Chapra download

The approximate error at this point is 6. These results are all displayed as message boxes. For example, the solution check is displayed as 5. Here is a VBA Sub procedure to implement the modified false position method. It is set up to evaluate Example 5. However, as it approaches the root, it approaches quadratic convergence. The approximate error at this point is 9.

This is much smaller than the number of function evaluations in the standard false position method 5n. The approximate error at this point is 2. The approximate error at this point is 4.

The approximate error at this point is 3. The approximate error at this point is 1. The program from Prob. Using a stopping criterion of 0. Due to the concavity of the slope, the next iteration will always diverge. The sketch should resemble figure 6. The partial solutions for each iteration intersected the x-axis along its tangent path beyond a different root, resulting in convergence elsewhere.

Notice that we have named the cells containing the parameter values with the labels in column A. The result is Notice that we have rearranged the two functions so that the correct values will drive them both to zero.

We then drive the sum of their squared values to zero by varying x and y. The result is 7. The result is For guesses of 1. Select Range "a3". Offset 0, 1. Select Next j ActiveCell. Offset 1, -n. Select Next i Range "a1". Results may be inaccurate. Select Next i Range "a3". However a sample of initial guesses spanning the range yield the following roots: 6 -2, -4 Although some follow the pattern, others jump to roots that are far away.

For example, the guess of -6, 0 jumps to the root in the first quadrant. This underscores the notion that root location techniques are highly sensitive to initial guesses and that open methods like the Solver can locate roots that are not in the vicinity of the initial guesses. However, if Set 1 and 3 are reordered so that they are diagonally dominant, they will converge on the solution of 1, 1, 1. Therefore, a tridiagonal solver is well worth using.

Notice that we create some of the unprofitable z2 while producing none of the profitable z3. This occurred because we used up all of Y in producing the highly profitable z1. Thus, there was none left to produce z3. The conversion factors range between 0 and 1. In addition, the cost function can not be evaluated for certain combinations of XA1 and XA2. The problem is the second term, 0. This problem can be set up on Excel and the answer generated with Solver: The solution is The solution is Our goal is to minimize the wetted perimeter by varying the depth and width.

We apply positivity constraints along with the constraint that the computed area must equal the desired area. The result is Thus, this specific application indicates that a 45o angle yields the minimum wetted perimeter. The verification of whether this result is universal can be attained inductively or deductively. The inductive approach involves trying several different desired areas in conjunction with our solver solution.

As long as the desired area is greater than 0, the result for the optimal design will be 45o. The deductive verification involves calculus. The minimum wetted perimeter should occur when the derivative of the perimeter with respect to one of the primary dimensions i. That is, the slope is zero. To formulate P in terms of w, substitute Eqs. Inspection of Eq. Our goal is to minimize the wetted perimeter by varying the depth, width and theta the angle.

The result is Thus, this specific application indicates that a 60o angle yields the minimum wetted perimeter. Because of the presence of 1 — s in the denominator, the function will experience a division by zero at the maximum. Select Range "a5". Row Selection. End xlDown. Row - n Range "a5". Offset 1, Select Next i Range "d5". Row - nint Range "d5". Select Next i Range "e5".

Select ActiveCell. Select Next i Range "a5". Note that the line is also displayed on the plot. The result along with a plot of the model versus the data estimates are shown below. Trendline can be used to fit each range separately with the exponential model as shown in the second plot. We did this so that the sum of the squares of the residuals would not be miniscule. Sometimes this will lead the Solver to conclude that it is at the minimum, even though the fit is poor.

The solution is: Although the fit might appear to be OK, it is biased in that it underestimates the low values and overestimates the high ones. The poorness of the fit is really obvious if we display the results as a log-log plot: Notice that this view illustrates that the model actually overpredicts the very lowest values. The third and fourth models provide a means to rectify this problem. Because they raise [S] to powers, they have more degrees of freedom to follow the underlying pattern of the data.

However, if we assume that the function has one maximum and no minima within the interval, a check can be included. Here is a VBA program to implement the Golden section search algorithm for maximization and solve Example Then the objective function line can be superimposed for various values of P until it reaches the boundary. Notice also that material and storage are the binding constraints and that there is some slack in the time constraint.

Excel spreadsheet solution: least squares fit 3. Linear regression gives 0. An alternative approach would to assume that the physically-unrealistic non-zero intercept is an artifact of the measurement method. Therefore, use the exponential solution. It is set up to solve Example Select Next i Range "e3".

