = Specific gravity of a gas, referred to air as unity
p (rho) = Density of fluid
p. (mu) = Absolute viscosity of fluid
v (nu) = Kinernatic viscosity of fluid
p. = s p:
g = Force of gravity per unit of mass, 32.2 for English units.
n, rr, rr, rr, rr, I,R, S, T, = Dimensionless parameters defined in
Table 1.
LlTERATURE ClTED
'S. P, Johnson, A Survey of Flow Calculation Methods, ASME .Summer
Meeting of Aeronautic and Hvdraulic Divisions, Stanford University, June
_ 19, 20, 21, 1934. •
• Hunter Rouse, Evaluaton of Boundary Roughness, Proceedings of the Second
Hydraulic Conference, University of Iowa BulIetin No. 27, Published by the
urnversrtv, 1943. I
3 Buckingham , "The TI Theorem." The Ptvysicai Review (London), Vol. IV,
page 345, October 1914.
• Osborn Reynolds, Philosophicol Transaetions of lhe Royol Socie/y of London,
Vol. CLXXIV, 1883, page 975.
'Howard Moore. Analvsis and Control of Hvdraulic Surge (page 32), pub-
lished by Magnilastic Division oí Cook Electric Co., Chicago, lIlinoi s,
• Stanton and Pannell, "Similarity of Motion in Relation to Surface Friction
of Fluids," Phllosophical Transactions of Royal Soeiety, A·214. 199-191'l.
1 Lewis F. Moody, "Friction Factors for Pipe Flow," ASME Transaetlons,
November 1944.
'William H. McAdams, Heat Transfer, 3rd Edition, Chapter V-Dimen·
sional Analysis, McGraw·HilI Book Co.
• Piggott, R.V.S .• ASME Transactions, Vol. 55, l!ln.
'o AWWA Steel Pipe Manual MIl, Chapter 7, Water Hammer and Surge. •
201