# Learn Transport Phenomena with Balance Equations and Numerical Solutions: A Review of Plawsky's Book

## Transport Phenomena Fundamentals: A Comprehensive Guide

Transport phenomena are the processes that describe how matter, energy, and information move from one place to another. They are essential for understanding and designing many engineering systems, such as heat exchangers, reactors, separators, membranes, microfluidics, and biomedical devices. In this article, we will introduce the basic concepts and principles of transport phenomena, the main topics and methods of study, and some useful resources for further learning.

## transport phenomena fundamentals plawsky pdf 31

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## What are transport phenomena?

Transport phenomena can be defined as the study of the rates of transfer of mass, energy, momentum, and charge in physical and chemical systems. They are often classified into three types:

Heat transfer: the movement of thermal energy due to a temperature difference.

Mass transfer: the movement of chemical species due to a concentration difference.

Momentum transfer: the movement of fluid particles due to a pressure or velocity difference.

Charge transfer is a special case of mass transfer that involves electrically charged particles or ions. Transport phenomena can occur in different phases (solid, liquid, gas) or across phase boundaries (solid-liquid, liquid-gas, etc.). They can also be influenced by external forces (gravity, electric field, magnetic field) or interactions (chemical reactions, phase changes).

Some examples of transport phenomena are:

Conduction: heat transfer through a solid or a stationary fluid by molecular collisions.

Convection: heat or mass transfer by the bulk motion of a fluid.

Radiation: heat transfer by electromagnetic waves.

Diffusion: mass transfer by random molecular motion.

Osmosis: mass transfer across a semipermeable membrane due to a solvent concentration difference.

Viscosity: momentum transfer by internal friction in a fluid.

Laminar flow: fluid flow with smooth and orderly streamlines.

Turbulent flow: fluid flow with chaotic and irregular fluctuations.

Transport phenomena are important for many engineering applications, such as:

Thermal engineering: design and optimization of heat engines, refrigerators, heat pumps, boilers, furnaces, etc.

Chemical engineering: design and optimization of reactors, separators, distillation columns, absorption towers, etc.

Bioengineering: design and optimization of artificial organs, drug delivery systems, tissue engineering scaffolds, etc.

Nanotechnology: design and optimization of nanoscale devices and materials with enhanced thermal, electrical, optical, or mechanical properties.

## How to study transport phenomena?

The study of transport phenomena requires a combination of mathematical modeling, physical reasoning, and experimental validation. A common approach is to use the balance equation method, which consists of four steps:

Modeling: identify the system of interest and define a control volume (a fixed or moving region in space) that encloses it.

Differential equations: derive the governing differential equations that describe how the transported quantity (mass, energy, momentum, charge) changes with time and space within the control volume.

Boundary conditions: specify the values or relations of the transported quantity at the boundaries of the control volume or at any internal surfaces.

Solution: solve the differential equations with the boundary conditions to obtain the distribution of the transported quantity in time and space.

The governing differential equations are usually based on conservation laws (mass conservation, energy conservation, momentum conservation) or constitutive relations (Fourier's law for heat conduction, Fick's law for diffusion, Newton's law for viscosity). They can be ordinary differential equations (ODEs) if they involve only one independent variable (usually time) or partial differential equations (PDEs) if they involve more than one independent variable (usually time and space).

The solution of the differential equations can be analytical or numerical. Analytical solutions are exact expressions that can be obtained by using mathematical techniques such as separation of variables, similarity transformations, or Laplace transforms. However, analytical solutions are only possible for simple geometries, boundary conditions, and transport mechanisms. Numerical solutions are approximate values that can be obtained by using computational methods such as finite difference, finite element, or finite volume. Numerical solutions are more flexible and can handle complex situations, but they require more computational resources and may introduce errors or instabilities.

## What are the main topics in transport phenomena?

The main topics in transport phenomena can be divided into two parts: diffusive transport and convective transport. Diffusive transport deals with the transfer of mass, energy, momentum, or charge due to molecular motion or random fluctuations. Convective transport deals with the transfer of mass, energy, momentum, or charge due to fluid motion or external forces. We will briefly introduce each topic below.

