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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.
You can access various problem topics by clicking on the following targets:
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. 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 
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 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 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. 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. 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. ANIMATIONS
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
Design of a rapid mix basin - The procedure with explanations and justification for the choices made.
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.
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 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
More
contaminant transport type problems can be found in the chemistry section Derivation
of advection equation with reaction Derivation of oxygen usage model, batch system, no reaeration 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 mean velocity gradient, G Development of finite difference solution to advection dispersion equation 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
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 !) 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 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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