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Radial Thermal Conductivity in Cylindrical Beds Packed by Shaped Particles
0263–8762/04/$30.00+0.00 # 2004 Institution of Chemical Engineers Trans IChemE, Part A, February 2004 Chemical Engineering Research and Design, 82(A2): 293–296
www.ingentaselect.com=titles=02638762.htm
RADIAL THERMAL CONDUCTIVITY IN CYLINDRICAL BEDS PACKED BY SHAPED PARTICLES E. I. SMIRNOV*, V. A. KUZMIN and I. A. ZOLOTARSKII Boreskov Institute of Catalysis SB RAS, Novosibirsk, Russia
E
xperimental data on radial thermal conductivity of beds formed of solid cylinders, Rashig rings, six-spoke wheels, four- and 52-hole cylindrical pellets are presented. The use of shaped particles in packed beds can significantly improve radial heat transfer. Thermal conductivity of the bed increases with the increase of the channel cross-section in the particle. The hydrodynamic approach to the description of convective radial thermal conductivity in packed beds of cylindrical particles with arbitrary number and form of channels is proposed. The approach is based on the account of mean fluid velocities around and through the hollow cylinders in the packed bed. Keywords: heat transfer; pressure drop; fixed bed; shaped catalyst; modelling
INTRODUCTION
controversial. Bahurov and Boreskov (1947) carried out experiments with porcelain rings with diameter 15 mm, length 15 mm and hole diameter 11 mm and glass tubes with diameter 4.5–5.8 mm, length 16.2–25.8 mm and hole diameter 3.4–4.7 mm and found that for rings the radial thermal conductivity is 80% higher than for regular particles, and for glass tubes even higher. Yagi and Kunii (1957) presented the data on the dependence of radial thermal conductivity on RePr for Rashig rings with diameters 6.74, 15 and 25 mm and shown the increase of radial thermal conductivity in comparison with spheres from 40 to 70%. England and Gunn (1970) studied radial diffusion in beds formed of Rashig rings with diameter 6 mm, length 6 mm and different diameters of holes, and found that the radial diffusion coefficient does not depend on the diameter of the hole and is approximately 30% higher than that for solid cylinders of the same dimension. Bauer and Schlunder (1978) have carried out many experiments with different Rashig rings and have obtained a general formula in which radial thermal conductivity of hollow particles is higher than that of solid cylinders and depends on the relative diameter of the hole and the ratio of length of particle to its diameter. At that time Peters et al. (1988) studied mass transfer in beds packed by Rashig rings 7.8 mm in diameter and found the radial diffusion coefficient to be the same as that for cylinders. Landon et al. (1996) investigated heat transfer in beds formed of five-spoke wheels with diameter 8.5 mm, height 7.5 mm and spoke thickness of 1.3 mm, and obtained radial thermal conductivity of 80% lower than that for cylinders. In present work experiments on radial heat transfer of beds packed with solid cylinders and shaped particles were performed. As shaped particles Rashig rings, sixspoke wheels, four-hole and 52-hole cylindrical pellets
Enhancement of radial heat transfer in catalytic tubular reactors with fixed beds has been the subject of intense scientific research for the last 50 years. The prominent factors that influence on the effectivity of packed bed reactors are: (1) overall activity of the catalyst; (2) heat transfer properties of fixed bed; (3) pressure drop over the catalyst bed. It is conventional opinion that optimal combinations of these factors can be achieved by choosing correctly the catalyst dimensions and the reactor tube diameter. The shape of catalyst particles is seldom taken into consideration. Usually solid spheres or pellets are used. At the same time for some processes, a number of companies have started marketing catalysts with improved shape. These are processes of methanol oxidation into formaldehyde and steam reforming. The best known catalysts for steam reforming are produced by Johnson Matthey Catalysts, UK (four-hole pellet, quadralobe), Haldor Topsoe, Denmark (seven-hole pellet), BASF, Germany (clover leaf), Sud-Chemie, Germany (wheels), but the optimal catalyst shape for a given process is still debatable, as demonstrated by the competition between the above-mentioned catalysts. The problem has not been studied systematically on a fundamental level. Literature data about effective radial thermal conductivities of packed beds of shaped particles are limited and *Correspondence to: Dr. E. I. Smirnov, Boreskov Institute of Catalysis, pr. Akad. Lavrentieva, 5, Novosibirsk 630090, Russia. E-mail: [email protected]
293
294
SMIRNOV et al.
were taken. A comparison of obtained results with literature data is performed. The hydrodynamic approach to the description of convective radial thermal conductivity in packed beds of cylindrical particles with arbitrary number and form of channels is proposed.
