ENGR37926D
Modeling and Simulation
Lecture 5: Fluid and
Thermal Systems
MAZIAR HEIDARI
FALL 2022
Disclaimer
The examples in this course are mainly intended for the illustration of the course concepts, and
they should not be treated as actual engineering designs.
Course Outline
Lecture 1: Introduction to Dynamic Systems and Control.
Lecture 2: Translational Mechanical Systems.
Lecture 3: Rotational Mechanical Systems.
Lecture 4: Electrical and Electromechanical Systems.
Lecture 5: Fluid and Thermal Systems.
Lecture 6: State space representation and analysis.
Course Outline
Lecture 7: Numerical Simulation and Analytical Solution of Linear Dynamic Systems.
Lecture 8: Analysis of Dynamic Systems Using Laplace Transforms.
Lecture 9: Transient and steady-state response analysis.
Lecture 10: Introduction to Control Systems
Lecture 11: Case Studies in Dynamic Systems and Control
Types of Systems
Hydraulic systems (liquid)
Pneumatic systems (gas)
Thermal systems.
Hydraulic Systems
Main variables in Hydraulic systems are:
Pressure
Mass-flow rate
Volumetric flow rate
Hydraulic Systems - Resistance
Hydraulic resistance is any component that resists flow and dissipates energy.
Main types of flows in Hydraulic systems: Laminar or Turbulent
Hydraulic resistance with Laminar flow:
Kluever, 2019
Hydraulic Systems - Resistance
Hydraulic resistance with
Turbulent flow:
Kluever, 2019
Hydraulic Systems - Capacitance
Hydraulic capacitance: the ability of a reservoir to store
energy due to pressure.
In case of a reservoir:
Kluever, 2019
Hydraulic Systems - Inertance
Hydraulic inertance: the ratio of change in pressure due to the change in the
time rate of volumetric flow.
Hydraulic Systems - Sources
Two types of sources are considered in this course:
Pressure source and flow source
Modeling of Hydrohalic Systems
From Conservation of mass for a control
volume we have:
Kluever, 2019
Modeling of Hydrohalic Systems
If the fluid is incompressible ( =0)
Example 4.1 (Kluever, 2019)
a) Derive the mathematical model of the system assuming laminar flow
through the valve.
Kluever, 2019
Example 4.1 (Kluever, 2019)
Conservation of mass for an incompressible
fluid:
For laminar flow we have:
Kluever, 2019
Assuming atmospheric pressure at the output:
Example 4.1 (Kluever, 2019)
Class Participation:
b) Derive the mathematical model of the system assuming turbulent flow
through the valve.
Kluever, 2019
Example 4.1 (Kluever, 2019)
Conservation of mass for an incompressible
fluid:
For turbulent flow we have:
Kluever, 2019
Note that for turbulent flow the equation
becomes non-linear.
Example 4.1 (Kluever, 2019)
Class Participation:
c) Derive the mathematical model of the system assuming laminar flow in
terms of h (instead of P)
Kluever, 2019
Example 4.1 (Kluever, 2019)
Conservation of mass for an incompressible
fluid:
For turbulent flow we have:
Kluever, 2019
Hydromechanical Systems
Conservation of mass:
Kluever, 2019
Assuming single input mass
flow and no mass flow out.
Hydromechanical Systems
Fluid bulk modules is a measure of fluid resistance to compression:
Where
is the reference fluid density and nominal pressure and temperature.
What is the equation for incompressible fluids?
Example
Videoplayer (wiley.com)
Example 4.2 (Kluever, 2019)
Derive the mathematical model o the hydromechanical system.
Kluever, 2019
Example 4.2 (Kluever, 2019)
Hydraulic model:
Is this a linear or non-linear model?
Example 4.2 (Kluever, 2019)
Mechanical model:
How would the model change if the fluid was incompressible?
- With constant input pressure of P?
- With constant flow of Qin?
Pneumatic Systems - Resistance
Pneumatic resistance.
Kluever, 2019
Pneumatic Systems - Resistance
Kluever, 2019
The flow is said to be “choked”
when it achieves sonic conditions
(speed of sound) at the throat.
What is the resistance when the
flow is choked?
Pneumatic Systems - Capacitance
Pneumatic capacitance:
Ideal gas law and the polytropic process model:
Where n is 1 for isothermal process and
the above it can be shown that:
for isentropic process. Using
Modeling of Pneumatic Systems
For every control volume:
It can be shown that:
Where:
Example
Videoplayer (wiley.com)
Thermal Systems
Kluever, 2019
Thermal Systems
From conservation of energy:
Where is the net rate of heat energy stored in
the system
Kluever, 2019
Thermal Systems - Resistance
Heat transfer:
- Conduction
- Convection
- Radiation
Radiation heat transfer is highly non-linear with respect to temperate
difference, but conduction and convection can be approximated as linear
functions of temperature difference, i.e.:
Thermal Systems - Capacitance
Thermal capacitance is a measure of a body’s ability to store heat energy
with respect to temperature change:
Where m is the mass and cp is the specific heat capacity at constant
pressure.
Modeling of Thermal Systems
From conservation of energy:
Kluever, 2019
Example
Videoplayer (wiley.com)
Next Session
Lecture 6: State space representation and analysis.
Questions?
References
[1] Craig A. Kluever, “Dynamic Systems: Modeling, Simulation, and Control”,
Second Edition, Wiley, 2019