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Three-phase mass transfer: theoretical development of the overall effectiveness factor for cylindrical geometries

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Abstract

This work presents a theoretical analysis for predicting the overall effectiveness factor (η) for cylindrical geometries in a three-phase catalytic reaction system. Implicit expressions for \(\eta\), taking into account all the transport effects, gas–liquid mass transfer, liquid-solid mass transfer and intraparticle diffusion as well the chemical reaction were obtained. Various types of kinetics were considered; namely, Langmuir–Hinshelwood, power-law and zeroth order. The obtained results are presented as a useful tool to be used in field of reactors design and laboratory kinetics data interpretation.

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Abbreviations

a B :

\({\text{Gas-liquid}}\,{\text{interfacial}}\;{\text{area}}\;{\text{per}}\;{\text{unit}}\;{\text{volume}}\;{\text{of}}\;{\text{reactor}}\;\left[ {{\text{cm}}^{2} \;{\text{cm}}^{-3} } \right]\)

a P :

\({\text{External}}\;{\text{area}}\,{\text{of}}\,{\text{particles}} \;{\text{per}}\;{\text{unit}}\;{\text{volume}}\;of\;{\text{reactor}}\;\left[ {{\text{cm}}^{2} \;{\text{cm}}^{-3} } \right]\)

A :

\({\text{Species}}\;{\text{A}}\;\left[ {\text{dimensionless}} \right]\)

AS :

\({\text{Species}}\;{\text{A}}\;{\text{adsorbed}}\;{\text{on}}\;{\text{the}}\;{\text{site}}\;{\text{S}}\;\left[ {\text{dimensionless}} \right]\)

B :

\({\text{Species}}\;{\text{B}}\;\left[ {\text{dimensionless}} \right]\)

BS :

\({\text{Species}}\;{\text{B}}\;{\text{adsorbed}}\;{\text{on}}\;{\text{the}}\;{\text{site}}\;{\text{S}}\;\left[ {\text{dimensionless}} \right]\)

C A :

\({\text{Concentration}}\;{\text{of}}\;{\text{species}}\;{\text{A}}\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{*}\) :

\({\text{Concentration}}\;{\text{of}}\;{\text{A}}\;{\text{in}}\;{\text{the}}\;{\text{liquid}}\;{\text{in}}\;{\text{equilibrium}}\;{\text{with}}\;{\text{the}}\;{\text{gas}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{*,Critical}\) :

\({\text{Critical}}\;{\text{concentration}}\;{\text{of}}\;{\text{A}}\;{\text{in}}\;{\text{the}}\;{\text{liquid}}\;{\text{in}}\;{\text{equilibrium}}\;{\text{with}}\;{\text{the}}\;{\text{gas}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{G}\) :

\({\text{Concentration}}\;{\text{of}}\;{\text{A}}\;{\text{in}}\;{\text{the}}\;{\text{bulk}}\;{\text{gas}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{L}\) :

\({\text{Concentration}}\;{\text{of}}\;{\text{A}}\;{\text{in}}\;{\text{the}}\;{\text{bulk}}\;{\text{liquid}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{S}\) :

\({\text{Concentration}}\;{\text{of}}\;{\text{A}}\;{\text{at}}\;{\text{the}}\;{\text{catalyst}}\;{\text{surface}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{A}}^{S,Critical}\) :

\({\text{Critical}}\;{\text{concentration}}\;{\text{of}}\;{\text{A}}\;{\text{at}}\;{\text{the}}\;{\text{catalyst}}\;{\text{surface}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

C AS :

\({\text{Concentration}}\;{\text{of}}\;{\text{adsorbed}}\;{\text{A}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} } \right]\)

C A0 :

\({\text{Concentration}}\;{\text{of}}\;{\text{species}}\;{\text{A}}\;{\text{at}}\;{\text{center}}\;{\text{line}}\;{\text{of}}\;{\text{catalyst}}\;{\text{particle}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

C B :

\({\text{Concentration}}\;{\text{of}}\;{\text{species}}\;{\text{B}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

\(C_{\text{B}}^{L}\) :

