488
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3,
1979
R' = Vaq/ Vorg,dimensionless Vag, Vorg,V, = volumes of aqueous phase and organic phase in the total volume of the reactor, cm3
Acknowledgment
We wish to thank Dr. T. J. Hardwick, Mr. G. B. Bell, and Mr. S. Krishnamurthy for their contributions to this work.
Greek Letter = residence time (VT/Faq),s
T
Nomenclature
Literature Cited
CA, CB = concentrations of species A and B, respectively, mol/cm3 CA,eq= equilibrium concentration of species A (uranium ion) in the aqueous phase mol/cm3 CAi = reactor inlet concentration of species A, mol/cm3 CAo = reactor outlet concentration of species A, mol/cm3 CBi = reactor inlet concentration of species B, mol/cm3 CB, = reactor outlet concentration of species B, mol/cm3 C, = concentration of extracting agent, mol/cm3 Faq= flow rate of aqueous phase, L/min Forg= flow rate of organic phase, L min K = equilibrium constant as define by eq 6, dimensionless KLa = volumetric chemical mass transfer coefficient, min-* NA = transfer rate of A, mol/min-cm3 P = power input, ft-lbf/s R = Fag/Forg,dimensionless
Baes, C. F., Jr., J. Inorg. Nuci. Chem., 24, 707 (1962). Baes, C. F., Jr., Nucl. Sci. Eng., 16, 405-412 (1963). Beenackers, A. A. C. M., ACS Symp. Ser., No. 65, 327 (1978). Blake, C. A., Jr., Baes, C. F., Jr., Brown, K. B., Ind. Eng. Chem., 5 0 , 1763 (1958). Blake, C. A., Baes, C. F., Jr., Brown, K. B., Coleman, C. F., White, J. C., ORNL-2259 (Feb 10, 1959). Carr, N. L., Shah, Y. T., Can. J. Chem. Eng., (1979), in press. Dyksira, J., Thompson,B. H., Cbuse, R. J., I&. Eng. Chem., 50(2), 161 (1958). Merritt, R. C., "The Extractbe Metallurgy of Uranium", Colorado School of Mines Research Institute, 1971. Olander, D.. AIChf J., 6(2), 233 (1960). Ryon, A. D., Daley, F. L., Lowrle, R. S., Chem. Eng. Prog. 55(10), 70 (1959). Rushton, J. H., Costich, E. W., Everett, H. J., Chem. Eng. Prog., 46, 467 (1950). Scott, L. S., Hayer, W. B., Holland, C. D., AIChf J., 4, 346 (1958).
d
Received for review June 5 , 1978 Accepted March 12, 1979
The Axial Laminar Flow of Yield-Pseudoplastic Fluids in a Concentric Annulus Richard W. Hanks Department of Chemical Engineering, Brigham Young University, Provo, Utah 84602
Useful engineering design charts are presented which make the design of concentric annular ducts for transporting yield-pseudoplastic non-Newtonian fluids and slurries in laminar motion an easy matter. These charts were computed from theoretical solutions of the equations of motion for the concentric annular geometry using the Herschel-Bulkley or yield-pseudoplastic rheological model. Practical examples are presented illustrating the use of the charts to compute volume throughput given pressure drop and also to compute pressure drop given volume throughput for a slurry of known rheological properties in a given concentric annulus. The design charts make such computations very simple to perform. Complete mathematical results from which the design charts were derived are also given. Thus, computations may easily be made for specific systems.
Aziz, 19721, generalized Bingham (Cheng, 1971, 1975), yield power-law (Torrance, 1963; Hanks and Ricks, 1974; Hanks, 1978), or Herschel-Bulkley model (Herschel and Bulkley, 1926; Skelland, 1967). The flow problems that have been solved for this rheological model include laminar flow in pipes (Govier and Aziz, 1972; Cheng, 1975; Skelland, 1967), laminar flow between parallel planes (Skelland, 1967), laminar-turbulent transition in pipes (Cheng, 1971; Hanks and Ricks, 1974), and turbulent flow in pipes (Torrance, 1963; Hanks, 1978). To date, however, no solution for flow in the important concentric annulus geometry has been published. The purpose of this paper is to present the theory of laminar flow of such fluids in concentric annuli together with appropriate design charts. The design curves presented below should be useful in a number of practical problems including drill-bore cementation, drilling mud pumping, extrusion of viscous polymeric suspensions, etc.
