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مقدمه اي بر رسانش

Conduction: transfer of thermal energy from the more energeticparticles of a medium to the adjacent less energetic ones

Unlike temperature, heat transfer has direction as well as magnitude,and thus it is a vector quantity

Chapter2: Conduction Dr. Afsaneh MojraChapter2: Conduction Dr. Afsaneh Mojra

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مقدمه اي بر رسانش

راستاي بردار انتقال حرارت


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رسانایی متغیري از زمان و مکان

Heat transfer problems are often classified as being steady (also calledsteady-state) or transient (also called unsteady)

For example, heat transfer throughthe walls of a house will be steady whenthe conditions inside the house andthe outdoors remain constant for several hours.


هدایت پایدار و گذرا وابستگی یا استقلال نرخ انتقال حرارت از زمان

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زمان و مکانرسانایی متغیري از


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The cooling of an apple in a refrigerator, on the other hand, is a transientheat transfer process since the temperature at any fixed point within theapple will change with time during cooling

temperature varies with time as well as position


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Lumped System: In case of variation with time but not with position, thetemperature of the medium changes uniformly with time.

Example: A small metal object such as a thermocouple junction or a thincopper wire

Real Problems: Steady-State? Transient? Lumped?


به مکان وابستگی یا استقلال نرخ انتقال حرارت

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انتقال حرارت چند بعدي


در دستگاه هاي مختلف مختصات

با توجه به هندسه و ماهیت فیزیکی مسأله

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Solar-panel simulation9/86

معادلات حاکم بر انتقال حرارت رسانایی

Fourier’s law of heat conduction for one-dimensional heat conduction


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Fourier’s law of heat conduction for three-dimensional heat conduction

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Most engineering materials are isotropic in nature, and thus they havethe same properties in all directions. For such materials we do not needto be concerned about the variation of properties with direction.


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But in Anisotropic materials such as the fibrous or composite materials,the properties may change with direction. In such cases the thermalconductivity may need to be expressed as a tensor quantity to account forthe variation with direction.

Here, we will assume the thermal conductivity of a material to beindependent of direction.


Orthotropic materialAnisotropic material13/86


A medium through which heat is conducted may involve the conversionof electrical, nuclear, or chemical energy into heat (or thermal) energy. Inheat conduction analysis, such conversion processes are characterized asheat generation.


Heat Generation

Thermal Simulation forCPU Cooling

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heat generation is a volumetric phenomenon. That is, it occursthroughout the body of a medium. Therefore, the rate of heat generationin a medium is usually specified per unit volume.


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مختصات کارتزین


ONE-DIMENSIONAL HEAT CONDUCTION EQUATION

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Consider a thin element: thickness ∆x, density of the wall is , specific heat is C, area of the wall normal to the direction of heat transfer is A

Energy balance on this thin element during a small time interval ∆t:


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= : thermal diffusivitySpecial cases:

کارتزینمعادلات کلی انتقال حرارت یک بعدي در مختصات

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partial and ordinary derivatives of a function are identical when thefunction depends on a single variable only [T = T(x) in this case]

اياستوانهمختصات:


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Energy balance on this cylindrical shell element during a small timeinterval ∆t:


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or

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معادلات کلی انتقال حرارت یک بعدي در مختصات استوانه اي


Special cases:

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کرويمختصات:


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کرويمعادلات کلی انتقال حرارت یک بعدي در مختصات


Special cases

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کارتزین، استوانه اي و کرويانتقال حرارت یک بعدي در مختصات شکل عمومی معادله


n=0 for a plane walln=1 for a cylindern=2 for a sphere

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ASSUMTIONS Resistance wire can be considered to be a very long cylinder since its

length is more than 100 times its diameter. Heat is generated uniformly in the wire and the conditions on the outer

surface of the wire are uniform. Therefore, it is reasonable to expect the temperature in the wire to vary

in the radial r direction only and thus the heat transfer to be 1D. Then we will have T=T(r) during steady operation.


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GENERAL HEAT CONDUCTION EQUATION

Rectangular CoordinatesRectangular Coordinates

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Energy balance on this element during a small time interval ∆t:


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Constant thermal conductivity:


Fourier-Biot EquationFourier-Biot Equation

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Cylindrical CoordinatesCylindrical Coordinates

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Spherical CoordinatesSpherical Coordinates

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شرایط مرزي و اولیه حاکم بر معادلات


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heat flux and the temperature distribution in a medium depend on theconditions at the surfaces

The mathematical expressions of the thermal conditions at theboundaries are called the boundary conditions (BCs)


Boundary & Initial Conditions

Solving of PDE (heat transfer equation)

arbitrary constants: BCs

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Condition at the beginning of the heat conduction process

medium is initially at a uniform temperature of Ti:

steady conditions: heat conduction equation does not involve anytime derivatives, and thus we do not need to specify an initialcondition.


