2004:325 CIV
MA S T ER’S T H E SI S
Simulation Model for the
Climate at the Windshield of
a Passenger Car Compartment
NADEJDA TCHERTOVSKAIA
MASTER OF SCIENCE PROGRAMME
Luleå University of Technology
Department of Applied Physics and Mechanical Engineering
Division of Physics
2004:325 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 04/325 - - SE
Simulation model for the climate
at the windshield of a passenger car
compartment
NADJA TCHERTOVSKAIA
Department of Physics
LULEÅ UNIVERSITY OF TECHNOLOGY
Department of Building Physics
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden, 2002
FOREWORD
This thesis work, equivalent to 30 ECTS credits, is the final part of the Engineering
Physics Master of Science programme. The thesis work was performed at the
Department of Building Physics, Chalmers University of Technology, for Volvo Car
Corporation in Göteborg.
It was executed under supervision of Carl-Eric Hagentoft from the Department of
Building Physics, Chalmers University of Technology, and Anneli Högberg from the
Department of Computational Fluid Dynamics, Volvo Car Corporation. Examiner of
this thesis work was Björn Graneli from the Department of Physics, Luleå University
of Technology.
I would like to thank them all for their invaluable help and for finding the time to
answer all of my questions. I would also like to thank Zenitha Chronéer from the
department of Computational Fluid Dynamics, Volvo Car Corporation for her support
and help, and Christer Svärd from the Group of Climate Analysis and Verification at
Volvo Car Corporation for providing help with experimental data.
Stockholm, October 2002
Nadja Tchertovskaia
1
ABSTRACT
In this thesis a simulation model is developed for the evaluation of the energy balance
at the inside of a motorcar windshield, including evaporation and condensation of
humid air. A model is derived from applications of the physical processes to a
simplified geometry. The model is implemented in a Matlab-Simulink environment
with the aid of the Building Physics Toolbox developed at the Department of Building
Physics at Chalmers. This study is the second part of the development of a fast
computational tool for use with estimations of the climate of a passenger car
compartment. The simulation model is validated with tests performed in a climatic
chamber at Volvo Car Corporation, Torslanda, Göteborg. The computational model
appears to yield satisfactory results.
2
TABLE OF CONTENTS
FOREWORD ..........................................................................................................................................1
ABSTRACT ............................................................................................................................................2
TABLE OF CONTENTS .......................................................................................................................3
1
INTRODUCTION ........................................................................................................................4
1.1
1.2
2
THE MODEL OF THE ENERGY BALANCE..........................................................................5
2.1
2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.4
2.5
2.6
2.7
3
BACKGROUND .......................................................................................................................4
AIM ........................................................................................................................................4
DEW AND ICE ON THE WINDSHIELD – A LITERATURE SURVEY ............................................5
PHENOMENA ASSESSED .........................................................................................................6
HEAT TRANSFER ...................................................................................................................6
Conduction .......................................................................................................................7
Convection ........................................................................................................................8
Radiation ..........................................................................................................................9
Combined mechanisms of heat transfer..........................................................................10
THERMAL NETWORK ANALYSIS .........................................................................................10
MASS TRANSFER .................................................................................................................13
MOISTURE ...........................................................................................................................14
THE MODEL AND ITS ALGORITHM.....................................................................................16
IMPLEMENTATION OF THE MODEL ................................................................................21
3.1
TOOLS .................................................................................................................................21
3.2
IMPLEMENTATION OF THE MODEL IN SIMULINK ...............................................................22
3.2.1
Input blocks ....................................................................................................................22
3.2.2
Main part of the model. ..................................................................................................24
4
VALIDATION OF THE MODEL.............................................................................................29
4.1
4.2
4.3
CLIMATIC CHAMBER ..........................................................................................................29
DEFROSTER TEST FOR THE WINDSHIELD ACCORDING TO 78/317/EEC ............................29
COMPUTATIONAL MODEL OF THE WINDSHIELD ................................................................31
5
RESULTS ....................................................................................................................................34
6
CONCLUSIONS .........................................................................................................................37
7
RECOMMENDATION FOR FUTURE WORK .....................................................................38
8
LIST OF SYMBOLS ..................................................................................................................39
APPENDIX A .......................................................................................................................................40
REFERENCES .....................................................................................................................................43
3
1
1.1
INTRODUCTION
Background
Human factors have been considered as central issues in vehicle design for more than
thirty years. One of the important factors that strongly affect driving safety is the
visibility through the windshield. Because of this, climate control systems have been
developed and are widely used nowadays for windshield defrosting and for
elimination of such problems as window fogging and condensation. The climate
control system also increases the sensation of climate comfort in the car compartment
and may thereby affect greatly the driver’s concentration and enhance safety of the
passengers. Thus it is important in vehicle design to understand and describe the heat,
air and moisture conditions.
1.2
Aim
The aim of this thesis work was to develop a model in MATLAB-Simulink
environment for the energy balance at the windshield of a passenger car, including
evaporation and condensation of humid air. The computational model was compared
with full-scale measurement results. This study is the second part in the development
of a fast computational tool for use in estimations of the climate of a passenger
compartment.
4
2
2.1
THE MODEL OF THE ENERGY BALANCE
Dew and ice on the windshield – a literature survey
The difficulty of removing dew and ice from the windshield is one of the issues, along
with the thermal comfort inside passenger cars, which has lately received much
attention from the automotive industry. Thermal comfort has a very strong influence
on humans. This fact created a need for studying it, and ways to implement it.
Apparently thermal comfort is a safety factor, as it is much easier for a person to drive
safely when being alert. On the other hand thermal comfort alone is not enough.
Reducing the amount of dew and ice on the windshield plays a great role as well. This
in turn influences thermal comfort, making it harder to achieve a good balance.
