Sizing Pipe for Flow of Cellulose F. L. SYMONDS', A. J. ROSENTHAL, AND E. H.SHAW CELANESE CORPORATION OF AMERICA, SUMMITT.
N. J.
F o r solutions o f cellulose acetate in acetone, w h i c h exhibit anomalous viscosity in t h e viscous flow range, shear stress (7) a t t h e pipe wall has been f ound t o be related t o t h e rate of shear ( R ) a t t h e pipe wall by t h e expression T a R 1 - k ( l )wherein a and k are constants f o r a particular solution. A modification of t h e Poiseuille equation can be developed f r o m (I) t o express pressure drop per unit length of pipe in terms of t h e volumetric flow rate, t h e pipe diameter, a, and k. By measurement of viscosities a t t w o rates of shear w i t h a rotational viscometer, t h e values of a and k can be established. Close agreement has been observed between calculated a n d measured pressure drop over a range of rate of shear f r o m 0.06 t o 25 sec. Nomographs are illustrated f r o m w h i c h can be read t h e pressure drop per unit length of pipeline f o r a n assumed volumetric flow rate (or vice versa) of a solution f o r wh i c h t h e values of a and k have been determined.
=
P
RINCIPLES of the design of pipelines for the flow of nonNewtonian liquids in general have been well described elsewhere ( I , 2). Application of these principles has facilitated the development of a relatively simple procedure for sizing pipes for the viscous flow of cellulose acetate solutions. The ease and rapidity with which the procedure can be applied, together wit,h a minimal requirement for experimental data, recommend it for use by plant engineering departments. Although the procedure has been tested specifically only for the rather restricted area of cellulose acetate solutions, pseudoplastic in the viscous flow range investigated, it would be expected to be applicable to other liquids of similar nature.
For a Kewtonian liquid, Equation 4 becomes ( - d V , / d r ) , = 2w = Ri
(5)
By determining instrument viscosities a t various spindle speeds and plotting (2W,4&Rf) versus w , values of d ( 2 u p ~ f ) / d W can be measured for substitution in Equation 4 and calculation of the corresponding rates of shear.
Correlation of Pipeline and Rotational Viscometer Data
In the absence of thixotropy or rheopexy, a logarithmic plot of rate of shear versus shear stress for a particular pseudoplastic liquid, whether obtained by rotational or pipeline viscometer techniques, should result in a single curve such as illustrated in Figure 1 (I, 6). Calculations of the rates of shear and shear stresses from data obtained with the two types of viscometers can be made from the equations that follow. For B viscometer of the rotating-spindle type, the rate of shear a t the surface of the spindle can be expressed as (-dV*/dr), =
Ti/pi
(1)
LOG
O R LOG 7-w
Figure 1. Representative combined loga r i t h m i c shear diagram f o r pseudoplastic liquid
The shear stress a t the surface can be expressed as Ti
= 2mPR.f
(2)
where j is an appropriate factor for converting the indicated instrument viscosity to units of (lb. force) (sec.)/(sq. ft.). For a ratio of spindle diameter to cup diameter which approaches aero,
For flow in a pipeline, the rate of shear a t the pipe wall can be expressed, after Mooney ($), as
(3)
where the shear stress a t the pipe wall is (4) dw 1
Present address, Celanese Corporation of America, Charlotte, N. C.
December 1955
rW =
DAp/4L
(7)
By determining simultaneous pressure drops and volumetric flow rates through a pipe of fixed length and diameter and by
INDUSTRIAL AND ENGINEERING CHEMISTRY
2463
ENGINEERING, DESIGN, AND EQUIPMENT 1000
0 ROTATIONAL-VISCOMETER DATA 0 TUBE-VISCOMETER DATA
10
Figure 2.
