Class 13 - Mathematical Modeling Of Thermal System
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System Modeling Coursework
Class 13: Modeling of Thermal systems
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. • Of late, this has becoming supplement to what is taught in the class. So BEWARE! You are on two tracks!
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of a Thermal system • Model of the Thermal system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
Introduction • Thermal systems are those that involve the transfer of heat from one substance to another. • Thermal system may be analyzed in terms of thermal resistance and thermal capacitance although they may not be represented as lumped parameters. • But by making some assumptions, they can be represented as distributed parameters, which make the analysis simple. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
A Thermal System
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Assumptions for the system • • •
Fluid in the tank is perfectly mixed so that it is at uniform temperature The tank is insulated to eliminate heat loss to the surrounding air. There is no heat storage in the insulation.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Definitions for variables of the system • θi = Steady state temperature of inflowing liquid, • θ = Steady state temperature of out-flowing liquid, • H = Steady state heat input rate from heater.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
What happens when you move from steady state?
•
Let ∆H be a small change in the heat input rate from its steady state value. This change in H will result in the following changes. – – –
Change in heat output rate by an amount ∆H1. Change in heat storage rate of liquid in the tank by an amount ∆H2. Change in temperature of out-flowing liquid by an amount ∆θ.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Thermal Resistance • Change in outflow heat rate is given by ∆H1 = Q Cs ∆θ • Where Q = Steady state liquid flow rate Cs = Specific heat of liquid
• ∆H1 = ∆θ/R – If R = 1/QCs which is defined as the Thermal Resistance July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Thermal Capacitance • Change in heat storage rate is given by ∆H2 = MCs d∆θ/dt • Where – M = mass of the liquid in the tank – ∆dθ/dt = rate of rise of temperature in the tank
• ∆H2 = C d∆θ/dt – Where C = MCs which is defined as Thermal Capacitance. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
Transfer function for the system ∆H= ∆H1 + ∆H2 • The mathematical model of a thermal system shown in figure is ΔH=
d Δθ Δθ +C R dt
• Applying Laplace transform
Δθ ( s ) + CsΔθ ( s ) R ⎡1 ⎤ = ⎢ + Cs ⎥ Δθ ( s ) ⎣R ⎦
ΔH(s)=
⎡1 + RCs ⎤ Δθ ( s ) = ⎢ ⎥ R ⎣ ⎦ July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
Transfer function of the system
Δθ ( s ) ⎡ R ⎤ = ⎢ ⎥ ΔH(s) ⎣1 + RCs ⎦
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Summary • Thermal systems are simple systems like level and do not yield to complexities like pneumatic or hydraulic systems. • Hence, it is easy and possible to associate analogous situations and derive the transfer function.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
References 1. Advanced Control Systems Engineering, Ronald Burns 2. Modern Control Engineering, Ogata 3. Control Systems, Nagoor Kani 4. A course in Electrical, Electronic Measurements and Instrumentation, A.K. Sawhney …amongst others
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
14
And, before we break… • Love has the patience to endure the fault we cannot cure.
Thanks for listening…
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
15
Class 13: Modeling of Thermal systems
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. • Of late, this has becoming supplement to what is taught in the class. So BEWARE! You are on two tracks!
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of a Thermal system • Model of the Thermal system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
Introduction • Thermal systems are those that involve the transfer of heat from one substance to another. • Thermal system may be analyzed in terms of thermal resistance and thermal capacitance although they may not be represented as lumped parameters. • But by making some assumptions, they can be represented as distributed parameters, which make the analysis simple. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
A Thermal System
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Assumptions for the system • • •
Fluid in the tank is perfectly mixed so that it is at uniform temperature The tank is insulated to eliminate heat loss to the surrounding air. There is no heat storage in the insulation.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Definitions for variables of the system • θi = Steady state temperature of inflowing liquid, • θ = Steady state temperature of out-flowing liquid, • H = Steady state heat input rate from heater.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
What happens when you move from steady state?
•
Let ∆H be a small change in the heat input rate from its steady state value. This change in H will result in the following changes. – – –
Change in heat output rate by an amount ∆H1. Change in heat storage rate of liquid in the tank by an amount ∆H2. Change in temperature of out-flowing liquid by an amount ∆θ.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Thermal Resistance • Change in outflow heat rate is given by ∆H1 = Q Cs ∆θ • Where Q = Steady state liquid flow rate Cs = Specific heat of liquid
• ∆H1 = ∆θ/R – If R = 1/QCs which is defined as the Thermal Resistance July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Thermal Capacitance • Change in heat storage rate is given by ∆H2 = MCs d∆θ/dt • Where – M = mass of the liquid in the tank – ∆dθ/dt = rate of rise of temperature in the tank
• ∆H2 = C d∆θ/dt – Where C = MCs which is defined as Thermal Capacitance. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
Transfer function for the system ∆H= ∆H1 + ∆H2 • The mathematical model of a thermal system shown in figure is ΔH=
d Δθ Δθ +C R dt
• Applying Laplace transform
Δθ ( s ) + CsΔθ ( s ) R ⎡1 ⎤ = ⎢ + Cs ⎥ Δθ ( s ) ⎣R ⎦
ΔH(s)=
⎡1 + RCs ⎤ Δθ ( s ) = ⎢ ⎥ R ⎣ ⎦ July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
Transfer function of the system
Δθ ( s ) ⎡ R ⎤ = ⎢ ⎥ ΔH(s) ⎣1 + RCs ⎦
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Summary • Thermal systems are simple systems like level and do not yield to complexities like pneumatic or hydraulic systems. • Hence, it is easy and possible to associate analogous situations and derive the transfer function.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
References 1. Advanced Control Systems Engineering, Ronald Burns 2. Modern Control Engineering, Ogata 3. Control Systems, Nagoor Kani 4. A course in Electrical, Electronic Measurements and Instrumentation, A.K. Sawhney …amongst others
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
14
And, before we break… • Love has the patience to endure the fault we cannot cure.
Thanks for listening…
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
15
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