List of Solved Examples
Chapter 1 Examples
1.3.1
Example of a phenomenological model – the Soave-Redlich-Kwong (SRK) equation of state
1.3.3
Example of reactors-in-series
1.3.2
1.4.1
1.4.2
1.4.3
1.4.4
1.4.5
1.5.1
1.5.2
1.5.3
1.6.1
1.6.2
1.6.3
1.6.4
1.6.5
1.6.6
1.6.7
Example of a stage-wise separation process
The concept of the compartmental model
1.4.1.1 One-compartment model for drug metabolism and elimination
1.4.1.2 The two-compartment model
Two-pool urea-kinetic model for hemodialysis
Reactors-in-series
Lumped parameter modelling of a bio-artificial membrane device for controlled insulin release
Liquid drainage from a two-tank assembly
Heat diffusion with generation in a composite cylinder – the shell balance technique
Modeling of biofiltration
Modeling of a differential contactor for extractive fermentation
Unsteady state heat transfer through a rectangular fin
Unsteady state heat conduction in a rectangular solid
Oxygen transport in tissues – the Krogh cylinder
Absorption of a drug through the skin – a combination of lumped and distributed parameter
models
Bioheat transfer – the Pennes equation
Slow release of a fertilizer through a polymer coating
Controlled release of a drug from a drug-eluting stent (DES) to prevent restenosis
Chapter 2 Examples
2.1
Solution of an ordinary differential equation (ODE) – use of integrating factor
2.3
Cooling of a ball by natural convection
2.2
2.4
Solution of a Bernoulli’s equation
Cooling of a metal ball in a well-stirred liquid
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
Modeling of two continuous stirred tank reactors (CSTRs) in series – variable volume, first order
reaction
Two CSTRs in series – first order reaction
Mathematical modeling of cancer stem cell hypothesis
Drug administration at a regular interval – one-compartment model
Solution of a linear second order non-homogeneous ODE
Determination of the particular integral of an ODE – variation of parameters method
Heat flow in a rod connecting two walls
Temperature distribution in a current-carrying conductor
Heat conduction with generation in a composite wall
Microwave heating of a slab
A simple theoretical analysis of solute transport in a hemodialyzer
Two-pool urea kinetic model for hemodialysis
Effectiveness factor of a spherical catalyst pellet
Mathematical modeling of oxygen-limited growth of follicles
Biofiltration of a volatile organic compound (VOC)
Oxygen transport in tissues – the Krogh cylinder
Axial dispersion in a tubular reactor
Diffusion with reaction in a membrane
Diffusion and metabolic consumption of oxygen absorption in a small organism
2.24
Gas absorption with chemical reaction in a slurry of fine catalyst particles
2.26
Theoretical analysis of an extractive fermenter with axial dispersion
2.25
2.27
2.28
2.29
2.30
Facilitated transport through a membrane – instantaneous reaction with the carrier
Diffusion-controlled sensitivity of a semiconductor gas sensor
Modeling of solidification of a liquid at its freezing point – pseudo-steady state analysis of a
moving boundary problem
Slow release of a fertilizer – pseudo-steady state approximation
Diffusion-reaction in a liquid drop – pseudo-steady state analysis of a moving boundary problem
2.31
Modeling of a diffusion-limited hollow fiber membrane reactor
2.33
Pennes bioheat equation and burn depth
2.32
2.34
2.35
2.36
2.37
2.38
2.39
2.40
2.41
2.42
2.43
2.44
2.45
Temperature distribution in a layer of tissue under microwave heating
Mathematical modeling of wound healing – the critical size defect
Diffusion controlled growth of a solid tumor
Neutron flux distribution in a slab
Neutron flux distribution in a sphere
Inversion of L -transforms
Solution of a linear, second order non-homogeneous ODE using the L -transform technique
Radio-active decay of a species
Solution of a set of ODEs using the L -transform technique
Liquid flow from a two-tank assembly
Indoor air pollution due to release of VOCs from a washing machine
Two CSTRs in series – the matrix method of solution
Multi-compartment solute kinetics in hemodialysis – matrix solution
Chapter 3 Examples
3.1
Gamma, beta, and error functions – evaluation and identities
3.2
Similarity solution of heating of a solid
3.4
Series solution of an ODE around an ordinary point
3.3
3.5
3.6
Regular singularity
Series solution around an ordinary point
Frobenius case (i) – the roots of the indicial equation are real and distinct and do not differ by an
integer
3.7
Frobenius case (ii) – roots of the indicial equation are equal
3.9
Proof of an integral relation of the Bessel function
3.8
3.10
Frobenius case (iii) – roots of the indicial equation are equal
Expansion of an arbitrary function in terms of orthogonal functions
3.11
Expansion an arbitrary function in terms of a set of Bessel functions
3.12
Heat transfer from a radial or circular fin of uniform thickness
3.14
Bio-heat transfer in a tumor
3.13
3.15
3.16
3.17
3.18
3.19
Modeling of microwave heating of a slab
An example involving the confluent hypergeometric function
Modeling of the diffusion-reaction problem in a hollow fiber bio-reactor
Theoretical analysis of a parallel plate countercurrent dialyser with a flat membrane
Oxygen transport to plant roots
Mathematical modeling of healing of a disk-shaped bone wound
