Generalized correlations for mass transfer and pressure drop in fiber-based catalyst supports

Chemical Engineering Journal 325 (2017) 655–664

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Generalized correlations for mass transfer and pressure drop in fiber-based catalyst supports Erik Reichelt a,⇑, Matthias Jahn b a b

Technische Universität Dresden, Institute of Materials Science, 01062 Dresden, Germany Fraunhofer IKTS, Fraunhofer Institute for Ceramic Technologies and Systems, Winterbergstrabe 28, 01277 Dresden, Germany

h i g h l i g h t s
Mass transfer in fiber-based supports can be described by a packed bed correlation.
The Ergun equation can be applied to fiber-based supports.
The Sauter diameter allows the applicability of the correlations to both structures.
The correlations are validated for Re = 0.01–10,000.

a r t i c l e

i n f o

Article history: Received 22 February 2017 Received in revised form 17 May 2017 Accepted 18 May 2017 Available online 19 May 2017 Keywords: Mass transfer Pressure drop Fiber-based catalyst supports Low Reynolds numbers

a b s t r a c t Fiber-based catalyst supports are of increasing interest for different catalytic applications. So far, no reliable correlations for mass transfer and pressure drop with a wide range of applicability are available. The paper shows that in both cases correlations for packed beds of spheres can be applied if the Sauter diameter is used as characteristic length. The findings are confirmed with help of a comparison to literature data and own experimental results on mass transfer and pressure drop. The generalized correlations simplify the design of novel fiber-based supports and of the corresponding reactors. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Due to the shortage of important resources and an increasing competition on the world market, process intensification is of increasing importance for the chemical industry. The major goal is the development of cleaner, smaller and more efficient novel processes or the improvement of established processes. Therefore, several process-intensifying technologies have been developed in industry as well as in academia. Very often these developments are focused on the minimization of mass and heat transfer limitations in different process units. Due to the high importance of heterogeneous catalysis and the great potential for process intensification in this field, several novel reactor concepts and catalyst support structures have been proposed in the last decades. A large number of these concepts are based on structures built from fibers or cylinders. These structures can be summarized by the term

⇑ Corresponding author at: Fraunhofer IKTS, Fraunhofer Institute for Ceramic Technologies and Systems, Winterbergstrabe 28, 01277 Dresden, Germany. E-mail address: [email protected] (E. Reichelt). http://dx.doi.org/10.1016/j.cej.2017.05.119 1385-8947/Ó 2017 Elsevier B.V. All rights reserved.

fiber-based catalyst supports, including well-known structures like wire meshes, but also metallic fiber filters, glass fiber fabrics, ceramic mats and microfibrous entrapped catalysts [1–3]. In recent years additive manufacturing is becoming increasingly interesting in different fields of application, also for the preparation of catalyst supports. Thus, techniques like robocasting [4,5] and selective electron beam melting [6,7] were applied for the manufacturing of fiber-based catalyst supports. Besides these novel structures also foams [8] can be regarded as fiber-based catalyst supports, with the struts being the basic structure. Except for wire meshes and to a limited extent foams, most of the fiber-based catalyst support structures proposed in literature are not industrially applied. A step in this direction could be the availability of reliable correlations for mass transfer and pressure drop. So far no generalized correlations for fiber-based catalyst support structures exist, as Section 2 shows. The aim of this work is to show that the mass transfer correlation developed by Reichelt et al. [9] for packed beds can be applied to fiber-based catalyst supports if the Sauter diameter is used as the characteristic length. Following the same approach, the well-known Ergun

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Nomenclature d dh dS D Dax f fS

diameter (m) hydraulic diameter (m) Sauter diameter (m) diffusion coefficient (m2 s1) axial dispersion coefficient (m2 s1) Fanning friction factor (–) Fanning friction factor with dS as characteristic length (–) 2 Hg Hagen number ¼ 2fRe (–) HgS Hagen number with dS as characteristic length ¼ 2f S Re2S (–) km mass transfer coefficient (m s1) L length (m) M mesh number (m1) NRMSD normalized route-mean-square deviation (–) p pressure (Pa) Pe Péclet number ¼ ud D (–) Péclet number with dS as characteristic length ¼ udDS (–) PeS Peax axial Péclet number ¼ Dudax (–) Peax,S

axial Péclet number with dS as characteristic length S ¼ ud Dax (–)

Sc

Reynolds number ¼ udgq (–) Reynolds number with dS as characteristic length ¼ udgS q (–) Schmidt number ¼ Dgq (–)

Sh

Sherwood number ¼ kmDd (–)

ShS

Sherwood number with dS as characteristic length ¼ kmDds (–) geometric surface area (m1) time (s)

Re ReS

SV t

u: V

superficial velocity (m s1) volumetric flow rate (m3 s1)

Greek letters c coefficient from Eq. (9) e porosity (–) g dynamic viscosity (Pa s) 0 temperature (K)

q s u

v x

density (kg m3) tortuosity (–) volume fraction (–) trade-off index (–) mass fraction (–)

Subscripts AP active particle app apparent B bed corr calculated by a correlation F fiber i inner max maximum value meas measured value min minimum value Re = 0 at stagnant conditions SC single cylinder Sph sphere SS single sphere St strut Str structure

equation [10] can be applied for the description of pressure drop characteristics.

2. Literature review Considering the large number of publications on different fiberbased catalyst supports [3], the amount of works on mass transfer is rather low. Nevertheless, some correlations were reported [3]. A drawback is that the applicability of these correlations is most often only confirmed for a limited range of Reynolds number. A summary of experimental results on mass transfer at different fiber-based catalyst supports is given in Fig. 1. As characteristic lengths for Sherwood and Reynolds number the fiber or strut diameter (dF, dSt) were chosen. The results are not corrected for the influence of bed porosity eB on mass transfer. For packed beds of spheres with bed porosities in the range of eB = 0.26–0.80 the [11–14]. However, dependence is known to be about Shapp  e1 B the influence of porosity on mass transfer at fiber-based catalyst supports is less considered in literature. Bed porosities for fiberbased structures are generally in the range of eB = 0.70–0.95 [3]. Considering this rather low range of variation, the results depicted in Fig. 1 indicate the possibility to describe the mass transfer of fiber-based catalyst supports by a single correlation. The results of Richardson et al. [15] on foams and of Groppi et al. [16] on fiber filters differ from the bulk of experimental results. They show considerably lower apparent Sherwood numbers. In general, the choice of a suitable characteristic length for foams is difficult. Often pore or window diameters are used [17– 19]. However, these lengths are difficult to compare with other

