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Mechatronics

Thermal Systems

K. Craig

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Thermal Systems

Introduction to Thermal Systems

Introduction to Heat Transfer

What and How? Physical Mechanisms and Rate Equations

Conservation of Energy Requirement

Control Volume

Surface Energy Balance

Thermal Resistance Thermal Capacitance

Thermal Sources: Temperature and Heat Flow


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Introduction to Thermal Systems

Thermal systems transfer or store thermal energy by

virtue of temperature and heat flow rate. The

thermal effects are conduction, convection,radiation, and heat storage capacity.

Thermal systems have a static and dynamic behavior

similar to mechanical, electrical, and fluid systems,but in some ways they are quite different.

Thermal systems exhibit resistance and capacitance

effects, can be analyzed by circuit analysis, and have

dynamic responses that can be characterized by time

constants, etc.


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However, they often require nonlinear, variable-coefficient,

or distributed-parameter models. Also there is no thermal

inductance.

The analysis of thermal systems often requires the

combination of three technologies: Thermodynamics, Heat Transfer, Fluid Mechanics

Effort (Across) Variable: Temperature (C or K)

Flow (Through) Variable: Heat Flow Rate (W = J/sec)

Conduction, convection, and radiation all display an

algebraic relation between temperature and heat flow,i.e., no dynamic effects; these effects represent

thermal resistance.


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Introduction to Heat Transfer

Energy can be transferred by interactions of a

system with its surroundings:

Heat Work

Thermodynamics deals with equilibrium end states

of the process during which an interaction occurs. Itprovides no information concerning the nature of the

interaction or the time rate at which it occurs.

Heat Transferis inherently a non-equilibrium

process and we study the modes of heat transfer and

develop relations to calculate heat transfer rates.


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What is heat transfer?

Heat transfer (or heat) is energy in transit due to a

temperature difference.

Whenever there exists a temperature difference in a

medium or between media, heat transfer must occur.

Different types of heat transfer processes are called

modes:

Conduction: When a temperature gradient exists in a stationary

medium (solid or fluid), heat transfer occurs across themedium.

Convection: Heat transfer occurs between a surface and a

moving fluid when they are at different temperatures.

Radiation: All surfaces of finite temperature emit energy in the

form of electromagnetic waves. In the absence of an

intervening medium, there is heat transfer by radiation between

two surfaces at different temperatures.


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Heat Transfer Modes: Conduction, Convection, and Radiation


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Physical Mechanisms and Rate Equations

Conduction

Processes at the atomic and molecular level sustain this mode

of heat transfer.

Conduction may be viewed as the transfer of energy from the

more energetic to the less energetic particles of a substance dueto interaction between the particles.

Consider a gas with no bulk motion in which there exists a

temperature gradient. We associate the temperature at any

point with the energy of the gas molecules in the vicinity of thepoint: random translational motion as well as internal

rotational and vibrational motions of the molecules. Higher

temperatures are associated with higher molecular energies.

When particles collide, a transfer of energy from the more

energetic to the less energetic particles must occur. When a

temperature gradient is present, energy transfer by conduction

must occur in the direction of decreasing temperature.


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Association of Conduction Heat Transfer with Diffusion of

Energy due to Molecular Activity


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The situation is much the same in liquids, although in liquids

molecules are more closely spaced and the molecular

interactions are stronger and more frequent. In a solid, conduction may be attributed to atomic activity in

the form of lattice vibrations (exclusively for a non-conductor),

as well as translational motion of free electrons when the

material is a conductor.

Conduction Rate Equation

For heat conduction, the rate equation is known as Fouriers

Law. For a one-dimensional plane wall having a temperature

distribution T(x), the rate equation is:

x

2

x x

dT

q kdx

q heat flux (W/m ) q / A

k thermal conductivity (W/m K)

= = =

=

Heat flux is the heat transfer rate in

the x direction per unit area

perpendicular to the direction of

transfer


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Thermal Systems

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Convection

Convection heat transfer mode is comprised of two

mechanisms:

Energy transfer due to random molecular motion

(diffusion)

Energy transferred by the bulk (macroscopic) motion ofthe fluid. Large numbers of molecules moving collectively

in the presence of a temperature gradient gives rise to heat

transfer.

Total heat transfer is due to a superposition of energy transportby the random motion of molecules and by the bulk motion of

the fluid. The term convection is used to refer to this

cumulative transport, while the term advection is used to refer

to transport due to bulk fluid motion.