A lot of research on this subject has been done within automotive industry. The recent
developments in Computational Fluid Dynamics (CFD) and experimental diagnostic
techniques have encouraged a number of researchers to examine the climatic
environment in vehicles. These studies range from those reporting general flow
observations to those attempting to model the prevailing environment within the
compartment, resulting in recommendations for optimum climatic conditions and
modifications. Among the articles on the subject are "Comparison of CFD Simulation
Methods and Thermal Imaging with Windscreen Defrost Pattern” [2] , "Effects of
Vehicle Windshield Defrosting and Demisting Process on Passenger Comfort" [3]
and "Progress in the Optimized Application of Simulation Tools in Vehicle Air
Conditioning” [4].
In the first one of these articles two CFD simulation methods are compared with
thermal imaging, which provide confidence in the methods since all of them gave
good results. The authors recommend, though, the use of CFD methods rather than
thermal imaging, since CFD calculations offer significantly more information and can
be performed without an actual vehicle. They also point out that the best procedure
would be to use CFD for developing the design with input from thermal imaging
during the prototyping phase.
The second article describes an investigation of the influence on passenger comfort
from the fluid flow and heat transfer at the windshield, including the effect of the air
discharge from the defroster vents. During the investigations, both full-scale tests and
numerical CFD simulations have been performed, showing good agreement between
numerical predictions and experimental results. The last of the above mentioned
articles focuses on the use of simulation tools during the development of vehicle airconditioning systems.
5
2.2
Phenomena assessed
The intention of this work was to describe the following physical phenomena, which
may be observed under various conditions cf. Figure 2.1:
™ Condensation occurs when a vapour is in contact with a surface, which is at
a temperature below the dew point temperature. When the liquid
condensate forms on a surface, it will flow under the influence of gravity.
Normally the liquid wets the surface, spreads out, and forms a film. Such a
process is called film condensation. Another kind of condensation called
dropwise condensation is also possible, but it is not relevant for this work.
™ Freezing or solidification is the process that transforms a liquid substance
into solid phase. It takes place under equilibrium conditions at the same
temperature as melting Tm, and is accompanied by release of latent heat.
™ Frost is a material composed of a mixture of ice and air. Frosting is the
vapour-to-solid phase change in which airborne water vapour becomes frost
on a cooled surface. Frost forms on surfaces when the surface temperature
is below 0oC and below the dew point of the water vapour contained in a
surrounding body of air.
Figure 2.1
2.3
The three phases of water.
Heat transfer
Heat transfer denotes energy in transit as the result of a temperature difference. It can
occur between any two bodies by one or more of three possible modes: conduction,
convection and radiation. Thermal conduction refers to the direct transfer of energy
between particles at the atomic level. Thermal convection may include some
conduction, but refers primarily to energy transfer by eddy mixing and diffusion, i.e.
by fluids in motion. Thermal radiation describes a complex phenomenon which
includes changes in energy form: from internal energy at the source, to transmitted
electromagnetic energy, and then back to internal energy at the receiving surface.
6
2.3.1
Conduction
In an effort to understand the process of conduction one should approach the concepts
of atomic and molecular processes. That is, the conduction can be thought of as
energy transfer from more energetic to less energetic particles due to interaction
between the particles.
In order to explain the mechanisms of conduction we take as an example a gas that
occupies the space between two surfaces kept at different temperatures. This means
that we have a temperature gradient. We further assume that there is no bulk motion.
Temperature at any point is represented by the energy stored in the molecules in the
vicinity of the point. This energy is related to random translational, vibrational and
rotational motions of the molecules. Furthermore, higher energies are associated with
higher temperatures; and due to a steady rate of collision between molecules, transfer
of energy occurs from faster, more energetic molecules to slower, less energetic ones.
If we are dealing with the case where the temperature gradient is not zero, then energy
transfer by conduction must take place in the direction of decreasing temperature. For
the cases of liquids and solids, the situation is much the same, but much more
complicated due to simultaneous macroscopic motion – see convection below.
Figure 2.2
Illustration of the heat conduction process in solids.
Thus energy transfer by conduction is accomplished in two ways:
™
™
by molecular interaction
by “free” electrons in material with electronic conduction
Heat conduction is primarily a molecular phenomenon and can be described by the
one-dimensional Fourier equation:
qx
dT
= −λ
A
dx
(2.1)
where qx is the heat transfer rate in the x-direction, A denotes the area normal to the
dT
direction of heat flow,
is the temperature gradient in the x-direction and λ the
dx
7
thermal conductivity of the material in question. A more general expression for the
three-dimensional case is represented by the following equation:
q
= −λ ⋅ ∇T
A
(2.2)
stating proportionality of the heat flux to the temperature gradient. The thermal
conductivity λ in this case is assumed to be independent of direction. Strictly
speaking the last equation applies only to isotropic media.
2.3.2
Convection
Heat transfer by means of convection may be regarded as a combination of two
mechanisms: energy transfer due to random molecular motion, equivalent to
conduction, as treated above, and energy transfer due to bulk or macroscopic motion
of the fluid, i.e. when large numbers of molecules are moving collectively. The
former dominates near the surface where the flow velocity is low. At the interface
between the surface and the fluid it is the only mechanism working since fluid
velocity there is zero. The latter mechanism originates from the fact that the boundary
layer grows as the flow progresses in the direction parallel to the surface. The
boundary layer is a region in the fluid over which the velocity varies from zero at the
surface to a steady value U∞ associated with the flow. The heat that is conducted
through this layer is swept downstream and eventually transferred to the fluid outside
the boundary layer.
