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D.M.Griffin Jr.,PhD, P.E.
 
 Civil Engineering
Louisiana Tech University

MathCad File Distribution Page

 

I am using Mathcad 11 as of 11/2002. So, over time, most of the files here will be saved as version 11 files.  If you download a file and find you can't open it, e-mail me and I will resave it in the requested format or as a PDF file.

Since starting this web site I have become overwhelmed with requests from folks all over the world asking for various types of help with Mathcad, and with problems done in Mathcad. Unfortunately, I can no longer provide this type of assistance to persons other than my students due to time constraints and other responsibilities, sorry...

I have done workshops using Mathcad on a variety of topics for the Louisiana Transporation Research Center, the University here and local consulting firms. If this is of interest...contact me.

                                                           

 
Current attractions - I ran across a book - Numerical Methods for Engineers with Mathcad, by Khyruddin Akbar Ansar. I got it off Amazon.com but it's published by Ulyssian Publications.  It appears to be a collection of Mathcad files put together by the author .
It's a bit  pricey in my opinion ($90.00) but  the author has some interesting material dealing with function approximation of data as well as matrix treatment of linear and nonlinear regression.
 

                    You can access various problem topics by clicking on the following targets:

Course Materials for Contaminant Transport (CE 531)
Notes, Derivations, Discussions for Contaminant Transport
Air Pollution
Chemistry
Probability and Statistics
Civil Engineering
Misc. Problems
Numerical Methods
Fluid Mechanics
Dynamics
Water Resources I and II -  CE 310 and 411
Mathcad Procedures
 

CHEMISTRY

File

 

Description of File

Strong acid strong base titration Numerical solution of a strong acid strong base titration with associated plots 
Chemistry problem Total dissolved carbon in a solution in contact with solid calcite
PDF version of file above PDF file of file above
Chemistry Problem pH of rain water from an acid atmosphere
Chemistry Problem partial pressure of CO2 in solution
Chemistry A discussion of alkalinity in natural waters 
Sodium bicarbonate addition for pH control and buffering This problem arose as a result of the need to predict the pH and alkalinity resulting from the addition of sodium bicarbonate to a low alkalinity well water.  We defined a function,  fct(Na), which calls a solve block for each dosage. Resulting pH and alkalinity plotted vs dosage. 
Master Variable Diagram for H2S Species Species concentrations as function of pH
Ammonia Speciation Problem deals with computing the ammonia/ammonium distribution in water
H2S concentration in a section of Sewer Calculations to determine the speciation of H 2 S in section of sewer. Comments on the practical use of such calculations
Solvent Spill Calculations relating to evaparation of solvents after a spill
Fog Calculations dealing with pollutants in fog vapor 
Alkalinity Equations Numerical and analytical calculations for computing alkalinity
Development of alkalinity equations Development of concepts and equations used to quantify alkalinity in natural waters
pH of weak acid solution - 2 weak acids Numerical solution of the salts of 2 weak acids in water
Chemistry Alkalinity of a sodium bicarbonate-sodium carbonate solution - solve block used
Examination of alkalinity variation in solution Carbon at varying pH-alkalinity levels
System with constant carbon concentration Find the partial pressure of an acidified system where the aqueous C concentration remains constant
NaOH added to CO2/water system NaOH is added to a CO2 /water system to produce NaHCO 3
Sodium hypochlorite added to water The effect on pH of the addition of the salt of a weak acid to water
Magnesium precipitation Effect of buffer on requirements for precipitation. Based on problem from Snoeyink and Jenkins text
How much Freon in the bottle? A neat applied problem using a little chemistry and a little thermodynamics. From the Environmental Organic Chemistry text by Schwarzenbach et al.

Methyl Bromide on lettuce leaves

A very practical little proble from Schwartzenbach et al. but I'm not sure if my solution is correct. Anyone seeing a mistake is asked to email me. Posted 2/25/00 8:30 am
Stumm&Morgan/7-1_solubility_of_FeCO3.mcd Solving for the soluble Fe+2 concentration when the system contains a buffer system. 
9-6_iron_oxidation_kinetics.mcd Problem illustrates that equilibrium calculations don't tell the whole story.

8_1_biochemical_calculations_M@E.mcd

A neat little problem from Metcalf and Eddy which draws together material from different areas to compute the oxygen requirement for metabolism of casein, posted 1/18/02, 1:45pm

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AIR POLLUTION
Air Pollution Problem Air  flow at non-standard temperature and pressure. Notice that this problem is basically a fluids and gas law application
Air Pollution Emissions Particulate emission calculations
Air Pollution - Puff Model  A simple steady-state air pollution model - uses a Mathcad surface plot for illustration
Air Pollution Flue gas flow calculations
Coal combustion analysis Stoichiometric analysis of coal combustion
Unit conversion for concentration unit in air pollution Converting ppm to micrograms per cubic meter- a discussion and some examples posted 11 am, 6/20/00
PDF version of previous file PDF version of unit conversion file with discussion and explanation, posted 11 am, 6/20/00

 

