Modelling of a Compression Refrigeration Plant for Fault Simulation

Modelling of a

Compression Refrigeration Plant

for Fault Simulation

the shell-and-tube evaporator: theory

-

2nd Interim Report

-Report DEMO 95/15

H.T. Grimmelius

Preface

This is an interim report, describing a part of the research project "Condition Monitoring and Fault Diagnosis for compression refrigeration plants on board ships", which in turn is a part of the ICMOS (Intelligent Control and MOnitoring Systems) research project of the Department of Marine Technology of the Delft University of Technology in cooperation with Dutch Industry. The ICMOS-project started in 1989, with an emphasis on propulsion (diesel)engines. Since 1991 compression refrigeration plants have also become part of the research field.

In this report, the detailed model for the shell-and-tube evaporator is described. The model is based on process physics, combined with empirical descriptions of complex phenomena such as heat transfer and pressure drop. The necessary properties of the refrigerant and water are approximated with regression equations.

In the context of this research project five other reports have been published:

"Metingen cran een koelinstallatie

(Measurements carried out on a refrigerant plant)"

Report OEMO 95/11: K.T. van der Heiden (in Dutch)

'Modelling of a Compression Refrigeration Plant Ibr Fault Simulation (the reciprocating compressor)"

Report OEMO 94/11: H.T. Grimmelius

"Diagnostic Techniques and Simulation Modelling (focused on compression refrigeration plants)"

Report OEMO 93/19: H.T. Grimmelius

" On(wikkeling van een prototype storings-diagmosesysteem voor compressor koelinstallaties

(Development of a prototype .failure diagnostic system for compression refrigeration plants)"

Report OEMO 92/13: H.T. Grimmelius (inDutch)

"Haalbaarheidsonderzoek gecnitomatiseerde storingsanalyse by compressokoelinstallaties

(Feasibility study on automated failure analysis fin- compression refrigeration plants)"

Report OEMO 92/08: H.T. Grimmehus (in Dutch)

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Introduction

Compression refrigeration plants are widely used for cooling in air conditioning and refrigeration of victuals. Frequently failure of the plant is not recognized before a breakdown occurs. Symptoms that could be used to diagnose a beginning of failure are difficult to discern by operating personnel on site. This lack of diagnostic knowledge is the result of the closed cycle character of the process and the strong dependency of measurable variables on operating conditions. The lack of knowledge is amplified by the high reliability and maintenance free operation of these plants.

Especially on board ships, personnel lacks specialist knowledge and experience because of the wide variety of machine systems that have to be controlled and monitored. Much attention is given to the propulsion system, and its diesel engines. Auxiliary systems. like a compression refrigeration plant, get less attention, but are very important for the operation of the ship. On board experience in controlling, monitoring and fault diagnosis will decrease in the near future with the decreasing of the crew size and with the introduction of Maritime Officers, who have both nautical and technical duties. For these reasons expert systems may play an important role by supporting the engineer in machinery monitoring

and fault diagnosis at an early stage, resulting in improvements in safety, economy and

cargo conditioning on board ships

An expert system for fault diagnosis generally consists of two types of knowledge. First, knowledge concerning the healthy plant behaviour, in order to determine wether or not malfunctioning is occurring. Second, knowledge necessary to determine a probable cause, given a deviation from healthy behaviour. for instance a priori knowledge of the behaviour of the malfunctioning plant. In a prior research project [Grimmelius, 1992; Grimmelius c.s., 199511 this second type of knowledge proved to be the main obstacle in developing a reliable expert system.

This report describes some results of the research currently conducted. The aim of this research is to determine the behaviour of a malfunctioning plant using computer simulation. In a preliminary study it was concluded that for the purpose of simulation of fault introduction, modelling based on process physics was the only suitable technique [Grimmelius, 1993]. The compression refrigeration plant is modelled per component; this report describes the theoretical model for the shell-and-tube evaporator.

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This research is made possible through the cooperation of the Delft University of Technology and Van Buuren-Van Swaay By. Zoetermeer, and is financially supported by the Dutch Foundation for the Coordination of Marine Research CMG.

I would like to thank professor J. Klein Woud and professor H. van der Ree, both of Delft University

of Technology, and G. Been and L.J. van Wees, both of Van Buuren-Van Swaay B.V., for their

continuing support and criticism Throughout this research.

Acknowledgements

Leiden, April, 11995

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Table of Contents

Nomenclature ix

1 Symbols _ix

2 Subscripts xi

3 Prefixes xii

4 Dimensionless Numbers xiii

1

General Modelling Concept

1

Li

Introduction 1

1.2 Goals and requirements 3

1.3 Modelling components 5

1.4 Modelling a compression refrigeration plant 7

1.4.1 Overall characterization of components 7

1.4.2 Choice of in and output variables 8

2 Evaporator Model

11

2.1 Description of the evaporator in the test plant 11

?.? _{General concept and structure 13}

2.3 The evaporator model ₁₅

2.3.1 Choice of modelling concept 15

2.3.2 Shell wall segment ₁₆

2.3.3 Water segment 16

2.3.4 Wall segment 18

2.3.5 Refrigerant segment 1 8

2.4 Parameters in the evaporator model 73

Appendix A

Derivation of Equations for the Evaporator Model

27

A.1 Heat transfer coefficient: chilled water - tubes 27

A.1.1 General calculation concept 77

A.1.2 Calculation of correction factors ₂₉

A.2 Equations for vessels in refrigerant segment 31

A.2.1 Two phase flow region ₃₁

A.2.2 Solving the algebraic loop in the equations for the two phase vessel ₃₄

A.2.3 One phase flow -gas- region ₃₅

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A.3 Equations for resistance parts in refrigerant segment 37

A.3.1 Two phase flow region 37

A.3.2 One phase flow -gas- region 39

A.4 Heat transfer coefficient: tube - refrigerant 40

A.4.I Two phase flow region 40

A.4.2 One phase flow -gas- region 41

Appendix B

Properties of Freon 22 43

Appendix C

References

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Properties of water

53 57

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11

Symbols

Letters

A cross sectional area

specific heat [Me IChil

d diameter

acceleration due to gravity [ms-2 ]

f

factor, defined locally _[

-G mass velocity . [kgm-2 ]

specificenthalpy electric current

heattransfer coefficient; roughness

[ain-J-2]; [m]

K

'factor, definedlocally length,level m mass [kg] n revolutions, number of

_{b [ -]}

p pressure power _[W]

Nomenclature

This is a general list of symbols used throughout this research; it may includesymbols, subscripts or prefixes that are not used in the present report.

lix

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time _[s]