It is possible to classify convection heat transfer with respect to the nature of the flow:
™
forced convection, when the flow is driven by external means (e.g. a
fan or a pump)
natural or free convection, when the flow is induced by buoyancy forces, which arise
from density differences due to temperature variations in the fluid (e.g. convection
heat transfer from hot components in still air)
Typically, the energy transferred concerns the sensible or internal thermal energy of
the fluid, which may be understood as the energy related to molecular motion. But
there also exist convection processes with latent heat exchange, which is generally
associated with a phase change between the liquid and vapour states of the fluid, see
Figure 2.3.
8
Figure 2.3
Illustration of the process of heat transport by convection.
Convective heat transfer may be described by Newton rate equation:
q
= α c ⋅ ∆T
A
(2.3)
where q is the rate of heat transfer due to convection, A the area normal to the
direction of the heat flow, ∆T the temperature difference between surface and fluid
and αc is the convective heat transfer coefficient. This coefficient depends, in general,
on system geometry, the properties of the fluid, and the magnitude of the temperature
difference ∆T.
In the literature a multitude of formulae have been suggested for evaluation of the
convective heat transfer coefficient. For instance, in the case of forced convection
with known air speed u (m/s) parallel to the surface, the following expressions have
been suggested:
αc = 6 + 4 ⋅ u
α c = 7.41⋅ u 0.78
u ≤ 5m s
u ≥ 5m s
(2.4)
(2.5)
2.3.3 Radiation
Thermal radiation is electromagnetic energy emitted by matter at a finite temperature.
Regardless of the form of matter, the emission may be attributed to changes in the
molecular state of the constituent atoms or molecules. The energy of the radiation is
conveyed by electromagnetic waves (or photons); hence radiation does not require the
presence of a material medium, as is the case for conduction and convection.
9
Figure 2.4
Radiative energy transfer at condensed matter surfaces.
The rate of energy emission from a black body is given by:
q
= σ ⋅ T4
A
(2.6)
where q is the rate of radiant energy emission, A the area of the emitting surface, T is
the absolute temperature, and σ the Stefan-Boltzmann constant, a fundamental
constant of nature with the approximate value:
σ = 5.6705 ⋅10−8
W
m 2K 4
(2.7)
Modifications to this equation may be required if the emitting and receiving surfaces
deviate from black-body behaviour, and if geometrical factors associated with radiant
exchange between a surface and its surroundings are of importance.
2.3.4
Combined mechanisms of heat transfer
It is important to note that actual situations with only one mechanism involved in the
transfer of energy are exceedingly rare, and it is therefore important to understand
combined mechanisms of heat transfer. In order to simplify thermal design issues it is
possible to use network analysis.
2.4
Thermal network analysis
In order to transform actual thermal systems into a solvable problem thermal network
analysis may be utilized. A thermal model is a simplified description of a thermal
system. It is used for studying characteristical properties, and for making predictions
about the behaviour of the system. Thermal network analysis allows steady-state and
periodic problems to solved. The technique may also be used with problems
concerning the heat balance of surfaces or ventilated spaces, etc.
10
Often the purpose of a thermal analysis is to determine unknown temperatures, for
instance the air temperature of a room or the interface surface temperature between
two materials in contact. These surfaces or air volumes are represented in the network
analysis by nodes (graphically, as below, a full dot). There is no net inflow of heat to
a node, that is, no heat is stored in the node. A thermal network represents a set of
equations for the determination of the node values and the flows. The basic
components of the network and reduction rules for simplifying the analysis can be
found in appendix A.
Let us take as an example the case of steady-state conduction through a plane wall
with its surfaces held at constant temperatures T1 and T2, see Figure 2.5.
L
T1
x
T2
Figure 2.5
Temperature change through a plane wall with constant surface
temperatures.
Using the Fourier rate equation for the x-direction and solving for qx, subject to the
boundary conditions T(x = 0) = T1 and T(x = L) = T2, we get
qx
dT
= −K
A
dx
L
(2.8)
T
2
qx
dx
=
−
K
∫T dT
A ∫0
1
qx =
(2.9)
KA
(T1 − T2 )
L
(2.10)
The last equation resembles the Newton rate equation
q x = hA∆T
(2.11)
This similarity can be utilized in problems involving both heat conduction and
convective energy transfer. Let us consider a composite wall consisting of three
different materials in different layers, with gas at temperature Th (hot) on the left side
of the wall and gas at temperature Tc (cool) on the right, as shown in Figure 2.6,
11
where the vertical temperature axis is not shown.
Th
k1
k3
L2
T3
L3
T2
T1
L1
Figure 2.6
k2
T4
Tc
Composite wall consisting of three layers and two convective surfaces.
The problem is to express the steady-state heat transfer rate per unit area for this
system. Using equations (2.10) and (2.11):
qx = hh A (Th − T1 ) =
K A
K1 A
K A
(T1 − T2 ) = 2 (T2 − T3 ) = 3 (T3 − T4 ) = hc A (T4 − Tc ) (2.12)
L1
L2
L3
It is easy to see that:
Th − T1 = q x
1
hh A
(2.13)
T1 − T2 = qx
L1
K1 A
(2.14)
T2 − T3 = qx
L2
K2 A
(2.15)
T3 − T4 = qx
L3
K3 A
(2.16)
T4 − Tc = q x
1
hc A
(2.17)
Adding the last five equations we obtain:
⎛ 1
L
L
L
1 ⎞
Th − Tc = qx ⎜
+ 1 + 2 + 3 +
⎟
⎝ hh A K1 A K 2 A K3 A hc A ⎠
And finally:
12
(2.18)
qx =
Th − Tc
L
1
1
L
L
+ 1 + 2 + 3 +
hh A K1 A K 2 A K 3 A hc A
(2.19)
We now start with a simple electrical circuit with resistors R in series and potential U1
and U2 at the ends:
R
R
R
U1
Figure 2.7
resistors.
