Problem Setup

So far, the examples have assumed that the differential equations are given. However, as chemical engineers, we know that more often than not, the main problem is in the derivation of the differential equation and the associated conditions. To address that aspect of mathematical methods in this chapter, a problem setup section follows. 

The traditional approach of outlining the theory and presenting some supporting examples has been followed up to now. However, a needed deviation from traditimi is a how to or a problem setup section. This section is included to demraistrate one approach to formulating a physically applicable first-order ordinary differential equation. 

Consider the continuous extraction of benzoic acid from a mixture of benzoic acid and toluene, using water as the extracting solvent [4]. Both streams (acidic mixture and water) are fed into a tank where they are stirred efficiently and the mixture is then pumped into a second tank where it is allowed to settle into two layers. The upper organic phase and the lower aqueous phase are removed separately, and the problem is to determine what proportion of the acid has passed into the solvent phase. 

A list of simplifications for the idealized problem (model) follows  

Express stream-flow rates on solute-free basis. 

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If the only units remaining after canceling are the units to be used in the answer, you are finished with the problem setup and need only to do the calculator work. 

Problem setup. Section 4.1.1 describes the variables that are usually specified. All of these, except the separation specification, are straightforward. 

The computer is now the preferred vehicle for solution of many heat-transfer problems. Personal computers with either local software or communication links offer the engineer ample power for the solution of most problems. Despite the ready availability of this computing power I have resisted the temptation to include specific computer programs for two reasons (1) each computer installation is somewhat different in its input-output capability and (2) a number of programs for microcomputers in a menu-driven format are already on the scene or soon to be available. The central issue here has been directed toward problem setup which can be adapted to any computational facility. 

Use the problem setup in Problem 10.12 and deduce the wall effect when measuring the terminal velocity by dropping a sphere into a fluid. Compare with Perry and Green (1997, p. 6-54). 

Before this section is concluded, there is one key point that should be emphasized. The ultimate goal of the calculation was to calculate the effectiveness factor rjcat- In the present case, this came down to solving Eq. (4-170) for the concentration distribution in the boundary-layer near the particle surface. Because of all the assumptions that we made in the original problem setup, Eq. (4-170) is simple and easy to solve. However, even if the equation we achieved in the boundary-layer region had been much more complicated so that we could not solve it, the process of setting up the asymptotic framework, by means of nondimensionalization and rescaling, provides most of the important information about rjcat, and would do so even if we had not been able to solve (4-170). If we go back to the definition of rjcat in Eq. (4-191), we see that the effectiveness factor depends on (dc/dr) =. However, we see from fhe rescaling process that... 

Check As we predicted, V2 < Vj. Let s think about the relative values of P and L as we check the math. P more than doubled, so V2 should be less than Wi (0.0105/0.0248 < 5). Comment Predicting the direction of the change provides another check on the problem setup To make V2 < V, we must multiply V by a number less than 1. This means the ratio of pressures must be less than 1, so the larger pressure (P2) must be in the denominator, P /P2-... 

Here we will review a few methods for solving the first-order ordinary differential equations. Following each method are examples demonstrating the application of that method. Also, the notion of translating prose into mathematical symbolism is introduced as Problem Setup in Section 2.4. 

It should also be noted that each example problem was stated in prose and required transformation to mathematical symbolism. This transformation or problem setup is an important step and is usually where most students are left behind. However, in this book, whenever the demonstration involves physical phenomena such as those encountered in chemical engineering, the formats of Examples 2.1 and 2.2 will be followed. As an aid to this step, it is suggested that the student invest some time in reviewing the laws of conservation of mass and energy, as well as the unit operations principles discussed in undergraduate chemical engineering courses. 

As in Section 2.4, some problem setup as well as solution techniques will be demonstrated. Eor example, consider the following problem in heat transfer through a cylindrical conductor [15]. 

Chapter 2 deals with select first order ordinary differential equations and provides chemical engineering examples that demonstrate the use of solution techniques. A section addressing the formulation of some physically applicable first-order ordinary differential equations (problem setup) is included. 

Many new and revised charts, diagrams, and plots contain data, curves, and waveforms calculated from theory or obtained from the original literature to provide an accurate and realistic representation. Throughout the text, we have attempted to present material in a student-friendly style that is active and engaging. Examples are sprinkled throughout each chapter to aid in solving relevant and interesting problems. The solutions to the problems in each example are indicated so that students can easily separate the problem setup from the problem solution. 

A Per expression can be used as a conversion factor. In a problem setup, a conversion factor is written as a fraction. Each Per expression yields two conversion factors, one the reciprocal of the other. For example, 7 days Per week produces... 

Once again, to check our problem setup we ll use a reasoning approach, but this... 

The problem setup looks all right. How about the numerical answer Is it reasonable ( ) or not reasonable ( ) ... 

Strategy Use Equation 14.11 to solve for fe- Pay particular attention to units in this type of problem. Setup Solving Equation 14.11 for 2 gives... 

New material was added that illustrates how to count significant figures in equalities and in conversion factors used in a problem setup. 

FIG. 4. Flow chart for SETUP. The initial problem setup (or re-setup) is contained in this procedure, and is initiated by the statement "RUN, SETUP" entered on the terminal. A dialog, structured by the computer, enables the user to enter information about the problem to be solved. 

In summary, if the CFD analyst is careful when addressing the issues of problem setup and solution convergence, the potential benefits that can be extracted from the simulation are numerous. Furthermore, the computational resources available today, in terms of both speed and power, should encourage engineers to make use of high density grids and complex models so as to achieve results of the best possible quality. 

Figure 2. Problem setup for one-dimensional diffusion. The computational domain is an RVE for a membrane randomly filled with aligned flakes.

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