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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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    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.