# 1. Derivation of the Internal Effectiveness Factor for a Spherical Catalyst Pellet Consider a first-order reaction \( A \rightarrow \text{Products} \) occurring inside a spherical pellet of radius \( R \), with reaction rate limited by pore diffusion. **Step 1: Shell Mass Balance** Take a thin spherical shell at radius \( r \) and thickness \( dr \). At steady state, the rate of accumulation is zero, so: \[ \frac{d}{dr} \left( 4 \pi r^2 N_A \right) + 4 \pi r^2 r_A = \] where \( N_A \) is the molar flux of \( A \) by diffusion, and \( r_A \) is the rate of reaction per unit volume. Assuming Fick’s law and first-order kinetics, \( N_A = -D_{eff} \frac{dC_A}{dr} \) and \( r_A = -k C_A \): \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dC_A}{dr} \right) = \frac{k}{D_{eff}} C_A \] **Step 2: Thiele Modulus** Define the Thiele modulus: \[ \phi = R \sqrt{\frac{k}{D_{eff}}} \] Rewrite the differential equation in dimensionless form, letting \( \eta = r/R \): \[ \frac{1}{\eta^2} \frac{d}{d\eta} \left( \eta^2 \frac{dC_A}{d\eta} \right) = \phi^2 C_A \] **Step 3: Boundary Conditions** At the center (\( \eta = \)), symmetry implies \( \frac{dC_A}{d\eta} = \). At the surface (\( \eta = 1 \)), the concentration is equal to the external surface value, \( C_A(1) = C_{As} \). **Step 4: Solution** The general solution is: \[ C_A(\eta) = \frac{\sinh(\phi \eta)}{\eta \sinh(\phi)} C_{As} \] **Step 5: Effectiveness Factor Definition** The effectiveness factor \( \eta_{eff} \) is the ratio of the actual overall reaction rate to the rate if the entire pellet were at surface concentration: \[ \eta_{eff} = \frac{\text{Actual total rate}}{\text{Rate if } C_A = C_{As} \text{ everywhere}} \] The actual rate is the integral of the local rate over the pellet: \[ \text{Actual rate} = \int_^R 4 \pi r^2 k C_A(r) dr \] The rate if \( C_A = C_{As} \) everywhere is \( k C_{As} \cdot \frac{4}{3} \pi R^3 \). **Step 6: Final Effectiveness Factor Expression** Evaluating the integral and simplifying, the effectiveness factor for a sphere is: \[ \eta_{eff} = \frac{3}{\phi^2} \left[ \phi \coth(\phi) - 1 \right] \] This expression shows how diffusion limitations (via the Thiele modulus) decrease the internal utilization of catalyst. --- # 2. Sketch of Residence Time Distribution for Tank-in-Series with Bypass A tank-in-series (TIS) model with a bypass stream consists of a fraction \( f \) of the flow that skips the reactor, while the remainder, \( 1-f \), passes through \( N \) ideal CSTRs in series. The E-curve is given by: \[ E(t) = f \delta(t) + (1-f) \cdot \frac{N}{\tau} \frac{(N t/\tau)^{N-1}}{(N-1)!} \exp(-N t/\tau) \] where \( \tau \) is the mean residence time of the main path. **Description of the Sketch:** - The plot starts with a sharp spike at \( t= \), representing the immediate exit of tracer through the bypass (\( f \delta(t) \)), whose area equals the bypass fraction. - After the spike, the curve rises and falls more slowly, representing the distributed residence times of the flow through the series of tanks. This part is skewed to longer times as \( N \) increases. - Contrasted with an ideal CSTR, which has a decaying exponential \( E_{CSTR}(t) = \frac{1}{\tau} \exp(-t/\tau) \), the TIS+bypass has a split: an early peak and a delayed, narrower distribution. **[Sketch: E-curve for tank-in-series with bypass]**  --- # 3. Thermal Runaway Criteria in a Tubular Reactor with Jacket Cooling Consider a tubular reactor with axial coordinate \( z \), cooled by a jacket, with a first-order exothermic reaction. Let \( T(z) \) be the temperature profile along the reactor. **Step 1: Mass and Energy Balances** The mass balance for reactant \( A \): \[ \frac{dC_A}{dz} = -\frac{r_A}{u} \] The energy balance (assuming constant density and specific heat): \[ \frac{dT}{dz} = \frac{-\Delta H}{\rho C_p u} r_A - \frac{U a}{\rho C_p u} (T - T_c) \] where \( r_A = k_ \exp\left(-\frac{E}{RT}\right) C_A \), \( T_c \) is the coolant temperature, and \( U a \) is the heat transfer per unit volume. **Step 2: Dimensionless Variables** Define dimensionless variables: - \( \theta = \frac{T - T_c}{T_c} \) - \( X = \frac{C_{A} - C_A}{C_{A}} \) - Damköhler number \( Da \), dimensionless activation energy \( \beta = \frac{E}{R T_c} \) **Step 3: Hot Spot and Critical Trajectory** Thermal runaway occurs when the rate of heat generation overtakes removal, leading to an uncontrolled temperature rise. The "hot spot" is the location \( z^* \) where \( T(z) \) is maximum. At the hot spot, \( dT/dz = \). For a maximum, \( d^2T/dz^2 = \) is the critical trajectory. From the energy balance: \[ \frac{dT}{dz} = \implies \text{heat generated} = \text{heat removed} \] Take the derivative: \[ \frac{d^2T}{dz^2} = \frac{d}{dz} \left[ \frac{-\Delta H}{\rho C_p u} r_A - \frac{U a}{\rho C_p u} (T - T_c) \right ] \] Set \( \frac{d^2T}{dz^2} = \) at the hot spot: \[ = \frac{-\Delta H}{\rho C_p u} \frac{dr_A}{dz} - \frac{U a}{\rho C_p u} \frac{dT}{dz} \] At the maximum, \( \frac{dT}{dz} = \), so: \[ \frac{dr_A}{dz} = \] Since \( r_A = k_ \exp\left(-\frac{E}{RT}\right) C_A \), this gives a relationship between \( C_A \) and \( T \) at the hot spot. **Step 4: Parametric Sensitivity** The sensitivity of the hot spot temperature to operating parameters is governed by: - The dimensionless activation energy \( \beta \): Large \( \beta \) means a small increase in \( T \) causes a large increase in reaction rate and heat generation, increasing runaway risk. - The heat of reaction \( -\Delta H \): Large exothermicity increases the heat generated, also raising runaway risk. An increase in either parameter makes the system more prone to runaway, shifting the critical trajectory so that the hot spot occurs earlier or is more pronounced. --- **Summary of Findings** - The internal effectiveness factor for a spherical pellet under diffusion-limited conditions depends on the Thiele modulus and is given by \( \frac{3}{\phi^2} [\phi \coth(\phi) - 1] \). - The RTD for a tank-in-series with bypass shows an immediate peak at zero time (bypass) and a delayed, distributed peak (main flow), contrasting with the exponential decay of an ideal CSTR. - The criterion for thermal runaway in a cooled tubular reactor is the point where the second derivative of temperature along the reactor vanishes, corresponding to the hot spot. The sensitivity of the hot spot temperature to process parameters is strongly influenced by the dimensionless activation energy and reaction exothermicity.