100
IO
b
A N D Z , LB-FORCE/SQ
1000
FT
Combined logarithmic shear diagrams for cel Iu lose acetate solutions
plotting 8 q / r D s versus DAp/4L, values of d ( 8 q / r D 3 ) / dD ( Ap/4L) can be measured for substitution in Equation 6 and calculation of the rates of shear corresponding to shear stresses from Equation 7. For a Newtonian liquid, Equation 6 becomes (-dVJdr)-
=
3 2 q / ~ D a= E ,
(8)
and substitution from Equation 7 into the general expression (-dV,/dr) =
(9)
r/p
yields the familiar Poiseuille equation D A p / 4 L = 32q/rD3
(10)
After a shear diagram, such as shown in Figure 1 , has been prepared for a particular pseudoplastic liquid from measurements made with a rotational viscometer and after it has been determined that the diagram correlates with pipe-flow data, the curve can be employed in conjunction with Equation 6 for the design of pipelines for the liquid. For‘ example, one procedure recommended by Alves and coworkers ( 1 ) involves expressing the rate of shear as a function of the shear stress, substituting the function in Equation 6, then integrating formally to obtain an expression which can be solved for pipe diameter or pressure drop. Shear Diagram for Viscous Flow of Cellulose Acetate Solutions
Solutions of cellulose acetates produced from different source materials and of varying degrees of acetylation and polymerization were prepared a t concentrations of 15 to 27% by weight in several solvents, principally 95% acetone. Instrument viscosities of the solutions were measured with a Brookfield Synchrolectric viscometer a t spindle speeds of 2, 4, 10, and 20 r.p.m., corresponding to nominal rates of shear of 0.419, 0.838, 2.09, and 4.19 sec.-I, respectively. The nominal viscosities of the 2464
solutions were in the region of 200 to 3000 poises. In initial work, pressure drop versus volumetric flow rate data were obtained for the flow of solutions from a pressurized vessel through tubing ranging in diameter from 0.08 to 0.80 cm. and in length from 0.2 to 25 cm. In later work of larger scale, similar data were obtained by the use of a pipeline viscometer in which solutions were metered by a gear pump through a “quieting section” of pipe, thence through a continuing, %-foot straight run of 1or 2-inch nominal diameter pipe equipped with terminal pressure gages. By Equations 4 and 6, the rates of shear a t spindle surface and tube or pipe wall were calculated for the applied shear stresses of Equations 2 and 7 . The nominal rates of shear, Ri and R,, were also calculated from Equations 5 and 8, and logarithmic shear diagrams were prepared. The logarithmic shear diagrams for rotational viscometer data employing nominal rate of shear (2w ranging from 0.419 to 4.19 sec.-l) were straight lines having slopes exceeding unity by variable departures. The correlation between rotational-viscometer diagrams and tube-flow diagrams for the nominal rates of shear was found to be essentially as good as the correlation for rates of shear calculated from Equations 4 and 6 within the limits of experimental error. The extrapolation of logarithmic shear diagrams from rotational data to higher rates of shear and comparison with shear data obtained with tube viscometers is illustrated in Figure 2. For the solutions examined, extrapolation from rotational-flow data to tube-flow data a t high rates of shear (>200 sec.-l) was found generally to result in only fair correlation. At such rates of shear, the log-log relationship of shear rate to shear stress was obviously deviating appreciably from linearity. This would indicate that the portion of a logarithmic shear diagram, as shown in Figure 1, which would characterize the diagrams obtained by experimental data would be that portion of the lowest section near the point a t which the line begins t o curve rapidly upward. At rates of shear of 25 sec.-l, the approach of the logarithmic shear diagram to linearity and the correlation of rotational-flow data to tube-flow data resulted in a difference of approximately 25y0 (or less) between rates of shear a t equal shear stress. The correlation between rotationrl-flow and tube-flow shear diagrams improved with decreasing rate of shear. In the normal transport of cellulose acetate solutions and other liquids of similar viscosity, the rates of shear encountered are generally considerably less than 25 sec.-l In the frequently encountered range of less than 5 sec.-1, the correlation is good, expected deviation being less than 10%. Flow Equations for Cellulose Acetate Solutions
The relationship of shear stress to rate of shear for the viscous flow of cellulose acetate solutions in the range of shear rates of practical interest can be expressed as Ti,w
=
a(R)?ik
(11)
where a and k are constants for a particular solution. The units of coefficient a depend on the value of the constant, k . The deviation from Newtonian behavior increases as the magnitude of constant k increases from zero toward unity. From the general equation for the expression of the coefficient of viscosity and from Equation 11 a
=
p g ( R ) Aor f i e = a ( R ) 3
(12)
One measurement with a rotational viscometer is then sufficient for expressing coefficient a as a function of constant k . For example, if an instrument viscosity of poises be measured a t a spindle speed of 10 r.p.m. a = 2 (2.094)h 478.9
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 12
HIGH POLYMER ENGINEERING
g'-GAL. PER M IN.
