CRE | Chemical Reaction Engineering – External and internal diffusion effects

Steps in a heterogeneous catalytic reaction

Diffusion effects

Industrial reactions at high temperature
- Limited by the rate of mass transfer between bulk fluid and the catalytic surface.
Mass transfer
- Any process in which diffusion plays a role.
- Rate laws cannot be applied directly.
- Need to consider fluid velocity and fluid properties while writing molar balances.

External mass transfer

Internal mass transfer

Diffusion: definitions

Molar flux with no diffusion

$W_{Az} = \frac{C_A \upsilon}{A_C} = \frac{F_{Az}}{A_C}$

$\frac{\text{mol}}{\text{m}^2 \cdot \text{s}} \qquad \frac{\text{mol}}{\cancel{\text{m}^3}} \cdot \frac{\cancel{\text{m}^3}}{\text{s}} \cdot \frac{1}{\text{m}^2}$
- $F_{Az}$ : Uniform at a given cross section (plug flow assumption)
Diffusion
- Spontaneous intermingling or mixing of atoms or molecules by random thermal motion.
- Gives rise to the motion of species relative to the motion of the mixture.
- From regions of high concentration to regions of low concentration in the absence of other gradients.
- Diffusion results in a molar flux of the species (A), $W_A$ (moles/area time), in the direction of the concentration gradient.
$W_A = i W_{Ax} + j W_{Ay} + k W_{Az} \qquad \text{[Rectangular coordinates]}$

$\Rightarrow$ Need to develop mole balance that incorporates both diffusion and reaction.

Mole balance in rectangular coordinates

$\begin{aligned} & \left[ \text{molar flow rate in} \right]_z -& \left[ \text{molar flow rate out} \right]_{z+\Delta z} \\ +& \left[ \text{molar flow rate in} \right]_y -& \left[ \text{molar flow rate out} \right]_{y+\Delta y} \\ +& \left[ \text{molar flow rate in} \right]_x -& \left[ \text{molar flow rate out} \right]_{x+\Delta x} \\ +& \left[ \text{rate of generation} \right] \\ =& \left[ \text{rate of accumulation} \right] \end{aligned}$

$\begin{aligned} & \Delta x \Delta y W_{Az} \bigg|_z & -& \Delta x \Delta y W_{Az} \bigg|_{z+\Delta z} & +\Delta x \Delta z W_{Ay} \bigg|_y & -& \Delta x \Delta z W_{Ay} \bigg|_{y+\Delta y} \\ & +\Delta z \Delta y W_{Ax} \bigg|_x & -& \Delta z \Delta y W_{Ax} \bigg|_{x+\Delta x} & +r_A \Delta x \Delta y \Delta z & =& \Delta x \Delta y \Delta z \frac{\partial C_A}{\partial t} \end{aligned}$

Differential Form:

Dividing by $\Delta x \Delta y \Delta z$ and taking the limit as they go to zero:

$\frac{\partial W_{Ax}}{\partial x} + \frac{\partial W_{Ay}}{\partial y} + \frac{\partial W_{Az}}{\partial z} + r_A = \frac{\partial C_A}{\partial t}$

Molar Flux

$W_A$ : Total molar flux of species $A$
- $J_A$ : Molecular diffusion flux relative to bulk motion of fluid due to concentration gradient
- $B_A$ : Flux resulting from bulk motion of fluid

$W_A = J_A + B_A$

$B_A$ : $= y_A \left( \sum W_i \right)$
- $\sum W_i$ : Total flux of all molecules
- $y_A$ : mole fraction of $A$
For $A \rightarrow B$ (two-component system):

$W_A = J_A + y_A \left( W_A + W_B \right) = J_A + C_A U$

Fick’s law

Describes how molar diffusive flux is related to the concentration gradient.
Fourier’s Law (thermal conduction)

$q = -k_t \nabla T;\quad k_t: \text{thermal conductivity}$

$\nabla = i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}$
For Mass Transfer: Fick’s Law

$J_A = -D_{AB} \nabla C_A$
- $D_{AB}$ : Diffusivity of $A$ in $B$ $(m^2/s)$
$W_A = -D_{AB} \nabla C_A + C_A U$
In One Dimension

