4.3 Effective Heat and Mass Transport Properties

Transcription

1 4.3 Eectie Heat and Ma Tranport Propertie Where aailable you hould alway ue experimentally meaured alue or thermal conductiity and diuiity. For irt etimate and where experimental data are not aailable, eeral correlation are uggeted below THERMAL CONDUCTIVITY Suppoe you hae a lurry o olid particle haing a thermal conductiity,, and a continuou luid phae with conductiity. At zero low condition, a a irt approximation, you might expect that the thermal energy will pa through the lurry with an eectie thermal conductiity that i proportional to the olume raction o the material. Thi i analogou to aying that in the ideal model, a hown in Figure 4- that the total heat lux, Q i the um o the total heat luxe through the eparate phae, Q and Q, a gien by Q A Q + Q T A x ( ε + ( ε) ) T x + A T x (4-3) It i aumed in thi idealized cae that the temperature proile through the dierential element are linear and that the heat lux i only in the x-direction. Hence we conclude that the eectie thermal conductiity i gien by ε + ( ε) T Surace Temperature T Direction o Heat Flow (4-3) Area o Fluid Phae A Fluid Phae Area o Solid Phae Solid Phae A x Figure 4-. A thin dierential element o a lurry with dierential thicne x. The temperature change acro the dierential element i gien by T T T. The urace o the two phae in contact with the boundarie at each ide o the dierential element are A and A. The ratio o luid area to total area equal the poroity, ε A. A 4-7

2 I the olid phae i non-conducting then one would expect the eectie thermal conductiity to be related to the poroity by o ε (4-33) a deduced rom Eq.(4-3) by etting equal to zero. Maxwell (J.C. Maxwell, A Treatie on Electricity and Magnetim, Vol., 3 rd ed., Doer, New Yor, 954) experimentally teted the analogou electrical conductiity problem and deried the correlation o ε 3 ε (4-34) Equation (4-33) and (4-34) are plotted in Figure 4-. From the plot we ee that the idealized cae rom Eq.(4-33) ollow the ame trend a determined rom Maxwell and oer predict by about % in the.4 to.6 poroity range. In many engineering application Eq.(4-33) may be adequate / Ideal Maxwell Poroity, ε Figure 4-. Eectie conductiity eru poroity baed on Eq.(4-33) and (4-34). 4-8

3 4.3. MASS DIFFUSIVITY Ma diuiity in a multiphae ytem i analogou to thermal conductiity. By replacing the thermal conductiity term with diuiity term in Eq.(4-3) and (4-34) you hae the analogou expreion relating eectie diuiity to the poroity and the diuiitie o the indiidual phae HEAT TRANSFER COEFFICIENT For ga phae low through a paced bed o olid, the wall heat traner coeicient, h, i etimated by the ollowing empirical equation (McCabe and Smith, ibid). hd p.5.33 N u.94rep Pr (4-35) g Gd p In Eq.(4-35) the Reynold number i deined a, R ep, where G i the ma lux µ (ma per area o bed per time). Unortunately, thi expreion i limited to paced bed o porou inorganic material uch a alumina and ilica gel with a poroity o approximately.3. Other correlation are needed to tae into account the wide range o poroitie that are poible with lurrie. For dilute lurrie, the Sieder Tate equation can be ued a an approximation, or turbulent low in a tube. In the Sieder Tate equation, the bul lurry propertie are ubtituted or the luid propertie: N u.3r.8 e P.333 r µ µ w hd ρ VD C pµ where N u, R e, and P r. The bul denity and icoity were µ dicued preiouly. The bul heat capacity i gien by C εc + ( ε) C p p p.4 (4-36) (4-37) MASS TRANSFER COEFFICIENT By analogy, the lurry ma traner coeicient at the wall are analogou to thoe gien by Eq.(4-35) and (4-36) or heat traner coeicient. The heat traner Nuelt number Nu i replaced by the ma traner Sherwood number Sh, and the Prandtl number Pr i replaced by the Schmidt number Sc. Thee number are deined by S D x h (4-38) cdab 4-9