T temperature [K]

specific inner energy [I kg' ]

voltage; volume [V]; [in3

IV velocity (generally of one phase in two phase flow), [ms' ]

W work _[J]

In

mass fraction of more volatile component (quality of two-phaseflow

pistondisplacement ,?l+ m11 [m]

Z real gas factor [

-Greek symbols

a heat transfer coefficient; flow coefficient

polytropic exponent

void fraction (volume equivalent of x:

C flow resistance coefficient

ri efficiency; dynamic viscosity

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temperature isentropic exponent _[ -A thermal conductivity [ Wm' ] specific volume

[m3 kg]

if

pressure ratio ( > I) - I

p

specific weight [kgm' ]

a

surface tension [Nm' ] q5 flow [kgs-1], [Js"' ] 49 angle of rotation [ heat flow

[Wm ]

Q heat _[J]

specific latent heat, radius

[Re

[m]

R gas constant Pkg-1 ]

specific entropy; distance [Jkg-1K-1]; [in]

I - 1 ] rwm-2 - I [ -l[kgrnil V ,./t

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2

Subscripts

general subscripts av averaged value bends clear clearance

comp compressible medium

cr critical value

discharge (in combination with compressor variables)

env environmental variable

eq equivalent value

FCB forced convective boiling

fr friction

inlet, interior dimension leakage, liquid

Iv interface between liquid and vapour phase

mass (in combination with 0: mass flow)

NB nucleate boiling

outlet, exterior dimension

at constant pressure (combined with c) heat (in combination with 0: heat flow)

suction (in combination with compressor variables)

sub subcooling (combined with prefix A); subcooling zone in condenser

sup superheat (combined with prefix A); superheating zone in evaporator

surf surface

at constant volume (combined with c), vapour

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subscripts indicating components

compressor chw chilled water cond condenser crk crank crkc crankcase cw cooling water cyl cylinder

cylw cylinder wall

ev evaporator

exp expansion valve

fd filter/drier element head head of the evaporator

oil oil

of the nth element

plenum (in combination with compressor variables)

pi piping (e.g. Vpi(c-cond) depicts the volume of the piping between compressor and condenser)

pist piston refrigerant

rod piston rod

tu tube

window in baffle plates

3

Prefixes

A difference

6 infinitesimal small difference, partial derivative

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_{Dimensionless Numbers}

pc vi

- P c Pr = Re = v I p Bo =

G r

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Nusselt number; a measure of the ratio of the total heat transfer to conductive heat transfer.

Peclet number; combines the Prandtl and Reynolds numbers (Pe = Pr. Re),

Prandtl number; indicates the specific ratio between energy conveyed through impulse transport and energy conveyed through conduction.

Reynolds number; indicates the ratio between the influence of inertia and the influence of the internal friction on the flow.

Boiling number; gives a measure of indicates the amount of evaporation per unit length.

Nu

a.

in. if

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The literature provides several concepts for modelling a compression refrigeration plant, ranging from detailed descriptions - some partially - based on process physics [Hiller, 1976; Touber, 1976; James, 1980; Yasuda, 1981; Van der Meer, 1987; Hamilton c.s., 1990; Wang, 1991; Conde, 1992] to black-box models [Zorzini, 1975; Peitsman, 1992; Van Galen, 1993]. In a preliminary study [Grimmelius, 1993] it was concluded that for the purpose of simulation of dynamic behaviour and fault introduction, modelling based on process physics was the only suitable approach. An overview of possible applications of models and the requirements in the various fields of application are given in Table 1.1 and Table 1.2,

while Table 1.3 gives an indication of the suitability of modelling based

on process physics for simulation purposes. The tables give information about application, requirements and suitability for two kinds of application: healthy system behaviour and fault symptom prediction; for two time domain demands: off-line and on-line operation. In an off-line simulation the time needed for calculation are not critical, where as in an on-line situation the results have to be available almost simultaneous with the actual measurements. In the next section these general descriptions are translated into _{more detailed} requirements for the modelling in this research.

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General Modelling

Concept

1.1 Introduction

a. Healthy system behaviour b. Fault symptom prediction

1. Off-line simulation Enabling component selection and

system tuning during system design.

Basis for reduced on-line model to predict healthy system behaviour.

Prediction of symptom patterns for various failure modes.

Support for service department.

2. On-line simulation Generation of reference values for

healthy system behaviour.

Failure mode hypotheses testing.

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Table 1.2 Main requirements for, simulation modelling

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source:1Grimmelfus, 19931

...

a. Healthy system behaviour i.b. Fault symptom prediction

'F. Off-line simulation Accuratedescriptionof transient

as well as static behaviour

Easy system building (use of

component models).

Unique representation of

-relevant-physical faults.

Flexible as to what faults can be

introduced.

i

2. Online simulation Fast enough ta follow system

dynamics.

Accurateenoughto enable' the

!required fault detection.

1

A

.Fast enough to make diagnosis

',possible in time for compensation. Flexible as to what faults can be

introduced (if hypotheses testing is to be used).

Table 1.3 Suitability of models based on process physics for simulation. source: Primarelius, 1.99311

a. Healthy system behaviour b. Fault symptom prediction

1. Off-line simulation + + I

+/-1.2 Goals and requirements

The mathematical model to be designed consists of a number of component models, interconnected to represent the refrigerant cycle.

The goals of the model are:

GI - To gain knowledge about the steady-state and dynamic behaviour of compression refrigeration plants.

G2 - To gain knowledge about the steady-state and dynamic behaviour of compression refrigeration plants after the introduction of faults in the plant.

G3 - To enable prediction of steady-state and dynamic plant behaviour of individual plants _{- both in} healthy condition and after the introduction of a fault - based on a limited amount of measured data.

In order to reach these goals the resulting model should have the following _{characteristics::}

Cl - Quantitatively correct description of the steady-state behaviour within the working range of the components, with an accuracy that is the same as could be accomplished with measurements (GI). C2 - Quantitatively correct description of dynamic behaviour in terms of settling times and oscillating

frequencies (G1).

C3 - Qualitatively correct responses to fault conditions, both for dynamic and steady-state behaviour (G2).

C4 - Built-in possibilities for changing of components and refrigerant (G3).

C5 - Possibility of model verification and validation both with measurements and with manufacturer's datasheets (resulting in filament of requirements Cl through C3).

These characteristics result in the following requirements for the overall structure and functional design of themodel:

RI - _{The model should be modular, since it should be possible to replace or add components, within}

the same overall structure, enabling the simulation of the behaviour of different refrigeration plants. This also enables the use of different approaches to the modelling of the components, as long as in and output variables are the same (C4).

