14. 3.3 The Transport Equation
3.3.1 Mass Transport
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
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
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.
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.