Equation problems involving

One of the most common solution techniques applicable to linear homogeneous partial differential equation problems involves the use of Fourier series. A discussion of the methods of solution of linear partial differential equations will be the topic of the next chapter. In this chapter, a brief outline of Fourier series is given. The primary concerns in this chapter are to determine when a function has a Fourier series expansion and then, does the series converge to the function for which the expansion was assumed Also, the topic of Fourier transforms will be briefly introduced, as it can also provide an alternative approach to solve certain types of linear partial differential equations. 

The specific enthalpies ia equation 9 can be determined as described earUer, provided the temperatures of the product streams are known. Evaporative cooling crystallizers operate at reduced pressure and may be considered adiabatic (Q = 0). As with of many problems involving equiUbrium relationships and mass and energy balances, trial-and-error computations are often iavolved ia solving equations 7 through 9. 

The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundaiy conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

Because of the presence of Q g and Q2e in Equation (2.84) and of 3 g and 326 f Equation (2.87), the solution of problems involving so-called generally orthotropic laminae is more difficult than problems with so-called specially orthotropic laminae. That is, shear-extension coupling complicates the solution of practical problems. As a matter of fact, there... 

Applications of Newton s Second Law. Problems involving no unbalanced couples can often be solved with the second law and the principles of kinematics. As in statics, it is appropriate to start with a free-body diagram showing all forces, decompose the forces into their components along a convenient set of orthogonal coordinate axes, and then solve a set of algebraic equations in each coordinate direction. If the accelerations are known, the solution will be for an unknown force or forces, and if the forces are known the solution will be for an unknown acceleration or accelerations. 

The last definition has widespread use in the volumetric analysis of solutions. If a fixed amount of reagent is present in a solution, it can be diluted to any desired normality by application of the general dilution formula V,N, = V N. Here, subscripts 1 and 2 refer to the initial solution and the final (diluted) solution, respectively V denotes the solution volume (in milliliters) and N the solution normality. The product VjN, expresses the amount of the reagent in gram-milliequivalents present in a volume V, ml of a solution of normality N,. Numerically, it represents the volume of a one normal (IN) solution chemically equivalent to the original solution of volume V, and of normality N,. The same equation V N, = V N is also applicable in a different context, in problems involving acid-base neutralization, oxidation-reduction, precipitation, or other types of titration reactions. The justification for this formula relies on the fact that substances always react in titrations, in chemically equivalent amounts. 

Practical problems involving variable-density PFRs require numerical solutions, and for these it is better to avoid expanding Equation (3.4) into separate derivatives for a and u. We could continue to use the molar flow rate, Na, as the dependent variable, but prefer to use the molar flux,... 

Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

We can use the ideal gas equation to calculate the molar mass. Then we can use the molar mass to identify the correct molecular formula among a group of possible candidates, knowing that the products must contain the same elements as the reactants. The problem involves a chemical reaction, so we must make a connection between the gas measurements and the chemistry that takes place. Because the reactants and one product are known, we can write a partial equation that describes the chemical reaction CaC2(. ) +H2 0(/) Gas -I- OH" ((2 q) In any chemical reaction, atoms must be conserved, so the gas molecules can contain only H, O, C, and/or Ca atoms. To determine the chemical formula of the gas, we must find the combination of these elements that gives the observed molar mass. 

In any stoichiometry problem, work with moles. This problem involves gases, so use the ideal gas equation to convert P-V-T information into moles. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

This section deals with problems involving diffusion and heat conduction. Both diffusion and heat conduction are described by similar forms of equation. Pick s law for diffusion has already been met in Sec. 1.2.2 and the similarity of this to Pourier s law for heat conduction is apparent. 

Equations (13), (14), and (15) can be used to solve three types of problems involving first-order processes. 

Equation-oriented. The equation-oriented or equation-based approach solves the set of equations simultaneously. If the problem involves n design variables, with p equations (equality constraints) and q inequality constraints, the problem becomes one of ... 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

In Illustrations 8.3 and 8.6 we considered the reactor size requirements for the Diels-Alder reaction between 1,4-butadiene and methyl acrylate. For the conditions cited the reaction may be considered as a pseudo first-order reaction with 8a = 0. At a fraction conversion of 0.40 the required PFR volume was 33.5 m1 2 3, while the required CSTR volume was 43.7 m3. The ratio of these volumes is 1.30. From Figure 8.8 the ratio is seen to be identical with this value. Thus this figure or equation 8.3.14 can be used in solving a number of problems involving the... 

Chapters 3 to 7 treat the aspects of chemical kinetics that are important to the education of a well-read chemical engineer. To stress further the chemical problems involved and to provide links to the real world, I have attempted where possible to use actual chemical reactions and kinetic parameters in the many illustrative examples and problems. However, to retain as much generality as possible, the presentations of basic concepts and the derivations of fundamental equations are couched in terms of the anonymous chemical species A, B, C, U, V, etc. Where it is appropriate, the specific chemical reactions used in the illustrations are reformulated in these terms to indicate the manner in which the generalized relations are employed. 

To solve problems involving calibration equations using multivariate linear models, we need to be able to perform elementary operations on sets or systems of linear equations. So before using our newly discovered powers of matrix algebra, let us solve a problem using the algebra many of us learned very early in life. 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

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