ChE306: Heat and Mass Transfer

1
Chapter 4
Numerical Differentiation and Integration
2
Numerical Differentiation

• Two major reasons for considering numerically
  approximations of the differentiation process
• Approximation of derivatives in ordinary
  differential equations and partial differential
  equations. This is done in order to reduce the
  differential equation to a form that can be
  solved more easily than the original differential
  equation.
• Forming the derivative of a function f (x) which
  is known only as empirical data (xi, yi) i
  1, . . . , m. The data generally is known only
  approximately, so that yi f (xi), i 1, . . .
  , m.

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Numerical Differentiation

• Two methods are used for numerically
  approximations of the differentiation process
• Taylors theorem as approximation of the
  derivative.
• Differentiation of interpolation approximation of
  the function.

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Differentiation by Backward FD

• First-order derivative in terms of ? with O(h)
• Develop relationship between BFD and ?
• Apply to y at i
• Truncate all but first term

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Differentiation by Backward FD

• Truncation error
• i.e., truncation error of higher-order terms
  O(hn) is smaller as the step size h decreases

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Differentiation by Backward FD

• Second-order derivative in terms of ? with O(h)
• Apply to y at i, truncating all but first term

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Differentiation by Backward FD

• First-order derivative in terms of ? with O(h2)

(3.32)
Combine both equations
(3.36)
truncate
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Differentiation by Backward FD

• Second-order derivative in terms of ? with O(h2)

(3.36)
combine
(3.37)
truncate
9
Differentiation by Backward FD

• Error of order h

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Differentiation by Backward FD

• Error of order h2

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Differentiation by Forward FD

• First-order derivative in terms of ? with O(h)
• Develop relationship between FFD and ?
• Apply to y at i
• Truncate all but first term

(3.53)
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Differentiation by Forward FD

• Second-order derivative in terms of ? with O(h)
• Apply to y at i, truncating all but first term

(3.57)
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Differentiation by Forward FD

• First-order derivative in terms of ? with O(h2)

(3.53)
combine
(3.57)
truncate
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Differentiation by Forward FD

• Second-order derivative in terms of ? with O(h2)

(3.57)
combine
(3.58)
truncate
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Differentiation by Forward FD

• Error of order h

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Differentiation by Forward FD

• Error of order h2

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Differentiation by Central FD

• First-order derivative in terms of ? with O(h2)
• Apply to y at i, truncating all but first term

(3.78)
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Differentiation by Central FD

• Second-order derivative in terms of ? with O(h2)
• Apply to y at i, truncating all but first term

(3.81)
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Differentiation by Central FD

• First-order derivative in terms of ? with O(h4)

(3.78)
combine
(3.82)
truncate
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Differentiation by Central FD

• Second-order derivative in terms of ? with O(h4)

(3.81)
combine
(3.83)
truncate
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Differentiation by Central FD

• Error of order h2

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Differentiation by Central FD

• Error of order h4

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Example 4.1

• Mass transfer flux from an open vessel
• Flux
• unsteady-stateconcentration
• where ? isdetermined by

Step 1 determine ? Step 2 evaluate X Step 3
calculate N (requires evaluation of derivative)
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Example 4.1 Input data

• Mass transfer flux from an open vessel
• time eps3600
• axial positions vector 0.1, 0.2, 0.3
• diffusion coefficient vapor in air 2.2e-5 m2/s
• temperature 25C
• pressure 101325 Pa
• vapor pressure at temp 3161 Pa
• File containing equation for ? 'Ex_4_1_phi'
• File containing concentration profile
  'Ex4_1_profile'

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Example 4.2

• Numerical differentiation (1st to 4th-order
  derivatives) of a series of numerical data.
• deriv.m is applied in this example to calculate
  first- to fourth-order derivatives of a matrix of
  data. Central finite difference is used to
  calculate the derivatives.

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Numerical Integration
27
Numerical Integration
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Numerical Integration
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Numerical Integration

• We also want to choose the approximates
  of a form we can integrate directly and easily.
  Examples are polynomials, trig functions,
  piecewise polynomials, and others. If we use
  polynomial approximations, then how do we choose
  them. At this point, we have two choices
• Taylor polynomials approximating f (x)
• 2. Interpolatory polynomials approximating f (x)

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Newton-Cotes Integration

• Accomplished by replacing function yf(x) with a
  polynomial, such as the Gregory-Newton forward
  interpolation formula.
• First three Newton-Cotes Integration formulas
  are Trapezoid, Simpson's 1/3, and Simpson's 3/8
  rules

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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule
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Trapezoid Rule

• Use one segment with h and fit polynomial through
  (xo,yo) and (x1,y1).
• Retain first 2 terms of Gregory-Newton

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Simpson's 1/3 Rule

• Fit polynomial through two segments of h (i.e.,
  through points (xo,yo), (x1,y1), (x2,y2)).
• Retain first 3 terms of Gregory-Newton

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Simpson's 3/8 Rule

• Fit polynomial through three segments of h.
• Retain first 4 terms of Gregory-Newton

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Summary of Newton-Cotes Integration Formula
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Example 4.3

• Numerical integration using Trapezoidal and
  Simpsons 1/3 Rules.
• trapz.m and Simpson.m are used for nunmerical
  integration.

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Gauss Quadrature

• Applied to unequally-spaced data
• Lagrange polynomial used to approximate function.
  Orthogonal polynomials applied to locate the
  loci of base points.

areaabove areabelow
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Gauss Quadrature

• Transform interval a,b to -1,1

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Gauss Quadrature

• Equivalent to Trapezoid to this point
• Gauss Quadrature eliminates error term by
  expanding error in terms of 2nd degree Legendre
  polynomial.
• Values of z0 and z1 chosen as root of 2nd degree
  Legendre polynomial, z0,1 ??1/3½which causes
  the R term to vanish.

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Higher-Point Gauss-Legendre

• Replace yf(x) with Lagrange polynomial

Identical to Newton-Cotes, except that Lagrange
polynomial used in place of Gregory-Newton
interpolation
where
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Higher-Point Gauss-Legendre

• substitute
• convert interval

where
polynomials of degree n and n1, respectively
52
Higher-Point Gauss-Legendre

• Differs from Newton-Cotes in that error term
  eliminated by expanding polynomials in the error
  term with Legendre orthogonal polynomials zi
  chosen as roots of (n1)th degree Legendre
  polynomial. Accuracy increased from n to 2n1.

0
error in Gauss-Legendre(Seinfeld Lapidus)
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Spline Integration

• Sum terms of integrated cubic splines across all
  intervals.

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Multiple Integrals

• inner integration first, e.g., Trapezoid Rule
• m ? number of divisions
• k ? integration step in y-direction
• x ? constant

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Multiple Integrals

• Plug result into double integral
• Apply Trapezoid rule to each integral
• n ? number of divisions
• h ? integration step in x-direction
• y ? constant

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Multiple Integrals

• Combine

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Example 4.4

• Use Gauss-Legendre Quadrature to solve the
  temperature profile in a vertical falling film.

T profile
yltlt? near wall
vz profile
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Example 4.4

• Combine
• Boundary conditions for short contact time

where
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Example 4.4

• Analytical Solution
• Must integrate using Gauss-Legendre

where
and