4 S c µ (4-39) ρd AB DISPERSIVITY (FLOW EFFECTS) Numerou correlation are aailable in reerence and textboo. For example, in McCabe and Smith (ibid) or a porou inorganic material uch a alumina, ilica gel, or an impregnated catalyt, the eectie thermal conductiity o the bed i proportional to the ga phae thermal conductiity, g, gien by g 5 +.R P (4-4) ep r Thi how that the eectie thermal conductiity i aected by the low rate, due to a diperion mechanim. Diperion i caued by the luid ollowing tortuou path and becoming intermixed in the lateral direction normal to the preailing direction o low. 4.4 Hindered Settling What happen when particle ettle in concentrated olution? A each particle all it diplace luid which in turn mut moe upward. In a concentrated ytem thi caue an upward luid motion which interere with the motion o other particle. Thi i hown in Figure 4-3. DISPLACED FLUID FALLING PARTICLES Figure 4-3. Hindered ettling: a a particle all it diplaced luid moe upward and low the obered ettling rate o neighboring particle. 4-

5 The net eect i a lower, hindered, ettling rate or the group o particle a compared to the ree ettling terminal elocity o one particle by itel. Coe and Cleenger (Tran. Am. Int. Min. Met. Eng. 55, 356, 96) obered that during a batch ettling operation, the edimenting luid deelop eeral zone (Figure 4-4). In zone A the particle are in low concentration and ettle at their terminal elocity. In zone B and C the particle are in hindered ettling. In zone D the ediment ha particle in contact with each other; the particle are no longer ettling though the ediment may compact due to the weight o the oerburden. The concentration o the particle in zone D near the C - D interace i approximately that o looe pacing a gien by the correlation in Figure 4-5. Not all our zone are preent in all ettling procee. A Zone A clear liquid zone. Zone B contant compoition zone. Zone C ariable compoition zone. Zone D ediment. B C D Figure 4-4. Zone o ettling obered by Coe & Cleenger. T. Allen (Particle Size Meaurement, Volume, 5 th ed, Chapman & Hall, London, 997, page 4) note that zone B ettle in ma and the relatie motion o luid to particle i analogou to low through a paced bed, hence Eq. (4-5) could be ued here to model the motion o zone B. Maude & Whitmore (Br. J. Appl. Phy. 9, , 958) modeled the hindered ettling proce a a power law in the concentration (olume raction o the liquid phae) u n u ε t (4-4) where or dilute olution ε and u u t. Here u t i calculated a in Chapter 3 or a n ingle particle alling through a clear luid and ε account or the hindered ettling eect. The parameter n i determined experimentally. Unortunately n i not a contant but arie a a unction o the particle geometry and the Reynold number. Perry Handboo (6 th ed, pg 5-68) how that n arie rom.3 to 4.5 or pherical particle and ha a dramatic eect on the calculated alue or the hindered ettling elocity. 4-

6 In the ection that ollow a rational approach to hindered ettling i decribed in which the particle ettle through the lurry intead o the clear luid. Thi approach i a preerred alternatie to the Maude & Whitmore approach RATIONAL ANALYSIS OF HINDERED SETTLING The primary reaon or the phenomena o hindered ettling are: (a) Large particle all at a dierent rate relatie to a upenion o maller particle o that the eectie denity and icoity o the luid are increaed, (b) In high concentration, larger olume o luid are diplaced cauing an upward luid elocity. The ettling elocity to an external oberer i dierent than the eectie elocity dierence be the two phae, and (c) Velocity gradient in the luid near the particle urace are increaed a a reult o the concentration o the particle. Terminal elocity o a ingle particle i correlated through the drag coeicient C D a gien by the deining equation relating the inetic orce acting on the particle and the particle inetic energy, F Cd A KE π d p C D ut ρ 4 (4-4) where u t i the obered elocity o the particle relatie to the tationary eel wall. actually repreent the elocity dierence between the particle and the tationary luid phae, u. (4-43) t When ettling occur in a large eel o cro-ectional area A the diplaced luid elocity i negligible. Let the z-direction be the direction o graity; then the particle hae a poitie elocity in the z-direction and the luid ha a negatie elocity oppoite to the direction o graity. At teady tate the olume rate o low o particle downward mut equal the olume rate o low o luid upward. We can write thi a πd 4 p A where A i large and hence i mall compared to. In hindered ettling the olume rate o low o particle i related to the luid phae elocity through the olid phae olume raction ( ε ) and the eel cro ectional area, A, by or u t (4-44) ( ε ) A εa (4-45) 4-

