Performance prediction of HVAC systems

(1)

Preface

This report is represents the final thesis for the completion of my study Maritime Technology at the Delft University of Technology. I specialized in Marine Engineering. The final year of my study did consist out of two projects of which the first one was in Hoofddorp at

“Bluewater Energy Services” and the second one at “Imtech Marine & Offshore” in Rotterdam. The latter is described in this report.

The report describes several issues that have to do with HVAC (Heating, Ventilation & Air Conditioning) installations on board of ships, such as the HVAC components like heat exchangers, heat transmission in spaces on board and control systems.

I would like to thank the people who made it possible for me to make this thesis. The people from “Imtech Marine & Offshore”, in particular Martin van Holsteijn, they gave me more insight in the HVAC systems. From the University I would like to thank Professor Klein Woud for his advices and Mister Grimmelius for his expertise in Matlab Simulink.

Furthermore, I would like to thank all the people who have contributed to my work and that have supported me in all possible manners.

Delft, August 2005 Bart Bouthoorn

(2)

Nomenclature

Roman variables

A A c C d e E E area constant specific heat constant diameter

energy per unit mass emissive power energy 2 2 [ ] [ ] [ /( )] [ ] [ ] [ / ] [ / ] [ ] m J kg K m J kg W m J − ⋅ − F g h H k L m m shape factor gravitational constant specific enthalpy enthalpy

heat transfer coefficient length mass mass flow 2 2 [ ] [ / ] [ / ] [ ] [ /( )] [ ] [ ] [ / ] m s J kg J W m K m kg kg s − ⋅ N O p q Q r R t number of elements controller output pressure heat flux heat flow evaporation/melting heat thermal resistance thickness 2 [ ] [ ] [ ] [ / ] [ ] [ /( )] [ / ] [ ] Pa W m W J kg K W K m − − ⋅ T u U v V W x z temperature

specific internal energy internal energy velocity space volume work coordinate in x-direction height 3 [ ] [ / ] [ ] [ / ] [ ] [ ] [ ] [ ] K J kg J m s m W m m

(6)

Greek variables

α ε φ λ ν θ ρ σ τ

heat transfer coefficient error efficiency thermal conductivity specific volume temperature density Stefan-Boltzmann constant time constant 2 3 3 2 4 [ /( )] [ ] [ ] [ /( )] [ / ] [ ] [ / ] [ /( )] [ ] W m K W m K m kg C kg m W m K s ⋅ − − ⋅ ° ⋅

(7)

Subscripts

a a,in a,out b c conv cond da air entering air leaving air blackbody constant convective conductive dry air e ha i in ins m at end time humid air integral incoming inside mean value out outs p rad sv sur outgoing outside proportional radiative saturated vapour outside surface w w,m w,out wall, out wv wl 0 water

water mean temperature leaving water

external wall water vapour water liquid at time t=0 s

(8)

List of figures

Figure 1-1: HVAC component ... 12

Figure 1-2: Storage and resistance elements with their causalities ... 13

Figure 1-3: Volume-Resistance Network with causalities ... 13

Figure 1-4: Control volume... 18

Figure 1-5: Duct with constant diameter... 20

Figure 1-6: Physical model ... 23

Figure 1-7: Matlab Simulink model ... 23

Figure 1-8: Central Air-handling Unit ... 24

Figure 1-9: Central Air-handling Unit in Matlab Simulink ... 24

Figure 1-10: Plug flow ... 25

Figure 1-11: Time delay... 27

Figure 1-12: Simulation results for a different number of elements ... 30

Figure 1-13: Simulation results for a different number of elements (zoom in) ... 30

Figure 2-1: Heat conduction... 31

Figure 2-2: Convective heat transfer ... 33

Figure 2-3: Convex black object in a black isothermal enclosure ... 34

Figure 2-4: Radiation enery exchange between tow finite surfaces... 36

Figure 2-5: Time constant ... 37

Figure 2-6: Time delay and time constant... 38

Figure 2-7: Elementary control system ... 39

Figure 3-1: Space ... 42

Figure 3-2: External wall... 43

Figure 3-3: Matlab Simulink model of the external wall ... 55

Figure 3-4: Matlab Simulink model for internal walls, floor and ceiling ... 56

Figure 3-5: Matlab Simulink model of the entire space... 57

Figure 3-6: No inside walls, no furniture ... 58

Figure 3-7: Including inside walls, no furniture... 59

Figure 3-8: Including furniture, no side walls... 60

Figure 3-9: Including sidewalls and furniture ... 61

Figure 3-10: Step change every 12 hours... 62

Figure 3-11: Space without walls... 65

Figure 3-12: Space with walls ... 67

Figure 3-13: Space with walls (zoomed in) ... 68

Figure 3-14: Heat flow from walls and furniture ... 69

Figure 3-15: Heat flows... 70

Figure 4-1: Tube and fin of the simulated heat exchanger... 73

Figure 4-2: Element, tube and tube row ... 82

Figure 4-3: Matlab Simulink model of one element of a tube ... 83

Figure 4-4: Matlab Simulink model of one tube ... 84

Figure 4-5: Matlab Simulink model of one tube row... 85

Figure 4-6: Visualisation of the simulated heat exchange ... 86

Figure 4-7: Step change... 88

Figure 5-1: Control of the after heater ... 91

Figure 5-2: Control in the CAU ... 92

Figure 5-3: Space in Matlab Simulink ... 93

(9)

Figure 5-7: PI-controller in Matlab Simulink ... 95

Figure 5-8: Design condition... 96

Figure 5-9: Disturbance condition... 97

Figure 5-11: Data design condition ... 99

Figure 5-12: Disturbance condition I ... 100

Figure 5-13: Disturbance condition I ... 101

Figure 5-14: Disturbance condition II ... 102

Figure 5-15: Disturbance condition II ... 102

Figure 6-1: System model ... 105

Figure 6-2: Matlab Simulink model of the system... 109

Figure 6-4: Results of the design condition ... 112

Figure 6-5: Results of the design condition ... 112

Figure 6-6: Mass flow changes ... 113

Figure 6-7: Results of disturbance condition I ... 114

Figure 6-8: Results of disturbance condition I ... 114

Figure 6-10: Results of disturbance condition II... 116

Figure 6-11: Results of disturbance condition II... 117

Figure 6-13: Results of disturbance condition III ... 119

Figure 6-14: Results of disturbance condition III ... 120

Figure 6-16: Results of disturbance IV ... 122

Figure 6-17: Results of disturbance IV ... 122

Figure 6-19: Results of disturbance V... 124

Figure 6-20: Results of disturbance V... 124

Figure 6-22: Results of design condition (winter) ... 126

Figure 6-23: Results of design condition (winter) ... 126

Figure 6-25: Results of disturbance (winter)... 128

Figure 6-26: Results of disturbance (winter)... 128

Figure 0-1: Opposite rectangles ... 136

Figure 0-2: Adjacent rectangles ... 137

Figure 0-1: Sideview ... 140

(10)

List of tables

Table 1-1: Simulation time... 29

Table 3-1: Material properties... 44

Table 3-2: Internal walls ... 44

Table 3-3: Floor/ceiling... 44

Table 3-4: Furniture material ... 45

Table 3-5: Time constants of the layers in the external wall ... 61

Table 3-6: Time constants ... 64

Table 3-7: Time delays... 64

Table 4-1: Heat exchanger dimensions ... 86

Table 4-2: Air temperature comparison ... 87

Table 4-3: Water temperature comparison... 87

Table 5-1: Mass flows ... 100

Table 5-2: Mass flows ... 101

Table 6-1: Channel properties ... 106

Table 6-2: Resistances... 107

Table 6-3: Heat loads ... 107

Table 6-4: Heat loads ... 107

Table 6-5: Space supply air flows ... 108

Table 6-6: Channel air flows ... 108

Table 6-7: Mass flow changes... 116

Table 6-8: Mass flow changes... 118

(11)

Introduction

Imtech Marine & Offshore designs and installs heating ventilation and air-conditioning (HVAC) systems. To improve their capabilities to predict the performance in off-design conditions, Imtech Marine & Offshore requested the assistance of the Delft University of Technology. In September 2003 Roely Ruissen started to develop a simulation tool on which he graduated in 2004. Because the work was not yet finished it had to be continued. The new commission is in “Appendix I: Commission”. This report and accompanying simulation tool are the results.

