OF A
REFRIGERANT EVAPORATOR
ffl
D
föïfaLX0
n a
Df
HDD
D
a D D
D O a " D D
DODDDD
aaan
D
□ D U D D D
OF A
REFRIGERANT EVAPORATOR
OF A
REFRIGERANT EVAPORATOR
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus, prof.dr. J.M. Dirken,
in het openbaar te verdedigen ten overstaan van een commissie
daartoe aangewezen door het College van Dekanen
op 27 oktober 1987 te 14.00 uur
door
JAKOB STEFANUS VAN DER MEER § Prom i-.-pjcin 1 g
geboren te Delft,
Werktuigkundig Ingenieur
?/
y Sdr.ir. S. Touber
heeft als begeleider in hoge mate bijgedragen
aan het totstandkomen van het proefschrift.
Samenvatting i Summary v Conclusions and recommendations viii
Chapter 1 Introduction
1.1 Scope of the investigation 1
1.2 Literature 2 1.3 Conclusions from the literature 6
Chapter 2 Modelling the evaporator
2.1 Evaporator modelling 9 2.2 Model equations 10. 2.3 Simplifying the model 11
2.1) Extending the model 13 2.4.1 The dynamic behaviour of the evaporation region 13
2.4.2 Transportation times in the evaporation region 15 2.4.3 Extending the modelling of the boundary between the
evaporation and the superheat region 16 2.4.4 Superheat region, steady situation 17 2.4.5 Superheat region, dynamic situation 18
2.5 Pressure drop 19 2.5.1 Pressure drop evaporation region 19
2.5.2 Pressure drop superheat region 20 2.6 Heat transfer correlations 21 2.6.1 Heat transfer water side 21 2.6.2 Heat transfer superheat region 21
2.6.3 Heat transfer evaporation region 21 2.6.4 Validation of the heat transfer correlation 23
2.7 Model of the suction pipe 25 2.8 Liquid refrigerant in the suction pipe 28
2.8.1 Flow pattern 28 2.8.2 The quality of the refrigerant leaving the evaporator 29
2.8.3 The temperature behaviour of the pipe wall at
different superheats 31 2.8.4 Liquid refrigerant in the superheated vapour 31
Chapter 3 Modelling the thermostatic expansion valve
3.1 Why a thermostatic expansion valve 35 3.2 Construction and action of a TEV 36
3.3 Modelling the TEV 37 3.4 The equilibrium pressure of the bulb contents 37
3.5 Diaphragm and springs 39 3.6 The throttle process 41 3.6.1 Throttling of a flow of boiling or nearly boiling liquid 41 •
3.6.2 Influence of the subcooling 42 3.6.3 Mass flow as a function of the opening of the valve 42
3.7 The throttled mass flow of refrigerant for other
3.7.3 Conclusions about throttling 48 3.8 Steady state characteristics of the TEV 49
3.9 Dynamic properties of the TEV 51 3.9.1 Dynamic behaviour of the TEV without the bulb 52
3.9.2 Dynamic behaviour of the bulb 52 Chapter 4 Validation of the water chiller model
4.1 Experimental setup 57 4.2 Validation of the water chiller model for steady state
conditions 60 4.2.1 Measurements steady state conditions 60
4.2.2 Conclusions steady state.validation 62 4.3 Validation of the water chiller model for dynamic conditions 63
4.3.1 Measurements dynamic conditions 63 4.3.2 Conclusions dynamic measurements 64 Chapter 5 Modelling an evaporator with
parallel refrigerant circuits
5.1 P a r a l l e l c i r c u i t s 67
5.2 Mathematical model of an evaporator with two p a r a l l e l
groups of c i r c u i t s 69
5.2.1 Model assumptions 69 5.2.2 Mathematical equations 67 5.3 Distributing the mass flows of refrigerant 71
5.3.1 Model for the distribution of the refrigerant 72
5.4 Pressure drop 73 5.4.1 Correlations for the pressure drop 73
5.4.2 Correlations for the pressure drop in the two
phase flow according to Brauer 74 5.4.3 Influence of the evaporation on the pressure drop 75
5.4.4 Influence of the bends on the pressure drop 75 5.4.5 Influence of the heights of the circuits 76 5.4.6 Influence of a change in flow through area 76
5.4.7 Influence of the valves 77 5.5 Steady state model for the distribution of the refrigerant 77
5.5.1 Modelling the two circuit water chiller test stand
for the validation of the distribution model 77 5.5.2 Influence of the subcooling on the refrigerant 79
5.6 About using Brauers correlation 80 5.7 Modelling the distributor in the dynamic evaporator model 81
5.8 Modelling the air cooler 83 5.8.1 Configuration of the pipes 83 5.8.2 The fins of the air cooler 84 5.8.3 Heat transfer air side 85 5.8.4 Heat leakage through the fins 85
5.9 Correlations refrigerant side 87 5.-9.1 Pressure drop refrigerant side 87 5.9.2 Heat transfer evaporation region 88 5.10 Influence of the collector on the refrigerant flow to
the suction pipe 89 5.11 Interpretation of measuremenst on the distributor
5.11.2 About the position of the distributor 94 5.11.3 Influence of the distribution of the refrigerant 95
5.12 Measurements on the distribution of the air flow 96
5.12.1 Distribution of the air flow 96 5.12.2 Influence of the distribution of the air flow 97
5.12.3 Frosted cooler 98 Chapter 6 Validation of the air cooler model
6.1 Air cooler test stand 101 6.2 Steady state validation of the air cooler 101
6.2.1 Measurements steadystate conditions 101 6.2.2 Conclusions steady state validation 104 6.3 Dynamic validation of the air cooler 104 6.3.1 Measurements dynamic conditions 104 6.3.2 Conclusions dynamic validation 108 Chapter 7 Using the steady state implementation
7.1 Start program 109 7.2 Separate steady state system 109
7-3 Examples of using the steady state program 110 7.4 Simulation results with a TEV with integrating action 111
7.5 Simulation results without integrating action of the TEV 113 7.6 About using a constant minimum length of the superheat
region as optimal setpoint to control an expansion valve 115
7.7 Influence of the superheat on the COP 116
7.8 Conclusions 118 Chapter 8 Using the dynamic model
1 Liquid sensing expansion valve 119 1.1 Working of the expansion valve 119 1.2 Liquid sensing expansion valve 120
1.3 Simulation LSEV and TEV 121 1.4 Conclusions relatinf to the LSEV 121
2 Time constant of the bulb 121 Chapter 9 Conclusions
9.1 Modelling 127 9.1.1 About the model of the TEV 128
9.1.2 About the distribution of the refrigerant 129
9.1.3 About the transferred heat 130 9.2 Influences on the stability 131 9.3 Ability and usefulness of the model 133
9.4 Control of the superheat 134 Appendix Compressor and condenser
A.1 Modelling of other components of the cycle 135
A.2 Compressor model 135 A.2.1 Open compressor 135
A.3.1 Heat t r a n s f e r a r e a s 139 A.3.2 Dynamic e q u a t i o n s of the c o n d e n s e r 140
A.4 Implemented v e r s i o n s of t h e e v a p o r a t o r model 142
Nomenclature 143 Literature 145
Samenvatting Doel en onderwerp van het onderzoek.
Het voornaamste doel van het verrichte onderzoek is geweest het verkrijgen van een hulpmiddel dat bij het ontwerpen en het optimalizeren van een compressie-koelmachine gebruikt kan worden, speciaal met het oog op het dynamische gedrag. Dit hulpmiddel is een computermodel geworden.