7 ( ε) ε. (4-46) Since the terminal elocity i the relatie elocity dierence between the olid particle downward motion and the luid phae upward motion, then u t + ε ( ε) ε and the elocity obered by an external oberer i (4-47) ε u. (4-48) t Thi i conitent with the extreme cae o dilute olution. In the limit when only one particle i preent, ε, the obered elocity approache the terminal elocity, u t. For more concentrated olution the particle interere with the drag coeicient on each other. Dai and Hill (J. Fluid. Mech, 36, , 99) tudied hindered ettling with phere alling through lurrie o neutrally buoyant particle. Their wor aume Brownian motion and interparticle attractie/repulie orce are negligible. The reult o their wor how the elocity eect are nearly independent o particle ize ratio. Geanopli (Tranport Procee and Unit Operation, 3ed, Prentice Hall, Englewood Cli, 993, pg 8) ugget that we replace the luid phae denity and icoity, ρ and µ in the hindered ettling correlation with the lurry bul denity and bul icoity, ρ o and µ o, where ρ ε ρ + ( ε ) ρ p ( ) (4-49) and µ µ ε (4-5) i a unction o the luid phae olume raction, ε, a related through Eq.(4-3) to (4-9). Eectiely we are aying that the all o a ingle particle in a lurry i the ame a i all o the other particle in the lurry are part o the urrounding luid phae. where ( ε ) For neutrally buoyant particle in the lurry (but the alling particle i not neutrally buoyant), in the Stoe Law range the obered elocity i gien by the modiying Stoe Law, Eq.(3-6) to be ( ρ ρ ) gd o p p ut. (4-5) 8µ I all o the particle in the urrounding lurry are alo etting, then we mut tae into account the upward motion o the luid phae a done in Eq.(4-4) which gie 4-3

8 ( ρ ρ ) ε gd o p p εut. (4-5) 8µ Thi aume all o the particle are approximately the ame ize and denity. How can we approach thi problem i the particle hae a ariation in ize and denity? I there i a ariation in the ize or denity o the particle, then the dierent type o particle will ettle at dierent rate. Let i be the obered elocity o the i th type o particle (o ize d pi and denity ρ i ) which occupy a olid phae olume raction total olume). ε i (olume o all i th particle diided by The bul denity become ρ ε ρ + ε i ρi (4-53) where ε i ε ( ε) i the total olume raction occupied by the olid phae. We hae no additional inormation on the bul icoity o we ue the ame model a gien in Eq.(4-3) through (4-9). Since the elocitie are dierent, we mut relate all o the elocitie to the olumetric rate o diplacement, Q Aε A εi The luid phae diplacement i gien by i (4-54) Q A ε Q (4-55) We are not intereted in a ma aerage olid phae elocity. The drag coeicient correlation relect the act that we are accelerating the luid around the particle, hence we are actually intereted in the olumetric low rate o we can relate it to the luid phae ma. The terminal elocity o the i th ize particle i which can be manipulated a uti i (4-56) 4-4

9 u ti Q i + Aε A εi i + Aε ε i i + + ε i j i ε ε j j (4-57) or upon rearrangement, we get the expreion i εu ti j i ε ( ε + ε) i j j (4-58) which i the expreion that applie to ettling Zone C. In etimating alue or u ti in Eq.(4-58) one may ue the modiied orm (ubtitute bul denity and bul icoity or the luid propertie) o Stoe Law, Newton Law or intermediate range, depending upon the Reynold number or the particle ize. Since Eq.(4-5) i the mot general orm we can apply it to eeral example cae to demontrate it utility CASE STUDY COMPARISONS IN HINDERED SETTLING CASE : One particle in ree ettling. Thi i the implet cae. Since there i only one particle, there i only one particle ize. The luid occupie a igniicantly greater olume hence ε and ε, the luid denity i the ame a the bul denity, ρ ρ, and the luid icoity i the ame a the bul icoity, µ µ. Equation (4-58) reduce to where the ummation term () u u (4-59) t t ( + ) ε j j i j i zero becaue no term exit or j. Hence Eq.(4-58) reduce to the olid elocity equal the terminal elocity, a expected. CASE : One particle ettling in a lurry o neutrally buoyant particle. In thi cae, the neutrally buoyant particle aect the bul icoity o the lurry, but not the bul denity. The neutrally buoyant particle are o a concentration repreented by olume raction ε which i non zero. The ettling particle, a in Cae, ha a olume raction o eentially zero, ε. 4-5