Select Range "d5:f25". ClearContents Range "d5". The following shows the solution for Prob. Offset 2, 0. Select Next i Range "e4". Value Call Spline x , y , n, xu, yu, dy, d2y ActiveCell. It is set up to solve Prob. Note that even though we used a slow PC, we had to call the function numerous times to obtain measurable times. These times were then divided by the number of function calls to determine the time per call shown below N time s 32 0. Obtain the R — squared value to determine the goodness of fit.

Dye Concentraion vs. Time 4. Row - n 'Input data from sheet Range "a6". The results of a and b are not exact because they include trapezoidal rule evaluations. Substituting these values into Eqs. Select Range "a1". Select Sheets "Sheet1". Range "a5:d25". However, we can try to fit different order polynomials to see if we can get a decent fit to the data. This yields the surprising result that a 4th-order polynomial results in almost a perfect fit. Thus it approximates the quartering of the error that we would expect according to Eq.

One approach is to set up Excel with a series of equally-spaced x values from 0 to Then one of the formulas described in this Part of the book can be used to numerically compute the derivative. For example, I used x values with an interval of 1 and Eq.

The resulting plot of the function and its derivative is 1 There might be a slight discrepancy due to roundoff. The results are: h I 0. Thus, there is a negative root. However, if we pick a value that is slightly higher a per machine precision , it will gravitate towards the positive root. In addition, a graph of the entire solution is also displayed. Thus, the use of a simple explicit Runge-Kutta scheme would involve using a very small time step in order to maintain a stable solution.

A solver designed for stiff systems was used to generate the solution shown below. Two views of the solution are given. The first is for the entire solution domain. The resulting linear system can be solved with an approach like the Gauss-Seidel method.

The following table and graph summarize the results. As can be seen, the person died 1. The non-self-starting Heun yielded the following time series of temperature: 80 60 0 1 2 3 4 This result is reinforced when a state-space plot of the calculation is displayed.

This result is also reinforced when a state-space plot of the calculation is displayed. In fact, both the time series and state-space plots are indistinguishable from each other. The 4th-order RK method yields a stable solution. The first few values are shown in the following table. A plot of the result for x is also shown below. Then we formed a column containing the squared residuals between our predictions and the measured values.

Coverage includes the necessary issues of surface water modeling, such as reaction kinetics, mixed versus nonmixed systems, and a variety of possible contaminants and indicators; environments commonly encountered in water-quality modeling; model calibration, verification, and sensitivity analysis; and major water-quality-modeling problems.

Although the book points toward numerical, computer-oriented applications, strong use is made of analytical solutions. In addition, the text includes extensive worked examples that relate theory to applications and illustrate the mechanics and subtleties of the computations. About the Author Steven C. I entered the course with several undergraduate calculus courses under my belt and directly after taking a graduate-level refresher course in advanced calculus.

No prerequisites exist for the course using the textbook and the professor is using the textbook because he heard Chapra lecture on the subject.

Even the professor has admitted the book is unclear at numerous points and reads like Chapra's lectures are orally presented. The material in the book is presented as if the reader has had extensive mathematical training especially in integration.

Clear, step-wise demonstration of equation derivations are lacking at best or completely absent. It may be helpful in clarifying the mathematical equations behind QUAL2K output, for someone with extensive mathematical training and who is already very familiar with the functionality of QUAL2K. Taking Dr. Chapra's Course now By Anthony The book is based on having a working knowledge of diff eq and builds upon it sequentually through a series of lectures as chapters in the book.

Written by one of the top water quality modelers in the world, this is a book that's easy to follow and also serves as great reference for common mass balances found in nature and water treatment processes. OK, but hard to read By Mark C The text covers a lot of worthwhile material, but it is not easy to understand.

The written sections and examples don't always pertain to the homework problems. The examples don't explain the solution step by step, but often skip key steps with phrases like "the following can be derived". Chapra Kindle. Posting Komentar. Senin, 25 Juni [R Chapra Just for you today! Chapra New upgraded!

Most helpful customer reviews 0 of 0 people found the following review helpful. Written by one of the top water quality modelers in the world, this is a book that's easy to follow and also serves as great reference for common mass balances found in nature and water treatment processes 2 of 2 people found the following review helpful.

See all 16 customer reviews Chapra Kindle [R Chapra Doc [R Chapra Doc. Diposting oleh Jimmy Dodson di Label: Ebooks. Tidak ada komentar:. Langganan: Posting Komentar Atom.