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PROBABILITY AND STATISTICS
Probability problem A probability problem which computes the odds of winning a lottery
An interesting probability problem A cool probability problem from the best seller "Innumeracy" by John A. Paulos
Statistics-Chemistry Problem A statistics problems using pH data. Uses many of Mathcad's statistical functions
Linear Regression A simple but insightful problem using linear regression
Linear and non linear parameter estimation Parameter estimation in an environmental problem
Statistics using pH and DO lab data Shows what can be done with even simple data sets - A PDF file
Kruskal_Wallis.pdf Based on a file from a Mathsoft E-book on statistics. Illustrates the Kruskal-Wallis procedure and explains some of the more esoteric Mathcad statements used. The test is a nonparametric version of ANOVA.  A PDF file, reads external data sets. Posted 6/21/00 7:05 am
Mann_Whitney.pdf Based on file from Mathcad ebook. Nonparametric version of the t test. A PDF file. Posted 6/21/00 7:40 am
Monte Carlo Methods using a normal distribution Based on a file taken from the Mathcad Treasury. I tried to dress it up a little and add some explanation and a possible application so that the user can learn a little statistics as well as Mathcad. posted 11/24/00, 12:55 pm
Monte_Carlo_probabilities_lognormal_distribution.mcd This illustrates the fact that sample means are normally distributed even when the underlying population is not. However you must collect alot of samples with a few observations per sample or fewer samples with many observations per sample. Posted 2/23/01, 9:15 am
Haan - binomial_distribution_ example.mcd
A hydrologic example of the use of the binomial distribution, posted 3/5/01, 5:10 pm
 
 
A problem illustrating the value of a bivariate normal distribution in making predictions, even when data sets are only minimally correlated , posted 7:35 am, 8/30/02

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CIVIL ENGINEERING

Specific Energy plot and calculations for pipe

From V. T. Chow's book, a simple specific energy problem with some bells and whistles added

BOD problem

Problem uses real data collected from a malt plant in Moorhead Minn.

Uniform open channel flow

A simple problem computing normal depth in a channel

Clear water transfer capacity of aerator ANSI/ASCE 2-91

Shows both linear and nonlinear regression procedures

PDF version of the file above

Here it is - let me know if you want assistance 
interpreting it. The second part of the document is the ASCE standard procedure.

Normal depth in a triangular channel

A discussion of 2 methods for dealing with Manning's equation when looking for flow depth, posted for Water resources II class, 8/16/00, 10 am

Hydrogeology - groundwater plume

2 dimensional solution - long time frame. Uses a surface plot to present results

Model of an Ant colony

Mathematical Model of an ant colony using data from
an American Scientist article

Fluids Problem Illustrating a Cross Connection

Must reading for municipal engineers

PDF file for the problem above

Same file as above but in PDF format 

A PDF file for the Ant colony problem

PDF File

Prob_3-24_sluice_gate-stilling_basin.mcd

A neat little problem from Henderson's classic open channels text. Enlightening computations regarding the conditions for hydraulic jump formation.

An optimization problem - Sewer pipe

 Based on a problem from the text by Ozzenbrugen

PDF version of sewer optimization problem above

PDF FILE

Acrobat version of reservoir computations file

PDF File

Hydrology/land use problem

An enlightening problem showing what happens when urbanization occurs. Required reading for developers and local officials

2 solution techniques for a falling body problem Solve the same problem analytically and numerically

Chlorination/chemistry problem 

An extension of a simple chlorination problem from the text by Davis and Cornwell, 3rd edition. We included the ion product of water as one of the equations to be solved.   PDF format

Same chlorine chemistry problem as above

Same problem as above but we neglected the ion product of water in order to examine the effect of leaving it out PDF Format

An all purpose trickling filter design file using the NRC equations

Inputs needed are the influent BOD, flow and desired effluent BOD

Tray settler analysis

An extension of a problem from the text by
"Metcalf and Eddy" based on type I clarification principles

PDF version of tray settler problem 

An extension of a problem from the text by "Metcalf and Eddy"

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 Fluid Mechanics

A simple pump cavitation problem

Extended for a variety of water temperatures

       Mays_12-2-4_NPSH_available.mcd

Computing the net positive suction head required for a pump

A neat little discharge problem

An observation that needed explaining. Based on a small consulting assignment

Control_valve_selection.mcd

An algorithm for contolling flow in a pumped system using a valve. Originally posted by Andrés Felipe Ortega Montoya, Chemical Engineer - Universidad Pontificia Bolivariana - Medellín, Colombia.
E - Mail: aortega@janua.upb.edu.co. 
I just added some comments and dressed it up a little, posted 7/3/2000

Control_valve_selection.PDF

PDF version of the file above, posted 7/3/2000

pump_station_design.PDF

PDF version of problem 9-1 from Wastewater Engineering, collection and pumping of wastewater. posted 7/16/00 

Headloss in a horizontal pipe, f not known a priori

Solution uses a solve block. Problem illustrates the fact that pressure gages can have same reading in flow through an inclined pipe. Could not happen if pipe was horizontal. posted 11/19/00, noon

Measuring flow with American Sigma equipment

In this file (a pdf file) we read in 2 external data sets. One was logged using an American Sigma 950 data logger attached to a 4" McCrometer water meter with transmitter. The other was logged using an American Sigma ultrasonic level sensor attached to a second 950 unit. We then carry out the necessary calculations to compute flow volumes. This data comes from an Interstate rest area on I-55 at the Louisiana/Mississippi border. Real world stuff....posted 12/24/00, 2:15 pm

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MISC AND UNCLASSIFIED PROBLEMS

An Ion Exchange problem Computation of the capacity of an Ion exchange resin and other relationships. Uses the spline algorithm in Mathcad

A worksheet for BOD analysis    Worksheet computes reaction rate constants as well as BOD5 for three different, seeded dilutions. Compares fitted BOD equation to observations for each dilution.

