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2. HEAT TRANSFER
Temperature( T) : expressed either as the thermodynamic temperature T, K, or the Celsius
temperature
T(K) = T(°C)+273.15
Quantity of Heat (Q) : measured in Joules(J)
Heat flow rate(Φ) : measured in Watts(W = J/s)
Heat capacity(C) : quantity of heat required to raise the temperature by dT
C= (J/K)
Specific heat capacity(c): for a homogeneous material
c= (J/kg.K) whereby m is the mass of the object
Modes of heat transfer:
1. Conduction
2. Convection
3. Radiation
1. HEAT CONDUCTION
If we have two systems or bodies at different temperatures T1 and T2 which are in some way
thermally connected, heat will flow from the warmer to the colder
Φ = (T1 - T2).Λ ,W
Whereby, Λ, (Big lambda), is the thermal conductance, W/K.
If within a body of an isotropic material there exists a temperature gradient, grad T, the density
of heat flow rate q can be calculated as
q = - λ. grad T , W/m2
λ is the thermal conductivity of the material, W/mK.
The heat flow rate Φ through a surface with area A given by x = constant with a uniform
temperature gradient then becomes
𝐓
Φ = −λ ⋅A⋅
𝐱
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Steady state calculations
The temperatures of the system do not vary with time. Heat is not being stored in or removed
from any part of the system since this implies a change in temperature.
dQ = C dT
For a homogenous wall slab, with a temperature gradient in the direction normal to the
surface, the consequence is, that if no heat is being stored at any point in the wall, the
temperature gradient has to be constant. This also implies that the temperature is linearly
distributed between the surfaces.
dT/dx = constant = (T2-T1)/d
The density of heat flow rate q, W/m2, can
accordingly be expressed as
q = −λ ⋅(T2 − T1)/d = ⋅(T1 − T2 )
This can also be written as
q = (T1 − T2)/R W/m2
whereby R = d/λ m2K/W
is known as the thermal resistance of the wall slab.
Steady state heat flow and temperature distribution in a multilayer wall with no internal heat
sources.
From the steady state condition it follows that the
heat flow is constant through the construction
q1 = q2 = q3 = q4 =. . . . ….. = qn
(T1-T2)=q.R1 , (Tk-Tk+1)=q.Rk
(T1 − Tn ) = ∑ (𝐓𝑘 − 𝐓𝑘 + 1) = ∑ (𝐪. 𝐑𝑘 ) =q ∑ (𝐑𝑘 )
Rtot = ∑ (𝐑𝑘 )
𝐓 𝐓
q=
𝐑
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Surface resistance
The heat transfer from a construction surface to the surroundings with a given temperature is
taking place by radiation to surrounding surfaces and due to heat conduction and air
movements close to the surface
The coefficient of surface heat transfer hs , W/m2K, is defined as
q = hs (Ts - Ta )
In practical applications, these complicated processes are often approximated by a fictive
material layer between the surface, Ts, and an ambient temperature, Ta, which is often chosen
as the air temperature.
These fictive resistances can be chosen on the inside towards a heated room with normal
indoor climate
Rsi = 0.13 m2K/W
and on the outside towards average external temperature and wind conditions
Rse = 0.04 m2K/W
Thermal transmittance
The ratio between the density of heat flow rate q, W/m2, through the construction and the
temperature difference between the ambient temperatures on both sides
U=
( )
For a construction with n layers the U-value then becomes
𝟏
U= ∑
𝑹𝒔𝒊 𝑹𝒋 𝑹𝒔𝒆
EXAMPLE
An outer wall is from the outside made of 120 mm brick, cellulose fiber insulation and a 22 mm
gypsum plasterboard. Calculate the thickness of the insulation layer in order to reach a U-value
of 0.2 W/m2K.
Input data
λ gypsum =0.25 d gypsum= 0.025
λ brick = 0.6 R si = 0.13
λ cellfib =0.042 R se = 0.04
d brick= 0.12
Rearranging the U-value equation, we can calculate the insulation thickness
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2. CONVECTIVE HEAT TRANSFER
Heat flow due to air flow between two systems at temperatures T1 and T2 can be expressed in
terms of the mass flow and temperature difference. If a volume dV1 is removed from system 1
and added to system 2 and the same mass of air from system 2 at temperature T2 is added to
system 1 the resulting quantity of heat removed from system 1 is for constant c given by:
And quantity of heat added to system 2:
If we have a continuous flow of air between the systems
L=dV1/dt
the heat flow rate from system 1 to system 2 due to air flow is
Φ12=L.ρ1.c(T1-T2)
2.1. Convective surface heat transfer
Refers to heat transfer between the boundary surfaces and the air within a given volume
If the air in the volume is standing still, heat is transferred with conduction in the air only, but at
higher temperature differences between the surface and the air the density differences will
generate air movements and the heat transfer will be a combined process of conduction and
convection. When referring to convective surface heat transfer we usually mean the combined
process including conduction in the air as well.
