Direct Solution of Isothermal An Improved Method Flow in Pipes B. F. RUTH Iowa State College, Ames, Iowa
A
MONG the large number of possible plots which may
be constructed from appropriate pairs of dimensionless ratios for the solution of the pipe flow problem, the set obtained by plotting:
for velocity unknown, T’ = Va/fp
(7)
Less usual cases take quite as simple forms. For example, if the maximum permissible horsepower P = Ap &/550, and if the diameter and the nature of the fluid are specified, the rate of flow turns out to be:
aa ordinates against the quantities Re/fp, R e / d & Re, R e a R e 6 and Refp as abscissas most nearly satisfies the following desirable objectives:
Q
=
d550
(&)
128IrgD4 Lp
=
Qm/flp
(8)
I n each of these cases the quantities with the subscript s are readily calculable because all the variables in the HagenPoiseuille formula except that on the left-hand side of the equation are known. From Equation 3 i t follows that
1. The solutions for all important quantities should be ob-
tainable from a single chart. 2. The numerical magnitude of the dimensionless ratios should not be so large as to hinder calculation. 3. The various dimensionless groups should be so related that formidable and complicated expressions are avoided. 4. The same procedure should be employable in the solutions for all unknowns. The group fp may be called the “streamline friction factor” because it is actually the factor by which the pressure drop as computed by the Hagen-Poiseuille law must be multiplied in order to obtain the true pressure drop. I n other words, f p is the ratio of the Fanning friction factor, f, to the value fa obtained for the same Reynolds number by extrapolating the line f = 16/Re. I n the streamline regime the value of f, is evidently unity; in the turbulent regime its value is greater than one. If Equation 1A is solved for the viscosity p ,
Therefore this group is merely the Reynolds number with p a written in place of p. If, therefore, a plot of f P against Re/fp is available, fp is easily found from the chart when p is unknown, and p can be calculated from Equation 3. Analogously, plotting f p against
which may be written !-4
(3)
= PJfP
if pais used to denote the value of p computed on the assumption that streamline flow actually obtains for the conditions a t hand. Similarly, one finds for diameter unknown with velocity given, D = DdvdZ
permits the direct determination of fp and consequently of the required unknown for cases corresponding to Equations 4,4A, 6, and 8. The case of V unknown is the same as that of Q unknown. For the construction of the chart shown in Figure 1. the equations of the von Karman type,
(4)
for diameter unknown with rate of flow given, D =
1 = 4.0 (log Re& (44
(12)
for smooth tubing (lines B ) and
for pressure drop unknown, AP = APa
fP
1 = 3.2 (log R e d j j 4
(5)
for rate of flow unknown,
Q
- 0.4
d f
Qdfp
+ 1.2
(13)
for commercial iron pipe (lines A ) , were adopted, and fp was computed by means of the relation:
(6)
1 Presented as part of the symposium on fluid dynamics at Carnegie Institute of Technology, Pittsburgh. Other papers appeared in April, pages 408-486,in May, pages 618429,and on pages 972-984 of this issue.
fP
985
=
f1 Re 6
VOL. 31, NO. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY
986
UNKNOWN
I
/08
I
M9
Rk-HVOLDS MODULUS nvD, AND SUBS7/7U7-,ED S7-RE.M.L/NF MODUL L /u
FIGURE1.
STREAMLIND FRICTION FACTOR-CHART
Equation 12 is that found by Nikuradse; Equation 13 is that recommended by Drew and Genereaux (1). The use of Figure 1 is extremely simple. Although a tabular key of the most useful relations has been placed for convenience in the lower right-hand corner, such a key is actually unnecessary, since all of the required relations are easily memorized or may be derived direct from the definition of the streamline friction factor given in Equation 1. The direct solution for any unknown may be carried out in four simple steps : 1. By means of the Hagen-PoiseuilIe law, the theoretical streamline value of the desired unknown is calculated. 2. A streamline modulus is evaluated by employing in Reynolds modulus the theoretical streamline value obtained in step 1 in place of the actual (unknown) value. 3. The corresponding value of fp is next read from the appropriate line on the streamline friction chart illustrated in Figure 1. 4. The actual value of the unknown quantity is obtained by multiplying or dividing the streamline value secured in step 1 by jp, or fias indicated by Equations 3 to 8.
qz
Problems illustrating the use of the chart are given at the end of this paper.