### Diffusive transport of momentum, energy, mass, and charge

The diffusive transport of momentum, energy, mass, and charge is governed by similar differential equations that have the form: $$\frac\partial \phi\partial t + \nabla \cdot \mathbfJ = S$$ where $\phi$ is the transported quantity per unit volume (density, internal energy, concentration, charge density), $\mathbfJ$ is the flux vector (momentum flux, heat flux, mass flux, current density), and $S$ is the source term (external force per unit volume, heat generation per unit volume, reaction rate per unit volume, charge generation per unit volume). The flux vector $\mathbfJ$ is usually proportional to the gradient of $\phi$, $$\mathbfJ = -D \nabla \phi$$ where $D$ is the diffusivity (kinematic viscosity, thermal conductivity, diffusion coefficient, electrical conductivity). The diffusivity $D$ depends on the properties of the medium and the transported quantity. The source term $S$ depends on the external conditions and the interactions within the system. The solution of these equations gives us information about how $\phi$ varies with time and space. Some examples of diffusive transport are:

Poiseuille flow: laminar flow of an incompressible fluid through a circular pipe driven by a pressure gradient. The momentum balance equation gives us the velocity profile and pressure drop along the pipe.

Fouling: accumulation of unwanted material on solid surfaces due to mass transfer from a fluid. The mass balance equation gives us the concentration profile and fouling rate on the surface.

Joule heating: generation of heat due to electric current passing through a conductor. The energy balance equation gives us the temperature profile and heat generation rate in the conductor.

### Convective transport and fluid mechanics

### The convective transport and fluid mechanics deal with the transfer of mass, energy, momentum, or charge due to fluid motion or external forces. They are governed by coupled differential equations that have different forms depending on whether the fluid is compressible or incompressible, Transport coefficients and boundary layer theory

The transport coefficients are the proportionality constants that relate the flux vector to the gradient of the transported quantity. They measure how easily a fluid can transfer mass, energy, momentum, or charge. They depend on the properties of the fluid, such as viscosity, thermal conductivity, diffusion coefficient, and electrical conductivity. They can also vary with temperature, pressure, or composition.

The boundary layer theory is a simplified analysis of the fluid flow near a solid surface, where the effects of viscosity are significant. It assumes that the fluid flow can be divided into two regions: a thin boundary layer near the surface, where the velocity changes rapidly from zero to a free stream value; and an outer region away from the surface, where the velocity is nearly constant and inviscid. The boundary layer theory allows us to calculate the transport coefficients and the drag force on the surface.

Some examples of transport coefficients and boundary layer theory are:

Nusselt number: dimensionless ratio of convective to conductive heat transfer across a boundary layer.

Sherwood number: dimensionless ratio of convective to diffusive mass transfer across a boundary layer.

Reynolds number: dimensionless ratio of inertial to viscous forces in a fluid flow.

Prandtl number: dimensionless ratio of momentum diffusivity to thermal diffusivity in a fluid.

Schmidt number: dimensionless ratio of momentum diffusivity to mass diffusivity in a fluid.

### Radiative heat transfer

Radiative heat transfer is the transfer of thermal energy by electromagnetic waves. It does not require a medium to propagate and can occur in vacuum. It depends on the temperature and emissivity of the radiating body, as well as the geometry and view factors of the system.

Some concepts and equations related to radiative heat transfer are:

Blackbody: an idealized body that absorbs all incident radiation and emits radiation according to Planck's law.

Graybody: a real body that absorbs and emits radiation according to its emissivity, which is constant and independent of wavelength.

Stefan-Boltzmann law: the total power emitted per unit area by a blackbody is proportional to the fourth power of its absolute temperature: (14.S.18) P = Ïƒ T 4 where Ïƒ is the Stefan-Boltzmann constant.

Wien's displacement law: the wavelength at which a blackbody emits maximum radiation is inversely proportional to its absolute temperature: (14.S.19) Î» m a x = b T where b is Wien's displacement constant.

Radiation shield: a thin layer of material that reflects or absorbs radiation and reduces the heat transfer between two bodies.

Radiation enclosure: a closed surface that completely surrounds one or more radiating bodies and exchanges radiation with them.