This report deals with the extension of created modules. After further development the modules will be used for a test of a part of the RNLN ship Launching Platform Dock two (LPD2). Besides the modules that will be extended, a control system will be created and included in the test to be able to compare the results with the control system on board LPD2. The first chapter describes the model developed by Ruissen. At the end of the chapter an improvement will be described. The second chapter gives theoretical information that is used in the third, fourth and fifth chapter. The third chapter describes the development of a model to predict the air condition in a space. The influence of heat accumulation in walls on the space temperature will be investigated. Chapter four gives a model for a heat exchanger, which can be used to control the temperature of the air. Then in chapter five the control of the LPD2 will be described and an optimisation will be investigated. In chapter six, a part of the LPD2 will be simulated using the developed simulation tool. Finally, conclusions and recommendations are given in chapter seven.

(12)

1 Description of HVAC model

This chapter gives a summary of the HVAC model that is developed by R.Ruissen and

described in [Ruissen, 2004]. In the first section the structure of the model and its relations are given. In the second section the model in Matlab Simulink is described and finally in the third section an extension is described. Next to this model a heat exchanger model for a heater and a cooler, an accumulation model of walls for the spaces and a control system for the entire system will be developed. These developments will be described in the following chapters.

1.1 Theoretical model

1.1.1 Model structure

The systems in the entire model are divided into smaller parts. Each part describes a process in the system. The parts are connected by mass and energy flows. To be able to connect the different modules (parts), each module must have the same number of inputs and outputs, which is visualized in Figure 1-1.

HVAC COMPONENT , , , , , , , , da da da da wv wv wv wv p m p m θ ρ θ ρ , , , , , , , , da da da da wv wv wv wv p m p m θ ρ θ ρ

Figure 1-1: HVAC component

with, da wv da wv da wv da wv m m p p ρ ρ θ θ

density of dry air density of water vapour temperature of dry air temperature of water vapour mass flow of dry air

mass flow of water vapour pressure of dry air

pressure of water vapour 3 3 [kg/m ] [kg/m ] [ C] [ C] [kg/s] [kg/s] [Pa] [Pa] ° °

To describe the airflow through a component, each component is either a resistance or a storage element. For the elements the direction of the information flows, or causalities, are fixed. Figure 1-2 shows the resistance and storage elements with their causalities.

(13)

STORAGE ELEMENT FLOW FLOW TEMPERATURE PRESSURE RESISTANCE ELEMENT PRESSURE PRESSURE FLOW

Figure 1-2: Storage and resistance elements with their causalities

Building the entire model using volume (storage) and resistance elements fixes the configuration as given in Figure 1-3.

Resistance i-j Resistance j-k Volume j Volume i i j m ₋ j j p T i i p T j k m ₋ Volume k j i m ₋ k j m ₋ k k p T

(14)

1.1.2 Psychrometrics

Basic terms

The model is developed to be able to predict the properties of air. The relations that give the properties of air incorporated in the model are described in [Ruissen, 2004] and can also be found in [ASHRAE, 1996].

The mass of humid air is defined as the sum of the dry air and the water damp, in formula: [ ] ha da wv m =m +m kg (1.1) with, ha da wv m m m

mass of humid air mass of dry air mass of water vapour

[ ] [ ] [ ] kg kg kg

Using the ideal gas law it can be obtained that the pressure of dry air and the pressure of water vapour can be described as:

[ ] [ ] da da da wv wv wd m R T p Pa V m R T p Pa V ⋅ = ⋅ = (1.2) with, da wv da wv R R T T V

gas constant for dry air gas constant for water vapour temperature of dry air

temperature of water vapour

control volume 3 [J/(kg K] [J/(kg K] [K] [K] [m ] ⋅ ⋅

The pressure of the water vapour in saturated condition for respectively from -100 to 0 º Celsius and from 0 to 200º Celsius can be determined with the equation:

2 3 4 1 2 3 4 5 6 7 lnp_sv C C C T C T C T C T C lnT [ ] T = + + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ − (1.3) 2 3 8 9 10 11 12 13 lnp_sv C C C T C T C T C lnT [ ] T = + + ⋅ + ⋅ + ⋅ + ⋅ − (1.4) with, 1 13 sv p C to C

saturated vapour pressure

constants from [ASHRAE, 2001]

[ ] [ ]

Pa −

(15)

The humidity ratio is described as the quotient of the mass of water vapour and the mass of dry air: [ / ] wv da m x kg kg m = (1.5)

The amount of moist in air is often expressed by the relative humidity; this is the quotient of the pressure of the water vapour and the pressure of the water vapour in saturated condition:

[ ] wv sv p p ϕ = − (1.6)

The relative humidity is often expressed in a percentage: 100 wv [%]

sv

p p

ϕ = ⋅ (1.7)

Enthalpy and internal energy of humid air

The enthalpy of air is defined as the sum of the internal energy and the product of the pressure and the volume:

[ ] H =U+ ⋅p V J (1.8) with, H U enthalpy internal energy [J] [J] The internal energy can now be formulated as:

[ ]

U =H− ⋅p V J (1.9)

For ideal gasses this is: [ ]

U =H−m R T J⋅ ⋅ (1.10)

For humid air two components can be extinguished, namely dry air and water vapour. The equation can also be translated to the specific enthalpy and specific internal energy:

[ ] [ ] ha da wv da da wv wv ha da wv da da wv wv H H H m h m h J U U U m u m u J = + = ⋅ + ⋅ = + = ⋅ + ⋅ (1.11) with,

(16)

da wv da wv da ha wv da ha wv h h u u H H H U U U

specific enthalpy of dry air specific enthalpy of water vapour specific internal energy of dry air specific internal energy of water vapour enthalpy of dry air

enthalpy of humid air enthalpy of water vapour internal energy of dry air internal energy of humid air internal energy of water vapour

[J/kg] [J/kg] [J/kg] [J/kg] [J] [J] [J] [J] [J] [J] This can be rewritten to:

[ / ] [ / ] ha da wd ha da wv da wv da da da ha da wd ha da wv da wv da da da H m m h h h h x h J kg m m m U m m u u u u x u J kg m m m = = ⋅ + ⋅ = + ⋅ = = ⋅ + ⋅ = + ⋅ (1.12) with, ha ha h u

specific enthalpy of humid air internal energy of humid air

[J/kg] [J/kg] For the specific enthalpy of humid air we assume that its value is 0 at a temperature of

0 C° and the water present in liquid form also with a temperature of 0 C° . In air handling systems, humid air can occur in three compositions:

