LIQUID FLOW IN A SUGAR CENTRIFUGAL

Transcription

1 LIQUID FLOW IN A SUGAR CENTRIFUGAL C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Khalique, A. Huchinson and M. C. Rademeyer Indusry represenaives Richard Loubser and Seve Davis Absrac Massecuie is a mixure of sugar crysals and molasses produced during he manufacure of sugar. A cenrifuge, which is a roaing cylindrical baske, is used o separae he sugar crysals from he molasses. During he laer par of he process waer and seam are inroduced ino he cenrifuge o furher faciliae drainage. The model developed indicaes ha a fify percen increase in molasses drainage can resul from he addiion of seam, whereas waer does no significanly affec drainage. 1 Inroducion The process of exracing molasses from sugar crysals is no opimal. In he early sages of he exracion process, a significan percenage of molasses is no drained from he cenrifuge and more ime is required o complee he process. Currenly, he addiion of waer or seam is used o aid in he efficiency of his process. I has Oxford Cenre for Collaboraive Applied Mahemaics, Universiy of Oxford, Oxform OX1 3LB, Unied Kingdom. please@mahs.ox.ac.uk Mahemaics Deparmen, Universiy of Wesern Ausralia, Crawley, WA 6009, Ausralia. e- mail: neville.fowkes@uwa.edu.au School of Compuaional and Applied Mahemaics, Universiy of he Wiwaersrand, Johannesburg, Privae Bag 3, Wis 2050, Souh Africa. David.Mason@wis.ac.za Deparmen of Mahemaical Sciences, Norh Wes Universiy, Privae Bag X2046, Mmabaho 2735, Souh Africa. Masoos.Khalique@nwu.ac.za School of Compuaional and Applied Mahemaics, Universiy of he Wiwaersrand, Johannesburg, Privae Bag 3, Wis 2050, Souh Africa. huchinson.ash@gmail.com School of Mahemaics, Universiy of he Wiwaersrand, Johannesburg, Privae Bag 3, Wis 2050, Souh Africa. maryke.rademeyer@sudens.wis.ac.za 59

2 60 Liquid flow in a sugar cenrifugal been proposed ha hydrogen peroxide may be added o ac as a bleach o reduce he colour of he sugar. In his paper, we use a simple mahemaical model in order o invesigae he mechanism of his process. We show ha he exracion process can be modified o improve he drainage of molasses from he sugar crysals and hus decrease he ime required for is compleion. According o his model, he addiion of seam leads o improving he drainage of molasses by fify percen. During he firs sage of he sugar manufacuring process juice is exraced from shredded sugar cane in a diffuser. The concenraion of sugar is hen increased o he poin of supersauraion by evaporaion. The resuling subsance ermed massecuie consiss of roughly 50 % sugar crysals suspended in he molasses which is a non-newonian fluid. The combinaion has an average viscosiy 0.8 Pa.sec. The massecuie is deposied ino a bach cenrifuge, consising of a cylindrical baske which is spun a high speed so ha he liquid surrounding he crysals, now referred o as molasses, is exraced hrough he walls of he baske as shown in Figure 1. The baske has a diameer of 2R = 1.2 m and is lined by a fine mesh which allows for he passage of liquid bu ensures ha he crysals remain rapped inside. While he massecuie is being deposied ino he baske, he roaional velociy of he cenrifuge is abou 200 rev/min. Afer some ime he roaional velociy is increased o 960 rev/min and by his sage a layer of massecuie of abou 100 mm has buil up on he inerior verical sides of he baske. This is known as he spin up sage of he exracion process. Afer molasses removal he cenrifuge is slowed down and he sugar crysals are removed by a plough-like mechanism. The addiion of waer and he washing off of he molasses regulaes he colour of he crysals. Furher bleaching may be achieved by means of an oxidising agen such as hydrogen peroxide. This invesigaion was promped by difficulies experienced in he sugar indusry during his process. Under sub-opimal loading condiions for eiher he composiion of he massecuie, is feeding rae, or he spin up process, he mixure will no uniformly coa he baske and his can lead o cenrifuge insabiliies eiher immediaely or laer during he spin cycle when spin speeds are large. Nonuniform drainage can also resul and blockages may occur. The spin up sage is briefly discussed in Secion 2. During he molasses sripping process, addiional difficulies can also arise during he waer and seam addiion sage of he process in ha nonuniform washing can resul in qualiy degradaion and he dissoluion of crysals. Based on experience and simple models pracical measures have been adoped o deal wih he above difficulies, however i is hoped ha a beer undersanding of he processes migh lead o beer operaional procedures. A useful reference o similar problems (bu in a geophysical conex) is Greenspan [1]. The parameer range is somewha differen in he presen conex and here are oher wo phase/lubricaion issues ha play a role. In Secions 3 o 6 a simple model for he molasses sripping process is