EXPERIMENTAL Temperature profiles of the air flowing through packed beds were measured in steady-state experiments. Experimental set-up for determining the heat transfer parameters in the packed bed (Figure 1) consisted of three tube sections with inner diameter of 84 mm: heating section (330 mm height), calming section (150 mm height) and cooled test section (650 mm height). The test section was cooled by water flowing through the annular jacket surrounding the inner tube. A water flow rate of about 25 l min1 provided almost constant temperature of the tube wall of the test section. The temperature of the air incoming the test section reached 80–100 C. The heating section and the calming section were filled with 4–5 mm porcelain balls to obtain more or less uniform radial temperature and velocity profiles. Superficial velocities of the air were in the range 0.2–2.0 m s1. Temperature fields of the air stream issuing from the top of the packing were measured by means of twelve 0.2 mm thermocouples that were held by a glass laminate cross (three thermocouples on each of four arms) in 12 different radial positions. The tips of the thermocouples protruded 4–5 mm out of the arms. At each bed height and air velocity the cross was rotated stepwise full turn with 30 angular intervals to obtain the radial temperature profiles averaged with respect to the angular coordinate. For determination of experimental radial heat transport parameters two-dimensional standard dispersion model (SDM) was used (experimental method and model details can be found in Smirnov et al., 2003). The inlet radial temperature profile was used as a boundary condition for the SDM. This model describes experimental temperature profiles of flowing gas very well in the bed core, but it fails to fit experimental points near the wall. Experimental heat transport parameters were determined by fitting only temperature points that were measured on the distance
more than equivalent hydraulic diameter of the packed bed d from the reactor wall. d¼
4 ebed a0 1 ebed
(1)
where ebed ¼ bed porosity without account of porosity of grains and a0 ¼ the volumetric surface area of one solid particle. Thus the effective radial thermal conductivity of the packed bed core ler,core and the wall heat transfer coefficient aw were determined. Dependences of these radial heat transport parameters on the superficial gas velocity (in terms of Reynolds number, Re) have been obtained. Bed porosities were measured by a weighting method. Physical properties of the air were considered to be constant in the test section. Prandtl number was accepted as Pr ¼ 0.71 for the air. RESULTS AND ANALYSES. HYDRODYNAMIC MODEL OF CONVECTIVE HEAT TRANSPORT Effective radial thermal conductivity in the fixed bed core is composed of the conductivity without flow and the convective part of the thermal conductivity: ler,core ¼ lbed þ K Re Pr
(2)
where ler,core ¼ ler,core =lf and lbed ¼ lbed =lf , lf ¼ fluid thermal conductivity and K ¼ the empirically determined parameter, depending on the ratio of diameters of the particle and the bed, the bed porosity and also of the particles shape. Reynolds number is expressed through superficial velocity u0 and the diameter of equivalent-volume sphere dp for given particles (the hole volume is included in the particle volume). For cylindrical particles rffiffiffiffiffiffiffiffiffi 3 3 L (3) dp ¼ d 2 d where d ¼ the diameter of the particles and L ¼ the length of the particles. For solid particles without channels the value of K is studied and presented in literature with good accuracy. The model proposed by Bauer and Schlunder (1978) is able to describe the convective thermal conductivity of beds packed by particles with more complex forms, particularly Rashig rings. Bauer and Schlunder suggested the following formula for the convective heat transport parameter: KBauer ¼
"
X
2 8 2 1 D=dp
!2 #
(4) dp
where D ¼ diameter of the fixed bed and the mixing length X for hollow cylindrical particles is ehole 1 ebed ebed Xcyl ¼ F d þ Fhole dhole ehbed ehbed cyl p (5)
Figure 1. Experimental set-up.
