Biomod/2013/NanoUANL/Reactor

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.¹

Enzymatic Reaction

The general reaction scheme is described as follows:

[math]\displaystyle{ E + S \leftrightarrow ES \rightarrow E + P }[/math]

With a reaction rate of:

[math]\displaystyle{ \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] }[/math] . . . 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(r_P)

Disappearance = V(-r_S)

Accumulation = [math]\displaystyle{ \tfrac{d[P]}{dt} }[/math]

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

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

[math]\displaystyle{ \frac{d}{dr}(r^2N_{Ar})=0 }[/math] . . . 1.3

where N_Ar represents molar flux. When N_Br=0 we obtain

[math]\displaystyle{ \frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0 }[/math] . . . 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 expression²:

[math]\displaystyle{ 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}} }[/math] . . . 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:³

• 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)⁴:

[math]\displaystyle{ D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|} }[/math] . . . 1.6

where

• F = Faraday's constant [A·s/g_eq]
• D_AB° = 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:⁵

[math]\displaystyle{ T_b=\mathbf{198} + \sum_{k} N_k(tbk) }[/math] . . . 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:⁵

[math]\displaystyle{ 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] . . . 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:⁵

[math]\displaystyle{ \bold{\lambda_L=\lambda_ba^m} }[/math] . . . 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

[math]\displaystyle{ m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n }[/math] . . . 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

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.⁶

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

[math]\displaystyle{ y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0 }[/math]

Choose a value [math]\displaystyle{ h }[/math] for the size of every step and set [math]\displaystyle{ t_n = t_0 + nh }[/math]. Now, one step of the Euler method from [math]\displaystyle{ t_n }[/math] to [math]\displaystyle{ t_{n+1} = t_n + h }[/math] is

[math]\displaystyle{ \bold y_{n+1} = y_n + hf(t_n,y_n) }[/math] . . . 1.11

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

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

[math]\displaystyle{ y^{(N)}(t) = f(t, y(t), y'(t), \ldots, y^{(N-1)}(t)) }[/math], . . . 1.12

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

[math]\displaystyle{ \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} }[/math] . . . 1.13

This is a first-order system in the variable [math]\displaystyle{ \mathbf{z}(t) }[/math] 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

[math]\displaystyle{ \frac{dC\lt sub\gt E\lt /sub\gt }{dt}=-k_1C_S+k_-1C_ES+k_2C_ES }[/math]

References

1. Fogler H. Scott, Elements of Chemical Reaction Engineering 4th Edition, Pretince Hall 2006, pages 37-45
2. Bird R. B, Stewart W.E. & Lightfoot E.N., Fenómenos de Trasporte, Reverté Ediciones SA de CV 2006, pages 17-9 – 17-11.
3. Geankoplis, Christie John, Procesos de transporte y principios de procesos de separación, 4th Edition. Continental 2006, pages 449-451
4. Anthony L. Hines & Robert N. Maddox, Mass Transfer Fundamentals and Applications, Prentince Hall PTR 1985, pages 34-35
5. Poling, Prausnitz & O’Connel, The Properties of Gases and Liquids, 5th Edition. McGrawl-Hill 2001, pages 2.26,10.45-10.46, C.2-C.4
6. Hoffman, Joe D., Numerical Methods for Engineers and Scientists, 2th Edition. Marcel Decker, Inc. 2001, pages 355-359