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**Chapter 7 Differentiation and Integration**

Finite-difference differentiation

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**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

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**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.

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**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

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**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

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**Differentiating an Interpolating Polynomial**

The derivative: Gregory-Newton interpolation polynomial: It is more difficult to evaluate the derivative:

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**Differentiation Using Taylor Series Expansion**

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**The second-order approximation:**

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

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**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:

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

How to derive (for reference only)

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**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:

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**Example: Evaporation Rates**

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

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**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):

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**Interpolation Formula Approach**

The Gregory-Newton interpolation polynomial:

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Trapezoidal Rule

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**Another way to get the trapezoidal formula**

The linear polynomial passing the data points:

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

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**Example: Trapezoidal Rule for Integration**

The trapezoidal rule provides

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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.

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**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.

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**For a pipe of diameter of 1 ft**

trapezoidal rule:

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**Simpson’s Rule where Simpson’s rule:**

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

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**Proof of Simpson’s Rule**

Using a second-order polynomial:

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**Passing through the three data points:**

Then, we can obtain the Simpson’s formula.

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

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**Example: Flow Rate Problem**

Applying Simpson’s rule to the data of Table.6

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**Romberg Integration Denoting the trapezoidal estimate as I01**

where a and b are the start and end of an interval. A second estimate I11:

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

and it can be rewritten as

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**Continuing this subdividing of the interval leads to the following recursive relationship**

The general extrapolation formula in recursive form is

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**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

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**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).

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We have a=0, b=1.

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**Table.7 Computations According to the Romberg Method**

j = 1 2 3 4 5 6 1

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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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