where ehbed ¼ ebed þ ehole (1 ebed ) is the bed porosity with account of porosity of grains, ehole ¼ porosity of one grain and Fcyl ¼ 1.75 and Fhole ¼ 2.8 are the form factors.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296
RADIAL THERMAL CONDUCTIVITY The characteristic length of the channel is dhole ¼
pffiffiffi 2L
(6)
However, the model of Bauer and Schlunder is not suitable for calculation of the parameter K for any arbitrary particle shape because of necessity experimental determination of empirical coefficients Fform for each of the shapes. In this work the original approach was used to characterize the hydrodynamics of gas flow through a packed bed. This approach has been developed to modify the method of the parameter K calculation for case of the bed packed with the shaped particles. The condition of the equivalence of pressure drop on the length of the channel at the external flow and at the flow inside the channel was the main idea. Using the equations for the hydraulic resistance in the bed packed with solid particles and in the channels makes it possible to calculate the velocities of the flow around the particle uout and in the particle channels uhole (uout and uhole ¼ interstitial velocities) at the given superficial velocity u0 from the set of equations: 8 <
u0 ¼ uout ebed þ uhole [ehole (1 ebed )] hcos ai DP ¼ f 1 uout þ f 2 u2out : g1 uhole þ g2 u2hole ¼ hcos ai L (7) Here DP ¼ the pressure drop over the distance hcos ai L and a ¼ the angle between the axis of the tubular reactor and the axis of the particle channels. The mean value of cos a is hcos ai ¼ 2=p. The first equation of the set (7) is the condition of mass conservation in the bed. The second equation of the set (7) is the condition of the pressure drop equivalence over the distance hcos ai L at the external flow and at the flow inside the channel. f1 and f2 are the coefficients of Ergun’s formulae (Ergun, 1952): 2 1 ebed m 2 f1 ¼ 150 dp ebed 1 ebed r f2 ¼ 1:75 dp ebed
295
The hydraulic diameter of the channel is 8 side of square if channel’s cross-section > > > > has a square form > < diameter of hole if channel’s cross-section dgidr ¼ > has a circular form > > Dsec t > > : if channel’s cross-section 1 þ Nhole =p has a sector form (12) Here Dsect ¼ the diameter of the circle that is formed by all sectors joined side by side and Nhole ¼ the number of channels in one particle. After the real velocities calculation from the set of equations (7), the pressure drop over the packed bed can be calculated. The radial thermal conductivity of the packed bed core can be defined as 1 ler,core ¼ lbed þ [ebed Reout þ ehole k (1 ebed ) Rehole ] Pr
(13)
where k is the modified coefficient of the model of Bauer and Schlunder (1978) for cylindrical particles: 2 !2 3 8 4 2 5 2 1 (14) k¼ 1:78 D=dp Reynolds numbers Reout and Rehole are expressed through calculated velocities. For Reout the diameter of equivalentvolume sphere dp is taken as characteristic dimension. For Rehole the characteristic dimension is: 1 dhole ¼ 2 hcos ai L þ d 1 pffiffiffiffiffiffiffiffiffiffi (15) Nhole From the equations (2) and (13) the convective heat transport parameter K is: " # dhole uhole 1 uout K ¼ ebed þ ehole 1 ebed k u0 dp u0 (16)
(8) (9)
where m and r are the fluid viscosity and fluid density respectively. The coefficients g1 and g2 are used to calculate the pressure drop in the particle channel (Vortuba et al., 1974): g1 ¼ 16p
m 2 dgidr
8 1 > > > <
if channel’s cross-section have a circular form
> 0:89 if channel’s cross-section is > > : square or sector p r e g2 ¼ 1:75 1 hole 4 L Nhole
(10)
(11)
Figure 2. Convective heat transport parameter K for cylinders and rings with outer diameter equal to length. Grey line ¼ model of Bauer and Schlunder (1978). Black line ¼ hydrodynamic model.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296
296
SMIRNOV et al. Table 1. Convective heat transport parameter K for cylinders and shaped particles.
No.
Particle
ebed
K (experimental)a
K (theoretical)b
1 2 3 4 5 6 7 8 9
Ceramic cylinder: diameter 10 mm, length 10 mm Ceramic cylinder: diameter 14 mm, length 9 mm Ceramic cylinder: diameter 9 mm, length 19 mm Ceramic cylinder: diameter 19 mm, length 19 mm Ceramic ring: outer diameter 14 mm, length 14 mm, wall thickness 3.5 mm Copper ring: outer diameter 14 mm, length 14 mm, wall thickness 1 mm Ceramic wheel: outer diameter 18mm, length 16 mm, six holes, wall thickness 2 mm Ceramic wheel: outer diameter 15 mm, length 7 mm, six holes, wall thickness 1 mm Ceramic 52-hole block: outer diameter 19 mm, length 17 mm, square holes 1.5 1.5 mm Ceramic four-hole pellet: outer diameter 14 mm, length 17 mm, hole diameter 4 mm
0.38 0.42 0.42 0.44 0.41 0.41 0.42 0.39 0.48
0.15 0.14 0.16 0.14 0.16 0.21 0.17 0.23 0.14
0.151 0.144 0.145 0.126 0.162 0.206 0.170 0.220 0.151
0.40
0.20
0.198
10 a
Estimated by regression of experimental data with equation (2). From equation (16).