\({\text{Concentration}}\;{\text{of}}\;{\text{B}}\;{\text{in}}\;{\text{the}}\;{\text{bulk}}\;{\text{liquid}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

C BS :

\({\text{Concentration}}\;{\text{of}}\;{\text{adsorbed}}\;{\text{B}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} } \right]\)

C E :

\({\text{Concentration}}\;{\text{of}}\;{\text{species}}\;{\text{E}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

C ES :

\({\text{Concentration}}\;{\text{of}}\;{\text{adsorbed}}\;{\text{E}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} } \right]\)

C S :

\({\text{Concentration}}\;{\text{of}}\;{\text{vacant}}\;{\text{active}}\;{\text{sites}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} } \right]\)

C T :

\({\text{Concentration}}\;{\text{of}}\;{\text{total }}\;{\text{active}}\;{\text{sites}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} } \right]\)

D E–A :

\({\text{Effective}} \;{\text{diffusivity}}\;{\text{of}}\;{\text{A}}\;\left[ {{\text{cm}}^{2} \;{\text{s}}^{-1} } \right]\)

E :

\({\text{Species}}\;{\text{E}}\;\left[ {\text{dimensionless}} \right]\)

ES :

\({\text{Species}}\;{\text{E}}\;{\text{adsorbed}}\;{\text{on}}\;{\text{the}}\;{\text{site}}\;{\text{S}}\;\left[ {\text{dimensionless}} \right]\)

H A :

\({\text{Henry's}}\;{\text{law}}\;{\text{constant}}\;{\text{defined}}\;{\text{as}}C_{A}^{G} /C_{A}^{*} \;\left[ {\text{dimensionless}} \right]\)

I 1 :

\({\text{First-order}}\;{\text{modified}}\;{\text{Bessel}}\;{\text{function }}\;{\text{of}}\;{\text{the}}\;{\text{first}}\;{\text{kind}}\;\left[ {\text{dimensionless}} \right]\)

I 0 :

\({\text{Zero-order}}\;{\text{modified}}\;{\text{Bessel}}\;{\text{function }}\;{\text{of}}\;{\text{the}}\;{\text{first}}\;{\text{kind}}\;\left[ {\text{dimensionless}} \right]\)

J :

\({\text{Gas}}\;{\text{constant}}\;\left[ {{\text{cal}}\;{\text{mol}}^{-1} \;{\text{K}}^{-1} } \right]\)

k :

\({\text{Rate}}\;{\text{constant}}\;\left[ {{\text{s}}^{-1} } \right]\)

k A :

\({\text{Species}}\;{\text{A}}\;{\text{adsorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} \;{\text{s}}^{-1} } \right]\)

k −A :

\({\text{Species}}\;{\text{A}}\;{\text{desorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{s}}^{-1} } \right]\)

k B :

\({\text{Species}}\;{\text{B}}\;{\text{desorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} \;{\text{s}}^{-1} } \right]\)

k −B :

\({\text{Species}}\;{\text{B}}\;{\text{desorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{s}}^{-1} } \right]\)

k E :

\({\text{Species}}\;{\text{E}}\;{\text{desorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{s}}^{-1} } \right]\)

k −E :

\({\text{Species}}\;{\text{E}}\;{\text{desorption}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} \;{\text{s}}^{-1} } \right]\)

k G :

\({\text{Gas - side}}\;{\text{mass}}\;{\text{transfer}}\;{\text{coefficient}}\;\left[ {{\text{cm}}^{3} \;{\text{cm}}^{-2} \;{\text{s}}^{-1} } \right]\)

k L :

\({\text{Liquid - side}}\;{\text{mass}}\;{\text{transfer}}\;{\text{coefficient}}\;\left[ {{\text{cm}}^{3} \;{\text{cm}}^{-2} \;{\text{s}}^{-1} } \right]\)

k m :

\({\text{Pseudo}}\;{\text{m - th-order}}\;{\text{rate}}\;{\text{constant}}\; \left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

k S :

\({\text{Liquid}}\;{\text{to}}\;{\text{solid}}\;{\text{mass}}\;{\text{transfer}}\;{\text{coefficient}}\;\left[ {{\text{cm}}^{3} \;{\text{cm}}^{-2} \;{\text{s}}^{-1} } \right]\)

