Mathematical Models for Modes of Heat Action

P M V SubbaraoProfessor

Mechanical Engineering Department

Tools to select Means of Heat Interactions between System and

Surroundings ….

The Pentium 4 Processor

Heat Sinks for Pentium 4

Pentium 4 While Performing

                                                           

                                         

Statement of Fourier’s LawThe (mod of) heat flux, q’’, (the flow of heat per unit area and per unit time), at a point in a medium is directly proportional to the temperature gradient at the point.

x

TTTThot

xxx

x

Temperature gradient across the slab of thickness x:

xxx

xTxxT hotcold

)()(

The heat flux across the slab

xxx

xTxxT

xxx

xTxxTq

hotcold

hotcold

)()(

)()(''

TTcold

xxx

xTxxTkq hotcold )()(

''

Local Heat flux in a slab:

xxx

xTxxTkq hotcold

xx

)()(lim ''

0

dx

dTkqx ''

Global heat transfer rate:

dx

dTkAqx

Mathematical Description

• Temperature is a scalar quantity.

• Heat flux is defined with direction and Magnitude : A Vector.

• Mathematically it is possible to have:

kqjqiqq zyxˆ''ˆ''ˆ'' ''

Using the principles of vector calculus:

Tkq ''

kz

Tj

y

Ti

x

Tkq ˆˆˆ''

Further Physical Description

• Will k be same in all directions?• Why k cannot be different each direction?• Why k cannot be a vector?

kkjkikk zyxˆˆˆ

Will mathematics approve this ?

What is the most general acceptable behavior of k, approved by both physics and mathematics?

Anisotropic Materials

x,y

z

Thermal Image of Laptop Casing

Graphite Covering

Thermal Image of Laptop Casing with Graphite cover

Most General form of Fourier Law of Conduction

dx

dTkqx ''

Tkq ''

kkjkikk zyxˆˆˆ

We are at cross roads !!!!!

Physically & mathematically Feasible Model

• Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is:

Tkq .~~

''

Thermal conductivity of a general material is a tensor.

z

T

y

Tx

T

kkk

kkk

kkk

q

q

q

zzzyzx

yzyyyx

xzxyxx

z

y

x

''

''

''

Surprising Inventions !!!

z

Tk

y

Tk

x

Tkq xzxyxxx ''

z

Tk

y

Tk

x

Tkq yzyyyxy ''

z

Tk

y

Tk

x

Tkq zzzyzxy ''

Radiative Mode of Heat Transfer

• Any body (> absolute zero) emits radiation at various wavelengths.

• Transparent bodies radiate energy in spherical space.

• Non-transparent bodies radiate energy in hemi-spherical space.

• The radiation energy emitted by a body is distributed in space at various wavelengths.

• This complex phenomenon requires simplified laws for engineering use of radiation.

Planck Radiation Law

• The primary law governing blackbody radiation is the Planck Radiation Law.

• This law governs the intensity of radiation emitted by unit surface area of a blackbody as a function of wavelength for a fixed temperature.

1

12,

5

2

kT

hcb

e

hcTE

h = 6.625 X 10-27 erg-sec (Planck Constant)

K = 1.38 X 10-16 erg/K (Boltzmann Constant)

C = Speed of light in vacuum

• The Planck Law can be expressed through the following equation.

Stefan-Boltzmann Law

• The maximum emissive power at a given temperature is the black body emissive power (Eb).

• Integrating this over all wavelengths gives Eb.

05

2

0 1

12,

d

e

hcdTE

kT

hcb

444

42

15

2TT

khc

hcTEb

• Driving forces: Heat transfer by radiation is driven by differences in emissive power (proportional to T4.

The total energy emitted by a real system, regardless of the wavelengths, is given by:

4syssurfacesyssysemitted TAQ

• where εsys is the emissivity of the system,• Asys-surface is the surface area, • Tsys is the temperature, and • σ is the Stefan-Boltzmann constant, equal to 5.67×10-8 W/m2K4. • Emissivity is a material property, ranging from 0 to 1, which

measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature

Radiation from a Thermodynamic System

Radiative Heat Transfer between System and Surroundings Consider the heat transfer between system surface with surroundings, as shown in Figure. What is the rate of heat transfer into system surface ?

4, sursursuremittedsur TAQ

This radiation is emitted in all directions, and only a fraction of it will actually strike system surface. This fraction is called the shape factor, F.

To find this, we will first look at the emission from surroundings to system. Surrounding Surface emits radiation as described in

The amount of radiation striking system surface is therefore:

4, sursursursyssurinceidentsys TAFQ

The only portion of the incident radiation contributing to heating the system surface is the absorbed portion, given by the absorptivity αB:

4, sursursursyssursysabsorbedsys TAFQ

Above equation is the amount of radiation gained by System from Surroundings. To find the net heat transfer rate for system, we must now subtract the amount of radiation emitted by system:

4, syssyssysemittedsur TAQ

The net radiative heat transfer (gain) rate at system surface is

emittedsysabsorbedsyssys QQQ ,,

44syssyssyssursursursyssursyssys TATAFQ

Similarly, the net radiative heat transfer (loss) rate at surroundings surface is

44sursursursyssyssyssursyssursur TATAFQ

What is the relation between Qsys and Qsur ?

44syssyssyssursursursyssursyssys TATAFQ