Chapter 2: Dynamic Models

1
Chapter 2 Dynamic Models

• Part E Fluid- Heat-Flow Systems

2
Material covered in the PRESENT LECTURE is
shown in yellow

• I. DYNAMIC MODELING
• Deriving a dynamic model for mechanical,
  electrical, electromechanical, fluid- heat-flow
  systems
• Linearization the dynamic model if necessary
• II. DESIGN OF A CONTROLLER Several design
  methods exist
• Classical control or Root Locus Design
• Define the transfer function Apply root locus,
  loop shaping,
• Modern control or State-Space Design
• Convert ODE to state equation Apply Pole
  Placement, Robust control,
• Nonlinear control Apply Lyapunovs stability
  criterion

3
Fluid-Flow Systems

• Pressure Force Exerted by Fluid on Rigid Surface
• Mass Flow Rate Continuity Equation
• Fluid Flow in Restricted Closed Systems
• Learning examples
• Hydraulic Piston, Water Tank Height,
• Hydraulic Actuator with Valve,

4
Pressure Force exerted by a Fluid

• By definition, the pressure force exerted by a
  fluid on a rigid surface of area A is given by
• If the pressure of the fluid, p, is uniform along
  the surface.

5
Example 1 Modeling a Hydraulic Piston

• Given
• There is a force FD acting on the piston of mass
  M
• The fluid pressure in the chamber is uniform and
  denoted by p.
• Required
• Determine the dynamic model for the piston

6
Ex 1 Modeling a Hydraulic Piston (contd)

• Dynamic Model
• where
• Therefore

FBD
7
Mass Flow Rate

• Given a fluid flowing at velocity v, the mass
  flow rate through a surface of area A is defined
  by
• If the fluid is incompressible (r C) and is
  the averaged velocity through the surface

8
Continuity Equation Statement of Conservation of
Matter

• where
• r is the fluid density, V is the volume of fluid
  thus, m is the mass of fluid
• is the mass flow rate into the prescribed
  portion of the system is the mass flow
  rate out the prescribed portion.

9
Example 2 Modeling Water Tank Height

• Required
• Determine the differential equation describing
  the height of the water in the tank.

?
Input
10
Continuity Equation Statement of Conservation of
Matter (contd)

• If the fluid is incompressible, and the system is
  closed (and entirely filled)

11
Mass Flow Rate Out Restricted Systems

• The general form of the resistance is given by
• where
• pressures at end of the
• path is occurring
• R, a constants whose values
• depend on the type of
• restriction, and the fluid

In many cases, the flow is resisted either by a
constriction in the path or by friction.
12
Recall Ex 2 Modeling Water Tank Height

• Required
• Determine the differential equation describing
  the height of the water in the tank.
• Given
• Assume there is a relatively short restriction
  at the outlet and that a 2.

13
Ex 2 Modeling Water Tank Height (contd)

• Known
• where
• since

14
Example 3 Modeling a Hydraulic Actuator
15
Example 3 Modeling a Hydraulic Actuator

• Given The fluid is assumed to be incompressible.
  The friction between the piston and the large
  chamber are characterized by b.

16
Example 3 Modeling a Hydraulic Actuator

• Given Both restrictions at the bottom of the
  small chamber are characterized by a 2 and
  R(x), where x is the input displacement of the
  valve.

17
Example 3 Modeling a Hydraulic Actuator

• Required Write the dynamic model relating the
  movement of the control surface, q, to the input
  displacement x of the valve, via the parameter
  R(x).