Gekozen is voor de ontwikkeling van mathematische modellen van componenten van koelsystemen, die geschikt zijn om met behulp van de computer het
verloop van de relevante procesvariabelen in het tijdsdomein te voorspellen. Omdat slechts een geringe hoeveelheid informatie uit de literatuur direct toe te passen was en ook uitgebreid experimenteel onderzoek noodzakelijk was om betrouwbare modellen te verkrijgen, is de studie beperkt tot de volgende basiscomponenten van een compressie-koelmachine of -warmtepomp:
- een zuigercompressor,
- een horizontale shell-and-tube watergekoelde condensor, - een smoororgaan,
- een droge verdamper met directe inspuiting.
De meeste aandacht werd hierbij geschonken aan de verdamper ofwel de koeler, met de bijbehorende regelkring voor de voeding. Veel problemen met
betrekking tot de dynamica blijken namelijk hieruit voort te vloeien,
terwijl het precieze mechanisme van de instabiliteit van deze regelkring tot nog toe onbekend was. Daarom is er met een literatuurstudie gepoogd een zo compleet mogelijk overzicht te verkrijgen van de elementaire eigenschappen van de verdamper en het smoororgaan, zodat dit in modellen verwerkt kon worden. Aanvullend waren hierbij uitgebreide metingen aan deze componenten van de cyclus noodzakelijk.
De hieruit resulterende eerste versies van de modellen werden verbeterd en verfijnd naar aanleiding van experimenten aan twee specifieke technische uitvoeringen: een 2 kW waterkoeler en een 10 kW luchtkoeler, beide met
gebruik van R-12 als koudemiddel, en met een thermostatisch expansie ventiel (TEV) als smoororgaan.
Bij het modelleren van de, pijpvormige, verdamper is speciale aandacht geschonken aan de mogelijkheid van de aanwezigheid van onverdampt vloeibaar koudemiddel in de oververhitte damp aan de uitlaat van de verdamper, en het storende effect hiervan op de stabiliteit van de regelkring. Met betrekking tot dit verschijnsel is ook een verdampermodel ontwikkeld met parallelle koudemiddelcircuits, die zich onderling verschillend kunnen gedragen. De testopstelling van de waterkoeler is met het oog hierop aangepast, en gebruikt voor bestudering van het mechanisme van de verdeling van het koudemiddel over deze parallelle verdampercircuits. De luchtkoeler was, zoals gebruikelijk, uit parallele circuits opgebouwd.
Modelvorming.
Behalve een dynamische versie van de modellen van de te simuleren processen werd er ook een statische versie ontwikkeld. Hiermee was het mogelijk
statische startwaarden te berekenen voor de procesvariabelen, als
uitgangspunt voor dynamische simulatie. Aanvankelijk werd zo'n startmodel niet nodig geacht omdat ook met ramingen kon worden gewerkt. Onder gegeven stationaire condities moet immers de dynamische simulatie in de statische condities uitmonden. Een afzonderlijk statisch model bleek echter bijzonder nuttig om rekentijd bij dynamische simulaties te besparen. Bovendien kan het zelfstandig gebruikt worden als hulpmiddel bij het ontwerpen van
Uit de in de literatuur gevonden verdampermodellen die op lineaire regeltheorie berusten, kan geconcludeerd worden dat lineaire modellen aantrekkelijk zijn door de mogelijkheid, in het frequentiedomein, een goed overzicht te verkrijgen van het dynamische gedrag van het te onderzoeken, gelineariseerde, systeem. Deze methode is echter minder aantrekkelijk bij een apparaat als de gekozen droge verdamper, waarbij het nog niet bekend is welke van de, meestal niet lineaire, eigenschappen werkelijk van belang zijn. Daarom is de nadruk gelegd op het ontwikkelen van structuurmodellen waarin, in het tijdsdomein, de noodzakelijke niet-lineaire eigenschappen voor de simulatie van het dynamische gedrag opgenomen kunnen worden. De voornaamste modelveronderstellingen zijn:
- Het model voor de toestand van het koudemiddel in de verdamper is gesplitst in deelsystemen: - de damp en de vloeistof in het
verdampingsgebied en - de damp in het oververhitings-gebied. Voor de simulatie van de commerciële luchtkoeler (Helpman LEX-18), waarin het
oververhittingsgebied uit twee ruimtelijk gescheiden delen kan bestaan, werd deze mogelijkheid tot splitsen ook in het model doorgevoerd. Afhankelijk van de hoeveelheid vloeibaar koudemiddel in het verdampermodel kan de grens tussen verdampingsgebied en oververhittingsgebied variëren, zodat de volumina (en dus de pijplengten) van beide gebieden variabel zijn. Dit is weer van invloed op de grootte van de warmteoverdragende oppervlakken en de
drukval.
De toestand van het koudemiddel wordt weliswaar voor de energie- en voor de massabalans ais homogeen beschouwd, maar een aantal effecten die niet in een homogeen model weer te geven zijn, worden daarnaast in rekening gebracht. De belangrijkste zijn: - de drukval in beide delen, - de temperatuurgradiënt in het oververhittingsgebied, i.v.m. de warmteoverdracht, - de aanwezigheid van onverdampt koudemiddel in het oververhittingsdeel.
De metalen constructie is analoog met het koudemiddel in, twee of drie, delen van variabele lengte gesplitst. Daarnaast is er nog een dynamisch model van de zuigleiding waarop de sensor van het TEV is bevestigd.
- Een relatief eenvoudig mechanisme, dat voornamelijk op empirie berust, draagt zorg voor het verrekenen van de invloed van de looptijden in het verdampingsgebied in het model. Plaatsafhankelijke beschrijving van de hier optredende processen zou kennis eisen van de tweefasen-stroming, in het bijzonder van de locale void-fraction of van de locale slipfactor. Het gecompliceerde karakter van deze stroming, speciaal gedurende de
niet-stationaire condities, verhindert de berekening van een meer gedetailleerde theoretische beschrijving van het verdampingsgebied.
Daarnaast dienen, in zowel de statische als de dynamische versie, de
correlaties voor drukval en warmteoverdracht gecorrigeerd te worden voor de specifieke configuratie van de gesimuleerde verdampers, omdat de in de literatuur beschikbare correlaties ontwikkeld zijn aan de hand van metingen aan specifieke, rechte pijpen. Metingen zijn daarom noodzakelijk, niet alleen wanneer het verdampingsgebied als een blok is gemodelleerd maar ook wanneer het, ondanks het hierboven genoemde bezwaar van de onbekendheid met de stromingspatronen, in meer delen opgedeeld zou zijn.
- Voor het oververhittingsgebied is de massabalans van het koudemiddel beschouwd als quasi statisch wegens de geringe massa van de damp.
De energiebalans houdt rekening met plaats-affiankelijke condities en wordt hier voorgesteld door de analytische oplossing van de partiële
differentiaalvergelijking die dit gebied beschrijft voor statische
condities. Voor de dynamische effecten is aan deze oplossing een correctie toegevoegd. Deze correctie is, voor alle invloeds-grootheden, uitgevoerd
volgens de dynamische analytische oplossing van de vergelijking voor het geval dat de verdampingstemperatuur de invloedsgrootheid zou zijn.
