Teaching Fluid Mechanics Using Mathcad

Students are taught that the laws of the conservation of mass and the conservation of momentum are fundamental in fluid mechanics analysis and design. These fundamental principles apply whether the flow is spatially varied or constant, temporally unsteady or steady, and closed conduit or open channel. Thus, the application of these basic principles to such a wide possibility of fluid flow problems presents the student with quite a bit of tasks to accomplish in his/her learning process. In the application of one or both of the basic principles, the student is faced with the need to learn and apply, at times, tedious and timeconsuming solution algorithms. As a result, both the teacher and the student are presented with auxiliary tasks, which many times, interrupt and hamper the teaching and learning process. Specifically, these auxiliary tasks include solution algorithms which use: iterative procedures; tedious numerical solutions; diagrams, charts, and nomographs; and other indirect and implicit solution procedures. The final result is that there is not much time left to focus on the modeling of the problem, formulation of the solution, interpretation of the results, changing the assumptions, and going back to modeling of the problem, etc., and thus be able to conduct a sensitivity analysis (or an experimental procedure) in order to find the optimum solution. The use of the mathematical software Mathcad to teach fluid mechanics has proven to greatly reduce the drudgery in solving fluid flow problems. As an illustration of this learning enhancement, Mathcad is used to model the occurrence of critical flow at a change of slope in an open channel flow situation. The Mathcad software performs efficient iterative and numerical solution procedures and direct solutions. The use of Mathcad has been made a requirement for all computational procedures in the Fluid Mechanics courses in the Department of Civil Engineering at Howard University. Introduction The laws of the conservation of mass and the conservation of momentum are fundamental in fluid mechanics analysis and design, whether the flow is spatially varied or constant, temporally unsteady or steady, and in a closed conduit or an open channel. The application of these basic principles to such a wide variety of fluid flow problems presents the student with a long list of tasks to accomplish in their learning process. In an effort to keep our students current with the fast-paced P ge 586.1 technological advances taking place in the scientific field of problem modeling and solution formulation, engineering educators are always in search of improved techniques to teach challenging subjects in Civil Engineering such as fluid mechanics. Because the solution of many problems in fluid mechanics and hydraulics requires repetitive calculations, using programmed procedures can save considerable time and tedious effort. There are various programming procedures available, which make use of advanced technology: 1) programmable scientific calculators and equation solvers, 2) spreadsheets, 3) mathematics software, 4) applications software, and 5) programming languages . While each procedure may provide certain advantages in varying circumstances, it appears that the mathematics software offers the most useful applications for solving engineering problems in general, as well as for fluid flow problems in particular. Solution of many fluid flow problems requires solving a set of simultaneous nonlinear equations and /or solving a set of linear or nonlinear ordinary or partial differential equations that may be boundary-value or initial value problems. Because Mathcad is indeed capable of handling such equations and is userfriendly, it was the chosen mathematics software used to teach fluid mechanics at both the introductory and intermediate levels to our undergraduate students. Prior to enrolling in this series of fluid mechanics courses, our students are taught Mathcad in the undergraduate courses Computer Essentials and Analysis Methods. In addition to the fluid mechanics course, currently there are three other courses in the Department of Civil Engineering at Howard University that integrate Mathcad in the teaching and learning process; these courses are statics, dynamics, and mechanics of materials . Typical Solution Procedure for Fluid Flow Problems Although the various types of fluid flow problems are vast in number, they each require the student to conduct a number of routine steps in order to reach a solution. The first step is to study the physical problem and determine the flow type: closed conduit or open channel flow, temporally unsteady or steady state flow, and spatially varied or constant flow. The second step is to apply the appropriate fundamental principles to the physical flow situation and thus accurately model the problem with the appropriate equations. The third step is to formulate the appropriate and efficient solution procedure in order to obtain accurate results. The fourth step is to interpret and possibly verify the achieved results. The fifth step is to potentially make changes in the assumed values for one or more of the known variables and repeat steps one through four above. The sixth step is to study the results of the sensitivity analysis (or experimental procedure) accomplished through a series of step five, draw intelligent conclusions. P ge 586.2 Traditional Techniques versus Mathcad Capabilities Whether the student is taught to use traditional or Mathcad techniques, he/she must still follow the typical procedure for fluid flow problems described above. Traditional techniques typically make use of other programmed procedures such as programmable scientific calculators and equation solvers or spreadsheets. These approaches used to solve fluid flow problems require that the student learn and apply tedious and time-consuming solution algorithms. As a result, both the teacher and the student are presented with auxiliary tasks that, many times, interrupt and hamper the teaching and learning process. Specifically, these auxiliary tasks include solution algorithms that use: iterative procedures; tedious numerical solutions; diagrams, charts, and nomographs; and other indirect and implicit solution procedures. The final result is that there is not much time left to focus on the modeling of the problem, formulation of the solution, interpretation of the results, changing the assumptions, and going back to modeling of the newly formulated problem, etc., and thus be able to conduct a sensitivity analysis (or an experimental procedure) in order to find either the optimum solution or various solutions corresponding to various assumptions. The use of the mathematical software Mathcad to teach fluid mechanics has proven to eliminate the drudgery and significantly enhance the solution of fluid flow problems for both closed conduit and open channel problems. Mathcad is used not only for the actual modeling of the problem and formulating of the solution, but also to derive the fundamental equations that apply to a given fluid flow problem. Applying the fundamental principles that describe the fluid flow, the Mathcad environment facilitates mathematical derivations of the appropriate equations through the symbolic integration and differentiation capabilities and arithmetical calculations. Mathcad is then used to apply the derived equations, to model the problem, using either analytical or numerical solve blocks or set up a differential equation solver. Finally, Mathcad allows a straightforward formulation and presentation of the solution for interpretation, and provides a high degree of ease in the possibility of modeling numerous related flow situations. The Role of Mathcad in Teaching the Undergraduate Fluid Mechanics Courses Mathcad has been extensively used to teach all of the topics covered in both the introductory and intermediate undergraduate fluid mechanics courses, which include fluid properties, fluid statics, fluid kinematics, and fluid dynamics. While the undergraduate topics include both spatially varied and constant flow, in both closed conduit and open channel flow, the undergraduate curriculum assumes steady state flow. Unsteady flow problems are addressed in a graduate course in open channel flow. Students are given Mathcad “worksheets” for lecture notes in addition to receiving detailed chalkboard instruction. Illustrated in the worksheets, Mathcad is used to derive the appropriate equations starting with the laws of the P ge 586.3 conservation of mass and the conservation of momentum. Numerous examples for each topic are also given in the worksheet. These examples illustrate how to use the appropriate derived equations to model a specific problem using either analytical or numerical “solve blocks” or set up a differential equation solver. Furthermore, the examples clearly show the Mathcad formulation and presentation of the solution for the unknown variables. The examples show how easy it is to make changes in the assumed values for the one or more of the known variables for which Mathcad presents a new solution. Students are assigned projects and homework problems similar to those done in class. Because of the significant amount of time and effort saved in using Mathcad as a teaching and learning tool, we are able to model a larger spectrum and more complex representations of the various fluid flow problems than previously permissible using the traditional techniques. Illustrative Example of Using Mathcad to Teach Fluid Mechanics There are a large variety of fluid flow problems from which we can choose an example in order to demonstrate the power of using Mathcad over traditional techniques to solve the problem. Assuming steady state flow, the two general categories are closed conduit flow and open channel flow. For each category we can further assume spatially varied or constant flow. Because there have been several authors [5,6,7] who have a