The Trapezoidal Rule

The first integral on the right-hand side is integrated with respect to x and the first forward difference is replaced with its definition of - 3 , to obtain 

The forward difference operators, A, A. . ., are replaced by their equivalent in terms of differential operators [Eqs. (3.57) and (3.58)], and the remainder term becomes 

The remainder series can be replaced by one term evaluated at 5, therefore. 

Addition of all these equations over the total interval gives the multiple-segment trapezoidal 

For simplicity, the error term has been shown as n(Xh )- Tliis is only an approximation because the remainder term includes the second-order derivative of v evaluated at unknown values of each being specific for that interval of integration. The absolute value of the error term cannot be calculated, but its relative magnitude can be measured by the order of the term. Because n is inversely proportional to h  

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. 

For / equally sized increments in the radial direction, the general form for the trapezoidal rule is 

Both F 0) and F R) vanish for a velocity profile with zero slip at the wall. The mixing-cup average is determined when the integral of F(r) is normalized by Q = 7tR u. There is merit in using the trapezoidal rule to calculate Q by integrating dQ = InrVzdr. Errors tend to cancel when the ratio is taken. 

The next few examples show the various numerical methods for a simple laminar flow reactor, gradually adding complications. 

Example 8.2 An isothennal reactor with L = 2m, = 0.01 m is being used 

Example 8.2 An isothermal reactor with L = 2m, R = 0.01 m is being used for a first-order reaction. The rate constant is 0.005s 1, and u = O.Olm/s. Estimate the outlet concentration, assuming piston flow. 

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. 

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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... 

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

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... 

The following Excel macro illustrates the use of the trapezoidal rule for evaluating both the numerator and denominator in this equation. 

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. 

Table 6 illustrates the steps involved in carrying out the Wagner-Nelson calculation. The third column (f 0 Cp dt) shows the area under the Cp versus time curve calculated sequentially from t = 0 to each of the time points using the trapezoidal rule (see Sec. VIII.A). The fourth column (kei f 0 Cp di) shows each of the preceding areas multiplied by k.ei (as estimated from the tail )... 

It should be readily apparent that the trapezoidal rule does not measure AUC exactly. However, it is accurate enough for most bioavailability calculations, and the segments are chosen on the basis of the time intervals at which plasma was collected. 

Table 7 Calculation of Area Under the Plasma Concentration Versus Time Curve (AUC) Using the Trapezoidal Rule...

It is not necessary to apply the trapezoidal rule to the entire plasma concentration versus time curve in order to calculate the total AUC. After the semilog plot becomes a straight line, the remaining area out to t = can be calculated using the following equation ... 

The correction to this expression involves multiple derivatives, although the basic equation, Eq. (60), does not The development of tins result, as above for the trapezoid rule, leads to the relation for the integral over the range a to b in the form... 

Computer Methods These methods are easily programmed in a spreadsheet program such as Microsoft Excel. In MATLAB, the trapezoid rule can be calculated by using the command trapz(x,y), where x is a vector of x values and y is a vector of values /(%.). Alternatively, use the commands... 

The trapezoid rule [see Numerical Integration (Quadrature) ] is applied to obtain... 

Equations (A), (B) and (C) are used in the algorithm to obtain the information required. Step (3) is used to calculate kA from equation (B), and step (4) is not required. Results are summarized in Table 12.2, for the arbitrary step-size in fA indicated G = tl[kA 1 -/A)j, and G represents the average of two consecutive values of G. The last column lists the time required to achieve the corresponding conversion in the second column. These times were obtained as approximations for the value of the integral in equation (A) by means of the trapezoidal rule ... 

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