Key Concepts in Transport Phenomena

Why This Matters

Transport phenomena sits at the heart of chemical engineering—it's the unified framework explaining how mass, momentum, and energy move through systems. Every reactor you design, every separation unit you optimize, and every heat exchanger you size depends on your ability to predict and control these three fundamental transport processes. You're being tested not just on equations, but on understanding why certain driving forces cause certain flows and how these principles connect across seemingly different applications.

The beauty of transport phenomena is its elegant analogy structure: diffusion of mass, conduction of heat, and viscous momentum transfer all follow mathematically similar forms. Master one, and you've got a head start on the others. Don't just memorize Fick's law or Fourier's law in isolation—know that they're parallel expressions of the same underlying physics. When you see an exam question about heat transfer, ask yourself: what's the mass transfer analog? That comparative thinking is what separates strong students from struggling ones.

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The Three Conservation Laws

These foundational principles govern every transport process. Nothing is created or destroyed—only transferred, transformed, or accumulated. Your ability to write correct balances depends on internalizing these laws.

Conservation of Mass

• Mass cannot be created or destroyed in a closed system—this is your starting point for every material balance
• Accumulation equals input minus output plus generation minus consumption; master this equation form for reactors and separators
• Flow rate calculations depend entirely on this principle—if mass isn't balancing, your process design is wrong

Conservation of Momentum

• Momentum is conserved in the absence of external forces—this governs how fluids accelerate, decelerate, and change direction
• Force balances on fluid elements derive directly from this law, enabling prediction of pressure drops and velocity profiles
• Leads to the Navier-Stokes equations, the governing equations for virtually all fluid flow problems

Conservation of Energy

• Energy transforms but never disappears—kinetic, potential, thermal, and work interactions must all balance
• First law of thermodynamics applied to flowing systems gives you the energy balance equation for process design
• Heat and work interactions in reactors, compressors, and turbines all require rigorous energy accounting

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Constitutive Laws: The Driving Force Framework

These laws describe how fast transport occurs in response to gradients. Each follows the pattern: flux = −(property) × (gradient). The negative sign indicates transport occurs down the gradient.

Fick's Law of Diffusion

• Mass flux is proportional to concentration gradient: $J_A = -D_{AB} \frac{dC_A}{dx}$ where $D_{AB}$ is the diffusion coefficient
• Diffusivity $D_{AB}$ depends on temperature, pressure, and molecular properties—gases diffuse faster than liquids
• Foundation for separation processes like absorption, distillation, and membrane separations where concentration differences drive mass movement

Fourier's Law of Heat Conduction

• Heat flux is proportional to temperature gradient: $q = -k \frac{dT}{dx}$ where $k$ is thermal conductivity
• Thermal conductivity varies dramatically—metals conduct well, insulators don't; this drives material selection
• Essential for insulation design and understanding heat loss through walls, pipes, and equipment shells

Newton's Law of Viscosity

• Shear stress is proportional to velocity gradient: $\tau = -\mu \frac{dv}{dy}$ where $\mu$ is dynamic viscosity
• Newtonian fluids follow this linear relationship; non-Newtonian fluids (polymers, slurries) require modified models
• Viscosity determines pumping costs and flow behavior—critical for equipment sizing and process economics

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Convective Transport Mechanisms

When fluids move, they carry mass and energy with them. Convection combines bulk fluid motion with molecular transport, and it's typically much faster than pure diffusion or conduction.

Convective Heat Transfer

• Heat moves between surfaces and flowing fluids according to $q = hA(T_s - T_\infty)$ where $h$ is the heat transfer coefficient
• Coefficient $h$ depends on flow conditions—turbulent flow gives higher $h$ than laminar flow due to enhanced mixing
• Dominates in heat exchangers, cooling towers, and any system where fluid motion contacts heated or cooled surfaces

Convective Mass Transfer

• Mass transfers between phases or surfaces via $N_A = k_c A (C_{As} - C_{A\infty})$ where $k_c$ is the mass transfer coefficient
• Fluid velocity dramatically affects rates—faster flow means thinner boundary layers and faster transfer
• Controls absorption, stripping, and evaporation processes where components move between gas and liquid phases

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Transfer Coefficients and Their Prediction

These coefficients quantify how fast convective transport occurs. They're not fundamental properties—they depend on geometry, flow conditions, and fluid properties.