Fig. 1. Literature data on apparent mass transfer in fiber-based catalyst support structures.

foams. Also the application of the strut diameter is problematic, because it is not constant over the whole cell length. It is therefore more reasonable to directly measure or calculate an average strut diameter [20–22]. For the results from Ref. [18] presented in Fig. 1, the average strut diameter was calculated by applying a simple cubic cell model:


¼ 4ð1  eB Þ : d St SV B

ð1Þ

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The geometric surface area of the bed SV B was determined by Incera Garrido [18] with help of magnetic resonance imaging. Richardson et al. [15] used the manufacturer specification for the pores per inch for the calculation of the characteristic length of the applied foam. This nominal value can considerably differ from the actual value, as it was shown e.g. by Groppi et al. [21]. This could explain the difference between the results of Richardson et al. [15] and the other results on foams. Unexpectedly low Sherwood numbers were also determined by Groppi et al. [16] for metallic fiber filters. The results were explained with help of the model of Nelson and Galloway [23]. However, the model could only be fitted to the experimental results by adding an empirical geometrical order index. A possible reason for the low Sherwood numbers measured by Groppi et al. [16] could be the coating procedure. The fiber filters from FeCrAl alloy were calcined at 0 = 950 °C for t = 10 h in air in order to promote the formation of an a-Al2O3 layer on the fiber surface. Our own experiments with fiber filters from FeCrAl alloy (see Section 4) at the same conditions revealed that two different morphologies of a-Al2O3 form on the surface. Besides a rough needle- or plate-like structure also a plane surface morphology exists (Fig. 2). Subsequent coating with xPt = 1% showed that Pt accumulates on the rough surfaces. On the plane surfaces Pt can barely be detected. The results of Groppi et al. [16] might therefore be explained by a not fully active structure. If only a part of the surface shows sufficient activity, low Sherwood numbers will be calculated from the experimental results. This possible explanation is supported by results of de Greef et al. [24]. They measured the mass transfer at metallic fiber filters by an electrochemical method that does not require an additional coating step. Plotting the correlation derived by these experiments in the same range as the results from Groppi et al. [16] shows that the correlation of de Greef et al. [24] lies closer to the other results depicted in Fig. 1. Besides mass transfer characteristics, the pressure drop of a catalyst support structure is an important property. It is well known that the Ergun equation allows a reasonable prediction of the pressure drop in packed beds of spherical particles. According to Ergun [10] the pressure drop can be calculated by:

Dp ð1  eB Þ2 gu ð1  eB Þqu2 qu2 þ 1:75 ¼ 2f  ¼ 150 2 3 L dSph eB dSph e3B dSph

ð2Þ

Even though there are is large amount of alternative pressure drop correlations [25]–empirical, semi-empirical [26,27] and theoretical [28,29]–the Ergun equation is most widely applied in literature.

Fig. 2. Scanning electron micrograph of a calcined FeCrAl fiber filter.

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For the calculation of pressure drop in fiber-based catalyst supports some correlations for specific structures were proposed [3]. In order to find a correlation with a wider applicability, the findings of Lacroix et al. [22] are valuable. They showed that Eq. (2) can also be applied to foams when the Sauter diameter

dS ¼

6 SV Str

ð3Þ

is used instead of the diameter of the sphere dSph. SV Str is the geometric surface area of the structure. From Eq. (3) follows in case of infinite fibers:

dS ¼ 1:5dF :

ð4Þ

This approach offers the possibility to describe the pressure drop for the vast majority of fiber-based supports by a single correlation. That mass transfer and pressure drop are linked is well known in terms of Reynolds [30] and Léveˆque analogy [31]. It is therefore interesting for the evaluation of novel structures to compare the trade-off between mass transfer and pressure drop to established supports (e.g. packed bed of spheres). Giani et al. [20] proposed a so-called trade-off index for that purpose:

v¼

SV B dShapp SV B qukm;app ¼ : 2fReSc Dp=L

ð5Þ

The correlation allows a classification of different support structures, but also has some drawbacks. The first one is, that the dependence on Schmidt number is not correct. While an increase in Schmidt number leads to an intensification of mass transfer [32], it has no impact on pressure drop. According to the definition in Eq. (5) v would decrease with increasing Schmidt number, indicating an opposing trend. Additionally, plots of v show nearly no dependence on Reynolds number [20]. However, there is a considerable influence due to the different dependencies of mass transfer and pressure drop on Reynolds number. But most important is the fact that a reliable comparison of different structures with help of Eq. (5) is only possible for an appropriate choice of the characteristic length d. 3. Development of the correlation 3.1. Mass transfer Several correlations describing the mass transfer in different fiber-based catalyst supports were published [3]. But so far there is no correlation allowing the prediction of mass transfer characteristics over a wide range of conditions. Reichelt et al. [9] developed a correlation for the description of mass transfer in packed beds of spheres. The correlation was derived on the basis of existing theoretical and semi-empirical correlations. It was shown that the resulting correlation is applicable with good accuracy for a wide range of Reynolds numbers, Schmidt numbers and bed porosity. It will be shown that this correlation can be applied to fiberbased supports by choosing the Sauter diameter as characteristic length. The correlation of Reichelt et al. [9] was derived on the basis of the mass transfer characteristics of a single sphere. Therefore, the mass transfer at a single cylinder in cross flow should be considered first. A comparison to literature data for single spheres is given in Fig. 3. It shows that Nusselt or Sherwood number for cylinders and spheres are comparable for a wide range of Reynolds numbers. Only for Re < 10 there is a difference between the results for cylinders and spheres. This can be explained by the influence of the different limiting values of the Sherwood number on low Reynolds number mass transfer. In case of a single sphere the