$W_{Az} = -D_{AB} \frac{dC_A}{dz} + C_A U_z$

Evaluating Molar Flux

$A$ diffusing in $B$

$W_A = -D_{AB} \nabla C_A + y_A (W_A + W_B)$
Equimolar Counter Diffusion (EMCD)

$W_A = -W_B \qquad W_A = J_A = -D_{AB} \nabla C_A$
Species $A$ Diffusing Through Stagnant Species $B$ ( $W_B = 0; W_A = J_A + y_A W_A$ )
- Solid boundary with adjacent stagnant fluid layer
$W_A = \frac{J_A}{1 - y_A} = -\frac{D_{AB} \nabla C_A}{1 - y_A}; \qquad W_A = -D_{AB} C \nabla \ln (1 - y_A)$
Bulk Flow of $A \gg$ Molecular Diffusion (plug flow model)

$W_A = B_A = y_A (W_A + W_B) = C_A U$

Evaluating Molar Flux

Small Bulk Flow $J_A \gg B_A$

$W_A = -D_{AB} \nabla C_A = J_A$
Knudsen Diffusion
- Occurs in porous catalyst where diffusing molecule collides more often with catalyst walls/pore walls than with each other.
  
  $W_A = J_A = -D_k \nabla C_A$
- $D_k$ : Knudsen diffusivity
Diffusion and convective transport: $F_{Az} = W_{Az} A_C$
- $F_{Az}$ : molar flow rate of $A$ in $z$ direction; $W_{Az}$ : molar flux in $z$ ; $A_C$ : cross-sectional area normal to flow
$W_{Az} = -D_{Az} \frac{dC_A}{dz} + C_A U_z$

$F_{Az} = W_{Az} A_C = \left[ -D_{Az} \frac{dC_A}{dz} + C_A U_z \right] A_C$
- Similar expressions for $F_{Ay}$ and $F_{Ax}$ .

Mole balance in rectangular coordinates

Substituting expressions for $F_{Az}$ , $F_{Ay}$ , and $F_{Ax}$ in mole balance

$D_{AB} \left[ \frac{\partial^2 C_A}{\partial x^2} + \frac{\partial^2 C_A}{\partial y^2} + \frac{\partial^2 C_A}{\partial z^2} \right] - u_x \frac{\partial C_A}{\partial x} - u_y \frac{\partial C_A}{\partial y} - u_z \frac{\partial C_A}{\partial z} + r_A = \frac{\partial C_A}{\partial t}$

In one dimension:

$D_{AB} \frac{\partial^2 C_A}{\partial z^2} - U_z \frac{\partial C_A}{\partial z} + r_A = \frac{\partial C_A}{\partial t}$

Boundary conditions
- Concentration at boundary: $@ z = 0 \rightarrow C_A = C_{A0}; @ z = L \rightarrow \frac{dC_A}{dz} = 0$
- Specify flux at the boundary
  - No mass transfer $W_A = 0$
  - Molar flux at surface = rate of reaction at surface
    
    $W_A|_{surface} = -r''|_{surface}$
  - Molar flux to boundary = convective transport across boundary layer
    
    $W_A|_{boundary} = k_c (C_{Ab} - C_{As})$

Temperature and pressure dependence of $D_{AB}$

Phase	$cm^2/s$	$m^2/s$	Temperature and pressure dependences
Gas
- Bulk	$10^{-1}$	$10^{-5}$	$D_{AB}(T_2, P_2) = D_{AB}(T_1, P_1) \frac{P_1}{P_2} \left( \frac{T_2}{T_1} \right)^{1.75}$
- Knudsen	$10^{-2}$	$10^{-6}$	$D_{AB}(T_2) = D_{A}(T_1) \left( \frac{T_2}{T_1} \right)^{1/2}$
Liquid	$10^{-5}$	$10^{-9}$	$D_{AB}(T_2) = D_{AB}(T_1) \frac{\mu_1}{\mu_2} \left( \frac{T_2}{T_1} \right)$
Solid	$10^{-9}$	$10^{-13}$	$D_{AB}(T_2) = D_{AB}(T_1) \exp \left[ \frac{E_D}{R} \left( \frac{T_2 - T_1}{T_1 T_2} \right) \right]$

$\mu_1, \mu_2$ are the liquid viscosities at temperatures $T_1$ and $T_2$ respectively.
$E_D$ is the diffusion activation energy.
Knudsen, liquid, and solid $D_{AB}$ are independent of $P$ .

