🌬️Heat and Mass Transport Unit 1 – Transport Phenomena Fundamentals

Key Concepts and Definitions

• Transport phenomena encompasses the study of momentum, heat, and mass transfer in systems
• Momentum transfer involves the transport of velocity and forces within a fluid or between a fluid and a solid surface
• Heat transfer refers to the transport of thermal energy from one location to another due to temperature differences
• Mass transfer deals with the transport of chemical species within a system or across system boundaries
• Fick's law describes diffusive mass transfer and states that the diffusive flux is proportional to the negative gradient of the concentration
  • Mathematically expressed as $J = -D \frac{\partial C}{\partial x}$, where $J$ is the diffusive flux, $D$ is the diffusion coefficient, and $\frac{\partial C}{\partial x}$ is the concentration gradient
• Fourier's law describes heat conduction and states that the heat flux is proportional to the negative temperature gradient
  • Mathematically expressed as $q = -k \frac{\partial T}{\partial x}$, where $q$ is the heat flux, $k$ is the thermal conductivity, and $\frac{\partial T}{\partial x}$ is the temperature gradient
• Newton's law of viscosity relates the shear stress to the velocity gradient in a fluid
  • Mathematically expressed as $\tau = \mu \frac{\partial u}{\partial y}$, where $\tau$ is the shear stress, $\mu$ is the dynamic viscosity, and $\frac{\partial u}{\partial y}$ is the velocity gradient

Transport Phenomena Basics

• Transport phenomena can be described using differential equations that relate the rate of change of a quantity to its spatial and temporal variations
• The continuity equation expresses the conservation of mass in a system
  • For an incompressible fluid, it is given by $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$, where $u$, $v$, and $w$ are velocity components in the $x$, $y$, and $z$ directions, respectively
• The momentum equation, derived from Newton's second law, describes the motion of fluids under the influence of forces
  • The Navier-Stokes equations are a set of partial differential equations that describe the conservation of momentum in a fluid
• The energy equation, based on the first law of thermodynamics, describes the conservation of energy in a system
  • It accounts for heat conduction, convection, and generation within the system
• Dimensionless numbers are used to characterize the relative importance of different transport mechanisms in a system
  • The Reynolds number ($Re$) represents the ratio of inertial forces to viscous forces in a fluid
  • The Prandtl number ($Pr$) is the ratio of momentum diffusivity to thermal diffusivity
  • The Schmidt number ($Sc$) is the ratio of momentum diffusivity to mass diffusivity
• Similarity analysis involves the use of dimensionless numbers to simplify complex transport problems and identify similar behavior across different systems

Conservation Laws and Equations

• Conservation laws form the foundation of transport phenomena and describe the balance of quantities such as mass, momentum, and energy in a system
• The conservation of mass equation states that the rate of change of mass within a control volume is equal to the net mass flux across its boundaries
  • For a steady-state, incompressible flow, it simplifies to $\nabla \cdot \vec{v} = 0$, where $\vec{v}$ is the velocity vector
• The conservation of momentum equation, also known as the Navier-Stokes equations, describes the balance of forces acting on a fluid element
  • For an incompressible, Newtonian fluid with constant properties, it is given by $\rho \frac{D\vec{v}}{Dt} = -\nabla p + \mu \nabla^2 \vec{v} + \rho \vec{g}$, where $\rho$ is the fluid density, $p$ is the pressure, $\mu$ is the dynamic viscosity, and $\vec{g}$ is the gravitational acceleration
• The conservation of energy equation describes the balance of energy within a system, accounting for heat transfer, work, and energy storage
  • For a system with constant properties and no internal heat generation, it is given by $\rho c_p \frac{DT}{Dt} = k \nabla^2 T$, where $c_p$ is the specific heat capacity and $k$ is the thermal conductivity
• The conservation of species equation describes the balance of chemical species within a system, considering diffusion, convection, and chemical reactions
  • For a binary system with constant properties and no chemical reactions, it is given by $\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = D \nabla^2 C$, where $C$ is the concentration and $D$ is the diffusion coefficient
• These conservation equations are often coupled and must be solved simultaneously to fully describe the transport phenomena in a system

Fluid Dynamics and Flow Regimes

• Fluid dynamics deals with the motion and behavior of fluids under various conditions
• Laminar flow occurs when fluid particles move in parallel layers without mixing
  • Characterized by low Reynolds numbers ($Re < 2300$ for pipe flow)
  • Velocity profile is parabolic in fully developed laminar flow
• Turbulent flow is characterized by chaotic and irregular motion of fluid particles
  • Occurs at high Reynolds numbers ($Re > 4000$ for pipe flow)
  • Velocity profile is flatter compared to laminar flow due to enhanced mixing
• Transitional flow exists between laminar and turbulent regimes, exhibiting characteristics of both
• The Hagen-Poiseuille equation describes the pressure drop in fully developed laminar flow through a circular pipe
  • $\Delta p = \frac{8 \mu L Q}{\pi R^4}$, where $\Delta p$ is the pressure drop, $L$ is the pipe length, $Q$ is the volumetric flow rate, and $R$ is the pipe radius
• The Darcy-Weisbach equation relates the pressure drop to the flow velocity in a pipe
  • $\Delta p = f \frac{L}{D} \frac{\rho v^2}{2}$, where $f$ is the Darcy friction factor, $D$ is the pipe diameter, and $v$ is the average flow velocity
• The Moody diagram is used to determine the friction factor based on the Reynolds number and the relative pipe roughness
• Boundary layers develop when a fluid flows over a surface, with velocity varying from zero at the surface to the free-stream velocity away from the surface
  • The thickness of the boundary layer increases in the flow direction