Introduction
In many industrial situations non-Newtonian fluids must be pumped and heated. A frequently encountered flow geometry is the annular space between two concentric pipes of radius R and u R (0 I u I 1). The parameter u is known as the aspect ratio of the annulus. A number of authors have considered the problem of laminar flow in this geometry for simple non-Newtonian fluid models such as the Bingham plastic (Olphen, 1950; Mori and Ototake, 1953; Laird, 1957; Fredrickson and Bird, 1958; Slibar and Paslay, 1957) and power-law (Fredrickson and Bird, 1958; Tiu and Bhattacharyya, 1973; Hanks and Larsen, 1979) and for some more complex models such as the Ellis model (McEachern, 1966) and the Eyring-Ree (Nebrensky et al., 1964) or Powell-Eyring model (Russell and Christiansen, 1974). Many non-Newtonian fluids, particularly slurries and suspensions, are not well approximated by either the Bingham plastic or the power-law models. They are, however, rather well represented by a combination model known variously as the yield-pseudoplastic (Govier and 0019-788217911118-0488$01.00/0
Theoretical Development
The rheological model to be used in the present development is the yield-power-law due originally to Her0
1979
American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 489
schel and Bulkley (1926). In terms of the stress tensor p and the rate of deformation tensor e ( h i s , 1962) this model may be expressed in a generalized Newtonian fluid form (Hohenemser and Prager, 1932; Bird, 1965) as p = 277e;
0 = e;
‘/,p:p >
If we define parameters To = 2ro/PR,A = PR/2k, and s = l / m , eq 7 and 9 may be formally integrated to give 6
ui = R A S l (A2 - x2 - Tox)sxx-s dx;
(1)
T~~
u,
= Ui(Eoi) =
uo = R A S i 1 ( x 2- X2 - T , X ) ~ dx; X-~
In eq 3, T ~ k, , and m are the three adjustable rheological parameters which must be determined by curve-fitting rheological data and might be called, respectively, the yield value ( r o ) ,the consistency index ( h ) , and the non-Newtonian flow index (m)by analogy with their counterparts in the simple Bingham and power-law models. The choice of algebraic sign for T~ is governed by the direction of the momentum flux in a given flow (Fredrickson and Bird, 1958). For the annular flow geometry being considered here the only surviving element of e in erz= 1/2(du,/dr)so that eq 1 and 2 reduce to the following
where we have converted to the momentum flux notation 7 = -p (Bird et al., 1960; Fredrickson, 1964), which is more commonly used by chemical engineers. The well-known solution of the equations of motion (Fredrickson and Bird, 1958; Bird et al., 1960) which gives the stress distribution for this geometry is
where 5 = r/R and P = ( p o- p L ) / L + pg, is the total head loss. The term X in eq 6 is an integration constant and represents physically the value of where 7, vanishes. A t present it is unspecified and its specification becomes one of the results to be derived below. From eq 6 it is evident that for [ < A, r,, < 0 while for t > ?, T,, > 0 which reflects the fact that a t [ = X the velocity maximizes and hence momentum flows to the boundary [ = 1 in the positive [-direction but to the boundary [ = u in the negative [-direction. Hence, for t < A, du,/dr > 0 and we must choose the (-1 sign for 7 0 in eq 4. Conversely, for [ > X,du,/dr < 0 and we must choose the (+) sign for T~ in eq 4. Thus, eq 4 and 5 are replaced by the following set
=
To
+ h(
toi5 [ 5 E,,,
tooI[ I1 (12)
-
To[oi = 0
(13)
too2 - X2 -
Toto0 = 0
(14)
from which one may easily show that
EOO - Eoi = TO X2 =
(15)
tootoi
(16)
These results are identical with those of Fredrickson and Bird (1958) for the Bingham plastic case showing that a change in the nature of the term involving du,/dr does not affect the breadth of the unsheared “plug”. However, the location of the zero shear surface is a strong function of this term. Equations 13-16 may be combined to give the equivalent but more useful results
(5)
y)
(11)
From eq 8 it follows that eq 7 and 9 respectively yield the results X2 - 50;
E( 2 5-
toi 5 E I500
uo(t00);
(10)
(2)
Y2p:p Iro2
where the “generalized” non-Newtonian viscosity, 7, is given by
0 = du,/dr;
u IE IEo;
+
[oi
= ‘/[(To2 4X2)’” - To]
(17)
too
= ‘/,[(To2+
(18)
4x211’2
+ To1
Now, from eq 11 combined with eq 10 and 1 2 it follows that 0 =
(A2 - x2 - T ~ X ) ’ dx X --~ &:(xz
-
X2 - Tox)sx-sdx (19)
Recognizing that tOi and tooare uniquely specified in terms of X and To by eq 17 and 18, we see that eq 19 in turn uniquely defines X in terms of To,u, and s = l / m . Since s is fixed by the rheology of the fluid, and u is fixed by the geometry of a particular annulus, Tobecomes a calculational working-parameter, essentially reflecting the effect of T ~ .It is interesting to note in passing that X is independent of k but depends rather strongly on ro and m. The volume flow through the annulus, Q, is given by
In terms of the velocity gradients and the quantity Q / r R 3 ,eq 20 may be written as
r
=
(8)
2)m; (00
I[ I1 (9)
where = toi(A) are the two positions at which T,, = ?To, respectively. These variables are analogous to X- and A+ as defined by Fredrickson and Bird (1958) for the Bingham plastic case.