## Where to find more resources on transport phenomena?

If you are interested in learning more about transport phenomena, there are many resources available for you. Here are some suggestions:

### Transport Phenomena Fundamentals by Joel L. Plawsky

This is a textbook that covers the basic concepts and principles of transport phenomena in a clear and concise way. It uses a balance equation approach to derive the governing differential equations for heat, mass, and momentum transfer. It also introduces topics such as convective transport, transport coefficients, boundary layer theory, and radiative heat transfer. It includes many worked examples, problems, and exercises that can help you practice your skills and test your understanding. You can find this book online at .

### Other books and online courses

There are many other books and online courses that can teach you different aspects of transport phenomena. Some examples are:

Bird, Stewart, and Lightfoot: Transport Phenomena: This is a classic textbook that provides a comprehensive and rigorous treatment of transport phenomena. It covers topics such as molecular theory, dimensional analysis, similarity solutions, macroscopic balances, shell balances, turbulence, interfacial phenomena, irreversible thermodynamics, and more. It is suitable for advanced undergraduate or graduate students who have a strong background in mathematics and physics. You can find this book online at .

Cussler: Diffusion: Mass Transfer in Fluid Systems: This is a textbook that focuses on mass transfer in fluid systems. It covers topics such as diffusion mechanisms, diffusion coefficients, multicomponent diffusion, convective mass transfer, mass transfer with chemical reaction, mass transfer in porous media, membrane separation processes, and more. It is suitable for undergraduate or graduate students who have some knowledge of fluid mechanics and thermodynamics. You can find this book online at .

Incropera et al.: Fundamentals of Heat and Mass Transfer: This is a textbook that covers the fundamentals of heat and mass transfer in engineering applications. It covers topics such as steady-state conduction, transient conduction, forced convection, natural convection, boiling and condensation, heat exchangers, radiation, diffusion, convection-diffusion, mass transfer equipment, and more. It is suitable for undergraduate or graduate students who have some knowledge of thermodynamics and fluid mechanics. You can find this book online at .

Kundu et al.: Fluid Mechanics: This is a textbook that covers the fundamentals of fluid mechanics in engineering applications. It covers topics such as fluid statics, kinematics, conservation laws, vorticity and circulation, potential flow, viscous flow, boundary layers, compressible flow, turbulent flow, geophysical fluid dynamics, and more. It is suitable for undergraduate or graduate students who have some knowledge of calculus and physics. You can find this book online at .

Coursera: Transport Phenomena for Chemical Engineers Specialization: This is an online course series that teaches you how to apply transport phenomena concepts and principles to chemical engineering problems. It consists of four courses: Momentum Transfer, Heat Transfer, Mass Transfer, and Transport Phenomena Applications. It covers topics such as fluid mechanics, heat conduction and convection, mass diffusion and convection, multicomponent systems, interphase transport, chemical reactors, separation processes, and more. It is suitable for undergraduate or graduate students or professionals who have some background in chemistry and mathematics. You can find this course series online at .

## Conclusion

In this article, we have introduced the basic concepts and principles of transport phenomena, the main topics and methods of study, and some useful resources for further learning. We hope that this article has sparked your interest and curiosity in this fascinating field of science and engineering.

## Frequently Asked Questions

Here are some frequently asked questions about transport phenomena:

What is the difference between heat transfer and mass transfer?

Heat transfer is the movement of thermal energy due to a temperature difference. Mass transfer is the movement of chemical species due to a concentration difference.

What is the difference between laminar flow and turbulent flow?

Laminar flow is fluid flow with smooth and orderly streamlines. Turbulent flow is fluid flow with chaotic and irregular fluctuations.

What is the difference between conduction and convection?

Conduction is heat transfer through a solid or a stationary fluid by molecular collisions. Convection is heat or mass transfer by the bulk motion of a fluid.

What is the difference between diffusion and osmosis?

Diffusion is mass transfer by random molecular motion. Osmosis is mass transfer across a semipermeable membrane due to a solvent concentration difference.

What is the difference between blackbody and graybody?

A blackbody is an idealized body that absorbs all incident radiation and emits radiation according to Planck's law. A graybody is a real body that absorbs and emits radiation according to its emissivity, which is constant and independent of wavelength.