Laplace_transforms_RHS=0_data_set_1.mcd Solution of two coupled second order ODE using Laplace transforms.  Uses a solve block for a symbolic solution - I had not seen this before.  Also uses keywords for symbolic manipulation.
 
 
 
 



A closer examination of statements and procedures in Mathcad

Unfortunately many of the most powerful Mathcad capabilities are not well documented, or the documentation is not in one place. The files below attempt to provide some better explanation of Mathcad statements and algorithms.  Some attempt to demonstrate how the more powerful features (programming, solve blocks, root solvers) in Mathcad can be combined. 

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Local assignments - A good start for those who want to learn to program - This file comes from the Mathcad Treasury. I doctored it a little to, hopefully, make it easier to understand. posted 11/15/00, 2:16 pm

An_interesting_statement.mcd    A close examination of a Mathcad statement that isn't well documented in the existing manuals

Iterating_a_solve_block.mcd I got this file off the Mathcad Web site. The file itself is essentially unchanged. I tried to add to and rewrite the explanation. Iterating a solve block is a very powerful tool that is not explained very well in most places. Posted 10/28/00 noon

Calling_solve_block_from_program.mcd - The original of this file came from the Mathsoft web site but, in my judgement, it is not documented or explained very well. I have added considerable explanation and tried to make the procedure easier to follow for the non-expert. Posted 10/30/00 11:45 am

Finding and extracting soln values meeting a condition - The original of this file also came from the Mathcad Web site. It involves finding a set of solutions with a solve block and then uses a program function to extract those values meeting a specified condition. I just dressed it up a little and added some explanation, posted 11/4/00, 11:20 am

Examples of 'while' loops - I have taken some simple examples of applications of while loops and added some explanation. I also note where the operation of the program seems confusing. posted 11/5/00, 11:05 am

Breaking out of a loop - A simple, annotated, example showing several ways to break out of a loop, posted 11/5/00, 11:10 am

Using the root function repeatedly and storing the output - Here we use the root function to solve a single equation many times. The results are stored in a matrix. posted 11/5/00, 11:20 am

Examination of the Conditional statement   - An example from the manual with a little bit more explanation -  posted 11/11/00 11:40 am

A program for extracting vector element locations - A minor change in an existing Mathcad quicksheet allows one to extract all locations (rather than values) of the maximum valued element in a vector. Posted 11/14/00

A file showing the definition and use of Boolean type functions - File originally from Mathcad Web site. I dressed it up a little and added some explanation. File also uses uniform random number generator, rnd(x). Added 11/22/00, 7:30 am

Elementary row operations - This file originally came from the Mathcad web site. I have tried to disect it a little and add additional explanation for the non-expert. posted 12/27/00, 7:25 am

Symbolic solution using keywords   - Here we solve a separable differential equation using Mathcad keywords, posted 2/1/01, 11:40 am

Multiple_SB_for_simultaneous_equations.MCD - This is an interesting file demonstrating the repeated use of a solve block coupled with a small piece of Mathcad code. The file solves two simple equations repeatedly. Posted 6/4/02, 7:30 am
 
 

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NUMERICAL METHODS
 
 
click on underlined portion to bring up file



ALGORITHMS FOR SOLVING SINGLE EQUATIONS

Secant Procedure    An illustration of the mechanics of the secant procedure for finding roots of equations

Numerical_methods/False_position_algorithm2.mcd A study of the number of iterations required for the family of functions: xn- 1

Newton Raphson Algorithm Here used to solve the Colebrook White equation. The C-W equation is used in fluid mechanics to determine the friction factor for flow in pipes. A graphical description of the procedure is included.  

Numerical_methods/Redlich_Kwong_Eqn.mcd  Explore the convergence properties of two forms of the Redlich Kwong equation of state, posted 9/22/02, 10:45 am

Falling_object_fixed_point_method.mcd  A simple problem exploring the fixed point solution procedure and its convergence properties, posted 9/26/02, 11 am.

Numerical Integration   Computing the area of a stream cross section. A common calculation when modeling streamflow. PDF version below. Uses several versions of Simpsons rule to compute the integral.

PDF file for numerical integration - A PDF file version of the problem above
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FINITE DIFFERENCE APPROXIMATIONS

Finite Difference Approximations of the First Derivative   A comparison of the forward, backward and centered difference approximations with the analytical derivative

Euler's Implicit Method for solving the Advection Equation     - A finite difference approximation . This file requires that the" Thomas" algorithm for solving the tri-diagonal matrix generated.  It is in the same directory and should work fine. File was produced using version 11.  Posted 11/27/02. 2:15 pm

A finite difference approximation for the solution of the steady state advection dispersion equation   - The solution is for a reactor. Could be a river also.

