1. CHAPTER 5 MESB 374 System Modeling and Analysis Thermal Systems
ME375 Handouts - Spring 2002
CHAPTER 5 MESB System Modeling and Analysis Thermal Systems

2. Thermal Systems Basic Modeling Elements Interconnection Relationships
ME375 Handouts - Spring 2002
Thermal Systems
Basic Modeling Elements
Resistance
Conduction
Convection
Radiation
Capacitance
Interconnection Relationships
Energy Balance - 1st Law of Thermodynamics
Derive Input/Output Models

3. Variables q : heat flow rate [J/sec = W] ( )
ME375 Handouts - Spring 2002
Variables
q : heat flow rate [J/sec = W] ( )
T : temperature [oK] or [oC] ( )
current,
power
voltage
Temperature in a body usually depends on spatial as well as temporal coordinates. As a result, the dynamics of a thermal system has to be described by partial differential equations. Moreover, nonlinearities are often essential in describing the heat transfer by radiation and convection. However, very few nonlinear PDEs have analytical (closed form) solutions. Usually, finite element methods (FEM) are used to numerically solve nonlinear PDE problems. Our purpose is to try to use lumped model approximations of thermal systems to obtain linear ODEs that are capable of describing the dynamic response of thermal systems to a good first approximation.

4. Basic Modeling Elements
ME375 Handouts - Spring 2002
Basic Modeling Elements
Thermal Resistance
Describes the heat transfer process through an element with the characteristic that the heat flow rate across the element is proportional to the temperature difference across the element, i.e.
Ex: Two bodies at temperatures T1 and T2 are separated by two elements with different thermal resistance R1 and R2. Heat flows through the two elements at a rate of q. Find the equivalent thermal resistance Req and solve for the interface temperature between the two elements. q R
+ DT - T1 T2 T1 T2 R1 R2 q q T1 T2
Req T1 q T2 R T1 T2 R1 R2 q T0

5. Three Types of Heat Transfer
ME375 Handouts - Spring 2002
Three Types of Heat Transfer
Conduction
Heat transfer through solid or continuous media via random molecular motion (diffusion).
a : thermal conductivity [W/m-oK]
Ex: Calculate the equivalent thermal resistance of a wall with a window.
Wall Window
Area AW AG
Thickness dW dG
a aW aG T1 T2 q d
Cross sectional area A x T
Resistors connected in Parallel RW RG

6. Three Types of Heat Transfer
ME375 Handouts - Spring 2002
Convection
Heat transfer between the interface of a solid material and a fluid material via bulk motion of the fluid.
Radiation
Heat transfer via electromagnetic waves
A : surface area [m2]
s : Stefan-Boltzmann constant [W/m2-oK4]
FE : effective emissivity
FV : view factor
Nonlinear! Will not be considered in this course T1 T2
Surface Area A q
A : surface area [m2]
h : convective heat transfer coefficient [W/m2-oK]
TS : surface temperature [oK]
TF : fluid temperature [oK]
h depends on surface geometry, fluid flow rate, temperature, flow direction, ...
Q: How would we model the process of storing thermal energy ?

7. Basic Modeling Elements
ME375 Handouts - Spring 2002
Basic Modeling Elements
Thermal Capacitance
The ability of a substance to hold or store heat is the heat capacity of the material and it behaves like a thermal capacitance. Since the specific heat cP can be interpreted as the heat storage capacity of the material per unit mass, the total heat storage capability of a material is:
If there is net heat flow into the material, the temperature of the material will change and the rate of temperature change is proportional to the net heat flow rate qstore:
Thermal Capacitance
Note:
+ TC -
qIN C TC
qOUT
Mass, M
Volume, V
Density, r
qIN - qOUT C
The above relationship holds only if we assume that the temperature is uniform across the entire material.

8. Interconnection Laws Energy Balance - 1st Law of Thermodynamics
ME375 Handouts - Spring 2002
Interconnection Laws
Energy Balance - 1st Law of Thermodynamics
Energy stored in the system is the sum of the net energy inflow, the energy generated within the system and the work done on the system:
Thermal EOMs are obtained by the balance of energy
Remember that there is NO inertia in macroscopic thermal system

9. Example (a simple first order system)
ME375 Handouts - Spring 2002
Example (a simple first order system)
Ex: A material with a thermal capacitance C is surrounded by an insulation material with thermal resistance R. Heat is added to the inner material at a rate of qi(t). Find the system model, if the inner material temperature TC is to be the output.
Equivalent electrical circuit C Tr Tc Ta + - R Ta
TC , C R
qi(t)

10. Example q2 R1 TC2 C2 TC1 C1 qc1 R2 qc2 Equivalent electrical circuit
ME375 Handouts - Spring 2002
Example
TC1 C1
Ta: R1
qi(t)
TC2 C2 R2
Equivalent electrical circuit C1 Tr
Tc1 Ta + - R1 C2 R2
qc1
qc2
Tc2 q2
State Space Representation:
EOMs:
Steady State (equilibrium)?

11. Example State Space Representation: EOMs for changes in temperatures
ME375 Handouts - Spring 2002
Example
State Space Representation:
EOMs for changes in temperatures
from equilibrium :
Steady State (equilibrium)?

12. ME375 Handouts - Spring 2002
In Class Example dP TA
Heat Sink dS
cP , rP , TP hA qi
Ex: The Pentium processor under normal operation will generate heat at a rate of qi(t). The processor itself has a specific heat of cP. The cross sectional area of the chip is AP with a thickness of dP. The average density of the processor is rP . To help dissipate the heat and reduce the processor temperature TP, a heat sink with the same cross sectional area and an average thickness of dS is added on top of the processor. The heat sink has a thermal conductivity of aS . The effective conduction area between the heat sink and processor is approximated by AP . To further improve heat dissipation, a fan is used to generate air flow on top of the heat sink, the effective convection coefficient is hA and the effective contact area between the heat sink and the air flow is AS . The temperature inside the computer is maintained at TA. Find the relationship between qi(t) and the temperature of the processor TP. AP C Tr Tp TA + - Rs Ra
1st law: