Biomod/2013/NanoUANL/Reactor

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== What is a reactor? ==
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{{Template:Nanouanl}}
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=== Introduction ===
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<p style="text-align: center;">
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<img alt="" src="http://openwetware.org/images/7/73/UANLReactor1.png" style="width: 475px; height: 402px;" /></p>
 +
<p>
 +
&nbsp;</p>
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<p>
 +
&nbsp;</p>
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<p>
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<strong><span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Theory</span></span></strong></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">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.</span></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">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 &ndash; 1 component balances and an overall mass balance expression are required. Alternatively a component balance may be written for each species.</span></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">In certain SCTR&acute;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.</span></span></p>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
<strong><span style="font-size:14px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Idea</span></span></strong></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">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.</span></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">The reactor&#39;s behavior is modeled by a Semi-Continuous Tank Reactor, assuming perfect mixing in the container.</span></span></p>
-
The CCMV capsid was considered as a continuous stirred-tank reactor with accumulation of the product.  
+
<h2>
 +
<span style="font-size:16px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><span class="mw-headline">Why is this a reactor? </span></span></span></h2>
 +
<h3>
 +
<span style="font-size:14px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><span class="mw-headline">Introduction </span></span></span></h3>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">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.</span></span></p>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The reactor&#39;s behavior is modeled by a Semi-Continuous Tank Reactor, assuming perfect mixing in the container.</span></span></p>
 +
<h4>
 +
<span style="font-size:14px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><span class="mw-headline">Enzymatic Reaction</span></span></span></h4>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The general reaction scheme is described as follows:</span></span></p>
-
For an enzymatic reaction of the type:
+
\begin{equation}
 +
E + S \leftrightarrow ES \rightarrow E^0 + P
 +
\end{equation}
-
<math>E + S \leftrightarrow ES \rightarrow E^0 + P</math>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">With a reaction rate of:</span></span></p>
-
with a reaction rate of:
+
\begin{equation}
 +
\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]
 +
\end{equation}
-
<math>\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]</math>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">This equation is affected by the constants k<sub>1</sub> , k<sub>-1</sub> and k<sub>2</sub>.</span></span></p>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
<strong><span style="font-size:14px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Mass balance</span></span></strong></p>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">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: </span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">ACCUMULATION = INPUT + APPEARANCE BY REACTION - DISAPPEARANCE BY REACTION </span></span></p>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">where </span></span></p>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">Input =F<sub>0,&nbsp;</sub>Appearance =V(r<sub>P</sub>), Disappearance =V(-r<sub>S</sub>) and Accumulation = \(\frac{d[P]}{dt}\)</span></span></p>  
-
We established the following in our system:
+
\begin{equation}
 +
\frac{d[P]}{dt}=F_0+V(r_P)-V(-r_S)
 +
\end{equation}
-
*Uniform distribution throughout the reactor
+
<p>
-
*K<sub>-1</sub> >> K<sub>1</sub> and K<sub>2</sub>
+
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The inlet flow was determined by diffusion. A mass balance, applied to a spherical envelope is described as:</span></span></p>
-
*One enzyme per reactor/VLP
+
-
*Tortuosity approaches zero during diffusion
+
-
Mass balance was presented as such:
+
\begin{equation}
 +
\frac{d}{dr}(r^2N_{Ar})=0
 +
\end{equation}
-
'''INFLOW= OUTFLOW- DISAPPEARANCE BY REACTION + ACCUMULATION
+
<p>
-
'''
+
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">where N<sub>Ar</sub> represents molar flux. When N<sub>Br</sub>=0 we obtain</span></span></p>
-
where
+
\begin{equation}
 +
\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0
 +
\end{equation}
-
Inflow= F<sub>0</sub>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">At a constant temperature the product (cD<sub>AB</sub>) is equally constant and x<sub>A</sub>=1-x<sub>B</sub>, the equation can be integrated into the following expression:</span></span></p>
-
Outflow= F<sub>0</sub>(1-X<sub>S</sub>)
+
\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}
-
Disappearance = V(-r<sub>S</sub>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">where &#39;&#39;x&#39;&#39; are the fractions, &#39;&#39;c&#39;&#39; is the concentration and &#39;&#39;r&#39;&#39; are the respective radius. </span></span></p>
-