Completely Algebraic Solution The streamline friction factor lines of Figure 1 all exhibit a slight curvature up to a n equivalent Reynolds number of about 200,000, after which they become practically straight. This behavior suggests that the plots might be straightened by plotting (f, - a ) against Re, where a is a suitable small constant found by trial and error. Line A in Figure 2 shows the data of Nikuradse (3) plotted as (f, 0.5) against Re. Although the plot is apparently straight, a slight double curvature is actually present, as may be seen by sighting along the foreshortened plot. T h e straight line drawn through the points has the equation:
-
f,
= 0.5
+ (12e/1700)0*817
(15)
Values of fp calculated by means of this equation do not differ from the experimental values by more than 6 or 7 per cent, even in the zone of maximum departure a t Reynolds numbers of 10,000 to 20,000. Throughout a considerable portion of the range the agreement is within 2 or 3 per cent. Perhaps the most interesting feature of this plot is the fact that i t apparently permits the safe extrapolation of fp out to values of Reynolds modulus far beyond the experimental limit of Re = 3,240,000. This conclusion is based upon the observation that the points above a Reynolds number of 100,000 lie on a very good straight line. Examination of those points in Figure 2 corresponding t o 10-cm. tubing only, shows that although the slope of the indicated line is not exactly the same as that drawn through all the points, the departure is nevertheless so slight as to make it appear that Equation 15 ought not to be in error by more than 2 or 3 per cent a t Reynolds numbers well above 10,000,000. To show the agreement between Equation 15 and the von Karman type equation for the copper line, the values which were calculated from Equation 12 for use in construction of the streamline friction chart are also shown in Figure 2 as the points consisting of solid circles centered upon crosses. A similar plot for the data calculated by means of Equation 13 for commercial iron pipe is given in line B of Figure 2. The equation of this line is f , = 0.5
+ (Re/1520)0.a38
(16)
and is in equally good agreement with the calculated values. B y substituting f R e / l 6 for fp, Equations 15 and 16 may be transformed into relations giving the Fanning friction factor in terms of Reynolds number, The resulting equations differ from the form, f = a b/Ren, introduced by Lees ( 2 ) only in the substitution of a variable term a/Re for the constant. The equation for copper tubing and smooth pipes becomes: f = 8/Re 0.0366/Re0.18a (17)
+
+
T h a t for 2-4 inch new commercial iron pipe is:
AUGUST, 1939
INDUSTRIAL AND ENGINEERING CHEMISTRY
f
= 8/Re
+ 0.03446/Re0*'62
(18)
The advantages of Equations 17 and 18 need not be stressed. It is clear that direct solutions for pressure difference may be obtained by using either the Fanning equation and a value off given by Equations 17 and 18, or by using Equations 15 and 16 instead of Figure 1 to evaluate f p in step 3 of the general solution procedure. Equations for the streamline friction factor as functions of the various substituted streamline moduli are also possible. To illustrate this point, the values of fp and Re obtained from Equation 12 have been plotted in Figure 2 (lines C and D)as (fp - 0.5) against both R e a and Refp. Except for the points corresponding to a Reynolds number range of Re = 4000 to 8000, the plots are quite straight and intersect the abscissa a t the points R e a = 1700 and RefP = 1700. The resulting equations are: fp fp
+ (4Qp/1700npD8)0.680 = 0.5 + (4Qap/17OOnpD)0**6~ 0.5
(1%
(20)
Equation 19 permits a direct solution for unknown pipe diameter without the aid of the streamline friction factor chart. Similarly, Equation 20 permits a direct solution for the rate of flow. Corresponding plots for the commercial iron pipe line have not been given in Figure 2 on account of the confusion that would have resulted. However, it is evident from the similarity of lines A and B that linear plots of f p - a against R e e P and Refp for the iron pipe line data are to be expected, and this is found to be the case. For the iron line data, better agreement a t low Reynolds numbers is secured if the value of a is taken somewhat less than 0.5. Recommended equations are: fp
fp
+ (4Qp/l52O?rpDa)O~~9~O = 0.2 + (4Q.p/1520~pD)0.455~
= 0.4
(21)
cal engineering purposes. Like Equations 15 and 16, Equations 19 to 22 appear to permit a safe extrapolation for fp out to Reynolds numbers far beyond the present limit of the experimental range. Because of their relative unimportance, plots of f p - 0.5 against Re/fp, R e / d g and R e d z have not been given. However, with the exception of the group Relf,, very fair linear agreement may be secured; it is thus possible to write Equations 15 to 22 in a generalized form as follows: fp
=a
+ (Re,/bP
(23)
The symbol Re, represents collectively the different streamline moduli that result when the streamline value of an unknown is substituted in Reynolds modulus for the unknown (actual) value of this quantity. It should be pointed out that Equation 23 is an empirical form having no apparent theoretical basis, and is proposed only because it affords a general correlation of all five of the fundamental dimensionless ratios a t present utilized in describing the isothermal flow of fluids through long pipes, with sufficient accuracy for most engineering purposes. At the same time it is simple and explicit in form, an advantage which is not shared by equations of the von Karman type.
Use of the Streamline Friction Factor Chart and Equations EXAMPLE I. UNKNOWN PRESSURE DROP.It is required to determine the pressure gradient necessary to transport water at
68" F. ( p = 1.0 centipoise or 0.000672 pound/foot X second, p = 62.3 pounds/cubic foot) through a horizontal standard 4-inch iron pipe (actual diameter 4.026 inches, D = 0.3354 foot) at the rate of 1.50 cubic feet/second. 1. The theoretical streamline pressure gradient A p . / L is
calculated:
APJL = ??%
rgD4
(22)
The values of fp given by these equations agree with the values of fp read from the corresponding lines in Figure 1 to within 3 or 4 per cent, and are thus sufficiently accurate for all practi-
987
2.