• Non-saturated humid air x<x_s

• Saturated humid air x≥x_s at a temperature θ ≥0, 01 C° , excess water in liquid form • Saturated humid air x≥x_s at a temperature θ ≤0, 01 C° , excess water in ice form with,

s

(17)

For ideal gasses and fluids we now obtain for the enthalpies: , , [ ] ( ) [ ] [ ] ( ) [ ] da da da da p da wd wd wd wd p wd w wl wl wl wl wl

ice ice ice ice ice ice

H m h m c J H m h m c r J H m h m c J H m h m c r J θ θ θ θ = ⋅ = ⋅ ⋅ = ⋅ = ⋅ ⋅ + = ⋅ = ⋅ ⋅ = ⋅ = ⋅ ⋅ − (1.13) with, , , ice p da p wv wl ice wl ice wl ice w ice wl c c c c h h m m r r H H

specific heat of ice specific heat of dry air specific heat of water vapour specific heat of water liquid specific enthalpy of ice

specific enthalpy of water liquid mass of ice

mass of water liquid melting heat of ice evaporation heat of water enthalpy of ice

enthalpy of water liquid

[J/(kg K] [J/(kg K] [J/(kg K] [J/(kg K] [J/kg] [J/kg] [kg] [kg] [J/(kg K] [J/(kg K] [J] [J] ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

Using c_p =c_v+R the relations for the internal energies can be derived:

, , [ ] ( ) [ ] [ ] ( ) [ ] da da da da v da wv wv wv wv v wv w wl wl wl wl wl wl wl

ice ice ice ice ice ice ice

U m u m c J U m u m c r J U m u m c p J U m u m c r p J θ θ θ ν θ ν = = ⋅ = ⋅ ⋅ = ⋅ = ⋅ ⋅ + = ⋅ = ⋅ ⋅ − ⋅ ⋅ = ⋅ ⋅ − − (1.14) with, , , v da v wv ice wl ice wl c c U U ν ν

specific heat of dry air (internal energy) specific heat of water vapour (internal energy) specific volume of ice

specific volume of water liquid internal energy of ice

internal energy of water liquid

3 3 [J/kg K] [J/kg K] [m / ] [m / ] [J] [J] kg kg ⋅ ⋅

(18)

1.1.3 Relations for the volume elements

According the Reynolds transport Theorem that is described in [White, 2003] a control volume is used. This control volume is visualised as in Figure 1-4.

Control Volume in v out v in A A_out in ρ ρout Q _W

Figure 1-4: Control volume

A flow with a certain velocity v and density _in ρ_inenters the control volume at A and leaves _in the control volume at A with a velocity _out v_outand densityρ_out, this flow is called the mass flow. Heat Q is added to the control volume and work W is extracted from the control volume. The relation for conservation of mass is:

] / [kg s m m dt dm out in cv ₋ =       (1.15) with, in out m m

incoming mass flow outgoing mass flow

[kg/s] [kg/s]

The energy per unit mass consists of internal, kinetic and potential forms of energy (other forms of energy are neglected). The energy per unit mass is:

] / [ 2 2 1 _v _gz _J _kg u e= + ⋅ + (1.16) with, e g z

energy per unit mass gravitational constant height 2 [J/kg] [m/s ] [m] The relation for conservation of energy now becomes:

] / [J s e m e m W Q dt dE out in cv ⋅ − ⋅ + − =       (1.17)

(19)

with, E Q W energy heat work [ ] [ ] [ ] J W W

Finally, three relations are used in the model. The relations are different for the situations mentioned in subsection 1.1.2.

The relation for the case of a non-saturated humid air mixture can be written into:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

θ θ θ θ − + +   =   +   − + + − + − + , , , , , , , , , , , , , , , , , , da in p da wv in p wv in cv da v da wd v wv

da out p da wv out p wv cv v da da out da in v wv wv out wv in cv da v da wv v wv Q W m c m c d dt m c m c m c m c c m m c m m m c m c (1.18)

The relation for the case of a saturated air mixture above 0,01°Ccan be written into:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

θ θ θ θ − + +   =   +   + + − +  ₋ ₊ ₊ ₋ ₊    + , , , , , , , , , , , , , , , , , , , da in p da wv in p wv in cv da v da wv v wv da out p da wv out wl wl cv da v da wv v wv v da da out da in v wv wv out wl wv in cv w wl da v da wv v wv Q W m c m c d dt m c m c m c m c m c m c m c c m m c m m m r m m c m c (1.19)

The relation for the case of a saturated air mixture below 0,01°Ccan be written into:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

θ θ θ θ − + +   =   +    ₊ ₊ ₊    − +  ₋ ₊ ₊ ₋ ₊    , , , , , , , , , , , , , , , , , da in p da wv in p wv in cv da v da wv v wv

da out p da wv out ice ice cv ice ice da v da wv v wv

v da da out da in v wv wv out ice wv in cv w ice da Q W m c m c d dt m c m c m c m c m c m r m c m c c m m c m m m r m m c

(

_{v da}_, +m c_wv _{v wv}_,

)

(1.20)

(20)

1.1.4 Relations for the resistance elements

Now we use the control volume for a duct with a constant diameter. See Figure 1-5 for the visualization of the duct.

v P1 P2 L D w τ z2 z1 v

Figure 1-5: Duct with constant diameter

A flow with a certain pressure p enters the control volume at a certain altitude ₁ z and leaves ₁ the control volume at a different pressure p and possibly a different altitude ₂ z . Inside the ₂ duct, with a length L and diameter D, the flow experiences a friction contribution τ_w. The relation for conservation of impulse now becomes:

2 1 2 1 2 1 1 4 2 1 2 2 1 2 2 2 4 0 ( ) ( ) _w [ /( )] p v g z D p v g z D D L kg m s ρ ρ π ρ ρ π τ π = + ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ − + ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ (1.21)

This can be re-written to:

1 2 1 2 4 ( ) [ ] w L p p g z g z Pa D τ ⋅ ⋅ = − +ρ⋅ ⋅ −ρ⋅ ⋅ (1.22)

The friction contribution τ_w can be described as a function of the friction factor f :

] [ 8 2 − ⋅ ⋅ = v f w ρ τ (1.23) By introducing the resistance

D L f

K = ⋅ the following relation is obtained:

2 1

1 2 1 2

2⋅K⋅ ⋅ρ v =(p −p +ρ⋅ ⋅g z −ρ⋅ ⋅g z ) [Pa] (1.24)

(21)

1 2 1 2 2 ( ) 2 ( ) [ / ] p p g z z v m s K ρ ρ ⋅ − + ⋅ ⋅ ⋅ − = ⋅ (1.25)

Finally, the mass flow can be determined using: [ / ]

(22)

1.2 Model in Matlab Simulink

In this section the components of heating ventilation and air conditioning are translated into Matlab Simulink models. The relations described in section 1.1 are implemented in the models. In subsection 1.2.1 the components are described. In subsection 1.2.2 the connected components in possible situations are described.

1.2.1 Components

The components are divided into four different groups: • Transport systems

• Air-handling components • Consumers

• External influences

The transport systems are ducting, T-piece, resistance, diffuser, nozzle and ventilator modules. The ducting and the T-pieces contain relations, which are described in subsection 1.1.3. These are the so-called “Volume elements”. The inputs of these elements are mass flows and temperatures. Lengths and diameters are also inputs given by the user. The outputs are pressures, temperatures, densities and humidities. Resistances, diffusers, nozzles and ventilators are using the relations that are described in the subsection 1.1.4. These are the “Resistance elements”. The resistance elements in contrary have pressures, densities and temperatures as input signals. The user inputs are resistance factors and diameters. The air-handling components are heaters, coolers, humidifiers and a recirculation valve. These components are all “Volume elements” and use the relations that are described in subsection 1.1.3. Inputs and outputs are equal to the ones described above. The user inputs are heating capacities, cooling capacities, steam supply, the volumes of the cooler, heater and humidifier and the position of the valve.

