Problem solving involving equations

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

When stated mathematically, each of these problems potentially involves many variables, many equations, and many inequalities. A solution must not only satisfy all of the constraints, but also must achieve an extremum of the objective function, such as maximizing profit or minimizing cost. With the aid of modem software you can formulate and solve LP problems with many thousands of variables and constraints. 

As discussed in Chapter 1, optimization of a large configuration of plant components can involve several levels of detail ranging from the most minute features of equipment design to the grand scale of international company operations. As an example of the size of the optimization problems solved in practice, Lowery et al. (1993) describe the optimization of a bisphenol-A plant via SQP involving 41,147 variables, 37,641 equations, 212 inequality constraints, and 289 plant measurements to identify the most profitable operating conditions. Perkins (1998) reviews the topic of plantwide optimization and its future. 

Most of the applications of artificial intelligence in chemistry so far have not involved numerical computation as a primary goal. Yet there are aspects of the AI approach to problem-solving which have relevance to computation. In scientific computation, one could view the knowledge base as the set of equations, input variable values, and unit conversions relevant to the problem, and the inference engine the numerical method used to solve the equations. This paper describes such a software system,... 

In this text all numerical problems involve integration of simultaneous ordinary differential equations or solution of simultaneous algebraic equations. You should have no trouble finding ways to solve algebraic equations with a calculator, a spreadsheet, a personal computer, etc. 

The previous work on this problem suggests an equation. You have two different distances represented by expressions involving a y. Write an equation and solve fory. What equation Why Pythagoras s, of course ... 

Number problems that need two answers or end up with two solutions have to be checked carefully. The equations used when solving these number problems frequently involve expressions relating one of the numbers to the other number using mathematical operations. 

Changing from words to equations involves identifying what the variables (the x s or y s or f s) represent and how to arrange them in an equation. Solving an equation requires algebraic know-how, but, if your equation is nonsense or doesn t fit the problem, then the answer to the equation will get you no closer to the answer to the problem than you were before you started. 

M My riting an equation to use for solving a story problem is more than half the battle. Once you have a decent equation involving a variable that represents some number or amount, then the actual algebra needed to solve the equation is typically pretty easy. 

Industrial problems are usually more complicated than the earlier problems in this book. But their solutions generally require the same steps, tools and procedures. Therefore, an engineer needs to learn how to handle these problems in both a direct and an integrated way. A typical industrial problem might involve solving one system of differential equations and then solving an algebraic equation (or another DE) at each point of the solution profile. 

Addition and subtraction are inverse operators of each other, as are multiplication and division. Usually, a problem will involve solving an equation for some variable. You want to convert the equation into the form x = some number (or some expression). Therefore, when you want to isolate a variable (i.e., solve the equation ), you need to undo any operations that are tying up the variable by applying the inverse operation. That is, if the variable is multiplied by a number, you need to divide both sides of the equation by that number. If a variable is divided by some number, you need to multiply both sides of the equation by that number. This eliminates that number from the variable by converting it into a 1. Since 1 is the multiplicative identity element, you can ignore l s that are involved in multiplication or division. If some number is added to the variable, you need to subtract that number from both sides of the equation. If some number is subtracted from the variable, you need to add that number to both sides of the equation. This eliminates that number from the variable by converting it into a zero. Since zero is the additive identity element, you can ignore zeros that are involved in addition or subtraction. 

Two graphical methods described here, a master variable (pC-pH) diagram and a distribution ratio diagram, are extremely useful aids for visualizing and solving acid-base problems. They help to determine the pH at which an extraction should be performed. Both involve the choice of a master variable, a variable important to the solution of the problem at hand. The obvious choice for a master variable in acid-base problems is [H+] [equations (2.9)—(2.12)], or pH when expressed as the negative logarithm of [H+]. 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

So far, we have been able to build a scenario which solves the horizon, flatness and monopole problems. As we already said, this scenario also explains the existence of an almost scale invariant spectrum in the cosmological perturbations. However, the derivation of this crucial result is significantly more involved than the previous one. First because we need to do a careful study of the cosmological perturbation in the context of general relativity and in an expanding universe. Second because we then need to solve these equations in the specific case of inflation. Third, because perturbation theory only tell of the evolution of cosmological perturbations, so that we need to specify the... 

Some of the commonly used methods for obtaining solutions to problems involving laminar external flows have been discussed in this chapter. Many such problems can be treated with adequate accuracy using the boundary layer equations and similar ity integral and numerical methods of solving these equations have been discussed. A brief discussion of the solution of the full governing equations has also been presented. 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

The degree-of freedom analysis tells us that there are five unknowns and that we have five equations to solve for them [three mole balances, the density relationship between V2 (= 225 Uh) and hj, and the fractional condensation], hence zero degrees of freedom. Hie problem is therefore solvable in principle. We may now lay out the solution—still before proceeding to any algebraic or numerical calculations—by writing out the equations in an efficient solution order (equations involving only one unknown first, then simultaneous pairs of equations, etc.) and circling the variables for which we would solve each equation or set of simultaneous equations. In this... 

The flowchart for the absorber involves seven unknown variables (mo and rha- having been determined in Problem 12.5). Write the seven equations you would use to calculate those variables, notii that the amounts of chlorine and water in the head space are constant and so do not enter into steady-state material balances. Then solve the equations. Note Trial-and-error will be required as part of the solution. An equation-solving program or spreadsheet is a convenient tool for performing the required calculations.)... 

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