b
The comparison of classical model of Bauer and Schlunder (1978) and proposed hydrodynamic model is shown in Figure 2. It shows the dependence of parameter K for Rashig rings on the ratio of the diameter of the channel to the outer diameter of the particle. As it is seen from Figure 2, the hydrodynamic model (black line) better describes the experimental data in particular for higher hole diameter to outer ring diameter ratios. In Table 1 the experimental and calculated from the hydrodynamic model values of the parameter K are presented. The table contains the bed porosities and geometrical dimensions of cylinders and shaped granules. Calculations of the convective heat transport parameter K were made for conditions when superficial velocity u0 was larger than 0.5 m s1 (Re 300). In this case value of parameter K practically does not depend on Reynolds number. The values of Kexperimental satisfy equation (2) for experimental dependence of SDM parameter ler,core on Reynolds number. It was estimated that lbed ¼ 10 for ceramic particles and lbed ¼ 20 for metallic particles. Maximal deviation of 13% was found for ceramic cylinders with diameter and length equal to 19 mm. The accuracy of experimental determination of Kexperimental was about 15% for cylinders and shaped particles. The Figure 2 and the Table 1 show that suggested hydrodynamic model is in a good agreement with the experimental data. CONCLUSIONS Use of shaped particles in fixed beds can significantly improve the radial heat transfer, reduce the pressure drop over the bed and enhance the volumetric surface area of the fixed bed. The proposed hydrodynamic model for the calculation of the convective radial thermal conductivity and the pressure drop in fixed bed has an advantage in comparison with literature. The main advantage of this model is that the radial
thermal conductivity for all cylindrical particles with arbitrary number and form of the channels is described by one formula with constant parameters. The model is very helpful for finding optimized dimensions of cylindrical grains of catalyst as well as numbers and form of the grain channels for design of tubular fixed bed reactors. ACKNOWLEDGMENTS The authors would like to thank NATO’s Scientific Affairs Division for the support (grant SfP-972557). REFERENCES Bahurov, V.G. and Boreskov, G.K., 1947, Effective thermal conductivity of contact beds, J Appl Chem (Russ), XX(8): 721–738. Bauer, R. and Schlunder, E.U., 1978, Effective radial thermal conductivity of packings in gas flow. Part I. Convective transport coefficient, Int Chem Eng, 18(2): 181–188. England, R. and Gunn, D.J., 1970, Dispersion, pressure drop, and chemical reaction in packed beds of cylindrical particles, Trans IChemE, 48: T265–T275. Ergun, S., 1952, Fluid flow through packed columns, Chem Eng Prog, 48: 89–94. Landon, V.G., Hebert, L.A. and Adams, C.B., 1996, Heat transfer measurement for industrial packed bed tubular reactor modeling and design, AIChE Symp Ser, 92(310): 134–144. Peters, P.E., Schiffino, R.S. and Harriott, P., 1988, Heat transfer in packedtube reactors, Ind Eng Chem Res, 27: 226–233. Smirnov, E.I., Muzykantov, A.V., Kuzmin, V.A., Kronberg, A.E. and Zolotarskii, I.A., 2003, Radial heat transfer in packed beds of spheres, cylinders and Rashig rings. Verification of model with a linear variation of ler in the vicinity of the wall, Chem Eng J, 91: 243–248. Vortuba, J., Mikus, O., Hlavacek, V. and Skrivanek, J., 1974, A note on pressure drop in monolithic catalyst, Chem Eng Sci, 29: 2128–2130. Yagi, S. and Kunii, D., 1957, Studies on effective thermal conductivities in packed beds, AIChE J, 3(3): 373–381. This paper was presented at ISMR-3–CCRE18, the joint research symposium of the 3rd International Symposium on Multifunctional Reactors and the 18th Colloquia on Chemical Reaction Engineering held in Bath, UK, 27–30 August 2003. The manuscript was received 7 July 2003 and accepted for publication after revision 24 November 2003.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296
www.ingentaselect.com=titles=02638762.htm
RADIAL THERMAL CONDUCTIVITY IN CYLINDRICAL BEDS PACKED BY SHAPED PARTICLES E. I. SMIRNOV*, V. A. KUZMIN and I. A. ZOLOTARSKII Boreskov Institute of Catalysis SB RAS, Novosibirsk, Russia
E