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single correlation. However, it is important to note that all correlations presented in this publication can be transferred to dh as characteristic length by applying Eq. (7). The accuracy and the applicability of the correlation to both fiber-based supports and packed beds is not affected by choosing between this two options. With dS as the characteristic length a generalized form of the mass transfer correlation of Reichelt et al. [9]–applicable to packed beds of spheres and to fiber-based supports–can be obtained:

"

#1=3

1  ð1  eB Þ5=3

ShS ¼ ShS;ReS ¼0 þ 1:26

2  3ð1  eB Þ1=3 þ 3ð1  eB Þ5=3  2ð1  eB Þ2 ! 0:037Re0;8 1=3 S Sc : ð8Þ  0:991PeS þ 1 þ 2:44Re0;1 ðSc2=3  1Þ S

Fig. 3. Comparison of the literature data on gas/solid heat and mass transfer on single cylinders [33] and single spheres [34–36].

limiting value is ShSS,Re=0 = 2. For long single cylinders it follows from theoretical considerations that ShSC,Re=0 ? 0, as for every structure with at least one infinite dimension [37]. The data in Fig. 3 confirms that it is reasonable to apply the same mass transfer correlation for single spheres and single cylinders. However, the application of the correlation of Reichelt et al. [9] to fiber-based catalyst supports leads to the problem of choosing an appropriate characteristic length. In literature different equivalent diameters were applied for that purpose. Often the diameter of a sphere with the same external surface area was chosen [38,39]. But also the reciprocal of the specific surface area SV B was used [3,15]. Two additional potential characteristic lengths are the Sauter diameter dS (Eq. (3)), as it was applied by Lacroix et al. [22], and the hydraulic diameter

dh ¼

4eB : SV B

ð6Þ

The hydraulic diameter is often chosen for mass transfer as well as pressure drop correlations [40–42] and is linked to the Sauter diameter via

dS ¼ 1:5

1  eB

eB

dh :

ð7Þ

From these two options the Sauter diameter was chosen as characteristic length in this publication, because it is more common to use particle diameters or in this case equivalent particle diameters for mass transfer and pressure drop correlations (except for honeycomb structures). The results in Section 5.1 indicate that the Sauter diameter is a proper choice, allowing the calculation of mass transfer in packed beds and fiber-based catalyst supports by a

In Eq. (8) ShS, ReS and PeS are the Sherwood, Reynolds and Péclet number with dS as characteristic length, respectively. For a packed bed of spheres these dimensionless numbers are identical to the ones without indices applied in Ref. [9]. The limiting value of the Sherwood number for packed beds of spheres is a function of bed properties [9]. For fiber-based catalyst supports the Sherwood number at stagnant conditions is ShS;ReS ¼0 ? 0. In contrast to spheres, in packed beds of fibers or cylinders different alignments are possible (Fig. 4). Based on its derivation the correlation is particularly applicable to randomly aligned fibers in cross flow. Supports with this kind of structure are fiber filters and wire meshes, in the latter case only if several layers are not arranged in-line by design. The dependence of mass transfer on a defined alignment cannot be described by Eq. (8). Such structures can be prepared by additive manufacturing techniques like robocasting and selective laser beam melting [4–7]. Ferrizz et al. [4] showed that different alignments of fibers at constant bed porosity influence the mass transfer characteristics. This influence and the applicability of the developed correlation to defined structures are subject of Section 5.1. 3.2. Axial dispersion For low Péclet numbers the influence of axial dispersion has to be accounted for. For the calculation of the resulting apparent Sherwood number the solution of Wehner and Wilhelm [43] to the mass balance of a flow reactor can be applied [9]. With dS as the characteristic length the apparent Sherwood number can be calculated by:

PeS 6ð1  eB ÞuAP   2 Pe 4cexp 2ax;S dLBS dS 4     ln cPeax;S LB cPe LB  ð1  cÞ2 exp  ax;S ð1 þ cÞ2 exp

ShS;app ¼ 

2

Fig. 4. Differently arranged packed beds of fibers: a) in-line, b) staggered, c) random.

dS

2

3 LB dS

5; ð9Þ

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Dp ð1  eB Þ2 gu ð1  eB Þqu2 qu2 þ 1:75 ¼ 2f S : ¼ 150 2 3 L dS eB dS e3B dS

with:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24ð1  eB ÞuAP ShS c¼ 1þ : Peax;S PeS

ð10Þ

The axial Péclet number can be calculated for packed beds of spheres with reasonable agreement by the correlation [44]:

1 eB 1 1 ¼ þ : Peax;S sB PeS 2

ð11Þ

For fiber-based catalyst supports the amount of published correlations for axial Péclet number is rather scarce [19,45–48]. Most works address axial dispersion in foams. As explained in Section 2 the definition of a proper characteristic length is difficult for this structure. The literature data were therefore converted to the characteristic length chosen in this work. The results are depicted in Fig. 5. It shows that dS as the characteristic length allows a proper comparison to Eq. (11). The experimental values are predominantly lower than predicted by the correlation. This is in contrast to packed beds of spheres, where in case of gas flow experimentally determined axial Péclet numbers are generally higher than predicted by Eq. (11). The lower axial Péclet numbers for liquid flow are well known from packed beds. Tsotsas and Schlünder [49] were able to explain this with the influence of microscopical flow nonuniformity on axial dispersion. Even though Fig. 5 shows a discrepancy between Eq. (11) and the experimental results, the database is too small to derive an empirical correlation. Therefore, Eq. (11) is used for the calculation of Peax,S in this work. The potential influence of this assumption on the accuracy of Eq. (9) is discussed in Section 5.1. Throughout this paper the apparent Sherwood numbers ShS,app,meas determined in the different experimental works are compared to the apparent Sherwood numbers calculated by Eqs. (8) and (9) in order to account for the influence of axial dispersion in the measurements. 3.3. Pressure drop and trade-off index As mentioned in Section 2, the correlation of Ergun [10] is established for the prediction of pressure drop in packed beds of spheres. As pointed out by Lacroix et al. [22], the correlation can also be applied to foams when the Sauter diameter is used:

