Mathematic Principle

The basic problems of the sciences and engineering fall broadly into three categories:

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.
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.

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:

Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process.
Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model.
Interpretation. Development of relations between the mathematical results and their meaning in the physical world.
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 combination 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

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