1. CONDUCTION HEAT TRANSFER gjv
2nd class

2. Recap Heat flows from hot to cold regions
Ease of movement of electrons in metals the reason for greater distribution of energy compared to other substances and explains the relationship between thermal and electrical conductivities
Conducting medium necessary for conduction

3. Basic law of conduction
Temperature gradient leads to energy transfer
Heat transfer per unit area (heat flux) proportional to normal temperature gradient

4. Fourier’s Law of heat conduction
For steady state one dimensional heat flow, the rate of heat flow in the x direction normal to the surface area, is directly proportional to the temperature gradient, the area of flow and inversely proportional to the distance.

5. Fourier’s Law Key features of the law
Not an expression that may be derived from first principle
A generalization based on experimental evidence
Defines the important material property of thermal conductivity

6. One dimensional steady flow
Isothermal surface T x1 dx x2 T1 x Q dT T2
Temperature profile

7. Assumptions
Uniform temperatures over the surface perpendicular to x which is the direction of heat conduction ( isothermal surface)
Steady flow ( Temperature does not vary with time)
Rate of heat flow constant
Consider an element of thickness dx with surfaces at temperature of T and T+dT

8. From Fourier’s Law:
Thermal conductivity of material

9. Thermal Conductivity
k, though not a function of temperature gradient, is a function of temperature and from experimental data:

10. Thermal Conductivity

11. Thermal Conductivity
For a nonlinear k the mean value is given by:
Thermal conductivities of metals are usually very high
Non-metallic solids and liquids lower

12. The Solid State
Modern view of solids highlights free electrons and atomic lattice structure
Thermal energy determined by Lattice vibrations which are additive:
k = ke+kl but we know that
ke= 1/ρe electrical resisticity
For pure metals with low ρe ke>>kl

13. The Solid State Hence contribution of kl to k is negligible
For alloys where ρe is large contribution of kl to k is no longer negligible
For non-metals, k is determined primarily by kl and depends on the frequency of interaction between atoms of the lattice
Chrystalline, well ordered substances such as diamond & quartz have high k values compared with amorphous substances (glass)

14. The fluid state
Larger intermolecular spacing and greater random motion lead to lower thermal energy transport
Thermal conductivity of gases and liquids more similar than solids
Kinetic theory of gases gives a good account of their thermal conductivities
Thermal conductivity of gases is directly proportional to the number of particles per unit volume, mean molecular speed and mean free path (Average distance travelled by a molecule before a collision)

15. Fluids
k α n c λ
Thermal conductivities of gases increase with increasing temperature and decreasing molecular weight since c increases accordingly
Thermal conductivities of gases are independent of pressure since n & λ are directly and indirectly proportional to gas pressure respectively

16. Thermal conductivity Conductivity of alloys less than the pure metals
Gases have very low conductivities and for ideal gases k is proportional to mean molecular velocity, mean free path and molar heat capacity

17. Liquid metals
Physical mechanisms of the thermal conductivity of liquid metals are still not well understood
Liquid metals are commonly used in high heat flux application such as in nuclear power plants.
Liquid metals thermal conductivities are much larger than non-metals

18. Temperature dependence of conductivity
Thermal insulation comprise low conductivity materials which when combined achieve even lower system thermal conductivity
Fiber, powder and flake type insulation have solid material finely dispersed in air spaces
The nature and volumetric fraction of the solid to void ratio characterizes the thermal conductivity of the insulation

19. Insulation
Cellular insulation – hollow spaces or small voids are sealed from each other and formed by fusion or bonding of solid material in a rigid matrix
Foam systems (plastic or glass)
Reflective insulation – thin sheets of high reflectivity foil spaced to reflect radiant energy
Evacuation of air from voids reduce effective thermal conductivity

20. Thermal conductivity of materials @ 0 oC
Metals
Silver W/mK
Copper W/mK
Aluminium W/mK
Iron W/mK
Lead W/mK
Chrome-nickel steel (18%Cr, 8%Ni) W/mK
Non-metallic solids
Diamond W/mK
Marble
Glass wool
Ice