Modeling diffusion with chemical reaction

Diffusion of species through stagnant film in which no reaction takes place
$A$ reacts instantaneously upon reaching the surface $C_{As} \approx 0$
Rate of diffusion through stagnant film = rate of reaction on the surface

Steps:

Perform differential mole balance $\rightarrow$ Equation for $W_{Az}$
Replace $W_{Az}$ by appropriate expression for concentration gradient
State boundary conditions
Solve for concentration profile
Solve for molar flux

Diffusion through a stagnant film

Flow Past a Single Catalytic Pellet
- Reaction takes place only on external catalyst surface and not in fluid surrounding it.
Hydrodynamic Boundary Layer: Distance from solid object to where the fluid velocity is 99% of the bulk velocity $U_0$ .
Mass Transfer Boundary Layer: Distance at which concentration of diffusing species is 99% of the bulk concentration.
- We cannot measure $\delta$ .

Diffusion through a stagnant film

Assumption:
- Fluid layer next to solid as stagnant film (hypothetical) of thickness $\delta$ .
- All the resistance to mass transfer is within the film.
- Properties at the outer edge of the film are identical to those of bulk fluid.
For film thickness $\ll$ radius of pellet:
- Curvature effects can be neglected
- Problem reduces to 1D diffusion.

$\begin{align*} &\text{In} &-& \text{Out} &+& \text{Generation} &=& \text{Accumulation} \\ &W_{Az}\big|_z &-& W_{Az}\big|_{z + \Delta z} &+& 0 &=& 0 \end{align*}$

Diffusion through a stagnant film

Dividing by $\Delta z$ and taking the limit as $\Delta z \to 0$ $\frac{dW_{Az}}{dz} = 0$
For diffusion through stagnant film at dilute concentrations,

$J_A \gg y_A (W_A + W_B)$

$W_{Az} = -D_{AB} \frac{dC_A}{dz} \quad \text{(also for EMCD)}$

Therefore, mole balance becomes

$\frac{d^2C_A}{dz^2} = 0$

Boundary Conditions (BCs)
- At $z = 0$ , $C_A = C_{As}$ ; At $z = \delta$ , $C_A = C_{Ab}$

Diffusion through a stagnant film

Integrating twice $C_A = k_1z + k_2 \implies C_{As} = k_2 \quad \text{and} \quad \left( \frac{C_{Ab} - C_{As}}{\delta} \right) = k_1$
Substituting BCs

$C_A = C_{As} + \left( C_{Ab} - C_{As} \right) \frac{z}{\delta}$

Surface Flux

$W_{Az}\bigg|_{\delta} = -D_{AB} \frac{dC_A}{dz} \bigg|_{\delta}$

$W_{Az} = \frac{D_{AB}}{\delta} \left[ C_{Ab} - C_{As} \right]$

At steady state: $\text{Flux of A to the surface} = \text{Rate of reaction of A on surface}$

The mass transfer coefficient

$k_c = \frac{D_{AB}}{\delta}$

Average molar flux from the bulk fluid to the surface

$W_{Az} = k_c (C_{Ab} - C_{As})$

$W_{Az} = \text{Flux} = \frac{\text{Driving force}}{\text{Resistance}} = \frac{C_{Ab} - C_{As}}{\left(\frac{1}{k_c}\right)}$

The mass transfer coefficient is found by experimentation
Correlations analogous to what one finds for a heat transfer coefficient.