3 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer 61 examined. Conclusions and suggesions are presened in Secion 7. Figure 1: Cylindrical polar coordinae sysem (r, θ, z) for a hin film of molasses draining from sugar crysals. The coordinae sysem moves wih he cenrifuge roaing wih angular velociy Ω. 2 Exracion process During he iniial sages of spin up he massecuie ha is fed ino he boom of he cenrifuge firs spreads ou in he radial direcion and hen creeps up he insides of he baske. Alernaively he massecuie may be sprayed direcly ono he baske. During his spin up sage of he process he mixure behaves like a homogeneous viscous liquid [1]. Afer he iniial spin up, he cenrifuge reaches a consan angular velociy of which i hen mainains. The liquid componen of he mixure, in oher words he molasses, filers hrough he crysal marix formed agains he walls of he baske. For sufficienly low liquid concenraion levels, he flow of molasses hrough he wall of he cenrifuge essenially ceases and he remaining liquid is flushed ou from he crysals via he addiion of waer or seam. Our main concern is he exracion of molasses from he sugar crysals once he flow of molasses has basically sopped and he cenrifuge has mainained a consan angular velociy for a sufficien lengh of ime. Waer or seam is required o faciliae furher molasses removal in his sripping sage of he exracion process.

4 62 Liquid flow in a sugar cenrifugal 3 Model descripion When considering he problem of removing molasses from he crysals, he heory of hin fluid films applies. The model for he drainage of molasses from he sugar crysals is illusraed in Figure 2. I is assumed ha he crysals form a fla wall wih iniially a uniform layer of molasses. I is also assumed ha he molasses can flow freely hrough he ouer wall of he cenrifuge. The ouer wall is made of a fine mesh which permis he passage of liquid bu no he sugar crysals. The iniial hickness of he molasses layer is of order m. In Figure 2 he horizonal lengh scale is exaggeraed for clariy. The model fixes he sugar crysals in he form of a wall of crysals 10 cm high. The molasses forms a hin fluid film on he crysal wall which drains from he surface. The flow is driven by he cenrifugal force of magniude Ω 2 R per uni mass where Ω is he angular velociy of he cenrifuge and R is is radius. The drainage of molasses is a coaing flow problem and hin fluid film heory may be applied. Alhough he molasses is a non-newonian fluid i will be modelled as a highly viscous Newonian fluid. Figure 2: Drainage of molasses from crysals The recangular coordinae sysem is shown in Figure 3. The flow is wodimensional and here is no dependence on x. The equaion of he free surface of he molasses is y = h(z, ). The componens of he fluid velociy are v x = 0, v y = u(y, z, ), v z = w(y, z, ). (1)

