Section 6. HEAT TRANSFER
Dr. Congxiao Shang
6.1 Definitions
Mechanisms of Heat (Thermal Energy) Transfer:
Conduction: transmission of heat
across matter, due to direct
physical contact, e.g. in solids,
liquids and gases.
Radiation: heat transfer due to
electromagnetic radiation across a
space, even in a vacuum.
Convection: heat transfer by “currents”
in a gas or liquid, due to temperature
differences or forced flow, an important
mechanism of energy transfer between
a solid surface and a liquid or a gas.
All mechanisms may be involved in practical heat transfer,
but the dominant mechanism differs in different cases.
(Source of illustrations: http://sol.sci.uop.edu/~jfalward/
heattransfer/heattransfer.html)
6.1 Definitions
Concepts & Terminologies:
THERMAL CAPACITY (of a system): Quantity of ENERGY required
to heat a whole system by 1 K (Kelvin). Unit: J·K-1
SPECIFIC HEAT (or SPECIFIC HEAT CAPACITY): Quantity of
ENERGY required to heat a unit mass of a system by 1 K.
Unit: J·kg-1·K-1
Kelvin = Degree Celsius (oC) + 273
J= joule
6.1 Definitions
d
THERMAL CONDUCTIVITY, k :
a measurement of heat flow through a Q
A
body. It is the heat transmitted in unit
time, in a direction normal to a surface
of unit area, through a distance, d,
T1
across a unit temperature difference
over the distance.
T2
Temperature profile
Q, heat flow per unit time (Js-1) × d, distance (m)
k=
A, area (m2) × (T1-T2) temperature difference (K)
Unit: Wm-1K-1
(error in handout
(or Wm-1 °C-1)
“m-3”
(Eq. 6a)
( note: 1W = 1 J s-1)
)
(analogous to electrical conductivity or hydraulic permeability)
6.1 Definitions
d
Another way of understanding the
THERMAL CONDUCTIVITY, k, is to
re-arrange the equation as :
Q
A
T1
Q, heat flow per unit time (Js-1)
T2
Temperature profile
A, area (m2)
=k
(T1-T2) temperature difference (K)
d, distance (m)
Heat flow per unit time per unit area is
proportional to the temperature gradient;
this proportionality is called thermal
conductivity, k.
(Eq. 6b)
The temperature
difference per unit
distance is called
temperature gradient
The higher the thermal conductivity, the faster the heat flows
6.1 Definitions
Why are diamonds so “cool”?
Material
Thermal conductivity
-1
-1
(298 K), W·m ·K
Diamond
895-2300 ( the highest k)
Carbon
Nanotubes
1400
Silver
429
Copper
386
Gold
317
Aluminium
237
Iron
80.2
Brick
0.15 – 0.6
0.12 -0.04
0.04 (very low k)
0.01
Wood
Wool
Styrofoam
(for building
insulation)
Why dose wool feel so “warm”?
6.1 Definitions
THERMAL RESISTIVITY, ρ : reciprocal of conductivity, 1/k
Unit: m K W-1
THERMAL Resistance of a system, R:
R = d = d/k
(error in handout, not divided by “A”)
where R is the resistance
d is the thickness
ρ is the resistivity
Note that the R-value above is a UNIT AREA THERMAL RESISTANCE
(or thermal insulance), because the resistivity, ρ, is related to the
conductivity, k, which is measured per unit area.
Unit for R: m K W-1 m = Km2 W-1
(The reason for defining the thermal resistance, R, is that the R values
are “additive” in multi-layer insulations and this makes calculations
simpler. This will be explained later)
6.2 Conduction
Q. How much heat is conducted
through a system ?
Q
d
A
We know:
-the larger the A, the larger the heat flow; T
-the larger the d, the smaller the heat flow. 1
Q
(T1  T2 )
k
A
d
T2
Temperature profile
Therefore
kA(T1  T2 ) A(T1  T2 ) A(T1  T2 )
Q


d
d
R
; or
Direct analogy with electricity:-
Q (T1  T2 )

A
R
( V1  V2 )
I
R
Current (I) is equivalent to Heat Flow per unit area (Q/A) ; & Potential
Difference (V1 - V2) or voltage is equivalent to temperature difference (T1-T2).
6.2 Conduction
In most situations we have composite materials to deal with - e.g. a wall
consisting of an outer skin (brick), a cavity, an inner skin and then plaster.
Since the thermal resistance of each
component has considered the thickness, the
Total UNIT AREA THERMAL RESISTANCE
is simply given by
brick
brick
cavity
R = R1 + R2 + R3 + ........
i.e. resistances in series
Electrical Analogue
6.2 Conduction
“Resistance in Parallel”: e.g. conduction through a wall with a window, which is
more complicated, as total (average) heat transfer depends on the thermal
resistances and the relative areas of both components.
A1 (T1  T2 ) A2 (T1  T2 ) ( A1  A2 )(T1  T2 )
Q  Q1  Q2 


R1
R2
R
1 1
A1
1
A2


R R1 ( A1  A2 ) R2 ( A1  A2 )

1
1
a1 
a2
R1
R2
Proportion of wall or window area to the
total area
The U - value is defined as 1/R, where R is resistance per unit area
6.2 Conduction
Thermal transmittance, the U value:
The U value is simply defined as 1/R; Unit: W·K-1·m-2
(Remember: R is resistance per unit area, so U is transmittance
per unit area as well)
Both the R-value and the U-value are used to grade the insulation
properties of a material or a system (e.g. a double-glazed assembly).
A higher R-Value means the materials
are more resistant to heat loss.
A lower U-Value means the system
will transmit less heat.
R
U
clear ?
Both mean the same thing & are desirable, from the insulation point of view.
6.2 Conduction
Simple Example:
Brick wall 105 mm thick
plaster
15 mm thick on inside
Internal temperature = 20°C
External Temperature= 0 °C
kbrick
= 0.84 W m-1 K -1
kplaster = 0.50 W m-1 K -1
Tbp
Brick
20°C
0 °C
105mm
What is the U - value of the construction and also
the temperature at the interface between the
brick and the plaster?
15mm
6.2 Conduction
Simple Example:
Brick
0.105
Rbrick  d d / k 
0.125m 2 CW 1
0.84
R plaster 
Tbp
0 °C
0. 015
 0. 030
0. 50
20°C
105mm
15mm
Total resistance, R = 0.125 + 0.030 = 0.155 m2 ºC W-1
and the U-value = 1/R = 1/0.155
=
6.45 W m-2 ºC-1
==============
Now heat flow in plaster = heat flow in brick=heat flow through the whole body
20Tbp Tbp 0 20 0 20 0
So if Tbp is temperature at interface



 129
R plaster Rbrick
R
0.155
Hence
Tbp
= 129 * 0.125
=
16 . 1ºC
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