TRANSLATING TE#
OR NOW-NEWTONIAN FLOW
several types of fluids in each of the two There major . areclasses. Methods described here cover: Time-Independent Flow R O B E R T S. B R O D K E Y
Non-Newtonian flow calculations ore not mysterious. Methods of handling non-Newtonianr parallel Newtonion flow technology. The changing viscosity no) only adds a variable to lhe physical problem, but gives the investigator a degree of freedom in reporting his rerulk.
The methods of comelation ond
nomenclature itself both vary.
You cannot
do a quick brush-up on non-Newtonian theory. It i s possible to simplify calculations by using a Reynolds number and average viscorily which apply to the speciflc problem. It Is also possible to mix unrelated terms and come up with a meaningless answer. For the most common problem, pipe flow, Table I shows you which terms can be used, and which authors used them. Toble II indicates whai they called the viscosities.
The
remainder of this article tells you how to handle non-Newlonians. There ore two major classes of non-Newlonian fluids. This article considers only timeindependent,
nonelastic fluids.
Viscosities
which vary with time and previous history of the material, such as thixohopic and rheopectic fluids, methods.
are not hondled by these
Newtonian. Ideal straight-line relationship between shear stres and shear rate. Bmgham Plastic. An offset straight-line relationship between shear stress and shear rate. An initial resistance must be overcome before flow starts. Pseudoplastic. Apparent viscosity (flow curve slope) decreases with increasing shear rate, often attaining a limiting slope. Dilatant. Apparent viscosity increases with increasing shear rate, often attaining a limiting slope. The correlation of flow data generally requires the use of a Reynolds number which can have many values, depending on the 'geometry and viscosity description used. However, any are usable as long as one appreciates the limitations, and realizes that each number can be related to the basic shear diagram. This basic diagram provides point relations, as contrasted to the average relations obtained in pipe or other gross flows. The Reynolds number so obtained is the only completely general one; the others are specific to some special case, and reduce to the basic number in the limit of a Newtonian. The more complex Reynolds numbers have a relatively simple form, when interpreted in terms of basic rheological terminology. RELATIONS B-EEN
SHEAR DIAGRAMS
The capillary flow diagram is the same as the basic shear diagram if the material is Newtonian. If not, a correction must be made. Metzner and Reed (70) modified the analysis of Rabinowiwh (77) to give a convenient relation between the shear rate at the wall and 6/ro:
(--do/dr).
=
[On'
+
0/W(6/re)
(Contimud on page 46) 44
I N D U S T R I A L A N D ENGINEERING C H E
(7)
BASIC SHEAR DIAGRAM
gainst the shear rate (dv/dr),'measured at the same point. The
mass and
a,c ton d ! U I
,
.,//
portionality constant of Equation 1
-p '(&/d,)
(3) where r > 7., This basic diagram is most important, since it relates point properties and thus is not dependent on any particular r -
If a tube or cap;uarY i s usEd to obtain rheological data, the basic shear diagram cannot be obtained directly; however, the "capillary flow diagram" can be used. In this diagram, rJP/2L is plotted ;' against 45/re or its equivalent, 4Q/nr2. The term, rJP/2L, is the 1 negative of the stress at the wall, rro. The term, 4Z/r0, has the form of 5 a shear rate. The figure shows this diagram for a non-Newtonian pseudoplastic material. As in the case of the basic shear diagr an apparent viscosity can be defined; this will be called the pipe apparent viscosity, pap. The corresponding equation is 7- = -rJP/2L = p0.(45/v.) For a Newtonian material, Equation 4 is the Hagen-Poiseuille law. If the flow occurs in an annular section, the resulting data can be treated in a similar manner. The annular apparent viscosity would ed from (rJP/2L) [l
~
+ kz +
I"-'
In k
= pa,,(4C/r0)
where k = r,/re. The correction for geometry can be associated with either the shear rate or shear stress in the figure. t
, :'.
I
,
ROTATIONAL FLOW DIAGRAM
L rotational viscometer (outer cylinder rotating, inner cylinder at
t) is often used to obtain rheological data.
5
In this case, T/2mlg I is plotted against 2rg%l/(rgg - rlz). The former is the negative of the 3 stress at the inner cylinder wall, while the latter is of the form of a c shear rate. The rotational apparent viscosity par,is given by a
. ..