R2 - The model should have fixed in and output variables for the_{component models, since only then} it will be possible to create different component models which_{are interchangeable (C4).}

R3 - The model should allow separate parametrization

of

the properties of the refrigerant to

accommodate the use of other refrigerants, since new legislation on CFKs brought about a

intensified search for new media. Future plants therefore could be operated

_{using other}

refrigerants, which changes the plants characteristic behaviour, thus reducing the value of the

experience now accumulated by service personnel. Added knowledge,

_{obtained through}

simulation, will be highly appreciated in these _{cases (C4).}

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R4 - The model should be based on process physics, this makes it possible to describe component behaviour reasonably accurate, especially outside the working range and under changing operating conditions, while enabling physically correct introduction of faults, as shown in the tables in the previous section (C2 & C3).

R5 - The model should be dynamic, since compression refrigeration plants are most often operating under continuously changing operating conditions, and furthermore toenable an increase in the available information for diagnosis without adding more sensors (C2 &C3).

R6 - The model should have built-in possibilities for .failure

mode simulation, because relations

between parameters in a model and a failure mode are not always

clear. Faults should be implemented in the model in a form closely representing their physical appearance (C3).

R7 - The model should supply measurable variables for in

and outputs when ever possible. Thus

temperature should be used instead of for instance enthalpy because the enthalpy cannot be measured directly (C5).

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1.3 Modelling components

In modelling a dynamic system, !it is divided into smaller parts, which are interconnected through energy

flows. The energy flow is given as the product of two separate variables representing a flow of

information:

A tension variable; for 'instance a pressure*, a force" or a voltage.

A flow variable for instance mass flow`, velocity" or electrical current.

The parts are categorized in three main groups, depending on their influence on the energy flow: Inertia elements, in which accumulation of a tension variable takes place, resulting in a value for the corresponding flow variable; for instance a mass (in long thin pipe), a mass" or a self-induction. Storage elements, in which accumulation: of a flow variable takes place, resulting :in a value for the corresponding tension variable; for instance a vessel', a spring" or a

capacitor.

Resistance elements, in which either variable determines the value of the corresponding lone; for

instance a valves., friction" or a resistor.

The direction of the information flow, or the causality, is fixed for both inertia and storage elements. In' nature, processes are governed by integration: an inertia element generates a flow variable, given one or more tension inputs and a storage element generates a tension variable, given one or more flow inputs. For instance, the velocity of an object - an inertia element - is the result of an integration of forces working on it over a period of time, and the initial velocity. Similarly, the reaction force of a spring - a storage element - is the result of the compression, which is an integration of the compression velocity over a period of time, and an initial compression. This fixed direction of information flows implies the impossibility to connect two inertia or storage elements with each other without an intermediate element with 'inverted causality.

This concept of modelling for dynamic systems is most stricktly applied with the Bondgraphpinethod [Karnop c.s., 1975], but the concept is the basis for all models based on process physics.

When focusing on a system 'involving mainly the low velocity flow of a gas or liquid, usually only storage: and resistance elements are present, since, as long as the physical dimensions of the system are not

extremely large, the inertia of the flow can be ignored without introducing large errors. A storage

element in such a system is referred to as a vessel. A vessel is an integrating function for both mass and energy and consists in fact of two interlinked storage elements. Its inputs are in- and outgoing mass flows and the thermodynamic properties of the inflow, The thermodynamic properties of the output are 'usually taken to be those of the contents of the vessel'. This implies that the contents of the vessel is ideally stirred. The vessels in a system determine the pressure levels.

II I In a thermodynamicsystem. In a mechanical system. In an electrical system, 4.A4:0/

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A resistance element is an algebraic function ruling the flow between two storage elements in the system. It determines the mass flow given two pressure levels, but could also be used vice versa, in order to calculate a pressure drop given a mass flow. The resistance elements in a system with only vessels (when neglecting inertia) are always used for mass flow calculation, however. If more than one resistance elements are present between two vessels, an equivalent resistance has to be ascertained in order to calculate the overall mass flow, with which -afterwards-intermediate results for both pressure and mass flow are determined algebraically, if required.

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1.4 Modelling a compression refrigeration plant

1.4.1 Overall characterization of components

For this research, the compression refrigeration plant is divided into five major components. The general

structure is that of the available laboratory plant as

shown in Figure 1.1. Components to be taken into

account are:

1 - A reciprocating compressor, compressing gas

from the low evaporation pressure level to the higher condensation pressure level.

2 -

A condenser, conveying heat from the refrigerant

to the cooling water, resulting in the

condensation of the refrigerant.

3 - A filter/dryer, preventing small -wear- particals

and water to enter the expansion valve.

4 -

An expansion valve, creating the necessary

pressure difference between condenser arid

evaporator and used to control the refrigerating Figure 1.1 Basic components of a compression capacity.

5 An evaporator, conveying heat from the chilled

water to the refrigerant, resulting in the evaporation of the refrigerant

The overall characteristic of a component is translated into one of two basic forms: a vessel or a resistance element, as described in the previous section. Inertia of the gas or liquid flow is neglected. This leads to the following classification:

Compressor: A compressor determines the mass flow given the input condition of the gas (pressure, temperature, void fraction after the evaporator) and the discharge or condenser pressure. It is therefore a resistance element, even though its 'resistance' is negative as the direction of the mass flow is from low pressure to high pressure.

Condenser:

A condenser determines the properties of the liquid refrigerant. A balance is

established between inflowing gas, outflowing liquid, level of the fluid in the

condenser and the temperatures of incoming refrigerant and cooling water. A

condenser as a whole can be described as a vessel, even though the contents are never in a thermodynamic equilibrium.

Filter/drier: A filter/drier is purely a resistance element as long as contamination of the refrigerant with dirt or water is very low.

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Expansion valve An expansion valve regulates the mass flow. It is a resistance element, accordingly., The fact that its resistance changes 'in order to control the superheat does not alterthe

overallcharacter of the valve.

Evaporator: An evaporator determines the thermodynamic properties of the refrigerant, much like, the condenser. It acts as a vessel in the system.