\(k_{R}^{DS}\) :

\({\text{Forward}}\;{\text{reaction}}\;{\text{rate}}\;{\text{constant}}\;\left( {{\text{dual-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{g}}^{3} \;{\text{mol}}^{-1} \;{\text{s}}^{-1} } \right]\):

\(k_{ - R}^{DS}\) :

\({\text{Backward}}\;{\text{reaction}}\;{\text{rate}}\;{\text{constant}}\;\left( {{\text{dual-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{s}}^{-1} } \right]\)

\(k_{R}^{SS}\) :

\({\text{Forward}}\;{\text{reaction}}\;{\text{rate}}\;{\text{constant}}\;\left( {{\text{single-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\; \left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} \;{\text{s}}^{-1} } \right]\):

\(k_{ - R}^{SS}\) :

\({\text{Backward}}\;{\text{reaction}}\;{\text{rate}}\;{\text{constant}}\;\left( {{\text{single-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\,\left[ {{\text{s}}^{-1} } \right]\):

k 0 :

\({\text{Zero-order}}\;{\text{rate}}\;{\text{constant}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\(k_{LH}^{DS}\) :

\({\text{Rate}}\;{\text{constant}}\;\left( {{\text{dual-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{cm}}^{3} \;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\(k_{LH}^{SS}\) :

\({\text{Rate}}\;{\text{constant}}\;\left( {{\text{single-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{cm}}^{3} \;{\text{g}}^{-1} \;{\text{s}}^{-1} \;({\text{Eq}} .\;{\text{S}}14),\quad {\text{s}}^{-1} \;({\text{Eq}} .\;65)} \right]\)

K A :

\({\text{Adsorption}}\;{\text{equilibrium}}\;{\text{constant}}\;{\text{of}}\;{\text{species}}\;{\text{A}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} } \right]\)

K B :

\({\text{Adsorption}}\;{\text{equilibrium}}\;{\text{constant}}\;{\text{of}}\;{\text{species}}\;{\text{B}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} } \right]\)

K E :

\({\text{Adsorption}}\;{\text{equilibrium}}\;{\text{constant}}\;{\text{of}}\;{\text{species}}\;{\text{E}}\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} } \right]\)

\(K^{\prime}_{E}\) :

\({\text{Desorption}}\;{\text{equilibrium}}\;{\text{constant}}\;{\text{of}}\;{\text{species}}\;{\text{E}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} } \right]\)

K L :

\({\text{Overall}}\;{\text{gas}}\;{\text{to}}\;{\text{liquid}}\;{\text{mass}}\;{\text{transfer}}\;{\text{coefficient}}\;\left[ {{\text{cm}}^{3} \;{\text{cm}}^{-2} \;{\text{s}}^{-1} } \right]\)

\(K_{R}^{DS}\) :

\({\text{Reaction}}\;{\text{equilibrium}}\;{\text{constant}}\;\left( {{\text{dual-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{g}}\;{\text{mol}}^{-1} } \right]\)

\(K_{R}^{SS}\) :

\({\text{Reaction}}\;{\text{equilibrium}}\;{\text{constant}}\;\left( {{\text{single-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{cm}}^{3} \;{\text{mol}}^{-1} } \right]\)

L :

\({\text{Thichness}}\;{\text{of}}\;{\text{a}}\;{\text{catalyst}}\;{\text{slab}}\;\left[ {\text{cm}} \right]\)

m :

\({\text{Order}}\;{\text{of}}\;{\text{reaction}}\;{\text{with}}\;{\text{respect}}\;{\text{to}}\;{\text{species}}\;{\text{A}}\;\left[ {\text{dimensionless}} \right]\):

M A :

\({\text{Overall}}\;{\text{gas}}\;{\text{to}}\;{\text{solid}}\;{\text{mass}}\;{\text{transfer}}\;{\text{coefficient}}\;{\text{for}}\;{\text{species}}\,{\text{A}}\;\left[ {{\text{s}}^{-1} } \right]\)

P :

\({\text{Total}}\;{\text{pressure}}\;\left[ {\text{atm}} \right]\)

r :