$Sh = 2 + 0.6 Re^{1/2} Sc^{1/3} \qquad \text{ Frössling correlation }$

$Sh = \frac{k_c L}{D_{AB}} ;\quad Sc = \frac{\nu}{D_{AB}} = \frac{\mu}{\rho D_{AB}} ;\quad Re = \frac{UL}{\nu} = \frac{UL\rho}{\mu}$

Mass transfer–limited reactions in packed beds

$\left[ \text{Molar rate in} \right] - \left[ \text{Molar rate out} \right] \\ + \left[ \text{Molar rate of generation} \right] = \left[ \text{Molar rate of accumulation} \right]$

In one dimension at steady state

$F_{Az} \bigg|_z - F_{Az} \bigg|_{z+\Delta z} + r''_A c (A_c \Delta z) = 0$

Dividing by $A_c \Delta z$ and taking the limit as $\Delta z \to 0$

$-\frac{1}{A_c} \left( \frac{dF_{Az}}{dz} \right) + r''_A c = 0$

Mass transfer–limited reactions in packed beds

Express $F_{Az}$ and $r''$ in terms of concentration

$F_{Az} = A_c W_{Az} = (J_{Az} + B_{Az}) A_c$

In almost all situations involving flow in packed-bed reactors, the amount of material transported by diffusion or dispersion in the axial direction is negligible compared with that transported by convection (i.e., bulk flow)
Neglecting dispersion as $J_{Az} \ll B_{Az}$

$F_{Az} = A_c W_{Az} = A_c B_{Az} = U_c A_c$

$-\frac{d(CU)}{dz} + r''_A c = 0$
For constant superficial velocity

$-U\frac{dC}{dz} + r''_A c = 0$

Mass transfer–limited reactions in packed beds

The boundary condition at the external surface is

$-r''_A = W_{Ar} = k_c (C_A - C_{As})$

$-U\frac{dC}{dz} - k_c a_c (C_A - C_{As}) = 0$

$-U\frac{dC}{dz} = k_c a_c (C_A - C_{As})$
In most mass transfer-limited reactions, the surface concentration is negligible with respect to the bulk concentration ( $C_A \gg C_{As}$ )

$-U\frac{dC}{dz} = k_c a_c C_A$
At $z = 0, C_A = C_{A0}$

$\frac{C_A}{C_{A0}} = \exp \left( - \frac{k_c a_c}{U} z \right ) ;\qquad \ln \frac{1}{1-X} = \frac{k_c a_c}{U} L$

Diffusion and reactions in homogeneous systems

When reactants must diffuse inside a catalyst pellet in order to react
- The concentration at the pore mouth must be higher than that inside the pore.
- Entire catalyst surface is not accessible to the same concentration
- The rate of reaction to vary throughout the pellet.

Diffusion and Reaction in Homogeneous Systems

Mole balance on species $A$ for 1D diffusion at steady state

$-\frac{dW_{Az}}{dz} + r_A = 0$

$W_{Az} = -D_{AB} \frac{dC_A}{dz}$

$D_{AB} \frac{d^2 C_A}{dz^2} + r_A = 0$

Applications
- Medicine
- Cancer treatment using drug-laced particulates
- Tissue engineering

Effective diffusivity

$D_e = \frac{D_{AB} \phi_p \sigma_c}{\tilde{\tau}} ;\qquad W_{Az} = -D_{e} \frac{dC_A}{dz}$
The effective diffusivity accounts for the fact that:
- Not all of the area normal to the direction of the flux is available (i.e., the area occupied by solids) for the molecules to diffuse, (porosity, $\phi_p$ )
- The paths are tortuous (tortuosity, $\tilde{\tau}$ )
- The pores are of varying cross-sectional areas (Constriction factor, $\sigma_c$ )

Diffusion and reactions in spherical catalyst pellets

Consider irreversible isomerization reaction $\ce{A -> B}$
The reaction occurs on the surface of pore walls within a spherical pellet of radius $R$ .
Assumptions
- Steady state; spherical coordinates
- Constant pressure and temperature conditions

Steady state mole balance on a spherical shell $r \rightarrow r + \Delta r$

$\begin{align*} (\text{In at } r) &-& (\text{Out at } r + \Delta r) &+& (\text{Generation within } \Delta r) &=& 0 \\ (W_{Ar} 4 \pi r^2 |_r) &-& (W_{Ar} 4 \pi r^2 |_{ r + \Delta r }) &+& (r'_A \rho_c 4 \pi r_m^2 \Delta r) &=& 0 \end{align*}$