If this characterization is maintained for all modules, integrating causality of the cycle is automatically reached if the resistance of the filter/drier and the expansion valve are joined The character of a module can be changed by adding or removing (line-)resistance. For instance, a detailed model of the compressor will accommodate for several vessels within the compressor, such as the suction and discharge plenum and crankcase. The overall causality, however, is maintained through the addition of resistance elements both on the suction and on the discharge side of the compressor model.

ptl

1.4.2 Choice of in

and output

variables

As stated before, the main objective of the models developed in this research is to gain knowledgeabout

measurable behaviour. For this reason directly measurable variables are chosen to describe the

thermodynamic state of the refrigerant, whenever possible. When describing the thermodynamic state two kinds of quantities of state are distinguished: measurable quantities (pressure, temperatureand

density), and non-measurable quantities (specific entropy, enthalpy and internal energy). Two quantities

have to be measured to know the state of the refrigerant. Pressure and temperature are directly

measurable, and will suffice in all cases, except when both vapour and liquid exist within the same vessel.. In such a saturated two phase environment both temperature and pressure contain the same information

and are no longer independent variables. Usually a special quantity is used to avoid the use of

non-measurable quantities: the void fraction or quality, but this quantity is also 'not measurable and has to be determined using refrigerant properties.

In order to enable development of independent modules, the connecting variables are now defined including their direction, according to the rules describe in the previous section. This is illustrated in Figure 1.2 and Table 1.4.

Table 1.4 Connecting and continuity conditions or plant model' in Figure 11.2_

1

11:

0,,,

= 2: Pco = 13 cond 1:

T,,

= 700

i

1:

4: °In.condo = Okm. I di 3: Pict/ = P tend = icon, I' 3: X mi = X cond 5: :0,,,, . = 6: 5: T,, =

T

5: = )(expo

0 = &L., 7: = Pr.,. 17: To, = T,,,, _117:

3: T,,

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<

Chilled water

IntelW in Report Evaporator Model

Evaporator 5. '1 41,74* '4Expansion ;:yetve 9 Electric power consumption, Lleutd - Ins

Figure 1.2 General conceptual model of the refrigeration plant.

In this general concept, control' 'lines are not considered. In most systems, the expansion valve is controlled through the pressure and temperature after the evaporator. Also capacity control on the compressor, based most often on the chilled water temperature, is frequently present. Depending on the choice of the system 'boundaries for the modules, these controls will be modelled within the _respective component model or will be taken as input variables from the environment.

Compressor Condenser 2 3

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Cooling water Environment

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2.1 Description of the evaporator in the test plant

The evaporator is a heat exchanger of the tube-and-shell type. Refrigerant flows through the tubes in two passes, from the lower half of the inlet piece through the pipes and back to the upper half of the inlet. The water flows in short turns in generally the opposite direction as the refrigerant on the outside of the pipes. The shell is divided in small sections with baffle plates, one horizontal and several vertical as illustrated in Figure 2.1

The main dimensions are:

Water side: Internal diameter: dx.h.i

=210 mm

Diameter across tubes _(1,h.th

= 197 mm

Number of tubes: n ru = 22 (per pass)

Number of tubes in window: n ru. = 6 (per baffle plate)

Baffle plate tube hole diameter: d ha

=20

mm

Baffle plate distance: bile = 50 mm

First section length: _{ba, 1}

= 150 mm

External tube diameter: _dru.ol = 19.1 mm ( = 3/4")

Tube distance vertical': S

=20.5 mm

Tube distance horizontal': ₅₂ = 12 mm

Minimum pipe distance*: = 4.7 mm

Distance between horizontal baffle

plate and first row of tubes _'ha.bar In

The definition of s, , s,and e is illustrated in Appendix A.1.2, Figure A.I.

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Figure 2.1 Tube-and-shell evaporator.

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IM/ . MIMI MI II= =I MN

ril

=WAIL' MB IMI I 1.1111 =I MI MI MINMNIMMIIffir MII=MIIMIIMMII

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tt EM MI MI MIMIMI MI

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Refrigerant side: Internal pipe diameter: dric = 1 7.5 min

Head diameter: "had

= 215 mm

Head depth: = 70 mm

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concept and structure

The evaporator has been characterized as a vessel in the overall model: it determines the pressure level both on the suction side of the compressor and on the discharge side of the expansion valve. This defines the overall causality,

inside the evaporator model

resistance elements are defined, however, to enable modelling of the pressure drop over the evaporator.

The following connecting variables result, as illustrated in Figure 2.2

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Figure 2.2 Evaporator model, highest level.

Entrance side:

input: (provided by expansion valve model) refrigerant mass flow

temperature

quality (void fraction)

output: pressure

The line between expansion valve and evaporator is considered as a part of the expansion valve. Discharge side: input: (provided by compressor model)

refrigerant mass flow

output: pressure

temperature

quality (void fraction; usually zero)

The discharge line between evaporator and compressor is considered as a part of the compressor.

Environment:

input: chilled water entrance temperature chilled water mass flow

environmental temperature output: chilled water outlet temperature

The water flow is considered incompressible, and the specific heat is considered independent of the pressure.

13

-2n 2n-1

2n-2

n+1

C

0

CD

h

-1 2 3

0

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2.3 The evaporator model

2.3.1 Choice ofmodelling concept

The evaporator model is used to predict both steady state and dynamic behaviour when faults occur in the system. Therefore the model should meet the following requirements:

It should describe the accumulation of mass in the evaporator.

It should give estimates of the heat transfer under transient conditions. It should enable calculation of superheat under transient conditions. It should enable introduction of faults in an unambiguous manner.

In general, three approaches to the modelling of a dry evaporator are found in literature. The least complicated models are black-box models, based purely on measured data, The relations between the input and output variables are described with transfer functions [Van Galen, 1993; Zorzini c.s., 1975]. In these models correct introduction of failure modes is difficult because most of the parameters in these transfer functions have no direct physical interpretation. A second category models are the lumped zone models, which give a representation of the evaporator using two or three zones: one zone for the evaporation, one zone for the superheating of the refrigerant gas, and sometimes one for the intermediate 'dry-out zone [Van der Meer, 1987; Bonte c.s., 1983]. The accuracy of these models depends strongly on measurements and empirical equations. The third category models is formed by distributed models [James, 1980; Wang, 1991], which describe the change of the conditions of the refrigerant, the wall, and chilled water or air with differential equations, both in time and place. The set of partial differential equations that results is solved numerically. From a survey of available models, only distributed models were found to be capable of meeting the requirements.

Interim Report Evaporator Model

The distributed model is approximated with a finite element approach is. The evaporator is divided into a

number of elements, with finite dimensions. These

elements are so connected that they represent the flow pattern through the evaporator; see Figure 2.3. In turn,

each element is subdivided into four segments, as

illustrated in Figure 2.4: one segment representing the shell wall,

one representing

the refrigerant, one representing the tube wall, and one representing the chilled water. The necessary equations are derived for

each segment within an element in the following

sections, resulting in a mathematical model and a set of assumptions. The segments are coupled, both with the corresponding segments in the neighbouring elements

and - within one element - with the neighbouring

segments. Since 'place' has thus been eliminated from

Tv,.,, prim xr, MEM MM. (1)h,env-d1W,11 )414

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Om.enw Ochw1,11

Figure 2.4 One element from the evaporator

model.

