External and internal diffusion effects
Chemical Reaction Engineering
Steps in a heterogeneous catalytic reaction
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External mass transfer
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External mass transfer
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Internal mass transfer
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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
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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}
\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
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}
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}
Temperature and pressure dependence of D_{AB}
| 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
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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.
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\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
C_A = C_{As} + \left( C_{Ab} - C_{As} \right) \frac{z}{\delta}
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
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\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
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- 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
Diffusion and reactions in spherical catalyst pellets
- 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
\frac{d^2 \psi}{d \lambda^2}
+ \frac{2}{\lambda} \left( \frac{d \psi}{d \lambda} \right)
- \phi^2 \psi^n = 0
\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 )
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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}
}
\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})
The internal effectiveness factor
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The internal effectiveness factor
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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}}
| External diffusion |
U^{1/2} |
(d_p)^{-3/2} |
\approx Linear |
| Internal diffusion |
Independent |
(d_p)^{-1} |
Exponential |
| Surface reaction |
Independent |
Independent |
Exponential |
