A Model of the Air Temperature in a Truck Cabin

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A Model of the Air Temperature in a Truck Cabin

M A T T I A S B J Ö R K L U N D

Master's Degree Project Stockholm, Sweden 2004

IR-RT-EX-0426

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Abstract

This thesis project was carried out at Scania in S¨odert¨alje. The aim was to develop a model for the air temperature in a truck cabin. The model should simulate the truck temperature as it would have been measured by the Automatic Climate Controller’s (ACC’s) temperature sensor. Apart from the actual cabin temperature model, models for the heater and cooler systems also had to be developed.

The model was developed in Simulink and is intended to be used in real-time ”hardware-inloop”

simulations at Scania’s integration laboratory. For that purpose, simplicity of the model is given priority over high accuracy.

Both black- and white-box techniques to model the cabin air temperature where developed and evaluated. Both techniques resulted in a model with acceptable performance. The main problem with the white-box model was a non-symmetry in the cabin temperature dynamics. This problem was solved by using variable parameters in the white-box model.

The results of this thesis project shows possible approaches to do more accurate models of the cabin air temperature. The white-box model, in particular, is suitable to enhance in order to improve the performance.

Acknowledgements

This master thesis project was the last part in my Master of Science education in Engineering Physics. There have been many people who in one way or another have helped me during this thesis project. I would specially like to thank my advisor at Scania, Emil Axelson and my advisor at KTH, Assoc. Prof. Elling W. Jacobsen. I would also like to thank the people at RESA on Scania for all the help during this time.

I would also like to thank my family and friends “back home” for the support during my time at KTH. Special thanks to Entomed.

To my mother and father - you will always be in my heart.

Stockholm, November 2004

Mattias Bj¨orklund

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Notations

All units in this thesis are SI-units, except for temperature which is in units of Celsius.

A Area

c_p Isobaric heat capacity c_p,air Isobaric heat capacity for air c_p,cwt Heat capacity for the cooling water

m Mass

m_cab Cabin air mass

˙

m Mass ﬂow

˙

m_ATA Air (mass) ﬂow through the ATA

˙

m_cwt Cooling water (mass) ﬂow

˙

m_HVAC Air (mass) ﬂow through the HVAC-system

Q Heat

Q_spec Speciﬁc heat Q˙ Heat ﬂow (power)

t Time

T Temperature

T_AC Air temperature directly after the evaporator in the HVAC-system T_ATA Air temperature from the ATA

T_cab Cabin air temperature

T_cwt,c Cooling water temperature out from the heat exchanger T_cwt,i Cooling water temperature in to the WTA

T_cwt,o Cooling water temperature out from the WTA and/or in to the heat exchanger T_eng Engine compartment temperature

T_mix Mixed air temperature - air temperature directly after the heat exchanger in the HVAC-system T_out Outside air temperature

T_walls Temperature of cabin walls v Vehicle speed

σ Heat transfer coeﬃcient

σ_cab Heat transfer coeﬃcient through the cabin walls

σ_in Heat transfer coefficient from the outside air to the cabin walls σ_out Heat transfer coefficient from the cabin air to the cabin walls σ_eng Heat transfer coefficient from the engine compartment to the cabin

λ Heat conductivity coeﬃcient (thermal conductivity)

ρ Density

All time derivatives in this thesis are noted by a dot. For example, the time derivative of the heat Q - the heat transfer - is noted as ˙Q.

Abbreviations

AC Air Conditioner

ATA Air To Air heater

ACC Automatic Climate Control

CAN Controller Area Network

cwt Cooling water

ECU Electronic Control Unit

HVAC Heater Ventilation Air Condition

SITB System Identiﬁcation Toolbox

WTA Water To Air heater

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List of Figures

2.1 Overview of the heating and cooling system in a truck . . . 3

2.2 Basic diagram over the HVAC-system . . . 4

3.1 The ﬁrst law of thermodynamics . . . 7

3.2 Heat ﬂow through a wall . . . 8

3.3 A system with input u, output y and disturbance e. . . . 10

4.1 The general structure of the model . . . 16

4.2 The AC-model . . . 18

4.3 The speciﬁc power Q_spec for a heat exchanger. . . 19

4.4 The heat exchanger model . . . 19

4.5 The WTA-model . . . 20

4.6 The ATA-model . . . 21

4.7 Poles for the two black-box models . . . 22

4.8 Zeros for the two black-box models . . . 23

4.9 Simulation of the cabin temperature at 0 ambient temperature. . . 24

4.10 Plot of the heat conduction coeﬃcient σ_eng . . . 27

5.1 Simulation with the ATA-model . . . 29

5.2 Simulation with the ATA-model . . . 30

5.3 Simulation of the temperature after the heat exchanger at 0 ambient . . . 31

5.4 Model error of the mixed air-simulation with 0 ambient temperature . . . 32

5.5 Simulation of the temperature after the heat exchanger at -20 ambient . . . 33

5.6 Model error of the mixed air-simulation with -20 ambient temperature . . . 34

5.7 Simulation of the AC and the model error . . . 35

5.8 Cabin temperature simulation and the model error with the black-box model and -20 ambient temperature . . . 36

5.9 Cabin temperature simulation and the model error with the black-box model and 0 ambient temperature . . . 36

5.10 Step responses in the black-box cabin temperature model . . . 37

5.12 Simulation of the cabin temperature at -20 ambient temperature. . . 38

5.14 Simulation of the cabin temperature at 0 ambient temperature using the “complete” model. . . 39 B.1 The top-level block for the complete model . . . B-2 B.2 The HVAC-model block . . . B-3 B.3 The AC-model block . . . B-4 B.4 The “evaporator temperature to air temperature”-model block . . . B-5

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B.5 The heat exchanger model block . . . B-6 B.6 The ATA model block . . . B-7 B.7 The WTA model block . . . B-8 B.8 Block for cooling water temperature after WTA . . . B-9 B.9 The cabin model box . . . B-10 B.10 Calculation block for parameters . . . B-11

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1 Introduction

1.1 Background and Purpose

The next generation of Scania trucks contains up to 25 control units (ECU:s). These are connected together via a CAN-network. For example, there are control units for the engine, the gearbox, the break system and the automatic climate control system. The functionality of the control units when they are connected together are tested in an integration laboratory. A dynamic model reads the CAN-network and generates appropriate signals to the diﬀerent ECU:s. At this point there are only a few signals generated by the model. The other signals are generated by various hardwares that are built specially for each ECU.