- De bij het onderzoek gebruikte smoororganen waren thermostatische
expansieventielen (TEV), hetgeen de gebruikelijke keuze is in combinatie met een droge verdamper. De modellering van een smoororgaan van dit type vereist specifieke empirische gegevens, die niet verkregen kunnen worden uit
catalogi van commercieel beschikbare componenten. Daarom waren uitgebreide metingen van het ventielhuis, de temperatuurvoeler en het smoorproces noodzakelijk, om gegevens te verkrijgen voor verscheidene werkgebieden. Het dynamische gedrag van het TEV, dat alleen van de temperatuursensor
afhankelijk is, bleek beschreven te kunnen worden met een eerste-ordesysteem.
Het model kan aangepast worden voor het gebruik van andere typen sensoren. Zo is in deze studie het model van een "ideale" sensor beproefd, die niet gebaseerd is op de oververhitting van de damp, maar op de kwaliteit ervan
(het massa aandeel vloeistof in de koudemiddelstroom). Aangetoond is dat de instelling van het smoororgaan dan geen correcties nodig heeft bij
verandering van het werkgebied, zoals bij een TEV wel het geval is. Een dergelijke sensor is echter nog niet beschikbaar.
- De compressor kon beschreven worden als quasi-statisch omdat de snelheid waarmee de processen hier gepaard gaan, van een hogere orde van grootte is dan die bij de overige processen van een koelcyclus.
Om bij de validatiemetingen aan de verdamper de invloeden van buitenaf te verminderen, werd de verdamper ten opzichte van zulke invloeden zoveel mogelijk afgeschermd. Zo waren de eigenschappen van de compressor niet van belang vanwege een tussen de compressor en de verdamper geplaatste
restrictie, waarin het koudemiddel met superkritische snelheid stroomde. De procescondities na deze restrictie hadden daarom geen invloed op de grootte van de massastroom koudemiddel die de verdamper verliet.
- Om de verdamper ook aan de intrede van zijn omgeving af te schermen, werd de condensor gedurende de validatiemetingen van de verdamper op constante druk geregeld, terwijl de onderkoeling met een aparte warmte-wisselaar werd geregeld. De condensor kon daardoor aanvankelijk op eenvoudige wijze
beschreven worden, namelijk door het TEV van een koudemiddelstroom bij constante procescondities te voorzien.
Bij de latere modellering van de condensor zijn, evenals bij die van de verdamper, de massa's in twee gebieden gesplitst.
- De validatiemetingen werden aan een niet berijpende luchtkoeler verricht, om de gecompliceerde processen bij zulk een berijping buiten beschouwing te kunnen laten. Om de berijping te voorkomen werd met een tweede luchtkoeler de lucht in de koelruimte gedroogd.
- De eigenschappen van het koudemiddel (R-12) zijn, vanwege de rekentijd bij een dynamische simulatie, beschreven in het werkgebied met behulp van
polynomen volgens getabelleerde waarden uit de literatuur ([A2] en [V1]), in plaats van gebruik te maken van standaard pakketten van stofgegevens.
Summary Purpose and scope of the investigation.
The main purpose of the investigation was to obtain a tool for the design and the optimization of the compressor refrigeration cycle, especially with respect to the dynamic behaviour. This tool has become a computer model. The way chosen has been the development of mathematical models of the
components of a refrigerating system, suitable to predict with the computer the behaviour of the relevant process variables in the time domain.
As only little material could be extracted directly from the literature and extensive experimental validation was envisaged to produce reliable models, this study was limited to the following basic components of a compression refrigerator or heat pump:
- a reciprocating compressor,
- a horizontal shell and tube water cooled condenser, - an expansion valve, and
- a direct feed dry evaporator.
Most attention was paid to the dry evaporator, or the cooler, and its liquid feed control loop. Many problems concerning the dynamic behaviour are
originated here while the exact mechanism of for instance the instability of this control loop was still unknown. Consequently, completeness was aimed at in the study of the literature in order to find basic data for the
construction of the evaporator and expansion valve models. Also the own measurements on specimens to obtain the necessary empirical information had to be extensive.
The resulting first versions of the models have been improved and refined with the aid of experiments on two specific cases: a 2 kW water chiller and a 10 kW air cooler, both using R-12 as refrigerant, and with a thermostatic expansion valve (TEV) as expansion device.
With the modelling of the, tubular, evaporator special attention has been paid to the possibility of the presence of liquid refrigerant in the superheated vapour at the outlet of the evaporator, and to the possible negative effect of this on the stability of the control loop. With respect to this phenomenon an evaporator model with parallel refrigerant circuits has been developed also. The water chiller test stand has, originally with a single refrigerant circuit, has been re-arranged to study the processes of the distribution of the refrigerant. The air cooler was, as usual,
constructed with parallel circuits. Description of the model:
In addition to the dynamic version of the models of the processes to be simulated, a steady state version has been developed. With this version it is possible to calculate values of the process variables in steady
conditions, to use for the start of a dynamic simulation. First such a start program was not thought to be necessary because starting a dynamic
simulation with estimated values would result in steady conditions too. A separate program however proved to be useful to save calculation time for dynamic simulations. Moreover it can be used separately as a useful tool in design practice.
From the literature relating to evaporator models based on linear control theory, it could be concluded that the linear approach is attractive by its possibility, using the frequency domain, to obtain a good insight in the dynamic properties of a linearized system. This method however is less attractive in the case of the chosen dry evaporator, in which case it was
unknown which of the, mostly non linear characteristics, were really of influence. Therefore the stress was laid on building structure based models which non-linear phenomena take place. It was a priori unknown which of
these phenomena had to be taken into account. Therefore it was decided to make use of structure based models which could contain the inevitably non linear equations for the simulation of the dynamic behaviour of specific equipment in the time domain.
The most important model assumptions are:
- The model describing the refrigerant in the evaporator model has been split up in lumps: the evaporation region and the superheat region. For the simulation of the commercial air cooler (a Helpman LEX-18), where the
superheat region could consist of two separated parts due to the special construction, the possibility of an extra superheat region has been modelled. According to the amount of liquid mass in the evaporator, the boundary of these regions could vary continuously and so could the volumes and the pipe lengthes of these regions. This influences the size of the heat transfer areas and the pressure drop.
The refrigerant has been assumed homogeneous for the energy and the mass balance, but some effects which cannot be described with a homogeneous model are described separately: - the pressure drop in both parts, - the
temperature gradient in the superheat region, because of the heat transfer, and - the presence of unevaporated refrigerant in the superheat area.
The metal construction was split up in lumps like the refrigerant regions in two or three parts. Also the suction pipe on which the sensor of the TEV has been clamped, has been modelled dynamically.
- A rather simple mechanism, mainly based upon empirics, corrects the model for the influence of the transportation times in the evaporation region. A place dependent description of the local processes would require knowledge of the two phase flow, specially of the local void fraction or of the local slip factor. The complex character of these processes, especially during non steady conditions, prevents a more detailed theoretical description of the evaporation region.
Besides, the influence of the pressure drop and the heat transfer have to be calculated according to correlations which are corrected for the evaporator configuration to be described, in both the steady and the dynamic
implementation. This is necessary because the correlations in literature are based upon measurements on specific, straight tubes. Measurements will be necessary, not only in the case that the evaporation region has been modelled as one lump, but even with a detailed local description.
- The mass balance of the refrigerant in the superheat region has been regarded as quasi static, because of the small mass of the vapour.