ICMOS CRP

T U Delft

the equations, it is no longer necessary to solve complex partial differential equations, and only ordinary differential equations (in time) remain.

This approach differs from approaches found in literature for the detailed modelling of evaporators. Wang [1991] for instance develops a set of continuous equations, which are mostly partial differential equations both in time and place, and simplifies these before discretization of the model. Because of the discretization in place at an early stage, the conservation laws derived are relatively simple differential equations in time only. The otherwise complex calculation of the heat transfer coefficient and refrigerant side pressure drop (through integration to place or to void fraction) is replaced by algebraic equations with which local values are calculated. Within one element the local value is taken as mean value. The use of standard model building software makes it possible to program these relatively simple equations without complex reordering or pre-calculation.

Note: All the equations derived inthis section are validfor the nth element, so the subscript

his

omitted.

2.3.2 Shell wall segment

The shell wall represents a relatively small thermal mass, compared to the mass flow of water and the

mass of the tubes. Furthermore, the temperature gradient over the wall is small due to the small

temperature difference between environment and chilled water, and the foam isolation on the outside of the shell. The shell wall is supposed to have the same temperature as has the tubes and therefore the shell mass is added to the tube mass. The heat flux from the environment through the wall to the chilled water is supposed to be small and constant for all segments.

2.3.3 Water segment

The water segment has as inputs the constant flow of chilled water Cs., with an inlet temperature Oc, ( = , and the tube walltemperature On,. Outputs are the heat flux to the tube wall A. and

the water outlet temperature

Due to the construction of the evaporator, cross-flow is the

predominant flow pattern.

Assumptions:

Water is incompressible.

The mass flow is constant and known. van Buuren-van Swaay

-van Euuren--van Swaay

CI114,411, QV

-2

a chw-N 1G.i ft A, fz, tt

Report Evaporator Model

The process within the water segment is therefore described with one law of conservation:

Conservation of energy for the chilled water

CI e_{chw 4)h. -chw h, chw-tv 4)m, chw(0chw, - chw)}

dt In chwC

w Inchw

The heat flux from the water to the tube is calculated with:

A A 0 chw-w.av eta.. = With: (6chw, n-1 - tu (0chw 8 t,,/ A t3 K d 2 17

TUDelft

(2.1) (2.2) Where

Parameters dependent only on the dimensions

of

the evaporator:

fce Geometry factor; to account for the changing of flow direction in the window. Leakage factor; to account for the water flow through the gap between the tubes and the holes in the baffle plate, and the gap between the baffle plate and the shell.

iBy

By-pass factor; to account for the water flow along the shell, which is not

participating in the heat exchange

fLa Lay-out factor; which describes the difference between the Nusselt number fora

bundle of tubes as compared to a single tube in cross-flow. Outer diameter of the tubes.

Parameters dependent on the temperature

of

the water: Nun, : Nusselt number for a single tube in cross-flow.

AH, Heat conductivity of water at the average water temperature.

Correction factor for the difference in Prandt1 number between water at average temperature and water at tube wall temperature.

The calculation of the heat transfer coefficient is described in detail in Appendix A. I according the methods of the VDI-Warmeatlas [1977, Chapter Ge; 1984, Chapter Gf, Gg].

The logarithmic temperature difference is more accurate, but calculating the logarithm could give numerical problems if the temperature differences differ only slightly, see Appendix

- V- Interim

chw-tu

2.3.4 Wall segment

The wall segment has two inputs: heat flux from the chilled water to the tube Oh, and the heat flux

form the tube wall to the refrigerant Oh. The only output parameter is the tube wall temperature 0,, . Assumptions:

Radial heat resistance of the tube is neglected. Axial conductivity of the tube is neglected.

The first assumption is based on the relatively high conductivity of the copper of the tube walls as compared to the conductivity of the boundary layer on both the chilled water and the refrigerant side.

The second assumption is based (I) on the relatively small temperature differences between the wall of subsequent segments and (2) on the relatively small contact area in axial direction as compared to the

available area for heat transfer in radial direction.

Again, as in the water segment, the whole process in the wall is described through one law of

conservation.

Conservation of energy for the tube wall

d _{4)h.chw-tu (1) tu-r}

d r

cm

_tu

The respective heat flows are calculated within the water segment and therefrigerant segment.

2.3.5 Refrigerant segment

Each refrigerant segment is divided into two parts: a vessel and a resistance part. In the vessel, besides the incoming and outgoing mass flows, the heat necessary for the evaporation is supplied, resulting in an evaporating mass flow, pressure, and mass-storage of refrigerant. In the resistance part the pressure difference between two vessels is used to determine the mass flow. The evaporator as a whole is divided into two main zones: the evaporating zone, where two phase flow occurs, and the superheating zone, where one phase -gas- flow occurs. The border between the two zones is always located between two elements and itslocation is determined during the simulation. Because of the completely different flow

regime, these two zones are modelled separately. Two phase flow:

ICMOS - CRP

,rkt,

TUDelft

(2.3) Buuren-van Swaay

-Li 61. van Buuren-van Swail

ks "A t$

23.5.1

Two phase flow vessel

In the two phase flow vessel four mass flows with there respective properties are input:

incoming vapour C. and liquid mass flow Om, fi , known from the resistance part of the previous element, and outgoing vapour Om, ,,, and liquid mass flow C.,, known from the

resistance part within the element. The only other input is the tube wall temperature O,,. Outputs and unknown variables within the vessel are: the stored vapour mass my, the stored liquid mass ml, the pressure p. the evaporating mass flow and the heatflux

Assumptions:

Height and velocity changes are neglected.

Both liquid and vapour are in saturated conditions. Both liquid and vapour are ideally stirred

The first assumption only gives cause to very small deviations since internal energy changes

due to height en velocity changes are very small in comparison to the internal energy

changes due to evaporation. The second assumption implies a fundamental choice: the

evaporation is not modelled as a process but instead as the result of energy flows.

Physically, if both liquid and vapour were saturated, they would be in an equilibrium and no transfer of mass from one phase to the other would occur. The third assumption implies that the both outgoing flows have the properties of the bulk of that phase in the vessel.

With the pressure, both vapour and !liquid - saturated - properties are known. There are three conservation laws:

Conservation of mass for the liquid phase

di' - (2.4):

Conservation of mass for the vapour phase.