The consumers are the different cabins or mess rooms. These are “Volume elements” which have as user inputs, the dimensions of the spaces, the sensible heat load and the latent heat load.

Finally, the external influence is the condition of the outside air. This module has only outputs and user inputs. The outputs are the pressure, temperature, density and humidities. The input from the user is the temperature outside, the pressure outside and the relative humidity outside.

(23)

1.2.2 Entire model

The discussed components can be connected to build a system that can occur in reality. In Figure 1-6 the model is shown and in Figure 1-7 a picture of a connected system in Matlab Simulink is visible. The components described in subsection 1.2.1 are hidden in the visible blocks.

Cabin 1 Cabin 2 Mess room

Air Handling Unit

Supply flow

Re-circulation flow Ambient air flow

Extraction flow

Figure 1-6: Physical model

outside w orld outside w orld 2 Supply Discharge Output mess room Air supply from CAU

Pressure Cabin 1 Pressure Cabin 2 Pressure Messroom

Pressure Air supply Supply Mess room Supply Cabin 2 Supply Cabin 1 Supply ducting A-deck

In1 Subsystem2 In1 Subsystem1 In1 Subsystem 4 In1 Subsystem Pressure Cabin 1 Pressure Cabin 2

Outside Air Pressure

Discharge Cabin 1

Discharge Cabin 2

Discharge ducting san. A-deck

Pressure Messroom Pressure Cabin 2 Pressure Cabin 1 Pressure CAU Discharge Cabin 1 Discharge Cabin 2 Discharge Messroom Retrun Air to CAU Discharge ducting + Corridor A-deck

Supply Pressure A-deck

Return Air A-deck

Pressure Outside Air

Supply Air A-deck

Return Air Pressure

Central Airhandling Unit

Supply Discharge1 Discharge2 Output Cabin 2 Supply Discharge1 Discharge2 Output Cabin 1

(24)

The block “Central Air-handling Unit” contains a heater, a cooler, a steam humidifier, a nozzle, a ventilator and a recirculation valve. The unit in reality is shown in Figure 1-8.

Fan Heater Cooler Steam

humidifier Nozzle Mixing box Re-circulation Valve Re-circulation air Exhaust air Ambient air Supply air

Figure 1-8: Central Air-handling Unit

In Figure 1-9 the same air-handling unit is shown in a Matlab Simulink model, here the extra resistances are included to be able to run the model in Matlab Simulink.

(25)

1.3 Extensions

1.3.1 Introduction

The goal of this study is to extend the model made by R.Ruissen. Heat exchangers can be developed and implemented in the cooler and heater. In the existing heater and cooler it was only possible to give a heat capacity. The spaces can be simulated in order to acquire a variable and more realistic temperature, which on its turn gives the opportunity to develop a control system. Besides this development of components, the ducting created by R.Ruissen needs some improvement. When air with a certain condition enters a duct, the air condition leaves the component at the same moment. In other words, the time delay is missing in the Matlab Simulink channel modules.

1.3.2 Flow pattern

Two types of flow pattern can be distinguished. These flow patterns are: • Ideally mixed flow

• Plug flow

An ideal mixed condition is often used as assumption for the simulation of a vessel. The assumption of an ideally mixed fluid implies that is assumed that each element in a volume (duct) will vary in time in exactly the same fashion. In a duct of a certain length this is not a realistic assumption.

In case of plug flow (see Figure 1-10), it is assumed that each element x+ ∆xdoes not

necessary vary in time in the same fashion as another element. When the number of elements are taken infinite we can speak about plug flow.

θ(x) θ(x+∆x)

v v

x = 0 x x+∆x x = L

Figure 1-10: Plug flow

The reality lies in between the two flow patterns. However, to determine what the real number of elements should be to acquire the most realistic result, the results should be compared with tests.

(26)

1.3.3 Optional solutions

In Matlab Simulink three options (Tak, 2005) to create the dynamic behaviour in the ducts are compared, these are achieved with (a):

• variable transport delay • S-function

• Integrators

Variable transport delay

A Variable transport delay is a block in Matlab Simulink, which can simulate a time delay with plug flow. For a system with a constant velocity the following is valid:

( ) ( ) [ ] out t in t delay C θ =θ ⋅ −τ ° (1.27) with, in out delay θ θ τ incoming temperature outgoing temperature time constant C C t ° °

the time delay τ_delay can be determined with the quotient of the length of a duct and the air speed: [ ] delay L t v τ = (1.28) with, L v length velocity [m] [m/s]

The variable time delay is a good option when the airspeed remains constant. When the airspeed is variable, the Matlab Simulink block will use the airspeed at the end of the simulation to calculate the time constant and will apply it over the entire length of the duct. Example: Simulation of a duct

0 1 1 3 0 1 1 3 s s s s L v v θ θ − − − − 10 2 4 20 25 [m] [m/s] [m/s] [ C] [ C] ° °

(27)

The time constant can be calculated with: 1 0 1 2 1 3 0 1 : 2 1 2 1 3 : 4 2 8 s s s L v t m s L v t m − − − = ⋅ = ⋅ = − = ⋅ = ⋅ = (1.29) 1 2 0 1 1 3 2 8 3 2 4 delay s s L L s v v τ − − = + = + = (1.30)

Figure 1-11: Time delay

In Figure 1-11 the results of this simulation is presented. It can be seen that the time delay is 2,5 s instead of the calculated 3, 0 s . Matlab Simulink has calculated the time delay with:

1 3 10 2,5 4 delay s L s v τ − = = = (1.31) S-function

A S-function is user-definable block in Matlab Simulink that may be written in M, C, Fortran or Ada and must confirm to S-function standards. t, x, u and flag are automatically passed to the S-function by Matlab Simulink. Extra parameters may be specified in the S-function parameter field. [Matlab Simulink, 2002]

The differential equation that gives the compartmental model for a flow is given by:

1( ) [ ( ) 1( )] k k k d v N t t t dtθ + L θ θ + ⋅ = ⋅ − k=0,1,...,N−1 (1.32)

(28)

1 1 2 2 0 0 ( ) ( ) ( ) 0 0 ( ) 0 ( ) 0 0 ( ) ( ) 0 0 0 in N N v N v N L t t L v N v N t t L L d t dt t t v N v N L L θ θ θ θ θ θ θ ⋅   −   _ _⋅ _   _ _   _ _ ⋅ ⋅   _ _   _ _ −   _ _   _ _  = ⋅ =_ _⋅   _ _   _ _   _ _   _ _   _ _   _ _   _⋅ _⋅      ₋      (1.33)

[

]

1 2 ( ) ( ) ( ) 0 0 1 ( ) out N t t t t θ θ θ θ         = ⋅         (1.34) in out v L N θ θ velocity

length of the duct number of elements incoming temperature outgoing temperature [m/s] [m] [-] [ C] [ C] ° ° [Bosgra, 2002]

When the number of elements is infinite, the plug flow model is valid.

