Catalyst effectiveness factor distributions in isothermal packed bed reactors

Chemical Engineering Science 66 (2011) 3003–3011

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Catalyst effectiveness factor distributions in isothermal packed bed reactors D.A. Graf von der Schulenburg a, M.L. Johns b,n a b

Department of Chemical Engineering and Biotechnology, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK School of Mechanical and Chemical Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

a r t i c l e i n f o

abstract

Article history: Received 8 October 2010 Received in revised form 28 March 2011 Accepted 1 April 2011 Available online 9 April 2011

Lattice Boltzmann (LB) techniques are used to simulate a first order reaction occurring in a variety of porous catalyst pellet shapes; namely spheres, cylinders and trilobes. This enables a relationship between the effectiveness factor and the Thiele modulus for individual pellets to be determined. In the case of single pellets for an infinite flow rate, good agreement is produced between the simulations and analytical/semi-analytical descriptions of this relationship. The effect of flow rate on this relationship is then explored and quantified. Finally the simulations are applied to 3D random packings of the various pellet shapes. Reasonably good agreement with respect to the mean effectiveness factor is produced when the analytical/semi-analytical description is applied to the respective pellet size distributions. By comparison with the simulations however, the analytical/semi-analytical descriptions tended to underpredict the standard deviation of the effectiveness factor distributions, suggesting an additional influence due to local random packing and hence flow heterogeneity. & 2011 Published by Elsevier Ltd.

Keywords: Packed beds Simulation Thiele modulus Catalyst pellets Lattice Boltzmann Effectiveness factor

1. Introduction Fixed bed reactors are often composed of a random packing of discrete solid catalyst particles or pellets. Integral to reactor modelling and subsequent reactor design, is a quantitative description of the effect of internal diffusion of both reactants and products within these porous pellets on overall reaction rate. This is often and conveniently expressed using the catalyst effectiveness factor, defined as the ratio of the reaction rate averaged across the pellet volume to the reaction rate prevalent on the pellet surface (i.e. as dictated by the mean pellet surface concentration). The effectiveness factor is then expressed as a function of pellet geometric characteristics and intrinsic reaction rate, which is usually collectively quantified using the Thiele Modulus. Considerable work has been conducted defining, quantifying and verifying this functional dependency for single pellets (e.g. ¨ Thiele, 1939; Zeldovitch, 1939; Damkohler, 1937, Rao et al., 1964) or a large number of pellets being stirred to avoid external mass transfer limitations (e.g. Xu and Chuang, 1997). There has however been little consideration as to how the relationship translates to random, and hence geometrically complex, packings of the pellets. There is a substantial body of work in the literature (e.g. Kutsovsky et al., 1996; Sederman et al., 1998; Yarlagadda and Yoganathan, 1989) indicating velocity field heterogeneity in packings of modest aspect ratio, generally showing pronounced

n

Corresponding author. E-mail address: [email protected] (M.L. Johns).

0009-2509/$ - see front matter & 2011 Published by Elsevier Ltd. doi:10.1016/j.ces.2011.04.001

flow rates adjacent to the wall. It is often postulated, and has been showed in both simulations (Sullivan et al., 2005) and experiments (Yuen et al., 2003), that this velocity heterogeneity results in pore scale conversion heterogeneity. In the work presented here, we apply Lattice Boltzmann (LB) techniques to simulate a first order irreversible reaction occurring in a single liquid phase on and in various porous pellet shapes (spherical, cylindrical and (cylindrical) trilobe) as a function of reactant feed velocity. This is initially applied to the case of single pellets and the dependency of effectiveness factor upon pellet and reaction characteristics is compared to analytical and semianalytical predictions. The simulation methodology is subsequently applied to 3D random packings of the various catalyst pellet shapes in cylindrical containers operating under continuous flow. The spatial distribution and variance of the effectiveness factor distribution is quantified and compared to that predicted by the appropriate analytical and semi-analytical descriptions applied to the pellet size distribution.