Scania are currently building a new laboratory that will enable automatic integration tests. In this lab, most of the signals will be generated by dynamic models and circuit boards. There is currently no complete dynamic model to generate signals for the automatic climate control system (ACC).

The purpose with this thesis is to develop a dynamic model that will generate appropriate signals to the ACC depending on the control output from the ACC and other systems. That is, we want to “dry run” the ACC. The main task is to model the temperate in the truck cabin and how it depends of factors such as auxiliary heaters, sun intensity, et cetera.

The air temperature depends on the heat ﬂow from the heating and cooling system (the HVAC- system) and auxiliary heaters. No complete models for these heaters and coolers were available.

A major task in this thesis project was therefore to model the heating and cooling system in a truck.

1.2 Previous Works

In a previous thesis work on Scania, Andersson [1] tried to model the temperature in the truck by setting up a complete model over the heat transfers through the different parts of the cabin walls and so on. This white box approach proved to get difficult to work well. The approach in this thesis is to model the temperature with as few as possible equations. For example, we can try to use only one heat transfer equation and then try to determine the coefficient in this equation from measurements of the cabin temperature.

There is also a thesis work by Hammarlund et al [2], where the climate in a passenger car was modelled. This was done as a black-box modelling using System Identiﬁcation Toolbox [10].

Eriksson [3] developed a model of the evaporator temperature in the AC-system. A signiﬁcant part of that model will be used in this thesis project.

1.3 Delimitations

One can spend an inﬁnite amount of time trying to model the temperature in a truck cabin. The aim of this thesis work is to develop a model suﬃciently good for integration testouts.¹ Because of that there are some things that we not will consider:

There are some diﬀerent types of engines and cabins. Since there is a lack of available measurement to use in system identiﬁcation, we have to limit the number of models to, perhaps, one general model. The aim is to show a fairly straight forward procedure to adapt the parameters to other types of engines and cabins.

1The definition of “sufficiently good” in this context is a little vague, but it would require a thesis of its own to define it...

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We will only consider the major contributions to the heat in the cabin. Factors such as heat generated by people in the cab, et cetera, will not be considered.

The system identiﬁcation and the model validation will rely on existing measurements. There was no time in this thesis project to do new experiments in the climate chamber

Some work was also done with implementation of the model into the existing integration laboratory at Scania.

1.4 Sources

The main sources for the heat transfer part in the theory section was Holman [4] and Pierre [5].

Other sources that are used are referred to in the text.

1.5 Readers Guide

Chapter 2 gives an overview of how a Scania truck cabin is constructed from a “heat transfer point of view”.

Chapter 3 gives a brief introduction of the diﬀerent theories that applies to this thesis project.

In chapter 4 the actual modelling work is presented.

Chapter 5 discusses the validation of the diﬀerent models that are developed in the thesis.

The overall results of the thesis work and a discussion over these results are presented in chapter 6.

Appendix A gives a more detailed description of the diﬀerent inputs and outputs in the Simulink- model. This description is intended for the people who are going to use and maintain the model.

Appendix B contains ﬁgures over the most important Simulink-blocks in the model.

Appendix C contains the physical parameters and other constants used in the simulations.

Appendix D gives a more detailed description over the measurement data used for modelling and validation in this thesis project.

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2 Description of the Air Heating and Cooling System in a Truck

This chapter gives a description of the diﬀerent types of heaters and coolers that are used in a Scania truck.

Depending of the conﬁguration of the truck, we can have diﬀerent heaters in the cabin. A truck can be equipped with auxiliary heaters, such as the Water to Air heater or the Air to Air heater.

Those auxiliary heaters are intended to be used at long stops, such as overnight stops, when the driver wants to keep the temperature in the cabin at a comfortable level. We also have the

“ordinary” heating system, the HVAC-system, which itself can consist of a heater and possibly a cooler (the AC-system). See ﬁgure 2.1 for a overview.

HVAC−system

AC Outside or recirculated air

Heat exchanger

Air after AC Mixed air to cabin

Cooling water after heat exchanger WTA

Cooling water from engine Cooling water after WTA Truck

cabin

Ventilated air Recirculated air

Recirculated air ATA

Air from cabin Heated air to cabin

Figure 2.1: Overview of the heating and cooling system in a truck

2.1 Water to Air Heater (WTA)

The WTA is a device that uses diesel to heat the engine’s cooling water. Depending of its temperature the cooling water are pumped in a short or a long circulation. At low water temperatures it ﬂows in the short circulation, which includes the WTA and the HVAC-system. At higher temperatures the water ﬂows in the long circulations, which apart from the short circulation also includes the engine. This means that the engine is already warm at startup, which reduces emissions and wear on the engine.

Since the WTA heats the cooling water the heating of the cabin works in the same way as it would

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if the engine were running. The WTA only controls the temperature of the cooling water. The cabin temperature is controlled by the ACC.

2.2 Air to Air Heater (ATA)

The ATA works in a similar fashion as the WTA but it heats air instead of the cooling water. A fan forces air from the cabin through the ATA, where it gets heated, and then it is led back to the cabin.

Contrary to the WTA, the ATA does control the cabin temperature. Because of this the ACC control of the cabin temperature is disengaged when the ATA is running.

There is only either a WTA or an ATA (or none) installed in a truck. Never both at the same time.

2.3 The Ventilation System (HVAC)

Air from the outside is passed from the front of the truck, through an evaporator and then through a heat exchanger (see the following sections for a description). The airﬂow can be induced by the vehicle’s speed, but it can also be forced by a fan. It is also possible to recirculate the cabin’s air.

