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---
name: academic-chemical-engineering-reactors
description: "Specializes in Chemical Engineering, Reactors, Transport Phenomena, and CFD building on H. Scott Fogler (Elements of Chemical Reaction Engineering), R. Byron Bird, Warren E. Stewart & Edwin N. Lightfoot (Transport Phenomena), and John D. Anderson (Computational Fluid Dynamics). Covers Molar and Energy Balances in Batch, CSTR, PFR, and Catalytic PBR, Heterogeneous Catalysis (Langmuir-Hinshelwood, Thiele Modulus, Effectiveness Factor η), Non-Isothermal Reactors and Multiple Steady States, Vapor-Liquid Equilibrium (VLE with NRTL/UNIQUAC and McCabe-Thiele), Fluid Mechanics (Navier-Stokes Equations for Laminar and Turbulent Flow), Heat Transfer (Fourier Conduction, Convection, Stefan-Boltzmann Radiation), Mass Transfer (Fick's Laws and Diffusive Convection), Boundary Layer Theory, and Computational Fluid Dynamics (Finite Volume Method FVM and SIMPLE Algorithm)."
---
# Chemical Engineering, Reactors, Transport Phenomena, and CFD
This skill establishes the rigorous differential and integral balances of mass, momentum, and energy applied to industrial reactor sizing, chemical separation processes, and numerical simulation of fluid, heat, and mass transport building on the works of **H. Scott Fogler**, **Bird, Stewart & Lightfoot**, and **John D. Anderson**.
---
## 🧪 1. Design Equations for Ideal Chemical Reactors (Fogler)
```mermaid
flowchart LR
A["General Molar Balance:<br/>F_j0 - F_j + ∫ r_j dV = dN_j/dt"] --> B["Batch:<br/>t = N_A0 ∫ dX/(-r_A V)"]
A --> C["CSTR (Perfect Mixing):<br/>V = F_A0 X / (-r_A)"]
A --> D["PFR (Tubular Plug Flow):<br/>V = F_A0 ∫ dX/(-r_A)"]
A --> E["PBR (Catalytic Bed):<br/>W = F_A0 ∫ dX/(-r_A')"]
```
| Reactor | Operation Type | Differential Design Equation | Integrated Equation |
| :--- | :--- | :--- | :--- |
| **Batch** | Transient, uniform mixing | $N_{A0}\frac{dX}{dt} = -r_A V$ | $t = N_{A0} \int_0^X \frac{dX}{-r_A V}$ |
| **CSTR** | Steady state, homogeneous | $F_{A0} - F_A + r_A V = 0$ | $V = \frac{F_{A0} X}{-r_A(X_{outlet})}$ |
| **PFR** | Steady-state tubular | $\frac{dF_A}{dV} = r_A \iff F_{A0}\frac{dX}{dV} = -r_A$ | $V = F_{A0} \int_0^X \frac{dX}{-r_A}$ |
| **PBR** | Fixed bed with mass $W$ | $F_{A0}\frac{dX}{dW} = -r_A'$ | $W = F_{A0} \int_0^X \frac{dX}{-r_A'}$ |
### 1.1 Heterogeneous Catalysis, Thiele Modulus ($\phi$), and Effectiveness ($\eta$)
- **Thiele Modulus for a Spherical Catalyst Pellet**:
$$\phi_1 = R \sqrt{\frac{k}{D_e}}$$
- **Internal Effectiveness Factor ($\eta$)**:
$$\eta = \frac{\text{Actual Rate with Diffusional Limitation}}{\text{Rate at the Surface}} = \frac{3}{\phi_1^2} (\phi_1 \coth\phi_1 - 1)$$
- If $\phi_1 \ll 1 \implies \eta \approx 1$ (Chemical reaction control).
- If $\phi_1 \gg 1 \implies \eta \approx \frac{3}{\phi_1}$ (Strong intraparticle diffusional limitation).
---
## 🌡️ 2. Non-Isothermal Reactors and Thermal Stability
- **Differential Energy Balance in a PFR**:
$$\frac{dT}{dV} = \frac{U a (T_a - T) + (-r_A)(-\Delta H_{Rx}^\circ)}{\sum F_i C_{pi}}$$
- **Multiple Steady States in an Exothermic CSTR**:
The intersection of the heat generated curve $Q_g(T) = (-r_A V)(-\Delta H_{Rx})$ with the heat removed line $Q_r(T) = (\sum F_{i0} C_{pi} + U A)(T - T_c)$ can give rise to 3 steady-state operating points (two stable and one unstable ignition/extinction point).
---
## 🌊 3. Transport Phenomena (Bird, Stewart & Lightfoot)
### 3.1 Fundamental Navier-Stokes (Momentum) Equations
For an incompressible Newtonian fluid with constant dynamic viscosity $\mu$:
$$\begin{aligned}
\nabla \cdot \mathbf{v} &= 0 \quad &\text{(Mass Conservation/Continuity)} \\
\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) &= -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \quad &\text{(Momentum Conservation)}
\end{aligned}$$
### 3.2 Heat Transfer (Energy)
General energy conservation equation with thermal generation $\dot{q}$:
$$\rho C_p \left( \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \right) = k \nabla^2 T + \dot{q} + \mu \Phi$$
where $k$ is the thermal conductivity and $\Phi$ is the viscous dissipation function.
### 3.3 Mass Transfer and Fick's Laws
- **Fick's 1st Law (Steady-State Molecular Diffusion)**:
$$\mathbf{J}_A = -D_{AB} \nabla C_A$$
- **Convective-Diffusive Chemical Species Transport Equation with Reaction**:
$$\frac{\partial C_A}{\partial t} + \mathbf{v} \cdot \nabla C_A = D_{AB} \nabla^2 C_A + r_A$$
---
## 💻 4. Computational Fluid Dynamics (CFD - Anderson)
- **Finite Volume Method (FVM)**: Integration of the differential equations over discrete control volumes $\Omega_P$ of a structured or unstructured computational mesh.
- **SIMPLE Algorithm (Semi-Implicit Method for Pressure-Linked Equations)**:
1. Solve the discretized momentum equation using an initial estimate of the pressure field $p^*$.
2. Compute the pressure correction $p'$ through the discretized continuity equation.
3. Correct the face velocities and the pressure field ($p = p^* + \alpha_p p'$).
4. Solve the energy ($T$) and species ($C_A$) transport equations.
5. Select and justify solver tolerances for the quantities of interest. Check residuals alongside mass and energy balances, and compare solutions across mesh and time-step refinements. Numerical convergence alone does not establish that the physical model or boundary conditions are valid.
### Measured velocity fields
For particle-image velocimetry, record image pairing, time separation, spatial calibration, coordinate conventions and preprocessing. Preserve flags for rejected vectors when replacing or interpolating them, and report validation and uncertainty before deriving flow statistics. A measured velocity field and a CFD prediction are distinct evidence; agreement requires a matched comparison and does not by itself validate every model condition. This guidance was informed by the [upstream PIV workflow](https://github.com/K-Dense-AI/scientific-agent-skills/blob/5f2bc0341354fd5590aef76a533ada1e1e5b4e62/skills/openpiv/SKILL.md).