2. Theoretical background The effectiveness factor, Z, of a catalyst pellet is defined as (Thiele, 1939): reff ¼ rsurf  Z,

ð1Þ

where reff is the reaction rate averaged across the entire pellet volume and rsurf is the average reaction rate on the surface of the pellet. In the case of no external mass transfer limitations for

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D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

a first order reaction, rsurf ¼k  CS(constant), where Cs(constant) is the constant surface concentration of reactant and k is the reaction rate coefficient. Due to mass transfer limitations imposed by internal diffusion of both reactants and products within the pellets, 0o Z r1, and often in practice Z 51. Thiele (1939) developed analytical solutions for first-order reactions describing the dependency of Z on particle geometry, k, and the internal diffusion coefficient, Dintra, for both spherical and rectangular pellet geometries respectively. These system parameters were later combined into a dimensionless Thiele modulus (Bischoff, 1965). For a first order irreversible reaction in a porous spherical catalyst, the Thiele modulus is defined as sffiffiffiffiffiffiffiffiffiffiffi r k ^¼ , ð2Þ 3 Dintra

to generate 3D simulation lattices for packed beds of these various pellet shapes, 3D images were acquired of real pellet packings using Magnetic Resonance Imaging (MRI). Table 1 summarizes both pellet and column characteristics. With respect to the MRI measurements, the individual packings were flooded with water and consequently the pore space imaged using a 3D RARE MRI sequence (Henning et al., 1986). A 400 MHz (1H frequency) Bruker AV spectrometer was used, equipped with a vertical bore 9 T magnet and a 25 mm inner diameter r.f. coil. The isotropic imaging resolution of 100 mm adequately resolved the pore space in the various packings, with 25–30 voxels representing the main axis of the various particle shapes considered. 3D rendered images of the 3 packings are presented in Fig. 1. More details on the principles of MRI can be found elsewhere (Callaghan, 1991).

where r is the pellet radius. The analytical solution for Z is   1 1 1  Z¼ : F Tanhð3FÞ 3F

3.2. Lattice Boltzmann (LB) reaction simulations ð3Þ

As noted by Thiele (1939) the assumption of a constant reactant surface concentration, Cs(const), (equal to the inlet concentration) and thus the complete removal of external mass transfer limitations, corresponds strictly to an infinite flow rate over the pellet surface. Eq. (3) thus effectively represents an upper limit. For first order reactions in irregular particle shapes, ^ can be approximated as sffiffiffiffiffiffiffiffiffiffiffi V k ^¼ , ð4Þ S Dintra where V is the pellet volume and S the pellet surface area (Aris, 1957), Z can then be determined using Eq. (3) for a sphere and Eq. (5) for other shapes (Aris, 1957) I ð2FÞ Z¼ 1 FI0 ð2FÞ

ð5Þ

In(x) is the modified Bessel function of order n. A more comprehensive treatment of the effectiveness factor can be found in Aris (1975). Analytical solutions are also possible for reversible reactions (e.g. Schneider and Mitschka, 1966; Xu and Chuang, 1997), where the Thiele modulus, ^, depends on the equilibrium concentration. Effects of temperature gradients have also been considered in this relationship (Copelowitz and Aris, 1970; Hlavacek and Kubicek, 1970; Bischoff, 1968).

3. Methodology 3.1. Systems studied 3D simulation lattices for a single sphere, cylinder and trilobe pellet were artificially created. Trilobes are a clover shaped overlap of three identical cylinders. These lattices were represented by at least 50 grid points along the major dimension of the respective pellet, sufficient grid independence was demonstrated by re-performing the simulations at a coarser resolution. In order

