Download Lecture notes on macroscopic balances

CHBE 386
Mathematical Modeling
Instructor:
Assist. Prof. Dr. M. Oluş Özbek
Teaching Assistant:
İlayda Acaroğlu Degitz
Course objective
Building the mathematical models of chemical and
physical processes, as well as developing necessary
differential equations and their solutions for the
problems of heat & mass transfer and chemical
reaction engineering.
M. Oluş Özbek CHBE386 Lecture Notes
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Instructor:
Assist. Prof. Dr. M. Oluş Özbek
office: A814
e-mail: [email protected]
Teaching Assistant:
İlayda Acaroğlu Degitz
Web-page
http://chbe.yeditepe.edu.tr/courses/chbe386
M. Oluş Özbek CHBE386 Lecture Notes
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TOPICS
•
•
•
•
•
•
•
•
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Introduction & basic concepts in chemical engineering
Mathematical modelling using macroscopic balances
Mathematical modelling using microscopic balances
Equation of motion and equation continuity
Solution techniques of 1st order ODE’s for chemical
engineering problems
Solution techniques of 2nd order ODE’s for chemical
engineering problems
Solution techniques of PDE’s for chemical engineering
problems
Introduction to MATLAB
Applications of numerical methods
M. Oluş Özbek CHBE386 Lecture Notes
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Attendance: 80% Compulsory
Grading:
Attendence/HW/Projects/Quiz
2 Midterm Exams (20% each)
Final Exam
10%
50%
40%
Exam Dates:
To be announced
M. Oluş Özbek CHBE386 Lecture Notes
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Source Materials
M. Oluş Özbek CHBE386 Lecture Notes
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INTRODUCTION
Mathematical Modeling
Mathematics (from Greek máthēma, “knowledge, study, learning”) is
the study of topics such as quantity (numbers), structure, space, and
change.
Model (n): a miniature representation of something; a pattern of
something to be made; an example for imitation or emulation; a
description or analogy used to help visualize something (e.g., an atom)
that cannot be directly observed; a system of postulates, data and
inferences presented as a mathematical description of an entity or state
of affairs
A mathematical model is a description of a system using mathematical
concepts and language. The process of developing a mathematical
model is termed mathematical modeling.
M. Oluş Özbek CHBE386 Lecture Notes
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Why do we need a mathematical model for?
Mathematical modeling and simulation is a cost effective
method of designing or understanding behavior of
engineering systems (such as chemical plants) when
compared to study through experiments.
Mathematical modeling cannot substitute experimentation,
however, it can be effectively used to plan the experiments or
creating scenarios under different operating conditions.
Thus, best approach to solving most chemical engineering
problems involves judicious combination of mathematical
modeling and carefully planned experiments.
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Basis for mathematical modeling
Basis of science and engineering are:
• Mass (chemical species),
• Energy (momentum)
conserved quantities!
For any quantity that is conserved, an inventory rate equation can be written to
describe the transformation of the conserved quantity. Inventory of the conserved
quantity is based on a specified unit of time, which is reflected in the term rate. In
words, this rate equation for any conserved quantity ϕ takes the form
Rate of
Input
of ϕ
Rate of
Output
of ϕ
Rate of
Generation
of ϕ
M. Oluş Özbek CHBE386 Lecture Notes
Rate of
Accumulation
of ϕ
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Basis for mathematical modeling
Rate of
Input
of ϕ
Rate of
Output
of ϕ
Rate of
Generation
of ϕ
Rate of
Accumulation
of ϕ
Solutions of engineering problems are based on the rate equations for
the:
•
•
•
•
Conservation of chemical species,
Conservation of mass,
Conservation of momentum,
Conservation of energy
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Characteristics of the Basic Concepts
• Independent of the level of application,
• Independent of the coordinate system to
which they are applied,
• Independent of the substance to which they
are applied.
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Levels of application of the basic concepts
Microscopic Level → Equations of Change
At the microscopic level, the basic concepts appear as partial differential equations in three
independent space variables and time. Basic concepts at the microscopic level are called the
equations of change, i.e., conservation of chemical species, mass, momentum, and energy.