xperimental data on radial thermal conductivity of beds formed of solid cylinders, Rashig rings, six-spoke wheels, four- and 52-hole cylindrical pellets are presented. The use of shaped particles in packed beds can significantly improve radial heat transfer. Thermal conductivity of the bed increases with the increase of the channel cross-section in the particle. The hydrodynamic approach to the description of convective radial thermal conductivity in packed beds of cylindrical particles with arbitrary number and form of channels is proposed. The approach is based on the account of mean fluid velocities around and through the hollow cylinders in the packed bed. Keywords: heat transfer; pressure drop; fixed bed; shaped catalyst; modelling
INTRODUCTION
controversial. Bahurov and Boreskov (1947) carried out experiments with porcelain rings with diameter 15 mm, length 15 mm and hole diameter 11 mm and glass tubes with diameter 4.5–5.8 mm, length 16.2–25.8 mm and hole diameter 3.4–4.7 mm and found that for rings the radial thermal conductivity is 80% higher than for regular particles, and for glass tubes even higher. Yagi and Kunii (1957) presented the data on the dependence of radial thermal conductivity on RePr for Rashig rings with diameters 6.74, 15 and 25 mm and shown the increase of radial thermal conductivity in comparison with spheres from 40 to 70%. England and Gunn (1970) studied radial diffusion in beds formed of Rashig rings with diameter 6 mm, length 6 mm and different diameters of holes, and found that the radial diffusion coefficient does not depend on the diameter of the hole and is approximately 30% higher than that for solid cylinders of the same dimension. Bauer and Schlunder (1978) have carried out many experiments with different Rashig rings and have obtained a general formula in which radial thermal conductivity of hollow particles is higher than that of solid cylinders and depends on the relative diameter of the hole and the ratio of length of particle to its diameter. At that time Peters et al. (1988) studied mass transfer in beds packed by Rashig rings 7.8 mm in diameter and found the radial diffusion coefficient to be the same as that for cylinders. Landon et al. (1996) investigated heat transfer in beds formed of five-spoke wheels with diameter 8.5 mm, height 7.5 mm and spoke thickness of 1.3 mm, and obtained radial thermal conductivity of 80% lower than that for cylinders. In present work experiments on radial heat transfer of beds packed with solid cylinders and shaped particles were performed. As shaped particles Rashig rings, sixspoke wheels, four-hole and 52-hole cylindrical pellets
Enhancement of radial heat transfer in catalytic tubular reactors with fixed beds has been the subject of intense scientific research for the last 50 years. The prominent factors that influence on the effectivity of packed bed reactors are: (1) overall activity of the catalyst; (2) heat transfer properties of fixed bed; (3) pressure drop over the catalyst bed. It is conventional opinion that optimal combinations of these factors can be achieved by choosing correctly the catalyst dimensions and the reactor tube diameter. The shape of catalyst particles is seldom taken into consideration. Usually solid spheres or pellets are used. At the same time for some processes, a number of companies have started marketing catalysts with improved shape. These are processes of methanol oxidation into formaldehyde and steam reforming. The best known catalysts for steam reforming are produced by Johnson Matthey Catalysts, UK (four-hole pellet, quadralobe), Haldor Topsoe, Denmark (seven-hole pellet), BASF, Germany (clover leaf), Sud-Chemie, Germany (wheels), but the optimal catalyst shape for a given process is still debatable, as demonstrated by the competition between the above-mentioned catalysts. The problem has not been studied systematically on a fundamental level. Literature data about effective radial thermal conductivities of packed beds of shaped particles are limited and *Correspondence to: Dr. E. I. Smirnov, Boreskov Institute of Catalysis, pr. Akad. Lavrentieva, 5, Novosibirsk 630090, Russia. E-mail: [email protected]
293
294
SMIRNOV et al.
were taken. A comparison of obtained results with literature data is performed. The hydrodynamic approach to the description of convective radial thermal conductivity in packed beds of cylindrical particles with arbitrary number and form of channels is proposed.