R
R
U2
Graphical layout of an electrical circuit with serially connected
From this it is possible to write an expression for the current I through the circuit:
I=
∆U
R1 + R2 + R3 + R4 + R5
(2.20)
Comparing equations (2.19) and (2.20) we note an analogy such that each term in the
denominator of equation (2.19) can be thought of as a thermal resistance due to
convection or conduction.
2.5
Mass transfer
Mass transfer can be defined as mass in transit as the result of a species concentration
difference in a mixture. It is a transport process that depends on atomic and molecular
activity.
In connection with this work moisture transfer is a good example of mass transfer.
The following mechanisms may cause a transfer of moisture:
o
o
o
o
o
diffusion (net transfer of water molecules in the direction of decreasing
vapour concentration)
convection (net transfer of water molecules, water droplets or snow
crystalls by air flow)
capillary suction (due to differences in pore water pressure)
wind pressure
gravity
Only the first two of these apply to this work. The density of moisture flow rate is
denoted by g (kg/m2s).
13
2.6
Moisture
In this work moisture is assumed to come from indoor and outdoor air humidity.
Indoor humidity may be caused by several reasons such as washing and drying
clothes, cooking, and so on. Humidity by volume, or water vapour content of air, is
denoted by v (kg/m3). According to the ideal gas law the partial pressure of water
vapour contained in air can be found as:
pv = 461.4 ⋅ (T + 273.15) ⋅ v
(2.21)
where pv is the partial pressure of water vapour and T temperature in degrees Celsius.
Air can only contain a certain maximum amount of water vapour, which is denoted by
vs and is known as humidity by volume at saturation. Another useful and widely used
term when talking about water vapour content of air is relative humidity. It is denoted
by ϕ and is usually expressed as per cent.
ϕ=
v
⋅100
vs
(2.22)
Moisture transfer in air or by air may be accomplished by a number of processes:
™ diffusion
In this case Fick’s law of diffusion is appliable. It can be written in the
following standard form:
g = −D ⋅
dv
dx
(2.23)
where D is the diffusivity of water vapour in air (25•10-6 m2/s at 20 oC)
™ convection
The convective moisture flow rate g (kg/m2s) between the ambient air and a
surface may be found according to:
(
g = β ⋅ v a − v surf
)
(2.24)
where β is the moisture transfer coefficient (it depends on the air velocity
close to the surface), va and vsurf are humidities by volume of ambient air and
air that is close to the surface respectively.
A good approximation for estimating β is known as Lewis’ formula:
14
β=
αc
(2.25)
ρ a ⋅ c pa
As seen from the Lewis formula the β - value is related to the convective heat transfer
coefficient αc. Analyzing formula (2.24) we easily find
™ if humidity by volume of the ambient air is greater than or equal to the
humidity by volume at saturation at the surface then condensation will
take place. Considering a surface at temperature T we obtain:
g = β ⋅ (v a − v s (T))
(2.26)
Condensation of water on the surface will influence the heat balance,
and might result in an increase of the surface temperature and a
reduction of condensate until energy balance is reached.
™ analogically for evaporation from a wet surface, we will get:
g = β ⋅ (v s (T) − v a )
(2.27)
thus eventually providing cooling of the surface.
In analogy to thermal network analysis it is possible to use network analysis for
solving problems regarding mass transfer. In the majority of situations one has to
consider mixed problems, where both heat and mass transfer take place.
15
2.7
The Model and its Algorithm.
The model for the windshield is rather simple, see Figure 2.8. The glass pane is
assumed to be 0.0056 m thick with an area of 1 m2. It is divided into four layers of the
same thickness for making more precise calculations. Thickness of the film is
calculated with a precision of 10-9 m, if thickness of the film is less than this value
steady-state calculations are performed. With the thickness of the film larger than 10-9
m a transient solution is attempted. The model calculates relevant temperatures for the
glass, and uses them in order to find the temperature of the film.
Figure 2.8
Cross sectional model of the windshield glass pane.
The model of the windshield is implemented in a MATLAB-Simulink environment,
see Figure 2.9. First, all temperatures, values for mass of the film and density of
moisture flow rate are initiated.
m f = 0, g = 0
(2.28)
Then the time loop starts. In the beginning the equivalent temperature in the
compartment is calculated.
Te = Ta +
g ⋅r
,
αe
where α e = α c + α r
(2.29)
Here the effective heat transfer coefficient αe is used, which is equal to the sum of the
heat transfer coefficients for convection αc and radiation αr.
16
If the mass of the water film is zero, then the surface temperature at the inside of the
windshield is calculated and taken as the film’s temperature.
Ts =
where K g =
8Aλg
dg
K g ⋅ T ( 4 ) + α e ⋅ A ⋅ Te
K g + αe ⋅ A
,
T f = Ts
(2.30)
is conductance of the half of the layer of the windshield glass.
The density of moisture flow rate is then calculated using the equivalent temperature
in the car compartment and the surface temperature of the windshield.
g = ( ve − vs (Ts ) ) ⋅ β ⋅ A
(2.31)
Here β is a coefficient for surface water vapour transfer and v the humidity by
volume.
The change of mass of the water film is then calculated and finally the value for the
mass of the film and the time mark are updated.
m f = m f + ∆m f
(2.32)
t = t + ∆tstat
where ∆tstat is the time step that is small enough for the solution to converge.
Next a new value of the equivalent temperature is calculated (see Formula 2.29) and
the model inspects the value for the mass of the film. If the mass of the water film is
not zero the thickness of the film is calculated:
df =
mf
(2.33)
ρf
Note that the density of the film is a function of temperature. If the film’s temperature
is above 0oC then water is present and thus the density for fluid water is used; if the
temperature of the film is below – 0.1oC, then ice must be present, and the density for
ice is used. If the temperature of the film is between 0oC and – 0.1oC then we assume
that we have either freezing or thawing and interpolate a value for the density that lies
between the values for the densities of ice and water. The same procedure is executed
for the thermal conductivity λf of the film and the latent heat associated with phase
change.