Modified Lax Method applied to the advection equation - Can change flow velocity to show what happens when stability criteria are violated. Useful for predicting bulk motion.

Modified_lax_method_applied_to_advection_equation.PDF   - PDF version of the file above


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EULER'S METHOD

The Basics of Euler's Method   A simple problem illustrating Euler's Method for solving ordinary differential equations.

Predictor corrector modification for Euler's Method   This file uses subscripts instead of Mathcad programming to carry out the procedure.  This requires that the procedure be cast using a single variable name subscripted for time.  Solves the radiation equation.

Numerical solution of the radiation equation  Use the modified Euler predictor-corrector method.  This version uses Mathcad programming.  As a result, differing variable names can be used.  Solves the same radiation equation as the file above.

Numerical solution of a stiff ODE   Uses the Euler explicit solution technique

A PDF version of a stable and unstable solution to a stiff ODE  Same as the problem above. PDF version   Note - One definition of a "stiff ODE" is an ODE in which the solution function exhibits rapid and extreme changes in the dependent variable with small variations in the independent variable.  As a result a plot of the solution function over long time frames (time is the independent variable) will look quite different from a plot over short time frames.

An example showing progressive instability in Euler's Method   A nice example showing progressive instability as a function of the size of the time step.

A comparison of 1. analytical 2. Euler's explicit and 3. Leapfrog method   Three ways to solve the same problem.   An examination of the relative error for the numerical methods.

Solution of a 2nd order boundary value ODE   In this problem the derivative of the dependent variable is assumed, then adjusted until the right hand BC is met. Consideration of mesh size is examined.  "Odesolve" also using for a comparative solution. 

PDF solution for the problem above   Same problem, in Adobe for non-Mathcad types.


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PREDICTOR CORRECTOR METHODS

Second order boundary value differential equation - heat loss along a rod - A comparison of (1) the internal algorithm "odesolve" (based on 4th order R-K procedures), (2) a Mathcad programming solution in which the second order equation is reduced to a system of first order equations, these are then solved using the modified Euler predictor corrector procedure and (3) the analytical solution to the second order differential equation describing heat loss by convection and conduction along a rod.

A PDF version of the heat loss from a rod problem - Same problem as above,  in Adobe for those who don't have Mathcad.

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RUNGE KUTTA PROCEDURES

The basic difference between the various numerical procedures used to solve ODEs is the way in which the functional relationship BETWEEN the points on the solution function is approximated. While Euler's technique is one of the simplest, using a linear approximation, the Runge Kutta procedures use more sophisticated relationships.

A comparison of 2 numerical procedures- Euler explicit and 4th order RK    A comparison of these methods used to solve the radiation equation. Here we use the Mathcad function for the 4th order R-K procedure rather than code it directly.

A discussion of R-K 2nd order solution technique and an example   - A Mathcad file.   Here I try to give some introduction and background regarding the solution procedure.  The procedure is demonstrated on a badly behaved function.

The Runge Kutta Fehlberg Method   - This procedure allows for efficient determination of local truncation error.  A Mathcad file.   

Numerical_methods/gradually_varied_flow_Runge_Kutta.mcd  - Mathcad programming used to implement the 4th order R-K procedure to plot a water surface profile , posted Oct 13, 2002 , 10:45 am

2 Lakes in series - This file uses both the R-K fixed algorithm in Mathcad as well as a Mathcad program to solve ODEs related to contaminant concentrations in a lake.  


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ANIMATIONS


An animation file - This is an avi file showing an animation of the advection equation. A square wave is used as the initial condition.
 
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 COURSE MATERIALS FOR CE 314 - Environmental Engineering
Click on the underlined item to bring up the PDF or Mathcad file
 

Assorted Problems and notes - click on the underlined portion to open the file, most are in PDF format


Chlorination problem ion product of water not included  Problem 3-17 3rd edition Davis and Cornwell PDF file


Chlorination Problem - includes ion product of water  Problem 3-17 3rd edition Davis and Cornwell. In this version the ion product of water was included in the solution.  PDF file


 A simple alkalinity problem  Problem 3-23 A simple alkalinity calculation makes use of the mathematical definition of  alkalinity - PDF file


3-37 Davis and Cornwell - alum addition   A problem from Davis and Cornwell illustrating calculations relating to alum addition.


Hardness_calc.PDF  Problem 3-31 Davis and Cornwell dealing with hardness calculations - PDF file

Design of a rapid mix basin - The procedure with explanations and justification for the choices made. 


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Mg_precipitation.PDF  A precipitation problem showing the effect of a buffer.  In this particular problem the presence of a bicarbonate buffer increases the amount of base needed to raise the pH of the solution sufficiently to precipitate magnesium hydroxide by a factor of 16 !  PDF file.


A primary clarifier design problem  Design of a primary clarifier.


All purpose 2 stage trickling filter design file based on NRC equations  PDF File - replaces 2 earlier trickling filter files.
 