The single value relaxation Bhatnagar–Gross–Krook based LB method (Chen and Doolen, 1998) was used to simulate the 3D pore-scale flow field. The mass transport/reaction LB method is based on the model proposed by Flekkøy (1993), with simulations being conducted using the simulated flow field as a template. Appropriate and different diffusion coefficients were chosen for the inter- and intra-pore spaces with respect to simulations performed on the packings. More details on the LB method used in this study and on its experimental, via direct comparison with 3D MRI velocity and concentration mapping, and analytical validation can be found in the literature for both hydrodynamics (Manz et al., 1999; Mantle et al., 2001; Gladden et al., 2003; Yuen et al., 2003) and mass transport/reaction (Sullivan et al., 2005, 2006). 3.3. Simulations conducted Using the generated model simulation lattices of single pellets, simulations were initially conducted assuming no external mass transfer limitations for a first order reaction (A ) B) with an arbitrary reaction rate constant of k¼0.06 s  1. This hence assumes a constant reactant (A) concentration, CS(constant), at the surface of the pellets and effectively corresponds to an infinite flow rate adjacent to the pellet surface. Under these conditions, Eqs. (2) and (3) are strictly valid. Dintra was varied from 1  10  11– 1.45  10  9 m2 s  1 in order to probe a suitable range of the Thiele modulus (^). This set of simulations is hereafter referred to as Simulation Set A. The complexity of these single pellet simulations was then increased by including a pressure driven flow field over the pellet surface with Cinlet being set equal to the constant surface concentration, CS(constant), used in the previous set of simulations. These simulations were conducted for various flow rates to study the effect of velocity on the relationship between Z and ^. Pellets were orientated with their major axis perpendicular to the direction of superficial flow and the superficial velocity was varied between 0.05 and 3.12 mm s  1. This set of simulations is hereafter referred to as Simulation Set B. The full 3D pellet packing images (Fig. 1) were then used as simulation lattices. In this case, due to computational requirements, only two values of Dintra: 1.45  10  9 and 5.0  10  10 m2/s

Table 1 Pellet and column characteristics. Pellet shape

Material

Dimensions mean

Stan. dev. (V/S) (mm)

Glass column diameter (mm)

e

Sphere Cylinder Trilobe

Glass Z-alumina Z-alumina

3 mm (Dia.) 3 mm (Dia.)/3 mm (Leng.) 2.5 (Dia.)/7 mm (Leng.)/1.4 mm (Dia. of one lobe)

0.015 0.021 0.038

20 20 20

0.50 0.45 0.55

D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

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Fig. 1. Cutaway view of the 3D packings of (a) spheres, (b) cylinders and (c) trilobe shaped pellets, as imaged using MRI.

and a superficial velocity of 0.03 mm s  1 were considered. For comparison purposes, all other parameters were identical to those used in the single particle simulations. This set of simulations is hereafter referred to as Simulation Set C. 3.4. Effectiveness factor determination The value of Z was calculated for every pellet as follows: P k  ni¼ 1 Ci ð6Þ Z¼ rsurf  n where Ci is the concentration of reactant A in voxel i and n is the total number of voxels in each pellet. In the case of Simulation Set A, Z can be trivially calculated using rsurf ¼k  CS(constant). In the case of Simulation Set B, rsurf was assumed to equal either k  C S or k  CS(constant) where C S is the mean surface concentration of reactant A on the pellet surface. With respect to Simulation Set C, the situation is more complex. Initially, a pore partitioning algorithm (Baldwin et al., 1996) was used to identify all individual pellets in the packing structure. This systematically defines boundaries between pellets as minima in hydraulic radius. Geometric characteristics can be computed for each pellet, namely the particle radius, r (in the case of the spherical particles), surface area, S, and volume, V. Whilst the total

surface area of the pellets can participate in reaction/mass transfer when considering single catalyst pellets, this is not true of pellet packings where surface area in contact will other pellets will effectively not allow mass transfer into the pellets. S thus corresponds only to the pellet–liquid interface. This allows the calculation of ^ for each particle according to Eq. (2) (for spheres) and 4 (for cylinders and trilobes). For each pellet, all surface voxels are then identified and the mean surface reactant concentration, C S , calculated. rsurf is then calculated for each particle as: k  C S . In a continuous packed bed reaction, surface reactant concentration, CS, will naturally decrease with depth into the packing, which renders the use of the constant inlet concentration ( ¼CS(constant)) in calculating rsurf, and hence Z, for each pellet as being dominated by depth into the bed and hence worthless. rsurf, as calculated using C S , accounts for the variation in CS across each pellet’s surface and thus corresponds directly to the mean pellet surface reaction rate.