- TIz)
- r l = T/2nrla = p, 2rzzQ/(rzz
where 0 is the angular velocity of the outer cylinder. For a Ne tonian material, Equation 6 is the Margules equation (7, 72).
'
"2.
'~
:
.i
c
tame
For a non-Newtonian power-law material:
where n' is defined from the relation, I, =
I
!
l
1I.
~
(-r.AP/ZL)
= K'(4drY
(8)
The primed term are used to indicate that they are based on the capillgry shear diagram. In this definition, n' is mf 11cccssm'~y II corutut#. An alternate and better definition (70) would be
-
Inspection of Equations 7 and 8 show that for n' 1, &/r. is equal to the shear rate at the wall, and Equation 8 reduces to Newton's law with K' = fi.
When the material M any type of a non-Newtonian, the following procedure Win give the baaic shear diagram from capillary data. -At a given value of the shear stress at the wall (-roAp/2L), obtain n' from the slope of a log-log plot of (-r,,Ap/ZL) us. (4#/re) (Equation 9). -Using this value of n' and the corresponding value of (4f!/rJ, calculate the shear rate at the wall from Equation 7. -The value of shear stress at the wall and this value of shear rate at the wall give one point on the basic shear diagram. Of course, the calculations are simplified considerably if n' is constant. The rotational flow diagram is the same as the basic shear diagram if the material is Newtonian. For nonNewtonian materials, the rotational viscometer can be designed to give the basic shear diagram directly. If the viscometer has a narrow gap, fie, will be equal to the true apparent viscosity k,obtained from the basic shear diagram. In a poorly designed unit, a correction must be made if the basic shear diagram is desired. For rotational flow, where angular acceleration can exist,
Newton's law is
or
I
46
The unprimed term denote that they are from a basic shear diagram. Combining with the expression Cor shear stress:
Integration gives
! !=
- (T/ZnK)""
(n/Z)r-""
+c
(14)
Evaluation of the comtant at r = I, where = 0, noting that at r = rs, n = u8/r, rearranging to the form of the shear rate, combining with Equation 13 written at the inner cylinder wall, and finally rearranging once again gives an equation somewhat analogous to Equation 7:
-
n [d(dd/drl I (dob/dr)l = [1 (rdrd*/n( 1 ( t d r P ) ] [ Z i t n/G?
-
-
- r13 ]
(15)
A similar equation can be derived for the case of the inner cylinder rotating, and the outer cylinder at rest (I). The only difference will be the boundary conditions used to evaluate the constant in Equation 14. This equation, when contrasted to Equation 7, can be seen to have a geometric dependency. As in Equation 7, for n = 1, the shear rate at the inner cylinder is identical to the term in the brackets on the right. For n # 1, the geometrical dependence can be used to advantage. The limit as rx +. r? is:
-
Thus, as r, +. rs, (duo/&), approaches 212 n/(r? 11%) for any value of n. An equation can also be derived for the casc of the Bingham plastic (7). Here Equation 3 in polar coordinates would replace Equation 11, and the development would be the same. The starting equation is (17)
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
and the finalintegrated equation is
Thi Article
Other Articles
In the limit of n + tt, the last term gwa to m, and thus the basic shear diagram can be obtained directly.
When the rotational &meter has not been designed to take advantage of a narrow gap, then corrections are necessary to obtain the basic diagram. Usually, the variation in the shear rate across the gap is such that n can be considered a constant over this limited range of shear rates. Then according to Equation 15, the true shear rate is directly proportional to the apparent shear rate as given by the second bracketed term. Thus, a log-log plot of the terms of Equation 6 would still give the correct value of n, even though the graph is not exactly the basic shear diagram under the conditions of shear rate being considered. The necessary correction can now be obtained from Equation 15. If n varies over the entire range of shear rates being considered, then the calculations will have to be made at a number of points. Other procedures have been suggested to accomplish the same correction; for example, Krieger and Maron (7) have described a similar method based on the apparent fluidity (l/par).