4 in

Interim Report Evaporator Model'

'Conservation of energy within a two phase segment

Rewriting the energy equation, taking into account the assumption listed above and using the two conservation of mass equations, the following relation for the pressure is found (see Appendix A.2):i IV (2.6)1 dp kvi-70 , cOimish,11-di' _{ii dui} pn sat 4),,,,v0 (h - mi(h4-10 - ( Iv dl, in dp dr vi 50,, (2.5)) dm m. Io -

-TUDelft

1 ( v- v) u1) -sat V

van 8 uuren.va n Swaay

W : 4

ICMOS - CRP.

Two more equations are needed to calculate the five output variables. A fourth equation describes the evaporating mass flow as a function of known process variables. This equation is derived from the boundary condition posed by the fact that there is only one :control volume, while two separated phases occur, resulting in the coupling :of the equations

of state of the two phases. This gives (see Appendix A.2):

1 dp j

dic

d t dp v1, - vs kl, .(12 dV sat dp

TODelft

The fifth and final equation gives the energy flow from the tube wall tothe refrigerant and

is formulated as:

14)1Ltu-r ttnsrAtu,036/ - r) 2 . 8

The refrigerant temperature is directly known from the pressure because of the saturated condition. The heat transfer coefficient a,_, is calculated with Dhar's correlation

[Dhar c.s.,, 19831:

( a

2

/

44

a pcji = 0r.1 15 i(x 411( 1 -x2 )Til (Pr; f.7

g pI a , 0

I

For qualities above a certain dry-out value xd,r a transition equation' for the heat transfer coefficient is used because the wall is no longer completely wetted with liquid refrigerant.

This equation describes a parabolic transition between the two phase and one phase heat transfer coefficient, the latter being considerably lower. This is described in more detail in Appendix A.4

23.5.2' Two phase flow resistance part

In the resistance part of the refrigerant segment the mass flow between two vessels is :established based on the pressure drop, similar to one phase flow. In two phase flow,

however, a difference in phase velocity occurs, adding a second variable to the resistance model: the velocity ratio.

Overall pressure drop is commonly :described in a simplistic form, since it is used primarily for fine tuning in global models of the evaporator. Pierre [1964a, I 96413] gives a more elaborate description for the overall pressure drop, accounting for evaporation speed and

X2.7)

van Suuren-van Swaay

The second variable in the resistance model - the velocity ratio between the two phases -'is

based the results of detailed flow simulation, given by Wang [1991]. Wang approximated the simulation results for the velocity ratio with a polynomial function of the void fraction:

- 10

Interim Report Evaporator Model

TUDelft

(2.11)

Combining this ratio with the total mass flow gives the vapour mass flow and liquid mass flow, However, the _{coefficients ft} are in the order of magnitude of 104, whereas the void fraction (and all its powers) vary between zero and unity. The resulting velocity ratio lies between unity and four. This means that the ratio is the result of adding of ten numbers that are four orders of magnitude larger. Because of this, equation (2. 1 1 ) has been replaced here with a more linear description, as described in the Appendix, to prevent numerical errors, without much change in the resulting velocity ratio.

One phase flow:

The border between the two phase and one phase flow region is determined through the quality of the flow. If in a vessel the quality becomes almost unity, then the flow in the next resistance part and all following elements is considered as one phase flow.

2.3.5.3 One phase .flow vessel

In the one phase flow vessel two mass flows with there respective properties are input: incoming vapour massflow known from the resistance part of the previous element, and outgoing vapour mass flow 0 ,, known from the resistance part within the element.

The only other input is the tube wall temperature Oh,. Outputs and unknown variables within the vessel are: the stored vapour mass mi,, the pressure p, and the heat flux

Assumptions:

Height and velocity changes are neglected. The vessel is ideally stirred.

Vapour is considered an idealized gas.

There are two laws of conservation applicable for the vessel, as described in Appendix A.3.2; Conservation of mass dm d t = cf. - (l) _(2.12) Conservation of energy

dp\ lc- 1

As P v )..voi Wk w-r (2.13) d t

The heat transfer coefficient a calculated using the Dittus and Boelter correlation:

= 0.023 Re "Pr 0.4 (2.14)

The energy flow from the tube wall to the refrigerant is modelled with equation (2.8), but the refrigerant temperature is no longer directly known from the pressure. It has become a function of both the pressure and the specific volume, in accordance with the gas law

-=

,

Z - P P TR

!CMOS - CRP

= CA,_v/2_{P (P -P,,.1)}

With resistance coefficient C

1.63

Re -I Re <2320

C 0.316Re 3.25 2320 Re 8.104

0.0054 .0.3964 Re -03 for Re > 8.10 4

TUDelft

because the superheated gas is modelled as an idealized gas. Two corrections are made to prevent large errors. The first correction influences the calculated temperature directly through the use of the real gas factor Z, which is defined as [KUttner, 1991]:

(2.15)

By calculating the real gas factor at the actual pressure in the vessel, the error

in the resulting temperature is reduced.

The second correction influences the ratio between the specific heat at constant pressureand the specific heat at constant volume (K), which is calculated again on the basis of the actual pressure.

Both Z and rare considered constant because both the pressure and temperaturechanges in the superheat zone are relatively small.

One phaseflow resistance

part

Input variables for the one phase resistance part are the pressures on both sides of the resistance: pi and p. The only output is the resulting mass flow .

Assumptions:

Vapour is incompressible. Pressure ratio is sub-critical.

With these assumptions, the mass flow is calculated in accordance with the VDI-Warmeatlas [1984, Chapter Lb], as a function of the pressure drop over an element:

(2.16)

(2.17)

23.5.4

van 13uurenvan Swaay

Interim Report Evaporator Model

2.4 Parameters in the evaporator model

The following parameters are given as input parameter file for the evaporator model:

Constants:

g gray

=9.81;

Arbitrary constants: eta fin = 0.25; phi_env_ch = 0; 1 el

=0.2;

Dimensaional parameters:

n tu

d tu_o

=0.0191;

d tu

i nom= 0.0175;

d tu

i = 0.005;

d sh

i = 0.21; 1 ba = 0.05; delta ba = 5e-3;

n ba

= l_el/l_ba; 1 tu = I el; l_eq = pi/2*d_tu o; psi

=0.418;

f Ge

=2.061;

f Le

= 0.886;

f By

= 0.642;

f La

= 0.741;

Al

= 2.280e-4; A_tu_o tu*l_el*pi*d_tu o,

A tu

i = n_tu*l_tu*(pi*d i_tu_nom+16*0.008*eta_vin); A i = n_tu*(pi/4*d_i_tu_nom**2-52.5e-6);