In that case air from the cabin, rather than air from the outside, is passed through the HVAC system. This is used to reduce the heat-up time of the cabin air at cold days. See ﬁgure 2.2 for a diagram over the HVAC-system.

AC

evaporator Outside or

recirculated air

Refrigerant

Heat

exchanger Cooling water from the engine

Cooling water returning to the engine

Cabin

Ventilated or recirculated air

Figure 2.2: Basic diagram over the HVAC-system

2.3.1 The Air Conditioning (AC) system

The principles of the AC-system is basically the same as for an ordinary refrigerator. Low pressure and temperature refrigerant enters the evaporator where it evaporates as a consequence of the warmer air that passes through the evaporator. The heat needed for the evaporation is taken from the air, which therefore is cooled down.

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The AC-system is mainly used to cool down the air during hot days before it enters the cabin. It can also be used to dry the air when needed. The air is ﬁrst cooled down by the evaporator. Since cool air cannot contain as much water as warmer air, water condenses on the evaporator and the air is hence getting dryer. The air is then heated up again by the heat exchanger.

2.3.2 The Heat Exchanger

Heated cooling water from the engine is passed through the circuit of the heat exchanger. The cooling water can also be heated by the WTA, as discussed in section 2.1. Fresh or recirculated air is passed through the ﬂanges of the heat exchanger where it gets heated. The air is then passed out to the cabin.

The cooling water temperature in to the heat exchanger is intended to be at a constant level at about 70 to 80 when the engine is warm and running. The power of the heat exchanger is controlled by the cooling water ﬂow through the heat exchanger. The cooling water ﬂow in its turn is controlled by a water valve.

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3 Theory

The intention with this chapter is to give the reader a brief overview of the theories that this thesis is based on.

Section 3.1 gives a brief overview of the concept of heat transfer and heat ﬂow. Most of the things covered here should be known for those who are familiar with basic thermodynamics. However, one thing that is not usually covered in basic thermodynamics is the speciﬁc power for a heat exchanger. Read more about this in section 3.1.8 on page 10.

Section 3.2 gives brief overview of systems theory and model building.

3.1 Thermodynamics and Heat Transfer

3.1.1 Thermodynamics

What is “heat”? “By ’heat’ we mean the energy exchange between systems caused by a differ- ence in temperature” [6]. This means that a system never can contain heat. Heat is not a state of a system. The only thing we can determine is heat flow, which is when heat flows from one system to another.

From this it follows that two connected systems will exchange energy with each other until their temperatures are equal.

What will happen when we, in some way, increase the temperature of a system? The internal energy (which in some sense the temperature is a measure of) of that system will increase, but the system will possible perform a work depending on the conditions. That follows from the first law of thermodynamics, which defines the internal energy E of a system to be the difference of the heat transfer Q into the system and the work W done by the system. That is

E₂− E1= Q− W . (3.1)

The work W can for example be the work needed to do an isobaric expansion of a gas. A common example is if we heat a gas conﬁned in a cylinder (see ﬁgure 3.1 on the facing page), that heat will then increase the internal energy of the gas. It will also perform a work against the surrounding gas. Since the gas is free to expand, it will be kept at a constant pressure.

Heat capacity. One of the interesting things to know in this thesis project is how much the temperature of a substance (for example air) increases when we add a certain amount of energy.

Since we know the specific heat capacity c for that substance, and we also know the amount of energy we add to the system, we can calculate the increase in temperature.¹ This is due to that the specific heat capacity c is defined as

E = cm∆T , (3.2)

where E is the added energy, ∆T is the change in the temperature in the substance, m is the mass of the substance and c is the heat capacity.

Isobaric and isochoric heat capacity. One diﬀers between the isobaric heat capacity c_pand the isochoric c_v heat capacity. As we saw earlier, heat (energy) added to a system can cause the system to do a work on something. The isobaric heat capacity of a substance is the heat capacity when the substance’s pressure is free to remain constant. On the other hand, the isochoric heat capacity is the heat capacity when the substance’s volume is kept constant.

1The convention used here is that a positive sign on the heat, is heat that ﬂowsto the system.

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Pambient

Pinternal,1

V1, T 1, m

Pambient

Pinternal,2

V2, T 2, m Heat added

Q

P_ambient=P_internal,1=P_internal,2

Figure 3.1: Sketch over the ﬁrst law of thermodynamics. When heat is added to the system, the gas expands and hence perform a work, aside from that the internal energy increases.

Latent and sensible heat. In some cases we have to differ between latent heat and sensible heat. Sensible heat is heat that causes the temperature of a system to change. However, during phase-transitions the heat flowing from or to a system will not cause the temperature of that system to change. For example - if we boil water, we have to add a certain amount of heat just to boil off the water. This heat are called latent heat and will not increase the temperature of the water/steam.

3.1.2 Heat Transfer

Dynamic systems. The science of heat transfer is a supplement to the thermodynamics. Ther- modynamics deals with systems in equilibrium and we can therefore not use it to model dynamic systems. To model how the heat in a system varies over time, we have to use the results from the heat transfer science. There are three types of heat transfer: conduction, convection and radiation.

There will only be a brief discussion of those four types in this thesis. For a more comprehensive discussion about the heat transfer theory, cf. Holman [4] and Pierre [5].

3.1.3 Conduction

Biot (1804) and Fourier (1822) came up with the equation describing the fundamentals for heat transfer in a material. In the one-dimensional case the heat ﬂow ˙Q through a wall with area A is

Q =˙ −λA∂T

∂x , (3.3)

where λ is the thermal conductivity.

This equation is called Fourier’s law of heat conduction. If we assume that there are no heat sources or sinks in the material, ˙Q has to be equal through the whole wall. If we assume λ to be constant in the wall material and the wall face temperatures to be T₁ and T₂we get

Q = λA˙ T₁− T2

x₁− x2 . (3.4)

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Figure 3.2 shows a sketch over the one-dimensional heat ﬂow.

x1 x2

Q Q

A T₁ T₂

Figure 3.2: A sketch over a one-dimensional through a wall with area A and wall-face temperatures T1 andT2.

