*-_‘~‘_‘A4 THE EFFECT OF THE COLUMN TO , PARTICLE DIAMETER RATIO ON THE , g CORRELATION or THE PRESSURE DROP _ o- 0v...” 0-..”... .p- y... "VERSUS FLOW RATE osrwl-Ds' 5.- _ THROUGH PACKED BEDS Tim {0. 1|... 0me of M. 5.. _- _ MICHIGAN STATE UNIVERSITY- Dévendra Mehté — - 1966 ........ .......... 93 310128 0182 _ 3: I , IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 1J1, ] I boa-mg] ABSTRACT THE EFFECT OF THE COLUMN TO PARTICLE DIAMETER RATIO ON THE CORRELATION OF THE PRESSURE DROP VERSUS FLOW RATE OF FLUIDS THROUGH PACKED BEDS by Devendra Mehta Ergun has established a correlation between the pressure drop in a packed column and the flow rate of a fluid through it. This correlation is considered valid only when the diameter of the column is much larger than the diameter of the packed particles; the Ergun correlation does not include the wall effect on the pressure drop versus flow rate correlation. In this project, the Ergun correlation was modified to in- clude the wall effect which eliminated the assumption that the diameter of the column should be much larger than that of the packed particles. Experiments to obtain data for the pressure drop in packed beds were performed for cases where the wall effect was important. The investigations were performed with an one-half inch diameter column packed with Spherical glass beads. Water was used as a fluid flowing through packed beds. Pressure drop - flow rate data were plotted on a log-log Apgp D graph in the form of the packed friction factor, ( 2c )(EB) X G 3 D G E . _L_ (1-6 versus Reynold 3 number, (”(1_€)). It was observed that, when the wall effect was significant, the plot of the above groups deviated from the plot of the Ergun correlation. However; Devendra Mehta Apgcp D after the friction factor and Reynold's number, (-—E-)(EE) X G ( E3)( 1 ) and ( DpG )( 1 ) resPectively l-E AD + l p(l-E) AD + l” ’ .___B___ 6Dc(l-E) 6DC(l-E) were modified to include the wall effect as a parameter, a log- log plot of the above groups coincided with the Ergun correlation. THE EFFECT OF THE COLUMN TO PARTICLE DIAMETER RATIO ON THE CORRELATION OF THE PRESSURE DROP VERSUS FLOW RATE OF FLUIDS THROUGH PACKED BEDS By Devendra Mehta A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1966 TO MY PARENTS AND BROTHERS ii ACKNOWLEDGMENT The author gratefully acknowledges the Department of Chemical Engineering for the financial support of this work. Sincere appreciation is also extended to Dr. Martin C. Hawley for his invaluable assistance and guidance during the course of the project. iii TABLE OF CONTENTS P_a_g£ ACKNOWLEDGMENT .. ----------------------------------------- 111 LIST OF TABLES ........................................... v LIST OF FIGURES ------------------------------------------ vi INTRODUCTION ............................................. 1 HISTORY .................................................. 4 SCOPE OF THE PROBLEM ---------------------------------- —-- 12 THEORY ................................................... 14 EXPERIMENTAL EQUIPMENT ................................... 24 GLASS BEADS .............................................. 24 MANOMETER FLUID .......................................... 24 MEASUREMENT OF VOID FRACTION ----------------------------- 26 PACKING PROCEDURE ........................................ 27 RUN PROCEDURE ............................................ 27 RESULTS -------------------------------------------------- 29 CONCLUSIONS .............................................. 32 APPENDIX A ---------------------------------------------- 33 APPENDIX B .............................................. 40 APPENDIX C .............................................. 49 NOMENCLATURE ............................................. 50 BIBLIOGRAPHY ............................................. 52 Table Data Data Data Data Data Data for for for for for for DC/Dp Dc/Dp DC/Dp DC/Dp DC/Dp D /D LIST OF TABLES 7. 91: 45: 1 ..................... 1 ..................... 36:1 ..................... 25: 18: 7: 1 ..................... 1 ..................... 