 Chapter 7 Differentiation and Integration

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Chapter 7 Differentiation and Integration
Finite-difference differentiation

Example: Evaporation Rates
Table: Saturation Vapor Pressure (es) in mm Hg as a Function of Temperature (T) in °C T(°C) es(mm Hg) 20 17.53 21 18.65 22 19.82 23 21.05 24 22.37 25 23.75

The slope of the saturation vapor pressure curve at 22°C (3 methods) :
The true value is 1.20 mm Hg/°C, so the two-step method provides the most accurate estimate.

Differentiation Using a Finite-difference Table
Example: Finite-difference Table for Specific Enthalpy (h) in Btu/lb and Temperature (T) in ºF T h Δh Δ2h Δ3h Δ4h 800 1305 155 1000 1460 -30 125 25 1200 1585 -5 -20 120 5 1400 1705 1600 1825

For example, at a temperature of 1200 ºF, the forward, backward, and two-step methods yield:
The rate of change of cp at T= 1200 ºF is

Differentiating an Interpolating Polynomial
The derivative: Gregory-Newton interpolation polynomial: It is more difficult to evaluate the derivative:

Differentiation Using Taylor Series Expansion

The second-order approximation:
The second-order approximation of the first derivative with forward difference:

The second-order approximation of the second derivative with forward difference:
The first-order and second-order approximation of the first derivative with backward difference:

The first-order and second-order approximation of the second derivative with backward difference:
How to derive (for reference only)

The first-order and second-order approximation of the first derivative with the two-step method:
The first-order and second-order approximation of second derivative with the two-step method:

Example: Evaporation Rates
Second-order with forward, backward, two-step: The true value at T = 22ºC is 1.2 mm Hg/ºC

Numerical Integration
The area under the curve f(x) between x=a and x=b: Example: the volume rate of flow (Q) of water in a channel or through a pipe is the integral of the velocity (V) and the incremental area (dA):

Interpolation Formula Approach
The Gregory-Newton interpolation polynomial:

Trapezoidal Rule

Another way to get the trapezoidal formula
The linear polynomial passing the data points:

The absolute value of the upper bound on the error for the Trapezoidal rule is:

Example: Trapezoidal Rule for Integration
The trapezoidal rule provides

Example: Flow Rate The flow rate (Q) of an incompressible fluid is given by the integral. in which V is the velocity and A is the area. For a circular pipe of radius r, the incremental area dA is equal to 2πrdr.

Table 6: The Data for Estimating the Flow Rate of a Fluid in a Circular Piple
ri (ft) Vi (fps) 1 10.000 2 1/12 9.722 3 1/6 8.889 4 1/4 7.500 5 1/3 5.556 6 5/12 3.056 7 1/2 0.

For a pipe of diameter of 1 ft
trapezoidal rule:

Simpson’s Rule where Simpson’s rule:
Simpson’s rule can only be applied when there are an even number of subintervals:

Proof of Simpson’s Rule
Using a second-order polynomial:

Passing through the three data points:
Then, we can obtain the Simpson’s formula.

The absolute value of the upper bound on the error for the Simpson’s rule is estimated by

Example: Flow Rate Problem
Applying Simpson’s rule to the data of Table.6

Romberg Integration Denoting the trapezoidal estimate as I01
where a and b are the start and end of an interval. A second estimate I11:

A third estimate I21 can be obtained using three equally spaced intermediate points m1, m2, and m3
and it can be rewritten as

Continuing this subdividing of the interval leads to the following recursive relationship
The general extrapolation formula in recursive form is

The values of Iij can be presented in the following upper-triangular matrix form:
I0,N-1 I0,N I0,N+1 I11 I12 I13 . I1,N-1 I1,N I21 I22 I2,N-1 I31 IN-2,3 IN-1,2 IN1

Example: Romberg Method for Integration
For the function f(u) = ueku, the integral is Let k = 2, we want to calculate The true value of the integral is (seven significant digits).

We have a=0, b=1.

Table.7 Computations According to the Romberg Method
j = 1 2 3 4 5 6 1

Similarly for i = 5, I51= I02 is computed as following: I03 is computed as following: The Romberg method yields an exact value to six significant digits for the integral of

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