The document discusses environmental modeling and transport phenomena. It covers topics like Fick's laws of diffusion, the transport equation, and dimensionless formulations. Numerical methods for solving transport equations like the finite difference method are presented. Case studies on models for river water quality and biodegradation kinetics are analyzed. Redox sequences and their importance in environmental systems are also introduced.

1. 환경문제 연구 6th Week (4/16)
Fundamentals of
Environmental Modeling (3)

2. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

3. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

4. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

5. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

6. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

7. 3.2 Fick’s Law and Generalizations
3.2.1 Diffusion

8. 3.2 Fick’s Law and Generalizations
3.2.2 Dispersion

9. 3.2 Fick’s Law and Generalizations
3.2.2 Dispersion

10. 3.3 The Transport Equation
3.3.1 Mass Transport

11. 3.3 The Transport Equation
3.3.1 Mass Transport

12. Concentration Polarization

13. Concentration Polarization

14. 3.3 The Transport Equation
3.3.1 Mass Transport
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Transfer of material
Distinctly different from fluid flow
Interplay between mass transfer and fluid flow
Transport of species within continuum
Continuum could be stationary or itself mobile
Mass transfer can take place across an interface,
i.e. from one continuum to another
Mechanisms
 Purely diffusive (molecular diffusion)
 Convective mass transfer

15. 3.3 The Transport Equation
3.3.1 Mass Transport

16. 3.3 The Transport Equation
3.3.1 Mass Transport

17. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

18. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

19. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

20. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

21. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

22. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

23. 3.3 The Transport Equation
3.3.2 Fourier’s Law and Heat Transport

24. 3.4 Dimensionless Formulation

25. Correlations of Mass Transfer Coefficient
You have a method for analyzing the results of mass transfer
experiments… but
For example
k = a ua rb nc Ld dhe …….
25

26. Dimensionless Groups
26

27. 27

28. Fluid-Fluid Interface
28

29. Solid-Fluid Interface
29

30. 3.4 Dimensionless Formulation

31. 3.5 Boundary and Initial Conditions

32. 3.5 Boundary and Initial Conditions

33. Transport Solutions

34. 4.1 1D Transient Solution for the Infinite Domain

35. 4.1 1D Transient Solution for the Infinite Domain

36. 4.1 1D Transient Solution for the Infinite Domain

37. 4.1 1D Transient Solution for the Infinite Domain

38. 4.1 1D Transient Solution for the Infinite Domain

39. 4.2 A Simple Numerical Model

40. 4.2 A Simple Numerical Model
• In the next row the concentration is shifted by one position to the right.
For such an operation MATLAB® offers the circshift command.
Parameters of the command are the vector and the number of positions
to be shifted. One has to use the transpose-', because circshift operates
on column vectors, only.
• Additionally the circular shift puts the concentration from the last cell into
the first cell, which is not the intention here. The concentration in the first
cell should be the inflow value cin. This setting is performed in the next
command, which overrides the preceding value in that cell.

41. 4.2 A Simple Numerical Model
simpletrans.m

42. 4.2 A Simple Numerical Model

43. 4.2 A Simple Numerical Model
diffusion.m
simpletrans2.m

44. 4.2 A Simple Numerical Model

45. 4.2 A Simple Numerical Model

46. 4.2 A Simple Numerical Model

47. 4.2 A Simple Numerical Model

48. 4.2 A Simple Numerical Model

49. 4.2 A Simple Numerical Model
Stability of Numerical Solution

50. 4.3 Comparison Between Analytical and
Numerical Solution
N=50

51. 4.3 Comparison Between Analytical and
Numerical Solution
N=100

52. 4.4 Numerical Solution Using MATLAB® pdepe

53. 4.4 Numerical Solution Using MATLAB® pdepe

54. 4.4 Numerical Solution Using MATLAB® pdepe

55. 4.4 Numerical Solution Using MATLAB® pdepe

56. 4.4 Numerical Solution Using MATLAB® pdepe

57. 4.4 Numerical Solution Using MATLAB® pdepe

58. 4.4 Numerical Solution Using MATLAB® pdepe

132. Introduction
•
Ordinary differential equations (ode) are differential equations
for functions which depend on one independent variable only.
• These ‘odes’ are simpler than partial differential equations which
contain more than one independent variable.
• In almost all models or simulations independent variables are
either time and/or space.
• In environmental modeling, two situations can be distinguished
in which odes appear.
• The first situation deals with systems in which spatial
differences can be neglected and the temporal development
is questioned.
• In chemistry, the continuously stirred reactor is an often
used concept for which an approach is allowed with time t
as the only independent variable.

133. Introduction
•
Aside of analytical solutions, two numerical solvers of
MATLAB® are introduced in this chapter, one (ode15s)
designed for the solution of initial value problems, the other
(bvp4c) for the solution of boundary value problems (bvp).
• In initial value problems boundary conditions are formulated
for one value of the independent variable only (typically: t =
0),
• In boundary value problems there are conditions required at
both ends of the interval of the independent variable.

134. 9.1 Streeter-Phelps Model for River Purification
The Streeter-Phelps equation is used in the study of water pollution as a water
quality modeling tool. The model describes how dissolved oxygen (DO) decreases
in a river or stream along a certain distance by degradation of biochemical oxygen
demand (BOD). The equation was derived by Streeter and Phelps in 1925, based
on field data from the Ohio River. The equation is also known as the DO sag
equation.

135. 9.1 Streeter-Phelps Model for River Purification

136. 9.1 Streeter-Phelps Model for River Purification

137. 9.1 Streeter-Phelps Model for River Purification

138. 9.2 Details of Michaelis–Menten or Monod Kinetics

139. 9.2 Details of Michaelis–Menten or Monod Kinetics

140. 9.2 Details of Michaelis–Menten or Monod Kinetics

141. 9.2 Details of Michaelis–Menten or Monod Kinetics

142. 9.3 1D Steady State Analytical Solution

143. 9.3 1D Steady State Analytical Solution

144. 9.3 1D Steady State Analytical Solution

145. 9.3 1D Steady State Analytical Solution

146. 9.3 1D Steady State Analytical Solution

147. 9.4 Redox Sequences

148. 9.4 Redox Sequences

149. 9.4 Redox Sequences

150. 9.4 Redox Sequences