Some Lecture Notes

1. Heat Transfer by Conduction

2. Heat
• Heat is a form of Energy
• A material becomes hotter when it gains sensible heat energy
• Gain in heat energy can lead to a change of phase
• Heat energy can transfer from one point to another in space
• The SI unit is Joules [J]
• Commonly used symbol is 𝑄

3. 𝑄 =
𝑑𝑄
𝑑𝑡
Heat Flow Rate
𝑄 =
𝑆
𝑞. 𝑑 𝑠Heat Flux is given as
[W]
Heat Transfer
Where Heat flux is 𝑞 [W.m-2]
[W]
Heat energy is effectively transferred from one point in space to
another under a temperature gradient.

4. Heat Transfer: First Law of Thermodynamics
• Principle of Conservation of Energy
𝐸𝑖𝑛 − 𝐸 𝑜𝑢𝑡 = ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚
Steady State of Heat Transfer:
𝑄𝑖𝑛 = 𝑄 𝑜𝑢𝑡
Energy balance in a closed system with constant mass and only
involving heat transfer (no work):
𝑄 = 𝑚𝐶 𝑣∆𝑇
Energy balance in a closed system with steady flow and only
involving heat transfer (no work):
𝑄 = 𝑚𝐶 𝑝∆𝑇

5. Heat Transfer by Conduction
Thermal Conduction is the transfer of Heat energy from the more energetic
particles of a substance or material medium to adjacent less energetic ones as a
result of interactions between the particles.
Fourier’s Law of Heat Conduction
𝑞 𝑐𝑜𝑛𝑑 = −λ𝛻𝑇
In one dimensional form:
𝑞 𝑐𝑜𝑛𝑑 = −λ
𝑑𝑇
𝑑𝑥

6. Thermal Conductivity
• Thermal conductivity is the property of the material medium’s
ability to conduct heat. It is expressed in SI units as [Wm-1K-1].
• It is also defined as the rate of heat transfer through a unit
thickness of the material per unit area per unit temperature
difference.
• A high value of λ indicates good thermal conductor and a low
value indicates thermal insulator.

7. Thermal Conductivity of Materials

8. Thermal Resistance
𝑅 =
∆𝑥
λ
[m2KW-1]
Using above relation, Fourier’s heat conduction equation can be
written as
𝑞 =
∆𝑇
𝑅
Analogy with Ohm’s Law of electrical conduction
Fourier’s Law Ohm’s Law
∆𝑇 = 𝑞𝑅 𝑉 = 𝐼𝑅

9. Network of Thermal Resistances
∆𝑥1 ∆𝑥2
𝑞
𝑇1 𝑇2 𝑇3
λ1 λ2
𝐴 𝑠
∆𝑥
𝐴1
𝐴2
λ1
λ2
𝑞1
𝑞2
𝑇1 𝑇3
𝑅1 𝑅2
𝑅1
𝑅2
𝑅𝑡𝑠 =
𝑖=1
𝑛
𝑅𝑖
𝑅𝑡𝑝 =
1
𝑖=1
𝑛 1
𝑅𝑖

10. Calculation of Thermal Resistances
Example #1: Thermal Resistance of a circular cylinder
𝑟1
𝑟2
𝑄 = λ 2𝜋𝑟𝐿
𝑑𝑇
𝑑𝑟
1
2𝜋𝐿λ
𝑟1
𝑟2
𝑄
𝑟
𝑑𝑟 =
𝑇1
𝑇2
𝑑𝑇
𝑄 =
2𝜋𝐿λ∆𝑇
ln
𝑟2
𝑟1
𝑅 𝑐𝑦𝑙 =
ln
𝑟2
𝑟1
2𝜋𝐿λ

11. Thermal Contact Resistances
𝑇1 𝑇2
𝑞 𝑎𝑣𝑔
𝑅 𝑐 =
∆𝑇
𝑞 𝑎𝑣𝑔

12. Transient Heat Conduction
One dimensional transient heat conduction equation:-
Variable Conductivity:
Constant Conductivity:
𝜕
𝜕𝑥
λ
𝜕𝑇
𝜕𝑥
+ 𝜀 𝑔𝑒𝑛 = 𝜌𝐶
𝜕𝑇
𝜕𝑡
𝜕2
𝑇
𝜕𝑥2
+
𝜀 𝑔𝑒𝑛
λ
=
1
𝛼
𝜕𝑇
𝜕𝑡
where
𝛼 =
λ
𝜌𝐶

13. Thermal Diffusivity
• It is the property of the material medium which indicates how
fast the heat diffuses through that material.
• It can also be explained as the ratio of heat conducted to the
heat stored in that material per unit volume.
• The higher thermal diffusivity means faster propagation of
heat through the material.
• It’s unit is [m2s-1]

14. Thermal Effusivity or Thermal Absorptivity
𝑏 = λ𝜌𝐶
Thermal absorptivity of textile materials indicates the warm-cool feeling to
touch by human fingers. A higher value indicates cooler feeling.
ALAM-
BETA
EFFECT OF FABRIC STRUCTURE, COMPOSITION AND TREATMENT ON THE LEVEL OF
THERMAL ABSORPTIVITY b [Ws1/2/m2K], contact pressure 200 kPa
20 - 40 Micro-fibre or fine PES fibre non-woven insulation webs
30 - 50 Low density raised PES knits, needled and thermally bonded PES light webs
40 - 90 Light knits from synthetic fibres (PAN) or textured filaments, raised tufted carpets
70 - 120 Light or rib cotton RS knits, raised light wool/PES fabrics, brushed micro-fibre weaves
100 - 150 Light cotton or VS knits, rib cotton woven fabrics
130 - 180 Light finished cotton knits, raised light wool woven fabrics
150 - 200 Plain wool or PES/wool fabrics with rough surface
180 - 250 Permanent press treated cotton/VS fabrics with rough surface, dense micro-fibre knits
250 - 350 Dry cotton shirt fabrics with resin treatment, heavy smooth wool woven fabrics
300 - 400 Dry VS or Lyocell or silk weaves, smooth dry resin-free heavy cotton weaves (denims)
330 - 500 Close to skin surface of wetted (0,5 ml of water) cotton/PP or cotton/spec. PES knits
450 - 650 Heavy cotton weaves (denims) or wetted knits from special PES Fibres (COOLMAX)
600 - 750 Rib knits from cotton or PES/cotton or knits from micro-fibres, if superficially wetted
750 Other woven and knitted fabrics in wet state
1600 Liquid water (evaporation effect not considered)
• It is the rate at which a material can absorb heat.
• It is the property that determines the contact temperature of
two bodies that touch each other.