# Biomod/2013/NanoUANL/2

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## Why is a reactor?

### Introduction

For an ideal approach, the CCMV capsid could be considered as reactor with an accumulation of the product inside the capsid. An analysis of a reactor is a common in chemical engineering. The reactor proposed is a complete opposite type from tubular plug-flow and stirred batch reactors, or a continuous stirred tank reactor and can be very useful when studying the behavior of a gas, liquid or solid.

The reactor's behavior is modeled by a Semi-Continuous Tank Reactor, assuming perfect mixing in the container.

#### Enzymatic Reaction

The general reaction scheme is described as follows:

$E + S \leftrightarrow ES \rightarrow E + P$

With a reaction rate of:

$\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]$ . . . 1.1

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

#### 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 describe 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.

Applying a mass balance to our system we obtained:

ACCUMULATION = INPUT + APPEARANCE BY REACTION - DISAPPEARANCE BY REACTION

where

Input = F0

Appearance = V(rP)

Disappearance = V(-rS)

Accumulation = $\tfrac{d[P]}{dt}$

$\frac{d[P]}{dt}=F_0+V(r_P)-V(-r_S)$ . . . 1.2

The inlet flow was determined by diffusion. A mass balance, applied to a spherical envelope is described as:

$\frac{d}{dr}(r^2N_{Ar})=0$ . . . 1.3

where NAr represents molar flux. When NBr=0 we obtain

$\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0$ . . . 1.4

At a constant temperature the product (cDAB) is equally constant and xA=1-xB, the equation can be integrated into the following expression:

$F_A=4\pi r_1^2N_{Ar}|_{r=r1}= \frac{4\pi cD_{AB}}{1/r_1-1/r_2} \ln\frac{x_{B2}}{x_{B1}}$ . . . 1.5

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

This equation defines the nanoreactor inflow.

We neglect the possibility of an outflow of species because:

• The gradient of concentration of S tends to stay inside the capsid
• The positive charge in the outside of the VLP made a repulsion of the specie S
• An evident agglomeration of specie P will increase it size and remain inside

#### Diffusion coefficient

For S and P being ionic silver and reduced silver respectively. The ionic silver diffusion coefficient in solution is described by Nerst's equation (1888):

$D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|}$ . . . 1.6

where

• F = Faraday's constant [A·s/geq]
• DAB° = Diffusion coefficient at infinite dilution [m2/s]
• λ+° = Cationic conductivity at infinite dilution
• λ-° = Anionic conductivity at infinite dilution
• Z+ = Cation valence
• Z- = Anionic valence
• T = Absolute temperature [K]

#### Boiling temperature

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

$T_b=\mathbf{198} + \sum_{k} N_k(tbk)$ . . . 1.7

in which Nk is the number of times that the contribution group is present in the compound.

#### Critical temperature

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

$T_c=T_b\Bigg[ \mathbf{0.584}+\mathbf{0.965} \bigg\{\sum_{k} N_k(tck) \bigg\} - \bigg\{\sum_{k} N_k(tck) \bigg\}^2 \Bigg]^{-1}$ . . . 1.8

Joback's Method Contributions (Table C1. Prausnitz)

Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K
CH3 23.58 0.0141 CH2(ss) 27.15 0.01 ACOH 76.34 0.0240 NH 50.17 0.0295
CH2 22.88 0.0181 CH(ss) 21.78 0.0122 O 22.42 0.0168 NH(ss) 52.82 0.0130
CH 21.74 0.0164 C(ss) 21.32 0.0042 O(ss) 31.22 0.0098 N 11.74 0.0169
C 18.25 0.0067 =CH(ds) 26.73 0.0082 C=O 76.75 0.0380 =N-(ds) 74.60 0.0255
=CH2 18.18 0.0113 =C(ds) 31.01 0.0143 C=O(ds) 94.97 0.0284 =NH X X
=CH 24.96 0.0129 F -0.03 0.0111 CH=O 72.20 0.0379 CN 125.66 0.0496
=C 24.14 0.0117 Cl 38.13 0.0105 COOH 169.09 0.0791 NO2 152.54 0.0437
=C= 26.15 0.0026 Br 66.86 0.0133 COO 81.10 0.0481 SH 63.56 0.0031
≡CH 9.20 0.0027 I 93.84 0.0068 =O -10 0.0143 S 68.78 0.0119
≡C 27.38 0.0020 OH 92.88 0.0741 NH2 73.23 0.0243 S(ss) 52.10 0.0019

(ss) indicates a group in a nonaromatic ring, (ds) indicates a group in an aromatic ring, X indicates a non-available parameter

#### Conductivity

The conductivity of the compound is determined by the Sastri method:

