Biomod/2013/NanoUANL/Reactor: Difference between revisions

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<html>  <script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\$','\$']]}}); MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script> <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <head> <title>HTML Editor Sample Page</title> </head> <body> <img alt="" src="http://openwetware.org/images/7/73/UANLReactor1.png" style="width: 475px; height: 402px;" />     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. <h2> Why is this a reactor? </h2> <h3> Introduction </h3> 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. <h4> Enzymatic Reaction</h4> 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} </body> </html>

Biomod/2013/NanoUANL/Reactor: Difference between revisions

Revision as of 16:57, 26 October 2013

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@@ Line 76: / Line 76: @@
 			<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 = </span></span></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>
-		<p>
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">F<sub>0</sub> </span></span></p>
+\begin{equation}
-		<p>
+\frac{d[P]}{dt}=F_0+V(r_P)-V(-r_S)
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">Appearance = </span></span></p>
+\end{equation}
-		<p>
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">V(r<sub>P</sub>) </span></span></p>
+<p>
-		<p>
+			<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>
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">Disappearance = </span></span></p>
-		<p>
+\begin{equation}
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">V(-r<sub>S</sub>) </span></span></p>
+\frac{d}{dr}(r^2N_{Ar})=0
-		<p>
+\end{equation}
-			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">Accumulation =&nbsp;</span></span></p>
+<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>
+\begin{equation}
+\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0
+\end{equation}
+<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>
+\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}
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