Macroscopic Level → Design Equations
Integration of the equations of change over an arbitrary engineering volume exchanging mass
and energy with the surroundings gives the basic concepts at the macroscopic level. The
resulting equations appear as ordinary differential equations, with time as the only independent
variable. The basic concepts at this level are called the design equations or macroscopic
balances. For example, when the microscopic level mechanical energy balance is integrated over
an arbitrary engineering volume, the result is the macroscopic level engineering Bernoulli
equation.
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Model Scales
Macro Scale
Mezo Scale
Micro Scale
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DEFINITIONS
The functional notation
ϕ = ϕ(t,x,y,z)
indicates that there are three independent space variables, x, y, z, and one independent time
variable, t. The ϕ on the right side represents the functional form, and the ϕ on the left side
represents the value of the dependent variable, ϕ.
Steady-State
The term steady-state means that at a particular location in space the dependent variable does
not change as a function of time:
∂ϕ
=0
∂t x,y,z
Uniform
The term uniform means that at a particular instant in time, the dependent variable is not
a function of position. This requires that all three of the partial derivatives with respect to
position be zero, i.e.,
∂ϕ
∂ϕ
∂ϕ
=
=0
=
∂yCHBE386
14
∂z x,y,t
∂x y,z,tM. Oluş Özbek
x,z,tLecture Notes
DEFINITIONS
Equilibrium
A system is in equilibrium if both steady-state and uniform conditions are met simultaneously.
An equilibrium system does not exhibit any variation with respect to position or time.
Flux
The flux of a certain quantity is defined by:
Flux =
Flow of quantity / Time
Area
=
Flow rate
Area
Inlet and Outlet Terms
A quantity may enter or leave the system by two means:
i. by inlet and/or outlet streams,
ii. by exchange of a particular quantity between the system and its surroundings through
the boundaries of the system.
(Flux)x(Area)
if flux is constant
Inlet / Outlet rate =
∫∫(Flux)dA
if flux = f(x,y,z)
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DEFINITIONS
Flux =
Flow of quantity / Time
Area
A1 = A
=
Flow rate
Area
A2 = 2A
F1 = (2 Soldier) / A
F2 = (4 Soldier) / 2A
F1 = F 2
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DEFINITIONS
Rate of Generation
The generation rate per unit volume is denoted by R and it may be
constant or dependent on position. Thus, the generation rate is
expressed as
Generation rate =
(R)x(Volume) if flux is constant
∫∫∫(R)dV
if R = f(x,y,z)
Also:
Depletion Rate = ─ Generation Rate
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DEFINITIONS
Rate of Accumulation
The rate of accumulation of any quantity ϕ is the time rate of change of that
particular quantity within the volume of the system. Let ρ be the mass density and ϕ
be the quantity per unit mass. Thus,
Total quantity of ϕ = ∫∫∫ ρϕdV
and the rate of accumulation is given by
Accumulation rate =
d
dt
∫∫∫ ρϕdV
d
dt
(mϕ)
If ϕ is independent of position, then
Accumulation rate =
where m is the total mass within the system.
The accumulation rate may be positive or negative depending on whether the
quantity is increasing or decreasing
with time within the volume of the system.
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SIMPLIFICATIONS OF THE RATE EQUATION
Rate of
Input
of ϕ
Rate of
Output
of ϕ
Rate of
Generation
of ϕ
Rate of
Accumulation
of ϕ
Steady-State Transport Without Generation
St-st:
Accumulation rate =
No Generation:
d
(mϕ) = 0
dt
R=0
Rate of
Input
of ϕ
Rate of
Output
of ϕ
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SIMPLIFICATIONS OF THE RATE EQUATION
Steady-State Transport With Generation
St-st:
Accumulation rate =
Rate of
Input
of ϕ
d
(mϕ) = 0
dt
Rate of
Generation
of ϕ
Rate of
Output
of ϕ
∫∫(Inlet flux of ϕ)dA + ∫∫∫( R )dV = ∫∫(Outlet flux of ϕ)dA
Inlet flux
of ϕ
Inlet
Area
R
System
Volume
M. Oluş Özbek CHBE386 Lecture Notes
Outlet
flux of ϕ
Outlet
Area
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Building / analyzing / solving a model:
• Define your system: A system is any region that occupies a
volume and has a boundary.
• If possible, draw a simple sketch: A simple sketch helps in
the understanding of the physical picture.
• List the assumptions: Simplify the complicated problem to
a mathematically tractable form by making reasonable
assumptions.
• List the available data: Write down the inventory rate
equation for each of the basic concepts relevant to the
problem at hand.