5 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer 63 Figure 3: Coordinae sysem for a hin film of molasses draining from sugar crysals. For he drainage problem considered here he fluid pressure is consan. The fluid sicks o he wall and here is no shear force on he surface of he molasses, y = h(z, ). The boundary condiions are w(0, z, ) = 0, w (y, z, ) = 0. (2) y In he lubricaion approximaion of hin fluid film heory, he ineria erm is negleced. The viscous force and he cenrifugal force herefore balance and he hin film equaions reduce o µ 2 w y 2 = ρω2 R. (3) Inegraing (3) wice wih respec o y and imposing he boundary condiions in (2), we obain w(y, z, ) = ρω2 R y(2h y). (4) 2µ Now, by considering he conservaion of mass equaion and he free surface boundary condiion i can be shown ha where h + (h < w >) = 0, (5) z < w >= 1 h h 0 w(y, z, ) dy. (6)

6 64 Liquid flow in a sugar cenrifugal Using (4) i follows ha and we have ha < w >= ρω2 R 3µ h2 (7) h + ρω2 R h h2 µ z = 0. (8) This equaion corresponds o he exac equaion wihin hin film heory describing wo-dimensional drainage of a hin layer of viscous incompressible fluid down a verical surface due o graviy if Ω 2 R is replaced by g [2]. In order o non-dimensionalise (8) we define: h = h 0 h, = T, z = H z, (9) where h 0 is he iniial widh of he hin film, and he characerisic ime T is o be deermined. When (9) is subsiued ino (8) we obain and he iniial condiions become, h + ρ Ω2 R h 2 0 T h 2 h µh z = 0 (10) = 0, z = 0 : 0 h 1, (11) = 0, 0 < z 1 : h = 1. (12) The characerisic ime, T, which is an esimae of he ime aken for he molasses o drain from he sugar crysals, is herefore Using (13), (10) becomes T = µh ρω 2 Rh 2. (13) 0 h 2 h + h = 0. (14) z We will suppress he overhead bars in he subsequen analysis i being undersood ha dimensionless variables are being used. Consider now an esimae of he characerisic ime T. For he cenrifuge an esimae of he parameers in (11) a 60 C is: h 0 = m ρ = kg/m 3 H = 0.1m R = 0.5m µ = 100Pa s Ω = 100π 3 radians/sec Wih hese values of he parameers, T = 25 sec. This esimae for T is wihin he expeced range of values for he characerisic ime for filraion.

7 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer 65 4 Mahemaical soluion Consider he model of drainage of molasses from sugar crysals as shown in Figure 3. The hickness h(z, ) of he molasses beween z = 0 and z = 1 is described by he firs order quasi-linear parial differenial equaion subjec o he iniial condiions h + h2 h z = 0, (15) = 0, z = 0 : 0 h 1, (16) = 0, 0 < z 1 : h = 1. (17) The condiions, (16) and (17), which are illusraed in Figure 4, assume ha here is iniially a uniform layer of molasses on he sugar crysals. The molasses escapes from he cenrifuge for z > 1. The differenial equaions of he characerisic curves of (15) are and he characerisic curves are hus d 1 = dz h 2 = dh 0 (18) h = c 1, (19) z c 2 1 = c 2, (20) where c 1 and c 2 are consans. The general soluion of (15) is h = F (z h 2 ), (21) where F is an arbirary funcion. Consider firs 0 < 1. From he iniial condiion (17), and herefore F (z) = 1, 0 < z 1, (22) h(z, ) = 1, < z 1. (23) To deermine he soluion for 0 z, consider he iniial condiion (16). From (20), c 2 = 0 and since 0 h 1, h(z, ) = [ z ] 1 2, 0 z. (24)

8 66 Liquid flow in a sugar cenrifugal For > 1, h(z, ) = [ z ] 1 2, 0 z 1. (25) The spaial gradien is infinie a z = 0: h 1 (z, ) = z 2(z) 1 2 as z 0, > 0. (26) Figure 4: The iniial condiions (16) and (17) for he parial differenial equaion (15). Graphs of h(z, ) ploed agains z (0 z 1) for a range of values of are presened in Figure 5. The graphs illusrae he evoluion of he hickness of he layer of molasses on he crysals. For > 1 he hickness of he layer of he molasses on he crysals is less han uniy and channels sar o form in he molasses which is illusraed in Figure 5. Since his firs occurs a = 1, i shows ha he ime T defined by (13) is a suiable characerisic ime for drainage of he molasses. 5 Analysis of resuls The objecive is o drain as much molasses as possible from he sugar crysals in he leas amoun of ime. Le V () be he volume of he molasses remaining on he crysal wall per uni widh afer draining for ime. The volume V () is a useful