Vet

= el*A i;

V water

= pi/4*d sh i**2*1 e1/2-A tu o*d o/4;

circ tu

= pi/2*d_tu_o;

TO Delft

(indicates the efficiency of the surface of the finning inside the tubes) (indicates the amount of heat exchanged with the environment) (indicates the length of a calculational element)

23 = 22;

van Buurenvan Swaay

Parameter arrays used within the model Ipar_Nu_ev = [f Ge f Le f By f la]; par_Re_psi = [Leg psi Al'];

par_a_fcb = [g_grav d_tu_i]; par_Re_l

= [d_tu_i Ai];

par_Re_v = par_Re_1;

par_Bo = [A_tu_o

par_phi_m = [A_ii

Parameters not linked with dimensions:

par_slip_x = [00.2 .84 .96 1]; (see Appendix A.3. I)

par_slip_y =11 3.5 3.5 2.7 1]; (see Appendix A.3.1)

par_dry =11.159057 7.881472e-3 2.979559e-2 -.4928893 -6.458085e-21; i(see Appendix A.4.1)

f Pierre

=0,0185;

par_DitBoe = [0.023 0.8 0.4 d_ittLi] par Ji_nsat =196.147 273];

Water properties (equations described in Appendix C): 11

par_Pr_w = [1.1138 -1.5702e-2 I.1675e-4 -6.8705e-7 2.1712e-9]; par_eta_w = [0.2431 -1.3865e-2 9.6340e-5 -5.0885e-7 1.3432e-9]; par c_p_w = [4.2170 -3. 1815e-3 8.8133e-5 -9.9843e-7 4.3969e-9] par_lbd_w = [-0.2444 1.3588e-3 -5.9669e-6 0 0];

par_rho_w = [999.85 5.5104e-2 -7.7091e-3 4.5268e-5 -1.5326e-7I Refrigerant properties R22 (equations described in Appendix D):

par_satv_u =[3.654078e5 4.688288e3 -3.392784e2 1.290613e1 -1.954222e-1]I;

par_satv_v = [-6.633963e-I -9.250949e-1 -3.717648e-3 -7.579142e-71;

par sally = [-3.1554119 4.444014e-2 3.146921e-3

2.470363e-7];

par_v_dudp = [6.2881e-2 -1.2177e-1 1.1011e-2 -5.5765e-4 1.7034e-5 -2.I742e-7]; par_l_dudp = [1.7496e-I -2.8890e-1 2.4796e-2 -1.2030e-3 3.5695e-5 -4.3582e-7]; par_v_dvdp = [2.302617-8 8.8901I8e-7 -1.922984];

par_l_dvdp = [-1.539759-8 1.646824e-6 -6.245685e-1 1.574657e-2]i,

par_satt = [2.3596 I.8665e-3 -4.1353e-5 6.5005e-7 -4.4839e-9 3.9563e-3 9.6765e-2];

!CMOS - CRP

legx

TUDelft

par_satl_u = [1.490655e5 1.357263e4 -8.441795e2 3.210527e1 -4.769057e-I]; par_satv_h = [3.870158e5 5.309016e3 -4.013617e2 1.532014e1 -2.323523e-1]; par_satl_h = [11.490601e5 1.364602e4 -8.429830e2 3.20861 6e1 -4,764917e-1],

A_i]; l_tu d_tu_i];

van &tyre n-van Swaay

Interim Report Evaporator Model

Cupper properties:

m_cu = 8900*(n_tu*1_el*(d_tu o**2-d_tu_i nom**2)*pi/4);

c cu

= 3900;

lambda_cu = 390;

par_tu = [c_cu m_cu];

25

van Suuren-van 5waay

The average heat transfer coefficient on the chilled water side of the evaporator (ac.,,) is estimated using the method as described in the VDI-Warmeatlas [1977, Chapter Ge., 1984, Chapter Gf, Gg].

A.1.1

General calculation concept

The heat transfer coefficient a is based on the general description of heat transfer:

= _A

However, instead of usint2, this logarithmic-mean temperature difference, the arithmetic mean temperature difference is used:

°°

f

(°

chwo)

A Oav.chw-tu

2

Interim Report Evaporator Model

kw

TUDelft

(A.1)

Using this arithmetic expression prevents possible instability of the calculations during iteration which result when the actual temperature differences are small and_{zero or negative values can occur. The error} introduced by this simplification are discussed in the box below.

chw-tu chw-tu tu, In, chtv-tu

With: A _chw-to In o_to

-)

_{chwi (A.2)}

e-6

to chwo

Appendix A

Derivation of

Equations for the

Evaporator Model

A.1

Heat transfer coefficient: chilled water

-

tubes

van Buurenvan Swaay

In the shell of the evaporator the length I to be used is the frontal flow length of a tube, which equals halve the circumference:

/_eq

d.

(A.5)

2

The basis for the Nusselt number calculation is the following relation for one pipe in cross flow:

2 \ Nu = K (0.3 ?Nu 123, Nu hi,) With: Nu = 0.664 1/Re 11./ Nil _curb 1 + 2.443 Re (Pr2/3- 1) 0.037 Re _{1, tp} Pr

ICMOS - CAP

TUDelft

(A.6) Pr < 0.11 1 =O. Pr 6,

(Pr

Pr61 Pr Pr )0.25

The error introduced by using the arithmetic-mean temperature difference is less than four percent if the inlet temperature difference is twice the outlet temperature difference. Because here small elements are used, the actual error will be much smaller. This is demonstrated if the logarithmic expression in equation (A.2) is developed in a series:

( fir diwi)`( 0111- echwo) 2

A_{0log. chw4u}

, 1 Ow- eichwi)-03tu-6c.hwo 2 1 Om- ec )-(e)tu-echwo)

4 (A.4)

3 (etu-echwd (0111eeh ) 5 ( en- chin). (0tu- ch )

From this expression the relation between the logarithmic-mean and the arithmetic-mean temperature difference is easily determined. As the denominator approaches unity, the difference between the two diminishes.

K

+

van 8uuren-van Swaay

To arrive at the Nusselt number for a tube bundle with baffle plates, four correction factors are

introduced:

fLu Lay-out factor; which describes the difference between the Nusselt number for a bundle of

tubes as compared to a single tube in cross-flow.

frie Geometry factor; to account for the changing of flow direction in the window.

Leakage factor; to account for the water flow through the gap between the tubes and the holes in the baffle plate, and the gap between the baffle plate and the shell.

flip By-pass factor; to account for the water flow along the shell, which is not participating in

the heat exchange.