21. Thermal conductivity of materials @ 0 oC
Liquids
Mercury W/mK
Water
Lube oil, SAE
Freon 12, CCl2F
Gases
Hydrogen
Helium
Air
H2Og CO

22. Example
One face of a copper plate 3.0 cm thick is maintained at oC and the other face is kept at oC. How much heat is transferred through the plate?
KCu = oC
Estimate the heat loss per m2 through a brick wall 0.5 m thick when the inner surface is at K and the outside surface is at K.
Kbrik = oC

23. Thermo physical properties
Important ant properties for heat transfer calculations:
Kinematic viscosity, ν (m2/s)
Density, ρ (kg/m3)
Heat capacity, cp, cv (J/m3K)
Thermal diffusivity, α (m2/s)

24. Thermal diffusivity
The ability to conduct thermal energy relative to the ability to store it:
Materials with large α respond quickly to changes in their thermal environment
Accuracy of engineering calculations depend on the accuracy of determining the thermo-physical properties

25. Example Use tables to calculate α for the following:
Pure 300K $ 700 K
Silicon 1000 K
300 K

26. Steady state conduction
Heat flow into & from tank T T
Tank wall
Boiling H2O
refrigerant
Air
Air
Insulation
Insulation x x
Consider a flat walled insulated tank containing a refrigerant at oC with outside air at 28.0 oC

27. Steady flow conduction
For x distance from the hot side:
x2-x1=B Thickness of insulation layer…T1-T2= ∆T Temperature drop across insulation layer
Thermal resistance

28. Compound resistance in series
Consider a flat wall with three layers, A,B & C
Let thicknesses be BA, BB & BC and average thermal conductivities be kA, kB & kC for the layers respectively.
Temperature drop
∆TA
∆TB
∆TC Q RB RA RC As
dTa, dTb, dTc BA BC BB T x

29. Compound resistance in series
Assume that the layers are in excellent thermal condition so no dT across the interface…but Q=Qa=Qb=Qc

30. Compound resistance in series
Ra,b.c individual layer resistances…rate of heat flow additive as with electricity

31. Example
An exterior wall of a house consists of a 4.0 cm of common brick [k=0.7 W m-1 oC-1] and a 1.5 cm layer of gypsum plaster [k =0.48 W m-1 oC-1]. What thickness of loosely packed rock-wool [k=0.065 W m-1 oC-1] insulation should be added to reduce the heat loss (or gain) through the wall by 80.0 %?

32. Radial Systems Assumptions:
Internal & external temperatures are constant
Heat flow is in radial direction only
Area exposed to heat flow proportional to the radius
Cylindrical shape (Thick walled tube) T1 T dT
T+dT T2
At a distance of r from the centre, A=2∏rl, so from Fourier’s law r2 T1 T2 r1 r dr

33. Thick walled tube
Heat flow at any given radius is given by……for a tube l metres long with inner and outer radii, r1 &r2

34. Thick walled tube For multiple cylindrical layers:
The thermal resistance would then be:
Logarithmic mean radius …for thin walled tubes use ra-arithmetic mean radius
For multiple cylindrical layers:

35. Example
A tube of 60.0 mm OD is insulated with a 50.0 mm layer of silica foam [k=0.055 W/moC] and a 40.0 mm layer of cork [k=0.05 W/moC]. Calculate the heat loss per unit length of pipe given that the temperature at the inner surface of the pipe is oC while the outer surface of the cork is kept at 30.0 oC .

36. Conduction through a spherical shelldr
r+dr r
Q=-kAdT/dr=-k4πr2dT/dr
Q∫ dr/r2 =-4kπ∫dT
Q= 4kπ(T1-T2)/(1/r1-1/r2) r2 T1
Iintegrating bet r1 & r2..where T=T1 & T2

37. Critical Thickness of insulation
For a pipe covered with a layer of insulating material
Inner temp of insulation fixed at T1 outer temp at T2 T2 r1 r2 T1

38. Example
What is the critical radius of insulation for asbestos [k=0.17 W/moC] insulation on a pipe in a room at ambient temperature of 20.0 oC, where h=3.0 W/m2oC and what would be the heat loss from a 5.o cm diameter pipe kept at oC when covered with this critical radius of insulation as well as without?