Combining the above with Equation 4 gives
Pip. Flow
Most of the work on non-Newtonian Bow has been directed toward obtaining a means of estimatiug the pressure loss in the system. Both laminar and
Combining Equations 2 and 3 gives
turbulent pressure drop correlations have been presented in the literature (3, 6, 8, 70, 74, 77, and 78). For laminar flow, the wrrelations are essentially equivalent, since they are based on the same fundamental theory. However, for turbulent flow, Thomas (76)has suggested that the three main correlations are in reality three different correlations for three different types of materials. The correlation of Dodge and Metzner (3) is for asymmetrically-shaped partides and non-viscoelastic polymers; that of Shaver and Merrill (74) is for viscoelastic polymer solutions; and that of Thomas (17) is for symmetrically-shaped particles. In all of the correlations cited, some form of a Reynolds number is used. Every Reynolds number is in some way related to the others, as illustrated in Table I. It is important to note that each Reynolds number, including the one first derived by Metzner and Reed (70),has a basic rheological definition. Unfortunately, the notation used in the literature is not consistent; Table I1 compares the notations used by the various authors cited. In pipe flow, viscosities pLa,k Vand , p' are important. For each viscosity there is a Reynolds number; as:
The Reynolds number based on the plastic viscosity
NFW p a p
(19)
= d8P/Pa,
If the relations between the viscosities are known, then the relations between the Reynolds numbers follow directly. The Buckingham-Reiner equation (72) for a Bingham plastic with T , > roris: Q = (--m,'hp/8p'L)
+I(">'] 3
[l- '3 ( Tw 2)
Tw
(20)
P' =
k[1
-
(To/T)l
(22)
has been used by Hedstrom (Si, McMdlen (a), Thomas (75, 77) Weltman (78),and Winning (20). The relation between k and papfor a material defined by Equation 8 is derived from Equation 2 written at the wall, and Equations 4 and I : paw =
~.,[4n'/13n'
+ 111
(23)
where the sub w on fl0 denotes that it is to be evaluated at the wall shear ratr. Of course, if the material is an ideal plastic, one need only equate Equations 21 and 22 to obtain the corresponding relation. The Reynolds number based on the pipe apparent viscosity is identical to the modified Reynolds number of Metzner and Reed (10).
NR., = d/'$-"'p/r where
=
(24)
K'V'-'
Combining Equations 4 and 8 gives pap =
K'(48/ro)n'-1
(25)
or, in terms of diameter, k~ =
K,8~~-~gm~-l/do"'-l
(26)
Substituting this into the Reynolds number of Equation 19 gives exactly Equation 24. (Continued m mt page) VOL 5 4
NO. 9
SEPTEMBER 1962
47
The dependency on the wall shear rate as shown by Equation 23 is the reason for the general validity of the Dodge and Metzner ( 3 ) pressure drop correlation. This correlation is valid for materials with a variable n ' , if n' is evaluated at the wall stress T ~ .Even more recently, Bogue (2) was able to predict the friction factor us. Reynolds number plot for high Reynolds numbers by assuming that the flow is characterized by the apparent viscosity at the wall, barn. The Reynolds number of Equation 19 has been used by others (70, 74, 75, 78, 20).