We are especially interested in convection heat transfer

between a fluid in motion and a bounding surface when the

two are at different temperatures.


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Boundary Layer Development in Convection Heat Transfer


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Hydrodynamic, or velocity, boundary layer: region in the fluid

through which the velocity varies from zero at the surface to a

finite value u associated with the flow. If the surface and flow temperatures differ, there will be a

region of the fluid through which the temperature varies from

Ts at the surface to T in the outer flow. This region is called

the thermal boundary layerand it may be smaller, larger, orthe same size as that through which the velocity varies.

The contribution to heat transfer due to random molecular

motion (diffusion) generally dominates near the surface where

the fluid velocity is low.

The contribution to heat transfer due to bulk fluid motion

originates from the fact that boundary layers grow as the flow

progresses in the x direction. Nature of the Flow:

Forced Convection: flow is caused by some external

means, e.g., fan, pump, wind.


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Free (or natural) Convection: flow is induced by buoyancy

forces in the fluid, which arise from density variations

caused by temperature variations in the fluid.

Convection heat transfer is then energy transfer occurring

within a fluid due to the combined effects of conduction and

bulk fluid motion. In general, the energy that is being

transferred is the (sensible) internal thermal energy of the fluid. However, there are convection processes for which there is, in

addition, latentheat exchange. This latent heat exchange is

generally associated with a phase change between the liquid

and vapor states of the fluid, e.g., boiling and condensation.

Convection Rate Equation

Regardless of the particular nature of the convection heat

transfer mode, the rate equation is of the form:( )s2

x

2

q h T T

q convective heat flux (W/m )

h convection heat transfer, or film, coefficient (W/m K)

=

=

=

Newtons Law of Cooling


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The film coefficient, h, encompasses all the effects that influence

the convection mode, e.g., boundary layer conditions, surface

geometry, nature of fluid motion, fluid thermodynamic andtransport properties.

Range of Values for h (W/m2K):

Free Convection: 5 25 Forced Convection: 25 250 (Gases), 50 20,000 (Liquids)

Convection with Phase Change (boiling or condensation):

2500 100,000

Radiation

Thermal radiation is energy emitted by matter (solid, fluid, or gas)

that is at a finite temperature, attributable to changes in the

electron configurations of the constituent atoms or molecules. The energy of the radiation field is transported by electromagnetic

waves and does not require the presence of a material medium, in

fact, it occurs most efficiently in a vacuum.


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Radiation Rate Equation

The maximum flux (W/m2

) at which radiation may be emittedfrom a surface is given by the Stefan-Boltzmann Law:

Such a surface is called an ideal radiator. The heat flux

emitted by a real surface is less than that of the ideal radiatorand is given by:

Determination of the net rate at which radiation is exchanged

between surfaces is quite complicated.

4

s

s

2 4

q T

T absolute surface temperature (K)Stefan-Boltzmann Constant = 5.67E-8 (W/m K )

=

= =

4

sq T

emissivity (radiative property of the surface)

=

=


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Special Case: net exchange between a small surface and a much

larger surface that completely surrounds the smaller one. The

surface and surroundings are separated by a gas that has no effecton the radiation transfer. The net rate of radiation heat exchange

between the surface and its surroundings is:

Note that the area and emissivity of the surroundings do not

influence the net heat exchange rate in this case.

( ) ( )( )( ) ( )4 4 2 2

s surr s surr s surr s surr r s surr

q

q T T T T T T T T h T TA = = = + + =

Radiation Exchange between a Surface and its Surroundings

total conv radq q q= +


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Conservation of Energy Requirement

In the application of the conservation laws, we firstneed to identify the control volume, a fixed region of

space bounded by a control surface through which

energy and matter pass.

Energy Conservation Law:

The rate at which thermal and mechanical energy enters a

control volume through the control surface minus the rate at

which this energy leaves the control volume through thecontrol surface (surface phenomena) plus the rate at which

energy is generated in the control volume due to conversion

from other energy forms, e.g., chemical, electrical,

electromagnetic, or nuclear, (volumetric phenomena) mustequal the rate at which this energy is stored in the control

volume (volumetric phenomena).

in g out stE E E E+ =


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The inflow and outflow rate terms are surface phenomena in

that they are associated exclusively with processes occurring atthe control surface, and the rate at which they occur is

proportional to surface area. Usually they involve energy flow

due to heat transfer by the conduction, convection, and/or

radiation modes; they may also include work interactions. In asituation involving fluid flow across the control surface, these

terms also include energy transported by the fluid into and out

of the control volume in the form of potential, kinetic, and

thermal energy. Example Problem

A long conducting rod of diameter D and electrical resistance

per unit length Re is initially in thermal equilibrium with the

ambient air and its surroundings. This equilibrium is disturbed

when an electrical current I is passed through the rod. Develop

an equation that could be used to compute the variation of the

rod temperature with time during passage of the current.