The energy balance here accounts for a distributed model and is represented by the steady state solution of the partial differential equation which describes this area for the steady conditions. A correction for the dynamical effects has been added to this solution, for all influencing
parameters, according to the analytical dynamic solution for the case of the evaporation temperature as input parameter.
- The expansion device model has been worked out for the usual type of device in combination with a dry evaporator, the thermostatic expansion valve (TEV). The modelling of throttle valves of this type requires specific empirical data which cannot be extracted from catalogue presentations of commercially available valves. Therefore, extensive measurements of the properties of the valve body, the temperature sensor bulb and the throttle process were required to acquire data for simulation at different operating ranges. It was found that the dynamic behaviour of the TEV, which va3
concentrated in its feeler bulb, could be represented by a first order system.
The model can be converted to make use of other sensor types. In this study an "ideal" sensor, not based on vapour superheat, but on mixture quality
(liquid/total mass flow ratio) has been worked out. It could be demonstrated that such a device does not require any pre-adjustment as necessary for the TEV. Such a device is not yet available.
- The compressor could be described as quasi static because of the higher order of speed of its processes compared with those of the other components of the refrigerating cycle.
To minimize the influences from outside during the validation measurements of the evaporator, the evaporator was screened from them as much as
possible. The influence of the compressor properties on the evaporator were eliminated by placing a restriction with supercritical flow in the suction line of the compressor. The process conditions after this restriction had for this reason no influence on the amount of mass flow which left the evaporator.
- The condenser was during the validation measurements for the evaporator controlled at a constant pressure, to screen off the evaporator also at its entrance, while the subcooling was controlled with a separate heat
exchanger. Therefore the condenser could be modelled in a simple way, supplying the TEV with a refrigerant flow of constant process conditions. Later on with the modelling of the condenser, the condenser was split up into two regions, like the evaporator.
- The validation measurements were held with a non-frosted cooler, to avoid the complex phenomena which would result from frost formation. To prevent the frosting, the air in the cooling room was dried with a second cooler. i
- The properties of the refrigerant (R-12) were, because of the calculation time for a dynamic simulation, represented on the working range by polynoms which were based upon tabulated properties in literature ([A2] and [V1 ]), instead of using standard libraries.
Conclusions and recommendations
The developed model was able to predict stable as well as instable operation of the evaporator/expansion valve loop according to mechanisms which agreed with the validation measurements.
The correlations in the model for the two phase flow,phenomena (heat
transfer, pressure drop, void fraction) have been corrected for the specific geometry of the test evaporators and expansion valves according to own
measurements. This proved to be necessary in order to obtain values accurate enough for the purpose of modelling. With the corrections for the two phase flow correlations the model is able to describe an evaporator and TEV
quantitatively rather accurate. Without these corrections only a qualitative simulation is possible.
The model can be useful to study the influences of different control methods as has been done in this study. Also it can give an understanding of the complex processes and the importance of the influencing parameters and process conditions.
With respect to the processes and parameters it was found that:
- even a TEV which suits well to the evaporator dimensions, cannot work stable at low superheat conditions when waves of unevaporated liquid
refrigerant leave the evaporator, because of the disturbing effect of this liquid on the measurement of the superheat temperature.
- the resulting description of the region of heat loads where stability exists agrees, according to this mechanism, with the Minimum Stable Superheat lines as proposed by Huelle already in 1967.
- the results obtained do not support the often suggested rule "the lower the superheat, the better the COP". A superheat lower than the one where instability starts will normally not result in a better COP of the
refrigerating cycle, sometimes even in a worse one.
- the dominating influence of the liquid at the outlet on the stability of the evaporator with the expansion valve makes that their steady state characteristics have to be described carefully. A good steady model would be able to produce the necessary information on this subject.
- the combination of a special TEV and evaporator alone is not useful. A good choice of the TEV and its setting is not possible without information about the process conditions which can be expected.
- stability cannot be guaranteed by only influencing the time constant of the feeler bulb of the TEV. With a slow bulb the oscillations during
instability of the mass flow through the TEV will be smaller, but those of the temperature at the evaporator outlet might increase.
- a slower bulb, which is sometimes even recommended in the literature for the case of stability, will also result in a lower protection against overflow of liquid refrigerant out of the evaporator.
- stability and optimal setpoint of the expansion valve could be obtained by using another signal than the superheat for the control of the valve. A signal based upon the presence of liquid refrigerant at the outlet could be able to guarantee safety as well as a good COP.
- a very low subcooling temperature can have under special conditions a negative effect on the stability of the system. The superheat at which liquid starts to leave the evaporator increases when a too low quantity of flash gas results in a low pressure drop in the distributor pipes. The then relatively more important influence of the gravity makes the distribution of the refrigerant uneven.
Both the evaporator and the TEV showed a strong non linear behaviour. For the modelling a combination of theory and empirics was necessary. The
problems encountered with the modelling were mainly due to the complexity of the two phase flow processes:
- the available correlations for heat transfer and pressure drop in the evaporation region are not capable to be used for the aim of modelling. Being developed according to measurements on straight pipes their
inaccuracy of some 20% will increase for the case of more complicated configurations.
- the slip factor and consequently the void fraction cannot be calculated in the case of short horizontal pipes connected with U-bends.
- the oscillations of the transition point or the distance necessary for evaporation of liquid waves which left the evaporation region are the result of complex flow phenomena in the two phase flow region.
- transient process conditions even increase the complexity of the above mentioned problems.
- the throttling of the refrigerant is influenced by the formation of flash gas and by metastability.
- the influence of the position of the distributor on the distribution of the refrigerant along the parallel pipes of the evaporator is due to the inhomogeneity of the throttled refrigerant.
Also the time available for a dynamic simulation has to be regarded. A model therefore should be as simple as possible, without losing accuracy.
Remarkable is that the non ability of describing the two phase flow
processes has in a way a positive effect on the calculation time, because an ever increasing precision of modelling by dividing the model in smaller lumps will be without influence on the reliability of the model.
The reported corrections for the correlations for the two phase flow
phenomena and the heat transfer coefficients of the water side in the water chiller according to measurements on the test stands make for each new configuration to be described new measurements necessary. For small changes of the configuration however the, changed, model could still be able to supply good results.
Further investigation on the subject of modelling of the refrigerating cycle should be:
- a better look at the condenser. The model has to be checked with validation measurements and probably changed or extended.
- the modelling of other versions of the components of the refrigeration cycle. For instance a wet evaporator or an air cooled condenser.
- the implementation and validation of series of configurations of
components, for instance dry evaporator air coolers, to gain information about the necessary fit factors for the model as function of the
configuration.
- the extension of the air cooler model with condensation or even frost formation on the air side.
- the use of the produced cold should be studied. Based upon the developed steady state model of the evaporator the dynamic behaviour of the cold store room and the products stored in it can be modelled for instance.
Introduction
1.1 Scope of the Investigation
The subject of this study is the dynamic behaviour of a compression refrigeration cycle.
A refrigerating circuit in its most simple form consists of a compressor, a condenser, a throttle device and an evaporator. Vapour refrigerant is compressed, cooled, and condensed. This liquid is throttled, normally with the formation of flash gas. The temperature of the refrigerant decreases due to the energy necessary for this partial evaporation. At this lower
temperature the refrigerant can receive energy from the surroundings, which will be cooled. Receiving this energy, the refrigerant evaporates and can even be superheated before it is compressed again.