ð12Þ

The Fanning friction factor with dS as characteristic length can be calculated by:

fS ¼

    1  eB 1  eB þ 1:75 : 150 ReS 2e3B

ð13Þ

The applicability of Eqs. (12) and (13) to other fiber-based catalyst supports than foams is discussed in Section 5.2. Just like for mass transfer the alignment of fibers has an influence on pressure drop. This influence is also discussed in that section. The disadvantages of the trade-off index definition proposed by Giani et al. [20] were already outlined in Section 2. An alternative definition of the trade-off index is proposed here:

v¼

ShS Hg S Sc1=3

:

ð14Þ

The choice of dS as characteristic length allows a good comparison of mass transfer and pressure drop characteristics of different catalyst supports. The Hagen number

Dp qdS L g2

3

Hg S ¼ 2f S Re2S ¼

ð15Þ

allows the dimensionless description of pressure drop. The dependence of mass transfer on Schmidt number is considered by the exponent 1/3 [32]. The apparent trade-off index can be calculated by:

vapp ¼

ShS;app Hg S Sc1=3

:

ð16Þ

4. Experimental As Fig. 1 shows, there is only a small amount of publications on mass transfer in fiber-based catalyst supports. These experimental works are mainly focused on mass transfer at Re > 1. In order to get reliable data for Re < 1, new experiments were conducted. Sherwood numbers were measured in the external mass transfer limited regime of a heterogeneously catalyzed CO oxidation. Details on experimental setup and procedure are described elsewhere [9]. Different fiber-based catalyst supports were applied for the mass transfer measurements. Because the experimental results from Groppi et al. [16] showed unexpectedly low Sherwood numbers for fiber filters, own experiments on coating were conducted. For this purpose commercial filters made from FeCrAl alloy were used (dF = 25 lm, LB = 0.7 mm, NV Bekaert SA). The prepared filters were calcined at 0 = 950 °C in air for t = 10 h. Catalytic coating was achieved by incipient wetness impregnation [9]. The mass fraction of Pt was xPt = 1%. Measurements on mass transfer at wire meshes were conducted with commercial platinum meshes (Chempur Feinchemikalien und Forschungsbedarf GmbH). The mesh number was M = 3200 m1 and the fiber diameter dF = 60 lm. The porosity was calculated according to Ref. [50]:

eB ¼ 1 

Fig. 5. Literature data on axial Péclet number in fiber-based catalyst supports in comparison to Eq. (11). All measurements were conducted on foams, except for the results from Ref. [48] (packed bed of cylinders).

659

pMdF 4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ ðMdF Þ :

ð17Þ

The wire meshes were applied without any further coating steps. However, during the experiments the meshes showed a fast deactivation, probably due to the diffusion of impurities to the fiber surface [40]. By treatment in concentrated HCl the activity could be restored. The fiber diameter was measured prior and

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subsequent to the experiments on scanning electron micrographs. No change in surface morphology or fiber diameter was detected due to catalytic corrosion. Ceramic fibers were prepared by wet phase inversion spinning. The preparation and characterization of the fibers made from LaMnO3 was described by Kaiser et al. [51]. Even though the perovskite is active for the oxidation of CO an additional coating with Pt was necessary to achieve sufficient activity for measurements in the external mass transfer limited regime. The fibers were also coated by incipient wetness impregnation. The loading of Pt was xPt = 1%. A summary of the properties of the fiber-based supports is given in Table 1. Inside the reactor, the fiber-based supports were placed between inert beds. In case of fiber filters and wire meshes alumina paper (LB  1 mm) was applied. The coated perovskite fibers were placed between quartz glass wool. In addition to the large amount of literature data on pressure drop over fiber-based catalyst supports [7,20,22,41,52–54] some measurements on pressure drop in fiber filters in laminar flow were conducted. The sample was placed between the flanges of two glass tubes with an inner diameter of di = 16 mm. The flange was sealed with a clamp and a gasket, with the gasket holding the sample in place. Pressure drop over the filter was measured at 0 = 20 °C by a differential pressure sensor (Kalinsky Sensor Elektronik GmbH & Co. KG). The measurements were conducted with nitrogen and corrected for the pressure drop of the empty tube.

amount of publications as well as to the covered range of eB, ReS and Sc. As explained in Section 2, the less reliable results from Refs. [15,16] were omitted for the comparison. Because the literature data mainly covers the range PeS > 1, new measurements at lower Péclet numbers were carried out. A comparison of the correlation to literature results and to our own experiments for mass transfer in fiber-based catalyst supports is given in Fig. 6. It demonstrates a suitable agreement between the correlation and the experimental results. However, for our own experimental results on mass transfer at wire meshes higher deviations occur. These measurements were conducted at Reynolds or Péclet numbers in the range of ReS  1 or PeS  1. This range is barely covered by literature. A possible explanation for the higher deviations is the influence of axial dispersion on mass transfer. For PeS  1 axial dispersion can be neglected, while for PeS 1 the convective component of axial dispersion can be neglected (Eq. (11)). In this case it follows:

Peax;S ¼

sB Pe : eB S

ð18Þ

For the range PeS  0.1–10 a correlation is necessary that describes axial dispersion in fiber-based catalyst supports. As it was shown in Section 3.2, such a correlation is not available. The deviation between the experimental results given in Fig. 5 and Eq. (11) is larger than the one for packed beds of spheres [44]. This

:

The flow was varied in the range of V = 0.08–85 L min1 (0 °C, 101.325 kPa). Two mass flow controllers with different flow ranges were applied in order to cover the applied flow range.

5. Comparison to experimental data 5.1. Mass transfer The applicability of the mass transfer correlation was confirmed by a comparison to literature data and our own experimental results. As Table 2 demonstrates, reliable data for mass transfer in fiber-based catalyst supports is scarce. This applies to the

Table 1 Properties of the applied catalyst supports.

dF/lm dS/lm

eB

y

Fiber filter

Wire mesh

Perovskite fibers

25 37.5 0.73

60 90 0.85

1100 1650 0.70y

Mean value of several packed beds.

Fig. 6. Comparison of Eq. (8) to literature data and to our own experimental results on mass transfer in different fiber-based catalyst supports. The influence of axial dispersion was accounted by Eq. (9). For further information on the references see Table 2.