10 Since all o the olume raction mut um to, we get ε ( ε ) gie the etimated elocitie to be εu ε u t ( + ε) t. Equation (4-58) (4-6) Where the elocity o the neutrally buoyant particle i zero becaue the remain motionle with the luid phae. Thi tell u that the particle ettle at it modiied terminal elocity rate where u t u t ( ρ, µ ). A a conitency chec, Eq.(4-58) gie the elocity o the neutrally buoyant particle to be zero,. CASE 3: Group o particle o ame ize all ettling at the ame rate. Thi cae i more complex than Cae becaue now there are many particle ettling, not jut one particle. There are no neutrally buoyant particle preent in thi ytem. The olume raction, ε occupied by the particle i not zero. Thi reult in ε + ε. Since only one type o particle i preent in the ytem, the ummation term in the numerator o Eq.(4-5) ummed oer j i i identically zero. The obered elocity become ε ut (4-6) where the terminal elocity i a unction o the bul denity and icoity. CASE 4: Two ize o particle ettling at dierent elocitie. Thi i the mot complex cae that will be conidered here. In thi cae there are two particle ize, denoted and. Neither o the particle are neutrally buoyant. The olume raction are related by ε + ε + ε. (4-6) The obered elocitie are determined rom Eq.(4-58) to be and εu ε (4-63) t ( ε ) εu ε (4-64) t ( ε ) where we ee that the two elocitie are interdependent. I we ue Eq.(4-64) to eliminate the elocity o particle rom the right ide o Eq.(4-63) then we get the elocity o particle a a unction o the terminal elocitie o both particle, ( ) ε u ε u. (4-65) t t 4-6

11 With Eq.(4-65) we can calculate the obered ettling elocitie or the two type o particle preent in the lurry. The limitation o the reult are reiterated here or emphai. Thee reult aume laminar, low low condition. I there are any diturbance in the lurry then the eddy current will dirupt the low pattern. Alo, thee reult only hold a long a the olume raction are contant. The ettling proce i inherently unteady hence thee reult only hold at the moment in time that the concentration are thoe ued in the equation; the concentration will ary with time and poition. Sedimentation o multicomponent mixture ha been the ubject o numerou paper in literature. See or example: J. F. Richardon, and R.A. Meile, Sedimentation and Fluidization Part III, The Sedimentation o Uniorm Fine Particle and o Two- Component Mixture o Solid, Tran. Int Chem. Engr, 39, , 96. EXAMPLE 4-3. Calculate the obered ettling elocitie o a mixture o latex phere and and in water. Suppoe a mixture o latex phere (39 micron, intrinic denity o.8 g/cm 3 ) and and ( mm, intrinic denity o.5 g/cm 3 ) are ettling in water at room temperature. Etimate the obered elocitie o the latex and the and. The mixture i uniormly mixed (initially) with ε latex. and ε and 5. a the olume raction occupied by each type o particle. The water olume raction i gien by ε εlatex εand 85.. The water icoity i. cp and uing Eq.(4.4) the lurry bul icoity i etimated to be.49 cp. With the water denity o g/m 3, the bul denity i calculated uing Eq.(4-) to be 93 g/m 3. Auming Stoe law range, the terminal elocity o the latex particle i calculated to be 4.394x -5 m/ uing Eq.(3-6) modiied with the bul denity and icoity ( ρ ρ ) gd latex latex u latex. (4-66) 8µ The Stoe Law aumption i checed by the Reynold number which calculate to be.3 uing ρ u latexd latex R e. (4-67) µ Similarly, the terminal elocity or the and i etimated rom Stoe Law to be.46 m/ but the Reynold number i 4 which place the and in the intermediate range. A chec o Newton Law range, gien by Eq.(3-6) modiied with the bul denity and icoity gie a terminal elocity o.649 m/ and Reynold number o 5, alo indicate the intermediate range. Hence, the intermediate range Eq.(3-6) mut be ued, which i modiied a 4-7