4. Results and discussion 4.1. Single pellets—Simulation Sets A and B Fig. 2(a) shows the 3D spatial distribution of reactant concentration, C, (and hence in effect Z as they scale linearly) as

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Upstream edge

Downstrem edge

0.04 0.035 0.03

r [s-1]

0.025 0.02 0.015 0.01 0.005 0 0.0005

0.0015 0.001 position along the particle [m]

0.002

Fig. 2. Concentration variation in 3D for a spherical pellet simulated in Simulation Set A (a) and Simulation Set B (b). The variation in reaction rate across the pellet diameter is shown in (c) in the direction of superficial flow. ^ ¼ 1.03 and the superficial velocity is 0.26 mm s  1.

produced by Simulation Set A, for the case of ^ ¼1.03. The effect of flow (a superficial velocity of 0.26 mm s  1 was imposed) is shown for the same geometry in Fig. 2(b), extracted from Simulation Set B. A trailing concentration wake is evident downstream of the pellet. Fig. 2(c) shows the corresponding local reaction rate across the diameter of the pellet in the direction of superficial flow. The expected asymmetry is evident due to variations in Cs, this will become more prominent for a lower flow rate or larger pellet. Fig. 3(a–c) shows the variation in Z with ^ for the three pellet shapes, respectively, for Simulation Set B, when rsurf is calculated using k  CS(constant). Also shown is the data produced by Simulation Set A; agreement between the analytical solution (Eqs. (2) and (3)) and the simulations for spherical pellets (Fig. 3(a)) is excellent, disagreement between the analytical solution and the simulations are consistently smaller than 0.4%. The agreement

between the simulations (Set A) and the semi-analytical solution (Eqs. (3)–(5)) for cylindrical and trilobe pellets (Fig. 3(b) and (c)) is reasonably good and acceptable given the simplifications required by Eq. (4). The corresponding data for Simulation Set B predictably lies below that of both the analytical/semi-analytical solutions; the surface concentration of reactant, Cs, will always be less than the inlet concentration of reactant, Cinlet. The dependence of Z upon superficial velocity (expressed here in more general terms as the Pe (v  2d=Dinter ) number, where d is a characteristic length scale) is quite distinct; as velocity (and Pe) increase the discrepancy between the simulation and analytical/ semi-analytical solution diminishes, consistent with these solutions being strictly applicable to an infinite velocity. The discrepancy also diminishes as ^ is increased, for the range of conditions simulated for all pellet shapes. The relative discrepancy is however always less than 50% (which is the value

D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

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1 anal 1 semi-anal Set A Set B: Pe 1012 670 336 η

η

Set A Set B: Pe 3100 1550 1033 516 258 52

0.1

0.1

0.1

1

10

100

0.1

10

1

100

Φ

Φ

1 semi-anal Set A Set B: Pe 896 594 η

298

0.1

0.1

1

10

100

Φ Fig. 3. Variation of Z with ^ as produced by both Simulation Sets A and B for (a) spherical, (b) cylindrical and (c) trilobe shaped catalyst pellets. The prediction of the analytical/semi-analytical solution is shown by the solid lines. In Simulation Set B, rsurf was calculated using k  CS(constant). The superficial velocities was varied from 0.05 to 3.12 mm s  1 and simulations were conducted for k¼ 0.06 s  1 and Dintra between 1  10  11 and 1.45  10  9 m2 s  1. Additional simulations were also performed for Set A with Dintra set to 1.1  10  12 and 5  10  9 m2 s  1, respectively.

corresponding to the lowest flow rate considered at the lowest value of ^ considered). The velocities (and hence values of Pe) simulated are significantly lower than what would be typical in reactors, the discrepancy diminishes as velocity is increased, and thus the data suggests that the analytical/approximate literature correlations are reasonably accurate with higher velocities (albeit for a single pellet). Whilst the data in Fig. 3 is reasonably accurate when rsurf is set equal to k  CS(constant), where CS(constant) ¼Cinlet, as mentioned previously this is not applicable to a packed bed as an erroneous reduction in Z would occur with distance into the bed. We note that reaction engineers would never estimate bed performance using the inlet concentration to calculate Z. Consequently the data is re-plotted in Fig. 4 for Simulation Set B, with rsurf set equal to k  C S , where C S is the means surface concentration of reactant for the particular pellet. Agreement between the simulations and the

analytical/semi-analytical description is reasonably good although slightly lower values of Z are produced by the simulations compared to the analytical and in particular the semi-analytical solutions. The simulation results are effectively independent of the velocity (and hence value of Pe number) employed. 4.2. Packed beds The superficial (z) velocity component of the flow fields through the packed beds are shown in Fig. 5(a–c), a significant level of heterogeneity is evident. The subsequent mass transport simulations are shown in Fig. 5(d–f) in the form of maps of local conversion of the reactant. The effectiveness factor was calculated for each pellet in the simulations using Eq. (6) and the calculated value of C S for each pellet. Number distributions of Z are presented in Fig. 6(a–c) for the three pellet shapes with