If we have a multi-layer wall, as in the case with the walls in a truck cabin, ˙Q must still be the same in all part of the wall. For a three layer wall that is

Q = λ˙ _AAT₁− T2

∆x_A = λ_BAT₂− T3

∆x_B = λ_CAT₃− T4

∆x_C . (3.5)

We can solve (3.5) for ˙Q, and we then get

Q = A˙ T₁− T4

∆x_A/λ_A+ ∆x_B/λ_B+ ∆x_C/λ_C . (3.6) What it essential here is that the heat ﬂow is proportional to the wall face temperatures. It will not introduce a non-linearity. For the purposes in this thesis project, we need to know the thermal conductivity in the layers and the thickness of the layers. It is also possible to estimate the denominator in (3.6) by measurements of the heat ﬂow.

3.1.4 Convection

When heat is transported away due to, for example, air flowing pass a surface we call it convection heat transfer. It is easy to realize that the theory of convection heat transfer is all but trivial. We have factors such as the orientation of the surface, the viscosity of the flowing media, the velocity of the flowing media and many other things which influence the rate of heat transfer.

To keep things simple we will only consider Newton’s law of cooling (1701)

Q = αA(T˙ _w− T∞) , (3.7)

where T_wis the surface temperature of the wall and T_∞is the media’s temperature. The quantity α is called the convection heat-transfer coefficient. This quantity is dependent on a large number of factors. For example, we intuitively know that the convection heat transfer depends on such things as the speed of the flowing media, the type of media (water, air, etc.) and so on. Equation (3.7) is in fact the defining equation for α.The quantity α can be determined analytically, but it is normally determined experimentally.

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3.1.5 Radiation

From Stefan-Boltzmann’s law (1879; 1884) we have the thermal radiation from a blackbody

Φ = σAT⁴, (3.8)

where Φ is the total radiation energy emitted per unit time, and T is the temperature in Kelvin.

This equation is valid for emitted radiation from a blackbody. We note that the net exchange between two blackbodies is proportional to the absolute temperatures to the fourth power,

Φ_net∝ σA(T1⁴− T2⁴) . (3.9)

This expression is still only valid for blackbodies. Other types of surfaces do not radiate as much energy as a blackbody, but they generally follow the T⁴proportionality. However, to analyze the heat exchange for real bodies we also have to consider things such as shape factor, et cetera. It is beyond the scope of this thesis project to do a thorough analysis of this kind of heat transfer.

What is interesting is that the heat transfer is close to proportional to the fourth power of the absolute temperatures.

3.1.6 Composite Heat Transfer

The heat transfer in real systems is a composite of all types of heat transfer. For example - we might have a heat transfer by convection from the air inside a truck cabin to the wall, then a heat conduction through the wall and one for heat transfer by convection from the outside of the wall to the outside air. We also have a heat radiation from the walls in the cabin to the ambient. The composite heat transfer has the following components

Q˙_tot= ˙Q_c,i+ ˙Q_r,i+ ˙Q_con,i+ ˙Q_c,o+ ˙Q_r,o, (3.10) where ˙Q_c,iand ˙Q_c,oare the heat transfer due to convection on the cabin’s inside and outside wall respectively. Q˙_r,i is the net radiation heat ﬂow between the inside cabin air and the cabin wall.

Q˙_r,o is the same for the outside air and the cabin wall. Finally, ˙Q_con,i is the heat transfer by conduction in the cabin’s wall.²

In this thesis project the heat transfer by radiation was neglected. That is ˙Q_r,i= ˙Q_r,o= 0. If we replace the expressions for the different modes of heat flow, except for heat transfer by radiation, in (3.10) with the full expressions for the respective heat flows, as we have derived in the previous sections, we get

Q˙_tot= σ (T_cab− Tout) , (3.11)

where σ is the heat-transfer coeﬃcient. Equation (3.11) is the expression for heat transfer that was commonly used in this thesis project.

3.1.7 Heat Exchangers

To express a heat exchanger in a mathematical way, i.e. to give a physical description of it is not easy. The amount of heat exchanged between the medias depends of many things, such as the material used in the heat exchanger, the area of the heat exchanger and so on. We will not go into details here. See Holman [4] for details.

2The wall of the cabin is build of diﬀerent layers of materials, so this is in some sense a composite heat ﬂow too.

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The heat ﬂow from a heat exchanger can be expressed as

Q = U A∆T˙ _m, (3.12)

where U A are a material parameter and ∆T_mis the logarithmic mean temperature. The logarithmic mean temperature is

∆T_m= (T_h2− Tc2)− (Th1− Tc1)

ln [(T_h2− Tc2) / (T_h1− Tc1)], (3.13) where T_h1and T_h2is the temperature of the cooling water ﬂowing in and out from the heat ex- changer respectively. T_c1and T_c2is the temperature of the air before and after the heat exchanger.

Without going any further, we can see that we might have a problem here. We want to use this expression to calculate the air temperature after the heat exchanger, but we have (at least) two unknowns in this expression - ˙Q and T_c2. A more easy way is to use the speciﬁc power for a heat exchanger.

3.1.8 Speciﬁc Power for a Heat Exchanger

As seen in the previous section, it is hard to do a physical model of a heat exchanger unless we know all of the material parameters. Even if we know the parameters, we still have two unknowns in the expression for the heat ﬂow.

It is better to use empirical data over the heat exchanger. Using the specific power of a heat exchanger is such a way. The specific power is basically the power generated by a heat exchanger normalized with differential of the inlet temperatures of the medias in the heat exchanger (in this case the inlet air temperature and the cooling water temperature). That is

Q_spec = Q

T_cwt,in− T_air,in . (3.14)

Figure 4.3 on page 19 shows an example of the specific power as a function of cooling water flow and air flow.

3.2 Systems Theory and Models

3.2.1 Dynamic Models

By models we mean something that describes the relationship between diﬀerent measured signals.