Table 2 Summary of the mass and heat transfer data for fiber-based catalyst supports used for the comparison with the developed correlation. Structure

Reference

Measurement method

eB

ReS

Sc

PeS

Wire mesh

[40] [55] Own experiments [56] [18] [21] Own experiments [57] [4] [33]

Hexene/toluene oxidationà CO oxidationà CO oxidationà SCR of NOà CO oxidationà CO oxidationà CO oxidationà Heat transfer CO oxidationà Heat transfer

0.71–0.91 0.85* 0.85 0.85* 0.75–0.81 0.84–0.94 0.70 0.5–0.83 0.43 1

0.7–14 2–165 0.009–12 1–176 5–686 13–151 0.8–230 112–33456 273–481 0.4–566612

1.38–1.46 0.77y 0.77 0.64 0.77y 0.77y 0.77 0.72 0.77y 0.72

1–19 1–190 0.007–9 0.91–113 4–528 10–116 0.6–180 81–24089 210–370 0.3–407960

Wire mesh honeycomb Foam Packed bed of fibers

Cylinder * y à

Porosity value assumed. Sc calculated for the applied fluids. Mass transfer limited.

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Ferrizz et al. [4] applied robocasting for the manufacturing of fiber-based catalyst supports with in-line and staggered arrangement of fibers (Fig. 4a, b). Random packed beds of fibers were prepared from platinum coated perovskite fibers (Section 4). The comparison between the experimental data and the correlation shown in Fig. 7 indicates that Eq. (8) is able to describe the mass and heat transfer in random and staggered packed beds of fibers. The data from Kays [57] for beds with in-line arrangement show only minor deviations at lower Sherwood or Reynolds numbers, respectively. In contrast, the results from Ferrizz et al. [4] differ considerably from the calculated values. This is in accordance to the pressure drop results for in-line arranged beds of fibers (Section 5.2). Therefore, it can be concluded that the developed correlation should preferably be applied to random or staggered arranged fiber-based catalyst supports. The normalized route-mean-square deviations

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X Fig. 7. Comparison of Eq. (8) to literature data and to our own experimental results on mass transfer in differently arranged packed beds of fibers. The influence of axial dispersion was accounted for by Eq. (9). For further information on the references see Table 2.

Table 3 Normalized route-mean square deviation and slope for the comparison of the mass transfer correlation to experimental results (Figs. 6 and 7). Structure

Reference

NRMSD

Slope

Wire mesh

[40] [55] Own experiments [56] [18] [21] Own experiments [57] [4] [33]

0.3082 0.0849 0.0975 0.1946 0.1127 0.1450 0.0780 0.0223 0.0137 0.0269

0.7719* 0.9658 1.0619 0.9286 1.1001* 0.9658* 1.0686 1.0087* 0.6344* 1.0749

Wire mesh honeycomb Foam Packed bed of fibers

Cylinder *

Average slope of all data sets presented in the reference.

could explain the lower accuracy in the intermediate range of PeS. Further studies on axial dispersion in fiber-based catalyst supports may lead to a clarification of this point. In Section 3.1 the possibility of influencing the mass transfer by different fiber arrangements was mentioned. This effect was investigated with help of heat and mass transfer data from literature and from our own experimental results. Kays [57] measured the heat transfer in differently arranged beds of fibers (Fig. 4a–c).

ðShS;app;corr ShS;app;meas Þ2

n

NRMSD ¼

n

ð19Þ

ShS;app;meas;max  ShS;app;meas;min

as well as the slope for the data sets presented in the parity plots Figs. 6 and 7 are given in Table 3. As already stated, some correlations for mass transfer in fiberbased catalyst support structures were proposed in literature (Table 4). In order to compare the accuracy of Eq. (8) with these correlations, the normalized route-mean-square deviations of these different correlations were calculated for the experimental data given in Table 2. The influence of axial dispersion was accounted for by Eq. (9) for all correlations. The results presented in Table 4 show that Eq. (8) offers the lowest normalized routemean-square deviation over the covered range. Even though some of the other correlations also offer reasonable accuracies over a limited or even the whole range of experimental data, Eq. (8) has the additional advantage, that it can be applied to both fiberbased catalyst supports and packed beds [9]. The availability of a single correlation for these structures simplifies the comparison of both alternatives for a given application as well as the design of the corresponding reactor. As Table 4 shows, the application of dS as characteristic length allows also the application of other packed bed correlations to fiber-based supports. The well-known empirical correlation of Dwivedi and Upadhyay [14] offers a comparable accuracy to Eq. (8) at higher Reynolds numbers. For ReS < 1 the correlation proposed here offers a better prediction of mass transfer in fiberbased structures.

Table 4 Comparison of the normalized route-mean-square deviation (NRMSD) between different correlations for fiber-based catalyst supports. The correlations were compared to the experimental data from the references given in Table 2. The influence of axial dispersion was accounted for by Eq. (9). Correlation

Reference

Eq. (8) ShS ¼ 0:4548  Sh ¼ 0:91

Re0:5931 S

eB

Sc1=3 0:43

Re 2 ð1p2ffiffiffiffið1eB ÞÞ

Sc1=3 *

Originally developed for

NRMSD ReS = 0.009–566612

ReS = 0.009–1

ReS = 1–1000

ReS = 1000–566612

[9] [14]

Packed bed Packed bed

0.0137 0.0156

0.1380 0.3052

0.0593 0.0549

0.0231 0.0265

[21]

Foam

0.0719

0.1301

0.0703

0.1224

[40]

Wire mesh

0.1056

0.1134

0.1792

0.1795

[55]

Wire mesh

0.0721

0.1757

0.0858

0.1227

[56]

Wire mesh honeycomb

0.0230

0.1194

0.1042

0.0388

[24]

Fiber filter

0.0673

0.2650

0.0919

0.1144

3p

 0:283 Sh ¼ 0:94 Re Sc1=3 * eB  0:45 Sc1=3 * Sh ¼ 0:78 Re eB  0:5 Sc1=3 * Sh ¼ 1:08 Re eB 0:5

Sh ¼ 0:47 ReeB Sc1=3 * *

ReS S Sh ¼ Sh 1:5 ; Re ¼ 1:5 .