Heat Transfer Coefficients

• Defined as $h = q/(A \Delta T)$ with units of $\text{W/m}^2\text{·K}$—higher values mean more efficient heat transfer
• Correlations predict $h$ from dimensionless numbers; you'll use Nusselt number correlations extensively
• Limiting factor in heat exchanger design—the side with lower $h$ controls overall performance

Mass Transfer Coefficients

• Defined as $k_c = N_A/(A \Delta C)$ with units of $\text{m/s}$—quantifies how fast mass crosses an interface
• Sherwood number correlations predict $k_c$ analogously to how Nusselt correlations predict $h$
• Critical for absorber and distillation column sizing—determines required contact area and column height

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Dimensionless Numbers and Similarity

Dimensionless groups let you characterize transport behavior independent of scale. They're essential for correlations, scale-up, and understanding which physical effects dominate.

Dimensionless Numbers (Reynolds, Prandtl, Schmidt)

• Reynolds number $Re = \frac{\rho v L}{\mu}$ compares inertial to viscous forces—determines laminar vs. turbulent flow transition
• Prandtl number $Pr = \frac{\mu c_p}{k}$ compares momentum to thermal diffusivity—characterizes heat transfer behavior
• Schmidt number $Sc = \frac{\mu}{\rho D_{AB}}$ compares momentum to mass diffusivity—characterizes mass transfer behavior

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Governing Equations and Analysis Methods

These mathematical frameworks let you solve real transport problems by combining conservation laws with constitutive equations.

Navier-Stokes Equations

• Nonlinear PDEs governing fluid motion—derived from momentum conservation applied to a differential fluid element
• Include pressure, viscous, and body force terms: $\rho \frac{D\mathbf{v}}{Dt} = -\nabla P + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}$
• Exact solutions exist only for simple geometries—pipe flow, flow between plates, flow around spheres

Shell Balance Method

• Systematic approach for deriving differential equations: define a thin shell, write balances, take the limit as shell thickness approaches zero
• Produces velocity, temperature, or concentration profiles for laminar flow in simple geometries
• Builds physical intuition—you see exactly which terms represent which physical effects

Boundary Layer Theory

• Thin region near surfaces where velocity changes from zero (no-slip) to free-stream value
• Boundary layer thickness $\delta$ grows along the surface; thinner layers mean higher transfer rates
• Explains drag, heat transfer enhancement, and flow separation—crucial for external flow applications

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Design Applications

Transport phenomena principles come together in equipment design. This is where theory meets practice.

Heat Exchanger Design

• LMTD method uses log-mean temperature difference: $Q = UA \cdot \Delta T_{lm}$ where $U$ is overall heat transfer coefficient
• Overall coefficient $U$ combines resistances: $\frac{1}{U} = \frac{1}{h_i} + \frac{\Delta x}{k} + \frac{1}{h_o}$ plus fouling factors
• Flow arrangement matters—counterflow gives higher $\Delta T_{lm}$ than parallel flow for same inlet/outlet temperatures

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Quick Reference Table

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Self-Check Questions

1. Fick's law, Fourier's law, and Newton's law of viscosity all share a common mathematical form. What is it, and what role does the "transport property" play in each?

2. Which two dimensionless numbers compare momentum diffusivity to another diffusivity? What type of transport does each characterize?

3. Compare and contrast the heat transfer coefficient $h$ and mass transfer coefficient $k_c$: How are they similar in their role? How do their units differ?

4. If you're designing a heat exchanger and one fluid has a much lower heat transfer coefficient than the other, which side limits performance? How would you improve the design?

5. An FRQ gives you a pipe with fluid flowing through it and asks for the velocity profile. Which analysis method would you use, and what conservation principle does it apply?