1 ..................... Page 34 35 36 37 38 39 LIST OF FIGURES Figure Page 1. Velocity Distribution for the Flow in a Cylindrical Tube -------------------------------- 14 2. Schematic Diagram of the Equipment -------------- 25 3. Plot of the Friction Factor versus Reynold's Number of the Packed Bed for Dc/Dp = 91:1 ------ 41 4. Plot of the Friction Factor versus Reynold's Number of the Packed Bed for DC/Dp 2 45:1 ------ 42 5. Plot of the Friction Factor versus Reynold's Number of the Packed Bed for Dc/Dp = 36:1 ------ 43 6. Plot of the Friction Factor versus Reynold's Number of the Packed Bed for Dc/Dp = 25:1 ------ 44 7. Plot of the Friction Factor versus Reynoldls Number of the Packed Bed for Dc/Dp = 18:1 ------ 45 8. Plot Of the Friction Factor versus Reynold's Number Of the Packed Bed for Dc/Dp = 7.7:1 ----- 46 9. Plot of the Friction Factor versus Reynold's Number of the Packed Bed for Various DC/Dp, Wall Effect Neglected --------------------------- 47 10. Plot of the Friction Factor versus Reynold‘s Number of the Packed Bed for Various Dc/Dp’ Wall Effect Included ---------------------------- 48 11. Microphotograph of the Glass Beads -------------- 49 vi INTRODUCTION Ergun (5) has established a correlation between the pressure drop in a packed column and the flow rate of a fluid through it. The Ergun equation when stated in dimensionless groups is: Apg p D 3 ( c)(_2)(.1_§_€_)=(15_3_%&)+ 1.75 G2 L where Ap = pressure drop through a column having length L, gC = gravitational constant, p = density Of the fluid flowing thrbugh the column, G = mass flow rate per unit area, Dp = diameter Of the particles, L = length Of the packed column, 6 = void fraction, u = viscosity of the fluid. Many exPeriments have been performed by various investigators (2, 3, 4, 6, 7, 8, 9, 10), and it was found that the results from those experiments fit the Ergun equation well. However, Ergun's correlation has the limitation that the effect of the wall on the pressure drop through the bed is neglected. The deviations from the Ergun correlation are negligible if the ratio of the diameter of the column to the diameter of the packing particles is larger than 50:1. This assumes that the wall effect on the hydraulic radius -- the characteristic length dimension in the Reynold's number -- of the packed bed is negligible. Several investigators (3, 4, 5) have stated that the wall effect may be important for column to particle diameter ratios in the range of 8:1 to 20:1. In fact, it is safe to state that, if the ratio of the column to particle diameter is less than 50:1, the wall effect on the hydraulic radius is significant. Investigators failed to establish the magnitude of the wall effect. The purpose of this investigation was to examine the effect of the wall on the flow characteristics of fluids in packed beds and to determine the validity of the modified Ergun equation which includes the wall effect on the pressure drop through a packed column. EXperiments were performed at low flow rates with packed bed Reynold's numbers, ranging from 0.1 to 10.0. The apparatus was a column packed with Spherical glass beads. Nitrogen pressure was used to maintain constant flow in the bed. The pressure drOp between two positions in the bed was measured at various liquid flow rates through the packed column. The column diameter was kept constant, whereas the particle size was varied such that the column to particle diameter ratios ranged from 92:1 to 8:1. The glass beads used in packing the column were uniform and Spherical. Water was used as the fluid. Pressure drOp and flow rate data were obtained at various Apgcp D 63 D G diameter ratios. Values of ( 2 )(-LE) (-1_—€) and TIE-5:5- were G calculated and a log-log plot of these groups was made in order to compare it with the Ergun plot. It was found that at the smaller diameter ratios, the wall effect on the hydraulic radius or on the flow characteristics of fluids in packed beds cannot be neglected. This was observed from the deviation of the plot from the Ergun equation for the smaller ratio of column to particle diameter. If the corrected Ergun equation is used, there is no deviation Observed when the data for smaller diameter ratios, as well as for larger diameter ratios, are replotted. There- fore, if corrections are made by including the wall effect in the Ergun equation, it is not necessary to assume that the diameter of the column should