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D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

1

1

semi-anal anal Set A 3100

Set A 1012 670

1550 1033 516

336 packing semi-anal

η

η

258 packing anal packing LB

0.1

0.1

0.1

packing LB

1

10

0.1

100

1

10

Φ

100

Φ

1 semi-anal Set A 896 594 298

η

packing semi-anal packing LB

0.1

0.1

1

10

100

Φ Fig. 4. Variation of Z with ^ as produced by Simulation Set B, where rsurf was calculated using k  C S : (a) spherical, (b) cylindrical and (c) trilobe shaped catalyst pellets.

Dintra ¼1.45  10  9 m2/s. Also shown in Fig. 6 is the distribution of Z as predicted by the analytical solution for spheres (Eqs. (2) and (3)) and the semi-analytical solution for cylinders and trilobes (Figs. 3–5); in generating these data we used the S and V calculated for each pellet and as such we naturally include the effect of pellet size distribution (larger pellets will correspond to a relatively larger value of ^ and hence lower Z). The predictions of the analytical/semi-analytical theory are generally slightly larger than that produced by the simulations. However the general shape of the distribution is common. Table 2 shows the mean and standard deviation of Z as calculated by both the simulations and the analytical/semianalytical description for the various packed beds. There is reasonably good agreement between the simulation and analytical/semi-analytical mean although the simulation prediction is consistently slightly smaller, this disagreement is worse if we take the whole pellet surface area as opposed to the pellet–liquid surface area to determine S, as required by Eq. (4). Obviously particle–particle contact area would not be directly accessible for large-scale reactors; presentation of our data in a scaleable and applied format will be the focus of future work. The mean Z  ^

data from these simulations using the packings is superimposed into Fig. 4 and is clearly consistent, in terms of magnitude of under-prediction, with the single pellet results. The standard deviation in Z, as reported in Table 2, is partially a consequence of the pellets’ size distribution; the significantly larger variance in Z for the trilobe pellets reflects their comparatively broader size distribution (as presented in Table 1). The standard deviations for Z also tend to be narrower for a lower Dintra. At a higher Dintra a local reduction of CS provokes a higher loss of reactive volume of the pellet due to the smaller intra-pellet concentration gradient. However the spheres and cylinders have a comparable pellet volume variation. For this comparison, a normalized standard deviation in Z can be defined as the ratio of the standard deviation of the simulated Z, s(Z), and the standard deviation of the analytical/semi-analytical Z, s(Z0). s(Z0) will not include the impact of flow field heterogeneity on Z and will depend only on pellet size variations. s(Z) will included both effects and thus if s(Z)/s(Z0)  1 the heterogeneity in Z can be considered as being entirely due to pellet volume variation. For s(Z)/s(Z0)41 the flow field increases the Z heterogeneity and for s(Z)/s(Z0)o1 it decrease the heterogeneity in Z. Table 2 shows the s(Z)/s(Z0)

D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

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Fig. 5. Cutaway view of the superficial flow velocity component of flow through a 3D packed bed of sphere (a), cylinder (b) and trilobe (c) pellets as simulated by the LB method. The superficial flow direction is the positive z-direction. Corresponding cutaway views of the conversion of reactant (conversion ¼ 1 means the reactant has been fully converted into product) in the 3D packed bed of sphere (d), cylinder (e) and trilobe (f) pellets as simulated by the LB method.

ratio for Dintra,1 and Dintra,2 for all three packings. For Dintra,1 all packings have s(Z)/s(Z0) 41 indicating the significant effect of flow on the Z heterogeneity. Despite similar s(Z0) values, spherical pellets have less flow induced heterogeneity than cylindrical

pellets, potentially due a more homogenous packing structure. As expected s(Z0) is comparatively large for the trilobes due to the wider particle volume distribution. We note previous observations of the effect of flow heterogeneity in 2D beds on local Pe and

