CHEE 321 CHEMICAL REACTION ENGINEERING

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CHEE 222 CHEMICAL PROCESS DYNAMICS AND NUMERICAL
METHODS Module 4B Introduction to Laplace
Transform
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Topics to be covered in this Module

• Introduction to Laplace Transform
• As a method for solving differential equation
• As a tool for assessing dynamic response of a
  system
• Response to step input
• Response to pulse input
• Laplace Transforms of single-variable first order
  systems
• Transfer function analysis of first-order systems
• Conversion of state-space model to a transfer
  function model

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Laplace Transform

• Consider a time domain function f(t). The Laplace
  transform of f(t) is represented by Lf(t) and
  is defined as
• Laplace transform is a linear operator, i.e.
• The Inverse of the Laplace transform is denoted
  as L-1 and

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Laplace Transform of Exponential Function

• Exponential functions commonly arise in the
  solution of linear, constant coefficient, ODEs
• Laplace transform of the above exponential
  function is

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Laplace Transform of a Derivative

• One of the most powerful aspects of Laplace
  transforms is that it allows conversion of
  differential equations into algebraic equation.
  Laplace transform of a derivative, df(t)/dt is

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Laplace Transform of Step Function

• Step functions are employed to gain an insight
  into the dynamics of a system. A sudden change in
  input is applied and the response is monitored.
• Laplace transform of the above step function for
  tgt0 is

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Laplace Transform of Pulse Input Function

• Disturbances in process inputs may be described
  by pulse inputs. Such functions are described as
  follows
• Laplace transform of a pulse of magnitude A over
  a duration tp is

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Laplace Transform of Impulse Input Function

• Impulse function is a special case of pulse
  function with zero width (i.e. tp ? 0) and a unit
  pulse area (i.e. the pulse height or A1/tp).
  Impulse functions are denoted by symbol d.
• Laplace transform of a impulse input is given as
  follows
• Applying LHopitals rule

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First-order linear ODE single input-single output

• We have seen that a single-state variable
  first-order ODE with single input can be
  represented as
• where,
• time constant (units of time)
• k process gain (units of output/input)
• y output variable
• u input variable
• (More discussions in class)

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Response of First-Order Systems
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Laplace Transform of first-order linear ODE

• Let us start with the premise that the following
  form of a single-state variable first-order ODE
  was obtained by linearization of a non-linear
  ODE
• Therefore, we can specify u(0) 0 and y(0) 0.
  WHY?
• Taking the Laplace transform of the above
  equation, we get

(Derivation provided in class)
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Introducing the concept of Transfer Function

• The Laplace transform of 1st-order ODE can be
  arranged as follows
• where, g(s) is known as the transfer function.
• As symbolic-analytic people, we can represent the
  above equation by the following block diagram.

U(s)
Y(s)
g(s)