2.2. Natural convection
Air movements are generated due to density differences which occurs as a result of different
temperatures. Warmer air will always tend to rise upwards and cooled air will sink downwards
tills an equilibrium is gained.
2.3. Forced convection
Situations where air flow along the surface is generated by an external force independent of
the actual temperature difference between the surface and the medium.
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Examples are wind on an exterior surface, air gaps ventilated due to wind effects and air flow in
a fan coil unit.
2.4. Dimensionless numbers
Nusselt : Actual heat transfer coefficient hc in relation to that of still air over a
given characteristic length l. When the air is standing still the Nusselt
number will be equal to 1. With increasing air movements either due to
temperature differences or to external forces the Nusselt number will increase.
Grashof: Criterium for fluid movements due to natural convection.
As gravity g, volume expansion coefficient β and the kinematic viscosity ν
For air are fairly constant in the normal temperature range the Grashof
number will depend strongly on the geometry represented by the characteristic length l and
the temperature difference ΔT.
Reynolds: Properties of the flow at forced convection. For a given fluid and
geometry Reynolds number will increase linearly with the velocity of the fluid.
With increasing velocity, when the Reynolds number reaches a certain limit, we
will have transition from laminar to turbulent flow.
Prandtl: Properties of the flowing medium. For applications to building air flow
the Prandtl number can normally be set equalto 0.7. g = acceleration of gravity
9.81 m/s2
Whereby:
β = volume expansion coefficient K-1
l = characteristic length, m, that can be the hydraulic diameter in forced duct flow or the length
of the surface in the direction of the flow for air flow along exposed surfaces.
u = air velocity, m/s
ν = kinematic viscosity, m2/s
hc = the coefficient of surface heat transfer due to conduction and convection, W/m2K.
When the Nusselt number has been calculated for a given characteristic length l the convective
surface heat transfer coefficient can be calculated as
hc = Nu.λ/l
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2.5. Expressions for different flow types
a. Forced convection
For forced convection we usually have a system with a given air flow or given air velocity and
the Nusselt number is a function of the Reynolds number and the Prandtl number.
Nu = f(Re,Pr)
b. Natural Convection
For natural convection it is the temperature difference that generates the air movements and
thereby the conditions for surface heat transfer and the Nusselt number will be a function of
the Grashof number.
Nu = f(Pr,Gr)
2.6. Expressions for surface heat transfer coefficients
2.6.1. Forced laminar duct flow
The flow is generated by external forces and the criteria for laminar flow is
Pr > 0.6 Re < 2300
The characteristic length for calculation of the Reynolds number is the hydraulic diameter of
the duct which for non-circular geometries can be calculated as four times the section area A
divided by the length of the perimeter P of the interior duct section.
𝟒𝑨
dh =
𝑷
It follows that the hydraulic diameter for a rectangular duct with sides a and b will be
𝟒.𝒂.𝒃
dh =
𝟐(𝒂 𝒃)
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and if a>>b the hydraulic diameter will become 2b. This condition is typical for ventilated air
gaps in the exterior part of insulated constructions.
The following expression gives the average Nusselt number along the surface.
L is the length of the duct in the direction of the flow, m.
2.6.2. Forced turbulent flow of air in a duct
0.7 < Pr < 100
l/d > 60
Re > 10000
The characteristic length for calculation of the Reynolds number is the hydraulic diameter of
the duct which for noncircular geometries can be calculated as four times the section area
divided by the length of the perimeter of the interior duct section
n=0.4 if the surface is warmer than the air
n=0.3 if the surface is colder than the air
The expression can for air also be used approximately in the interval 2300 < Re < 10000 if n is
set equal to 0.4.
2.6.2 Forced flow along flat surfaces
The convective surface heat transfer at the exterior surfaces of outer walls and the roof is
usually governed by the wind generated air flow along the surfaces. The characteristic length
here is the length of the surface in the direction of the air flow at the surface i.e. the length or
width of a roof or a wall.
2.6.2.1. Forced laminar flow along a flat surface
0.6 < Pr < 2000
Re < 105
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And the expression for the Nusselt number becomes:
Nu = 0.664 Re1/2Pr1/3=( hc.l)/λ
l, which is the length of the surface in the direction of the flow, m, is also the characteristic
length to be used in the calculation of the Reynolds number.