IC distribution in an open system - Here I create a function which calls a solve block to find the inorganic carbon (IC) distribution in an open system for a range of partial pressures of CO2 posted 12/2/2000, 7:20 am. The plot is commonly called a Master variable diagram. In this case the partial pressure of CO2 is the "master variable"

PDF version of the IC distribution file immediately above - For those who don't have Mathcad.

A simple solubility product problem   Here I take a simple problem and beat it to death. Hopefully the annotations and explanations are useful.

Simple solubility product Mg(OH)2   Just a simple solubility product problem with a lot of explanation.

A second order rate Process   A simple second order rate problem. Uses some of MathCAD's linear regression algorithms (slope(x,y) intercept(x,y) corr(x,y)). Some explanation and examples of important rate dependent processes are given. 

A generalized Type I settling algorithm - using units - Uses an iterative solve block, with units, to compute terminal settling velocity of a spherical particle using Newton's law, Reynold's number equation, and CD -N R relationship.  This is done using Mathcad 2001.

Type I clarification - PDF file - Same file as above, in PDF format.

E qualization_basin_sizing_and_analysis.mcd - Based on problems from Chapter 5 of Davis and Cornwell, 3rd edition. Also a little discussion as to why the concept does not always work well in practice. 

5-20_problem_and_additional_HW.mcd     - A simple, complete mix activated sludge problem showing the interrelationship of the various equations used in design.

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NOTES, DISCUSSIONS, DERIVATIONS


Type 1 clarification notes  Notes on type 1 clarification-PDF file- see the Type I settling algorithm in the section above also. Posted 1/4/01, 3:30 pm


Alum coagualation at the Monroe WTP  A discussion of the basic chemistry of alum coagulation. A PDF file

A Discussion of Chemical Equilibrium and the Ion Product of Water

Derivation of G - velocity gradient    PDF file 

Using statistics on environmental data   An example of what simple statistics can be used for in environmental science/engineering - a pdf file.   Data set used is actual pH data collected in lab by students, winter quarter,  2000-2001
 
 

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COURSE MATERIALS FOR CE 531 - Contaminant Transport

Files to assist students in solving ordinary differential equations are provided in the section on numerical methods in this web page. click here to go to numerical methods section
 
 
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PROBLEMS

Volatization of an LNAPL

From Weber and DiGiano's new book

Monte_Carlo_Analysis_Streeter_Phelps.mcd

An example of Monte Carlo analysis applied to the Streeter Phelps Equation, from Schnoors text posted 8/5/01, 7:05 am
Ex_2-4_Toluene_leakage_into_pipe.mcd From Weber and Digiano's book. The book has an error in the solution however. I note it in my solution. Posted 10/22/00

Transient response - equalization basin

Weber and Digiano 2-5; mass balance, transient response. A PDF file

 Transient response for an equalization basin

Equalization basin with catalyst being added, problem 2-6 Weber and Digiano

 A simple mass balance on a conservative tracer

problem 2-7 Thomann and Mueller, a PDF file

 Material balance on color

problem 2-4 Thomann and Mueller, a PDF file

 Equalization basin

step feed, numerical solution required, PDF file

 Thomann and Mueller 2-8

A PDF file showing the use of conditional statements in Mathcad to screen and plot selected results

VOC movement in a river

Benzyl chloride, PDF file

 VOC (PER) in a river

tetrachloroethene, PDF file

Lake Model comparison of numerical and analytical solutions

Uses Mathcad programming language and imbedded matrices, PDF file

Concentration from a distributed source

T@M 2-13 Illustrates the use of conditional statements to plot results.

Prob 2-8 T@M

Compute concentration(s) resulting from multiple discharges in an advective system, posted 1/26/00, 7:15 am. Conservative and nonconservative cases considered.

A simple problem examining mixing in an estuary

3-4 Thomann and Mueller - Good use of conditional controlled plots

 3-5 T@M multiple contaminant sources in a river

Good use of principle of superposition

 Instantaneous spill calculations

 

 T@M 3-14 Mixing in an estuary

defines bulk dispersion coefficient.

 T@M 3-16 advective and dispersive transport

 

 T@M 3-17 mass balance, critical boundary condition

Use of solve blocks to obtain the critical downstream boundary condition for an estuary.

 Ex 4-1 T@M Lake response to pesticide discharge

Simple problem, makes use of Heaviside step function in Mathcad for describing input conditions

4-2 T@M mass balance on P lake Ontario

Uses principle of superposition, shows the effect of even small decay rates in systems with long detention times.

T@M 4-5 PCB Loading to Great Lakes 

Pieces together analytical solutions to describe the PCB concentration in Lake Ontario from 1950 to 12505.

Level 1 atrazine analysis

Development of equations for toxics, discussion of removal mechanisms.

 Level 1 DDT analysis

Compare the behavior of atrazine with that of DDT.  Look at the values of the respective partitioning coefficients

 Kinetics of phenanthrene solubilization

Mathematical treatment of a batch type dissolution experiment for a PAH compound. Here we determine the mass transfer rate as well as the maximum soluble phenanthrene concentration. 

 Contaminant_transport/Ex_7-2_PAH_solubilization.PDF

PDF version of the file above.

Multiple segment reservoir problem

Simple problem showing the effect of a sediment layer in a steady state problem.

