Biomod/2013/NanoUANL/Reactor

What is a reactor?

Introduction

The CCMV capsid was considered as a continuous stirred-tank reactor with accumulation of the product. This is a common ideal reactor type in chemical engineering. This is an open system that operates on a steady-state assumption, where the conditions of the reactor do not change with time. It is a complete opposite of tubular plug-flow and stirred batch reactors, and can be very useful when studying the behavior of a gas, liquid or solid.

The reactor is modelled by that of a Continuous Ideally Stirre-Tank Reactor (CISTR), assuming perfect mixing in the container.

Enzymatic Reaction

The general reaction scheme is as follows:

[math]\displaystyle{ E + S \leftrightarrow ES \rightarrow E^0 + P }[/math]

With a reaction rate of:

[math]\displaystyle{ \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] }[/math]

This equation is affected by the constants k₁ , k_-1 and k₂.

Considerations

We established the following considerations in our system:

• Uniform distribution throughout the reactor
• K_-1 >> K₁ and K₂
• One enzyme per reactor/VLP
• Tortuosity approaches zero during diffusion

Mass balance

Material balances are important, as a first step in devising a new process (or analyzing an existing one). They are almost always a prerequisite for all calculations for process engineering problems. The concept of mass balance is based on the physical principle that matter cannot be either created nor destroyed, only transformed. The law of mass transformation balances the mass of the inputs of the process with the output, as waste, products or emissions. This whole process is accounting for the material used in a reaction.

Mass balance was presented as such:

INPUT= OUTPUT- DISAPPEARANCE BY REACTION + ACCUMULATION

where

Input= F₀

Output= F₀(1-X_S)

Disappearance = V(-r_S

Accumulation = [math]\displaystyle{ \tfrac{d[P]}{dt} }[/math]

F₀ = F₀(1-X_S) - V(-r_S) + [math]\displaystyle{ \tfrac{d[P]}{dt} }[/math]

The intake and outflow were determined by diffusion, considering a spherical container. For the simplification of the diffusion phenomenon we considered:

• Constant temperature
• Constant pressure
• Species B stays in a stationary state (it does not diffuse in A)
• The container (VLP) has a spherical shape

A mass balance, applied to a spherical envelope is as follows:

[math]\displaystyle{ \frac{d}{dr}(r^2N_{Ar})=0 }[/math]

where N_Ar represents molar flux. For N_Br we obtain

[math]\displaystyle{ \frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0 }[/math]

At a constant temperature the product (cD_AB) is equally constant and x_A=1-x_B, the equation can be integrated into the following expression:

[math]\displaystyle{ F_A=4\pi r_1^2N_{Ar}|_{r=r1}= \frac{4\pi cD_{AB}}{1/r_1=1/r_2} \ln\frac{x_{B2}}{x_{B1}} }[/math]

where x are the fractions, c is the concentration and r are the respective radii.

This equation defines the nanoreactor inflow; a similar analysis yields the reactor outflow.

Diffusion coefficient

The ionic silver diffusion coefficient in function to the solution is described by Nerst's equation (1888)¹:

[math]\displaystyle{ D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|} }[/math]

where

• F = Faraday's constant
• D_AB°=Diffusion coefficient at infinite dilution
• λ₊°=Cationic conductivity at infinite dilution
• λ_-°=Anionic conductivity at infinite dilution
• Z⁺=Cation valence
• Z^-=Anionic valence
• T=Absolute temperature

Boiling temperature

Via Joback's method, we obtain the normal boiling temperature:

[math]\displaystyle{ T_b=\mathbf{198} + \sum_{k} N_k(tbk) }[/math]

in which N_k is the number of times that the contribution occurs in the compound.

Critical temperature

Using a similar approach, also by Joback, we estimated the critical temperature:

[math]\displaystyle{ T_c=T_b\Bigg[ \mathbf{0.584}+\mathbf{0.965} \bigg\{\sum_{k} N_k(tck) \bigg\} - \bigg\{\sum_{k} N_k(tck) \bigg\}^2 \Bigg]^{-1} }[/math]

Joback Method Contributions (C1 Prausnitz)

Group	tbk, K	tck, K
CH3	23.58	0.0141
CH2	22.88	0.0181
CH	21.74	0.0164

[etcetera]

Conductivity

Conductivity was determined by the Sastri method:

[math]\displaystyle{ \lambda_L=\lambda_ba^m }[/math]

where λ_L = thermic conductivity of the liquid [ W/(m·K)] λ_b = thermic conductivity at normal boiling point T_br= T/T_c = reduced temperature T_c = critical temperature, K

[math]\displaystyle{ m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n }[/math]

a=0.16 n=0.2

Contribución de Sastri (Tabla 10.5. Prausnitz 5a)

[TABLA]

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Silver density

{pendiente}

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References

1. Anthony L. Hines & Robert N. Maddox, Mass Transfer Fundamentals and Applications, Prentince Hall PTR 1985, pages 34-35
2. Bird R. B, Stewart W.E. & Lightfoot E.N., Fenómenos de Trasporte, Reverté Ediciones SA de CV 2006, pages 17-9 – 17-11.