• Use engineering correlations to evaluate the transfer
coefficients
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• Solve the algebraic equations.
Building / analyzing / solving a model:
Scale
Microscopic
Time Dependency
Steady state
Generation / Depletion
R=0
Macroscopic
Un-steady state
R≠0
Coordinates
Rectangular
Spherical
Conserved Property
Energy Momentum Mass
M. Oluş Özbek CHBE386 Lecture Notes
Cylindrical
Chem. Species
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Macroscopic Models
In macroscopic modeling, empirical equations that
represent transfer phenomena from one phase to
another contain transfer coefficients, such as the heat
transfer coefficient in Newton’s law of cooling. These
coefficients can be evaluated by using the engineering
correlations.
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Building / analyzing / solving a model:
Scale
Microscopic
Time Dependency
Steady state
Generation / Depletion
R=0
Macroscopic
Un-steady state
R≠0
Coordinates
Rectangular
Spherical
Conserved Property
Energy Momentum Mass
M. Oluş Özbek CHBE386 Lecture Notes
Cylindrical
Chem. Species
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Sample Problem
First order exothermic reaction
AB
ΔHrxn
rA = k.CA
takes place in a jacketed CSTR under steady-state conditions.
Calculate the mass flow rate of cooling water according to the
following data:
.
mw
Tw,in , Tw,out
.
mA,in
.
.
mA,out , mB,out
k
ΔHrxn
Cp,A = Cp,B = Cp,W
Vrxr
Trxr = TA,in
: mass flow rate of water
: inlet and outlet temperature of water
: mass flow rate of A into the reactor
: mass flow rate of A and B out of the reactor
: reaction rate constant
: heat of reaction
: heat capacity of A, B, and water
: reactor volume
: reactor andM.AOluş
inlet
temperature
Özbek
CHBE386 Lecture Notes
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Microscopic Models
In Macroscopic models the inventory rate equations are
written by considering the total volume as a system, and the
resulting governing equations turn out to be ordinary
differential equations in time.
If the dependent variables such as velocity, temperature, and
concentration change as a function of both position and
time, then the inventory rate equations for the basic
concepts are written over a differential volume element
taken within the volume of the system.
The resulting equations at the microscopic level are called
equations of change. M. Oluş Özbek CHBE386 Lecture Notes
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Differential volume elements
Cartesian
Cylindrical
Spherical
ΔX, ΔY, ΔZ
Δr, Δθ, ΔZ
M. Oluş Özbek CHBE386 Lecture Notes
Δr, Δθ, ΔΦ
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Example 1: Flow through a conical pipe
Consider the flow of water through the conical pipe as shown in the
figure. The inlet and the outlet velocity of the water are ûin and ûout
respectively. Total length of the pipe is L and its radius at any given point
(l) is r = r0 + αl, where α is a constant. Assuming steady state conditions
and constant properties,
a) Express the outlet velocity of water (ûout) in terms of r0, α and l.
b) Repeat part a for the velocity at any point, i.e. find ûl.
ûin
r = r0 + αl
ûout
L
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Example 2: Gravity flow tank
Consider a cylindrical gravity flow tank as shown in the figure. A liquid of constant density (𝜌) enters the
tank at a constant volumetric flow rate (𝑣𝑖𝑛 ) and leaves through a drain pipe at the bottom of the tank
with 𝑣𝑜𝑢𝑡 . The diameters of the tank and the drain pipe are 𝐷𝑡 and 𝐷𝑝 respectively. Using this information,
a)
b)
c)
d)
e)
Find the liquid height as a function of time.
Find the steady state height of the liquid (ℎ𝑠𝑡 ).
Starting from ℎ𝑠𝑡 , find the time required to drain the tank empty (h=0) when the inflow is stopped.
Find the time required to fill the empty tank to ℎ𝑠𝑡 with the drain pipe closed.
Repeat part-c with the drain pipe open.
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Example 3: Gravity flow tank
Repeat «Example 2: Gravity flow tank» for the conical tank as given in the figure.
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Example 4: Sublimating naphthalene
Consider a cubic naphthalene pellet in a non-permeable plastic package. At some point, top of
the package is opened and the naphthalene begins to sublimate into the surrounding
atmosphere. Assuming the naphthalene sublimates uniformly from the open surface, find the
change of naphthalene height with time if the package is opened in,
a) open air.
b) a wardrobe.
c)
Find the time when the naphthalene finishes for both cases.