Models of a refrigeration cycle that can be implemented on a computer can be used to simulate the behaviour of such a cycle. Optimization then will be possible because of the amount of influencing parameters that can be taken into account and the speed of the calculation that can be realized.
Steady state simulation models were already common use as an assist in design and optimization of installations and components . Dynamic models which can be used for the study of system or component behaviour during changes of operating conditions have been developed too on several places, but were not yet found able to be used in practice. Partly this was due toJ the tractability of the models, and partly to the uncertainty about the results produced by these models.
With the intention to develop a model that could really be used as
instrument, a project was started at the Delft University of Technology, which was made possible by financial support of the "Stichting Technische Wetenschappen". As part of this cooperation a "users committee" from
representatives of Dutch industry had followed the progress and the results of the research, and supplied useful suggestions regarding the scope of the investigation and the subjects to be covered.
Condenser l " ^ ^ ^ ^ ™ ^ " ™ ^ ^ ^ ™ ™ ^ ™ ™ " " * I compressor
figure 1.1 Compression
refrigeration cycle consisting of - a compressor,
- a condenser,
- a thermostatic expansion valve, - an evaporator.
TEV
Evaporator
1.2 Literature
In the field of refrigeration engineering the use of mathematical models has not yet found the wide application which can be observed in some other
fields of technology, such as electrical and control engineering, and
engineering mechanics. However, in the last two decades also in the field of refrigeration the literature relating to mathematical models and computer simulation is growing steadily. In a literature survey made in 1983 CT1] concerning refrigeration machinery and heat pumps, a number of 61 articles relating to modelling and simulation was made for the decade 1971-1981; for the two years following, 1882-1983, 160 articles were counted. Only a small number of the publications could be classified as related to dynamic models, respectively 7 for the decade and 13 for the two following years. The
remaining publications were related either to steady state simulation of complete systems or components and to specific models for the simulation of the cycle in reciprocating compressors.
The literature on dynamical modelling, as far as relevant for this thesis, will be reviewed below. Some of the material present in the literature could be used in this investigation, but none of the models was considered to be of such a nature that it could be used as a reliable basis for the model envisioned in this study: a dry evaporator interacting with an expansion device. Therefore in this thesis models have been newly developed starting from the basic physic equations.
In the following survey specially the evaporator combined with a
thermostatic expansion valve (TEV) will be envisioned, because here most problems concerning the dynamics are located.
A literature study has to make clear which methods of modelling could be used, and which problems and possibilities can be expected.
Chi and Didion [C5] described the transient behaviour of an air to air heat pump, where the components are described in lumped form. Both evaporator and condenser can consist of a series of elements. The TEV is modelled as having a linear relationship between the valve opening and the superheat pressure. The time step for the calculation process was 0.005 seconds. The model was reported to be in accordance with test results in the laboratory.
James [J1] mentions the lack of information of flow regimes in evaporators but good results could be obtained by methods used before in steam
generators. The evaporator can be divided in sections. Using three sections for the refrigerant and three for the medium to be cooled, also representing the thermal inertia of the metal, gave reasonable results for a liquid
chiller but not for an air cooler. The TEV can be approximated if no interest in the stability of the evaporator-expansion valve control loop exists. The mass flow through the valve then can be described as a linear function of the superheat with a first order dynamical response, in case the situation does not change much from the working point.
Dhar and Soedel [D2] published one of the first models for the transient behaviour of refrigeration systems. The presentation is more good looking than making the model clear. The evaporator consists of two refrigerant lumps with heat transfer between the saturated and the superheated
refrigerant. The model needs some empirical obtained data, as heat transfer coefficients and liquid flow out of the evaporator in case of sudden
pressure drop. The TEV is extensive modelled for the case that vapour flows through it, but poor for the case of liquid. The flow through area is
Hargreaves and James [H1] modelled a chilled water plant with a thermodynamical model. A transfer function model would depend on good measured data for its accuracy, while because of the non linearity of the system any given transfer function would only be valid on a small operating range. The use of variable time constants in the model could only be done safe when a full understanding of the non linearities exists.
The evaporator is like the one reported by James, but because the dynamics of the evaporator are much faster than those of the remaining system, two zones for the refrigerant and one for the chilled water were sufficient. The two lumps do not distinguish between the evaporation and the superheat
region. The heat transfer to the refrigerant is a fitted formula based upon experimental data obtained from the evaporators manufacturers. The TEV has a mass flow based upon its known capacity, using a square root equation. The time dependency of the TEV is modelled with a first order system with a time constant of 5 seconds. The dynamics of the water chilling system were well according to measured tests, but more extensive tests on the dynamics after load changes were still to be held.
Najork et al [N2] simulated a refrigeration cycle on an analog computer, using for each component a series of boxes. The non linear static
characteristics and the working point dependent time constants were derived from extensive measurements. The interdependence of the output values and the closed circuit made appropriate experimental manipulation necessary. The dependence of the evaporator on an increase in the mass flow had been
studied before with a separate model, for constant pressure condition. A dead time of 2 seconds was measured, but not used in the model. Extra tests made clear that the humidity from the air on the evaporator acted as an additional heat accumulator, and that a higher concentration of oil in the evaporator did not have a measurable influence on the dynamic behaviour. The TEV is modelled rather extensive, with different influences of process
conditions, based upon measurements.
Najork [N3] investigated also the possibilities to improve the stability of the dynamic behaviour of an evaporator with a thermostatic expansion valve with a block model on the analog computer.
De Bruijn, van der Jagt and Machielsen [B7] too implemented the model of. the refrigerating cycle on an analog computer. In order to see how the linearization of the thermodynamical relations around their working point affected the general properties of the original model, a back up model was implemented on a digital computer. When carefully applied, the
simplifications did not seem to have much influence.
The evaporator is modelled as one evaporation and three superheat sections. The TEV is modelled according to capacities from the catalogue, with a dynamical response of the bulb based on the bulb wall and filling.
Yasuda, Machielsen and Touber [Y2] used an evaporator model with two zones for the refrigerant, a two phase and a superheat zone, making use of a variable boundary. The metal of the evaporator wall can be divided into small subdivisions. The TEV was statically modelled according to capacity data from the catalogue, the dynamic behaviour of the bulb was represented with two non linear differential equations. The model was reported to be in good agreement with validation results, statically as well as dynamically.
Hoekstra [IH1] compared a lumped model of the same dry evaporator for a quasi static and a, one lumped, dynamic modelled superheat region. While the simulation results were more or less the same, the calculation time was reduced with a factor 20 by using the quasi static calculation. A factor 10 because of the smaller time constants in the dynamic equations describing
the region and a factor 2 because the differential equations then could be written in explicit form, with no need for a special solution method. The evaporator model is based on thermodynamic principles, with empirical correlations for pressure drop and heat transfer coefficient in the two phase flow region, based on measurements on the test stand. The model of the TEV is the same as Yasuda used. Validation measurements made clear that more research on the subject was necessary.
Bonte and van Veldhoven [B3], [IB1] modelled the dynamic behaviour of the evaporator after a step change in the compressor capacity, using transfer functions. The steady conditions, in between the dynamics are described, are calculated with a separate physical model. The model of the TEV as presented is not very surveyable because it is already rearranged for the method of sequential substitution used to solve the steady state equations. Enthalpy differences have been used instead of pressure and temperature differences because of available subroutines.
Zorzini, Panozzo and Fornasieri [Z1] made an experimental determination of the transfer functions for an evaporator of a vapour compression
refrigeration circuit. Found was that the measured superheat had a dead time of 3 seconds after an increase in the refrigerant flow to the evaporator, possible due to a transportation lag during which variations in mass inside the evaporator take place.