-
'v
W v)
I aa W
I
cn
LL
0 W
+ a a
-J
a 5 5
0
z
L
E
I "
2
2.5 3
4
5
6
7 8 . 9 1 0 12 14
.-
D'- INCHES Figure 3.
.1437 .-.
Nomograph for determination of shear correction factor and volumetric flow rate
where the factor 478.9 converts the units of viscosity from poises to (lb. force) (sec.)/(sq. ft.) and 2.094 is 2~ expressed in set.-' Substitution from Equation 13 in Equation 11 yields 7 = -
'la
478.9
(2.094)kB1'-k
(F)
a
/4 I O L
log 2 ~ and , k for the range found by subtracting from unity the slope of a straight line drawn through the points. Alternatively, instrument viscosities can be measured a t two spindle speeds, such as 2 and.20 r.p.m., and the value of k calculated by a simultaneous solution of Equation 11, which yields
(14)
which, for application t o flow in pipelines, can be rearranged to The value of pl0 can then be calculated by substitution in Equation 19 if it has not been measured. The combination of Equations 17a and 18 are especially useful for the solution of conventional piping problems.
where
Equation 15 is equivalent to a modification of the Poiseuille equation in which the right side is multiplied by a shear correction factor, F . This equation constitutes the basis for the construction of the nomographs of Figures 3 and 4. Equation 15 can be arranged to express the variables explicitly -for example
DLk 1-k
4 p = -(0.2056)k 11.75 Logq =
1 * 1 - k
log
(17%)
1 1 . 7 5 A ~ D ~ ~ ~ ~ (Lpio ( 1 7 ~ (0.2056)k
The determination of constants a and k of Equation 1 1 with a pipeline or tube viscometer, although less convenient than with a rotational viscometer, naturally provides better accuracy. Values of k for the solutions examined ranged between 0 and 0.4. The k values for cellulose acetates prepared from wood pulp were found generally to be greater than for those prepared from cotton linters. For a particular cellulose acetate, the k value was generally observed to decrease with decreasing concentration and with decreasing degree of polymerization. Conditions of manufacturing, degree of acetylation and solvent composition also affected the k value. Nomograph for Solving Pipe Flow Problems
For known values of p10 and k , expression of A p , q, or D as a function of one other of the three is relatively simple. Equation 17a, for instance, when D is known or assumed, reduces to the form p = bq"
(17d)
in which b and n are constants. Two methods can be employed for determining the value of constants k with a rotational viscometer. Instrument viscosities can be determined a t various spindle speeds, log T~ plotted against
December 1955
The nomographs of Figures 3 and 4 were constructed from Equation 15; however, units for the nomograph were changed to more convenient terms. Figure 3 is used to determine the value of the correction factor, F , to be applied to the Poiseuille equation. Figure 4 is used to apply the correction. Examples. Determine pressure drop for flow of 50 gallons per minute of solution through a 6-inch pipe. Instrument viscosities a t 2 and 20 r.p.m. are 810 and 406 poises, respectively. 1. k log 810 - log 406 = 0.30 2. log p10 = log 810 - 0.3 log 5 ; pi0 = 500
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND EQUIPMENT
D'- INCHESrt
v)
-cnw 0
n
0
T Q W
co W
LT LT
0 0
Figure 4.