Integrators

A third option in Matlab Simulink to determine the dynamic behaviour of the module is to use a model with integrators. Elements are placed in a row and on every element a time delay is applied. Every element has an incoming temperature and an outgoing temperature. The theoretical method is identical to the method of the S-function. The differential equation for every element is given by:

( ) [ ( ) ( )]

out in out

d v

t t t

dtθ =dx⋅ θ −θ (1.35)

(29)

1.3.4 Results of tests and conclusions

The three different options to simulate the time delay in a duct have been tested in Matlab Simulink. In Table 1-1 the simulation time for the different options is presented for equal conditions. The simulated time is 60 seconds.

Nr. of elements Variable transport delay S-function Integrator model

1 10 s 15 s -

10 - 35 s 40 s

50 - 60 s 75 s

100 - 100 s -

200 - - 270 s

Table 1-1: Simulation time

From Table 1-1 it is clear that the method of using a variable time delay is the fastest way for the calculation. However, as said before, when the airspeed is variable, this method is not applicable because the result is not correct.

The difference in simulation time between the S-function and the model with integrators is negligible. The S-function has the advantage that it is possible to change the number of element quite simple by giving another input. When it is necessary to change the number of elements in the integrator model, the model itself must be altered.

The number of elements for the S-function has an influence on the simulation time. When one element is used – the ideal mixing assumption – the simulation time is 15 seconds, however this assumption does not give the most realistic result. When hundred elements are used, the result is nearly equal to a plug flow model. The simulation time however is now hardly acceptable, about 100 seconds.

The number of needed elements to acquire the best results is dependent on the length and diameter of the duct, the airspeed, the flow profile, etcetera. It shall be clear that it is difficult to determine the number of elements that gives the most realistic result without measurement data. Since these data are not available, a choice should be made. A length-diameter ratio of 1 could give a good result. However, because the differences in the results are not too large it would be wise not use a large number of elements to avoid a long simulation time. The results of simulations with different numbers of elements are presented in Figure 1-12 and Figure 1-13. From the figures it can be concluded that the difference for the different numbers of elements is negligible for HVAC applications. Therefore for HVAC applications one element will be sufficient.

(30)

Figure 1-12: Simulation results for a different number of elements

Figure 1-13: Simulation results for a different number of elements (zoom in)

Note that for one element the response is not a response of a first order process. This is caused by the accumulating effect of the air in the ducting.

(31)

2 Theoretical background

This chapter gives a theoretical background for the chapters “Heat accumulation in walls”, “Heat exchanger” and “Control in the HVAC system”. The modes of heat transfer are discussed in section 2.1. Section 2.2 and section 2.3 deals with control.

2.1 Heat transfer

“In thermodynamics, heat is defined as energy transfer due to temperature gradients or differences. Consistent with this viewpoint, thermodynamics recognizes only two modes of heat transfer: conduction and radiation. These modes of heat transfer occur on a molecule and a subatomic scale.

A fluid, by virtue of its mass and velocity, can transport momentum. In addition, by virtue of its temperature, it can transport energy. Strictly speaking, convection is the transport of energy bulk motion of a medium. It is common engineering practice to use the term convection more broadly and describe heat transfer from a surface to a moving fluid also as convection even though conduction and radiation play a dominant role close to the surface, where the fluid is stationary. In this sense, convection is usually regarded as a distinct mode of heat transfer.” [Mills, 1999]

2.1.1 Conduction

On a microscopic level heat conduction is very complex. Therefore, heat conduction is described on a macroscopic level. Figure 2-1 shows the cross section of a plane wall area A and thicknessd.

(32)

At distance x=0 the temperature T =T₁ and at distance x=d the temperature T =T₂ .The heat flow through the wall is in the direction of decreasing temperature. If T₁ >T₂

then Q is in the positive direction of x . Equation

(2.1) gives Fourier’s law of heat conduction ~ Q dT q and q A = ∝ − dx (2.1) with, Q A q T x Heat flow Area of the wall Heat flux Temperature

coordinate in the x-direction 2 2 [ ] [ ] [ /( )] [ ] [ ] W m W m K m Now a constant of proportionality λ is introduced:

dT q dx λ = − ⋅ _(2.2) with,

λ thermal conductivity of the material [W /(m K⋅ )] [Mills, 1999]

So, the heat flow is determined with: [ ] A Q T W d λ⋅ = ⋅ ∆ (2.3) The relation can also be described with the thermal resistanceR, then it becomes:

[ ] T Q W R ∆ = _(2.4)

(33)

2.1.2 Convection

Convection or convective heat transfer is the term used to describe heat transfer to or from a surface to moving fluid, as shown in Figure 2-2. The surface may be inside a pipe, along a wall or other possible heat accumulating bodies. The flow may be forced, as in the case of water pumped through a pipe of a heat exchanger. On the other hand, the flow could be natural (or free), driven by buoyancy forces because of a density difference, as in the case of a wall inside an air conditioned space. Either type of flow can be internal or external. Also, both forced and natural flows can be either laminar or turbulent. Laminar flows are

predominant at lower velocities, for smaller sizes, and for more viscous fluids. Flow in a pipe becomes turbulent when the Reynolds number exceeds about 2300. Heat transfer rates tend to be much higher at turbulent flows.

Fluid flow (V,Te)

qs

Surface Ts

Figure 2-2: Convective heat transfer

Usually, heat transfer by convection is a complicated function of surface geometry and

temperature, the fluid temperature and velocity, and fluid thermo physical properties. The rate of heat transfer is approximately proportional to the difference between the surface

temperature T and the temperature of the stream fluid_s T . The constant of proportionality is _e called the convective heat transfer coefficientα_c:

2

[ / ]

s c

q =α ⋅ ∆T W m

(34)

2.1.3 Thermal radiation

The flux of radiant energy on a surface (see Figure 2-3) is its irradiation, G W m[ / 2]; the energy flux leaving a surface due to emission and reflection is its radiosity, J W m[ / 2]. A black surface is defined as a surface that absorbs all incident radiation, reflecting none. As a consequence, all of the radiation leaving a black surface is emitted by the surface and is given by the Stefan-Boltzmann law:

4 b J =E =σ⋅T (2.6) with, b E T σ

Blackbody emissive power Stefan-Boltzmann constant

absolute temperature of the surface 2 2 4 [ / ] [ /( )] [ ] W m W m K K ⋅

The net radiant heat flux through the surface is the radiosity minus the irradiation:

2 1 1 1 [ / ] q =J −G W m (2.7) or 4 4 2 1 1 2 [ / ] q =σ⋅T −σ⋅T W m (2.8) with, 1 2 T T

temperature of the object temperature of the enclosure

[ ] [ ] K K Convex black object, 1 A1 T2 Irradiation G Radiosity J Black isothermal enclosure, 2

Figure 2-3: Convex black object in a black isothermal enclosure

The described blackbody is an ideal surface. In reality surfaces absorb less radiation. The fraction is called the absorptance or absorbtivity, α [-]. A model that is used for a real surface is the gray surface, at which α is constant. The fraction of reflected radiation is called

(35)

1 ρ

= −

α

. Emission is also lower in reality. The fraction of the blackbody emissive power

4

T

σ ⋅ emitted is called the emittance or emissivity, ε. The emittance and absorbtance of a gray surface are equal. Typical values of ε can be found in literature. By using the equations

1 ρ

= −

α

and ε =α , absorbtance and the reflectance can be determined. The rate of heat flow can thus be determined with:

4 4 12 1 1 ( 1 2 ) [ ] Q =ε ⋅A ⋅ σ⋅T −σ⋅T W (2.9) with, 12 1 Q A

_{heat flow from 1 to 2}

the area of the object 2

[W] [m ]

The term T₁4 complicates calculations. Therefore it is useful to linearize equation (2.9) by factoring the term σ⋅T₁4−σ⋅T₂4 to obtain

2 2 12 1 1 ( 1 2 ) ( 1 2) ( 1 2) [ ] Q =ε ⋅A ⋅ ⋅σ T +T ⋅ T +T ⋅ T −T W (2.10) 3 1 1 1 2 ε ⋅A ⋅ ⋅σ (4⋅T_m ) (⋅ T −T ) [W] _(2.11) with, m

T mean value of and T₁ T ₂ [ ]K

if T₁T₂_and 1 2

2

m

T T

T + _{the result can be written as:}

12 1 r ( 1 2) [ ]

Q =A ⋅α ⋅ T −T W

(2.12) in this formula, α_r is equal to 4⋅ε σ₁⋅ ⋅T_m3 and is called the radiation heat transfer coefficient. At 25°C [298 K], 8 3 2 1 4 (5, 67 10 ) (298) [ /( )] r W m K α ε − = ⋅ ⋅ ⋅ ⋅ ⋅ _(2.13) or 2 1 6 [ /( )] r W m K α = ⋅ε ⋅ (2.14) For T₁ =320 K and T₂ =300 K the error by using the approximation is only 0,1 %; for

1 400 and 2 300

T = K T = K, the error is 2 %. [Mills, 1999]

Not all radiation leaving one surface will be intercepted by another surface. The fraction that is intercepted is given by the shape factor F₁₂ . How this factor is determined is explained in

(36)

2.1.4 Shape factor

“Consider radiation exchange between two finite black surfaces A₁ and A₂, as shown in Figure

2-4. By inspection, only part of the radiation leaving surface 1 is intercepted by surface 2 and vice versa. We define the shape factor (or view factor) F₁₂ as the fraction of energy leaving

1

A that is intercepted by A₂; likewise, F₂₁ is the fraction of energy leaving A₂ that is

intercepted by A₁. The shape factor is a geometrical concept and depends only on the size,

shape and orientation of the surfaces. Radiation leaves surface 1 at the rate of E A Wb₁ ₁[ ]; the

portion that is intercepted by surface 2 is then E A F_b₁ _{1 12}. Likewise, the radiation leaving surface

2 that is intercepted by surface 1 is E A F_b₂ ₂ ₂₁. Since both surfaces are black, all incident

radiation is absorbed, and the net radiant energy exchange is

12 b1 1 12 b2 2 21 Q =E A F −E A F (2.15) A1 Q1 Q2 Q12

Figure 2-4: Radiation enery exchange between tow finite surfaces

If both surfaces are at the same temperature, the second law of thermodynamics requires that there be no net energy exchange; that is, if E_b₁=E_b₂, Q12=0

_and

1 12 2 21

A F =A F

which is the reciprocal rule for shape factors. Note that since shape factors depend only on geometry, this relationship is true even when the surfaces are at different temperatures.” [Mills, 1999]

“Determining shape factors generally requires the evaluation of a double surface integral, which is not easy. However, shape factors have been determined for a great variety of configurations and are available in the form of formulas and graphs.”

[Mills, 1999]

For determining the radiation in a cubic space only the shape factor for opposite rectangles and for adjacent rectangles are needed. The formulas for these shape factors are in Appendix II: Shape factors.

(37)

2.2 Dynamic response

2.2.1 Time constant

In HVAC systems first order systems are used all the time. The time constant of a first order system gives important information about stability and speed of response. The time constant θ is defined as the time in seconds that is takes to increase the response to 0,632 (or 1-e-1) of its final value.

Figure 2-5: Time constant [Dijkstra, 2000]

2.2.2 Dynamic behaviour

Figure 2-6 shows the response of a typical HVAC subsystem to a sudden change in input. Many HVAC systems are characterized by a time delay (or dead time) and what is called a first-order response. There is a time delay after a controller output signal changes before the controlled variables changes, and that once the controlled variable starts to change, it

approaches its new value exponentially with time. One exponentional term serves to describe the system as first order.

0 1 0 250 - > t (s) 0, 632

θ

( / ) ( ) 1 t c t = −e− θ 0 1 0 250 - > t (s) 0, 632

θ

( / ) ( ) 1 t c t = −e− θ

(38)

50 % 75 % 25 % 63,2 % Time Delay Time Constant New steady state value Time T e m p e ra tu re Temperature change

Figure 2-6: Time delay and time constant

Figure 2-6 helps to define several terms used in control. The time delay is the difference in time between when the controller output is changed and when the response is observed. This time delay (dead time) is mainly caused by flow. It arises because the flow moves forward partially as a front without mixing. The time delay is often difficult to pick of a graph. A recommended manner to determine the delay time is to draw a straight line tangent to the steepest part of the exponentional response curve and noting where this intersects the initial value of he controlled variable. The time corresponding to this intersection is approximately the time delay. The time constant has already been explaned in the previous section.

(39)

2.3 Control

2.3.1 Elementary control system

Figure 2-7 illustrates an elementary control system. Air flows through a heating coil in a duct. The sensor measures the temperature of the air after the heating coil and passes this

information to the controller. The controller compares the temperature of the air with a set point value and sends a signal to the control valve. This signal is a signal to open or close the hot water valve in order to achieve a temperature of the air that matches the set point

temperature.

Heating coil Air flow T Temperature sensor C Controller Control valve

Figure 2-7: Elementary control system

This is a closed-loop system. All control systems may be reduced to combinations of these essential elements.

(40)

2.3.2 Controllers

Three kinds of control modes can be distinguished. The simplest mode is proportional control (P-controller). The proportional controller can be given in a mathematical expression:

p O=A K+ ⋅e (2.16) with, p O A e K controller output

a constant equal to the value of the controller output with no error the error, equal to the difference between the set point

and the measured value of the controlled variable proportional gain factor

Increasing the gain factor will make the controller more responsive, too high gain factor in contrary, will make the system unstable. It will cause a continuously oscillation or hunt around the set point. Decreasing the gain factor will improve the stability of the system, but the possible error will increase and the response and sensitivity will decrease. Usually, there will be an offset with proportional control because the error needed to generate the controller output will produce only enough capacity to match the load on the system.

The Proportional plus Integral (PI-controller) has an additional term in the mathematical expression, the control equation becomes:

p i

O=A K+ ⋅ +e K ⋅

∫

edt (2.17)

with,

i

K integral gain factor

The added term affects the output by the error signal integrated over time and multiplied by integral gain factor. Since the error sign may be positive or negative, the integral term may be plus or minus. The effect is that the controller will change as long as any error persists and the control offset will be eliminated.

Another term is added when a Proportional plus Integral Derivative control (PID-controller) mode is used. The control equation now becomes:

p i d

de

O A K e K edt K

dt

= + ⋅ + ⋅

∫

+ ⋅ (2.18)

The derivative term provides additional controller output that is related to the rate of change of the controlled variable. A rapid rate of change will increase the absolute value of the

derivative term. A small rate of change will decrease the value. Therefore, a derivative control can be used to reduce the overshoot when a rapid response in desired (requiring a high

proportional factor). [Haines, 1993]

(41)

3 Heat accumulation in walls

3.1 Abstract

This chapter deals with the influence of heat accumulation in walls that are composed out of different materials on board of ships. A model has been developed and has been translated to a Matlab Simulink simulation model. The model of the space includes convective heat transfer, thermal radiation and an internal heat load. With these heat capacities known, the influence on the space air temperature is determined. With step changes of the outside

temperature the time constant for the response temperature of the space air can be determined. It turns out that the time constant is such large that the dynamic influence of the outside temperature is very slow. However, the heat accumulation in the external wall is not negligible at all. It has a significant influence on the air condition in a space.