CA : Concentration of naphthalene in air
CN : Concentration of solid pellet
Mw : Molecular weight of naphthalene
Vair : Volume of the surrounding air.
ρ : Density of naphthalene pellet
L : Side length of the cube
<km> : overall mass transfer coefficient
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Example 5: Sublimating naphthalene
Repeat «Example 4: Sublimating naphthalene» for a cone shaped particle as shown
in the figure.
CA : Concentration of naphthalene in air
CN : Concentration of solid pellet
Mw : Molecular weight of naphthalene
Vair : Volume of the surrounding air.
ρ : Density of naphthalene pellet
H : Initial height of the cone
<km> : overall mass transfer coefficient
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Example 6: Dissolving solid
Consider a spherical salt pellet dropped in water. Assuming the spherical shape does
not change during the process, find the time dependency of the radius if the water is,
a) A very large tank.
b) A glass of water.
c) Find the time required to completely dissolve the salt for both cases (assume
infinite solubility)
C∞ : Concentration of salt in water
CA : Concentration of solid pellet
Mw : Molecular weight of salt
VT : Volume of the tank
r : radius of salt pellet [r(t)]
R: Initial radius of salt pellet
<km> : overall mass transfer coefficient
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Example 6: Dissolving solid
Repeat «Example 6: Dissolving solid» for a cubic salt
particle, whose side length is L. Consider that the salt
particle lies at the bottom of the tank on its side.
Repeat «Example 6: Dissolving solid» for a cubic salt
particle, whose side length is L. Consider that the salt
particle is suspended in water.
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Example 7: Dissolving solid
Repeat «Example 6: Dissolving solid» for a cylindrical
salt particle, whose radius is r and length is L (L>>r).
You may consider that the particle lies horizontal at
the bottom of the tank and length stays constant.
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Example 8: Cooling of a solid
Consider a spherical metal pellet of radius R, which is dropped in cold water.
Assuming temperature profile does not exist for the metal and the water, find the
time change of temperature if the water is in a,
a) Very large tank
b) Glass of water
c) Find the equilibrium temperature (Teqb) and the time needed to reach it.
Tm : Temperature of the metal sphere [Tm(t)]
Tw : Temperature of the water
Tm0 : Initial temperature of the metal sphere
Tw0 : Initial temperature of the water
ρm : Density of metal
ρw : Density of water
Cpm : Heat capacity of metal
Cpw : Heat capacity of water
VT : Volume of the tank/glass
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Example 9: Cooling of a solid
Repeat “Example 8: Cooling of a solid” for a cubic
metal particle with side length L that lies at the
bottom.
Repeat “Example 8: Cooling of a solid” for a cubic
metal particle with side length L that is suspended in
water.
Repeat “Example 8: Cooling of a solid” for a for a
cylindrical metal particle, whose radius is R and
length is L (L>>R).
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Example 10: Isothermal CSTR
Consider an isothermal CSTR (as shown in the figure) in which an elementary 1st
order reaction takes place. The physical properties of the liquids can be taken as
constant. For this process, find the exit concentration of the product (CB) under,
a) steady-state conditions
b) Unsteady-state conditions
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Example 10: Cooling an ideal mixture
It is required to cool a gas composed of 75 mole % N2, 15% CO2, and 10% O2 from 800
◦C to 350 ◦C. Determine the cooling duty of the heat exchanger if the heat capacity
expressions are in the form
CP(J/mol·K) = a + bT + cT2 + dT3 T [=] K
where the coefficients a, b, c, and d are given by
Species
N2
O2
CO2
a
28.882
25.460
21.489
b × 102
-0.1570
1.5192
5.9768
c × 105
0.8075
-0.7150
-3.4987
M. Oluş Özbek CHBE386 Lecture Notes
d × 109
-2.8706
1.3108
7.4643
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Example 10: Heating an ideal mixture
Consider a heated tank unit where two liquids with similar properties are mixed.
However these two input streams have different temperatures and flow rates. The
output stream is taken as overflow, thus the liquid volume in the tank stays constant.
Find the output temperature if the process is operated under
a) steady-state conditions
b) Unsteady-state conditions.
You may consider the physical properties of the liquids stay constant and uniform.
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