T film ≥ 0o C ⇒
λ film = λwater , ρ film = ρ water , rfilm = revaporation
17
(2.34)
−0.1o C < T film < 0o C
ρ film =
rfilm =
λ film =
ρ water − ρice
0.1
revap − rsubl
0.1
⋅ T film + ρice
(2.35)
⋅ T film + rsubl
λwater − λice
⋅ T film + λice
0.1
T film ≤ −0.1o C
⇒
λ film = λice , ρ film = ρice , rfilm = rsubl
⇒
(2.36)
This allows us to calculate the conductance for half of the film
Kf =
2⋅ A⋅λf
(2.37)
df
We now have to consider the error developing during calculations and we assume that
it is of order 10-9m. If the value of the film’s thickness is less than the assumed error,
we use steady-state calculations for finding the film’s temperature.
Tf =
K inf ⋅ T ( 4 ) + K out
f ⋅ Te
K inf + K out
f
1
where K inf =
(2.38)
d
1
+ f
Kg 2 ⋅ λf
1
and K out
f =
1
αe
+
df
2⋅λf
If the film’s thickness is greater than the assumed error limit we use a transient
solution. We calculate the surface temperature from the inside of the glass pane using
the equivalent compartment temperature and the previously calculated temperature of
the film.
Ts =
T f ⋅ K f + Te ⋅ α e ⋅ A
(2.39)
K f + αe ⋅ A
Then we calculate the amount of heat being transferred from the windshield’s fourth
layer to the air inside the car and determine the change of total energy content of the
film.
Q=
1
1
1
+
Kg K f
⋅ (T ( 4 ) − T f ) +
1
1
1
+
αe A K f
18
(T − T )
e
f
(2.40)
∆e =
Q ⋅ ∆t
mf
(2.41)
It is now possible to find the change of the film’s temperature and update the film
temperature.
∆T f =
∆e
cf
⎧
J
3
⎪ 4.2 ⋅10 kgK
⎪
⎪
J
where c f = ⎨3.3 ⋅106
kgK
⎪
⎪
J
3
⎪ 2.1 ⋅10
kgK
⎩
t > 0o C
− 0.1o C < t < 0 o C
(2.42)
t < −0.1o C
T f = T f + ∆T f
Then the density of moisture flow rate and change of the film’s mass are calculated,
the time step is updated and the next model loop starts. (See Formulas 2.29 and 2.30).
19
Initial & boundary
temperature
mf > 0
Calculating the equivalent
temperature:
mf = 0
Te = Ta +
g ⋅r
T film ≥ 0o C ⇒ λ film = λwater , ρ film = ρ water , rfilm = revaporation
αe
−0.1o C < T film < 0o C
If evaporation: r = rev
sublimation: r = rsu
ρ film =
rfilm =
Ts =
K g ⋅ T ( 4 ) + α e ⋅ A ⋅ Te
K g + αe ⋅ A
T film
T f = Ts
ρ water − ρice
0.1
revap − rsubl
⇒
⋅ T film + ρice
⋅ T film + rsubl
0.1
λ
−λ
λ film = water ice ⋅ T film + λice
0.1
≤ −0.1o C ⇒ λ film = λice , ρ film = ρice , rfilm = rsubl
df =
mf
Kf =
ρf
df > 10-9
2⋅ A⋅λf
df
df < 10-9
t = t + ∆tstat
df
Ts =
Q=
T f ⋅ K f + Te ⋅ α e ⋅ A
K f + αe ⋅ A
1
1
⋅ (T ( 4 ) − T f ) +
(T − T f
1
1
1
1 e
+
+
αe A K f
Kg K f
∆e =
Q ⋅ ∆t
mf
∆T f =
∆e
cf
)
T f = T f + ∆T f
Figure 2.9
K inf ⋅ T ( 4 ) + K out
f ⋅ Te
K inf + K out
f
where K inf =
and K out
f =
1
d
1
+ f
Kg 2 ⋅ λf
1
1
αe
g = (ve − v s (Ts )) ⋅ β ⋅ A
mf = mf +∆mf
Tf =
test: mf ≥ 0
if mf = 0 → Tf = Ts
Flow chart for the algorithm.
20
+
df
2⋅λf
3
3.1
IMPLEMENTATION OF THE MODEL
Tools
As tools for creating a model for simulation of the energy balance at the passenger car
windshield MATLAB and Simulink were used.
MATLAB combines numeric computation, advanced graphics and visualization, and
a high-level software language. It is used in a variety of application areas including
signal and image processing, control system design, financial engineering, and earth
sciences.
Simulink is an interactive tool for modeling, simulating, and analyzing dynamic
systems. It enables you to build graphical block diagrams, simulate dynamic systems,
evaluate system performance, and refine your designs. Simulink integrates seamlessly
with MATLAB, providing with immediate access to an extensive range of analysis
and design tools. These benefits make Simulink the tool of choice for control system
design, communications system design, and other simulation applications.
Simulink provides a complete set of modelling tools that can be used for quick
development of detailed block diagrams of various systems. Features such as
Simulink data objects, block libraries, hierarchical modelling, signal labeling, and
subsystem customization provide a powerful set of capabilities for creating,
modifying, and maintaining block diagrams.
Building physics toolbox [8], [9] is a Simulink library developed at the Department
of Building Physics at Chalmers University of Technology. It includes blocks for heat
transfers in layered structures, HVAC components, blocks for boundary conditions,
coupling to weather data, internal gains and blocks for room energy balance.
This toolbox, together with Simulink's comprehensive set of predefined blocks, makes
it easy to create concise representations of various systems, regardless of their
complexity.