Upon interchanging the order of integration in the iterated integrals and taking account of the algebraic signs of du,/dr this result may be written as
r
= J’(h2
-
x2)(
2)
dt
An alternative expression to eq 22 may be obtained by
490
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
Table I. Values of h for Various Values of
U,
T o ,and m m
To
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
u = 0.10
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85
0.3442 0.3427 0.341 2 0.3397 0.3382 0.3367 0.3352 0.3337 0.3321 0.3306 0.3290 0.3275 0.3259 0.3243 0.3 227 0.3215 0.3259 0.3349
0.3682 0.3656 0.3630 0.3604 0.3577 0.3550 0.3523 0.3495 0.3466 0.3438 0.3409 0.3379 0.3349 0.3319 0.3288 0.3257 0.3226 0.3194
0.3884 0.3851 0.3817 0.3782 0.3747 0.3710 0.3673 0.3635 0.3596 0.3556 0.3516 0.3474 0.3432 0.3389 0.3345 0.3301 0.3255 0.3209
0.4052 0.4014 0.3975 0.3935 0.3893 0.3850 0.3805 0.37 59 0.3711 0.3663 0.3612 0.3561 0.3508 0.3454 0.3398 0.3341 0.3283 0.3223
0.4193 0.4153 0.4110 0.4065 0.4019 0.3971 0.3920 0.3868 0.3813 0.3757 0.3699 0.3639 0.3577 0.3512 0.3446 0.3378 0.3308 0.3236
0.4312 0.4269 0.4225 0.4178 0.4128 0.4076 0.4021 0.3964 0.3904 0.3841 0.3776 0.3709 0.3639 0.3566 0.3491 0.3413 0.3332 0.3248
0.4412 0.4369 0.4323 0.4274 0.4222 0.4167 0.4109 0.4048 0.3988 0.3916 0.3846 0.3772 0.3695 0.3615 0.3531 0.3444 0.3354 0.3260
0.4498 0.4455 0.4408 0.4358 0.4305 0.4248 0.4187 0.4123 0.4055 0.3983 0.3908 0.3829 0.3746 0.3659 0.3568 0.3473 0.3375 0.3270
0.4572 0.4529 0.4482 0.4432 0.4377 0.4319 0.4256 0.4189 0.4118 0.4043 0.3964 0.3880 0.3792 0.36 99 0.3602 0.3499 0.3392 0.3280
0.4637 0.4594 0.4547 0.4496 0.4441 0.4381 0.4317 0.4248 0.4175 0.4097 0.4014 0.3926 0.3834 0.3736 0.3633 0.3524 0.3409 0.3289
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.4687 0.4674 0.4661 0.4649 0.4636 0.4623 0.4610 0.4596 0.4583 0.4570 0.4556 0.4542 0.4528 0.4525 0.4589 0.4700
0.4856 0.4835 0.4814 0.4792 0.4770 0.4748 0.4725 0.4701 0.4678 0.4653 0.46 29 0.4604 0.4578 0.4552 0.4526 0.4499
0.4991 0.4965 0.4938 0.4910 0.4882 0.4852 0.4822 0.4790 0.4758 0.4725 0.4692 0.4657 0.4622 0.4586 0.4549 0.4511
0.5100 0.5070 0.5040 0.5008 0.4974 0.4940 0.4904 0.4866 0.4828 0.4788 0.4747 0.4704 0.4660 0.4615 0.4569 0.4521
u = 0.20 0.5189 0.5157 0.5124 0.5089 0.5053 0.5014 0.4974 0.4932 0.4888 0.4842 0.4795 0.4745 0.4694 0.4641 0.4587 0.4530
0.5262 0.5230 0.5195 0.5158 0.5119 0.5078 0.5034 0.4988 0.4940 0.4890 0.4837 0.4782 0.4725 0.4665 0.4603 0.4539
0.5324 0.5291 0.5256 0.5218 0.5177 0.5133 0.5087 0.5038 0.4986 0.4932 0.4875 0.4815 0.4752 0.4732 0.4618 0.4546
0.5377 0.5344 0.5308 0.5269 0.5227 0.5181 0.5133 0.5082 0.5027 0.4969 0.4908 0.4844 0.4776 0.4705 0.4631 0.4553
0.5422 0.5389 0.5353 0.5313 0.5270 0.5224 0.5174 0.5120 0.5063 0.5002 0.4938 0.4870 0.4798 0.4723 0.4643 0.4560