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Table 5 Pressure drop data used for the comparison between experimental results and Eq. (12). Also given are the normalized route-mean square deviation and slope of the data sets presented in the parity plots in Figs. 8 and 9. Structure

Reference

eB

ReS

NRMSD

Slope*

Wire mesh

[41] [52] [20] [22] [53] Own experiments [7]

0.67–0.97 0.68 0.91–0.95 0.81–0.91 0.59–0.98 0.73 0.72–0.87

1–90 1–465 30–1084 4–149 0.03–0.09 0.02–19 44–782

0.1791 0.0539 0.1074 0.2721 0.1058 0.0602 0.1081

1.0344 1.0195 1.0154 0.9926 1.1423 1.0241 1.0033

Foam Fiber filter Periodic open cellular structure *

Average slope of all data sets presented in the reference.

5.2. Pressure drop For the comparison of the modified Ergun equation according to Lacroix et al. [22] (Eq. (12)) to experimental results, the data summarized in Table 5 was used. A comparison between Eq. (12) and the literature results on pressure drop is given in Fig. 8. It shows a satisfying agreement, although the scatter is quite large. This could be attributed to the strong dependence of pressure drop on bed porosity eB. For most structures the value for eB is difficult to determine and therefore influences the accuracy of the prediction. This may be confirmed by some of the results from Ref. [41]. For these pressure drop data a large deviation can be seen in Fig. 8. The measurements were carried out with a knitted wire mesh. In this case bed porosity is more difficult to determine than for woven wire meshes. In the latter case several correlations for the calculation of eB from mesh number and wire diameter exist [58]. The assumption of an inaccurate determination of eB in this case is supported by the fact that the other results from Ref. [41] and also the results for other structures with comparable high bed porosities show a better agreement with Eq. (12). The parity plot in Fig. 8 demonstrates that the modified Ergun equation is applicable for the prediction of pressure drop in several fiber-based catalyst support structures in laminar as well as in turbulent flow. But, as already mentioned in Section 3.3, the applicability might be limited for structures with defined fiber arrangements. Such structures can be prepared by additive manufacturing techniques. Pressure drop data for differently arranged fiber-based structures can be taken from the work of Klumpp et al. [7]. In this publication selective electron beam melting was

Fig. 8. Comparison of Eq. (12) to literature data for pressure drop in different fiberbased catalyst supports. For further information on the references see Table 5.

Fig. 9. Comparison of Eq. (12) to literature data for pressure drop in differently arranged periodic open cellular structures [7]. For further information on the references see Table 5.

applied for the manufacturing of periodic open cellular structures. The struts of the cells were arranged in-line or staggered. The offset was adjusted by tilting the cells in different angles to the longitudinal axis of the structure. Fig. 9 shows that for all structures with staggered arrangement of the struts a good agreement between experiment and correlation is achieved. For the additive manufactured structures the strut diameter and the bed porosity can be determined with higher accuracy, which leads to a smaller scatter in comparison to Fig. 8. However, Fig. 9 also shows that Eq. (12) is not suitable for structured supports with a defined in-line arrangement of fibers or struts. For these structures the correlation overpredicts the pressure drop. This is in agreement to the results on mass transfer given in Section 5.1. Besides novel additive manufactured structures a defined in-line arrangement of fibers is rare for fiber-based catalyst supports. Therefore, Eq. (12) should be applicable for design purposes for the vast majority of fiber-based structures. The normalized route-mean-square deviation between Dp/Lcorr and Dp/Lmeas for Eq. (12) was calculated to be NRMSD = 0.0245. For the calculation the experimental data from Table 5 was used. The less reliable data from Ref. [41] and the results for in-line arranged structures from Ref. [7] were omitted. Dietrich et al. [42] published an Ergun-type correlation for the pressure drop in foams by fitting the coefficients of the correlation to experimental results. Applying their correlation to the same data set gives a normalized routemean-square deviation of NRMSD = 0.0107. The comparison between Eq. (12) and the correlation of Dietrich et al. [42] shows that in case of pressure drop a fitted correlations is able to deliver a higher accuracy. However, the direct application of the Ergun

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equation with dS as characteristic length offers a reasonable accuracy. The important advantage is the possibility to use a single correlation for fiber-based supports and packed beds.

6. Trade-off index Section 5.1 showed that the mass transfer correlation from Ref. [9], originally developed for packed beds of spheres, can be applied to fiber-based catalyst supports if the Sauter diameter is used as characteristic length. The only difference between packed beds of spheres and fiber-based catalyst supports is the limiting value of the Sherwood number. For fiber-based catalyst support structures built from infinite fibers the limiting value always approaches ShS;ReS ¼0 ? 0. The limiting value for packed beds of spheres depends on the bed properties [9]. For the industrially most relevant case of uAP = 1 and LB/dS > 10 the limiting value also approaches ShS;ReS ¼0 ? 0. The pressure drop in packed beds of spheres and in fiber-based supports can also be described by a single correlation. Therefore, the trade-off indexes for both structures can be depicted together in one diagram. Fig. 10 illustrates the well-known fact that pressure drop has a stronger dependence on Reynolds number and on bed porosity than mass transfer. This means that an intensification of mass transfer by an increase in Reynolds number or a decrease in bed porosity leads to a disproportionate increase in pressure drop and to a decrease in v. An analogous design problem is well known from heat exchanger design [57]. Therefore, high values of v and vapp not necessarily indicate an advantageous structure. The right tradeoff between mass transfer and pressure drop depends on the specific application. In case of low Reynolds number gas/solid mass transfer also the influence of axial dispersion has to be considered. Fig. 10 shows that there is no intrinsic difference between mass transfer and pressure drop properties of fiber-based supports and packed beds as it is indicated by the trade-off index from Ref. [20]. The variety of different manufacturing technologies for fiber-based catalyst supports allows a wider variation of properties like fiber diameter and bed porosity. This allows to reach trade-offs between mass transfer and pressure drop that cannot be reached by conventional packed beds. Fiber-based catalyst supports allow to work at lower Reynolds numbers and higher bed porosity. This offers the possibility to significantly increase the trade-off index v.

Fig. 10. Influence of bed porosity eB on trade-off index v (dotted lines) and apparent trade-off index vapp (solid lines) for packed beds of spheres and fiberbased catalyst supports (uAP = 1, sB = 1.4, LB/dS = 10, Sc = 1).