21
3.2
Implementation of the model in Simulink
The previously described model of the windshield is implemented in Simulink
according to Figure 3.1. It is constructed by creating new blocks and using blocks
from the library BFTools1 [8]. The exterior and interior climate data are supplied to
the model from input MATLAB data files.
Figure 3.1
Graphical layout of the implemented windshield model.
As may be seen from Figure 3.1 the model consists of a main block that calculates the
temperature of a condensation film, its mass and thickness, and a few blocks that
supply required information to the main block.
3.2.1 Input blocks
There are three main input blocks that are used as sources of data in this model. The
first calculates the equivalent outside temperature and the convective heat transfer
coefficients (see Fig. 3.2). They are identified as “Fi, Teute, Tebil” in Figure 3.1.
22
Figure 3.2
The first input block and its graphical design.
The second input block is rather a number of blocks and is represented by the main
block called “4 Layers material new”, two blocks that are responsible for a supply of
boundary conditions named “BoundR” and “Subsystem1” and a number of blocks
that “translate” the answer of the main block to be plotted (Fig. 3.3). The main block
here divides the glass of the windshield into four layers and calculates the temperature
for each of the layers. It comes from the library BFTools1 [8]. The input block
“BoundR” comes from the same library. It takes care of the boundary between the
compartment and the windshield, while “Subsystem1” is responsible for the boundary
between the windshield and the exterior.
Figure 3.3
Second input block
The third and last of the input blocks is extremely simple. Here the thermal
conductance of any one layer of the glass is calculated (see Fig. 3.4). It is called “kg”
23
in Figure 3.1:
Figure 3.4
Graphical layout of the third input block.
3.2.2 Main part of the model.
Main part of the model is represented by the block named “Subsystem” in Figure 3.1,
and its graphical layout is as follows (Fig. 3.5):
Figure 3.5
Graphical design of the main block
Figure 3.5 shows six blocks that require further explanation. All other blocks are
standard Simulink blocks. These six blocks are named: “Film”, “g2”, “Subsystem”,
“dfilm, mf”, “Tsbil, mf = 0” and “Tfilm, Tsbil if mf > 10-6”. Let us see what these
blocks do, paying special attention to the last block.
The first block, called “Film”, defines properties of the eventual film and is the
equivalent of equations (2.34) – (2.36) of the algorithm. It looks as follows (see Fig.
3.6):
24
Figure 3.6
Graphical layout of block ”Film”.
The second block “g2” is responsible for the calculation of density of moisture flow
rate (see Fig. 3.7) according to equation (2.31):
Figure 3.7
Graphical design of the block “g2”
The third block “Subsystem” makes certain that the calculated density of moisture
flow rate is larger or less than zero (see Fig. 3.8):
Figure 3.8
Graphical layout of the block “Subsystem”
The fourth block “dfilm, mf” calculates the mass of the film by integration (see Fig.
3.9):
25
Figure 3.9
Graphical layout of the block ”dfilm, mf”
Prelast block “Tsbil, mf = 0” calculates surface temperature in the case when the mass
of the film is zero (see Fig. 3.10) in accordance with equations (2.30):
Figure 3.10
Graphical layout of the block ”Tsbil, mf = 0”
The last block called “Tfilm, Tsbil if mf > 10-6” handles calculations of the film and
surface temperatures if the mass of the film is larger than zero or equals to zero. It
represents the main part of the algorithm, and may be laid out graphically in the
following way (see Fig. 3.11):
Figure 3.11
Graphical design of the block ” Tfilm, Tsbil if mf > 10-6”.
26
Calculations are performed in accordance with equations (2.39) – (2.42) for the
algorithm. Blocks “Kfin” and “Kfut” compute constants for equation (2.40) from the
algorithm and look the following way:
Figure 3.12
Graphical layout of blocks “Kfin” (left) and “Kfut” (right).
Equation (2.40) is implemented in the next block called “Q” (see Fig. 3.13):
Figure 3.13
Block ”Q”
The next step in the model is the implemention of equations (2.41) and (2.42) from
the algorithm. It is executed in the block ”Tfilm, df > 10e-9” (see Fig. 3.14), and takes
care of calculations while the thickness of the film is larger or equal to 10-9 m.
Figure 3.14
Block ”Tfilm, df > 10e-9”
The case when the thickness of the film is less than 10-9 m is implemented in the
block named “Tfilm, df < 10e-9” (see Fig. 3.15). It is also described by equation
(2.38).
27
Figure 3.15
Block ”Tfilm, df < 10e-9”
The last block to be discussed is called “Tsbil, mf > 0” (see Fig. 3.16). It computes the
surface temperature in the case when the mass of the film is strictly larger than zero.
This block represents an implementation of equation (2.39).
Figure 3.16
Block ”Tsbil, mf > 0”.
28
4
VALIDATION OF THE MODEL
4.1
Climatic Chamber
An experiment was performed on a car model V70 at the Volvo Car Corporation’s
Climatic chamber KC-1, Torslanda, Göteborg in order to validate the previously
discussed model. The climate chamber is used for tests of the idling car’s climate
performance. It is intended for investigations of climatic conditions that do not
depend on wind speed or humidity, and tests that do not depend on the use of a
dynamometer. The climatic chamber is equipped with facilities for engine exhaust
fume removal. Dimensions of the climatic chamber are LxWxH are 7.5 x 3.3 x 2.7
m3. It is possible to simulate climates in the chamber with air temperatures that range
between –40oC / +50oC ± 0.2oC. The wind speed was about 0.8 m/s. Cooling capacity
is 35kW and heating capacity is 30kW. The test procedures are fully automatic
allowing real-time measurements.