be much larger than that of the particles. This correction involves including the effect of the tube wall on the hydraulic radius of the packed bed. Details of this correction are explained in the theory section. HISTORY The pressure drop due to the flow Of fluids through packed columns has been the subject of experimental investigation by many workers (2, 3, 4, 6, 7, 8, 9, 10) to determine the corre- lation between the pressure drop and the flow rate of fluid through packed columns. Those correlations differ in many reSpects; some are to be used for low flow rates, while others are to be used at high flow rates. Previous workers (2, 3, 4, 6, 9) derived relations using different assumptions and corre- lated the particular eXperimental data Obtained with or without some of the data published earlier. They agreed that the ex- pressions relating the pressure drop along the length Of bed and the flow of fluid through the bed contain the following factors: 1. Pressure drop along the length of bed and the flow rate of the fluid, 2. Density and viscosity of fluid, 3. Void fraction of the bed, 4. Shape and the surface of the particles. It was Reynolds (5), who first formulated the relation between the pressure dr0p and the flow rate. He stated that the resis- tance Offered by friction to the motion of the fluid was the sum Of two terms. He proposed that the first term was proportional to the first power of the velocity of the fluid and the second term was proportional to the product of the density of the fluid and the second power of the velocity. 2 A2: —— L a v0 + b 9 v0 (2) Here, Ap is the pressure drop along the bed of length L, V0 is the linear velocity of fluid, 0 is the density of the fluid, and a and b are constants. This relation was tested by Ergun and Orning (6) as well as by Morcom (9). They plotted the values of .%§ against 9 v0 which were obtained from their 0 investigation, and straight lines were Obtained as expected from equation 2, They noted that the values of a and b were different depending on other conditions such as the viscosity of the fluid, the closeness of particles, etc. Ergun and Orning (6) tried to develop relationships for the constants a and b in order to predict their experimental values. They were partially successful in deriving the mathe- matical model for those constants for the general case. This will be discussed in detail later. Morcom (9) used gases such as air, carbon dioxide, etc., as fluids and used different granular materials with various types of packing. He mentioned that pressure drop is also a function of closeness Of packing. He showed that the pressure drop is inversely prOportional to the cube of the void factor. This is true, but it will be shown later that pressure drop is also proportional to (1-6)2 for low flows and (1-6) for high flow rates, where E is the void fraction. It can be seen from equation 2 that the velocity approaches A . zero, the quantity IEE approaches a constant value, a. This 0 is the condition for viscous flow, and it can be seen that the above equation is similar to the Poiseulle equation (1, 4, 5); A -—B = constant Lvo and to Darcy's law (4); kA v0 = -E£, where k is a constant. If the velocity of the fluid is high, then in comparison with the term b 9 v0, the constant "a" is negligible. In other words, it is the condition for turbulent flow where the resis- tance to the flow is constituted by kinetic energy losses. So the resistance to the flow is the sum of the two factors - loss of viscous energy and loss of kinetic energy. The loss Of viscous energy is due to the friction between two layers of the fluid, and the magnitude Of viscous energy depends on local velocity gradient of the layers. The loss of kinetic energy is due to the motion Of the fluid as a bulk and the magnitude of it depends on the bulk average velocity. It can be seen that, if the constant "a" is replaced by a'p where a' is a constant, a' depends only on the charac- teristics of the bed. Kozeny (l, 4, 5) developed the correlation between the pressure drop and the flow rate as: 2 Apsc (1-6) n vO L 1 €3D 2 P where k1 is a constant. Comparing equations 2 and 3, it can be seen that equation 3 is similar to equation 2 for low flow 2 rates. Then the constant "a" is equal to k. Sl:§l.