CHEE 321 CHEMICAL REACTION ENGINEERING

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CHEE 222 CHEMICAL PROCESS DYNAMICS AND NUMERICAL
METHODS Module 3 Dynamic Lumped Parameter
System
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Topics to be covered in this module

• Origin of Ordinary Differential Equations
• Transient balance for lumped parameter systems
• Steady state balance for distributed parameter
  system
• Numerical Methods
• Numerical Integration of Ordinary Differential
  Equation
• Eulers Method
• Explicit and Implicit Methods Numerical
  Stability of Explicit Method
• Runge-Kutta Method
• Second- and Fourth-Order Runge-Kutta Methods
• Computer-Based ODE Solution Techniques
• Matlab Integration Routines
• Ode23 and Ode45Study of transient behavior
• Process Dynamics
• Case study CSTR with first-order reaction
  kinetics

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Types of System and Resulting Equations
Distributed Parameter System
Lumped Parameter System
Dynamic
Dynamic
Steady State
Steady State
X
Algebraic Equations
Differential Equations
Differential Equations
Partial Differential Equations
Linear
Non-Linear
Single Variable
Multi- Variable
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Transient Lumped Parameter System

• For a constant density (r) fluid of density being
  flowed in and out of a storage tank of uniform
  cross-section area (A), the transient mass
  balance (derived earlier in the class) takes the
  following form

h
Note The density parameter does not appear
because it cancels out
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Steady State Distributed Parameter System

• Consider viscous flow in a pipe.
• The differential equation to describe the local
  pressure drop takes the following form (youll
  learn more in CHEE 223)

D
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Numerical Methods for SolvingOrdinary
Differential Equations
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Eulers Method
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Numerical Solutions for ODEs The Euler Method

• Let us consider a single-variable ODE by the
  following equation
• with the initial
  condition
• Objective To find the solution x(t).
• Strategy Approximate or over a
  finite time length or
• time-step (Dt)
• The basic idea is to approximate by a
  difference scheme
• The value of expression f(x) is evaluated and is
  then used to estimate the value of x (tDt)

Graphical illustration will be provided in class
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Explicit Euler Methods

• In this method, the value of f(x) is evaluated at
  known value of x or at xk. The term dx/dt is
  represented by the following difference scheme
• The derivative is then equated to f(x)
• The value of x at ttDt is then calculated by
  the following equation
• Note that the RHS of the above equation contains
  all known values, therefore, the equation can be
  solved explicitly

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Eulers IMPLICIT Method
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Implicit Euler Methods

• This method differs from the implicit method in
  that the value of f(x) is evaluated at unknown
  value of x or at xk1. The dx/dt is represented
  by a difference scheme
• The derivative is equated to f(xk1) and we get
• The equation can be rearranged as follows
• Clearly, the equation is implicit in the unknown
  xk1 . Depending on the non-linearity of f(x), an
  iterative solution may be required

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Stability of Explicit Method
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Explicit Eulers Method Stability

• Tutorial5 problem illustrates that explicit
  Eulers Method may result in a solution that
  generates oscillatory solution.
• The oscillatory behavior is due to numerical
  approximation.
• The solution can also be unstable.
• For Eulers explicit method, both the instability
  and oscillation are the result of step-size
  chosen for integration.
• From mathematical theory, the conditions for
  stability can be found. Here, we provide you with
  that condition (the derivation is beyond the
  scope of this course).

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Explicit Eulers Method Stability

• The general form of solution for Explicit Euler
  Method is as follows
• Let us denote the RHS of the above equation as a
  function g(xk). The requirement for stable
  solution is as follows
• where, g?(xk) is the derivative of g(xk) with
  respect to xk
• In lecture, we will apply this criterion to the
  Tutorial6 problem

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Eulers Method Closing Remarks

• Eulers method is rarely used in advanced
  computational routines. Nonetheless, it is very
  useful in demonstrating the general concept on
  numerical integration
• Explicit Euler Method is easy to implement due to
  explicit nature of the equation but numerical
  stability can be a problem
• Stability problems can be overcome with an
  appropriate choice of step-size.
• Smaller steps sizes are always desirable because
  they lead to increased accuracy. However, small
  step-size come at the cost of increased
  computation time.
• Implicit methods are inherently stable.
• Implicit Euler Method for non-linear equations
  require an iterative solution such as Newtons
  method.
• Finally, stable does not mean accurate.

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Runge-Kutta Method
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Runge-Kutta Methods

• Runge-Kutta methods are variations of explicit
  methods.
• Runge-Kutta methods are used extensively in
  computational packages.
• MATLAB uses 2nd and 4th order methods ode23
  and ode45 solvers
• The basic idea is to estimate an improved guess
  for the derivative and, in turn, a better
  estimate for the integrated variable.