Dividing by $-4 \pi \Delta r$ and taking limit as $\Delta r \rightarrow 0$

$\frac{d(W_{Ar}r^2)}{dr} - r'_A \rho_c r^2 = 0$

Diffusion and reactions in spherical catalyst pellets

$W_{Ar} = -D_{e} \frac{dC_A}{dr}$

$\frac{-D_e (dC_A/dr) r^2}{dr} - r^2 \rho_c r'_A = 0$
Reaction rate { $r'_A = k_n C_A^n$ }
- per unit surface area $-r''_A$
- per unit mass of catalyst $-r'_A = S_a (-r''_A)$
- per unit volume $-r_A = \rho_c (-r'_A) = \rho_c S_a (-r''_A)$

$\frac{d^2 C_A}{dr^2} + \frac{2}{r} \left( \frac{dC_A}{dr} \right) - \frac{k_n}{D_e} C_A^n = 0$

Boundary conditions
- At the center of pellet: at $r = 0$ , $C_A$ is finite
- At the external surface: at $r = R$ , $C_A = C_{As}$

Diffusion and reactions in spherical catalyst pellets

Dimensionless form

$\psi = \frac{C_A}{C_{As}}; \quad \lambda = \frac{r}{R}$
Boundary conditions
- At $\lambda = 0$ , $\psi$ is finite
- At $\lambda = 1$ , $\psi = 1$

$\frac{d^2 \psi}{d \lambda^2} + \frac{2}{\lambda} \left( \frac{d \psi}{d \lambda} \right) - \phi^2 \psi^n = 0$

Thiele modulus

$\phi_n^2 = \frac{k_n R^{2} C_{As}^{n-1}}{D_e} = \frac{k_n R C_{As}^n}{D_e \left( \frac{C_{As} - 0}{R} \right)} \quad = \frac{\text{``a'' surface reaction rate}}{\text{``a'' diffusion rate}}$

When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction; when $\phi_n$ is small, the surface reaction is usually rate-limiting.

Diffusion and reactions in spherical catalyst pellets

Dimensionless concentration profile

$\psi = \frac{C_A}{C_{As}} = \frac{1}{\lambda} \left( \frac{ \sinh \phi_1 \lambda }{ \sinh \phi_1 } \right )$

The internal effectiveness factor

The magnitude of the effectiveness factor (ranging from 0 to 1) indicates the relative importance of diffusion and reaction limitations.

$\eta = \frac{\text{Actual overall rate of reaction}}{ \begin{array}{c} \text{Rate of reaction that would result if entire} \\ \text{interior surface were exposed to the external} \\ \text{pellet surface conditions } C_{As}, T_s \end{array} }$

Overall (observed) rate

$\eta = \frac{-r_A}{-r_{As}} = \frac{-r'_A}{-r'_{As}} = \frac{-r''_A}{-r''_{As}}$

Effectiveness factor for first order reaction spherical pellet

$\eta = \frac{3}{\phi_1^2} \left( \phi_1 \coth \phi_1 -1 \right)$

Reaction rate $-r_A = \eta (k_1 C_{As})$

If $\phi_1 < 2$ : $\eta \approx \frac{3}{\phi_1^2}[\phi_1 -1]$
If $\phi_1 > 20$ : $\eta \approx \frac{3}{\phi_1^2}$

The internal effectiveness factor

Determination of limiting situations from reaction- rate data

Reaction rate per unit mass of catalyst $-r'_A = k_c a_c C_A$

$k_c \propto \frac{U^{1/2}}{d_p^{1/2}} ; \qquad a_c \propto \frac{1}{d_p}$
For external mass transfer–limited reaction

$-r'_A \propto \frac{U^{1/2}}{d_p^{3/2}}$

Type of Limitation	Velocity	Particle Size	Temperature
External diffusion	$U^{1/2}$	$(d_p)^{-3/2}$	$\approx$ Linear
Internal diffusion	Independent	$(d_p)^{-1}$	Exponential
Surface reaction	Independent	Independent	Exponential