Accumulation = <math>\tfrac{d[P]}{dt}</math>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">This equation defines the nanoreactor inflow.<br />
 +
<br />
 +
We neglect the possibility of an outflow of species because:</span></span></p>
 +
<ul>
 +
<li>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The gradient of concentration of S tends to stay inside the capsid</span></span></li>
 +
<li>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The positive charge in the outside of the VLP made a repulsion of the specie S</span></span></li>
 +
<li>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">An evident agglomeration of specie P will increase it size and remain inside</span></span></li>
 +
</ul>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;"><strong><span style="font-size: 14px;">Diffusion coefficient</span></strong><br />
 +
For S and P being ionic silver and reduced silver respectively. The ionic silver diffusion coefficient in solution is described by Nerst&#39;s equation (1888):</span></span></p>
-
F<sub>0</sub> = F<sub>0</sub>(1-X<sub>S</sub>) - V(-r<sub>S</sub>) + <math>\tfrac{d[P]}{dt}</math>
+
\begin{equation}
 +
D_{AB}°=
 +
\frac{RT}{F^2}
 +
\frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-}
 +
\frac{|Z_-|+|Z_+|}{|Z_+Z_-|}
 +
\end{equation}
-
The intake and outflow flux were determined by diffusion , considering a spherical container.
+
<p>
 +
where</p>
 +
<ul>
 +
<li>
 +
<em>F</em> = Faraday&#39;s constant [A&middot;s/g<sub>eq</sub>]</li>
 +
<li>
 +
<em>D<sub>AB</sub>&deg;</em> = Diffusion coefficient at infinite dilution [m<sup>2</sup>/s]</li>
 +
<li>
 +
<em>&lambda;<sub>+</sub>&deg;</em>= Cationic conductivity at infinite dilution</li>
 +
<li>
 +
<em>&lambda;<sub>-</sub>&deg; </em>= Anionic conductivity at infinite dilution</li>
 +
<li>
 +
<em>Z<sup>+</sup></em>= Cation valence</li>
 +
<li>
 +
<em>Z<sup>-</sup> </em>= Anionic valence</li>
 +
<li>
 +
<em>T</em> = Absolute temperature [K]</li>
 +
</ul>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><span style="font-size:14px;"><strong>Boiling temperature</strong></span><br />
 +
Via Joback&#39;s method, we estimate the normal boiling temperature:</span></span></p>
 +
\begin{equation}
 +
T_b=\mathbf{198} + \sum_{k} N_k(tbk)
 +
\end{equation}
-
[[Image:CCMV2.jpg]]
+
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">in which &#39;&#39;N<sub>k</sub>&#39;&#39; is the number of times that the contribution group is present in the compound.</span></span></p>
-
For the simplification of the diffusion phenomenon we considered:
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><strong><span style="font-size: 14px;">Critical temperature</span></strong><br />
 +
<span style="font-size: 12px;">Using a similar approach, also by Joback, we estimate the critical temperature:</span></span></p>
-
*Constant temperature
+
\begin{equation}
-
*Constant pressure
+
T_c=T_b\Bigg[
-
*Species B stays in a stationary state (it does not diffuse in A)
+
\mathbf{0.584}+\mathbf{0.965}
-
*The container (VLP) has a spherical shape
+
\bigg\{\sum_{k} N_k(tck) \bigg\} -
 +
\bigg\{\sum_{k} N_k(tck) \bigg\}^2
 +
\Bigg]^{-1}
 +
\end{equation}
-
A mass balance, taking into account a spherical envelope leads to:
+
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Table 1. Joback&#39;s Method Contribution</span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/1/11/UANLReactortable1.PNG" /></span></span></p>
 +
<p>
 +
<span style="font-size:14px;"><strong><span style="font-family: trebuchet ms,helvetica,sans-serif;">Conductivity</span></strong></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">The conductivity of the compound is determined by the Sastri method:</span></span><math></math></p>
-
<math>\frac{d}{dr}(r^2N_{Ar})=0</math>
+
\begin{equation}
 +
\lambda_L=\lambda_ba^m
 +
\end{equation}
-
where N<sub>Ar</sub> represents molar flux. For N<sub>Br</sub> we obtain:
+
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">where *&#39;&#39;&lambda;<sub>L</sub>&#39;&#39; = thermic conductivity of the liquid [ W/(m&middot;K)] *&#39;&#39;&lambda;<sub>b</sub>&#39;&#39; = thermic conductivity at normal boiling point [ W/(m&middot;K)] *&#39;&#39;T<sub>br</sub>&#39;&#39; = &#39;&#39;T<sub>b</sub>/T<sub>c</sub>&#39;&#39; = normal boiling reduced temperature *&#39;&#39;T<sub>r</sub>&#39;&#39; = &#39;&#39;T/T<sub>c</sub>&#39;&#39; = reduced temperature *&#39;&#39;T<sub>c</sub>&#39;&#39; = critical temperature [K]</span></span></p>
-
<math>\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0</math>
+
<p>
 +
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">and</span></span></p>
-
At a constant temperature the product (cD<sub>AB</sub>) is equally constant and x<sub>A</sub>=1-x<sub>B</sub>, the equation can be integrated into the following expression:
+
\begin{equation}
 +
m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n
 +
\end{equation}
-
<math>F_A=4\pi r_1^2N_{Ar}|_{r=r1}=
+
<p>
-
\frac{4\pi cD_{AB}}{1/r_1=1/r_2}
+
<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">with
-
\ln\frac{x_{B2}}{x_{B1}}
+
a = 0.16 and n = 0.2 for the compound.</span></span></p>
-
</math>
+
-
where ''x'' are the fractions, ''c'' is the concentration and ''r'' are the respective radii.  
+
<p>
 +
<span style="font-size:14px;">Table 2. Sastri&#39;s Contributions</span></p>
 +
<p style="text-align: center;">
 +
<img alt="" src="http://openwetware.org/images/6/68/UANLReactortable2.PNG" style="width: 840px; height: 411px;" /></p>
 +
<p style="text-align: justify;">
 +
<strong><span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 14px;">Euler Method </span></span></strong></p>
 +
<p style="text-align: justify;">
 +
<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>
-
This equation defines the nanoreactor inflow; a similar analysis yields the reactor outflow.
 