£.. l 3 2 C D SO the Kozeny (4) equation is in partial agreement with equation 2. Carman (4), Lea (7), Nurse (7), and others have verified experimentally that constant "a” in the Reynold's equation is equal to: 2 k 1362 16D p However, most of their eXperiments were performed at low flow rates; so they failed to see the effects at high flow rates. Carman (4) did work in the high flow regions and found deviations in his results from those eXpected from Kozeny equation; however, he neglected it as an eXperimental error. Carman (4) did extensive work on the flow of fluids through granular beds. He developed the correlation between pressure drop and flow rate and arrived at the same equation that was mentioned by Kozeny. He used Darcy's law and included the void fraction as a parameter to describe packed bed data; v = v-B, (4) K is a constant. Furthermore, he also used the two dimension- less groupg, friction factor and Reynold's number for packed beds AngE DVD and 'ES— where s is the surface area Of packed as: vao s bed per unit volume of bed. He added further that there is a linear relation between the flow rate and the pressure drop through the bed. From this relation, he concluded that the di- 3 AngD 26 mensionless group 36uLvB(l-€) be designated as a constant j. 0 The above group is valid for a bed made up Of Spherical particles. For other shapes of particles, the dimensionless group Should be modified. Carman tried to calculate the values of j from the results Of the other investigations on the flow of fluids through packed beds and showed that the values of j for different sets of conditions varied. He argued that the variation was due to the expansion of the fluid when it passed through the bed and the expansion Of the fluid resulted in a change of viscosity of the fluid. In addition, the values of 6 may also change if the bed is under high pressure. The above variations were the important factors which caused the variation of j. Carman (4) reported in his papers that wall effect is also a factor in the variation of j. He used Coulson's modification which applies to Kozeny equation and which includes the wall effect as a parameter. However, the corrected values of j still varied. Blake (2) approached the problem of the fluid flow in a packed bed by comparing it with the fluid flow in a circular pipe. It has been established for the flow in pipes that a unique plot is obtained if the dimensionless groups, (pveDe/u) and R/pvez, are plotted against one another. Here, R = frictional force per unit area, D = pipe diameter, and v = actual e o velocity in the channel. Blake showed that the above groups can be modified for a packed bed if dimensional homogenity is used in comparing the flow in a packed bed with that in a circular pipe. He substi- tuted De by the hydraulic radius Rh where Rh = cross- sectional area per perimeter presented to fluid. For granular 6 beds, Rh = /s; S = surface area of particle / unit volume of bed. Again, R is defined in terms of Ap as: Apgce Ls R: 30 the dimensionless groups for packed beds are modified as: Ang63 9 vO (——-2—-> and ( us > L9 v0 8 These groups are known as the Blake dimensionless groups (4, 5, 6), since he was first to recognize the importance Of them. These groups can be plotted on log-log graphs, and a unique graph is obtained. Burke and Plummer (3) proposed that the pressure drop was due to the loss of kinetic energy. That is to say that Ap is prOportional to v2. They assumed that the granular bed was equivalent to a group of parallel channels; they regarded the bed to be made up of the sum of the separate resistances of the individual particles in it - as measured from the rate of free fall in the fluid. The force, F, acting on an isolated 3fipD v Sphere suSpended in a fluid stream is equal to -E-€B—. The C 10 number Of particles per unit of packing volume is 6(l-€)/UD:. Burke and Plummer stated that the rate of work W done due to the flow of the fluid is: 3nED v v o o 6 1-6 W - (‘713'6")(?)