18
Example 3 Modeling a Hydraulic Actuator
19
Ex 3 Modeling a Hydraulic Actuator (contd)

• Newtons law applied to piston
• Newtons law applied to control surface
• Relationship between y and q

20
Ex 3 Modeling a Hydraulic Actuator (contd)

• Newtons law applied to piston
• Newtons law applied to control surface
• Relationship between y and q

?
21
Ex 3 Modeling a Hydraulic Actuator (contd)

• Continuity equation
• Mass flow rate on the right-side
• Mass flow rate on the left-side

22
Ex 3 Modeling a Hydraulic Actuator (contd)

• Continuity equation
• Mass flow rate on the right-side
• Mass flow rate on the left-side

?
23
Ex 3 Modeling a Hydraulic Actuator (end)
24
Heat-Flow Systems

• One-Dimensional, Steady-State Conduction
• Heat Transfer Rate Fouriers Law
• Thermal Conductivity Thermal Resistance
• Unsteady-State Conduction Energy Balance
• Heat Capacity
• Learning examples Modeling Room Temperature

25
Fundamentals

• Heat transfer is thermal energy in transit due to
  a temperature difference.
• Whenever there exits a temperature difference in
  a medium or between media, heat transfer must
  occur.
• Heat transfers are classified with respect to the
  physical mechanism which underlies them
• There are 3 heat transfer processes.

26
Conduction, Convection Thermal Radiation
27
Conduction, Convection Thermal Radiation

• Conduction refers to the transport of energy in a
  medium due to a temperature gradient.
• In contrast, the convection refers to heat
  transfer that occurs between a surface and a
  fluid (at rest or in motion) when they are at
  different temperatures.
• Thermal radiation refers to the heat transfer
  that occurs between two surfaces at different
  temperatures. It results from the energy emitted
  by any surface in the form of electromagnetic
  waves.

28
Physical Mechanism in Conduction

• The conduction heat transfer results from
  diffusion of energy due to random molecular
  activity

29
Fouriers Law Thermal Conductivity

• For a plane wall having a temperature
  distribution T(x), and a cross section area A
  (perpendicular to the x-direction), the heat
  transfer rate by conduction through the wall in
  the x-direction is given by

k is the thermal conductivity (W.m-1.oK-1). It is
a transport property of the wall material.
30
Thermal Resistance Analogy between the
conduction of heat and electric charge

• Just as an electrical charge is associated with
  the conduction of electricity,

• a thermal resistance may be associated with the
  conduction of heat
• oK.W-1

31
Conduction Heat Transfer Rate versus Thermal
Resistance

• By definition
• Therefore

32
Relationship between Rconv and k

• By definition, the heat conduction is
• Therefore

33
Thermal Resistance

• For heat conduction, we saw that
• From Newtons law of cooling that governs the
  heat convection qconv hA (Ts-T8), it is easy
  to show that

34
Example 1 Steady Heat transfer Rate through
Composite Wall

• The total heat transfer is such as
• where

35
Relevant Properties of a Substance in Heat
Transfer Analysis

• Transport properties
• Diffusion rate coefficients
• Such as
• k, the thermal conductivity
• (for heat transfer)
• n, the kinematic viscosity
• (for momentum transfer)

• Thermodynamic properties
• Equilibrium state of a system
• Such as
• r, the density,
• cv, the specific heat,
• r cv, termed volumetric heat capacity (J.m-3.oK-1)

36
Heat Capacity into a Substance

• Just as the conservation of matter is governed by

• The conservation of energy in a solid substance
  of temperature T is governed by
• Defining the heat capacity, C (in J/oK) as C
  rcpV yields

37
Thermal Capacity Analogy between the storage of
heat and electric energy

• Just as a capacitor is able of storing energy in
  its electric field when a voltage varies across
  the element
• Coulombs per volt

• a substance may store heat when the temperature
  varies in it
• W.s.oK-1 ( J.oK-1)

38
Example 2 Modeling Room Temperature

• Given
• A room has all but two sides isolated (1/R 0).
  Assume that the room temperature, T, is uniform
  throughout the room.
• Required
• Find the differential equation that determine
  the temperature of the room.

39
Ex 2 Modeling Room Temperature (contd)

• Defining the heat capacity of the room as C, the
  energy equation is given by
• where
• therefore