Biomod/2013/NanoUANL/Reactor

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<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">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:</span></span></p>
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">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:</span></span></p>
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\(y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0\)
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<p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Choose a value <span class="texhtml"><i>h</i></span> for the size of every step and set <span class="texhtml"><i>t</i><sub><i>n</i></sub> = <i>t</i><sub>0</sub> + <i>n</i><i>h</i></span>. Now, one step of the Euler method from <span class="texhtml"><i>t</i><sub><i>n</i></sub></span> to <span class="texhtml"><i>t</i><sub><i>n</i> + 1</sub> = <i>t</i><sub><i>n</i></sub> + <i>h</i></span> is </span></span></p>
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\(y_{n+1} = y_n + hf(t_n,y_n)\)
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<p>
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The value of <img alt="\bold y_n" class="tex" src="http://openwetware.org/images/math/2/d/4/2d412236ddab9fd085d86d331fc41efd.png" /> is an approximation of the solution to the ODE at time <img alt="\bold t_n" class="tex" src="http://openwetware.org/images/math/2/0/c/20c21da56097aa97a9ad83c743109920.png" />: <img alt="y_n \approx y(t_n)" class="tex" src="http://openwetware.org/images/math/8/7/b/87bcb4988b614d2564dfa184f3a422c0.png" />. The Euler method is explicit, i.e. the solution <img alt="\bold y_{n+1}" class="tex" src="http://openwetware.org/images/math/3/a/6/3a6394ef8d7be438136b4ddb094e7d02.png" /> is an explicit function of <img alt="\bold y_i" class="tex" src="http://openwetware.org/images/math/c/5/4/c54efad69ef8398a457edc184511a782.png" /> for <img alt="i \leq n" class="tex" src="http://openwetware.org/images/math/0/6/4/064c678d55189edf8539d54cb383f358.png" />.</p>
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<p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">While the Euler method integrates a first-order ODE, any ODE of order <em>N</em> can be represented as a first-order ODE: to treat the equation:</span></span></p>
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\(y^{(N)}(t) = f(t, y(t), y'(t), \ldots, y^{(N-1)}(t)\)
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<p>
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<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">we introduce auxiliary variables <img alt="z_1(t)=y(t), z_2(t)=y'(t),\ldots, z_N(t)=y^{(N-1)}(t)" class="tex" height="18" src="http://openwetware.org/images/math/f/6/f/f6f962f62d0359b6b4acdc15835bcc12.png" width="298" /> and obtain the equivalent equation</span></span></p>
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\begin{equation}
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\mathbf{z}'(t)
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  = \begin{pmatrix} z_1'(t)\\ \vdots\\ z_{N-1}'(t)\\ z_N'(t) \end{pmatrix}
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  = \begin{pmatrix} y'(t)\\ \vdots\\ y^{(N-1)}(t)\\ y^{(N)}(t) \end{pmatrix}
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  = \begin{pmatrix} z_2(t)\\ \vdots\\ z_N(t)\\ f(t,z_1(t),\ldots,z_N(t)) \end{pmatrix}
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\end{equation}
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<p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">This is a first-order system in the variable <img alt="\mathbf{z}(t)" class="tex" src="http://openwetware.org/images/math/3/5/f/35f48cc203de93ed9cb8f7f095824943.png" /> and can be handled by Euler&#39;s method or, in fact, by any other scheme for first-order systems.</span></span></p>
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<p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Applying the method to our system, the differential equation are</span></span></p>
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\begin{equation}\frac{dC_E}{dt}=-k_1C_EC_S+k_{-1}C_{ES}+k_2C_{ES}\end{equation}
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\begin{equation}\frac{dC_S}{dt}=-k_1C_EC_S+\frac{F_S}{V_{reactor}}\end{equation}
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\begin{equation}\frac{dC_{ES}}{dt}=k_1C_EC_S-k_{-1}C_{ES}-k_2C_{ES}\end{equation}
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\begin{equation}\frac{dC_P}{dt}=k_2C_{ES}\end{equation}
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<h3>
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<strong><span style="font-size:14px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><span class="mw-headline">Results </span></span></span></strong></h3>
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<p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">For a <img alt="\bold k_1=1x10^7M^{-1}s^{-1}, k_{-1}=200s^{-1}" class="tex" height="16" src="http://openwetware.org/images/math/4/4/b/44bfa2eee3eb22cd70da118e5bd36c9d.png" width="206" /> and <img alt="\bold k_2=100s^{-1}" class="tex" height="17" src="http://openwetware.org/images/math/2/a/4/2a482255e383194d015e64611209e531.png" width="83" /> with <img alt="C_{Eo}=5.44x10^{-4} \frac {mol}{L}" class="tex" height="25" src="http://openwetware.org/images/math/a/7/f/a7fbf944891ea0b8505bc79880e02372.png" width="111" /> and <img alt="C_{So}=1.0x10^{-4} \frac {mol}{L} " class="tex" height="25" src="http://openwetware.org/images/math/5/6/3/5639d7021a88d1160cb0566dd02be0b5.png" width="104" /></span></span></p>
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<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Considering a volume of capsid and enzyme <img alt="\bold V_{capsid}=3.05x10^{-21}Lts, \bold V_{enzyme}=1.41x10^{-23}Lts" class="tex" height="19" src="http://openwetware.org/images/math/2/9/3/2931f720a329d6f8a887db0996c34735.png" width="338" /> respectively we obtained:</span></span></p>
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Revision as of 19:34, 26 October 2013