Josiassen [J6] developed an evaporator as well as a condenser model based upon a one lumped model with liquid and vapour in equilibrium conditions. Both superheating and subcooling are not delt with. The expansion valve is modelled extensive for the case that vapour leaves the condenser, while for liquid only a square root equation is used to describe the mass flow of throttled refrigerant.
A simple extension of models out of literature with the necessary number of partial derivatives with regard to time for the dynamic description of two phase flow processes in an evaporator is reported. Based on the size of the models and the amount of calculation time involved, such extensions were not thought to be useful.
However it is rather questionable if such derivatives of correlations developed for steady conditions are useful in transient conditions. Just using quasi static equations then would at least not give the impression of an ability to threat transient situations correctly.
MacArthurs model [M1] of the transient heat pump behaviour is a pure theoretical, heuristic, investigation, without experimental validation. It has been assumed that knowledge of the momentum balance is not necessary. The evaporator is divided in zones. The flow through the expansion valve was described by a square root equation, but this model will be extended and will increase more complex flow phenomena as choked, sonic, flow conditions.
Krug [K1] used a block model of the evaporator to study the behaviour of the TEV. The transfer functions even included a dead time for the evaporator response, depending on the evaporation temperature.
Cleland [C6] simulated an industrial refrigeration plant under variable load conditions using a model like James and Marshall [J2], however with disregarding the influence of the suction pipe. Later on, after being updated especially for the freezing product part, the model has been validated [C73.
Stolarski, Szarynger and Zak [S7] propose the use of a simplified model of a household compression refrigerator, to use for repeated optimization. A complicated mathematical model is developed, taking into account as much influencing parameters as possible. Linearities, intervals of variation and occurrence of extremes are found by calculations with the complex model.
With approximation of these results the simplified model is set up, and used for iterative calculations on the area checked already before with the
complex model.
Bullock, Wroblewski and Groff [B8] developed a dynamic simulation model for residential air-to-water heat pump systems. The heat pump itself is rather simple, where the heating capacity and the power input are expressed as functions of the outdoor air temperature and the heating system water temperature. Periodic defrosting of the outdoor coil is also included, and dynamic effects are modelled by time constants. The heat pump is part of a bigger system.
Stoecker [S3] studied the stability of the evaporator-expansion valve
control loop with a block diagram, containing three time dependent elements. The system has been envisioned with and without transportation lag in the evaporator dynamics. The assumed time lags were 1 or 2 seconds, but together with Sharon and Mumma [S4], Stoecker made more measurements on this subject. The transportation lag in the reaction of the superheat temperature then, after a step decrease or increase of the mass flow to the evaporator, was found in the order of 8 seconds. The model is reported to be rather
simplified and is posed more as example than as final method.
Domanski and Didion [D5] made a steady model of an air-to-air heat pump with a capillary tube. They used a first principle, or thermodynamical, model instead of one based on a sequence of regression analyses, because of the advantages in understanding the impact of local phenomena on the overall system. Besides, if designed correctly, the amount of input data is far less and easier to obtain. The model has been validated.
Beckey [B1] modelled a vapour compression heat pump. The evaporator and condenser are represented in the same mathematical fashion, being broken up into control volumes with conservation of mass and energy to express the relationship between the bulk parameter values for each volume. The for this method necessary assumption about the refrigerant velocities was a
homogeneity of the flow in each control volume. Each volume has one temperature, the condenser being divided into four, the evaporator into three volumes. Pure empirical correlations for the heat transfer have been used. Some simulations could be compared with measurements, showing a good agreement for the condenser and the evaporator pressure and the compressor mass flow. The pressure drop is neglected except for the throttle process. The mass flow through the expansion valve, modelled as a fixed valve with the mass flow as a square root function of the pressure difference, was not measured.
Rajendran and Pate [R1] modelled a vapour compression refrigeration system for the simulation of the startup transients. The refrigerant in each
component is described as a lumped system with a single node to represent each phase region. Heat transfer correlations from literature are used, and pressure drops are neglected except for the throttle process. The TEV is described well for the case of vapour leaving the condenser, but for liquid or liquid with vapour a simple correlation in used, neglecting any possible influence of metastability. The valve opening is linear with the superheat, and the dynamics of the bulb are represented by a first order system. No validation measurements have been made.
Tree and Weiss [T2] proposed a simple two time constant model for the
dynamic behaviour of residential heat pumps. The black box has as output the temperature change of the indoor coil. The two time constants are brought in relation to the mass of the coil and the time required to transport
refrigerant is supposed to leave the evaporator as liquid, the reaction of the model had to be quicker than under normal conditions.
The constants are obtained by regression fit of experimental data. In some cases the result could be better by using a time delay and/or. a third time constant.
This method is advised for control purposes, when a simulation model could and also should work faster than those normally presented in literature. With respect to this the model of Chen [CM] is taken as example. However, with a calculation time of 12 hours on a CDC 6500 computer for 6 minutes simulation time, Chen's model of a heat exchanger with all properties changing with both time and place, is not a representation of the models presented in literature. Nevertheless the idea of the two time constant approach can be very useful.
Wedekind, Bhatt and Beck [WM] used a mean void fraction model to predict various transient phenomena associated with two-phase evaporating and
condensing flows. Good results are reported in predicting the motion of the mean value of the transition point, or the end point of the evaporation region, and in the mass flow of the superheated vapour after a step change of the mass flow to an evaporator up to 20 % . The model itself has the
advantages of simplicity. The void fraction used is a mean value for the two phase flow region, and is time invariant. Without looking to specific
localized phenomena, the model does not contain empirical constants as such, and handles the two phase flow region as a lumped parameter system. From physical perspective the time invariance is explained with a quick
redistribution of the liquid and the vapour within the two phase region. The current rate of understanding on the redistribution mechanism is rather incomplete, but such a quick reaction should be due to the high vapour velocity instead of due to the liquid. The results are reported to agree well with validation measurements.
However, these measurements show a distinct dead time of a three seconds (also noted in [S4], figure 11 and 12), in contrary with the results of the model. This dead time is not envisioned by the authors in their comparison. Another disadvantage is that the model is only valid for constant
evaporation temperature situations, and needs inputed values of for instance the heat transfer coefficient.
1.3 Conclusions from the literature
The survey of literature made clear that several possibilities exist to model a refrigeration cycle, or components from this cycle.
- A first principle model or thermodynamical model is based on the
fundamental laws and the necessary (semi) empirical correlations for for instance heat transfer coefficients and pressure drop. Such a model can become rather extensive, while the accuracy is limited by the correlations used.
- A regression analyses model is based on measurements of a special test stand on a special working range. With such models for the components, the resulting cycle can be simulated, comparing different components or special situations for the process conditions. The results can be as accurate as the thermodynamical models, but then more input data are needed for the same working range.
- For dynamical situations the thermodynamical models contain the
the total behaviour. Parts that are relatively quick can then be regarded as quasi static, thus avoiding stiffness problems. Simplifications might be necessary to avoid partial differential or even implicit written
differential equations, to avoid long calculation times.
- The black box method in dynamical situations makes use of transfer functions, whose constants also have to be obtained from regression of results of measurement or, in the simple case, of analytical study.
This method should only be used if the processes are fully understood. If not, the results should be checked afterwards experimentally.