-
----
 
-
The ionic silver diffusion coefficient in function to the solution is described by Nerst's equation (1888)<sup>1</sup>:
+
\(y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0\)
-
<math>D_{AB}°=
+
<p>
-
\frac{RT}{F^2}
+
<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>
-
\frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-}
+
-
\frac{|Z_-|+|Z_+|}{|Z_+Z_-|}
+
-
</math>
+
-
where
+
\(y_{n+1} = y_n + hf(t_n,y_n)\)
-
*F = Faraday's constant
+
<p>
-
*D<sub>AB</sub>°=Diffusion coefficient at infinite dilution
+
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>
-
<sub>+</sub>°=Cationic conductivity at infinite dilution
+
-
<sub>-</sub>°=Anionic conductivity at infinite dilution
+
-
*Z<sup>+</sup>=Cation valence
+
-
*Z<sup>-</sup>=Anionic valence
+
-
*T=Absolute temperature
+
-
----
+
<p>
-
 
+
<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>
-
Via Joback's method, we obtain the normal boiling temperature:
+
-
 
+
-
<math>T_b=\mathbf{198} + \sum_{k} N_k(tbk)</math>
+
-
 
+
-
in which ''N<sub>k</sub>'' is the number of times that the contribution occurs in the compound.
+
-
 
+
-
----
+
-
 
+
-
Using a similar approach, also by Joback, we estimated the critical temperature:
+
-
 