Initial Conditions (ICs)

= : , , , ≡ , ,, , ?, , =

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Boundary Conditions (BCs)

Specified Temperature Specified Heat Flux

Convection Radiation

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The temperature of an exposed surface can usually be measured directlyand easily.

The specified temperatures can be constant, which is the case for steadyheat conduction, or may vary with time


Specified Temperature Boundary Condition

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When there is sufficient information about energy interactions at asurface, it may be possible to determine the rate of heat transfer and thusthe heat flux ̇ (heat transfer rate per unit surface area, W/m2) on thatsurface.

The heat flux can be expressed by Fourier’s law:


Specified Heat Flux Boundary Condition

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Some surfaces are commonly insulated in practice in order to minimizeheat loss (or heat gain) through them.

a well-insulated surface can be modeled as a surface with a specified heatflux of zero:


Special Case: Insulated Boundary

temperature function must be perpendicularto an insulated surface since the slope oftemperature at the surface must be zero

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For example, the two surfaces of a large hot plate of thickness Lsuspended vertically in air will be subjected to the same thermalconditions, and thus the temperature distribution in one half of the platewill be the same as that in the other half


Special Case: Thermal symmetry

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heat transfer problem in this plate will possess thermal symmetry aboutthe center plane at x=L/2

no heat flow across the center plane.


the center plane can be viewed as aninsulated surface

insulation or zero heat flux boundarycondition

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cylindrical (or spherical) bodies

thermal symmetry about the center line (or midpoint)

first derivative of temperature with respect to r (the radialvariable) be zero at the centerline (or the midpoint)

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most common boundary condition


Convection Boundary Condition

Surface Energy Balance

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1D heat transfer in the x-dir in a plate of thickness L

Note that a surface has zero thickness and thus no mass, and itcannot store any energy. Therefore, the entire net heat entering thesurface from one side must leave the surface from the other side.


heat continues to flow from a body to the surrounding medium atthe same rate, and it just changes vehicles at the surface from

conduction to convection (or vice versa)

T(0,t) & T(L,t) ??

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Radiation Boundary Condition

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In some cases, a heat transfer surface is surrounded by an evacuatedspace and thus there is no convection heat transfer between a surface andthe surrounding medium.

In such cases, radiation becomes the only mechanism of heat transferbetween the surface under consideration and the surroundings.

1D heat transfer in the x-dir in a plate of thickness L


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تعیین شرایط مرزي در ناحیه در تماس دو محیط به دو فرض زیر وابسته است:


Interface Boundary Condition

layers of different materialslayers of different materials

1. same temperature at the area of contact1. same temperature at the area of contact

2. an interface (which is a surface) cannot store any energy, and thus theheat flux on the two sides of an interface must be the same

2. an interface (which is a surface) cannot store any energy, and thus theheat flux on the two sides of an interface must be the same

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Perfect ContactPerfect Contact

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Imperfect ContactImperfect Contact

results in thermal contact resistance, which is considered in the next chapterresults in thermal contact resistance, which is considered in the next chapter

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مسائل انتقال حرارت رسانایی


Solution of steady 1D heat conduction problems

مسأله انتقال حرارتمسأله انتقال حرارت

معادله انتقال حرارت حاکم بر مسأله با در نظر گرفتن ترم هاي مختلف حرارتی) بدست آوردن معادله دیفرانسیل حاکم(

معادله انتقال حرارت حاکم بر مسأله با در نظر گرفتن ترم هاي مختلف حرارتی) بدست آوردن معادله دیفرانسیل حاکم(

حل عمومی معادله دیفرانسیلحل عمومی معادله دیفرانسیل

شرایط مرزياعمال شرایط مرزياعمال 61/86



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فرضیات:


noثابت kثابت kرسانایی پایارسانایی پایارسانایی یک بعديرسانایی یک بعدي heat gen.no heat gen.

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noثابت kثابت kرسانایی پایارسانایی پایارسانایی یک بعديرسانایی یک بعدي heat gen.no heat gen.فرضیات:

no radiationno radiation

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معادله توزیع دما در یک استوانه

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= lnln − + = ( ) ( − )

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معادله توزیع دما در یک کره

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= − ( − )76/86



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Composite Media

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Wire:Wire:

Twire(r=0)=? ButTI,wire=Twire(r=r1)??

Twire(r=0)=? ButTI,wire=Twire(r=r1)??

T1? Heat transfer equation on ceramicT1? Heat transfer equation on ceramic

Ceramic:Ceramic:

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Interface B.C.

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ضریب انتقال حرارت رسانایی متغیر

When the variation of thermal conductivity with temperature in aspecified temperature interval is large, however, it may be necessary toaccount for this variation to minimize the error.


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