Lecture 1: Introduction to
analytical methods
CHEE 3321
Reading: none
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Learning objectives (course)
1. Develop mathematical skills for analyzing problems in chemical
engineering
2. Apply skills to solve mathematical models of chemical
engineering systems
3. Develop a systematic method for modeling chemical
engineering systems
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Learning objectives (lecture)
1. Explain why mathematics is important for chemical engineers
2. Present a general method for generating engineering models
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Chemical engineers and math
Driving question: why do chemical engineers need mathematics?
• Describe relationships between quantities in chemical
engineering processes (mass, heat, momentum balances)
- Algebraic relationships
• Examples:
- Ordinary differential equations (ODEs): linear, nonlinear
• Examples:
- Partial differential equations (PDEs): linear, nonlinear
• Examples:
• Solve and analyze mathematical model to predict system
behavior under different conditions
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Method for generating models 1
1. Describe the problem
a. Identify relevant physical and chemical phenomena
b. Identify knowns and unknowns: dependent (state) variables,
independent variables, physical properties, speci ed
operating conditions
c. Draw a clear schematic
2. Construct governing equations (microscopic and/or macroscopic)
Balance equations:
a. Mass balance
b. Momentum balance
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c. Energy balance
Method for generating models 2
3. Construct governing equations (microscopic and/or macroscopic)
• Algebraic balances:
• Differential balances:
• Steady-state balances:
• Identify boundary and/or initial conditions
4. Apply constitutive relations (if any) for uxes, stresses, etc.
5. Apply simplifying assumptions to make the model solvable
How do we know how to make assumptions?
• Problem statement
• Experience in solving similar problems
• Intuition (to be developed!)
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Method for generating models 3
6. Solve the model
7. Analyze physical implications of the solution
8. Revise assumptions
• Better match experimental data
• Assess relevance of assumptions
Additional notes:
• A complete description is rarely possible
• Simple models provide useful insight
• Making use of models requires mathematical tools to solve them
• Focus of this course: developing these mathematical tools
- Detailed physical concepts are not the focus (developed in
other classes)
- Use simple physics to learn how to analyze model predictions
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Simple schematic model
Balance equation:
Schematic drawing:
Simple examples
• Mass balance:
• Momentum balance:
• Energy balance:
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Representative mathematical models
First-order reactions in a batch reactor
Transient continuous stirred-tank reactor (CSTR)
Steady-state conduction
Transient conduction
Conduction in heat exchange n
Conduction with heat generation
Reaction and diffusion in a porous catalyst
Multiple reactions in a batch reactor
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First order liquid phase batch reactor 1
Consider a rst-order liquid phase batch reactor (constant volume)
Rate of disappearance of A:
Reaction:
k: rst-order rate coef cient [1/s]
CA: concentration of species A [mol/L]
units:
Mass balance over time (assume: constant volume reactor):
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First order liquid phase batch reactor 2
Rewrite as a differential equation:
t!0 :
Note: if the volume V is constant, this is equivalent to:
Initial condition:
(
Solve: separate the variables
(
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First order liquid phase batch reactor 3
Apply initial condition:
Rearrange solution:
Physical dependence:
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Steady-state conduction in a slab 1
Physical examples: heat loss through insulation; heat transfer
through a thin metal or polymer lm
Assume that heat ux only depends on the variable z
Energy shell balance:
q: heat ux [J / m2 s]
z!0:
/ z,
Apply constitutive equation: Fourier’s law
(Empirical relationship between
temperature gradient and heat ux)
k: thermal conductivity [J/ m K s]
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Steady state conduction in a slab 2
Combine equations and assume constant k:
Boundary conditions:
1.
2.
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Conduction in a solid sphere 1
Physical examples: rapid quench of a metal ingot; cooling of droplets
Energy shell balance: accumulation = in - out + production
k: thermal conductivity [J/ m K s]
ρ: density [kg/m3]
Cp: speci c heat per unit mass [J/ kg K]
Boundary conditions:
1.
2.
3.
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Conduction in a solid sphere 2
De ne the thermal diffusivity:
Rewrite equation:
(linear partial differential equation)
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Conduction in a heat n 1
Physical examples: heat exchanges; ns (“osteoderms”) in stegosaurs
(see Farlow et al., Science 192, 1123-1125 (1976)!)
Assume that the n is thin, wide, and long:
Boundary condition: Newton’s law of cooling
(describes heat loss from a solid surface into a liquid)
h: heat transfer coef cient [J/m2 s K]
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Conduction in a heat n 2
Energy balance: accumulation = in - out + production
Apply Fourier’s law (along n) and Newton’s law of cooling (surface
losses):
/ x,
x!0:
Boundary conditions:
1.
2.
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