\({\text{Radial}}\;{\text{distance}}\;{\text{in}}\;{\text{the}}\;{\text{catalyst}}\;{\text{particle}}\;{\text{measured}}\;{\text{from}}\;{\text{its}}\;{\text{center}}\;\left[ {\text{cm}} \right]\)

r A :

\({\text{Rate}}\;{\text{of}}\;{\text{reaction}}\;{\text{per}}\;{\text{unit}}\;{\text{volume}}\;{\text{of}}\;{\text{catalyst}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} \;{\text{s}}^{-1} } \right]\)

r Maximum :

\({\text{Maximum}}\;{\text{rate}}\;{\text{of}}\;{\text{reaction}}\;{\text{in}}\;{\text{the}}\;{\text{catalyst}}\;{\text{particle}}\;{\text{based}}\;{\text{on}}\;C_{A}^{S} \;\left[ {{\text{mol}}\;{\text{cm}}^{-3} \;{\text{s}}^{-1} } \right]\)

R :

\({\text{Radius}}\;{\text{of}}\;{\text{the}}\;{\text{catalyst}}\;{\text{particle}}\;\left[ {\text{cm}} \right]\)

R A :

\({\text{Rate}}\;{\text{of}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;{\text{per}}\;{\text{unit}}\;{\text{volume}}\;{\text{of}}\;{\text{reactor}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} \;{\text{s}}^{-1} } \right]\)

S :

\({\text{An}}\;{\text{active}}\;{\text{site}}\;\left[ {\text{dimensionless}} \right]\)

S P :

\({\text{Catalyst}}\;{\text{particle}}\;{\text{geometric}}\;{\text{area}}\;\left[ {{\text{cm}}^{2} } \right]\)

t :

\({\text{Time}}\; \left[ {\text{s}} \right]\)

T :

\({\text{Temperature }}\;\left[ {\text{K}} \right]\)

ν :

\({\text{Reaction}}\;{\text{rate}}\;\left[ {{\text{mol}}\;{\text{cm}}^{-3} \;{\text{s}}^{-1} } \right]\)

V P :

\({\text{Catalyst}}\;{\text{particle}}\;{\text{volume}}\;\left[ {{\text{cm}}^{3} } \right]\)

w :

\({\text{Catalyst}}\;{\text{mass}}\;{\text{per}}\;{\text{unit}}\;{\text{volume}}\;{\text{of}}\;{\text{reactor}}\;\left[ {{\text{g}}\;{\text{cm}}^{-3} } \right]\)

Y:

\({\text{Preexponential}}\;{\text{factor}}\;\left( {{\text{Eq}}.\;62} \right)\;\left[ {{\text{s}}^{-1} } \right]\)

z :

\({\text{Coordinate}}\;{\text{oriented}}\;{\text{from}}\;{\text{the}}\;{\text{center}}\;{\text{line}}\;{\text{to}}\;{\text{the}}\;{\text{surface}}\;{\text{in}}\;{\text{a}}\;{\text{catalyst}}\;{\text{slab}}\;\left[ {\text{cm}} \right]\)

Z :

\({\text{Catalyst}}\;{\text{loading}}\;\left[ {{\text{g}}\;{\text{cm}}^{-3} } \right]\)

\(\eta\) :

\({\text{Overall}}\;{\text{effectiveness}}\;{\text{factor}}\;\left[ {\text{dimensionless}} \right]\)

\(\eta_{c}\) :

\({\text{Catalytic}}\;{\text{effectiveness}}\;{\text{factor}}\;\left[ {\text{dimensionless}} \right]\)

\(\eta_{C\infty }\) :

\({\text{Asymptotic}}\;{\text{catalytic}}\;{\text{effectiveness}}\;{\text{factor}}\;{\text{defined}}\;{\text{by}}\;{\text{Eq}} .\;{\text{S}}40\;\left[ {\text{dimensionless}} \right]\)

λ :

\({\text{Distance}}\;{\text{from}}\;{\text{the}}\;{\text{center}}\;{\text{of}}\;{\text{the}}\;{\text{catalyst}}\;{\text{at}}\;{\text{which}}\;C_{A} \;{\text{becomes}}\;{\text{zero}}\;\left[ {\text{cm}} \right]\)