EXPERIMENTAL Temperature profiles of the air flowing through packed beds were measured in steady-state experiments. Experimental set-up for determining the heat transfer parameters in the packed bed (Figure 1) consisted of three tube sections with inner diameter of 84 mm: heating section (330 mm height), calming section (150 mm height) and cooled test section (650 mm height). The test section was cooled by water flowing through the annular jacket surrounding the inner tube. A water flow rate of about 25 l min1 provided almost constant temperature of the tube wall of the test section. The temperature of the air incoming the test section reached 80–100 C. The heating section and the calming section were filled with 4–5 mm porcelain balls to obtain more or less uniform radial temperature and velocity profiles. Superficial velocities of the air were in the range 0.2–2.0 m s1. Temperature fields of the air stream issuing from the top of the packing were measured by means of twelve 0.2 mm thermocouples that were held by a glass laminate cross (three thermocouples on each of four arms) in 12 different radial positions. The tips of the thermocouples protruded 4–5 mm out of the arms. At each bed height and air velocity the cross was rotated stepwise full turn with 30 angular intervals to obtain the radial temperature profiles averaged with respect to the angular coordinate. For determination of experimental radial heat transport parameters two-dimensional standard dispersion model (SDM) was used (experimental method and model details can be found in Smirnov et al., 2003). The inlet radial temperature profile was used as a boundary condition for the SDM. This model describes experimental temperature profiles of flowing gas very well in the bed core, but it fails to fit experimental points near the wall. Experimental heat transport parameters were determined by fitting only temperature points that were measured on the distance
more than equivalent hydraulic diameter of the packed bed d from the reactor wall. d¼
4 ebed a0 1 ebed
(1)
where ebed ¼ bed porosity without account of porosity of grains and a0 ¼ the volumetric surface area of one solid particle. Thus the effective radial thermal conductivity of the packed bed core ler,core and the wall heat transfer coefficient aw were determined. Dependences of these radial heat transport parameters on the superficial gas velocity (in terms of Reynolds number, Re) have been obtained. Bed porosities were measured by a weighting method. Physical properties of the air were considered to be constant in the test section. Prandtl number was accepted as Pr ¼ 0.71 for the air. RESULTS AND ANALYSES. HYDRODYNAMIC MODEL OF CONVECTIVE HEAT TRANSPORT Effective radial thermal conductivity in the fixed bed core is composed of the conductivity without flow and the convective part of the thermal conductivity: ler,core ¼ lbed þ K Re Pr
(2)
where ler,core ¼ ler,core =lf and lbed ¼ lbed =lf , lf ¼ fluid thermal conductivity and K ¼ the empirically determined parameter, depending on the ratio of diameters of the particle and the bed, the bed porosity and also of the particles shape. Reynolds number is expressed through superficial velocity u0 and the diameter of equivalent-volume sphere dp for given particles (the hole volume is included in the particle volume). For cylindrical particles rffiffiffiffiffiffiffiffiffi 3 3 L (3) dp ¼ d 2 d where d ¼ the diameter of the particles and L ¼ the length of the particles. For solid particles without channels the value of K is studied and presented in literature with good accuracy. The model proposed by Bauer and Schlunder (1978) is able to describe the convective thermal conductivity of beds packed by particles with more complex forms, particularly Rashig rings. Bauer and Schlunder suggested the following formula for the convective heat transport parameter: KBauer ¼
"
X
2 8 2 1 D=dp
!2 #
(4) dp
where D ¼ diameter of the fixed bed and the mixing length X for hollow cylindrical particles is ehole 1 ebed ebed Xcyl ¼ F d þ Fhole dhole ehbed ehbed cyl p (5)
Figure 1. Experimental set-up.