0.5461 0.5429 0.5392 0.5353 0.5309 0.5261 0.5210 0.5155 0.5096 0.5032 0.4965 0.4894 0.4818 0.4738 0.4654 0.4565
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
0.5632 0.5621 0.5611 0.5601 0.5590 0.5579 0.5568 0.5557 0.5546 0.5535 0.5524 0.5520 0.5616 0.5765
0.5749 0.5732 0.5715 0.5698 0.5680 0.5661 0.5642 0.5623 0.5604 0.5584 0.5563 0.5542 0.5521 0.5499
0.5840 0.5819 0.5798 0.5775 0.5752 0.57 28 0.5704 0.5678 0.5652 0.5625 0.5597 0.5568 0.5539 0.5508
0.5912 0.5889 0.5864 0.5839 0.5812 0.57 84 0.5755 0.57 24 0.5693 0.5660 0.56 26 0.5590 0.5554 0.5516
0.5970 0.5946 0.5919 0.5892 0.5862 0.5831 0.5798 0.5764 0.5728 0.5690 0.5651 0.5610 0.5567 0.5523
0.6018 0.5993 0.5966 0.5936 0.5905 0.5871 0.5836 0.5798 0.5759 0.5717 0.5673 0.5627 0.5579 0.5529
0.6059 0.6033 0.6005 0.5974 0.5941 0.5906 0.5868 0.5828 0.5785 0.5740 0.5693 0.5643 0.5590 0.5535
0.6093 0.6067 0.6038 0.6007 0.5973 0.5936 0.5897 0.5854 0.5809 0.5761 0.5710 0.5656 0.5599 0.5540
0.6122 0.6096 0.6068 0.6036 0.6001 0.5963 0.5922 0.5878 0.5830 0.5779 0.57 26 0.5668 0.5608 0.5544
0.6147 0.6122 0.6093 0.6061 0.6025 0.5987 0.5944 0.5898 0.5849 0.5796 0.5740 0.5680 0.5616 0.5548
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.6431 0.6423 0.6414 0.6406 0.6397 0.6389 0.6380 0.6371 0.6362 0.6359 0.6488 0.6624
0.6509 0.6496 0.6483 0.6469 0.6454 0.6439 0.6424 0.6409 0.6393 0.6376 0.6359 0.6342
0.6570 0.6553 0.6536 0.6519 0.6500 0.6481 0.6461 0.6440 0.6418 0.6396 0.6373 0.6349
0.6617 0.6599 0.6580 0.6559 0.6538 0.6515 0.6491 0.6466 0.6440 0.6413 0.6385 0.6355
u = 0.40 0.6655 0.6636 0.6615 0.6593 0.6569 0.6544 0.6517 0.6489 0.6459 0.6428 0.6395 0.6360
0.6686 0.6667 0.6645 0.6621 0.6596 0.6568 0.6539 0.6508 0.6475 0.6440 0.6404 0.6365
0.6713 0.6692 0.6670 0.6646 0.6619 0.6590 0.6559 0.6525 0.6490 0.6452 0.6412 0.6369
0.6735 0.6715 0.6692 0.6667 0.6639 0.6608 0.6576 0.6540 0.6502 0.6462 0.6419 0.6373
0.6754 0.6734 0.6711 0.6685 0.6656 0.6625 0.6591 0.6554 0.6514 0.6471 0.6425 0.6376
0.6770 0.6750 0.6727 0.6701 0.6672 0.6639 0.6604 0.6565 0.6524 0.6479 0.6431 0.6379
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.7140 0.7134 0.7127 0.7121 0.7114 0.7107 0.7100
0.7191 0.7180 0.7170 0.7159 0.7147 0.7136 0.7124
0.7229 0.7216 0.7203 0.7189 0.7174 0.7159 0.7143
0.7259 0.7245 0.7230 0.7214 0.7196 0.7178 0.7159
u = 0.50 0.7283 0.7268 0.7252 0.7234 0.7215 0.7195 0.7173
0.7303 0.7287 0.7270 0.7251 0.7231 0.7209 0.7184
0.7319 0.7303 0.7286 0.7266 0.7244 0.7221 0.7195
0.7333 0.7317 0.7299 0.7279 0.7256 0.7231 0.7204
0.7345 0.7329 0.7311 0.7 290 0.7267 0.7241 0.7212
0.7355 0.7340 0.7321 0.7300 0.7276 0.7249 0.7219
a = 0.30
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
491
Table I (Continued) ~~~
m
To
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.35 0.40 0.45
