Multiphase reactors are crucial in chemical engineering, involving interactions between different phases like gas-liquid or gas-solid systems. These reactors use various models to describe mass transfer, reaction kinetics, and heat transfer, considering factors like film theory and surface renewal.

Designing and optimizing multiphase reactors requires understanding phase equilibria, interfacial phenomena, and mixing characteristics. Engineers must balance conversion, energy consumption, and costs while considering reactor performance in bubble columns, packed columns, and stirred tanks. Advanced simulation tools and experimental techniques aid in this process.

Multiphase Reactor Modeling and Analysis

Models for multiphase reactors

• Gas-liquid systems
  • Two-film theory assumes stagnant films on both sides of the gas-liquid interface with mass transfer resistance
  • Penetration theory considers unsteady-state diffusion of solute into a stagnant liquid phase (falling film reactors)
  • Surface renewal theory assumes continuous replacement of liquid elements at the gas-liquid interface (bubble columns, stirred tanks)
• Gas-solid systems
  • Shrinking core model describes reaction and diffusion in porous solid particles with a sharp interface between reacted and unreacted zones (gas-solid reactions, noncatalytic)
  • Grain model considers solid particles as a collection of small grains with reaction and diffusion occurring simultaneously (gas-solid reactions, catalytic)
  • Pore diffusion model accounts for diffusion and reaction in the pores of a catalyst particle (heterogeneous catalysis)
• Liquid-liquid systems
  • Droplet dispersion model describes mass transfer and reaction in dispersed liquid droplets (liquid-liquid extraction, emulsion polymerization)
  • Emulsion polymerization model considers the polymerization reaction in monomer droplets dispersed in an aqueous phase (production of latex, rubber)
• Conservation equations
  • Mass balance equations account for convection, diffusion, and reaction in each phase
  • Energy balance equations consider heat transfer between phases and heat of reaction
  • Momentum balance equations describe fluid flow and mixing in multiphase systems

Analysis of multiphase systems

• Mass transfer correlations
  • Gas-liquid: Higbie penetration theory for short contact times, Dankwerts surface renewal theory for continuous phase renewal (bubble columns, stirred tanks)
  • Gas-solid: Sherwood number $Sh$ relates mass transfer coefficient to particle size and diffusivity, Schmidt number $Sc$ represents the ratio of momentum to mass diffusivity, Reynolds number $Re$ characterizes the flow regime
  • Liquid-liquid: Sauter mean diameter describes the average droplet size, Weber number $We$ relates inertial to surface tension forces (emulsification, dispersion)
• Heat transfer correlations
  • Nusselt number $Nu$ relates heat transfer coefficient to thermal conductivity and characteristic length, Prandtl number $Pr$ represents the ratio of momentum to thermal diffusivity
  • Chilton-Colburn analogy relates heat and mass transfer coefficients based on the similarity between thermal and concentration boundary layers
• Reaction kinetics
  • Intrinsic kinetics describe the true reaction rate without mass transfer limitations
  • Effectiveness factor $\eta$ represents the ratio of actual to intrinsic reaction rate in the presence of diffusion limitations
  • Thiele modulus $\phi$ relates the reaction rate to the diffusion rate in a catalyst particle
• Dimensionless numbers
  • Sherwood number $Sh$: $Sh = \frac{k_c d_p}{D_{AB}}$ relates mass transfer coefficient $k_c$ to particle diameter $d_p$ and diffusivity $D_{AB}$
  • Schmidt number $Sc$: $Sc = \frac{\mu}{\rho D_{AB}}$ represents the ratio of momentum diffusivity $\mu/\rho$ to mass diffusivity $D_{AB}$
  • Reynolds number $Re$: $Re = \frac{\rho u d_p}{\mu}$ characterizes the flow regime based on fluid density $\rho$, velocity $u$, particle diameter $d_p$, and viscosity $\mu$
  • Weber number $We$: $We = \frac{\rho u^2 d_p}{\sigma}$ relates inertial forces $\rho u^2$ to surface tension forces $\sigma/d_p$

Multiphase Reactor Design and Optimization

Design of multiphase reactors

• Phase equilibria
  • Vapor-liquid equilibrium VLE determines the composition and properties of coexisting vapor and liquid phases (distillation, absorption)
  • Liquid-liquid equilibrium LLE describes the distribution of components between two immiscible liquid phases (extraction, separation)
  • Solid-liquid equilibrium SLE governs the solubility of solid components in a liquid phase (crystallization, precipitation)
• Interfacial phenomena
  • Surface tension is the force per unit length acting at the interface between two phases (gas-liquid, liquid-liquid)
  • Wetting and contact angle describe the interaction between a liquid and a solid surface (packed columns, trickle bed reactors)
  • Marangoni effect is the mass transfer along an interface due to surface tension gradients (mass transfer enhancement)
• Mixing characteristics
  • Mixing time is the time required to achieve a desired degree of homogeneity in a reactor (stirred tanks, bubble columns)
  • Circulation time represents the average time for a fluid element to complete one circulation loop in a reactor (loop reactors)
  • Power number $N_p$ relates the power consumption to the impeller dimensions and fluid properties in stirred tanks
• Optimization objectives
  • Maximize conversion or yield to achieve high reactor performance and product quality
  • Minimize energy consumption to reduce operating costs and environmental impact
  • Minimize capital and operating costs to improve the economic viability of the process

Performance of multiphase reactors

• Bubble columns
  • Gas holdup is the volume fraction of gas in the reactor, influencing mass transfer and mixing
  • Bubble size distribution affects the interfacial area and mass transfer rates (coalescence, breakup)
  • Axial dispersion coefficient quantifies the degree of mixing in the liquid phase (back-mixing, plug flow)
• Packed columns
  • Pressure drop is the driving force for fluid flow and affects the operating cost (permeability, particle size)
  • Liquid holdup represents the volume fraction of liquid in the column, influencing residence time and mass transfer
  • Wetting efficiency describes the fraction of packing surface area wetted by the liquid phase (mass transfer, catalyst utilization)
• Stirred tanks
  • Power consumption is the energy input required to maintain the desired mixing and mass transfer conditions (impeller type, size)
  • Mixing time characterizes the time required to achieve a desired degree of homogeneity (turbulence, circulation)
  • Gas-liquid mass transfer coefficient $k_L a$ quantifies the rate of mass transfer between gas and liquid phases (bubble size, interfacial area)
• Simulation tools
  • Computational Fluid Dynamics CFD simulates the detailed flow, mixing, and reaction behavior in multiphase reactors (velocity, concentration, temperature profiles)
  • Process simulators like Aspen Plus and HYSYS provide a framework for modeling and optimizing entire chemical processes, including multiphase reactors (mass and energy balances, unit operations)
• Experimental techniques
  • Particle Image Velocimetry PIV measures the instantaneous velocity fields in a fluid by tracking the motion of seeded particles (flow visualization, turbulence)
  • Laser Doppler Anemometry LDA determines the local fluid velocity by measuring the Doppler shift of laser light scattered by particles (non-intrusive, point measurements)
  • Radioactive particle tracking RPT tracks the motion of radioactive tracer particles to obtain the velocity and mixing patterns in a reactor (Lagrangian approach, opaque systems)