2.6.2.2. Forced convection with turbulent flow along a flat surface
The criteria for turbulent flow along a flat surface are as follows:
0.6 < Pr
6.105 < Re < 107
2.6.3 Natural convection on room surfaces
The characteristiclength used in the calculation of the Grashof and Nusselt
numbers is usually the height of the wall or the ratio Area/Perimeter of
the floor/ceiling, which is being considered. All cases can be expressed
with the same basic equation where A and B have different values for
different situations.
Nu = A(Gr⋅ Pr)B = hc ⋅ l/ λ
For laminar flow B=1/4 and for turbulent flow B=1/3.
2.6.3.1 Vertical walls
For vertical walls we can use the same expression independent on whether the wall is colder or
warmer than the room air. The mode of flow is determined from the Grashof number and the
characteristic length to be used in the calculation of the Grashof number is the wall height.
Laminar flow (Gr.Pr<109) A=0.59 B=1/4
Turbulent flow (Gr.Pr>109) A=0.13 B=1/3
2.6.3.2 Horizontal surfaces
For horizontal surfaces the heat transfer will depend not only on the temperature difference
but also on the thermal stability at the surface. For a ceiling colder than the room air the
density of the air at the surface will be higher than below which will generate turbulent air
movements at relatively low Grashof numbers. Similar instability will appear at warm floors
where the density of the air in the vicinity of the surface is lower than for the room air. The
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characteristic length will be the floor or ceiling area divided by the perimeter of the floor or the
ceiling. As an example consider a room 3x6x2.4 m. The floor area is 18 m2 and the perimeter is
18 m which will give the characteristic length 1 m.
For a relatively warm floor or a cold ceiling:
Laminar flow (Gr.Pr< 2.107) A=0.54 B=1/4
Turbulent flow (Gr.Pr>2.107) A=0.15 B=1/3
For a relatively cold floor or a warm ceiling we expect conditions to be stable up to high Grashof
numbers.
Laminar flow (Gr.Pr< 3.1010) A=0.27 B=0.25
2.7. Air flow through building components
The pressure difference as calculated above from various processes is the governing potential
which generates air flow through leakage paths in the construction. The leakage paths are
seldom desired or planned, which implies a high degree of uncertainty concerning the
geometry and other properties of the paths. Uncontrolled air leakage through building
constructions is usually linked to unwanted consequences such as moisture problems,
distribution of odors, energy loss and draught. Important exceptions are fresh air inlets and air
gaps for ventilation of the exterior parts of constructions.
It is however of great value to provide means to study the air leakage processes in order to
quantify crucial construction and material properties to ensure the required air tightness.
The air leakage path through a construction is usually of a complex nature.
As an example we have air entering from the inside through a crack in the
drywall, through an overlap in the vapor barrier, through a porous insulation
material to a crack in the wind barrier into the external ventilation gap and
out through the ventilation openings in the exterior brick cladding. The air
tightness however is often provided for in one layer in the construction such
as the vapor barrier, a concrete layer, a sheet metal deck or a wall layer of
aerated concrete. The rest of the construction can then often be considered
as non airtight and the openings in the air tight layer can be studied as
connecting directly the inside and the outside.
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EXAMPLE
A 50 mm air gap in the outer part of a construction is ventilated with air with velocity 0.5 m/s.
The average temperature in the gap is about 0 °C. We want to estimate the convective surface
heat transfer coefficient hc.
Basic input data see table 1
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3. RADIATIVE HEAT TRANSFER
All bodies emit energy by electromagnetic waves
which we call thermal radiation. The
characteristics of the radiation are depending
on the properties of the surface material and on
the surface temperature. Thermal radiation is
defined as radiation with wavelength between 10-5
and 10-8 m. This includes visible light as well as a
part of the infrared and of the ultraviolet spectrum.
At normal roomtemperature, surfaces emit radiation
far into the infrared spectrum which
we are not able to see. The visible light that we
detect with our eyes, as coming from a surface of a
body, is not emitted radiation but reflected radiation that originates from the sun or some
artificial light source.
3.1. Black body radiation
Black body is defined as a body with a surface that absorbs all incident radiation for all
wavelengths, directions and polarizations. The radiation from the real body can then be
expressed as the black body radiation multiplied by the emissivity of the real surface.
Black body total excitance is expressed by the Stefan Boltzman law
M°= σT4
3.2. Grey body properties
a) Emissivity
The emissivity of a surface is the ratio between the emitted radiation and the radiation of a
black body at the same temperature. The emissivity of surfaces can vary with the wavelength.