+
-
<math>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>
+
-
Tabla .- Joback Method Contributions (C1 Prausnitz)
+
\(y^{(N)}(t) = f(t, y(t), y'(t), \ldots, y^{(N-1)}(t)\)
-
[TABLA]
+
<p>
 +
<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>
-
----
+
\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}
-
Conductivity was determined by the Sastri method:
+
<p>
 +
<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>
 +
<p>
 +
<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>
-
<math>\lambda_L=\lambda_ba^m</math>
+
\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}
-
where
+
<p>
-
λ<sub>L</sub> = thermic conductivity of the liquid [ W/(m·K)]
+
<strong><span style="font-size:16px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">RESULTS</span></span></strong></p>
-
λ<sub>b</sub> = thermic conductivity at normal boiling point
+
<p>
-
''T<sub>br</sub>''= ''T/T<sub>c</sub>'' = reduced temperature
+
<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>
-
''T<sub>c</sub>'' = critical temperature, K
+
<p>
 +
<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>
-
<math>m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n</math>
+
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">A tendency analysis of graphs was made for each figure.</span></span></p>
 +
<ul>
 +
<li>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 1: It is a fact that a continuous inflow flux of substrate within time is represented; this fact will lead an accumulation of material inside the reactor.</span></span></li>
 +
</ul>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/4/4b/Reactorfig1.1.jpg" style="width: 498px; height: 310px;" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 1.- Substrate concentration within time</span></span></p>
 +
<ul>
 +
<li>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 2: It could be appreciated a decrement in concentration of pure enzyme because the matter agglomeration effect, which leads an increment of enzyme-substrate concentration inside the capsid, making more possible to find this last state of enzyme within time.</span></span></li>
 +
</ul>
 +
<p style="text-align: center;">
 +
<span _fck_bookmark="1" style="display: none;">&nbsp;</span><span _fck_bookmark="1" style="display: none;"> </span><span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/d/de/Reactorfig1.2.jpg" style="width: 498px; height: 310px;" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 2.-&nbsp; Enzyme and Enzyme-Substrate&nbsp; concentrations within time</span></span></p>
 +
<ul>
 +
<li>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 3: It is easily noticed that production of silver nanoparticles rise proportional with the inlet flux considering that all flux react forming the complex enzyme-substrate.</span></span></li>
 +
</ul>
 +
<p style="text-align: center;">
 +
<span _fck_bookmark="1" style="display: none;">&nbsp;</span><span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/3/3c/Reactorfig3.1.jpg" /><span _fck_bookmark="1" style="display: none;">&nbsp;</span></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 3.- Product concentration within time</span></span></p>
 +
<ul>
 +
<li>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 4: It describes the real percentage of volume required to &ldquo;Figure 1&rdquo; reach equilibrium. For a set time of 1.5 ms it requires 1061% of available volume.</span></span></li>
 +
</ul>
 +
<p style="text-align: center;">
 +
<span _fck_bookmark="1" style="display: none;">&nbsp;</span><span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/1/18/Reactorfig4.1.jpg" style="width: 512px; height: 310px;" /><span _fck_bookmark="1" style="display: none;">&nbsp;</span></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 4.- Percentage of occupied volume in reactor within time</span></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">A second analysis for the real reactor volume will be made, representing the theoretical maximum value.</span></span></p>
 +
<p>
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">In figures 5, 7 and 8 the same behavior as the previous figures 1, 3 and 4, respectively, analysis are appreciated. In figure 6 a constant relative concentration for enzyme and the complex enzyme-substrate, demonstrating that our system is an appropriate media for the reduction reaction of ionic silver. This phenomenon is explained by the approach of non-declined enzymatic activity for the available small volume (3053nm<sup>3</sup>).</span></span></p>
 +
<p style="text-align: center;">
 +
<br />
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/e/e6/Reactorfig2.1.jpg" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 5.- Substrate concentration within time at 100%volume capacity</span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/8/83/Reactorfig2.2.jpg" style="width: 498px; height: 310px;" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 6.-&nbsp; Enzyme and Enzyme-Substrate&nbsp; concentrations within time at 100% volume capacity</span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/f/f7/Reactorfig4.2.jpg" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 7.- Product concentration within time at 100%volume capacity</span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;"><img alt="" src="http://openwetware.org/images/d/de/Reactorfig3.2.jpg" style="width: 498px; height: 310px;" /></span></span></p>
 +
<p style="text-align: center;">
 +
<span style="font-size:11px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Figure 8.- Percentage of occupied volume in reactor within time at 100% volume capacity</span></span></p>
 +
</body>
 +
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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:

A tendency analysis of graphs was made for each figure.

  • Figure 1: It is a fact that a continuous inflow flux of substrate within time is represented; this fact will lead an accumulation of material inside the reactor.

Figure 1.- Substrate concentration within time

  • Figure 2: It could be appreciated a decrement in concentration of pure enzyme because the matter agglomeration effect, which leads an increment of enzyme-substrate concentration inside the capsid, making more possible to find this last state of enzyme within time.

 

Figure 2.-  Enzyme and Enzyme-Substrate  concentrations within time

  • Figure 3: It is easily noticed that production of silver nanoparticles rise proportional with the inlet flux considering that all flux react forming the complex enzyme-substrate.

  

Figure 3.- Product concentration within time

  • Figure 4: It describes the real percentage of volume required to “Figure 1” reach equilibrium. For a set time of 1.5 ms it requires 1061% of available volume.

  

Figure 4.- Percentage of occupied volume in reactor within time

A second analysis for the real reactor volume will be made, representing the theoretical maximum value.

In figures 5, 7 and 8 the same behavior as the previous figures 1, 3 and 4, respectively, analysis are appreciated. In figure 6 a constant relative concentration for enzyme and the complex enzyme-substrate, demonstrating that our system is an appropriate media for the reduction reaction of ionic silver. This phenomenon is explained by the approach of non-declined enzymatic activity for the available small volume (3053nm3).


Figure 5.- Substrate concentration within time at 100%volume capacity

Figure 6.-  Enzyme and Enzyme-Substrate  concentrations within time at 100% volume capacity

Figure 7.- Product concentration within time at 100%volume capacity

Figure 8.- Percentage of occupied volume in reactor within time at 100% volume capacity

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