where ehbed ¼ ebed þ ehole (1 ebed ) is the bed porosity with account of porosity of grains, ehole ¼ porosity of one grain and Fcyl ¼ 1.75 and Fhole ¼ 2.8 are the form factors.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296
RADIAL THERMAL CONDUCTIVITY The characteristic length of the channel is dhole ¼
pffiffiffi 2L
(6)
However, the model of Bauer and Schlunder is not suitable for calculation of the parameter K for any arbitrary particle shape because of necessity experimental determination of empirical coefficients Fform for each of the shapes. In this work the original approach was used to characterize the hydrodynamics of gas flow through a packed bed. This approach has been developed to modify the method of the parameter K calculation for case of the bed packed with the shaped particles. The condition of the equivalence of pressure drop on the length of the channel at the external flow and at the flow inside the channel was the main idea. Using the equations for the hydraulic resistance in the bed packed with solid particles and in the channels makes it possible to calculate the velocities of the flow around the particle uout and in the particle channels uhole (uout and uhole ¼ interstitial velocities) at the given superficial velocity u0 from the set of equations: 8 <
u0 ¼ uout ebed þ uhole [ehole (1 ebed )] hcos ai DP ¼ f 1 uout þ f 2 u2out : g1 uhole þ g2 u2hole ¼ hcos ai L (7) Here DP ¼ the pressure drop over the distance hcos ai L and a ¼ the angle between the axis of the tubular reactor and the axis of the particle channels. The mean value of cos a is hcos ai ¼ 2=p. The first equation of the set (7) is the condition of mass conservation in the bed. The second equation of the set (7) is the condition of the pressure drop equivalence over the distance hcos ai L at the external flow and at the flow inside the channel. f1 and f2 are the coefficients of Ergun’s formulae (Ergun, 1952): 2 1 ebed m 2 f1 ¼ 150 dp ebed 1 ebed r f2 ¼ 1:75 dp ebed
295
The hydraulic diameter of the channel is 8 side of square if channel’s cross-section > > > > has a square form > < diameter of hole if channel’s cross-section dgidr ¼ > has a circular form > > Dsec t > > : if channel’s cross-section 1 þ Nhole =p has a sector form (12) Here Dsect ¼ the diameter of the circle that is formed by all sectors joined side by side and Nhole ¼ the number of channels in one particle. After the real velocities calculation from the set of equations (7), the pressure drop over the packed bed can be calculated. The radial thermal conductivity of the packed bed core can be defined as 1 ler,core ¼ lbed þ [ebed Reout þ ehole k (1 ebed ) Rehole ] Pr
(13)
where k is the modified coefficient of the model of Bauer and Schlunder (1978) for cylindrical particles: 2 !2 3 8 4 2 5 2 1 (14) k¼ 1:78 D=dp Reynolds numbers Reout and Rehole are expressed through calculated velocities. For Reout the diameter of equivalentvolume sphere dp is taken as characteristic dimension. For Rehole the characteristic dimension is: 1 dhole ¼ 2 hcos ai L þ d 1 pffiffiffiffiffiffiffiffiffiffi (15) Nhole From the equations (2) and (13) the convective heat transport parameter K is: " # dhole uhole 1 uout K ¼ ebed þ ehole 1 ebed k u0 dp u0 (16)
(8) (9)
where m and r are the fluid viscosity and fluid density respectively. The coefficients g1 and g2 are used to calculate the pressure drop in the particle channel (Vortuba et al., 1974): g1 ¼ 16p
m 2 dgidr
8 1 > > > <
if channel’s cross-section have a circular form
> 0:89 if channel’s cross-section is > > : square or sector p r e g2 ¼ 1:75 1 hole 4 L Nhole
(10)
(11)
Figure 2. Convective heat transport parameter K for cylinders and rings with outer diameter equal to length. Grey line ¼ model of Bauer and Schlunder (1978). Black line ¼ hydrodynamic model.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296
296
SMIRNOV et al. Table 1. Convective heat transport parameter K for cylinders and shaped particles.
No.
Particle
ebed
K (experimental)a
K (theoretical)b
1 2 3 4 5 6 7 8 9
Ceramic cylinder: diameter 10 mm, length 10 mm Ceramic cylinder: diameter 14 mm, length 9 mm Ceramic cylinder: diameter 9 mm, length 19 mm Ceramic cylinder: diameter 19 mm, length 19 mm Ceramic ring: outer diameter 14 mm, length 14 mm, wall thickness 3.5 mm Copper ring: outer diameter 14 mm, length 14 mm, wall thickness 1 mm Ceramic wheel: outer diameter 18mm, length 16 mm, six holes, wall thickness 2 mm Ceramic wheel: outer diameter 15 mm, length 7 mm, six holes, wall thickness 1 mm Ceramic 52-hole block: outer diameter 19 mm, length 17 mm, square holes 1.5 1.5 mm Ceramic four-hole pellet: outer diameter 14 mm, length 17 mm, hole diameter 4 mm
0.38 0.42 0.42 0.44 0.41 0.41 0.42 0.39 0.48
0.15 0.14 0.16 0.14 0.16 0.21 0.17 0.23 0.14
0.151 0.144 0.145 0.126 0.162 0.206 0.170 0.220 0.151
0.40
0.20
0.198
10 a
Estimated by regression of experimental data with equation (2). From equation (16).