This is for instance utilized in window glazing with so called LE or low emittance coating where
the surface is treated to have low emissivity for the infrared spectrum while visible light is less
affected.
Total hemispheral emissivity is the total excitance of the considered surface M divided by the
total hemispheral excitance of a black body M° at the same temperature
ε =M/M°
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b) Total absorptance
Assume that we have an incident radiation with heat flow rate Φi towards a gray surface. A part
of this incoming radiation, Φa, is absorbed at the surface. The absorptance of the surface, α, is
defined as
α=Φa/Φi
c) Total reflectance
A part of the incident radiation, Φr, is reflected by the surface and the reflectance, ρ, of the
surface is defined as
ρ=Φr/Φi
d) Total transmittance
With transmitted radiation is meant incident radiation that is neither reflected nor absorbed in
the surface but passes further into or through the material layer. Even for so called transparent
materials the transmittance will vary dramatically with the wavelength. For most opaque
building materials the transmittance is zero for the whole spectrum. For normal window glass
the transmittance for visible light will be about 90 % but for long wave radiation as generated at
normal temperatures the transmittance is practically zero. The transmittance is defined as the
radiant heat flow rate transmitted through a surface, Φt, divided by the incident heat flow rate,
Φi
Ʈ =Φt/Φi
Therefore, incident radiation is either reflected, absorbed or transmitted and it follows that
α +ρ + Ʈ = Φt/Φi +Φr/Φi +Φa/Φi = 1
For opaque bodies where Ʈ is equal to zero it follows that
α +ρ = 1
3.3. Combined radiation and convective heat transfer
Normally we assume that the air does not absorb radiation. This means that radiation and
convective heat transfer can be regarded as two separate processes coupled through the
surface temperature. For a thin air gap the heat transfer coefficients can be calculated
separately and the overall heat transfer coefficient between the two surfaces, hcr, simply
calculated as the sum of these two.
q12 = (hc + hr ) ⋅(T1 − T2 ) = hcr ⋅(T1 − T2)
For the heat transfer at the surface in a room the situation is more complicated. The convective
heat transfer is taking place between the surface and the room air while the radiative heat
transfer is taking place between the surface and other surfaces of the room. A common
simplification when calculating the surface heat transfer for insulated building constructions is
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to assume that on the average the ambient air temperatures and the surrounding radiating
temperatures are the same.
Then a common surface heat transfer coefficient can be defined as
hs = (hc +hr )
and the surface resistances as
Rsi = 1/hsi
Rse = 1/hse
3.4. Solar radiation in building design
With increased insulation and air tightness of buildings the solar radiation through windows will
play a relatively larger role in the power- and energy balance for the building. The orientation
and design of the windows will influence the energy consumption for heating as well as the
level of comfort and the need for cooling in a building.
The performance of external solar collectors or photovoltaic solar cells is a function of their
orientation toward the sky. For real buildings it is often a question of making the best use of
available surfaces.
Solar radiation can heat up external exposed surfaces to temperatures around 80 °C and
surfaces with partly transparent coating to more than that. The temperature differences gives
rise to mechanical stress, dimensional changes and, together with moisture and chemical
reactions, cause a rapid decay of surface finishes.
Solar radiation is therefore an important factor in design for high durability, both in building
constructions and for infrastructural constructions like bridges and roads.
3.5. The elements of solar radiation
The total short wave radiation from the sun reaching an earthly surface, I W/m2 can be split
into three components
ID = direct solar radiation, W/m2
Rays directly from the sun to the surface
Id = diffuse solar radiation, W/m2
Secondary rays reradiated from the atmosphere
IR = reflected solar radiation, W/m2
Secondary rays reflected from surfaces on the ground
3.6. Heat balance on exterior surfaces
The components of heat transfer to be regarded at anexterior surface are
• Convective heat exchange between the surface and outdoor air
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• Solar radiation
• Long wave radiation
exchange with the sky
• Heat exchange between the
surface and the interior climate
Other important components of the surface
heat balance that are not treated here are
water transfer and phase change such as
evaporation of rain water and melting of
snow.
A heat balance for the surface gives
aI+hc(Te-Tes)+hr(Tsky-Tes)+U´(Ti - Tes) = 0
a is the absorbtance of the surface for shortwave solar radiation.
U´ is the modified U-value, i.e the heat transfer coefficient between the outer surface and the
ambient indoor air.
Rearranging the terms gives the equation for the temperature of the external surface
