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1.Selectivity by Diffusion. The Thiele modulus for a spherical catalyst of radius a is 2 ² = ka²/D such that the reaction is kinetically controlled for small Thiele modulus (<<1) and diffusion controlled for large Thiele modulus (>>1). The dimensionless concentration is C(r) = a. Show by applying the divergence theorem on the reaction-diffusion equation that the total reaction rate is determined by 4лаС D ·(r 1). The effectiveness factor is dc 00 dr defined by the ratio of the total flux by the reaction if the entire catalyst is at the bulk concentration, n($) 1 sinhor r sinho = A → B A →C 4ac_D(r=1) 4па kC 13 Derive this effectiveness factor and show that for small Thiele modulus, when there is no diffusion resistance, the effectiveness factor approaches unity. On the other hand, show for Thiele modulus much larger than unity, the effectiveness factor decays rapidly as n($)~² b. Consider two parallel reactions for the reactant A in the spherical catalyst, with reaction rates k, and k₂, Define the selectivity of first to the second reaction as the production rate of B divided by the production rate of C. Show that this selectivity goes from (ky/k₂) to (k₁/k₂)¹/² as the Theile modulus increases. Hence, if k₁<

1.Selectivity by Diffusion. The Thiele modulus for a spherical catalyst of radius a is 2 ² = ka²/D such that the reaction is kinetically controlled for small Thiele modulus (<<1) and diffusion controlled for large Thiele modulus (>>1). The dimensionless concentration is C(r) = a. Show by applying the divergence theorem on the reaction-diffusion equation that the total reaction rate is determined by 4лаС D ·(r 1). The effectiveness factor is dc 00 dr defined by the ratio of the total flux by the reaction if the entire catalyst is at the bulk concentration, n($) 1 sinhor r sinho = A → B A →C 4ac_D(r=1) 4па kC 13 Derive this effectiveness factor and show that for small Thiele modulus, when there is no diffusion resistance, the effectiveness factor approaches unity. On the other hand, show for Thiele modulus much larger than unity, the effectiveness factor decays rapidly as n($)~² b. Consider two parallel reactions for the reactant A in the spherical catalyst, with reaction rates k, and k₂, Define the selectivity of first to the second reaction as the production rate of B divided by the production rate of C. Show that this selectivity goes from (ky/k₂) to (k₁/k₂)¹/² as the Theile modulus increases. Hence, if k₁<

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
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1.Selectivity by Diffusion. The Thiele modulus for a spherical catalyst of radius a is 2 ² = ka²/D such that the reaction is kinetically controlled for small Thiele modulus (<<1) and diffusion controlled for large Thiele modulus (>>1). The dimensionless concentration is C(r) = a. Show by applying the divergence theorem on the reaction-diffusion equation that the total reaction rate is determined by 4лаС D ·(r 1). The effectiveness factor is dc 00 dr defined by the ratio of the total flux by the reaction if the entire catalyst is at the bulk concentration, n($) 1 sinhor r sinho = A → B A →C 4ac_D(r=1) 4па kC 13 Derive this effectiveness factor and show that for small Thiele modulus, when there is no diffusion resistance, the effectiveness factor approaches unity. On the other hand, show for Thiele modulus much larger than unity, the effectiveness factor decays rapidly as n($)~² b. Consider two parallel reactions for the reactant A in the spherical catalyst, with reaction rates k, and k₂, Define the selectivity of first to the second reaction as the production rate of B divided by the production rate of C. Show that this selectivity goes from (ky/k₂) to (k₁/k₂)¹/² as the Theile modulus increases. Hence, if k₁<<k₂, the selectivity of B actually increases as the Thiele modulus increases (with larger particles and higher diffusion resistance). Why?
Transcribed Image Text:1.Selectivity by Diffusion. The Thiele modulus for a spherical catalyst of radius a is 2 ² = ka²/D such that the reaction is kinetically controlled for small Thiele modulus (<<1) and diffusion controlled for large Thiele modulus (>>1). The dimensionless concentration is C(r) = a. Show by applying the divergence theorem on the reaction-diffusion equation that the total reaction rate is determined by 4лаС D ·(r 1). The effectiveness factor is dc 00 dr defined by the ratio of the total flux by the reaction if the entire catalyst is at the bulk concentration, n($) 1 sinhor r sinho = A → B A →C 4ac_D(r=1) 4па kC 13 Derive this effectiveness factor and show that for small Thiele modulus, when there is no diffusion resistance, the effectiveness factor approaches unity. On the other hand, show for Thiele modulus much larger than unity, the effectiveness factor decays rapidly as n($)~² b. Consider two parallel reactions for the reactant A in the spherical catalyst, with reaction rates k, and k₂, Define the selectivity of first to the second reaction as the production rate of B divided by the production rate of C. Show that this selectivity goes from (ky/k₂) to (k₁/k₂)¹/² as the Theile modulus increases. Hence, if k₁<<k₂, the selectivity of B actually increases as the Thiele modulus increases (with larger particles and higher diffusion resistance). Why?
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