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Surface Energy Balance

We frequently apply the conservation of energy requirement atthe surface of a medium. In this case, the control surface

includes no mass or volume. The generation and storage terms

of the conservation expression are no longer relevant. The

conservation requirement, holding for both steady-state andtransient conditions, then becomes:

Energy Balance for Conservation of Energy at the Surface of a Medium

in outE E 0 =

cond conv radq q q 0 =


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Example Problem

A closed container filled with hot coffee is in a room whose air

and walls are at a fixed temperature. Identify all heat transfer

processes that contribute to cooling of the coffee. Comment

on features that would contribute to a superior container

design.


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Thermal Resistance

Whenever two objects (or portions of the same object)

have different temperatures, there is a tendency for heat to

be transferred from the hot region to the cold region, in an

attempt to equalize the temperatures.

For a given temperature difference, the rate of heat transfer

varies, depending on the thermal resistance of the path

between the hot and cold regions.

The nature and magnitude of the thermal resistance depend

on the mode of heat transfer involved:

Conduction

Convection

Radiation


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( ) ( )1 2kA

q T t T tL

=

Through and Across Variables

Through Variable: heat flow rate q (J/s or W)

Across Variable: Temperature T (K)

Pure and Ideal Resistance Element

Instantaneous relation Conduction

Convection

Radiation

T(t)qR

=

( ) ( )1 2q hA T t T t =

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

4 4

1 2

2 2

1 2 1 2 1 2

q C T t T t

C T t T t T t T t T t T t

=

= + +


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Thermal Conductivity k

Material property found by experiments. Ideally it is aconstant, but in reality it may vary with temperature,

position in the body, and direction of heat flow.

Printed circuit boards are a good example of anisotropic(direction-sensitive) behavior of thermal conductivity.

Here the material is in the form of a sandwich with

layers of high-conductivity copper and low-

conductivity epoxy-fiberglass. Thermal conductivity of

the composite sandwich in a direction perpendicular to

the plane of the board may be only 0.05 times that for

the parallel direction.


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Contact Resistance

When heat flow occurs through the interface where twosolid bodies share a common surface, the phenomenon

of contact resistance is observed.

If the contact were perfectly smooth, the contactresistance would be zero and the temperature of the two

bodies would be identical at the contact surface.

Real objects always have some surface roughness,which causes essentially a step change in temperature

across the interface. This effect can be modeled with a

thermal contact resistance, which depends on the

roughness of the surfaces and the contact pressure for

any two given materials. Contact resistances are

difficult to measure and predict.


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For example, aluminum-to-aluminum joints may have

resistance values ranging from 8.3E-5 to 45.0E-5 C/W

for an area of 1 m2. The two aluminum piecesthemselves, taking 5 mm as a typical thickness, would

have a total resistance of about 5.0E-5 C/W for the

same 1.0 m2

area, showing clearly the large errorcaused by ignoring contact resistance.

Convection Film Coefficient h

The convection film coefficient depends on thegeometry of the solid bodies, the nature of the fluid

flow, the fluid properties, and temperature. It must be

found by experiment, but for many configurations theexperimental results have been generalized so that h

may be predicted with fair accuracy from calculations.


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Combined Heat Transfer Coefficient

Often conduction and convection are combined and wecan define an overall heat transfer coefficient and

thereby an overall thermal resistance.

Consider the automobile radiator (convector is a better

name!)

water air

t

water air

T TTq1 L 1R

h A kA h A

= =+ +

C t R di ti


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Comments on Radiation

Radiation often contributes a relatively small portion of

the total heat transfer unless the temperatures are quitehigh. However, if other modes are inhibited, then

radiation can be important even at low temperatures,

e.g., heat transfer at the outer surface of an orbitingsatellite must be entirely due to radiation since it is

exposed only to the vacuum of space, defeating any

conduction or convection.Note that emissivity values are usually more uncertain

than conductivities or convection coefficients, so highly

accurate calculations should not be expected.