ρ P :

\({\text{Density}}\;{\text{of}}\;{\text{the}}\;{\text{catalyst}}\;{\text{particle}}\;\left[ {{\text{g}}\;{\text{cm}}^{-3} } \right]\)

σ A :

\({\text{Parameter}}\;{\text{defined}}\;{\text{by}}\;{\text{Eq}} .\; 1 6\;\left[ {\text{dimensionless}} \right]\)

\(\upsilon_{A}\) :

\({\text{Stoichometric}}\;{\text{coefficient}}\;{\text{of}}\;A\;{\text{in}}\;{\text{the}}\;{\text{reaction}}\;{\text{presented}}\;{\text{in}}\;{\text{Eq}} .\;1\;\left[ {\text{dimensionless}} \right]\)

\(\upsilon_{B}\) :

\({\text{Stoichometric}}\;{\text{coefficient}}\;{\text{of}}\;{\text{B}}\;{\text{in}}\;{\text{the}}\;{\text{reaction}}\;{\text{presented}}\;{\text{in}}\;{\text{Eq}} .\;1\;\left[ {\text{dimensionless}} \right]\)

\(\upsilon_{E}\) :

\({\text{Stoichometric}}\;{\text{coefficient}}\;{\text{of}}\;{\text{E}}\;{\text{in}}\;{\text{the}}\;{\text{reaction}}\;{\text{presented}}\;{\text{in}}\;{\text{Eq}} .\;1\;\left[ {\text{dimensionless}} \right]\)

\(\phi\) :

\({\text{Generalised}}\;{\text{Thiele}}\;{\text{modulus}}\;\left[ {\text{dimensionless}} \right]\)

\(\phi_{0}\) :

\({\text{Thiele}}\;{\text{modulus}}\;\left[ {\text{dimensionless}} \right]\)

\({{\varOmega }}_{A}\) :

\({\text{Local}}\;{\text{rate}}\;{\text{of}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;{\text{per}}\;{\text{unit}}\;{\text{weight}}\;{\text{of}}\;{\text{catalyst}}\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{A}^{DSLH}\) :

\({\text{Local}}\;{\text{rate}}\;{\text{of}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;\left( {{\text{dual-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{A}^{PL}\) :

\({\text{Local}}\;{\text{rate}}\;{\text{of}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;\left( {{\text{power}}\;{\text{law}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{A}^{SSLH}\) :

\({\text{Local}}\;{\text{rate}}\;{\text{of}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;\left( {{\text{single-site}}\;{\text{Langmuir - Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{AA}^{SSLH}\) :

\({\text{Rate}}\;{\text{of}}\;{\text{adsorption}}\;{\text{of}}\;{\text{A}}\;\left( {{\text{single-site}}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{A}^{ZO}\) :

\({\text{Local}}\;{\text{rate}}\;{\text{of}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{of}}\;{\text{A}}\;\left( {{\text{zero-order}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

\({{\varOmega }}_{DE}^{SSLH}\) :

\({\text{Rate}}\;{\text{of}}\;{\text{desorption}}\;{\text{of}}\;{\text{E}}\;\left( {{\text{single-site }}\;{\text{Langmuir}{-}\text{Hinshelwood}}\;{\text{kinetics}}} \right)\;\left[ {{\text{mol}}\;{\text{g}}^{-1} \;{\text{s}}^{-1} } \right]\)

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Acknowledgements

The author thanks Prof. Dr. Santiago Capriotti for his advice and helpful discussions. This research was supported by the National Scientific and Technical Research Council (CONICET) and Universidad Nacional del Sur (UNS), Argentina.

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  1. Planta Piloto de Ingeniería Química (PLAPIQUI), Departamento de Ingeniería Química, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, CP 8000, Argentina

    Diego E. Boldrini

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Boldrini, D.E. Three-phase mass transfer: theoretical development of the overall effectiveness factor for cylindrical geometries. Reac Kinet Mech Cat 133, 17–41 (2021). https://doi.org/10.1007/s11144-021-01981-2

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  • DOI: https://doi.org/10.1007/s11144-021-01981-2

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