We diﬀer between input signals and output signals. The output signals are partially determined by the input signals. The output signals can also be aﬀected by signals we cannot measure, disturbance signals. Moreover, those signals are all functions of time. Figure 3.3 shows the relationship between the signals.

System

u y

e

Figure 3.3: A system with input u, output y and disturbance e.

The modelling problem is to describe how these three signals relate to each other. In the case of the temperature in a truck cabin, the input signals can be air temperature, wind speed and many

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other things. It might even be so that we have to consider signals that could be measured (such as the sun intensity) as disturbances, since this signal was not measured in the experiment data we have to our disposal. If you want a more comprehensive discussion about diﬀerent modelling techniques, please read Ljung et al [7].

The system itself can be described by diﬀerent methods. The perhaps most common way is to describe it on state-space form. Other ways to describe it is via transfer functions or on other forms. Read more about this further on.

3.2.2 Transfer Functions

The output y(t) of a causal linear time-invariant system can always be written as y(t) =

_∞

0

= g(τ )u(t− τ)dτ , (3.15)

where y(t)∈ R^kand the inputs u(t)∈ R^m. g(τ ) is a weighting function. We can Laplace-transform this expression into

G(s) =L(g) =

_∞

0

e^−stg(t)dt U (s) =L(u)

Y (s) =L(y)

. (3.16)

We can rewrite (3.15) to

Y (s) = G(s)U (s) , (3.17)

where G(s) is called the transfer function. It is in the scalar case common that G(s) is a rational function of s. That is

G(s) = B(s)

A(s) , (3.18)

where A(s) and B(s) are polynomials. In the multi-variable case, the corresponding expression to (3.18) is

G(s) = A⁻¹(s)B(s) . (3.19)

If G(s) is a rational function with A(s) and B(s) as polynomials, then the solutions to A(s) = 0 are the system poles and the solutions to B(s) = 0 are the system zeros. Read more about poles and zeros further on.

3.2.3 State Space Functions

It is always possible, via substitution, to rewrite a linear time-invariant diﬀerential equation like dⁿy

dtⁿ(t) + a_n−1dⁿ⁻¹y

dtⁿ⁻¹(t) +· · · + a0y(t) = dⁿu

dt^m(t) + b_m−1d^m−1u

dt^m−1(t) +· · · + b0u(t) . (3.20) to a system of first-order differential equations. It might also be that we have a set of first order differential equations that are dependent on each other. In either way we have a system of first order differential equations. A common way to describe this is with the state space form. That is

˙

x(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t), (3.21)

where x ∈ Rⁿ is the state vector, u ∈ R^m is the input signals and y ∈ R^k is the output signals.

Consequently A∈ R^n×n, B∈ R^n×m, C∈ R^k×n and D∈ R^k×m.

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We can take the Laplace-transform of the matrix expression (3.21), which is sX(s) = AX(s) + BU (s)

Y (s) = CX(s) + DU (s). (3.22)

Solving for X(s) in the ﬁrst expression and substituting X(s) in the second expression yields Y (s) =

C (sI− A)⁻¹B + D

G(s)

U (s) , (3.23)

where I is a n× n identity matrix. The function G(s) is again the transfer function.

3.2.4 Poles and Zeros

The poles of the system. The poles of a system are deﬁned to be the values s when the transfer function G(s) gets rank-deﬁcient. That is the same values as the solution to

det (λI− A) = 0 , (3.24)

if the system is on state space form.

The zeros of the system. For a square-matrix G(s) the zeros are deﬁned as the poles to G⁻¹(s). For a non-square G(s), the zero-polynomial are deﬁned as the greatest common divider to the nominators of the maximal sub-determinants of G(s), normalized so they have the pole- polynomial as denominators. The zeros of the system is the zeros of the zero-polynomial.

3.2.5 Physical (White-Box) Modelling

The most intuitive approach to model a system is perhaps to do a physical model over it. In that case we assume that we know all the describing equations for that system and all the parameters in the system. This is also sometimes called white-box modelling.

This way to model requires that we can describe the system in a mathematical way. It also requires that we know all the parameters in the equation or equations. It is also a fact that most of the physical equations are approximations. For example, the familiar equation

Q = mc_p(T₂− T1)

is an approximation. The isobaric heat capacity c_p is usually temperature dependent. One other example of diﬃculties is if we want use the expression above to calculate the increase of temperature of air when we add heat to it. “Ordinary” air consists of a certain amount of water. This means that if we add heat to air some of that heat will increase the temperature of the water in that air.

Water has a higher heat capacity than air, so the increase in temperature would not be as large as it would if the air was dry.

The bottom line is that it can be hard to derive physical equations for a complicated system.

Physical models are easy to overview for simple system, but could get hard to work for larger systems. It might even be that we do not know the mechanism in a system and we are then unable to derive the equations for that system.

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3.2.6 Black-Box Modelling

Black box modelling is a set of methods to model a system without the need of any physical insight. What one does is to find the parameters in a model that gives the best fit between the model and the measured data. Usually one has a set of model structures that are known to work for many different types of systems. The parameters are then estimated by using a least squares method.

The tool used in this thesis project for the black-box modelling was System Identification Toolbox (SITB) [10]. SITB gives the possibility to estimate models in different structures. By structures we mean that if the estimated model could be a state-space model, a ARX-model or something else. Different structures have different advantages. Some structures can be too advanced for certain purposes, and others too simple.

ARX-structures and similar. Very loosely speaking we can say that models such as the output y(t) from an ARX-model is the sum of the weighted inputs u(t) from a certain amounts of time steps. This means that the structure of these models are a bit diﬀerent from the structure of a state-space model. An ARX-model has the structure (in discrete time)

A(q)y(t) = B(q)u(t− nk) + e(t) , (3.25) where y(t) is the output, u(t) is the input and

A(q) = 1 + a₁q⁻¹+ . . . + a_naq^−na

B(q) = b₁+ b₂q⁻¹+ . . . + b_nbq^−nb. (3.26) The operator qⁱ is the lag-operator. That is qⁱu(t_k) = u(t_k+i). The parameters na, nb and nk deﬁnes the order of the model. The parameters a_i and b_i are estimated using the least squares method.