4.2
Defroster test for the windshield according to 78/317/EEC
The climatic chamber and the car were conditioned at -18 ± 3oC for at least 24 hours
before the test. The ice layer on the windshield was created by spraying 0.044cm3/cm2
of water with the help of a spray gun and was left for conditioning for 30 minutes.
[14] It was assumed that there was no sun and that the car was idle. The test was run
for 47 minutes untill all ice melted.
As output the experiment provided figures for the frost pattern at specified time
intervals and in this specific case also surface temperatures of the windshield. In
addition, temperature sensors were placed on the windshield glass as shown in Figure
4.1. Furthermore, no wipers were used during the test. Five photos of the frost pattern
on the windshield during the experiment at specified time intervals are shown in
Figures 4.2 - 4.6. They can be compared with the graph of the calculated thickness of
the water film depicted on Figure 5.2.
The point closest to the middle (1)
Point in the upper part of the windshield (lies
on the middle line) (2)
The point close to the
top left corner (4)
Driver’s field
of vision
The point closest to the
centre of passenger’s
field of vision (5)
Point in the lower part of the
windshield (lies on the middle line)
(3)
Point close to the bottom left corner
(6)
Figure 4.1
Positions of temperature sensors on the windshield
29
Figure 4.2
Defrost pattern 7 minutes after the start of the experiment
Figure 4.3
Defrost pattern 10 minutes after the start of the experiment
Figure 4.4
Defrost pattern 14 minutes after the start of the experiment
30
Figure 4.5
Defrost pattern 18 minutes after the start of the experiment
Figure 4.6
Defrost pattern 24 minutes after the start of the experiment
4.3
Computational model of the windshield
The previously discussed model of the windshield and eventual water film was
combined here with the model of the compartment implemented by Peter Ejdestig
[11]. The graphical layout of this combination can be seen on the Figure 4.7. The
output data from the model describes the compartment temperature and the relative
humidity of the air inside the compartment. These are in turn supplied to the model
describing the windshield and eventual water film.
31
Figure 4.7
Combined models for windshield and car compartment.
32
Values for the temperatures of the simulated air flow from the defroster, parallel to
the inside surface of the windshield, were according to Table 4.1.
Table 4.1
Variation in time of measured and simulated outlet temperatures from
the derfoster parallel to the inside of the windshield.
Time (minutes)
Temperature from measurements of V70 (K)
Temperature from simulation (K)
0
255
256
5
283
284
10
298
296
15
308
302
20
313
308
The model was tested for the top corner of the windshield on the passenger side. As
input the model uses ambient and inlet air temperature, humidity and the initial
amount of water on the windshield. For the computational model of the windshield it
was assumed that there was no sun and that the car was idle. Speed of the air flow
from the defroster in the passenger compartment was set to 2 m/s according to Figure
4.8. Speed of the outside air flow was set to 1 m/s. A simplified geometry of the
windshield was assumed, i.e. a flat pane of glass. As output the calculations return
thickness, mass and temperatures of the water film. Steady-state calculations are
performed if the thickness of the film is small enough (10-9 m in this case), otherwise
transient calculations are carried out. The simulation time was about 5 minutes on PII
350 MHz.
Figure 4.8:
Distribution of velocities over the windshield for V-70 model, by
courtesy of the group of Computational Fluid Dynamics, Volvo Car
Corporation.
33
5
RESULTS
The model yields the temperature of the water film and compares with the
temperature measured in the experiment at point A close to the top corner of the
windshield on the passenger’s side (Figure 5.1). It is also interesting to compare
photos of the defrost pattern shown in Figures 4.2 – 4.5 with the calculated thickness
of the water film presented in the Figure 5.2 below. The calculation model readily
gives information about how fast various thicknesses of the water film evaporates.
This is one of the advantages of an analytical model compared to experiment (the
calculation has not been not verified).
Figure 5.1
Comparison of measured and calculated results for the temperature of
the water film on the windshield.
Figure 5.2
Simulated thickness of the water film
34
In order to see if the model provides reasonable results, additional calculations were
performed. They are presented in Figures 5.3 - 5.6 below. The diagrams show results
for the situation with air flow speed from the defroster in the passenger compartment
equal to 2m/s (left top corner of the windshield) and 5m/s (central part of the
windshield). Initial conditions given for the water film thickness for both diagrams
were 0.01 cm3/cm2, 0.02 cm3/cm2, 0.044cm3/cm2, 0.05 cm3/cm2, 0.1 cm3/cm2. In the
Figures the measured temperature is plotted with the style of line different from those
used for the calculations. Also the curves for calculated temperatures have been
drawn using different colours: blue (0.01 cm3/cm2), magenta (0.02cm3/cm2), green
(0.044 cm3/cm2), red (0.05 cm3/cm2), cyan (0.1 cm3/cm2); calculated film thicknesses
are depicted in the corresponding colours.
Figure 5.3
Calculated water film thicknesses on the windshield with air flow
speed from the defroster in the passenger compartment set to 2 m/s
and different initial film thicknesses.
Figure 5.4
Calculated water film thicknesses on the windshield with air flow
speed from the defroster in the passenger compartment set to 5 m/s
and different initial film thicknesses.
35
Figure 5.5
Comparison of measured and calculated results for the water film
temperature on the windshield with air flow speed from the defroster in
the passenger compartment set to 2 m/s and different initial
thicknesses of water film.
Figure 5.6
Comparison of measured and calculated results for the water film
temperature on the windshield with air flow speed from the defroster in
the passenger compartment set to 5 m/s and different initial
thicknesses of the water film.
36
6
CONCLUSIONS
Good agreement between calculations and measurements was found, but there are still
a number of factors that can be improved. The real geometry of the windshield differs
from that used in the model rather substantially and the interior of the compartment is
not included in the calculations. Of course, since a combined model is used for
validation of the results all simplifications in the compartment model influence the
results as well, even though Peter Ejdestig’s model [11] provides very good results.