$\bold{\lambda_L=\lambda_ba^m}$ . . . 1.9

where

• λL = thermic conductivity of the liquid [ W/(m·K)]
• λb = thermic conductivity at normal boiling point [ W/(m·K)]
• Tbr = Tb/Tc = normal boiling reduced temperature
• Tr = T/Tc = reduced temperature
• Tc = critical temperature [K]

and

$m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n$ . . . 1.10

with a = 0.16 and n = 0.2 for the compound

Sastri's Contributions (Table 10.5. Prausnitz)

Hydrocarbon Groups Δλb Non-Hydrocarbon Groups Δλb Non-Hydrocarbon Groups Δλb
CH3 0.0545 O 0.0100 N(ring) 0.0135
CH2 0.0008 OH2 0.0830 CN 0.0645
CH -0.0600 OH3 0.0680 NO2 0.0700
C -0.1230 CO(ketone) 0.0175 S 0.0100
=CH2 0.0545 CHO(aldehyde) 0.0730 F4 0.0568
=CH 0.0020 COO(ester) 0.0070 F5 0.0510
=C -0.0630 COOH(acid) 0.0650 Cl 0.0550
=C= 0.1200 NH2 0.0880 Br 0.0415
Ring1 0.1130 NH 0.0065 I 0.0245
NH(ring) 0.0450 H6 0.0675
N -0.0605 3 member ring 0.1500
Ring7(other) 0.1100

1In polycyclic compounds, all rings are treated as separated rings, 2In aliphatic primary alcohols and phenols with no branch chains, 3In all alcohols except as described in2, 4In perfluoro carbons, 5In all cases except as described in 4, 6This contribution is used for methane, formic acid, and formates, 7In polycyclic non-hydrocarbon compounds, all rings are considered as non-hydrocarbon rings

#### Euler Method

Every time we propose a Matter Balance is quietly easy to assume a Steady-State, but in real life we could spec some disturbances in the out-flux stream caused by the Accumulation. The Accumulation of substance inside the reactor is highly common and when the capsid of the CCMV simulates a reactor it is not an exception. In this case an agglomeration of Silver Nano-particles of different sizes will be notorious and it is described by solving the differential equations of concentration of product in function of time, presented in the Mass Balance previously described.

A common method to approach the change within time is by the numeric method of Euler.

Suppose that we want to approximate the solution of the initial value problem

$y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0$

Choose a value h for the size of every step and set tn = t0 + nh. Now, one step of the Euler method from tn to tn + 1 = tn + h is

$\bold y_{n+1} = y_n + hf(t_n,y_n)$ . . . 1.11

The value of $\bold y_n$ is an approximation of the solution to the ODE at time $\bold t_n$: $y_n \approx y(t_n)$. The Euler method is explicit, i.e. the solution $\bold y_{n+1}$ is an explicit function of $\bold y_i$ for $i \leq n$.

While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a first-order ODE: to treat the equation

$y^{(N)}(t) = f(t, y(t), y'(t), \ldots, y^{(N-1)}(t))$, . . . 1.12

we introduce auxiliary variables $z_1(t)=y(t), z_2(t)=y'(t),\ldots, z_N(t)=y^{(N-1)}(t)$ and obtain the equivalent equation

$\mathbf{z}'(t) = \begin{pmatrix} z_1'(t)\\ \vdots\\ z_{N-1}'(t)\\ z_N'(t) \end{pmatrix} = \begin{pmatrix} y'(t)\\ \vdots\\ y^{(N-1)}(t)\\ y^{(N)}(t) \end{pmatrix} = \begin{pmatrix} z_2(t)\\ \vdots\\ z_N(t)\\ f(t,z_1(t),\ldots,z_N(t)) \end{pmatrix}$ . . . 1.13

This is a first-order system in the variable $\mathbf{z}(t)$ and can be handled by Euler's method or, in fact, by any other scheme for first-order systems.

Applying the method to our system, the differential equation are

$\frac{dC_E}{dt}=-k_1C_EC_S+k_{-1}C_{ES}+k_2C_{ES}$ . . . 1.14

$\frac{dC_S}{dt}=-k_1C_EC_S+\frac{F_S}{V_{reactor}}$ . . . 1.15

$\frac{dC_{ES}}{dt}=k_1C_EC_S-k_{-1}C_{ES}-k_2C_{ES}$ . . . 1.16

$\frac{dC_P}{dt}=k_2C_{ES}$ . . . 1.17

### Results

For a $\bold k_1=1x10^7M^{-1}s^{-1}, k_{-1}=200s^{-1}$ and $\bold k_2=100s^{-1}$ with $C_{Eo}=5.44x10^{-4} \frac {mol}{L}$ and $C_{So}=1.0x10^{-4} \frac {mol}{L}$

Considering a volume of capsid and enzyme $\bold V_{capsid}=3.05x10^{-21}Lts, \bold V_{enzyme}=1.41x10^{-23}Lts$ respectively we obtained:

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