- 1.0 Chapter Objectives
- 1.1 Classification of Transport Processes and Separation Processes (Unit Operations)
- 1.2 SI System of Basic Units Used in This Text and Other Systems
- 1.3 Methods of Expressing Temperatures and Compositions
- 1.4 Gas Laws and Vapor Pressure
- 1.5 Conservation of Mass and Material Balances
- 1.6 Energy and Heat Units
- 1.7 Conservation of Energy and Heat Balances
- 1.8 Numerical Methods for Integration
- 1.9 Chapter Summary
1.8 Numerical Methods for Integration
1.8A Introduction and Graphical Integration
Often, the mathematical function f(x) to be integrated is too complex and we are not able to integrate it analytically. Or, in some cases, the function is one that has been obtained from experimental data, and no mathematical equation is available to represent the data so that they can be integrated analytically. In these cases, we can use either numerical or graphical integration.
Integration of a function f(x) between the limits x = a to x = b can be represented by
By plotting f(x) versus x, the area under the curve is equal to the value of the integral.
1.8B Numerical Integration and Simpson’s Rule
It is often desirable or necessary to perform a numerical integration by computing the value of a definite integral from a set of numerical values of the integrand f(x). This, of course, can be done graphically, but in most cases numerical methods suitable for the digital computer are desired.
The integral to be evaluated is Eq. (1.8-1), where the interval is b – a. The most generally used numerical method is the parabolic rule, often called Simpson’s rule. This method divides the total interval b – a into an even number of subintervals m, where
The value of h, a constant, is the spacing used in x. Then, approximating f(x) by a parabola on each subinterval, Simpson’s rule is
where f0 is the value of f(x) at x = a; f1 is the value of f(x) at x = x1, ...; fm is the value of f(x) at x = b. The reader should note that m must be an even number and the increments evenly spaced. This method is well suited for digital computation with a spreadsheet, since spreadsheets often have advanced numerical calculation methods built into their programs. Simpson’s rule is a widely used numerical integration method.
In some cases, the available experimental data for f(x) are not at equally spaced increments of x. Then, the numerical integration can be performed using the sum of the single-interval rectangles (trapezoidal rule) for the value of the interval. This is much less accurate than Simpson’s rule. The trapezoidal-rule method becomes more accurate as the interval becomes smaller.
The experimental data for f(x) are sometimes spaced at large and/or irregular increments of x. These data can be smoothed by fitting a polynomial, exponential, logarithmic, or some other function to the data, which often can be integrated analytically. If the function is relatively complex, then it can be numerically integrated using Simpson’s rule.