The thermodynamic method could be recommended when the working of the
components are not yet fully understood, as a design analysis tool. Also in the case of bigger working ranges, which would require each time new
measurements and new regressions in the case of the regression method it can be better used. Finally the results of changes in a component can be
expected to be better predicted, even when the correlations used had been adjusted for the original configuration.
A model could also been built up as a mix of models according to the above described methods, in order to make optimal use of their possibilities. For instance if the working of a (group of) component(s) is rather complicated but has an almost neglectable or simple influence on the other components. Also if the properties of one special component are investigated, then the other components of the cycle can often be described in a more simple way. This study on the subject of optimization of the dynamic behaviour of
compression refrigeration cycles, had to deal with whether steady conditions can be reached and how they will be reached.
Therefore the subject of stability of the evaporator-TEV control loop had to be examined. Because the problems concerning this subject were not very clear, a thermodynamical model was to be preferred above a black box with regression obtained properties.
This opens also the possibility to use the model for design purposes, when for instance the influence of diameters or fin distances, according to the correlations used in the model, is to be studied.
The interest in the control of the evaporator makes also that a model with a variable boundary could be used better than one divided into fixed parts with one refrigerant temperature per part, because of the better calculation of the temperature of the superheated vapour.
None of the thermodynamical models presented in literature was reported to provide a fully understanding of the dynamic behaviour of a dry evaporator with a TEV. This together with the insight obtained with previous models developed in the Laboratory of Refrigeration and Indoor Climate Control ([B7], [Y2]), made clear that a new model should be developed, able to be extended if necessary according to validation measurements.
With respect to the correlations to be used, it has to be examined if they can directly be used like presented in literature or that each configuration in practice needs its own adjustments. If so, the design purpose of the model for a single component of the refrigerating cycle could be the subject of discussion.
Modelling the evaporator
2.1 Evaporator modelling
With the intention to make a model of a dry expansion evaporator capable of simulating the dynamic behaviour, it was thought to be wise to start with modelling a simple evaporator with good possibilities of measuring and controlling the process conditions. On the other h a n d , the model obtained should be capable of being extended to the model of a more complicated evaporator.
These wishes lead to a single pass liquid cooler as the evaporator to be modelled. The laboratory set-up comprised a block of ten horizontal tubes connected in series by u-bends in a vertical serpentine configuration. Each straight part of evaporator pipe w a s surrounded by an annular liquid
channel, with separate and controllable liquid feed. Without parallel pipes, problems with uneven distribution of the refrigerant and with a second
throttling in the distributor could be avoided. By using an oil separator after the compressor, the oil contents in the refrigerant was less than 0.5% and could be neglected in the mass and energy balances.
To isolate the evaporator from influences from other components, a water cooled condenser was chosen equipped with PID pressure controller and a separate control of the subcooling temperature. A l s o a throttle valve was placed at the outlet of the evaporator in the suction pipe: as long as choked flow was realized the compressor would have no influence o n the evaporator performance.
It was required that the model could predict instable behaviour of the evaporator liquid feed control loop, e.g. hunting. Because there was uncertainty about the importance of several factors influencing the stability, as many factors as practicable were taken into account.
On the other hand, for a model to be used in p r a c t i c e , the calculation time should be kept to a minimum. Therefore a rather simple model concept was chosen, leading to only a small number of differential equations, see figure 2.1 .
figure 2.1 Block diagram of the evaporator, with maas and energy flows. It contains three regions: saturated liquid, saturated vapour and superheated vapour. In the model equations the saturated regions will be taken together.
The model is a lumped o n e . With the saturated liquid and vapour taken together, it consists of two volumes, viz. an evaporation section and a superheating section, with a variable boundary between them.
In earlier research [ Y 1 ] , a multi-lumped model was already abandoned because of the lack of knowledge about the principles governing the two phase flow. Void fraction (the relative part of the volume of t h e vapour in the total two phase f l o w ) , heat transfer and pressure drop a r e influenced by the flow
M u jta. hn i A i
<w . r / M
9
f
V 1' Qv M U 9. 9 go go M1/ "» «Upattern. In steady situations the void fraction has no direct influence, on the contrary with the heat transfer and pressure drop. The exact influences are difficult to predict in the case of short horizontal pipes connected to each other with U-bends. In non steady situations this problem is even bigger because of the then still more complicated two phase flow pattern. Also the void fraction then is of dominating influence on the dynamic behaviour.
Without sufficient knowledge of values of void fraction or slip factor, a detailed modelling would only cost a lot of calculation time, without improving the capabilities of the model. Indeed, simplification of such a model into one consisting of two homogeneous control volumes of refrigerant was found not to have really influence on the quality of the simulation results while the profit on saved calculation time was worthwhile [Y1]. Therefore the basic model was kept simple as much as possible. Later on the model was extended if necessary, depending on the results of validation measurements.
2.2 Model equations
A homogeneous control volume requires thermodynamic equilibrium; liquid can only exist in the presence of saturated and not of superheated vapour. Neglecting the possibility of a subcooled region, the evaporator can be split up into three parts:
- a saturated liquid region, - a saturated vapour region, and - a superheated vapour region.
The mass balances for the refrigerant in these three parts are (figure 2.1): (1) (2) (3) The energy balances are:
d/dt (MlUl) = mi.(hli+0.5 cj+g Zj) + Qw l- Qlv
-~
mi v
(V
0'
5 cL
+ g Zm
)"
Al o
( hl o
+ 0-
5 cl V
g Zo
)<«>
d / d t (M u ) = m . ( h . + 0 . 5 c
2. + g z j + Q + Q, +
v v vi vi vi ° i wv lv
+ ft. (h +0.5 c
2+g z ) - m (h +0.5 c
2+ g z ) (5)
lv v l v ° m vg vo vo o
d/dt (M u ) = m (h +0.5 c2 +g z ) + Q - m (h +0.5 c2 +g z ) (6) g g vg vo vo B o wg go go go 6 oThese 6 equations contain 8 unknown variables : M, , M , M , u, , u , u , A, , and m
1' v' g 1 v' g' lv' vg
The necessity that the evaporator is filled completely with refrigerant gives an extra equation:
d/dt d/dt d/dt (
V
=*li-
mlv"
(
V
= A
vi
+
V
(M ) = m - m g vg go mn lo m vgM,v + M v + M v = V (7)
1 1 v v g g evap
Another equation is the description of the void fraction, which should
follow from the momentum balance of the two phase flow in the evaporation
region:
M v = (M v + M.v.
) a
p(8)
V V V V 1 1 Vf
The set of 8 equations with 8 unknown variables could be resolved, however
simplification is possible. When both the liquid and the vapour are
saturated, the evaporation region is in equilibrium and the temperatures of
liquid and vapour here are the same:
T = T = T (9)
v
i
e
The system will now be written as a set of 6 explicit differential equations
for the variables: M , M , M , A. , T and T . This is desired because
solving the set of equations then will need less calculation time during the
simulation than when they would have been implicitly written.