0.7095 0.7247 0.7314
0.7111 0.7098 0.7085
0.7126 0.7108 0.7090
0.7139 0.7117 0.7095
0.7149 0.7125 0.7099
0.7159 0.7131 0.7102
0.7167 0.7137 0.7105
0.7174 0.7142 0.7108
0.7181 0.7147 0.7110
0.7187 0.7151 0.7112
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.7788 0.7783 0.7778 0.7773 0.7768 0.7767 0.7865 0.7882
0.7818 0.7810 0.7802 0.7793 0.7785 0.7776 0.7766 0.7756
0.7840 0.7831 0.7821 0.7810 0.7798 0.7786 0.7774 0.7760
0.7858 0.7847 0.7836 0.7823 0.7810 0.7795 0.7780 0.7763
0.7872 0.7861 0.7848 0.7835 0.7820 0.7803 0.7785 0.7766
0.7884 0.7872 0.7859 0.7844 0.7828 0.7810 0.7790 0.7169
0.7893 0.7882 0.7868 0.7852 0.7835 0.7816 0.7794 0.7771
0.7902 0.7890 0.7876 0.7860 0.7841 0.7821 0.7798 0.7773
0.7909 0.7897 0.7882 0.7866 0.7847 0.7825 0.7801 0.7775
0.7915 0.7903 0.7888 0.7871 0.7852 0.7829 0.7804 0.7776
0.00 0.05 0.10 0.15 0.20 0.25
0.8389 0.8385 0.8382 0.8380 0.8412 0.8428
0.8404 0.8399 0.8393 0.8387 0.8381 0.8374
0.8416 0.8410 0.8402 0.8394 0.8386 0.8376
0.8426 0.8418 0.8410 0.8400 0.8390 0.8379
a = 0.70 0.8433 0.8425 0.8416 0.8405 0.8394 0.8381
0.8439 0.8431 0.8421 0.8410 0.8397 0.8382
0.8444 0.8436 0.8425 0.8413 0.8400 0.8384
0.8449 0.8440 0.8429 0.8417 0.8402 0.8385
0.8452 0.8444 0.8433 0.8420 0.8404 0.8387
0.8455 0.8447 0.8436 0.8422 0.8406 0.8388
0.00 0.05 0.10 0.15
0.8954 0.8954 0.8954 0.8960
0.8960 0.8957 0.8953 0.8949
0.8965 0.8961 0.8956 0.8950
0.8969 0.8964 0.8958 0.8952
a = 0.80 0.8972 0.8967 0.8961 0.8953
0.8975 0.8969 0.8962 0.8954
0.8977 0.8971 0.8964 0.8955
0.8979 0.8973 0.8965 0.8956
0.8980 0.8974 0.8966 0.8956
0.8981 0.8976 0.8967 0.8957
0.00 0.05
0.9489 0.9488
0.9491 0.9489
0.9492 0.9490
0.9493 0.9490
= 0.90 0.9493 0.9491
0.9494 0.9491
0.9495 0.9491
0.9495 0.9492
0.9495 0.9492
0.9496 0.9492
u = 0.60
a
using eq 19 and the specific forms for du,ldr in the two regions of flow t < A, t > A. Thus, an equivalent expression is
r
= -ASX"(A2
-
x2 -
T+)sx2-sdx
. -1
+
As &1(x2-A2-Tox)sx2-s 00 dx (23) where toi and too are given by eq 17 and 18, respectively. Equation 23 shows again that To is the calculational working-parameter, Unfortunately, the presence of Toin the argument functions of the integrals in eq 23 makes their analytical evaluation by the technique of Hanks and Larsen (1979) impossible and one must resort to numerical quadrature methods to evaluate X(To,u,s)and r(To,u,s,k) from eq 19 and 23, respectively. Everything in eq 19 is dimensionless. Equation 23 is easily rendered dimensionless as follows
$(To,u,s) = - X t a ( A 2 - x 2 - Tox)sx2-sdx
+ Tox)sx2-sdx (24)
0
Figure 1. Design chart for computation of flow rate Q given pressure drop Ap for flow of yield-pseudoplastic fluids having 0.70 I m 5 1.0 in concentric annuli having 0.10 I u 5 0.90.
where
4=
=
(:)T