We can see that with this kind of structure we perhaps do not have the same possibility to do a physical interpretation as with models in the state-space structure.

State-space structures For the state-space model we want to estimate the parameter vector θ in

˙

x(t) = A(θ)x(t) + B(θ)u(t)

y(t) = C(θ)x(t) + D(θ)u(t). (3.27)

This structure is in continuous time as oppose to the ARX-structure above.

The advantage with the state-space structure, in this thesis project anyway, is that the estimated model is easier to implement in Simulink than an ARX-model.

Examples of other structures. Other structures can use a moving average of the disturbance, such as the ARMAX-model

A(q)y(t) = B(q)u(t) + C(q)e(t) . (3.28)

3.2.7 Grey-Box Modelling

As the name implies grey-box modelling is a mix of black-box and white-box modelling techniques.

If we consider white-box modelling as a modelling technique based on pure theoretical knowledge of the system and black-box modelling as a modelling technique based on pure empirical knowledge of the system, then grey-box modelling is a technique where one combines theoretical and empirical

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knowledge of a system. In that sense the models developed in this thesis project were either black-box or grey-box models. Grey-box models because the describing diﬀerential equations for the systems were derived (theoretical), but the parameters in the equations where estimated from measurements (empirical).

3.2.8 Continuous and Discrete Time Systems

All the models in this thesis project were built as continuous time systems. The measurement data used in the system identification with System Identification Toolbox [10] is of course discrete time data. Therefore the models generated by System Identification Toolbox were also in discrete time. Those models were converted to continuous time models before there were validated or implemented into Simulink [12].

3.2.9 Disturbances and Disturbance Models

As mentioned in section 3.2.1 on page 10 systems normally have some kind of disturbances.

Such disturbances can be due to stochastic processes within the system or measurement errors.

Depending on the purpose of the model one could deal with disturbances in diﬀerent ways.

One way of “dealing” with the noise is simply to ignore it. One other way is to ﬁlter the measurement data before it is used in system identiﬁcation or validation. A third way is to model the noise, that is to do a disturbance model. In that case the characteristics of the disturbance is evaluated and included in to the model.

In this thesis project all the measurement data are filtered for high frequency noise before it is used in any system identification. However, in the validation of the model the unfiltered measurements are used as inputs to the model and in the calculation of model errors.

3.3 Model Validation

One major task in model building is the validation of the resulting model. We need to determine how “good” the model is. The way to do this is to feed the model with measurements of input signals to the system from experiments and compare the output from the model with the output from the actual system. It is of course a good thing if we have measurements derived under diﬀerent conditions to use in the validation.

The result from a validation can also give us a hint if there are other factors (inputs) that where not included in the measurements, but still aﬀects the output. For example, a model error which increases with time can be an indication of this. A model error that varies with one (or more) of the inputs might be an indication of a non-linearity in the system.

3.3.1 The RMS-value

A measure of the model error used in this thesis project is the Root-Mean-Square (RMS) value.

The RMS is deﬁned as

r =

1 n

n i=1

(˜x_i− xi)² _1/2

, (3.29)

where n is the number of measurements, ˜x_i is the simulated temperatures and x_i is the measured temperatures.

We can interpret this value as the mean deviation from the “real” values.

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3.3.2 Residual Analysis

Suppose that we have measured data y(t) from a system and a simulation ˆy(t) of that system.

Then the model error, or the residual,

(t) = y(t)− ˆy(t) (3.30)

should ideally be independent of the input signal u(t) to the system. If it is not there are some eﬀects from the input signal that the model are unable to catch. One can form

Rˆ_u(τ ) = 1 N

N t=1

 (t + τ ) u(t), |τ| ≤ M , (3.31)

and test if these values are close enough to zero. If{(t)} and {u(t)} is independent, then (3.31) should, for large N , be approximately normal-distributed with mean value zero.

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4 Models

4.1 Introduction

The goal with this thesis project is to model the cabin air temperature as it would be measured by the ACC’s temperature sensor. Both a black-box and a white-box model for the cabin air temperature were developed and tested. These two diﬀerent approaches have diﬀerent advantages and disadvantages, which are discussed further on.

To be able to model the cabin air temperature the heat flow from and to the cabin had to be modelled. Common for both the black-box and the white-box approach was that the HVAC- system¹and the ATA was modelled separately from the cabin air temperature model. This means that models for the HVAC-system and the ATA, both with air temperature and air flow as outputs, were developed. The outputs from those models served as inputs to the cabin air temperature model. Read more about the HVAC- and ATA-models in sections 4.2 to 4.3. The general structure of the complete model can be viewed in figure 4.1.

HVAC−system

AC−model Tout,mHVAC

Heat exchanger

model

TAC,mHVAC Tmix,mHVAC

Tcwt,c, m

cwt WTA−model

Tcwt,i,mcwt Tcwt,o,mcwt

Cabin

model

Tcab ATA−model

Tcab,mATA TATA,mATA

Figure 4.1: The general structure of the model

Black-box modelling. In the black-box approach, the cabin air temperature was modelled with help of System Identiﬁcation Toolbox [10]. Two sets of data, from the “0” and “-20 ”

experiments, were used for this. See appendix D for details about these two experiments. Two independent models were estimated with these data sets as a basis. That is, one model was estimated from the “0”-data and the other from the “.20 ”-data. Even though the models

1Since the WTA heat the cooling water to the heat exchanger in the HVAC-system, it is in some sense a part of the HVAC-system. The WTA is therefore included into the HVAC-model here.

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were independently estimated, there is a clear relationship between the poles in the two models.

Read more about the black-box modelling in section 4.4 on page 21.

The advantage with the black-box approach is that we do not need to know anything about the physics behind the heat transfer from and to the cabin. The disadvantages is that a black-box model is more diﬃcult to tune compared to a white-box model.