 Same problem as above - PDF version

 Coliform variation during/after a runoff event 

Multiple segment solution of coliform variation in a small cove during and after a runoff event. 

Treatment of coliform discharges required to meet permit limits

Principle of superposition applied to coliforms in an estuary, some discussion of dieoff coefficient

DO profile resulting from nitrification

A standard DO profile problem only the oxygen is used for nitrification instead of CBOD oxidation.

Parathion volatilization in a river

Good application of the 2 film theory of  gas transfer.

Prob 10-5 Chapra ; diffusion thru sediments

Good diffusion problem. I try to explore the similarity between the analytical solution used in the problem and the equation for a normal curve.

Euler's explicit method used to compute P concentrations in a lake 

A simple example showing how the explicit method can be used for environmentally related problems. 

Dissolved oxygen concentrations around Bogata, Columbia

Calculation of DO concentration with changing temperature and elevation above MSL. 

Finite difference solution for a mixed flow system

Development of a finite difference solution to the steady state advection dispersion equation.  Also listed in derivation section below. The resulting set of linear equations is solved by matrix inversion. This is inefficient because if you change the number of segments the matrices have to be manually reformulated.

Finite difference solution of mixed flow reactor using tridag

 

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I think this is a pretty neat file. Because the resulting matrices are tridiagonal I used the "tridag" algorithm to solve the set of linear equations. This allows the user to specify or change the number of segments in the solution without manually reformulating the matrices. Tridag does it automatically.  Saves much work. I made this a PDF file because you can't implement the tridag algorithm unless you have the numerical methods pack. 
 Acrolein_degradation_steady_state.mcd A relatively simple kinetics problem using data from Jerald Schnoor's contaminant transport text.
Fechner-Levy_ex_2-6.mcd TCE diffusion from a lake.

Fechner_Levy_2-27.pdf

Vapam, a soil fumigant, spill in the Sacramento River.
Analytical solutions to transport equations used. Good use of stoichiometry too.  From Fechner-Levy 2nd edition. A pdf file.

Problem_5-7_cascade_aerator.mcd

A gas transfer problem.  From Stumm and Morgan, 3rd edition
7-6_first_order_kinetics_for_Fe(II).mcd Rate constant determination for Fe(II) oxidation using the method of excess.   From Weber and DiGiano

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More contaminant transport type problems can be found in the chemistry section
 

LECTURE NOTES, DISCUSSIONS,  DERIVATIONS FOR CONTAMIANT TRANSPORT

Derivation of advection equation with reaction
A mass balance on a cube CV in space, reaction assumed to occur. A PDF file

Derivation of oxygen usage model, batch system, no reaeration
A simplified DO model for a batch system - a useful place to start - some discussion of parameters

A discussion of the S-F oxygen sag equation - A simple problem used to develop the Streeter Phelps equation, some discussion of the reaeration coefficient and the solution's sensitivity to it.

Stagnant 2 film model for phase exchange
A derivation of the basic model and a discussion of its implications, PDF file.

A Derivation of the mean velocity gradient, G

Development of finite difference solution to advection dispersion equation
Although this document contains an example problem, it is primarily a derivation.

Same file as above only in PDF format.   Same file as the one above, develops a simple finite difference  solution to the steady state advection, dispersion equation. 

Lake - Instantaneous spill - The derivation of the analytical solution for an instantaneous spill into a lake

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WATER RESOURCES I and II
Water Resources II is a computer oriented class taught at La. Tech. We make use of existing software packages from Haestad Methods and the Army Corps of Engineers (HEC-RAS, HEC-HMS) coupled with Mathcad to examine pipe networks, gravity drainage systems and hydrologic modeling. Students are taught open channel hydraulics as they are exposed to HEC-RAS.

Even though we have the HEC programs available, I like using Mathcad for much of the hydrology and hydraulics calculations because the mathematic manipulations required are input by and are visible to the student.

Mathcad is an excellent tool for carrying out UH construction (Snyder, SCS) and subsequent convolution with excess rainfall distributions. Minimizes tedious hand calculations.

Link to USACOE Hydrologic engineering center    http://www.hec.usace.army.mil/

Development of gradually varied flow equations   - Includes a qualitative discussion water surface profiles.

Govening_equations_for_gradually_varying_flow.PDF - Same as above PDF format.

2-6_unit_hydrograph_construction.mcd - Simple problem demonstrating some Mathcad techniques while developing a unit hydrograph for a gagged basin.

Modeling_IDF_curves.mcd - Mathematically modeling an intensity-duration-frequency curve using lsolve to evaluate the required parameters.  Taken from McCuen's hydrology text, 2nd edition.

Detention_basin_with_weir.mcd - This file explores the use of detention basins controlled by weirs located at the basin surface, the trade-off between plan area of the lake and the effect on the inflow hydrograph. 

Example_prob_SCS_handbook.mcd - This file is a rather comprehensive example problem illustrating how to use the SCS method and convolution to develop a DRH for a 6-hour rainfall event. 

SCS_dimensionless_unit_hydrograph_convolved_with_rain.mcd - This file is similar to the one above. An SCS UH is developed, excess rainfall is computed based on the UH curve number and the DRH is computed. Finally a mass balance check is performed. 

