Transcript Mathematics in the undergraduate chemical engineering program.

UNIVERSIDAD AUTONOMA DE ZACATECAS, MEXICO
Mathematics taught in chemical
engineering. What material should
be covered?
Benito Serrano, Juan José Mejía,
Víctor Javier Cruz, Alfonso
Talavera, Jesús Moreira.
1
Survey about the contents.
 To pass the self seminars.
 To solve ¨complicated problems¨
 Is this a risky task?
 A survey was performed.
 Several universities of Mexico
 Several universities of USA
 One university in Canada.
The AICHE-2008-Philadelphia PA
2
Common courses and topics.
 Differential and integral calculus, with one or




several variables,
Linear algebra
Vectorial calculus
Statistics and probability
Differential equations of first, second and nth order, linear, with constant coefficients,
homogeneous and non homogeneous.
The AICHE-2008-Philadelphia PA
3
Differences among the programs
 Differential equations with variable




coefficients,
Series of powers
Systems of differential equations of first
order.
Partial differential equations solved with the
method of variable separations.
This is a non definite frontier.
The AICHE-2008-Philadelphia PA
4
Structure of the Chem. Eng. Programs.
 Basic Area.
 Basic Professional Area.
 Professional Area.
 Complementary Area.
The AICHE-2008-Philadelphia PA
5
Focuses
 mathematical rigor
 Recipes.
 The collected documents did not show the
focus.
 Math in Chem. Eng. Taught by
Mathematicians?, Physics by Physicians?
 Is it correct to teach Math just as Formulas?
The AICHE-2008-Philadelphia PA
6
Math in Chem. Eng.
 A combination between the rigor and formulas.
 The basic professional and main nucleus of
subjects dictate the level of math.
 Analytical capacity because good formation in
mathematics.
 Math is the osseous skeleton of science.
The AICHE-2008-Philadelphia PA
7
Graduate Courses in Math.
 Matrices and eigen - values.
 Construction of mathematical models.
 Partial differential equations.
 Methods of discrete the equations.
 Numerical methods.
 Mathematical tools need for the thesis.
The AICHE-2008-Philadelphia PA
8
Graduate and Undergraduate Courses.
 The border is not clear and defined.
 Common courses: Ordinary diff. eqs. with
variable coefficients, systems of equations,
partial differential equations, etc.
The AICHE-2008-Philadelphia PA
9
Type of problems.
 Undergraduate: Convection + Chemical




Reaction: Ordinary Differential Equations.
Graduate: Diffusion + convection + chemical
reaction: Partial differential equations.
It is necessary to delimit the contents and focus
of all the subjects in Chem. Eng.
Certifying institutions request a number of
courses and hours.
Math provides a solid basis for graduate
studies.
The AICHE-2008-Philadelphia PA
10
Special Undergraduate Course.
 For those undergraduate students interested in
learning more about applied math.
 Special and optional course.
 Title: Applied Mathematics in Selected Topics.
 General Objective: To encourage the students to
pursue graduated studies.
The AICHE-2008-Philadelphia PA
11
Contents of the Course.
 A model of laminar flow through a lubricated





tube.
Slow flow over a solid sphere.
Transient flow inside a cylindrical tube.
No stationary heat conduction in a bar.
Potential flow on a cylindrical tube.
Multiple steady states of chemical reactors.
The AICHE-2008-Philadelphia PA
12
Transient Flow Equations.
u z
P
1   u z 



r

t
z
r r  r 
uz  0
at
du z
 0, at r  0
dr
rR
uz  0 at t  0
Using the following dimensionless variables:
r
s
R
uz
  max
uz
The AICHE-2008-Philadelphia PA
 t
 2
R
13
Transient Flow 1
  1   s  

s s  s 
0

  - 1- s 2
s 1
at


0
s
 0
Variable separation
s,   S  s   T    ST
The AICHE-2008-Philadelphia PA
14
along
s0
Transient Flow Equations 2.
d 2 S 1 dS
 
  2S  0
ds s ds
dT
  2T
d
T e
 2 t
A
S s   c1 J 0   s   c2Y0   s 

 1n  x  2 n
J 0 x   
 
2 2
n  0 n!

The AICHE-2008-Philadelphia PA
2
4
6
x
x
x
 1  
 .....
4 64 2304
15
J 0 0  1
Transient Equations 3
m1

2 




1
Hm 2


Y0  2 s  2    ln  s   J 0  2 s   2 m
 s
2
 
2


m!
m1 2
 
  0,


 

   1 s

 1  s   Bm  1  J 0  m  S 
2
n 1
Bm 
8
 m4
J1  m 
The AICHE-2008-Philadelphia PA
2
2m 
 
,



  Bm  J 0  m  s J 0  m  s sds
n 1

  8
1
0
J 0  n s 
3
n1  n J 1
16
 n 

exp   n2  

 2s  0
Results.
The AICHE-2008-Philadelphia PA
17
Conclusions.
 The undergraduate students in chemical




engineering have a big potential, if they are
motivated.
Special course to interests students.
Equilibrium between formulas and
mathematical rigor.
Border between undergraduate and graduate
courses should be indicated by the objectives of
the career.
Dominion of mathematics prepare graduate
students.
The AICHE-2008-Philadelphia PA
18
The AICHE-2008-Philadelphia PA
19