White-box modelling. The white-box approach requires more knowledge of the heat transfer physics than the black-box approach. It requires that we derive equations for the heat transfer from and to the cabin. These derived equations contains diﬀerent parameters, such as heat conduction coeﬃcients et cetera, that has to be either estimated or known.

It turned out during the project that the white-box model originally developed did not perform well. The problem, as it turned out, was a non-symmetry in the cabin air temperature dynamics.

It manifests itself by the results from experiments, where the cabin air temperature seems to have diﬀerent time constants depending on if the cabin air temperature is increased or decreased.

This non-symmetry is probably caused by the spatial air temperature distribution in the cabin.

Depending on the diﬀerence between the temperature of the air that enters the cabin and the cabin air, they will mix at diﬀerent rates. See section 4.5 on page 23 for further details about the white-box modelling.

4.1.1 Disturbances

Disturbances were never considered in this thesis project. When system identification were used to estimate black-box model, the data used in the identification was low-pass filtered to reduce noise.

4.2 The HVAC Model

As discussed in chapter 2, the HVAC-system basically consists of a heat exchanger and (possibly) an evaporator. The air ﬂows ﬁrst through the evaporator and then through the heat exchanger where it eventually gets heated.

In this thesis project the WTA was considered to be a part of the HVAC-system. The reasons for this ere that the WTA heats the engine’s cooling water as described in section 2.1 on page 3.

Therefore its in a modelling point of view convenient to include it in the HVAC-model.

In order to keep the modelling work easy, and to keep the diﬀerent sub-models as simple as possible, the HVAC-model was divided into three sub-models. The WTA, the heat exchanger and the AC-evaporator were modelled independent of each other.

4.2.1 Modelling the AC

In the thesis work by Eriksson [3] the temperature of the evaporator was modelled in Simulink.

Since that model only models the evaporator temperature a model of the air temperature after the evaporator, depending on the evaporator temperature and the air ﬂow through the evaporator, was developed.

A white-box as well as a black-box approach was tested to model the temperature after the evaporator. The most successful approach was the black-box approach. The black-box model was developed in System Identification Toolbox (SITB) [10]. A schematic figure of this model is viewed in figure 4.2 on the next page.

A set of measurement data with the diﬀerence between the evaporator temperature and the am- bient temperature, T_evap− Tout, used as input and the air temperature after the AC, T_AC, used as

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AC Tout

Tevap

TAC

Figure 4.2: The AC-model with the evaporator temperature Tevapand the outside air temperature T^outas inputs.

output was used in SITB. After the measurement data was low-pass ﬁltered, a best-ﬁt state space model was estimated. The estimated discrete time state space model was of the third order. It was transformed to continuous time before it was implemented into Simulink [12]. This gives us the parameters A, B, C and D in²

˙

x = Ax + B∆T

T_AC = Cx + D∆T ,

where T_AC is the air temperature after the evaporator and ∆T = T_evap− Tout is the temperature diﬀerence between the evaporator and the ambient air.

The reason for that the temperature diﬀerence was used as input and not the two temperatures T_evap and T_out separately as two inputs where that the ambient temperature T_out was almost constant during each of the measurement series. Using SITB with one constant, or almost constant, input can result in estimated models that has a very high gain for that input. In the case with the model developed here, that model proved to perform well with just ∆T as input.

The air flow is not considered explicitly in this model. However, the model for the evaporator temperature considers the air flow. This means that we do not have to care about the air flow in the model for the evaporator to mixed air temperature.

4.2.2 Modelling the Heat Exchanger

As discussed in the theory for heat exchangers, section 3.1.7 on page 9, the power generated by a heat exchanger depends on many factors. To do a physical model of the heat exchangers requires that we know all the material parameters as well as the fouling factor et cetera. Even if we know all these parameters, we still have to know the cooling water temperature after the heat exchanger.

One could use a black-box approach to model the heat exchanger. Since the heat generated by a heat exchanger is in its nature very un-linear, a un-linear black-box model would be required.

Such an approach was however never tested in this thesis project.

It is much more convenient to use empirical data over the heat exchanger, and that is the specific power who was discussed in section 3.1.8 on page 10. The heat exchanger was modelled by using a look up-table for the specific power for that heat exchanger. The data was from a heat exchanger that is commonly used in a Scania truck. Figure 4.3 on the facing page shows the specific power for the heat exchanger used in this thesis project.

The specific power Q_spec from a heat exchanger is a function of the cooling water flow ˙m_cwt and the air mass flow through the HVAC-system, ˙m_HVAC. The relationship between the cooling water temperature in to the heat exchanger T_cwt, the air temperature of the air before the heat exchanger T_AC, the specific power Q_spec and the power generated Q is

Q_spec = Q

T_cwt− TAC . (4.1)

2Since it is a three-dimensional system with one input and one output,A is 3 × 3, B is 3 × 1, C is 1 × 3 and D is a scalar. The actual values can be viewed in appendix C.

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0.05

0.1

0.15

0.2

0.1 0.2 0.3 0.4 20 40 60 80 100 120 140 160 180

Air flow [kg/s]

Cooling water flow [kg/s]

Qspec [W/K]

Figure 4.3: The speciﬁc power Q^specfor a heat exchanger.

This equation solved for Q and substituted into the equation for the speciﬁc heat capacity (Q = mc_p∆T ) yields the expression

T_air,out= Q_spec(T_cwt− TAC)

˙

m_HVACc_p,air + T_AC. (4.2)

From this expression we then have the air temperature after the heat exchanger, which is the output from this sub-model. Figure 4.4 shows a schematic ﬁgure of the heat exchanger model.

Heat exchanger Tcwt

TAC

m_cwt mHVAC

Tmix

Figure 4.4: The heat exchanger model with the temperature TAC, the cooling water temperature Tcwt, the cooling water ﬂow ˙mcwt and the air ﬂow ˙mHVAC as inputs.

