Derivation of intraparticle diffusion

Derivation of intraparticle diffusion - + = + S S r d S d r...

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Diffusion Effects in Enzymes Immobilized in a Porous Matrix At steady state, the intraparticle diffusion rate of substrate equals to the reaction rate in a spherical shell: r r v r r dr S d De r r r dr S d De = - + 2 4 2 4 ] [ 2 4 ] [ π v is the reaction rate per unit volume of support (mg/cm 3 -s). De is the effective diffusivity (cm 2 /s). R r r+ Δr
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Diffusion Effects in Enzymes Immobilized in a Porous Matrix r 2 ) 2 ] [ ( vr r dr S d De dr d = 2 ) ] [ 2 2 2 ] [ 2 ( vr dr S d r r dr S d De = + When → 0 Re-arrange this equation r π 4 2 2 ] [ 2 ] [ vr r r r dr S d De r r r dr S d De = - + yields Dividing the two sides of the equation by
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Diffusion Effects in Enzymes Immobilized in a Porous Matrix ] [ ] [ " ) ] [ 2 2 ] [ 2 ( S m K S m V dr S d r dr S d De + = + " m V Then is the maximum reaction rate per unit volume of support (mg/cm 3 -s). De is the effective diffusivity (cm 2 /s). v dr S d r dr S d De = + ) ] [ 2 2 ] [ 2 ( ] [ ] [ " S m K S m V v + = Dividing the two sides of the equation by r 2 , yields,
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] [ , , ] [ ] [ s S m K R r r s S S S = = = β + = + S S e D s S m V R r d S d r r d S d " 2 2 2 2 The above equation can be written in dimensionless form by defining the following dimensionless variables:
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Unformatted text preview: + = + S S r d S d r r d S d 2 2 2 2 e D s S m V R " = =Thiele modules With boundary conditions of r at , r / 1 r at , 1 = = = = d S d S + = + S S r d S d r r d S d 2 2 2 2 This differential equation can be solved numerically. Refer to H. Fogler, Elements of Chemical Reaction Engineering 1999, p746 for analytical solution for first order reaction. R r dr S d De R s N s r = = = ] [ 2 4 ] [ ] [ " s S m K s S m V s r + = At steady state, the rate of substrate consumption is equal to the rate of substrate transfer through the external surface of the support particle into the sphere. Under diffusion limitations, the rate per unit volume is usually expressed in terms of the effectiveness factor as follows:...
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Derivation of intraparticle diffusion - + = + S S r d S d r...

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