Heat Transfer Mechanisms

• Heat transfer occurs through three primary mechanisms: conduction, convection, and radiation
• Conduction is the transfer of heat through a material by molecular interactions
  • Governed by Fourier's law, which states that the heat flux is proportional to the negative temperature gradient
  • The thermal conductivity ($k$) is a material property that quantifies the ability to conduct heat
• Convection involves the transfer of heat between a surface and a moving fluid
  • Governed by Newton's law of cooling, which states that the heat flux is proportional to the temperature difference between the surface and the fluid
  • The convective heat transfer coefficient ($h$) depends on factors such as fluid properties, flow velocity, and surface geometry
• Radiation is the transfer of heat through electromagnetic waves
  • Governed by the Stefan-Boltzmann law, which relates the radiant heat flux to the fourth power of the absolute temperature
  • The emissivity ($\varepsilon$) is a surface property that quantifies the ability to emit and absorb radiation
• The overall heat transfer in a system often involves a combination of these mechanisms
  • The thermal resistance concept is used to analyze heat transfer through multiple layers or mechanisms in series
  • The Biot number ($Bi$) is a dimensionless parameter that compares the relative importance of conduction and convection in a system
• Heat exchangers are devices that facilitate the transfer of heat between two fluids without direct contact
  • The log-mean temperature difference (LMTD) method is used to analyze the performance of heat exchangers

Mass Transfer Principles

• Mass transfer involves the transport of chemical species within a system or across system boundaries
• Diffusion is the movement of species from regions of high concentration to regions of low concentration
  • Governed by Fick's first law, which states that the diffusive flux is proportional to the negative concentration gradient
  • The diffusion coefficient ($D$) is a measure of the ease with which a species can diffuse through a medium
• Convective mass transfer occurs when species are transported by the bulk motion of a fluid
  • Analogous to convective heat transfer, with the mass transfer coefficient ($k_c$) relating the mass flux to the concentration difference
• The Sherwood number ($Sh$) is a dimensionless parameter that characterizes the ratio of convective mass transfer to diffusive mass transfer
• The mass transfer Biot number ($Bi_m$) compares the relative importance of external and internal mass transfer resistances
• The effectiveness factor ($\eta$) is used to quantify the effect of internal mass transfer limitations on the overall reaction rate in porous catalysts
• Mass transfer can be enhanced through techniques such as agitation, turbulence, and the use of packed beds or membranes
• The analogy between heat and mass transfer allows the use of similar equations and dimensionless numbers in both fields

Boundary Layer Theory

• Boundary layer theory describes the behavior of fluids near solid surfaces, where viscous effects are significant
• The velocity boundary layer is the region near a surface where the fluid velocity varies from zero at the surface to the free-stream velocity
  • The thickness of the velocity boundary layer ($\delta$) is defined as the distance from the surface where the velocity reaches 99% of the free-stream velocity
• The thermal boundary layer develops when there is a temperature difference between the surface and the fluid
  • The thermal boundary layer thickness ($\delta_t$) is the distance from the surface where the temperature difference reaches 99% of the free-stream temperature difference
• The concentration boundary layer forms when there is a concentration difference between the surface and the fluid
  • The concentration boundary layer thickness ($\delta_c$) is the distance from the surface where the concentration difference reaches 99% of the free-stream concentration difference
• The Prandtl number ($Pr$) relates the relative thicknesses of the velocity and thermal boundary layers
  • For $Pr > 1$, the thermal boundary layer is thinner than the velocity boundary layer
  • For $Pr < 1$, the thermal boundary layer is thicker than the velocity boundary layer
• The Schmidt number ($Sc$) relates the relative thicknesses of the velocity and concentration boundary layers
• Boundary layer equations, derived from the Navier-Stokes equations, are used to analyze the flow and transport within the boundary layer
  • The Blasius solution provides the velocity profile for laminar flow over a flat plate
• Boundary layer separation occurs when the fluid flow detaches from the surface, leading to the formation of wakes and vortices

Applications and Real-World Examples

• Heat exchangers are widely used in various industries to transfer heat between fluids
  • Shell-and-tube heat exchangers are common in chemical processing and power generation
  • Plate heat exchangers are used in food processing and HVAC systems
• Cooling towers employ the principles of heat and mass transfer to reject heat from process fluids to the atmosphere
  • Used in power plants, chemical plants, and air conditioning systems
• Catalytic converters in automobiles rely on mass transfer and chemical reactions to reduce pollutant emissions
  • The porous catalyst substrate enhances the surface area for mass transfer and reaction
• Packed bed reactors are used in chemical processing to facilitate mass transfer and chemical reactions between fluids and solid catalysts
  • The design of packed beds considers factors such as particle size, bed porosity, and fluid flow distribution
• Membrane separation processes, such as reverse osmosis and ultrafiltration, utilize mass transfer principles to selectively remove components from a fluid
  • Used in water treatment, food processing, and biotechnology
• Heat pipes are passive heat transfer devices that use phase change and capillary action to efficiently transfer heat
  • Used in electronics cooling, solar thermal systems, and aerospace applications
• Microfluidic devices employ transport phenomena at small scales for applications in biomedical research, drug discovery, and lab-on-a-chip systems
  • The high surface-to-volume ratios in microfluidic channels enhance heat and mass transfer
• Convective drying is used in various industries to remove moisture from solids
  • Examples include food drying, paper production, and pharmaceutical manufacturing