Convolution_using_spline_to_pick_points_in_time .mcd - This is a comprehensive DRO problem. An SCS UH is developed first, then convolved with a rainfall hyetograph. An interesting wrinkle here is that, in order to get the same time interval for the UH readings as the rainfall hyetograph values prior to convolution, I use the interp(cspline) function to choose interpolated UH flows at 30 minute intervals, this is much faster than doing it manually. The technique was actually conceived by my students (and I thought they were asleep !) 


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Ebook/Prob4-2_Uniform_Flow.mcd - A straightforward uniform flow problem.

Ebook/Prob4-2_Uniform_Flow.PDF - Same file PDF format

Direct_Manning_equation_solver.mcd - Direct solution of Manning's equation for pipes partially full.   I used global definitions here so the user can change inputs and see the results without having to page back and forth.



 Haested_text_chap_1_prob_4.mcd - Another simple uniform flow problem.

Haested_text_chap_1_prob4.PDF - PDF version of the file above



Chow_3-7_specific_energy.mcd -  A proof and an example; specific energy at a critical section.

Chow_3-7_specific_energy.PDF - PDF version


Prob_2-15_channel_transition.mcd - An examination of specific energy and channel transitions.

prob_2-15_channel_transition.PDF - PDF version


Prob 1-2 Henderson surge in stream.mcd - This is a cross between an open channel flow problem and a dynamics problem. It involves motion relative to 2 different coordinate systems (observers).

Prob_1-2_Henderson_surge_in_stream.PDF - PDF version


Specific_energy_diagram_and_discussion.mcd - The classic specific energy diagram, plots for specific flow values, and some discussion as to what it all means.

Specific_energy_diagram.PDF - PDF version



Prob_5-6_canal_discharge.mcd - Problem considers the effect of downstream channel steepness on the discharge from a reservoir

Prob_5-6_canal_discharge.PDF - Same file in pdf format

Pipe_network_example.mcd - A problem from Hydraulics of Pipeline Systems by Larock et al. Illustrates what can be done using Mathcad. 



Larock_pg_66_7_pipe_network.mcd - A network problem showing how to deal with multiple reservoirs, a pump and a pressure reducing valve (PRV). Makes use of the Mathcad version of the file below to develop the equation of the pump curve. 

Larock_pg_66-7_pipe_network_vector_solution.mcd - This is EXACTLY the same problem as the one above only I have used vector notation where possible to reduce the amount of typing. 

Pipe_Network_Analysis_12-6-1-b_Mays.mcd - A simple network problem from the Water Resources text by Mays.



Developing_a_pump_characteristic_equation.mcd   A procedure for developing the equation of a pump curve from actual pump curve data. Also, an example problem showing how to solve a simple network containing a pump. Finally, a caution about using such techniques. posted 4/12/02 12:05 pm

Developing_a_pump_characteristic_Gould_12JMC.mcd Another pump characteristic problem, similar to the one above...with the same cautions. 

8-130_3_reservoir_problem.mcd -  A standard 3 reservoir problem along with some discussion regarding the details of solving such problems using computer software.

A simple detention basin simulation - Here is a simple detention basin simulation algorithm. Basin simulation is based on the principle of conservation of mass. Most textbooks provide only graphical procedures and, in my judgement, the connection between conservation of mass and the actual procedure gets lost in the details of tabulation.

A good pump/pipeline problem - This problem involves a number of issues dealing with pumping systems. 

Convex_channel_routing_2_reaches.mcd - Uses the Convex procedure to route a hydrograph through 2 reaches.

SCS_UH_with_convolution_using_D_as_the_time_increment.mcd - First we construct an SCS unit hydrograph. Then we convolve it with an excess rainfall hyetograph to obtain the DRH.

5-19_Wildcat_Creek.mcd   - Assessing the fit of hydrologic data to a normal distribution. Because I have Mathcad statistical functions I did not need to use normal probability paper. I compared the data directly to a cumulative normal distribution using the mean and standard deviation of the data. 

7-29_rational_method_pre_and_post_development.mcd - A pretty straightforward problem using the Rational Method to illustrate the increase in maximum flow rate as a result of basin development. 


Computer output from Haestad Software
The files below are in PDF format. They give an idea as to what can be obtained from the academic version of the "Flowmaster" software from Haestad. Reference: Computer Applications in Hydraulic Engineering , 3rd Edition, Haestad Methods, CD included. I found this to be quite cost effective for students, compared to the professional versions.

Haestad_rating_curve_problem_1-5.PDF   Flowmaster output

Haestad_Detailed_Report_prob_1-5.PDF   Flowmaster output.  I used the "notes" window to write in the problem statement. It's at the bottom of the page.


Mathcad/Visio/Hec-ras problems

Chow_9-8_water_surface_profile.mcd - Here I used Visio, a low end CAD package, to do the initial drawing. I used Mathcad to do some preliminary calculations. Then I used the current version of the Army corps of engineers open channel flow code, hec-ras version 2.2, to compute the water surface profiles. I then cut and pasted the graphical hec ras output into the Mathcad file. Note that I used the wrong channel width in the problem and had to revise the initial conditions (unrealistically). All of these packages are windows based, quite easy to use, and make it very easy to be quite productive and produce highly useful results.  Imagine how much work this problem would be to do by hand !