9 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer 67 Figure 5: Graph of h(z, ) agains z for = 0.5, 1, 1.5, 2 and 10. measure of he efficiency of he draining process. The cases 0 < 1 and > 1 need o be reaed separaely. For 0 < 1, V () = and using (23) and (24) we obain and hus Also, For 1 and using (25), which simplifies o V () = 0 h(z, )dz + 0 V () = z dz h(z, )dz, (27) 1dz (28) V () = 1 1. (29) 3 dv d = 1 3. (30) 1 0 V = 2 3 z dz, (31) (32)

10 68 Liquid flow in a sugar cenrifugal and we have dv d = (33) From (30) and (33) we see ha he molasses drains rapidly and a a consan rae for 0 < 1 bu for > 1 he rae of drainage decreases. A graph of he volume remaining, V, ploed agains is shown in Figure 6. Afer draining for ime = 15, approximaely 20 % of molasses is no drained and hus his percenage of molasses is no exraced. Consider now he hickness of he molasses layer on he crysals. A z = 1, h(1, ) provides an indicaion of he spacing beween adjacen fluid segmens a he wall of he cenrifuge as shown in Figure 2. I gives a measure of he growh of he channeling in he molasses. From (23) and (25) 0 < 1 : h(1, ) = 1, (34) 1 : h(1, ) = 1. (35) The graph of h(1, ) ploed agains ime is shown in Figure 7. We see ha a Figure 6: Graph of he volume remaining, V (), ploed agains drainage ime. channel is formed in he molasses a ime = 1. The rae of increase of he widh of he channel decreases as increases. Channelling of he wash waer is someimes observed where waer flows hrough par of he molasses layer and does no wash he remainder. We also see from Figure 7 ha afer ime = 15, abou 25 % of he molasses layer remains a he wall of he cenrifuge. Because of he slow rae of removal of he molasses from he crysal for > 1 mehods of enhanced molasses removal were analysed.

11 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer 69 Figure 7: Graph of h(1, ) ploed agains ime for Enhanced molasses removal The mehods used o enhance he removal of he molasses all reduce he viscosiy, µ, of he molasses. The characerisic drainage ime, T, defined by (13) is proporional o µ and will be reduced when µ is reduced. A mehod which is used o enhance molasses removal is waer spraying. The waer diffuses ino he molasses which is dilued and is viscosiy is reduced. The characerisic ime, T D, for waer diffusion ino he molasses is T D = h2 0 D W, (36) where h 0 is he iniial hickness of he molasses layer on he crysals and D W is he diffusion consan for diffusion of waer ino he molasses. Now and herefore, h 0 = m, D W = m 2 s 1, (37) T D = 80 sec. (38) Since he characerisic ime for he molasses o drain is T = 25 sec, waer akes longer o diffuse ino he molasses han he ime aken for he molasses o drain. Hydrogen peroxide is also sprayed on he molasses o bleach he molasses o a ligher colour [3, 4]. Spraying is done lighly for several minues. The hydrogen peroxide will remove he molasses in he same way as waer. The characerisic diffusion ime for hydrogen peroxide o diffuse ino he molasses is assumed o be