As mentioned in that report, no measurements of the humidity in the inlet air during
the performance tests were made, which complicated the validation of the value for
the humidity in the compartment model and also influenced the results.
The deviation between calculated and measured results might be a consequence of the
simplified description of the air flow (2 m/s parallell to the surface of the windshield).
The calculated results show that the speed of the air parallell to the inside of the
windshield is of significant importance. The temperature of the external surface of the
windshield was used for validation, but it was difficult to measure, since any
measurements of the temperature on the windshield is affected by the radiation
balance for the surrounding surfaces of the climatic chamber and the car.
The initial value for the water film thickness is not the same everywhere on the
windshield during the measurements. The calculated results show that the thickness of
the water film is of significant importance. Furthermore, the experimental data shows
that the car was not conditioned long enough. For validation of calculated results a
time delay of 2 minutes 38 seconds would have to be used.
37
7
RECOMMENDATION FOR FUTURE WORK
As future work the following is suggested:
™ Improving the model, in order to bring it closer to reality.
™ The present work concerned the windshield. It would, however, be interesting
to investigate what happens at other surfaces of the car with these climatic
conditions, e.g. defrosting of the side windows. This implies that circulation of
air flow in the entire car compartment need to be taken into account, which
probably can be done with the help of CFD calculations.
™ Combining the model with other fast computational tools for estimations of
the climate in a passenger compartment of the type of ADVISOR - advanced
vehicle research tool created by AVL [15].
38
8
T
αc
pa
K
ρf
λf
g
rf
q
Q
A
L
R
ϕ
v
df
c
LIST OF SYMBOLS
temperature, [oC]
convective heat transfer coefficient, [W/m2K]
partial water vapour pressure in the air, [Pa]
thermal conductance, [W/K]
film density, [kg/m3]
thermal conductivity of the film, [W/mK]
density of moisture flow rate, [kg/m2s]
latent heat associated with phase change, [J/kg]
density of heat flow rate, [W/m2]
heat flow rate, [W]
area, [m2]
length or thickness, [m]
thermal resistance, [K/W]
relative humidity, [-]
humidity by volume, [kg/m3]
thickness of the water film, [m]
specific heat capacity, [J/kg K]
39
APPENDIX A
Basic components of thermal networks:
Component
Mathematical
relation
Node
ΣnQn→i = 0
Resistance
T1 – T2 = R∗Q
Graphical
representation
R
K
Conductance
Q = K*(T1 – T2)
Prescribed
boundary value
T⏐boundary = T0
T0
Prescribed flow
Q⏐boundary = I0
I0
Other important elements of thermal engineering analysis are the reduction rules,
which simplifies the analysis greatly.
Reduction rules:
1. Resistances in series:
T1
R1
T12
R2
T2
T1
R1 + R2
T2
K1 ⋅ K 2
T1 K + K
1
2
T2
2. Conductances in series:
T1
K1
T12
K2
40
T2
3. Conductances in parallel:
K1
T1
T1
T2
K2
K1 + K2
T2
4. Resistances in parallel:
R1
T1
T1
T2
R2
R1 ⋅ R2
R1 + R2
T2
5. Reduction of N boundary nodes to a single one:
T1
K1
K0 = ΣKi
T2
T0 =
1
K0
N
∑K T
i i
1
KN
TN
6. Addition rule for sources at a node:
I1
I1 + I2
I2
41
7. Node coupled to boundary temperature and heat flow source:
T0
K0
K0
T
T0 + I0/K0
K
T
I0
42
References
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[2] 2001-01-1699 “Comparison of CFD Simulation Methods and Thermal Imaging
with Windscreen Defrost Pattern”, A.F. Skea, R.D. Harrison, A.J. Baxendale and D.
Fletcher from Fluids Group, MIRA.
[3] 2001-01-1729 “Effects of Vehicle Windshield Defrosting and Demisting Process
on Passenger Comfort”, A. Aroussi, A. Hassan from Flow Diagnostics Laboratory,
The University of Nottingham and B.S. AbdulNour from Ford Motor Company.
[4] 2001-01-1699 “Progress in the Optimized Application of Simulation Tools in
Vehicle Air Conditioning”, B. Taxis-Reischl, S. Morgenstern, F. Brotz from BEHR
GmbH & Co, and T.Mersch from BEHR Climate Systems.
[5] Incropera, Frank P. and DeWitt, David P. “Fundamentals of heat and mass
transfer” 2nd Edition. Singapore: John Whiley & Sons Inc. 1985
[6] “Using Simulink”. The Math Works Inc. 1999
[7] Siegel, R and Howell, John R. “Thermal radiation heat transfer” 3rd edition.
Austin, USA: Hemisphere Publishing Corporation. 1992
[8] Hagentoft, C-E. “Thermal system analysis using Simulink BFTools1”. Göteborg
CTH Report R-00: 8, 2000
[9] Kalagasidis, Angela S. “Thermal system analysis using Simulink BFTools2”.
Göteborg CTH, Report 2002:1
[10] Frank Kreith, Mark S. Bohn “Principles of heat transfer” 5th edition, PWS
Publishing Company. 1997
[11] Peter Ejdestig “Simulation model of the climate in a car passenger
compartment”, publication E-02:1
[12] ASHRAE handbook fundamentals, 1989.
[13] Eva-Lotta Karlsson “Modellering och simulering av kupeklimat i en personbil”,
examensarbete vid maskintekniklinjen, institutionen för Termo- och Fuiddynamik,
Chalmers Tekniska Högskolan
[14] Testing code ”Defroster test according to 78/317/EEC”, issue 3, Volvo
Personvagnar AB
[15] ADVISOR – advanced vehicle research tool; http://www.avl.com/advisor
43