General Mathematics
The basic problems of the sciences and engineering fall broadly into three categories:
1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems.
2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed.
3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column.
The mathematical treatment of engineering problems involves four basic steps:
1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process.
2. Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model.
3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world.
4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed.
The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form.
Input of conserved quantity - output of conserved quantity + conserved quantity produced = accumulation of conserved quantity
Rate of input of conserved quantity - rate of output of conserved quantity + rate of conserved quantity produced = rate of accumulation of conserved quantity
These statements may be abbreviated by the statement
Input - output + production = accumulation
When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, “material balance,” “energy balance,” and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as
Rate of energy in - rate of energy out + rate of energy produced = 0
Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. In mathematical language one such problem, the equilibrium problem,
is called a boundary-value problem (Fig. 3-1). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds and, on the boundaries of the region, by certain conditions (boundary conditions) that are dictated by the physical problem. The solution of the equation must satisfy the differential equation inside the region and the prescribed conditions on the boundary.
In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions.
The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”).
Prototypes are:
1. Elliptic. Laplace’s equation
Poisson’s equation
These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y).
2. Parabolic. The heat equation
describes nonequilibrium or propagation states of diffusion as well as heat transfer.
3. Hyperbolic. The wave equation
describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear.
The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical
equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available.
Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem.
The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas:
1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors.
The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable.
Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would
otherwise have been performed.
2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms
of probability statements.
Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers.
1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems.
2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed.
3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column.
The mathematical treatment of engineering problems involves four basic steps:
1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process.
2. Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model.
3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world.
4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed.
The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form.
Input of conserved quantity - output of conserved quantity + conserved quantity produced = accumulation of conserved quantity
Rate of input of conserved quantity - rate of output of conserved quantity + rate of conserved quantity produced = rate of accumulation of conserved quantity
These statements may be abbreviated by the statement
Input - output + production = accumulation
When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, “material balance,” “energy balance,” and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as
Rate of energy in - rate of energy out + rate of energy produced = 0
Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. In mathematical language one such problem, the equilibrium problem,
 |
| Equilibrium |
is called a boundary-value problem (Fig. 3-1). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds and, on the boundaries of the region, by certain conditions (boundary conditions) that are dictated by the physical problem. The solution of the equation must satisfy the differential equation inside the region and the prescribed conditions on the boundary.
In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions.
The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”).
Prototypes are:
1. Elliptic. Laplace’s equation
 |
| Formula-1 |
Poisson’s equation
 |
| Formula-2 |
2. Parabolic. The heat equation
 |
| Formula-3 |
3. Hyperbolic. The wave equation
 |
| Formula-4 |
The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical
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| Landfill Formula |
Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem.
The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas:
1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors.
The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable.
Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would
otherwise have been performed.
2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms
of probability statements.
Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers.
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