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2nd Order Runge-Kutta (RK2) Method

• Objective To numerically integrate the following
  ordinary differential equation.
• Integration Procedure
• Apply Eulers method to estimate the slope or
  derivative at xk
• Estimate the value of state variable at a
  mid-point, i.e. x(tDt/2).
• Calculate the slope or derivative at this
  mid-point
• Estimate the value of state variable at next time
  step, i.e. xk1 using slope at mid-point
• Illustrative examples to be discussed in lecture

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4th Order Runge-Kutta Method (RK4 Method)

• Integration Procedure
• Apply Eulers method to estimate the slope (m1)
  or derivative at xk
• Estimate the value of state variable at a
  mid-point, i.e. x(tDt/2).
• Calculate the slope or derivative (m2) at this
  mid-point to estimate new mid-point state
  variable x?k1/2
• Find corrected mid-point slope (m3) at this new
  mid-point x?k1/2
• A final slope (m4)is evaluated at the end of the
  step

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4th Order Runge-Kutta Method (Cont.)

• The new state variable at the integration step is
  calculated by a weighted average of three slopes

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Multi-variable RK method
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RK Methods Closing Remarks

• 2nd order Runge-Kutta Method is more accurate
  than Eulers explicit method
• 4th order Runge-Kutta Method is preferred when
  higher accuracy is required.
• Selection of step-size is crucial for accuracy
  and usually class problems are solved using a
  fixed step-size. However, Most of the integration
  routines use a variable step-size.

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Numerical Integration in Matlab

• So far, we have learned about Runge-Kutta methods
  for integration of ordinary differential
  equation.
• Runge-Kutta methods are used extensively in
  computational packages.
• MATLAB uses 2nd and 4th order methods ode23
  and ode45 solvers

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Study of Transient Behavior of a Chemical
ProcessCSTR with 1st-order Rate Kinetics
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CSTR Steady State and Transient Behavior

• CSTR Behavior and Transience
• CSTRs are designed to operate at steady-state
  and, as such, transient effects arise from the
  following situations
• start-ups or shut-down of reactor
• upsets or disturbances (fluctuations) of state
  and/or design variables during operation
• transience from one steady-state to the other.
• Role of a process engineer operating a CSTR.
• The primary interest is to analyze the steady
  state conditions. As well, to predict how the
  system behaves during transient period.
• How do we undertake this study of steady state
  and transient behavior?
• Derivation of Total Mass Balance
• Derivation of Mole Balance on Reactant A
• Calculation of Steady-State Concentration
• Determination of analytical solution for the
  problem, i.e. C(t) ?

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Case Study CSTR Transient Behavior
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Study of SS Transient Behavior of a CSTR

• Objective Apply differential mass and mole
  balance to a CSTR to study transient effects,
  i.e. to investigate the process dynamics of the
  system.
• Problem description Consider the following CSTR
  system wherein a chemical reaction following
  first-order rate kinetics is being carried out.

Initial Condition at t0, CA CA,initial 2
mol/L
Reaction A ? B Rate Kinetics (-rA) k?CA
where, k0.005 s-1
V 10 (L)
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Fundamental Balance Equations for a CSTR

• 1. Differential Mass Balance Equation
• Accumulation Input Rate Output Rate
• ? for constant density system (e.g. liquids)
• 2. Differential Mole Balance on Reactant A
• Accumulation Input Rate Output Rate Gen.
  Rate Cons.Rate

Rate of consumption is given by a constitutive
relationship reaction kinetics. Reaction
kinetics describes how reaction rate is related
to CA via a reaction rate constant
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Fundamental Balance Equations for a CSTR (cont.)

• 2. Differential Mole Balance on Reactant A
  (Cont.)
• For constant volume system, the following
  equation
• reduces to
• and can be rearranged as follows
• Introducing a new variable (tp,) below, the
  above equation can be re-written as

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Steady State Behavior of the CSTR

• 3. Calculation of the Steady-State Concentration
• The steady-state (SS) concentration (CA,s) can be
  calculated by setting dCA/dt term equal to zero
  in the following equation
• Thus, we have
• ?
• tp 1/(0.15/10 .005)-1 seconds 50 s
• CA,s 50 s ? (.15/10) s-1? 10 mol/L 7.5 mol/L

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Transient Behavior of the CSTR

• 4. Analytical Solution for Transient Mole Balance
  Equation
• The mole balance equation below represents a
  first-order ordinary differential equation.
• ?
• The equation can be re-written as follows
• ?
• The above equation can be solved to yield CA (t)
  by simple integration
• Applying the initial condition (at t0, CA CA,
  initial) to the above equation, we get

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Transient Behavior of the CSTR

• The solution can therefore can be derived as
  follows
• Dividing by -tp
• Rearranging, we get
• Finally,

DCA
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Case Study Graphical Representation of the
Solution
0.865 DCA
0.993 DCA
0.633 DCA
DCA
t 2tp
t 5tp
t tp
Can you see the relevance and utility of tp ?
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Where do we go from here ?
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Transient Effects of a CSTR

• Let us explore the dynamic behavior of the CSTR
  subjected to the following WHAT IF scenarios
• What if the initial concentration was varied.
• What if there was upset in feed concentration
  after SS operation was established.
• What if there was upset in feed volumetric rate
  after SS operation was established.
• The above scenarios will be explored via MATLAB
  simulations (to be demonstrated in class)

This exercise aims to demonstrate application of
computer-aided balance equations.