12 with the drag coeicient gien by rom Figure 3-. d 4 and g ρ ρ u and and 3 (4-68) CD ρ C D 85. / 35 / R ep (4-69) Since the terminal elocity in Eq.(4-68) depend upon itel, through the Reynold number, then an iteratie olution i needed. Uing ucceie ubtitution a gue or the terminal elocity i ued to calculate the Reynold number, calculate the drag coeicient, and calculate a new gue or the elocity. Thi et o calculation i eaily computed uing a computer preadheet. gue u Re Cd calc u Hence the terminal elocity o the and i u and.649 m/. Uing Eq.(4-57) the oberable elocitie o the latex and the and are and 66. m/ and latex. 578 m/. The negatie alue or the obered latex elocity indicate that the elocity i upward! The oberable elocity o the water may be determined rom Eq.(4-58) with ome manipulation a εi i ε (4.7). 67 m/ or which the minu ign indicate upward low. Flotat (Hungarian J. Indut. Chem, 3, 5, 995) argue that in edimentation the mallet particle may moe upward with thi luid motion a indicated by the negatie luid and latex elocitie. 4-8

13 4.5 Slurry Flow Cheremiino boo (N.P. Cheremiino and R. Gupta, Handboo o Fluid in Motion, M.C. Roco ed., Ann Arbor Science, Ann Arbor, Michigan, 983) and the Encyclopedia o Fluid Mechanic are two example o good reerence that dicu lurry low. Many concentrated upenion diplay non-newtonian low behaior, een when the upenion are pherical. Thereore the bul icoity correlation decribed preiouly in Section 4. mut be ued with ome caution. Some lurrie can be inluenced by electrical and magnetic orce that change their low behaior. Electrorheological luid (H. Conrad, Structure and Mechanim o Electro- Rheological (ER) Fluid, Chapter, in Particulate Two-Phae Flow, Butterworth, Boton, 993) are luid that change propertie under the inluence o electrical ield. With no electrical ield the luid low lie a Newtonian; with the electrical ield the luid low lie a yield-tre luid. The ER luid i made up o a non-conducting liquid phae and a conducting or emiconducting particulate phae. The mechanim by which thi wor i due to the particle lining up when an electrical ield i applied. Thi i hown in Figure 4-5. Thi i one example o how lurry behaior i not a imple extenion o the carrier luid propertie. Figure 4-5. Alumina particle in a ilicone oil line up in iber when an electrical ield o olt per centimeter i applied. Another way that lurry behaior dier rom the carrier luid i when phae eparation occur. Thi i epecially true or ga-liquid ytem. Kao (D.T.Y. Kao, Rheology o 4-9

14 Supenion, Chapter 33, in Handboo o Fluid in Motion, N.P. Cheremiino and R. Gupta ed., Ann Arbor Science, Ann Arbor, Michigan, 983) claiie upenion low behaior a: Single Phae Fine Diperion Coare Diperion Macro-mixed Stratiied Homogeneou Peudo-Homogeneou Heterogeneou Heterogeneou Heterogeneou A ingle phae by deinition i homogeneou, becaue it material content doe not ary with poition. Fine diperion are termed peudo homogeneou becaue een though two phae are preent, at a local cale the two phae are well among each other. Coare diperion are typically thoe with large particle (relatie to the ize o the lurry pipeline) and the larger particle may hae a tendency to ettle quicly. Macro-mixed diperion hae region o high concentration o particle and region o low concentration o particle, and thee region are located omewhat randomly throughout the lurry. Stratiied low hae region o high concentration o particle in a layer located at the bottom o a pipe, or example, and low concentration at the top o the pipe. 4-3