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1 1

0.8

0.8

ana. sol. for D1

0.6

cumulated freqeuncy

cumulated freqeuncy

D1 = 1.45 x 10-9 m2/s

0.4

0.2

0.6 D1 = 1.45 x 10-9 m2/s 0.4

0.2

0 0

0.2

0.4

0.6

η

0.8

semi-ana. sol. for D1

0

1

0

0.2

0.4

η

0.6

0.8

1

1

cumulated frequency

0.8

0.6

0.4 D1 = 1.45 x 10-9 m2/s semi-ana. sol. for D1 0.2

0 0

0.2

0.4

0.6

0.8

1

η Fig. 6. Analytical and simulated effectiveness factor distributions as extracted by the pore partitioning algorithm for different Dintra,1 ¼ 1.45  10  9 for 3D packings of spheres (a), cylinder (b) and trilobes (c).

Table 2 Simulated and analytical/semi-analytical mean effectiveness factor and effectiveness factor standard deviation for packings of spheres, cylinders and trilobes for two Dintra: 1.45  10  9 and 5.0  10  10 m2/s and a superficial velocity of 0.03 mm s  1. Shape

mean Z

s(Z)

s(Z0)

s(Z)/s(Z0)

Simulation Analytical Simulation Analytical Dintra,1 1.45  10  9 m2/s Sphere Cylinder Trilobe

0.61 0.19 0.33

0.66 0.25 0.39

0.029 0.082 0.119

0.020 0.012 0.063

1.45 6.83 1.89

Dintra,2 5.0  10  10 m2/s Sphere Cylinder Trilobe

0.42 0.13 0.23

0.46 0.15 0.25

0.012 0.058 0.072

0.019 0.008 0.047

0.63 7.25 1.53

Sh numbers (Guo et al., 2003), quantitative comparison with our results is not possible given the different dimensionalities, qualitative agreement is however evident. No correlation was however evident between Z and radial or axial position within the bed suggesting that ordered packing due to container walls and consequential velocity heterogeneity was not responsible for any increased variance in Z. Velocity heterogeneity due to local random packing is thus suspected, resulting in a reduction in effective surface area due to local stagnant regions. This relationship between distribution and variance of Z and local packing characteristics will be the feature of future work.

5. Conclusions Lattice Boltzmann (LB) techniques are used to simulate a first order reaction occurring in a variety of porous catalyst pellet

D.A. Graf von der Schulenburg, M.L. Johns / Chemical Engineering Science 66 (2011) 3003–3011

shapes, namely spheres, cylinders and trilobes. From these simulations on single pellets, the dependency of Z upon ^ was determined; this compared very well with the analytical solution for spherical pellets and reasonably well for cylindrical and trilobe shaped pellets. The effect of flow on these simulations was also explored—at relatively modest velocities; there was good agreement with the analytical/semi-analytical predictions. Using the average surface concentration on the pellets, Z was slightly over-predicted by the analytical/semi-analytical prediction. The simulation protocol was then applied to random packings of the various pellet shapes. Agreement between simulation and analytical/semi-analytical prediction was reasonable and consistent with the corresponding results for single pellets. The effects of flow heterogeneity and particle size distribution on the Z distribution were individually accounted for. The data suggests that a direct application of the analytical/semi-analytical descriptions of the Z  ^ relationship will produce a reasonably accurate description of the overall reactor performance under steady state, at least in terms of the narrow parameter space covered in the current simulation work. In future we will explore the sensitivity of overall bed performance to changes in local particle effectiveness factor.

Nomenclature CS(constant) constant surface concentration CS mean surface concentration Ci concentration in voxel i Dinter bulk diffusivity Dintra inner particle diffusivity d characteristic particle length scale In(x) modified Bessel function with order n k reaction rate constant Pe Pe´clet number r particle radius rsurf pellet surface reaction rate reff effective reaction rate S free particle surface area V particle volume Z effectiveness factor ^ Thiele modulus s(Z) standard deviation of simulated effectiveness factor s(Z0) standard deviation of analytical effectiveness factor

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