Nomograph for determination of pressure drop
3. On Figure 3, follow diagonal from 50 gallons per minute on marign t o intersection with vertical line for 6-inch pipe. Read horizontally t o right for rate of shear of 9.08 sec.-1 Continue t o right to sloping line representing k of 0.30. Read vertically t o find F factor of 0.644. 4. On Figure 4, from intersection of horizontal line for 500 poises and vertical line for 50 gallons per minute follow diagonal t o vertical line for 6-inch pipe. Read horizontally t o right t o vertical line for F of 0.644. Follow diagonal t o vertical line for F = 1, thence horizontally t o right to find pressure drop of 0.339 pound per square inch per foot of pipe length.
Although the nomographs were designed for determining pressure drop, the volumetric flow rate can be determined with only one simple calculation. Assume that a pressure drop of 0.15 pound per square inch per foot of pipe is available for pumping a solution with a 10 r.p.m.-viscosity of 1000 poises and a k value of 0.25 through an %inch pipe. What volumetric rate of flow can be expected? 1. From the following expression derived for the ratio of shear stress t o viscosity D'Ap' r W / ~ , = 1437 MOL calculate rul/puI= 1437 0*15 = 1.724 inches X lb./sq. inch poises X ft. 1000 x 1 ~
2. From point for rW/fiw of 1.72 at bottom right of Figure 3, follow diagonal t o inter olated value of 0.25 between sloping k-lines. Read horizontafiy to left t o vertical line for 8-inch pipe, then follow diagonal t o upper ordinate t o find flow rate of 21 gallons per minute. Nomenclature
a, b = constants D = inside diameter of pipe or tube, ft. D' = inside diameter of pipe or tube, inches
f
appropriate factor for converting indicated instrument viscosity to units of (lb. force)(sec.)/(s ft.) F = dimensionless shear correction factor for 3oiseuille equation L = length of pipe or tube, f t . k , n = exponents p = volumetric flow rate, cu. ft./sec. R = nominal rate of shear, sec.-l Ri = nominal rate of shear at surface of spindle, 2 ~ sec.-1 , R, = nominal rate of shear a t inner wall of pipe or tube, 32a/~DS. sea-1 -dV,/dr = rate of shear (ft./sec.)/(ft.) (-dV,/dr)i = rate of shear a t surface of spindle (ft./sec.)/(ft.) ( - d V r / d r ) , = rate of shear at inner wall of pipe or tube (ft./ sec.)/(ft.) A p = pressuredrop due to friction, lb. force/sq. ft. Ap' = pressure drop due t o friction, lb. force/sq. inch p = viscosit (lb. force)(sec.)/(s , ft.) = effect orviscosity (lb. forcegsec.)/(sq. ft.) /-~i = viscosity a t surface of spindle (lb. force)(sec.)/(sq. ft.) pw = viscosity a t inner wall of pipe or tube (lb. force)(sec.)/(sq. ft.) p~ = viscosity as indicated directly by rotating-spindle viscometer, poises pz,10.20 = viscosity as indicated directly by rotating-spindle viscometer a t spindle speeds of 2, 10, or 20 r.p.m., poises ?r = 3.1416 = shear stress, lb. force/sq. f t . T ri = shear stress at surface of spindle, lb. force/sq. ft. = shear stress a t inner wall of pipe or tube, lb. force/sq. f t . T, w = angular velocity, radians/sec. =
a,
Literature Cited (1) Alves, G. E., Boucher, D. F., and Pipford, R. L., Chem. Enq. Progr., 48, 385 (1952). (2) Lapple, C . E., "Fluid and Particle Mechanics," chap. 7, pp. 115-133, University of Delaware, March 1951. (3) Mooney, M., J. Rheol., 2, 210 (1931).
RECEIVED for review May 26, 1965.
ACCEPTEDSeptember 15, 1955.
E N D OF ENGINEERING, DESIGN, AND EQUIPMENT SECTION
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Vol. 47. No. 12