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Theory

To describe the dynamic behavior of a Semi-Continuous Tank Reactor (SCTR) mass, component and energy balance equations must be developed. This requires an understanding of the functional expressions that describe chemical reaction. A reaction will create new components while simultaneously reducing reactant concentrations. The reaction may give off heat or my require energy to proceed.

To develop a realistic SCTR model the change of individual species (or components) with respect to time must be considered. This is because individual components can appear / disappear because of reaction (remember that the overall mass of reactants and products will always stay the same). If there are N components, N – 1 component balances and an overall mass balance expression are required. Alternatively a component balance may be written for each species.

In certain SCTR´s (generally small vessels) the wall dynamics can have a significant effect on the thermal control and stability of a SCTR. If this is the case then an energy balance expression should be developed describing the rate of change of wall temperature with respect to time, assuming that the wall temperature is the same at any point.

 

Idea

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.

Why is this 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:

\begin{equation} E + S \leftrightarrow ES \rightarrow E^0 + P \end{equation}

With a reaction rate of:

\begin{equation} \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] \end{equation}

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) and Accumulation = \(\frac{d[P]}{dt}\)

\begin{equation} \frac{d[P]}{dt}=F_0+V(r_P)-V(-r_S) \end{equation}

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

\begin{equation} \frac{d}{dr}(r^2N_{Ar})=0 \end{equation}

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

\begin{equation} \frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0 \end{equation}

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

\begin{equation} 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}} \end{equation}

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):

\begin{equation} D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|} \end{equation}

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:

\begin{equation} T_b=\mathbf{198} + \sum_{k} N_k(tbk) \end{equation}

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:

\begin{equation} 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} \end{equation}

Table 1. Joback's Method Contribution

Conductivity

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

\begin{equation} \lambda_L=\lambda_ba^m \end{equation}

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

\begin{equation} m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n \end{equation}

with a = 0.16 and n = 0.2 for the compound.

Table 2. Sastri's Contributions

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

\(y_{n+1} = y_n + hf(t_n,y_n)\)

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:

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

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

\begin{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} \end{equation}

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

\begin{equation}\frac{dC_E}{dt}=-k_1C_EC_S+k_{-1}C_{ES}+k_2C_{ES}\end{equation} \begin{equation}\frac{dC_S}{dt}=-k_1C_EC_S+\frac{F_S}{V_{reactor}}\end{equation} \begin{equation}\frac{dC_{ES}}{dt}=k_1C_EC_S-k_{-1}C_{ES}-k_2C_{ES}\end{equation} \begin{equation}\frac{dC_P}{dt}=k_2C_{ES}\end{equation}

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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