0*00067i3
= 128 A X 32.2 X (0.3354)
=
0.1005 lb./sq. ft.
Reynolds modulus is determined: R e = -~ =Q
P
TIAD
n X
4 X 1.5 X 62.3 0.000672 X 0.3354 = 528'000
INDUSTRIAL AND ENGINEERING CHEMISTRY
988
3. The streamIine friction factor is read on the pressure-drop unknown line for iron pipe a t this value of Re as f p = 133,or it may be calculated by means of Equation 16: f p = 0.5 (Re/1520)0.838 = 0.5 (347.4)0.8ss = 135.4 4. The actual pressure gradient is determined by means of Equation 5: A p / L = Apafp = 0.1005 X 133 = 13.37 Ib./sq. ft./ft. of pipe EXAMPLE 11. UNKNOWN PIPEDIAMETER.It is required t o find the diameter of the iron pipe that will carry 1.5 cubic feet/ second of water at 68’ F. under a pressure gradient of 13.37 pounds/square foot/foot of pipe. 1. The diameter of the pipe that would carry this quantity of water under streamline flow conditions is calculated:
+
DsQ =
4-
4. The actual diameter is determined by Equation 4A: D = D s ~ i / f ’ ;= ; O.O9884/l33 = 0.3360 ft. or 4.030 in.
+
=
.\I
128 X 0.000672 X 1.5 ?r X 13.37 X 32.2
=
o,0988ft.
2. The streamline diameter modulus is determined:
3. The corresponding value of j p on the line for unknown diameter of iron pipe is read in Figure 1 as fp = 133, or it may be calculated by Equation 21 : f p = 0.4 (4Qp/152O.~pD,)~*~~~~ = 0.4 -I(1,790,000/1520)0~6s10 = 132.5
+
VOL. 31, NO. 8
Nomenclature All quantities are expressed in homogeneous metric or English engineering units : A p = pressure difference per unit area, lb./sq. f t . = length of pipe, f t . Q = volume rate of flow, cu. ft./sec. V = average fluid velocity, ft./sec.
L
horsepower fluid viscosity, lb./ft. X sec. fluid density, lb./ou. f t . D = pipe diameter, f t . = acceleration of gravity, ft./sec. = friction factor in Fanning equation, no units fp = streamline friction factor, no units Re, = general symbol for the dimensionlessratios which result from the substitution of streamline values of an unknown into Reynolds modulus. Subscript s = theoretical streamline value of a quantity as calculated from the Hagen-Poiseuille equation P
p p
= = =
Literature Cited (1)
Drew, T. B., and Genereaux, R. P., Trans. Am. Inst. Chem.
Engrs., 32, 17-18 (1936). ( 2 ) Lees, C.H., Proc. R o y . SOC.(London), A91, 45 (1915). (3) Nikuradse, J., Forsch. Cebiete Ingenieurw., Forschungsheft, 356 (1932).
Rates of Water Vapor Adsorption from Air by Silica Gel J. ELSTON AHLBERG‘ The U. S. Naval Research Laboratory, Washington, D. C.
S
UCH fields as catalysis, recovery of condensable vapors, air conditioning, and others equally as diverse are
concerned with adsorption under dynamic conditions. However, a search of the literature discloses a paucity of data concerned with rates of adsorption as contrasted to the large amount of information available on adsorption under static condition. It was therefore thought desirable to publish the results of an investigation of some of the more important factors which determine the rates of adsorption of water by silica gel. Lednum (9)and Miller (3) included a few such observations on silica gel in their publications, but the scope of their measurements is limited. They were concerned with practically complete drying of air. The measurements considered here were made a t air flow rates considerably above those ordinarily employed in air-drying applications. Silica gel has the property of adsorbing relatively large quantities of water at low as well as at moderate partial pressures. That is, the adsorption is relatively so large that the amounts of vapor adsorbed may be measured by simple weighing. For this and other reasons a study of the rates of adsorption of water vapor from air by granular silica gel makes a convenient subject for investigation, from which the more 1
Present address, Universal Oil Products Company, Chicago, Ill.
A n experimental study of the factors affecting the rates of adsorption of a condensable vapor contained in a “noncondensable” gas by a granular adsorbent was made. Silica gel and air containing variable amounts of water vapor were used. The data presented involve the following variables : temperature of the gas, partial pressure of the adsorbable vapor, rate of gas flow, amount of vapor already adsorbed, mesh size of the granular adsorbent, and thickness of the adsorbent beds. The effect of heat treatment is also considered.
important factors affecting rates of adsorption may be better understood.
Materials The silica gel used in these experiments was the standard commercial product of the Silica Gel Corporation. Silica gel is obtained (4) by the chemical interaction of a soluble silicate, of which water glass is typical, and an inorganic acid such as sulfuric. The hydrosol of silicic acid formed in the reaction “sets” in a definite time to a jellylike mass. The mass is washed with water, the temperature of which is controlled, to free it from excess electrolytes. It is then dried at a low temperature, crushed, sized, and again dried at 600” F. The resulting product is a hard, glassy, granular material, light in weight and highly porous. The following is