F u r t h e r simplification gives
T h i s is the Reynolds n u m b e r proposed by Fredrickson a n d Bird in a simplified f o r m for easy comparison. Combining Equations 30 a n d 6 gives
Substituting rhis into E q u a t i o n 27 a n d simplifying :
Annular Flow
The annular Reynolds number based on the apparent viscosity obtained from an annular tube viscometer (Equation 5) is
T h i s is identical to E q u a t i o n 31, which was obtained from ( Fredrickson a n d Bird is simply E q u a t i o n 28 ; thus: i V ~ e /of I V R ~ the , ~ ~Kewtonian ~, limiting form based o n pas. NOMENCLATURE
4
= pipe diameter = TJr"
k
Unfortunately, there are no ddtd for annular pseudoplastic non-Ne\\ tonian flow to allow testing of Equation 2 7 ; howeber, very recentlv Meter and Bird ( 9 ) have shown that its Newtonian form provides a satisfactory representation for Kewtonian annular data. For nonNewtonian pipe flow, Equation 27 reduces to dvRe,p,,. which has had excellent confirmation from the work of Dodge and Metzner ( 3 ) . It appears reasonable that this should correlate non-Xewtonian annular flow data for pseudoplastic materials. h analysis of annular flow has been made by Fredrickson and Bird (4, 5). This work has been commented on by IVilcox ( 7 9 ) and by- Savins ( 7 3 ) . The Reynolds number proposed b, these authors is n"-2-n" l'-Re''
=
[$,intf-r][
(1
+ k)
roAP
and exactly equal to the Rebnolds number defined in Equation 27. Equation 28 was derived in a manner analogous to the previously discussed modified Reynolds number of Metzner and Reed. E q u a t i o n 28 can-be simplified by using two definitions :
-yRe,,,= dont'j2-n"p/Kf l8n"-1
(29)
T h e latter term is a power-law type relation for a n n u l a r flow. E q u a t i o n 29 is of the same form as ~ Y R ~ , ~ ,however, ,~; the K" is defined differently ( E q u a t i o n 30 us. E q u a t i o n 8). Combining Equations 27 a n d 30 with 28, a n d using the identity from continuity (4Q,'nrO3) = ( 4 ~ / ~ ~ )(1 k2) gives #
(871"
- 1/ 2"" - 9
x In -k
S. Brodkey i s an Associate Professor of Chemical Engineering at Ohio State University.
A U T H O R Roberf
48
'?
Subscripts
- ,~)TZ~~T(-$)~"( ' ~ F Lw) ] (28)
dvRe,,=
K , K ' , K" = power-law constant L = length n, n', n" = power-law constant N R ~p ,a , ~I'R., p', ~ V RP~, ,,~ ,N R ~pan , = Reynolds numbers defined analogously to Equations 19 and 27 = pressure drop = $ 2 - P I AP = volumetric flow rate Q r = radius To = pipe radius = torque per unit height I' = velocity, time average a t a point 0, % V = velocity, average over the pipe 7 = shear stress 70 = yield stress TU = wall shear stress = inner wali shear stress 71 pa, pap, p par, paa = viscosities as measured on the various shear diagrams; Equations 1 to 3, 5 and 6 = density P R = angular velocity = defined by Equation 24 Y
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1 2
= = =
wall inner wall outer wall
REFERENCES
(1) Brodkey, R . S., "Notes on Fluid Transport," T h e Ohio State University, Columbus 10: Ohio, 1959. (2) Bogue: D. C., Ph.D. dissertation, Univ. of Delaware, 1960; A.1.Ch.E. Meeting, Tulsa, Okla., 1960. (3) Dodge, D. W., Metzner, A. B , A.Z. Ch. E. Journal 5 , 189-204 (1959). (4) Fredrickson, A. G., Bird, R . E.: IND.ENG.CHEM.50, 347-52 (1958). (5) Ibid.,pp. 1399-1600. (6) Hedstrom, B. 0. .4.,IND.ENG.CHEM. 44, 651-6 (1952). (7) Krieger, I. M., Maron. S. H., J . Appl. Phys. 2 5 , 72-5 (1954). (8) McMillen, E. E., Chem. Eng. P r q r . 44, 537-46 (1948). (9) Meter, D. M.. Bird, R . B., A.I.Ch.E. Journal 7, 41-5 (1961). (IO) Metzncr, A. B.; Reed, J. C., [bid.,1, 434-40 (1955). (11) Rabinowitsch, B., Z,p,$yJik. Chem 145A, 1 (1929). (12) Reiner, M., "Deformation and Flow," H. K. Lewis and Co , London, 1949. (13) Savins. J . G., A.Z.Ch.E. Journal 8, 272 (1962). (14) Shaver, R. G., Merrill. E. IV.,Zbid., 5 , 181-8 (1959). (15) Thomas, D. G., Ibid., 6 , 631 (1960). (16) Thomas, D. G., Paper 61 in "Progr. Intern. Research on Thermodynamic and Transport Properties," Amer. SOC. Mech. Engrs., New York, 1962. (17) Thomas, D. G., A.I. Ch. E. Journal 8, 266-71 (1962): 48, 386-7 (1956). (18) Weltman, R . N., IND. ENG.CHEM. (19) Wilcox, W.R., Zbid.,50, 1600 (1958). (20) i'v'inning. M . D.. M.S thesis, Univ. of Alberta, 1948