Chapter 4 Examples
4.1
4.2
4.3
Thermal transient in a large plane wall – the same constant temperature at the two
surfaces
Unsteady state temperature distribution in a large plane wall for given non-uniform initial
temperature
Unsteady heat conduction in a large plane wall with convective boundary condition at one
surface
4.4
Unsteady heat conduction in a large wall with different surface temperatures
4.6
Dynamics of an enzyme electrode – non-homogeneous boundary condition
4.8
Unsteady state heat conduction in a “long” cylinder – constant surface temperature, uniform
initial temperature
4.5
4.7
4.9
Method of eigenfunction expansion applied to a non-homogeneous partial differential equation
(PDE)
Steady state temperature distribution in a rectangular block
Thermal spreading in a semiconductor element
4.10
Unsteady state heat conduction in a sphere – convective heat transfer at the surface
4.12
Steady state concentration profile in a laminar flow tubular reactor for a first order reaction
4.11
4.13
4.14
Steady state reaction-diffusion in a short cylindrical catalyst pellet – the effectiveness factor
Unsteady state diffusional transport in a two-layer composite wall
Odor removal from an animal house
4.15
Indoor air quality – VOC emission from the walls or floor of a room
4.17
Bio-heat transfer – three-dimensional steady state temperature distribution in a burn
4.16
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
4.35
Bioheat transfer – unsteady state tissue temperature distribution for constant heat flux at the
skin
Slow release of potassium from a polymer-coated capsule
Cooling of a sphere in a limited volume of well-stirred liquid
Drug release from a cylindrical DES
Percutaneous drug absorption – a finite dose quickly absorbed by the skin followed by diffusion
Unsteady heat conduction in a semi-infinite solid – similarity solution
Unsteady heat conduction to a semi-infinite solid with a given constant surface heat flux
Higbie’s penetration theory – an example of similarity solution
Cooling of a stretching sheet in viscous flow
Freezing of a large pool of water – a moving boundary problem
Gas absorption with an instantaneous chemical reaction
Dissolution of a sphere in a large volume of stagnant liquid – a moving boundary problem
Evaporation of a droplet – analytical solution of a moving boundary problem in terms of
confluent hypergeometric functions
Solution of a non-homogeneous problem by superposition
Unsteady state heating of a sphere with a linearly time-varying surface temperature
Diffusion accompanied by a first order chemical reaction in a stagnant liquid
Modeling of CO2-diffusion in a leaf for photosynthesis
Unsteady state gas absorption with an accompanying first order reaction in a liquid pool of finite
depth
Mathematical modeling of a semiconductor gas sensor
4.36
Determination of the Green’s function for a boundary value problem (BVP)
4.38
Steady state 2-D heat diffusion with generation
4.37
Modeling of generation and decay of the growth inhibitor in a layer of tissue – use of the
Green’s functions
4.39
4.40
Bio-heat diffusion in a layer of tissue with time-dependent surface boundary condition – Green’s
function solution
Green’s function for unsteady heat diffusion in a long cylinder with time-dependent boundary
condition
Chapter 5 Examples
5.1
A proof on kernel function
5.3
Unsteady diffusion in a deep pool of liquid – use of Fourier sine transform
5.2
5.4
5.5
Determination of a few Fourier transforms and inverse
Unsteady state heat diffusion in a slab – use of finite Fourier sine transform
Determination of Laplace transforms and inverses
5.6
Controlled release from a drug-filled sphere
5.8
Diffusion of NO2 in a thin film of lead phthalocyanine with rate-governed surface concentration
5.7
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
Cooling of a slab in contact with a well-stirred liquid
Source-sink behavior in diffusion of VOC from a vinyl flooring in a room under imperfect
ventilation – Dirichlet boundary condition at the top surface
Plane wall transient – Robin boundary condition
Controlled release from a spherical capsule with simultaneous absorption at the wall of the
container
Convection and diffusion of drug from a stent into the arterial wall – inversion of an L transform with branch points
Diffusion in a bi-layer composite
Atmospheric dispersion of a contaminant instantaneously released at a point in a uniform flow
field
Dispersion and decay of a contaminant released from a continuous plane source in a flowing
stream
Concentration distribution in a plug flow tubular reactor with axial dispersion during start-up
Chapter 6 Examples
6.1
Determination of the roots of an algebraic equation
6.3
Regular perturbation solution of a second order ODE
6.2
Regular perturbation solution of a first order ODE
6.4
6.5
6.6
6.7
6.8
Non-isothermal first order reaction at steady state in a flat catalyst pellet – a regular
perturbation solution
Regular perturbation solution of the moving boundary problem of freezing of a deep pool of
water at the freezing point
Solution of a cubic equation using singular perturbation
Singular perturbation solution of a second order ODE, matched asymptotic expansion
Singular perturbation solution of simple enzyme-catalyzed reactions