(—%'§l) p But W in terms of Ap is also equal to '%E. Combining the above equations, they prOposed the following proportionality for the flow of fluids through packed beds: APE (1-€)P v 2 c o E D P This proportionality has been verified for high flow rates; however, it fails in low flow regions. This is due to the assumption that the pressure drop is due to only kinetic energy term. If the above equation is compared with equation 2, it can be shown that if the flow rates are high, equation 2 is similar to equation 5, as the constant a in equation 2 becomes negligible at high flow rates. The constant b is equal to: k-£%:§z . Using equation 2 and using the values of the con- 6 D P stants a and b, the following equation is obtained: 2 2 APS (1-6) H v (1-€)p v ——C=k °+k ° (6) L 1 63 D 2 2 63 D P P This equation is known as the Ergun equation (1,5). The above equation can be rearranged in terms of dimensionless groups: ll 4ng 3P. E3 Dp v0 ‘9 v 2)(L)(’1—-‘€) = k1 u(1-E) + k2 “’50 O This equation shows how the Blake dimensionless groups fit with the Ergun equation (5). Many workers (3, 4, 5) have stated that the wall effect is an important factor when considering the flow of fluids through packed beds. The magnitude of the wall effect has not been determined by past investigators. They assumed that the wall effect is negligible if the column to particle diameter ratio is very large. It will be proved in the later section that the Ergun equation can be modified to include the frictional effect due to the wall. SCOPE OF THE PROBLEM Ergun has developed an empirical relationship between the pressure drop and the flow rate in a packed bed. This relation- ship is represented in the form of a friction factor and Reynold's number. For a packed bed, the friction factor and the Reynold's number are defined as: Apg p 6 RhEZ Friction factor = ( 2C )( L ) G 6GRh and Reynold's number = -EEr-. Here, Rh is the hydraulic radius which is defined as the ratio of the cross—section available for flow to the wetted perimeter. For packed beds, Ergun used the expression for the hydraulic radius: 6D Rh = 6(l-é) (7) This expression does not include the wall effect. Hence, it shows no dependency on the column diameter; since, during the develOpment of equation 7, it was assumed that the packed particle diameter was much smaller in comparison with that of the column diameter. In this project, the diameter of the column is not considered too large when compared with the particle diameter. This results in modification of the Ergun equation. The object Of this research is to examine the validity of the modified Ergun equation which includes the wall effect on the hydraulic radius. The following expression is obtained for the hydraulic radius - when the wall effect is included: 12 13 E = (8) Rh egg-62 + 4/DC P Ach" 6 e3 1 Thus, friction factor = ( 2 )( L )(6EI-E) 4/D ) G D + c P and Reynold's number 6 1fEG 4/D EFL-1D + c} P where DC = column diameter, Dp = particle diameter, and E = void fraction. The validity Of the modified friction factor and Reynold's number was determined by making a log-log plot of the above groups calculated from experimental data and comparing that graph with a similar plot of the Ergun equation. Many chemical processes are carried out in packed beds where the wall effect is important. As the design of large scale equipment is based on small laboratory models, it is important to understand the flow characteristics so that large scale equipment can be reliably designed. With the help of the modified Ergun equation, laboratory data which are obtained under conditions where the wall effect is important can be Successfully scaled to size where the wall effect is not important. THEORY First, the derivation of the Ergun equation which is a corre- lation relating the pressure drop to the flow rate in a packed bed will be made with the assumption that the diameter of the packed particles is very small in comparison with that of a column. Then,this equaticn will be modified to take into account the effect of the column wall on the hydraulic radius. The problem is confined to Spherical particles in a cylindrical tube having a constant cross—section. First, consider the flow of an incompressible fluid with density 9 through a pipe. r 52 v = O . 2 v = max 2 _- .