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7. Conclusions The applicability of a mass transfer correlation originally developed for packed beds of spheres [9], was proven with help of a comparison to literature and own experimental results. It was shown that by applying the Sauter diameter as characteristic length the description of mass transfer for packed beds of spheres and fiberbased supports was possible by a single correlation. It was also shown that with help of the Sauter diameter the Ergun equation can be applied to fiber-based catalyst supports. The possibility to predict mass transfer and pressure drop characteristics of these support structures with the same correlations demonstrates that there is no general difference between these structures. The possibility to reach high bed porosities makes fiber-based catalyst supports interesting for applications that require low pressure drops, of course accompanied with poorer mass transfer. Besides that, fiber-based catalyst supports offer the advantage that small characteristic lengths and therefore low internal diffusion resistances can be achieved without the risk of catalyst loss or bed blocking. However, for a broad industrial application specific additional advantages like increased radial and axial heat conductivity or a defined inner structure of the fibers will be necessary. The correlations developed and applied here are valuable for structure and reactor design and might therefore support the industrial application of such novel fiber-based structures. Funding This work was supported by the German Research Foundation (DFG, MI 509/19-2). References [1] Y. Matatov-Meytal, M. Sheintuch, Catalytic fibers and cloths, Appl. Catal. A 231 (2002) 1–16. [2] A. Cybulski, J.A. Moulijn, Structured reactors, a wealth of opportunities, in: A. Cybulski, J.A. Moulijn, A. Stankiewicz (Eds.), Novel Concepts in Catalysis and Chemical Reactors, WILEY-VCH, Weinheim, 2010, pp. 189–209. [3] E. Reichelt, M.P. Heddrich, M. Jahn, A. Michaelis, Fiber based structured materials for catalytic applications, Appl. Catal. A 476 (2014) 78–90. [4] R.M. Ferrizz, J.N. Stuecker, J. Cesarano, J.E. Miller, Monolithic supports with unique geometries and enhanced mass transfer, Ind. Eng. Chem. Res. 44 (2005) 302–308. [5] J. van Noyen, A. de Wilde, M. Schroeven, S. Mullens, J. Luyten, Ceramic processing techniques for catalyst design: formation, properties, and catalytic example of ZSM-5 on 3-dimensional fiber deposition support structures, Int. J. Appl. Ceram. Technol. 9 (2012) 902–910. [6] T. Knorr, P. Heinl, J. Schwerdtfeger, C. Körner, R.F. Singer, B.J.M. Etzold, Process specific catalyst supports—selective electron beam melted cellular metal structures coated with microporous carbon, Chem. Eng. J. 181–182 (2012) 725–733. [7] M. Klumpp, A. Inayat, J. Schwerdtfeger, C. Körner, R.F. Singer, H. Freund, W. Schwieger, Periodic open cellular structures with ideal cubic cell geometry: Effect of porosity and cell orientation on pressure drop behavior, Chem. Eng. J. 242 (2014) 364–378. [8] M.V. Twigg, J.T. Richardson, Fundamentals and applications of structured ceramic foam catalysts, Ind. Eng. Chem. Res. 46 (2007) 4166–4177. [9] E. Reichelt, M. Jahn, R. Lange, Derivation and application of a generalized correlation for mass transfer in packed beds, Chem. Ing. Tech. 89 (2017) 390– 400. [10] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (1952) 89– 94. [11] G.F. Malling, G. Thodos, Analogy between mass and heat transfer in beds of spheres: contributions due to end effects, Int. J. Heat Mass Transfer 10 (1967) 489–498. [12] J.T.L. McConnachie, G. Thodos, Transfer processes in the flow of gases through packed and distended beds of spheres, AIChE J. 9 (1963) 60–64. [13] P.N. Rowe, K.T. Claxton, Heat and mass transfer from a single sphere to fluid flowing through an array, Trans. Inst. Chem. Eng. 43 (1965) 321–331. [14] P.N. Dwivedi, S.N. Upadhyay, Particle-fluid mass transfer in fixed and fluidized beds, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 157–165. [15] J.T. Richardson, D. Remue, J.-K. Hung, Properties of ceramic foam catalyst supports: mass and heat transfer, Appl. Catal. A 250 (2003) 319–329. [16] G. Groppi, E. Tronconi, G. Bozzano, M. Dente, Experimental and theoretical study of gas/solid mass transfer in metallic filters as supports for microstructured catalysts, Chem. Eng. Sci. 65 (2010) 392–397.