Problems by the nonlinearity in the specific power. The heat exchanger model is expected to have a relative poor performance at cooling water flows below 0.1 kg/s. The reason for this is clear from figure 4.3. The surface for the specific power is very steep for cooling water flows below 0.1 kg/s. This means that a small error in the cooling water flow will have a great impact on the specific power. Or, on the contrary, a small error in the table over the specific power will generate a large model error for cooling water flows below 0.1 kg/s.

Implementation of the heat exchanger model in Simulink. See ﬁgure B.5 in appendix B.

The model is basically implemented as described in equation (4.2), with the value of Q_spec from a look-up table. A ﬁrst order linear ﬁlter is also added to smooth the output.

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4.2.3 Modelling the WTA

As discussed in section 2.1 on page 3, the WTA heats the cooling water in the truck by using a diesel burner. The WTA was modelled by using a simple expression for the heat added to the cooling water. That is

Q˙_cwt= c_p,cwtm˙_cwt(T_cwt,o− T_cwt,i) , (4.3) where ˙Q_cwt is the heat added to the cooling water by the WTA, T_cwt,i is the cooling water temperature in to the WTA and T_cwt,ois the cooling water temperature out from the WTA.

The heat added to the cooling water, ˙Q_cwt, is given by the manufacturer of the WTA. Solving (4.3) for T_cwt,o yields

T_cwt,o=

Q˙_cwt

c_p,cwtm˙_cwt + T_cwt,i , (4.4)

which is the output from the WTA-model. Figure 4.5 shows a basic overview over this model.

WTA Tcwt,i

m_cwt PWTA

T_cwt,o

Figure 4.5: The WTA-model with the cooling water temperature Tcwt,iin to the WTA, the cooling water ﬂow ˙m^cwt and the applied powerP^WTA as inputs. The output from the model is the cooling water temperature after the WTA,Tcwt,o.

The reasons for this “simple” approach was a lack of appropriate measurement data. With an appropriate set of measurement data it might have been possible to do a black-box model of the WTA instead of this white-box approach. But it is nothing that says that a black-box model would have a better performance than a white-box model.

Implementation of the WTA-model in Simulink. A low-pass filter was added in the Simulink-model for the WTA after the calculation of T_cwt,o. This is to include some dynam- ics, since the temperature T_cwt,o certainly not will change with a time-constant zero when the WTA power is altered. The filter constant in this filter is used as a tunable parameter and it can be estimated by using measurement data from WTA tests. A low-pass filter has the Laplace- transform

Y (s) = 1

1 + τ sU (s) , (4.5)

where Y (s) is the filtered value, U (s) is the input to the filter and τ is the filter constant.

Since it was not possible to compare the WTA-model with measurement data³, the value of τ was set to 60 (seconds). This was considered to be a reasonable value, but further tests needs do be done to verify (or reject) this value.

4.3 The ATA Model

There was a lack of relevant measurement data from ATA tests. Therefore a white-box model was developed for the ATA. As with the WTA, the heating power generated by the ATA is known from the manufacturer. The heat equation

Q˙_ATA = ˙m_ATAc_p,air(T_cab− TATA) , (4.6) was used to calculate the temperature after the ATA. See ﬁgure 4.6 on the facing page for a overview over the ATA model.

3The lack of measurement data for the WTA will be discussed further on in section 5.2.3 on page 32.

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ATA Tcab

mATA PATA

TATA

Figure 4.6: The ATA-model with the cabin temperature Tcab, the air ﬂow ˙mATAthrough the ATA and the applied powerPATA as inputs. The output from the model is the air temperature after the ATA,TATA.

Implementation of the ATA-model in Simulink. With the save reasons as for the WTA- model, discussed in section 4.2.3 on the preceding page, a low pass ﬁlter was added to the ATA- model. The ﬁlter parameter was set to 60 (seconds).

Transport delays. There is a certain amount of transport delay in this system. This is due to that there is an air pipe from the cabin to the ATA, and then a pipe from the ATA and out to the cabin. The order of this transport delay is small relative to the other time constants in the system. It is implemented anyway, since it is easy to model a transport delay in Simulink. The transport delay is calculated based of the dimensions of the pipes from and to the ATA.

4.4 Black-Box Modelling of the Cabin Temperature

As stated in the introduction to this chapter, a black-box model of the cabin air temperature was one of the two type of models of the cabin air temperature developed and tested.

System Identiﬁcation Toolbox (SITB) [10] was used to do a Black-Box model of the cabin air temperature. The data set used for this was the “0” and “-20 ” data described in appendix D. From these data sets the measured ambient temperature T_out and the measured mixed air temperature T_mix were used as input signals and the measured seat temperature was used as output signal.

The air flow in to the cabin will also affect the temperature, and hence should be used as an input signal. However, the air flow was constant during the measurement. The best thing would be if the air flow had been changed during the experiment. In that case it would have been possible to establish a relationship between the air flow and the output signal, the cabin air temperature, as well. The approach used in this thesis project was to normalize the mixed air temperature T_mix with the mixed air mass flow ˙m_HVAC. That is

T_mix,norm− 273 = k m˙_HVAC

˙

m_HVAC,nom(T_mix+ 273) , (4.7)

where T_mix,normis the normalized temperature used as input to the black-box model and ˙m_HVAC,nom is the nominal mixed air mass flow. The nominal mixed air mass flow is the air flow in the exper- iment used in system identification. The parameter k is a tuning parameter and its nominal value is 1. Also note that absolute temperate was used here, hence the figure 273.

After the measured data was low-pass filtered for periods down to 0.1 rad/s⁴, a best-fit state-space model was estimated by SITB for each of the two measurement series. These models were of a fifth order for both of the measurement series. The ARX-models generated by SITB were also tested, but were found to have the same, or poorer, performance than the state-space models.

The state-space models generated by SITB were in discrete time, but were converted to discrete time before the implementation into Simulink.

4The major part of the noise in the measurement was found to have periods shorter than 0.1 rad/s. This corresponds to a wavelength of 6 seconds, and it would be of no interest to include such short periods in this model.

A Model of the Air Temperature in a Truck Cabin