‘___JUL____,L Figure 1: Velocity distribution for the flow in a cylindrical tube. Making a force balance on a shell of thickness Ar, the following differential equation is obtained: dvz Apghr d(-ru—d'; = L (9) dr Solving the above equation, and using the following boundary conditions: 1. at r = R, the velocity V2 is maximum; and 14 15 2. at r = 0, the velocity vZ is zero, the velocity distritution in z direction is defined as: 2 AngR Vz='TpL—{1 -r—2} (10) The average velocity is calculated by summing up all velocities over the crosscsection and then dividing by the cross-sectional area: 2n R f0 fovzrdrde = (11) f2” err d8 0 o ApRzgc = ————- 8uL (12) This is the well known Poiseulle equation. Developing the above equation, it was assumed that 1. the fluid was newtonian, 2. the end effects were neglected, 3. the temperature was constant, 4. the steady state was established when considering the force balance. This is the case for the column without any packing. All flow systems do not have same shape or same cross-section available for the flow of fluid. It will be assumed that the same equations which describe flow in a pipe describe flow in a packed bed. It is further assumed that the hydraulic radius is the characteristic length parameter in the Reynold's number. Now consider the steady flow of fluid through a cylindrical 16 tube filled with Spherical beads. Assume that the packed bed is a tube made up of very complicated cross-section with a hydraulic radius Rh. It can be shown that if void fraction iS one, four times the hydraulic radius is equal to the dia- meter of the tube. Equation 12 is transformed in terms of the hydraulic radius Rh by replacing R with 2Rh’ so that dimensional homogenity can be used to compare the flow properties in packed column with that in an empty cylinder. ApRng h <>=————- v ZuL (l3) Cross-section available for flow Now, Rh Wetted perimeter = Volume available for flow Total wetted surface = Volume of voids / Volume of bed Wetted surface / Volume of bed If the diameter of the column is considered too large when it is compared with that of the particle, the resistance Offered by the wetted surface of the column is negligible when it is compared with the wetted surface of the packed bed. Now, Volume of voids Volume of bed = E X Volume of bed Volume of bed l7 where 6 = void fraction. Also, Volume of bed = Volume of Sphere l - E In addition, Wetted surface Volume of bed Hence, the hydraulic radius Rh can be defined as: D E = __lL__. Rh 6(1-6) Substituting Rh in equation 13; Apg 62D 2 (v> = C 3 36(1-6) .2pL AngEZD 2 = 42’ (14) 72(1-6) pL v However, = fi?’ (15) where v0 = velocity of fluid if there was no packing in the column. Ang€3D 2 v = p (16) ° 72(1-E)2uL It is assumed that the path of liquid flowing through bed is l8 'L:ft:. But, this is not true Since - due to the bed - it makes a zigzag path which increases the effective length L. The emperfinmntal measurements indicate that the number 72 in the denominator be replaced by 150 (1, 5). Hence, the equation 16 changes as: 2 AngEBD = J (17) V 0 150(1-€)2uL This equation is known as the Blake — Kozeny (l, 4, 5) equation and is valid for low flow rates. For highly turbulent flow, the friction factor is only a function Of roughness when Reynold's number is high. The friction factor f is also called as a drag coefficent and it is a dimensionless quantity. It is approximately a constant at higher Reynold's number. New, for the flow Of a fluid through a bed of Spheres, the pressure drOp Ap is as follows: F/ Ap = ASC (18) where F Force exerted on the solid surfaces cross-sectional area. and .A Consider the fluid flowing through an empty column. The fluid will exert force F on the solid surfaces which is equal to: F = A'Kf (19) the surface area of column or wetted surface, 3> II where 19 K Kinetic energy / unit volume, and f friction factor. 30 for circular tubes of radius R and length L; 2 K = 16;) <.,~> (20) and A' = 271 RL F = (2n RL){%D 2)f (21) But Ap = F/ghA (18) = F 2 (22) gCTrR So substituting and rearranging the above equations, following equation is Obtained: Apg R f = .53 {___J;_l:} (23) 29 ‘ Again, the hydraulic radius R R.