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:
With a reaction rate of:
. . . 1.1
This equation is affected by the constants k_{1} , k_{-1} and k_{2}.
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 = F_{0}
Appearance = V(r_{P})
Disappearance = V(-r_{S})
Accumulation =
. . . 1.2
The inlet flow was determined by diffusion. A mass balance, applied to a spherical envelope is described as:
. . . 1.3
where N_{Ar} represents molar flux. When N_{Br}=0 we obtain
. . . 1.4
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:
. . . 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):
. . . 1.6
where
- F = Faraday's constant [A·s/g_{eq}]
- D_{AB}° = Diffusion coefficient at infinite dilution [m^{2}/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:
. . . 1.7
in which N_{k} 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:
. . . 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 |
---|---|---|---|---|---|---|---|---|---|---|---|
CH_{3} | 23.58 | 0.0141 | CH_{2}_{(ss)} | 27.15 | 0.01 | ACOH | 76.34 | 0.0240 | NH | 50.17 | 0.0295 |
CH_{2} | 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 |
=CH_{2} | 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 | NO_{2} | 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 | NH_{2} | 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:
. . . 1.9
where
- λ_{L} = thermic conductivity of the liquid [ W/(m·K)]
- λ_{b} = thermic conductivity at normal boiling point [ W/(m·K)]
- T_{br} = T_{b}/T_{c} = normal boiling reduced temperature
- T_{r} = T/T_{c} = reduced temperature
- T_{c} = critical temperature [K]
and
. . . 1.10
with a = 0.16 and n = 0.2 for the compound
Sastri's Contributions (Table 10.5. Prausnitz)
Hydrocarbon Groups | Δλ_{b} | Non-Hydrocarbon Groups | Δλ_{b} | Non-Hydrocarbon Groups | Δλ_{b} |
---|---|---|---|---|---|
CH_{3} | 0.0545 | O | 0.0100 | N_{(ring)} | 0.0135 |
CH_{2} | 0.0008 | OH^{2} | 0.0830 | CN | 0.0645 |
CH | -0.0600 | OH^{3} | 0.0680 | NO_{2} | 0.0700 |
C | -0.1230 | CO_{(ketone)} | 0.0175 | S | 0.0100 |
=CH_{2} | 0.0545 | CHO_{(aldehyde)} | 0.0730 | F^{4} | 0.0568 |
=CH | 0.0020 | COO_{(ester)} | 0.0070 | F^{5} | 0.0510 |
=C | -0.0630 | COOH_{(acid)} | 0.0650 | Cl | 0.0550 |
=C= | 0.1200 | NH_{2} | 0.0880 | Br | 0.0415 |
Ring^{1} | 0.1130 | NH | 0.0065 | I | 0.0245 |
NH_{(ring)} | 0.0450 | H^{6} | 0.0675 | ||
N | -0.0605 | 3 member ring | 0.1500 | ||
Ring^{7}_{(other)} | 0.1100 |
^{1}_{In 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
Choose a value h for the size of every step and set t_{n} = t_{0} + nh. Now, one step of the Euler method from t_{n} to t_{n + 1} = t_{n} + h is
. . . 1.11
The value of is an approximation of the solution to the ODE at time : . The Euler method is explicit, i.e. the solution is an explicit function of for .
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
, . . . 1.12
we introduce auxiliary variables and obtain the equivalent equation
. . . 1.13
This is a first-order system in the variable 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
. . . 1.14
. . . 1.15
. . . 1.16
. . . 1.17
Results
For a and with and
Considering a volume of capsid and enzyme respectively we obtained:
[INSERTAR GRAFICOS TENDENCIAS AQUI]