Substitution of (1), (2), (M) and (9) in (5) gives:
d/dt (T
e) = [Q
w l+Q
w v- C
Vu
±) m
l v +{ ( h
n-
U l> +0.5 < £ + g z ^ m ^
+
" \ r
uv
) +0'
5 cvi
+ g zi
}Ki'
{ ( hlo-
Ul>
+0-
5 CL
+ g Zo
} ftlo"
- {(h - u ) +0.5 c
2+ g z } m 1 / [M,d/dT (u, ) +M d/dT (u )] (10)
g v
g
o vg
J L1 1 v v
Jor:
d / d t (Te) = [ Qw l + Qw y- ( hy-h l) ml v+ pl V ; Ld / d t (Mx) + Pvvyd / d t <MV) + + { ( hv. - hv) +0 . 5 c2. + g z . } mv.+ { ( h ^ - h ^ +0 . 5 c2. + g zj m^-~ { ( hi o - V + 0'5 clo+ g zo} \o~ { (Vhv} +° -5 °g+ g Zo] % ] / / [Md/dT ( u , ) + M d/dT (u )] ( 1 0 ' ) 1 1 v vSubstitution of (2) in (3) gives:
Alv
=V "
Av
+ d / d t (Mv
+ Mg
}(1D
Rewriting of (8) and (7) gives:
V
avf
MlV «'-"vf*
\
) (12)M - (V - M,v -M v ) / v (13)
g evap 1 1 v v g
2.3 Simplifying the model
The dynamics of the refrigerant in the evaporator model have been described
with a set of six explicit written differential equations:
The major problems with the calculation of the heat flow are the coefficient of the heat transfer for the boiling two phase flow and the length of the evaporation region which is related to the mass of thé refrigerant M by the value of the mean void fraction. The heat transfer can vary 20% with values obtained with available correlations, while the void fraction cannot be described at all during non steady conditions. Using the quasi static value of the void fraction will give at least a 10? deviation from the real value. Errors in the calculated heat flow between the refrigerant and the wall will be of a higher order than the maximum possible influence of the work for decreasing or increasing the evaporation volumes. Also these will be of higher order than the influences of the pressure drop, the velocities and the differences in height. Therefore these influences could be neglected in the model
The denominator of equation (10) can be made easier to handle also by changing the terms du/dT to dh/dT. The values of the enthalpy h are better available as a function of the temperature than those of the energy u. The influence of this change is neglectable (in the order of one percent):
d/dT (u) = d/dT (h) - d/dT (p v) (14) The rewritten energy balance of the evaporation region (10) now can be
replaced by equation (15):
d/dt (Te) = [Qe-(hv-h1) ft ] / [M1d/dT(h1) +Myd/dT(hy)] (15) Where h. is the enthalpy of the refrigerant before the throttle device.
Figure 2.2 shows a diagram of the rewritten and simplified model.
T
* figure 2.2 Block diagram of the rewritten s e t of equations of the evaporator model as Implemented in the computer program. A subcooled l i q u i d region i s not handled separate. Only the unknown
parameters from the equations used are marked.
r?
M T v eE7Z
f
M. T j ■*- — 1 e ' *=-$2.4 Extending the model
2.4.1 The dynamic behaviour of the evaporation region
In steady conditions the length of the evaporation region and the
evaporation temperature have such values that the heat flow to the liquid is equal to the amount necessary for complete evaporation.
In dynamic conditions, after for instance an increase of the mass flow of refrigerant, to the evaporator, the evaporation temperature will increase because of two effects (see figure 2.3, case a) :
i
figure 2.3 Measured change of the evaporator pressure after a stepward change of the opening of the expansion valve. In case (a) the opening increased, in case (b) — I 1 1 r— it decreased.
0 \0 60 SO
i [»c.l
Firstly, less liquid will evaporate because of the extra vapour that enters the evaporator, while the amount of vapour that leaves stays the same. Consequently the pressure and the temperature will rise according to equation (15).
Secondly, the amount of extra liquid entering, in'combination with the lower quantity being evaporated, causes an increase of the liquid mass.
Accordingly, the length of the evaporation region will increase, and so the amount of heat transferred to this liquid. Both pressure and temperature, in equilibrium, then will increase a second time.
The, main, effect of the second mechanism will be delayed by the distance which the extra amount of liquid refrigerant has to travel before it reaches the end of the evaporation region and increases the length of this region. The void fraction will be different from the quasi steady value, during the time that this extra mass flow is travelling through the evaporation region. Correlations to describe the void fraction during transient situations are not available, and using a steady state correlation would give an
instantaneous change of the length of the evaporation region after a change of for instance the incoming liquid mass flow. This would give an immediate and too quick reaction of the evaporation temperature, compared with the measurements. To avoid this, a new way to obtain the length of the
evaporation region during non steady situations had to be developed.
Note that such a delay will not appear in the case of a rotational system as reported by Macken [M2]. In such an evaporator the liquid annulus and the vapour core can be envisioned as lumped systems. The transient response of such a lumped evaporator appeared to be quicker than that of a distributed one, where each portion of the two phase flow in the pipe has to communicate with the next. This introduces an extra process and a slower response.
The delay of all influencing parameters on the length of the evaporation region will be different for each parameter. The following assumptions have been made:
(a) The liquid flow to the evaporation region has to travel the whole length of the region before it influences the length of this region.
(b) The vapour flow entering the evaporator has a direct influence on the length of the evaporation region (see paragraph 2.4.2).
(c) The evaporating refrigerant evaporates along the whole length of the evaporation region. So the increase/decrease of thé liquid mass resulting from this changed vapour flow, is spread along the whole evaporation region. Depending on the distance to the outlet, each part of the evaporated flow has to be delayed with a different time;
(d) Liquid mass flow, as described in paragraph 2.4.3, leaving the evaporator has no direct influence on the length of the evaporation region. It would result in, non desired, extra oscillations of the
endpoint of this region. It still has an, indirect, influence by the liquid mass and the void fraction.
(e) The force that brings the two phase flow pattern after a disturbance in accordance again with the quasi static value of the void fraction is thought to work along the whole length of the evaporation region.
These assumptions were used in order to obtain a useful correlation for the calculation of the length of the evaporation region during non steady
conditions. The method is based on the differentiation to the time of the length of the evaporation region
1 = M./{p A (1-a)} (16) e 1 e
d/dt (1 ) - 1/{p A (1-a)} d/dt(M. ) + 1 /(1-a) d/dt(a) - 1 /p d/dt(p) (17)
e e I e e
The first term on the right side of the equal sign in equation (17)
describes the influence of the derivative of the liquid mass;> the second of the void fraction and the third of the density of the liquid. The change of this density per time step however can be neglected. The derivative of the mass can be calculated, with the mass flows to and from the evaporation region. The derivative of the void fraction, on the contrary, cannot be calculated. No correlations are available at all for this term.
The influence of a non homogeneous void fraction in dynamic conditions has therefore been described by using a derivative of the mass of liquid based on delayed mass flows instead of the real mass flows in the first t.erm of the equation. The used influences of the mass flows into and out of the evaporation region, see equation (1), on the length of this region are: 1 - Only the liquid component of the incoming mass flow has an influence on
the length and is delayed with a transportation lag and a first order system (a,b).
2 - The outgoing vapour flow is delayed with a first order system (c). 3 - liquid mass flow leaving the evaporation region has no direct influence
on the length of this region (d).
The values of the delay times and of the time constants of the first order systems have been taken proportional with the length of the evaporation region (see next paragraph).
Now, from the three terms in equation (17), only the first remained. However a second, extra term, is necessary, to describe a mechanism that pushes the void fraction back to its quasi static value after a disturbance. Without this correction the steady state void fraction after transient behaviour would be influenced by these transients instead of only by the new steady conditions. The shape of this extra term is based on the second term in equation (17) and on assumption (e). The length of the evaporation^