12 70 Liquid flow in a sugar cenrifugal he same as for waer. A hird way o enhance molasses removal is by spraying wih seam. The molasses is heaed by he seam which reduces he viscosiy. The characerisic ime for hea o diffuse ino he molasses is T H = h2 0 D H, (39) where D H is he diffusion consan for he diffusion of hea ino he molasses. Now, D H = k ρc p, (40) where k is he coefficien of hermal conduciviy, ρ is he densiy of he molasses and c p is he specific hea a consan pressure of he molasses. Now, k = 0.4 J m 1 s 1 K 1, ρ = kg m 3, c p = J kg 1 K 1 (41) and herefore T H = 0.6 sec. (42) Since T H << T, spraying wih seam is a pracical way o enhance molasses drainage and is a much beer way han diluing wih waer or hydrogen peroxide. The emperaure dependence of he viscosiy of molasses is given by [5] µ = µ 0 [ ] K , (43) K 2 where µ 0 is a consan and K is he absolue emperaure. The raio of he characerisic ime for he drainage a 100 C o he characerisic ime a 60 C is T (100 C) T (60 C) = µ(100 C) µ(60 = (44) C) Bu for > 1, he volume V of molasses remaining is, from (57), V = 2 3 where is he unscaled ime and herefore V (100 C) V (60 C) = [ ] 1 T 2, (45) [ T (100 ] 1 C) 2 = (46) T (60 C)

13 C.P. Please, N.D. Fowkes, D.P. Mason, C.M. Mmabaho, A Huchinson, M.C. Rademeyer T = 60 C T = 100 C Figure 8: Graph of he volume, V, of molasses remaining ploed agains ime a T = 60 C and T = 100 C. In Figure 8 he volume of he molasses remaining a 60 C and 100 C are ploed agains he scaled ime for 60 C for > 1. We see ha spraying by seam reduces by abou half he volume of he molasses remaining. We conclude ha heaing he molasses by spraying wih seam is a beer way o enhance he molasses removal han by diluing he molasses by spraying i wih waer or hydrogen peroxide. Spraying wih hydrogen peroxide of course has he oher use in he manufacuring process of bleaching he molasses o a ligher colour. 7 Conclusions The characerisic ime for he drainage of molasses, defined by (13), was calculaed o be 25 sec and is wihin he correc order of magniude. I is useful because i shows how he parameers µ, Ω and R can be varied o conrol he removal of he molasses from he crysals. The characerisic ime does no depend on he permeabiliy of he massecuie, he value of which would be difficul o esimae accuraely. The permeabiliy would occur in drainage models formulaed in erms of Darcy s law for flow hrough a porous medium. The spraying of hydrogen peroxide is used o bleach he molasses. The diffusion ime for hydrogen peroxide ino he molasses is abou 80 sec. The molasses layer

14 72 Liquid flow in a sugar cenrifugal herefore needs o be kep for several minues for he hydrogen peroxide o diffuse ino he molasses. The hydrogen peroxide will also remove he molasses from he crysals in he same way as waer. We found ha channels can occur in he bed of crysals a he ouer wall of he cenrifuge. This is due o he draining of he molasses from he crysals. Channelling is someimes observed when wash waer flows hrough par of he bed and does no wash he remaining par. Washing by waer was found no o be a good way o enhance he removal of molasses because he characerisic ime for diffusion of waer ino he molasses is abou 80 sec. Also he spraying wih waer dissolves some of he sugar crysals leading o losses in producion. By far a beer way o remove he remaining molasses is o hea he molasses by spraying wih seam. The characerisic ime for he diffusion of hea ino he molasses is abou 0.6 sec. Spraying wih seam can reduce by abou half he volume of he molasses remaining on he crysals. References [1] Greenspan, H.P. The heory of roaing fluids, Cambridge Universiy Press, Cambridge, [2] Acheson, D.J. Elemenary Fluid Dynamics. Clarendon Press, Oxford, 1990, pp [3] Jullienne, L.M.S.A. Washing sugar in bach A-cenrifugals. Proc. Souh African Sugar Technologiss Assoc. June 1983, pp [4] Saska, M. Reducing sugar colour by washing and bleaching wih dilue hydrogen peroxide. In. Sugar J., 109, 2007, [5] Barker, B. Massucuie consisency- A brief review. Sugar Milling Research Insiue, Durban, Souh Africa, Technical Repor No 1934, 25 March 2004.