• Thermal Contact Resistance• Log Mean Area of the Cylinder• Geometric Mean Area of the Sphere• Critical Radius of Insulation

• Thermal Contact Resistance• Log Mean Area of the Cylinder• Geometric Mean Area of the Sphere• Critical Radius of Insulation

Thermal Contact Resistance

gapcontact QQQ

erfacec TAhQ int

erfacec TAhQ int

erfacec T

AQh

int

/

(W/m2 0C)

(m2 0C/ W)AQ

T

hR erface

cc

/

1 int

hC: thermal contact conductance

Thermal contact resistance is inverse ofthermal contact conduction,

Depends on• Surface roughness,• Material properties,• Temperature and pressure at interface,• Type of fluid trapped at interface

Thermal contact resistance is inverse ofthermal contact conduction,

Depends on• Surface roughness,• Material properties,• Temperature and pressure at interface,• Type of fluid trapped at interface

CRITICAL RADIUS OF INSULATION

9. A 5-mm-diameter spherical ball at 50°C is coveredby a 1-mm-thick plastic insulation (k 0.13 W/m · °C).The ball is exposed to a medium at 15°C, with acombined convection and radiation heat transfercoefficient of 20 W/m2· °C. Determine if the plasticinsulation on the ball will help or hurt heat transferfrom the ball.

Subsidiary laws and Fundamental LawsSubsidiary laws

• FOURIER’S LAW OF HEAT CONDUCTION• NEWTON’S LAW OF COOLING FOR CONVECTION• STEFAN-BOLTZMANN LAW OF RADIATION

Fundamental Laws• The Laws of conservation of mass• The laws of conservation of momentum,• The Laws of conservation of energy

Subsidiary laws

• FOURIER’S LAW OF HEAT CONDUCTION• NEWTON’S LAW OF COOLING FOR CONVECTION• STEFAN-BOLTZMANN LAW OF RADIATION

Fundamental Laws• The Laws of conservation of mass• The laws of conservation of momentum,• The Laws of conservation of energy

GENERAL HEAT CONDUCTION EQUATION in Rectangular Coordinates

kC is called thermal diffusivity (m2/s).

• The product , which is frequently encounteredin heat transfer analysis, is called the heat capacityof a material. Both the specific heat Cp and theheat capacity represent the heat storagecapability of a material. But Cp expresses it per unitmass whereas expresses it per unit volume,

• Another material property that appears in thetransient heat conduction analysis is the thermaldiffusivity, which represents how fast heatdiffuses through a material and is defined asthermal diffusivity

• The product , which is frequently encounteredin heat transfer analysis, is called the heat capacityof a material. Both the specific heat Cp and theheat capacity represent the heat storagecapability of a material. But Cp expresses it per unitmass whereas expresses it per unit volume,

• Another material property that appears in thetransient heat conduction analysis is the thermaldiffusivity, which represents how fast heatdiffuses through a material and is defined asthermal diffusivity

Check whether FL satisfies or not

GENERAL HEAT CONDUCTION EQUATION in Cylinderical Coordinates

GENERAL HEAT CONDUCTION EQUATION in Spherical Coordinates

BOUNDARY AND INITIAL CONDITIONS• Mathematical expressions of the thermal conditions at the

boundaries are called the boundary conditions.• Initial condition, which is a mathematical expression for the

temperature distribution of the medium initially

1. Specified Temperature BoundaryCondition1. Specified Temperature BoundaryCondition

2. Specified Heat Flux Boundary Condition

3. Convection Boundary Condition3. Convection Boundary Condition

4. Radiation Boundary Condition

5. Interface Boundary Conditions5. Interface Boundary Conditions

VARIABLE THERMAL CONDUCTIVITY, k (T )

• The variation in thermalconductivity of a material withtemperature in the temperaturerange of interest can often beapproximated as a linearfunction and expressed as

where is called the temperaturecoefficient of thermalconductivity.

• The variation in thermalconductivity of a material withtemperature in the temperaturerange of interest can often beapproximated as a linearfunction and expressed as

where is called the temperaturecoefficient of thermalconductivity.

10. Consider a 1.5-m-high and 0.6-m-wide plate whose thickness is0.15 m. One side of the plate is maintained at a constanttemperature of 500 K while the other side is maintained at 350 K.The thermal conductivity of the plate can be assumed to varylinearly in that temperature range asWhere

Assuming steady one-dimensional heat transfer, determine therate of heat conduction through the plate.

10. Consider a 1.5-m-high and 0.6-m-wide plate whose thickness is0.15 m. One side of the plate is maintained at a constanttemperature of 500 K while the other side is maintained at 350 K.The thermal conductivity of the plate can be assumed to varylinearly in that temperature range asWhere

Assuming steady one-dimensional heat transfer, determine therate of heat conduction through the plate.

T2

k(T)

T1

K W/m24.34

2

K350)+(500)K10(8.7+1K) W/m25(

21)(

1-4-

120aveave

TTkTkk

The average thermal conductivity of the medium in this case issimply the conductivity value at the average temperature sincethe thermal conductivity varies linearly with temperature, and isdetermined to be

L

K W/m24.34

2

K350)+(500)K10(8.7+1K) W/m25(

21)(

1-4-

120aveave

TTkTkk

W30,820

m15.0

0)K35(500m)0.6mK)(1.5W/m24.34(21

ave L

TTAkQ

Then the rate of heat conduction through the plate becomes

1D conduction with internal heatgeneration PLANE WALL

Cylinder with heat source

Hence thermal resistance concept is notcorrect to use when there is internalheat generation

No more linear

11. A plane wall 10 cm thick generates heat at the rate of 4 × 104

W/ m3 when an electric current is passed through it. Theconvective heat transfer coefficient between each face of thewall and the ambient air is 50 W/m2 K. Determine (1) the surfacetemperature (2) the maximum temperature in the wall. Assumethe ambient air temperature to be 20°C and thermalconductivity of the wall material to be 15 W/m K.

12. An electrical transmission line made of a 25 mm annealedcopper wire carries 200 A and has a resistance of 0.4× 10-4

Ω/cm length. If the surface temperature is 200°C and ambienttemperature is 10°C, determine the heat transfer coefficientbetween the wire surface and ambient air and the maximumtemperature in wire. Assume k = 150 W/mK.

12. An electrical transmission line made of a 25 mm annealedcopper wire carries 200 A and has a resistance of 0.4× 10-4

Ω/cm length. If the surface temperature is 200°C and ambienttemperature is 10°C, determine the heat transfer coefficientbetween the wire surface and ambient air and the maximumtemperature in wire. Assume k = 150 W/mK.

13. The average heat produced by orange ripening is estimatedto be 300 W/m2. Taking the average size of an orange to be 8cm and assuming it to be a sphere with K = 0.15 W/m K,calculate the temperature at the centre of the orange if itssurface temperature is 10°C.

Applications:* current carrying conductors• * chemically reacting systems• * nuclear reactors

Applications:* current carrying conductors• * chemically reacting systems• * nuclear reactors