Chow_9-8_water_surface_profile.PDF - PDF version of the file

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DYNAMICS
COORDINATE SYSTEMS
Normal tangential coordinate system  
Radial Transverse coordinate system  
PARTICLE KINEMATICS
Particle kinematics A simple little problem requiring the chain rule. 

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PDF version of particle kinematics problem  
Falling particle with resistance  Prob. 11-21 B&J. I first published my version of this problem back in the early 90s' as part of an ASEE article. I've been working on it since, trying to develop an understandable study sheet to illustrate the usefulness of Mathcad for doing dynamics. 
Falling Particle with Resistance - PDF version
Numerical solution to the classic trajectory problem Turns out its not as easy to hit a target as it looks.
Web_site_dynamics/Numerical_trajectory_calculations.PDF PDF version of the previous file.
Vibrating Particle - A dynamics Problem Prob 11-85 B&J. An example of what can be gleaned from a simple little dynamics problem with MathCad
Particle trajectory problem - accelerating and retarding force used Problem demonstrates the versatility of a well chosen acceleration term.
Piston moving through oil 2 versions given, involves integrating the ODE for acceleration.
LINEAR MOMENTUM
Conservation of linear momentum An examination of the effect of an impulsive force on the velocity of an engine/flatcat.
Another simple momentum problem 3-196 M&K, simple but interesting, at least to me, posted 3/29/00, 8:50 am, A truck drives along a barge initially at rest in the water. Which way does the barge move and what's the name of the guy driving the truck ?
Analysis of a car crash Conservation of momentum used to find the initial speed of a car, 14-19 B&J 6th edition
PDF version of car crash problem 14-19 B&J 6th edition.
PDF file - M&K 3-196 - linear momentum Linear momentum 3-196 M&K. 
WORK ENERGY
Bullet.mcd Compute the work done firing a bullet - problem uses data and spline functions for numerical integration - A good example of integrating numerical mathematics into dynamics.
Analysis of ship fenders Requires some judgement in dealing with nonlinear materials, solution to MEMT 312 extra credit problem
Examination of a pile driver  
A simple problem combining thermodynamics and dynamics Combines ideal gas and definition of work.
SYSTEMS OF PARTICLES
Characteristics of a system of particles Problem shows the convenience of working directly with vectors in Mathcad.
Alpha particle - oxygen nuclei collision Good use of i-j-k vector notation.
Newton's Second Law
Connected Masses on a pulley Prob 12-19 B&J, 6th Ed. system 1
PDF version - connected masses over a pulley 12-19 B&J
Ship movement Newton's 2nd Law applied to ship motion to determine velocity
Crate slides on truck bed A discussion of the crates acceleration from different frames of reference.
PDF version - crates slides on truck bed
2 crates slide on an accelerating flatbed truck A discussion of friction forces and frames of reference.
 PLANE MOTION OF 2-D RIGID BODIES
USING VECTORS TO ANALYZE 2-D SYSTEMS
Basic application of angular velocity and angular acceleration applied to a plate-bar assembly Shows the value of the cross product and i,j,k notation for determining linear velocities in rotating systems, problem 15-10
3 dimensional motion - rotating bar A simple illustration of how to use vector algebra to solve a 3-D motion problem where visualization is difficult. A PDF file.
Simple 2-dimensional motion Visio drawings used
  SLIDER AND LINKAGE MOTION
Classwork Problem - collar and member on vertical bar The classwork problem for April 19.   pdf format , 15-32 B&J
Classwork Problem April 19, 2000 The classwork problem for April 19.  Mathcad version 7, 15-32 B&J
Velocities and accelerations for a piston Lots of pictures
Collar and bar mechanism Numerous drawings using Visio, saved as version 7 Mathcad
Problem using velocity center A discussion concerning how to locate a velocity center and what to do with it when you find it !
2-dimensional plane motion
Sample_problem_vehicle_motion.mcd Simple 2-dimensional plane motion problem
4-13_M_and_K_angular_momentum_of_spheres.mcd Angular momentum of system relative to inertial coordinate system, uses vector algebra.
16-1_board_in_back_of_pickup.mcd Translation
16-55_B&J_rolling_sprocket_wheel.mcd Make sure of your vantage point when thinking about acceleration
16-55_B&J_rolling_sprocket.PDF PDF version problem above
16-3_rod_in_slots.mcd Pretty straightforward
16-3_rod_in_slots.PDF PDF version of problem above
17-3_rotating_element.mcd work energy
17-3_rotating_element.PDF PDF version of problem above
Web_site_dynamics/P17-1_rotor_rotation.mcd simple work energy problem extended a bit with plots
Web_site_dynamics/P17-1_rotor_rotation.PDF PDF version of problem above
Web_site_dynamics/17-12_and_13_brake_pedal_assembly.mcd A neat braking assembly problem.
Web_site_dynamics/17-12_and_13_brake_pedal.PDF PDF version of problem above.
Web_site_dynamics/16-13_P-K_rotating_element.mcd Determining the velocity and acceleration of a point on a rotating element. Vector algebra used.
 

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