3.2 Introduction

A model has been made [R.Ruissen, 2004] that can determine the condition of the air in a space on board of ships. The model includes an HVAC system with a heater, a cooler, a humidifier, a mixing box, a duct and a space. In this chapter the space module will be

extended. The input possibilities for space modules are still limited. The only influence on the space temperature is the air from the HVAC system and an internal heat load.

In buildings, walls between the outside and inside accumulate a large heat capacity. The accumulation of heat in walls is described in detail for uniform walls [van Paassen, 1997]. This effect is often neglected on board of ships; therefore research will be done to the accumulation of heat in walls on board of ships. Because the thickness of the walls on board of ships is limited, the heat accumulation is also supposed to be small. However, because of insulation material it can take a time before the temperature of walls have changed. To be able to predict the influence of the walls on board of ships on the change of the internal

temperature a simulation model will be built. When the influences are clear, the model will be combined with the earlier developed space (Ruissen, 2004).

(42)

3.3 Modelling

3.3.1 Space

A simulation will be executed of a space inside the ship LPD2. The simulated space will consist of a ceiling, a floor, four walls and furniture material in the simulation. The ceiling, floor, furniture material and three of the four walls will be regarded as a uniform material, which will be dealt with in section 3.3.3. The fourth, external wall will be explained in section 3.3.2. In 3.3.1 the simulated space is presented. The dimensions of the simulated space are

5 5 2, 3x x meters. side wall 2 side wall 1 ceiling external wall floor opposite wall furniture Figure 3-1: Space

Heat transfer occurs on the external wall. By means of convection or solar radiation the temperature of the external wall increases or decreases. The external wall will transfer heat to the space air by means of convection and it will also transfer heat by means of radiation to the sidewalls, the opposite wall, the floor and the ceiling. Finally, the space air will exchange heat by means of convection with the walls (and to a lesser degree with ceiling and floor) and the furniture. Note that radiative heat transfer with the furniture is neglected. An explanation will be given in the discussion section.

(43)

3.3.2 External wall definition

Since the goal of the chapter is to investigate the influence of the accumulation of heat in walls on the temperature in a space, a precise definition of the walls is important to acquire reliable results. However, since external walls on board of ships are not homogenous, this is not straightforward.

The external wall has different materials inside the wall; steel, insulation material and navy board. The density and the specific heat of steel and for insulation materials differ much. The composition of an external wall can be found in [ISO 7547, 1985]. The model is given in Figure 3-2. Every part of the wall is regarded as a nodal point. It can be seen that some

simplifications have been made. The stiffeners and its insulation are not in the model. A factor for the mass of the stiffeners shall be introduced (see Appendix III: Factor for stiffeners). For the simulation a wall from the LPD-2 will be used. In Appendix IV: Walls in LPD2 walls of the LPD-2 are given.

Qconvection Qconvection Qconvection Qconvection

Qradiation

Qsolar

Qconduction

Qconduction Qradiation

air gap

inside

insulation

outside

insulation

steel

navy

board

10 75 500 25 10

Figure 3-2: External wall

The temperature of the steel part of the wall will be influenced by solar radiation, convection and heat conduction from the insulation part of the wall. The temperature of the insulation part of the wall will be influenced by heat conduction from the steel, heat convection from the air and by thermal radiation from the other insulation part. The temperature of the air gap on its turn is affected by convection from both the insulation parts. The temperature of the insulation part on the inside is affected in the same way as the other insulation part. Heat conduction from the insulation, convection from the space air and radiation from walls, the floor and the ceiling in the space finally influence the navy board temperature.

(44)

The properties of the materials are given in Table 3-1. Material _Density 3

[kg m Specific heat / ] [ /(J kg K⋅ )] Thermal cond. [W /(m K⋅ )]

Steel 7800 470 50

Insulation 100 1470 0,1

Air 1,2 1004 -

Navy board 5001) 26001) 0,08

Table 3-1: Material properties

1)

the density and the specific heat of navy board are estimations, the estimation is based on the densities and specific heats of wood (500 kg m/ 3 and 2800 J kg K/ ⋅ ) and paper

( 3

930 kg m/ and 2500 J kg K/ ⋅ ).

3.3.3 Internal walls, floor and ceiling

The internal walls, floor and ceiling can also have a large influence on the results of the simulation. Therefore, the walls inside the space are defined in this section in more detail. The internal walls are assumed to be homogenous walls. The used data for sidewalls are in Table 3-2: specific heat c 2000 [ /(J kg K⋅ )] thickness t 0,05 [ ]m thermal conductivity λ 0,5 [W/(m K⋅ )] density ρ 450 3 [kg m / ] Table 3-2: Internal walls

In the simulation, the floor and ceiling are also assumed as plain walls (by using a node). However, as in the external walls, the floor and ceiling are composed of different components which make it difficult to define the floor or ceiling, by only density and specific heat. In

Appendix V: Deck in LPD2 a deck is shown in a drawing and a calculation is made. The results of this calculation are in Table 3-3:

specific heat c 646,4 [ /(J kg K⋅ )] mass per square metre deck m 117 2

[kg m / ] thickness t 0,6 [ ]m

thermal conductivity λ 22) [W/(m K⋅ )]

Table 3-3: Floor/ceiling

(45)

2)

the thermal conductivity is an estimated value that is estimated with the help of the value for thermal conductivity of steel (50 W m K/ ⋅ ), navy board (0, 08W m K/ ⋅ ) and air.

3.3.4 Furniture material

The furniture material is regarded as one piece on which only convection has an influence. Input data that are necessary to determine the influence of the furniture are the surface of the furniture, the thickness, the mass, the specific heat and the thermal conductivity. In Table 3-4 these properties are presented.

Surface 2 [m ] Thickness [ ]m Weight [kg] Thermal cond. [W/(m K⋅ )] Specific heat [ /(J kg K⋅ )] Furniture 20 0,02 500 1 2000

(46)

3.4 Accumulation

3.4.1 Accumulation of heat in walls, floor and ceiling

Since the density and the heat capacity of the walls are known in a nodal point the heat accumulation can be determined:

[ / ] wall wall m ⋅c J K with, wall wall m c

mass of the wall

specific heat of the wall

[ ]

[ /( )]

kg J kg K⋅

3.4.2 Accumulation of heat in furniture material

Walls give the most significant contribution to the accumulation of heat. However, furniture material and other attributes can also have a contribution to the heat accumulation. The heat accumulation can be determined with an expression equal to the one that is used for the walls:

[ / ] furniture furniture m ⋅c ⋅ J K with, furniture furniture m c

mass of the furniture

specific heat of furniture material

[ ]

[ /( )]

kg J kg K⋅

3.4.3 Accumulation of heat in air

Although the heat capacity of the air is very small, it should not be neglected. The dimensions of a space have a large influence on the heat accumulation of the air. Small spaces will have little heat accumulation and if a small heat load is present in the space, the space air

temperature will rise quickly. Larger spaces in contrary will accumulate more heat and as a consequence the space air temperature will rise more slowly. When the dimensions of a space are known, the heat capacity of the space is expressed by:

space air spaceair [ / ]

m ⋅c J K with, space air spaceair m c

mass of the space air

Specific heat of air at constant volume

[ ]

[ /( )]

kg J kg K⋅

(47)

3.5 Heat transfer

As shown in subsection 3.3.2 three types of heat transfer occur in the external wall. In the following subsections, the heat transfer in this wall and the heat transfer to the space air and the other walls in the space will be described. Subsection 3.5.1 deals with conduction,

subsection 3.5.2 with convection and subsection 3.5.3 deals with thermal radiation. Finally in subsection 3.5.4 a commentary is given to the heat transfer coefficients.