=/2 So substituting the value of R in the terms of Rh in equation 23; Rh (APgc ) f = — —— L to 2 Apg L¢=é§p2£ (24) But = vo/e (15) ED and Rh = 6(l-E) 2 3(l-€)p v f Apgc = 3 o (25) L EDP Experimental data (1, 5) indicate that 6f = 3.5 20 Hence, ApgC 1.759 v:(l-E) T = 3 <26) 6 D P Equation 26 is known as the Burke-Plummer equation (1, 3, 5). When equation 17 and equation 26 are combined, following equation results: _ 2 A in - - I ng) —-_SOHJO (l-E)2 1.759 vo(l E) D E D E P P This is known as the Ergun equation (1, 5). This can be also written in terms of G, the mass flow rate and in the dimension- less groups: Apg p D 3 , r E lSOEgl-E) , ( C )ved)C-—-) = + 1.15 (28) 2 L 1-6 D G G P It can be seen that in the low flow regions where mosfi of the APngD 6 eXperiments are performed, the plot of log ( 2 ) versus G L(l-€) D G log 57%:ES will be a straight line with the slope of -1. At very low fig? rates or in laminar flow, S%;§ZE factor is dominant in the right side of the equation 28. At very high flow rates or in turbulent flow rates,$%:§AB becomes very negligible comparing to 1.75. So, 3 A D E PBCP p ) ( C2 L(l-€) remains constant at 1.75. The above derived relation does not include the wall effect. If the diameter of the column is very large compared to that of the packing particles, the pressure drop and the flow rates are 21 unaffected by the friction due to the wall. Assume now that the diameter of the column is not large when compared with that of the packing particles. Then there will be a correction required in the hydraulic radius relation in the final equation. Again, the ratio Cf :he wetted surface to the colume of the bed can be written as: Wetted s:rface of Spheres + Wetted surface of wall Volume of the bed = Wetted surface cf_§pheres + Wetted surface of wall Volume of bed volume of bed DD 2 nDPL = p + ‘ (29) 3 2 nDP /6(l-€) nDC L/4 where Dc diameter of the column. So, wetted surface volume of bed - I = 6D1 E + 33-3 (30) p C Rh = 6 1 66 (31) L). + 4/1) Dp c Substituting the values Of Rh in equation 13 and using the relation of equation 15, the following equation is written for v : o €3Apg v0 = C 2 (32) 211L(-—(----2-6 ll)-€ + L) D p c 22 3 2 EApgD or v0 = C P 2 (4 D 1 >2 (33) 72uL(1-€) /6 E + 1 D (1-6) C 4D Let M = Izfifzijgs + l (34) and 72 in the denominator be replaced by 150 as stated previously. €3AngD 2 v = P2 2 (35) 150(1-6) uL M This can be considered as modified Blake-Kozeny equation. Similarly, for the turbulent flow: APE C — p 2f (24) L ' 2Rh Again, substituting Rh from equation 31 and using the relation of equation 15, the equation changes to the modified Burke- Plummer equation: Apgc 1.75p v:(l-€) M L = 3 (36) D e p Hence, by adding these two modified relations, the corre- lation becomes: ADS 150A v (1-€)2M2 1.759 v2(l-E)M c o O ” 'IT"= 2 3 + 3 (37) D E D E P or in terms of G, the mass flow rate and in the dimension- less groups, equation 37 is transferred to as: APR 9 D 3 ___£t. .2. _§_ .1 = 150p§1e€)M . D G P 23 The above equation can be stated as the modified Ergun equation. When DP< 2:2 . v . . 1 I" II I. --~*-_ 16 "1 .- ‘1 o~~~—4H-.—g_ 1i . n .. D . . 1.1.-11 . h a a m 1m. ; a . 'dh.f .1). .. ...«......... L..— --4L . . ..:....>..1¢A+:|111 o 9-._ cq--. ..-- .—. 3:1. . 511 LET... --.—«.04 _ -wn . ...-.6441— . -. .. --....- .m ...n..u....::.s_.._r.._....—1,.-,~-.~. 3. ..: .. .1. ... nouwwnnuumu”HousemanaHnanwnn”Amvuno “mm“...mwfl.woufimmmmh.L.--I....-w.....1~iHAVANA: . .L . .1. I. .aaI. 1161 1|» I ....I.».. <4... ..— ‘5‘...“— --< ’safl -. c a —o . A o 3 A a . I“ DO“ - ... 4 o _. «o .4. + o -L __f, ..Q.;... ..s.... I..- .1.-- .6 6.96.0061. 70:16.... 1 7 6 5 um 1 a. W «h . nuaoru w n I ’ .¢.I.D!_ID¢I .OU 'UI“..HI.. J'Lla'x Itutsllli b). "0: Plot of the friction factor versus Reynold's number of the packed bed = 91 for Dc/D ‘1‘ Figure 3: Ergun' 3 line 42 0 y vs Y (without wall effect) .) 9 X Y vs X ' (with wall effect) 000 x X WY I... II I. I. A. "'— ...s xcurrcu'o' '63.". co. I x I even. a fo ""ff??? I‘VE L5 \, ' 10 . . : ,1 2 2.5 a xgxs 78 9190 1.5 2 2.5 a 4 5 o a D . Figure 4: Plot of the friction factor versus 4 Reynold's number of the packed bed for DC, a 45:.1. ‘ D P Ergun's line 43 (3 y vs x (without wall effect) 3 )( Y vs X (with wall effect) n’c KIUPPIL . Iona co. I I I Brett- 0 .. Hull 1";' -' . 2.5 96 7 8 9911(0 Figure 5: Plot of the‘friction factor versus 4 Reynold's number of the packed bed /D = 36:1 for D c ' P Ergun's line 44 x y vs (without wall effect) 0 Y vs X (with wall effect) X' ‘1‘..-