664

E. Reichelt, M. Jahn / Chemical Engineering Journal 325 (2017) 655–664

[17] F.C. Patcas, G. Incera Garrido, B. Kraushaar-Czarnetzki, CO oxidation over structured carriers: a comparison of ceramic foams, honeycombs and beads, Chem. Eng. Sci. 62 (2007) 3984–3990. [18] G. Incera Garrido, Mass and Momentum Transfer upon Flow through Solid Sponges (Dissertation), Universität Karlsruhe, 2009. [19] P. Parthasarathy, P. Habisreuther, N. Zarzalis, Evaluation of longitudinal dispersion coefficient in open-cell foams using transient direct pore level simulation, Chem. Eng. Sci. 90 (2013) 242–249. [20] L. Giani, G. Groppi, E. Tronconi, Mass-transfer characterization of metallic foams as supports for structured catalysts, Ind. Eng. Chem. Res. 44 (2005) 4993–5002. [21] G. Groppi, L. Giani, E. Tronconi, Generalized correlation for gas/solid masstransfer coefficients in metallic and ceramic foams, Ind. Eng. Chem. Res. 46 (2007) 3955–3958. [22] M. Lacroix, P. Nguyen, D. Schweich, C. Pham-Huu, S. Savin-Poncet, D. Edouard, Pressure drop measurements and modeling on SiC foams, Chem. Eng. Sci. 62 (2007) 3259–3267. [23] P.A. Nelson, T.R. Galloway, Particle-to-fluid heat and mass transfer in dense systems of fine particles, Chem. Eng. Sci. 30 (1975) 1–6. [24] J. de Greef, G. Desmet, G.V. Baron, Micro-fiber elements as perfusive catalysts or in catalytic mixers: flow, mixing and mass transfer, Catal. Today 105 (2005) 331–336. _ Demir, A revisit of pressure drop-flow rate correlations [25] E. Erdim, Ö. Akgiray, I. for packed beds of spheres, Powder Technol. 283 (2015) 488–504. [26] I.F. Macdonald, M.S. El-Sayed, K. Mow, F.A.L. Dullien, Flow through porous media-the Ergun equation revisited, Ind. Eng. Chem. Fund. 18 (1979) 199–208. [27] Y.S. Choi, S.J. Kim, D. Kim, A semi-empirical correlation for pressure drop in packed beds of spherical particles, Transp. Porous Media 75 (2008) 133–149. [28] J.P. du Plessis, S. Woudberg, Pore-scale derivation of the Ergun equation to enhance its adaptability and generalization, Chem. Eng. Sci. 63 (2008) 2576– 2586. [29] R.E. Hayes, A. Afacan, B. Boulanger, An equation of motion for an incompressible Newtonian fluid in a packed bed, Transp. Porous Media 18 (1995) 185–198. [30] O. Reynolds, On the extent and action of the heating surface of steam boilers, Proc. Literary Philos. Soc. Manchester 14 (1874) 7–12. [31] H. Martin, The generalized Lévêque equation and its practical use for the prediction of heat and mass transfer rates from pressure drop, Chem. Eng. Sci. 57 (2002) 3217–3223. [32] T.H. Chilton, A.P. Colburn, Mass transfer (absorption) coefficients prediction from data on heat transfer and fluid friction, Ind. Eng. Chem. 26 (1934) 1183– 1187. [33] W.H. McAdams, Heat Transmission, McGraw-Hill Book Company, New York, 1954. [34] N. Frössling, Über die Verdunstung fallender Tropfen, Gerlands Beitr. Geophys. 52 (1938) 170–216. [35] W.E. Ranz, W.R. Marshall Jr., Evaporation from drops, Chem. Eng. Prog. 48 (1952) 173–180. [36] S. Evnochides, G. Thodos, Simultaneous mass and heat transfer in the flow of gases past single spheres, AIChE J. 7 (1961) 78–80. [37] W. Paterson, A. Hayhurst, Mass or heat transfer from a sphere to a flowing fluid, Chem. Eng. Sci. 55 (2000) 1925–1927.

[38] R.G. Taecker, O.A. Hougen, Heat, mass transfer of gas film in flow of gases through commercial tower packings, Chem. Eng. Prog. 45 (1949) 188–193. [39] V. Gnielinski, Fluid-particle heat transfer in flow through packed beds of solids, in: VDI Heat Atlas, Springer, Berlin Heidelberg, Berlin, Heidelberg, 2010, pp. 743–744. [40] C.N. Satterfield, D.H. Cortez, Mass transfer characteristics of woven-wire screen catalysts, Ind. Eng. Chem. Fund. 9 (1970) 613–620. [41] A. Kołodziej, J. Łojewska, M. Jaroszyn´ski, A. Gancarczyk, P. Jodłowski, Heat transfer and flow resistance for stacked wire gauzes: experiments and modelling, Int. J. Heat Fluid Flow 33 (2012) 101–108. [42] B. Dietrich, W. Schabel, M. Kind, H. Martin, Pressure drop measurements of ceramic sponges—determining the hydraulic diameter, Chem. Eng. Sci. 64 (2009) 3633–3640. [43] J.F. Wehner, R.H. Wilhelm, Boundary conditions of flow reactor, Chem. Eng. Sci. 6 (1956) 89–93. [44] J.M.P.Q. Delgado, A critical review of dispersion in packed beds, Heat Mass Transfer 42 (2006) 279–310. [45] S. Zuercher, K. Pabst, G. Schaub, Ceramic foams as structured catalyst inserts in gas–particle filters for gas reactions—effect of backmixing, Appl. Catal. A 357 (2009) 85–92. [46] M. Saber, C. Pham-Huu, D. Edouard, Axial dispersion based on the residence time distribution curves in a millireactor filled with b-SiC foam catalyst, Ind. Eng. Chem. Res. 51 (2012) 15011–15017. [47] A. Montillet, J. Comiti, J. Legrand, Axial dispersion in liquid flow through packed reticulated metallic foams and fixed beds of different structures, Chem. Eng. J. 52 (1993) 63–71. [48] D. Brunjail, J. Comiti, Mass transfer and energy aspects for forced flow through packed beds of long cylindrical particles, Chem. Eng. J. 45 (1990) 123–132. [49] E. Tsotsas, E.U. Schlünder, On axial dispersion in packed beds with fluid flow, Chem. Eng. Process. 24 (1988) 15–31. [50] J.C. Armour, J.N. Cannon, Fluid flow through woven screens, AIChE J. 14 (1968) 415–420. [51] S. Kaiser, E. Reichelt, S.E. Gebhardt, M. Jahn, A. Michaelis, Porous perovskite fibers – preparation by wet phase inversion spinning and catalytic activity, Chem. Eng. Technol. 37 (2014) 1146–1154. [52] G. Ehrhardt, Flow measurements for wire gauzes, Int. Chem. Eng. 23 (1983) 455–465. [53] A.N. Karwa, B.J. Tatarchuk, Pressure drop and aerosol filtration efficiency of microfibrous entrapped catalyst and sorbent media: semi-empirical models, Sep. Purif. Technol. 86 (2012) 55–63. [54] B. Dietrich, Pressure drop correlation for ceramic and metal sponges, Chem. Eng. Sci. 74 (2012) 192–199. [55] A.F. Ahlström-Silversand, C.U.I. Odenbrand, Modelling catalytic combustion of carbon monoxide and hydrocarbons over catalytically active wire meshes, Chem. Eng. J. 73 (1999) 205–216. [56] H. Sun, Y. Shu, X. Quan, S. Chen, B. Pang, Z.Y. Liu, Experimental and modeling study of selective catalytic reduction of NOx with NH3 over wire mesh honeycomb catalysts, Chem. Eng. J. 165 (2010) 769–775. [57] W.M. Kays, A.L. London, Compact Heat Exchangers, McGraw-Hill Book Company, New York, 1984. [58] Z. Zhao, Y. Peles, M.K. Jensen, Properties of plain weave metallic wire mesh screens, Int. J. Heat Mass Transfer 57 (2013) 690–697.