Trapezoidal rule

Unless otherwise stated, all numerical integrations for the remainder of the book will be computed by the trapezoidal rule. It is accurate enough for most engineering applications, does not require equally spaced data, and is easy to implement in Excel and VBA. For information on more complex and accurate numerical integration methods, Google it  

Consider a tube with a circular cross section and suppose the radius has been divided into I equally sized increments in the radial direction. The general form for the trapezoidal rule is 

In this example, F(0) = 0 because r = 0 and F(R) = 0 because V iR) = 0. The mixing-cup average is determined when the integral of F(r) is normalized by 2 = There is merit in using the trapezoidal rule to calculate Q = AcM by integrating dQ = 2jtr dr. Numerical errors of integration tend to cancel when the ratio is taken. 

An isothermal pipe reactor with L = 2m, R = 0.01 m is being nsed for a first-order reaction. The rate constant is 0.005 s, and u = 0.01m s. Eqnation 8.1 gives the velocity profile. Molecnlar diffnsion is negligible. Determine the ontlet concentration from the reactor. 

SOLUTION Example 8.1 laid the gronndwork for this case of laminar flow without diffusion. The mixing-cup average is 

Code for Example 8.2 illustrates the use of the trapezoidal rule for evaluating both the numerator and denominator in this equation. The results are as follows  

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Several related rorles or algorithms for numerical integration (rectangular mle, trapezoidal rule, etc.) are described in applied mathematics books, but we shall rely on Simpson s mle. This method can be shown to be superior to the simpler rules for well-behaved functions that occur commonly in chemistry, both functions for which the analytical form is not known and those that exist in analytical form but are not integrable. 

This method approximates/(x) by a parabola on each subintei val. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J9 through Li are suitable, even though the base results (Z1-Z4) are not that close. 

Errors are proportional to At for small At. The trapezoid rule is a second-order method. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

The diserete data ean be analyzed from the residenee time distribution by using either the histogram method or the trapezoidal rule... 

In the trapezoidal rule method, the area under the response eurve is ... 

Table 8-4 shows tlie details of tlie suimuation, whieh is perfonued using the trapezoidal rule. The internal age distribution 1(6) is obtained from... 

Table 8-5 shows the details of the summation, whieh is performed using tlie trapezoidal rule. The internal age distribution 1(6) (Table 8-6) is obtained from 1(6) = 1 - E(6) = 1 - LE56. An Exeel spreadsheet (Example8-2.xls) was developed and Eigure 8-18 shows plots of E(6), E(6), and 1(6) versus 6. 

The total area, XA, under the curve may be obtained in several ways the rectangular or trapezoidal rules are generally quite satisfactory. The area concerned is between the original feed and the final bottoms composition for the particular component. 

The procedure is to start with the one-panel trapezoidal rule... 

Improper integrals of the other types whose problems involve both limits are handled by open formulas that do not require the integrand to be evaluated at its endpoints. One such formula, the extended midpoint rule, is accurate to the same order as the extended trapezoidal rule and is used when the limits of integration are located halfway between tabulated abscissas ... 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

Then the trapezoidal rule applied to the interval from 0 to 1 corresponds to... 

Two simple numerical methods are often used to determine the area under the curve that equals the desired integral. They involve the use of the trapezoidal rule and Simpson s rule. 

Figure A1.5 Graphical integrations using (a) the trapezoidal rule and (b) Simpson s rule.

Another approach is to use Simpson s rule instead of the trapezoidal rule for the integration. With Simpson s rule, three successive points, e.g,, n), are... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

The hnal step in the design calculations for a laminar flow reactor is determination of mixing-cup averages based on Equation (8.4). The trapezoidal rule is recommended for this numerical integration because it is easy to implement and because it converges O(Ar ) in keeping with the rest of the calculations. 

This modified density Is a more slowly varying function of x than the density. The domain of Interest, 0 < x < h, Is discretized uniformly and the trapezoidal rule Is used to evaluate the Integrals In Equations 8 and 9. This results In a system of nonlinear, coupled, algebraic equations for the nodal values of n and n. Newton s method Is used to solve for n and n simultaneously. The domain Is discretized finely enough so that the solution changes negligibly with further refinement. A mesh size of 0.05a was adopted In our calculations. 

Similarly we can estimate the specific secretion rate. It is obvious from the previous analysis that an accurate estimation of the average specific rates can only be done if the integral Jxvdt is estimated accurately. If measurements of biomass or cell concentrations have been taken very frequently, simple use of the trapezoid rule for the computation of Jxvdt may suffice. If however the measurements are very noisy or they have been infrequently collected, the data must be first smoothed through polynomial fitting and then the integrals can be obtained analytically using the fitted polynomial. 

The area under the PCP concentration-time curve (AUC) from the time of antibody administration to the last measured concentration (Cn) was determined by the trapezoidal rule. The remaining area from Cn to time infinity was calculated by dividing Cn by the terminal elimination rate constant. By using dose, AUC, and the terminal elimination rate constant, we were able to calculate the terminal elimination half-life, systemic clearance, and the volume of distribution. Renal clearance was determined from the total amount of PCP appearing in the urine, divided by AUC. Unbound clearances were calculated based on unbound concentrations of PCP. The control values are from studies performed in our laboratory on dogs administered similar radioactive doses (i.e., 2.4 to 6.5 pg of PCP) (Woodworth et al., in press). Only one of the dogs (dog C) was used in both studies. 

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