b
The comparison of classical model of Bauer and Schlunder (1978) and proposed hydrodynamic model is shown in Figure 2. It shows the dependence of parameter K for Rashig rings on the ratio of the diameter of the channel to the outer diameter of the particle. As it is seen from Figure 2, the hydrodynamic model (black line) better describes the experimental data in particular for higher hole diameter to outer ring diameter ratios. In Table 1 the experimental and calculated from the hydrodynamic model values of the parameter K are presented. The table contains the bed porosities and geometrical dimensions of cylinders and shaped granules. Calculations of the convective heat transport parameter K were made for conditions when superficial velocity u0 was larger than 0.5 m s1 (Re 300). In this case value of parameter K practically does not depend on Reynolds number. The values of Kexperimental satisfy equation (2) for experimental dependence of SDM parameter ler,core on Reynolds number. It was estimated that lbed ¼ 10 for ceramic particles and lbed ¼ 20 for metallic particles. Maximal deviation of 13% was found for ceramic cylinders with diameter and length equal to 19 mm. The accuracy of experimental determination of Kexperimental was about 15% for cylinders and shaped particles. The Figure 2 and the Table 1 show that suggested hydrodynamic model is in a good agreement with the experimental data. CONCLUSIONS Use of shaped particles in fixed beds can significantly improve the radial heat transfer, reduce the pressure drop over the bed and enhance the volumetric surface area of the fixed bed. The proposed hydrodynamic model for the calculation of the convective radial thermal conductivity and the pressure drop in fixed bed has an advantage in comparison with literature. The main advantage of this model is that the radial
thermal conductivity for all cylindrical particles with arbitrary number and form of the channels is described by one formula with constant parameters. The model is very helpful for finding optimized dimensions of cylindrical grains of catalyst as well as numbers and form of the grain channels for design of tubular fixed bed reactors. ACKNOWLEDGMENTS The authors would like to thank NATO’s Scientific Affairs Division for the support (grant SfP-972557). REFERENCES Bahurov, V.G. and Boreskov, G.K., 1947, Effective thermal conductivity of contact beds, J Appl Chem (Russ), XX(8): 721–738. Bauer, R. and Schlunder, E.U., 1978, Effective radial thermal conductivity of packings in gas flow. Part I. Convective transport coefficient, Int Chem Eng, 18(2): 181–188. England, R. and Gunn, D.J., 1970, Dispersion, pressure drop, and chemical reaction in packed beds of cylindrical particles, Trans IChemE, 48: T265–T275. Ergun, S., 1952, Fluid flow through packed columns, Chem Eng Prog, 48: 89–94. Landon, V.G., Hebert, L.A. and Adams, C.B., 1996, Heat transfer measurement for industrial packed bed tubular reactor modeling and design, AIChE Symp Ser, 92(310): 134–144. Peters, P.E., Schiffino, R.S. and Harriott, P., 1988, Heat transfer in packedtube reactors, Ind Eng Chem Res, 27: 226–233. Smirnov, E.I., Muzykantov, A.V., Kuzmin, V.A., Kronberg, A.E. and Zolotarskii, I.A., 2003, Radial heat transfer in packed beds of spheres, cylinders and Rashig rings. Verification of model with a linear variation of ler in the vicinity of the wall, Chem Eng J, 91: 243–248. Vortuba, J., Mikus, O., Hlavacek, V. and Skrivanek, J., 1974, A note on pressure drop in monolithic catalyst, Chem Eng Sci, 29: 2128–2130. Yagi, S. and Kunii, D., 1957, Studies on effective thermal conductivities in packed beds, AIChE J, 3(3): 373–381. This paper was presented at ISMR-3–CCRE18, the joint research symposium of the 3rd International Symposium on Multifunctional Reactors and the 18th Colloquia on Chemical Reaction Engineering held in Bath, UK, 27–30 August 2003. The manuscript was received 7 July 2003 and accepted for publication after revision 24 November 2003.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A2): 293–296