3.5.1 Conduction

In buildings walls are often composed out of a homogenous material. The entire heat transfer can then be described by heat conduction through the wall. By dividing the wall in more slices, nodal points, the result of the calculation will become more accurate. The disadvantage is that the calculation time will increase. In case of a wall in a ship the wall is constructed of different materials and a different composition as explained in subsection 3.3.2. However, conduction through the wall still occurs. The external wall is influenced by radiation and convection from outside, but it is also under influence of the conduction through the wall from the side of the insulation. The conductive heat transfer is described from the nodal point in the steel part of the wall to the nodal point in the insulation part of the wall. The conductive heat transfer Q is given by:

( ) [ ]

cond cond wall steel insulation

Q =k ⋅A ⋅ θ −θ W (3.1) with, cond steel insulation wall cond Q A k θ θ Conductive heat steel temperature insulation temperature area of the external wall total heat transfer coefficient

2 2 [ ] [ ] [ ] [ ] [ /( )] W C C m W m K ° ° ⋅ Because the layer consists out of two components, it is useful to describe the relation with resistances. The conductive heat transfer Q can be described as the quotient of the

temperature difference of the two nodal points and the sum of the resistances:

1 1 2 2 [ ] steel insulation cond steel insulation steel insulation

steel wall insulation wall

T Q W t t R R A A θ θ λ λ − ∆ = = ⋅ ⋅ + ₊ ⋅ ⋅ _(3.2) with,

(48)

steel steel steel insulation insulation insulation R t R t λ λ

conductive resistance of steel thickness of the steel layer thermal conductivity of steel conductive resistance of insulation thickness of the insulation layer thermal conductivity of insulation

[ / ] [ ] [ /( )] [ / ] [ ] [ /( )] W K m W m K W K m W m K ⋅ ⋅ On the inside of the external wall also heat conduction takes place. Heat transfer occurs between the nodal points of the insulation part and the navy board part of the wall. The heat transfer can be described as:

( ) [ ]

cond cond insulation navy board

Q =k ⋅ ⋅A θ −θ W (3.3) or 1 1 2 2 [ ]

insulation navy board cond

navy board insulation navy board insulation

insulation wall navy board wall

T Q W t R R t A A θ θ λ λ − ∆ = = ⋅ + ⋅ + ⋅ ⋅ _(3.4) with, navy board navy board navy board navy board R t λ θ

conductive resistance of navy board thickness of the navy board layer thermal conductivity of navy board temperature of the navy board

[ / ] [ ] [ /( )] [ ] W K d W m K C ⋅ °

(49)

3.5.2 Convection

Since the goal of this research is to investigate the influence of heat accumulation in walls on the space air temperature, convection is an important mode of heat transfer in this model. The only relation between heat accumulation in the wall and the space air temperature is

convective heat transfer.

In the model shown in section 3.3.2 a convective heat flow occurs on the external wall on the outer side. This will cause that the temperature of the wall will increase or decrease in

temperature. As a consequence a convective heat flow takes place on the external wall on the inside (i.e. inside the space). This convective heat transfer will thus have an influence on the inside space temperature. There is convection at walls inside the space, which are not exposed to the outside air and also on the furniture material. Their temperature will adapt a value that is nearly equal to the space air temperature. Finally, a convective heat flow occurs in between the external wall.

The three above described convective heat flows can be described by:

. ,

( )

conv out wall out ex wall steel

Q =k ⋅A ⋅θ −θ (3.5) with, . , conv out wall out ex wall steel Q k A θ θ Convective heat

total heat transfer coefficient area of the outside wall outside temperature

temperature of the steel in the wall exposed to the outside

2 2 [ ] [ /( )] [ ] [ ] [ ] W W m K m C C ⋅ ° °

See for calculation of heat transfer coefficients section 3.5.4.

The relation that describes the convective heat transfer inside the space on the external wall is:

. ,

( )

conv in wall in ex wall navy board

Q =k ⋅A ⋅θ −θ (3.6)

with,

. , in

ex wall navy board in

k

θ θ

total heat transfer coefficient

temperature of the navy board in the external wall temperature of the air inside the space

2 [ /( )] [ ] [ ] W m K C C ⋅ ° °

The convective heat transfers on the three walls that do not bound to the outside are:

( )

conv in inside wall in wall in

Q =k ⋅A ⋅ θ −θ (3.7) with, inside wall wall in A θ

area of the inside wall

temperature of the inside wall

2 [ ] [ ] m C °

(50)

The convective heat flow to the furniture is:

( )

conv furniture furniture in furniture

Q =k ⋅A ⋅ θ −θ (3.8) with, furniture furniture furniture A k θ

surface of the furniture

total heat transfer coefficient for furniture temperature of the furniture

2 2 [ ] [ /( )] [ ] m W m K C ⋅ °

Finally, convective heat transfer occurs between the air in the wall and its insulation. It happens as well on the inside as on the outside of the external wall. These convective heat transfers can be described with:

( ) [ ]

conv in wall inside walls insulation in wall

Q =k ⋅A ⋅ θ −θ W (3.9)

3.5.3 Thermal radiation

Besides the convective heat transfer, the temperature of walls is influenced by thermal radiation. Because of the relatively small difference in temperature of the walls, floor and ceiling (not exposed to the outside air) and the space air temperature, the radiation heat transfer will have a damping effect (slower response) on the temperature of the external wall. In contrary to the thermal radiation inside the space, the external radiation (from the sun) on the external wall will in summer not have a damping effect, but will certainly increase the wall temperature. However, in winter the solar radiation will cause a rise in temperature of the wall and will thus have a damping effect.

The thermal radiation on one wall, floor or ceiling can be described with:

/ / / / / /

( ( )

rad shape s walls floor ceiling walls floor ceiling wall floor ceiling

Q = Σ F ⋅k ⋅A ⋅ θ −θ (3.10) with, / / / / rad shape s

walls floor ceiling walls floor ceiling

Q F k A θ Radiative heat shape factor

total heat transfer coefficient area of the wall/floor/ceiling temperature of the wall/floor/ceiling

2 2 [ ] [ ] [ /( )] [ ] [ ] W W m K m C − ⋅ °

Radiation between insulation (in the wall) occurs and can be described with:

( )

rad shape s wall insulation in insulation out

Performance prediction of HVAC systems

Preface

Contents

Nomenclature

Roman variables

Greek variables

Subscripts

List of figures

List of tables

Introduction

1

Description of HVAC model

1.1 Theoretical model

(

)

(

)

(

)

(

)

(

(

)

(

)

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1.2 Model in Matlab Simulink

1.3 Extensions

[

]

2 Theoretical background

2.1 Heat transfer

1

ρ

= −

α

1

ρ

= −

α

2.2 Dynamic response