For radiation calculations, you must use absolute

temperatures!


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Rate of radiation heat transfer depends on the

emissivity of each body (surface property), geometrical

factors involving the portion of emitted radiation fromone body that actually strikes the other, the surface

areas involved, and the absolute temperatures of the

two bodies.


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Thermal Resistance Element

Electrical-Thermal Analogy

Voltage Temperature Difference

Current Heat Flux

However, when energy behavior is considered, the analogy

breaks down as heat flux is already power and current is not.Also, all the heat flux entering the thermal resistance at one

end leaves at the other end, and none is lost or dissipated;

whereas the electrical energy supplied to a resistor is all

converted into heat, and is thus lost to the electrical system.

In thermal system analysis and design, the overall

thermal resistance of hardware components, obtained

from lab testing, is widely used, particularly inelectronic and electromechanical applications.

Thermal Capacitance


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Thermal Capacitance

When heat flows into a body of solid, liquid, or gas, thisthermal energy may show up in various forms such as

mechanical work or changes in kinetic energy of a flowing

fluid.

If we restrict ourselves to bodies of material for which the

addition of thermal energy does not cause significant

mechanical work or kinetic energy changes, the added

energy show up as stored internal energy and manifests

itself as a rise in the temperature of the body.

For a pure and ideal thermal capacitance, the rise in

temperature is directly proportional to the total quantity of

heat energy transferred into the body: t0

0t

1T T qdt

C =


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We assume that, at any instant, the temperature of the body

is uniform throughout its volume. For fluid bodies, this

ideal situation is closely approached if the fluid is

thoroughly and continuously mixed. For solid bodies,

uniform temperature requires a material with infinite

thermal conductivity, which no real material has. Thusthere is always some nonuniformity of temperature in a

body during transient temperature changes.

A useful criterion for judging the validity of the uniform-temperature assumption for a solid body immersed in a

fluid is found in the Biot Number NB:

B

VolumehhLSurface Area

Nk k

=


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When the Biot number is less than 0.1, the assumption of

uniform temperature is acceptable, except for the early

times of a step change in fluid temperature.

Early vs. later times? The division is not precise but can be

estimated from another dimensionless group, the Fourier

number NF:

is the thermal diffusivity and this governs the diffusionof heat through a solid body. A large value of meansrapid diffusion of heat.

A conservative requirement on the Fourier number is that

it be greater than 10 for the uniform temperature

assumption to be accurate.

F 2 2

kt

ctN

L L

=

If the spatial variation of temperature rather than an


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If the spatial variation of temperature, rather than an

average temperature, in the solid body must be predicted,

we should use several lumps of thermal capacitance, ratherthan just one, in our model. Sometimes we must begin our

modeling with several lumps and let these results tell us if

we can simplify the model to fewer, or just one, lump ofthermal capacitance.

Thermal Capacitance Ct

The specific heat of real materials varies somewhat with

temperature, however, in many cases it is sufficientlyaccurate to use a constant value (average value for the

range of temperature covered).

Ct heat added mass specific heat Mctemperature rise = =

F fl id ( ti l l ) th ifi h t i ft


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For fluids (particularly gases) the specific heat is often

measured for two different situations: constant volume and

constant pressure. Since these values are quite different,be careful to use the value which corresponds most closely

to the actual application.

When heat is added to or taken away from a materialwhich is changing phase (melting or freezing, vaporizing

or condensing) the thermal capacitance is essentially

infinite, since one can add heat without causing anytemperature rise.

NOTE: Thermal Inductance is not necessary for the

description of thermal system behavior and is not definedor used! Thermal systems require only two elements, and

only one of these stores energy.

Thermal Sources:


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Thermal Sources:

Temperature and Heat Flow The ideal temperature source maintains a prescribed

temperature (either constant or time-varying) irrespective

of how much heat flow it must provide. Constant-temperature sources may often be quite well approximated

by utilizing materials undergoing phase change.

An ideal heat-flow source produces a prescribed (constantor time-varying) heat flow irrespective of the temperature

required. Perhaps the most convenient heat flow source

for many applications is electrical resistance heating. Aconstant or time-varying voltage applied to a resistance

heating coil produces an electrical heat generation rate

e2(t)/R if inductance is negligible.

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