With which the Nusselt number for the evaporator is found:

Nil'

swap fag "'Le 4 -ILa Nit tu (A.7)

And the heat transfer coefficient is:

Nu A.

Ct

chw-tu

A.1.2

Calculation of correction factors

The geometry and dimensions of the evaporator have

been given on page 12, in Figure 2.1, The

VDI-Warmeatlas uses the definitions as shown in

Figure A.1, which gives the following values for the parameters: 3 ,., b = a SI /do 11 1 1 1 1 1 Si _{41 b .32/do}

Figure Al Parameters for a tube bundle

2.147

19.1

12

0.628

19.1

The amount of free space is indicated with V/

h<1

- 1 31 - 1

-1 a b

3.142

III Report Evaporator Model

42.147 0.628

(As)

29

The distance between two tubes e is given through:

2

b<-1\12a+1 e = 1(1/2s

- d

= V20.52+122 - 19.1 = 4.654 mm

1 - o

2

Which results in a total minimal pass length of:

min r % (d d sh.h) n eiJbithor = 1/2(210 197 .)..3 -4.65 9.7 = 30.2 171177 The minimum pass-through area Au follows:

A min = Iha min = 1.508 .10-3 m 2

TUDelft

(A.8) 0.418 (A.10) d

van Buurenvan Semay

The total leakagearea At is the sum of the windowarea Athi, the area of the gap between the tubes and the tubeholes A,

It 2 2 A0. =

-(d- d0) (n

n .) 4.4 cm 2 4 A 7t,12

d2).

_1100-Y -- 2.3 cm 2 I,ba 4 sh,i ba 180 ° A, = A1,ba 6.7 cm 2

The by-pass area A by is defined as:

Aby 1/2 ba.[(d d e) + (1 ba,hor- e )] 3.3 cm 2

With these parameters, the correction factors are calculated:

A. 1.2. I Lay-out factor

Because the tubes are set alternately, the flow is forced around the tubes, increasing the effective frontal area. Subsequently, the lay-out factor is larger than unity:

2

fLa = 1

3b

-1 +

A.1.2.2 Geometry ,factor

Because the windows in the baffle plates contains less than half the total number of pipes, the geometry factor is close to unity:

n n 0.32 6 ( 6 ) 032 fGe = 1 (1/,w + 0.524

"

= 1 +0.524 = 0.89 (A.17) n_tu n tu 22 22 )

A.1.2.3 Leakage factor

Because the evaporator is a relatively small heat exchanger, the leakage factoris low:

A A s = 0.4 I. t4 + 1 - 0 . 4 131` e Anti" A 1. tot AI. tot 6.7 4 4

-Ls

0.4

1.4

e 15

ICMOS - CRP

- 2.061 (A.16) Al,tot = 0.64 444.

TUDelft

(A.18) 2 3 -0.628 and A4b, + = (A.I4)

-van Buuren--van Swaay

A.2

Equations for vessels in refrigerant segment

A.2.1 Two phase flow region

For a control volume within an open system as illustrated in Figure A.2, with saturated two phase

flow, the following conservation laws are

applicable

Conservation of mass for the vapour phase

The change in stored vapour equals the difference

between out and

in

flowing mass, plus the

evaporated mass.

dm,

lorcrini Report Evaporator Mock'

Conservation of energy

dt

dt

dr

(h..

p gl..

v - cOnt.0( h

dU

dQ

dW 2

) tu-r 4)In, vih1,1 4 4) hh ) vohvo (1)mit)h

CI t 31 Control volume V

4

441A.,

TUDelft

4: I vi (I) m. vo 4:nt.lv (A.20)

dr Figure A.2 Control volume in an opensystem

Conservation of mass for the liquid phase

The change in stored liquid equals the difference between out and in flowing mass, minus the evaporated mass:

dm

Which is a general description of the first law of thermodynamics. Because the control volume does not change in size, the work performed on the surroundings is zero. If differences in height and velocity are neglected, the first law becomes:

dU

dQ

di

dt

Assuming ideally mixed conditions for both phases coexisting within the vessel, and assuming the heat flux O the only supplied heat, this can be rewritten as:

Ii. - (tmoh. _(A.23)

(A.24)

dt

(1)m. ckmio (4)m, lv (A.21 )

Poglo*lPovol

(A.22)

--

-'

van ci uuren-van Swaay

w w

For the left hand term follows

din(ii yin v+u

lin)

d lc din

du1

dm ,

in +

ii, + _nit + _init

dt

dt

v'

di

v

di

di

Because of the saturated conditions for both vapour and liquid, which are ;ideally stirred, the differential of the internal' energy with time can be written as.

IICMOS - CRP

'Combined with the mass. eq lotions this gives

dp h.1.2 4),; h VTit.)+ Sinai h li -111)-(kmvo( h in 4)..10( h " 4)..tv(" v-10

di

duw du1 0.28)1

UI+

dp dp _sat

In which the following terms are recognized:

Heat transferred from the wall to the refrigerant (4. ).

Adjustment of incoming mass flow to vessel conditions and performed transport work (h.*-- ii4) Work needed to transport outgoing mass (hi - u).

Change of 'internal energy due to evaporation Or - ).

Change of stored internal energy within the control volume dp

id",

dt

dp d le iv + sat V dp

T unelft

Int].

at l(A.25}1

Note: The energy equation has. been rewritten to describe the change

of

pressure in the vessel

because the pressure is directly needed for the calculation

of

the mass.flOws.

In the

compressor model the same strategy has been used for the same reason., Necessity of volume being filled with refrigerant

'du du dp

(A.26)1

dt

dp

Which is valid for both liquid and vapour, thus:

dp

du,

dui

In Safi vp- mioh din dm 1(A.22)1

dtt dp

satV dp

in

di

di

u du I h h

)-v - )

-di n

V +

dt v

If differentiated to time, this (gives for a constant control volume: d (m vv) d (m iv 1)

- 0

di' di'

Note: Because of this differentiation is no longer ct parameter in the set of equations. Instead, equation (A.29) states a boundary condition ,for equation (A.30) and is necessary to calculate initial values for and nh, given the values /or the .specific volume (through the initial value of the pressure).

Equation (A.30) is written as:

dmv dv_, dm1 dv1

V, + III

c'1

V , +

in =0

di' ' cl t

di

' di'

Because of the saturated conditions for both vapour and liquid, which are ideally stirred, the differential of the internal energy with time can be written:

dv dv dp

(A.32)

di

dprat dt

Which is valid for both liquid and vapour, thus:

dm _dp

V +

di' I di'

Combined with the mass equations:

m. tn. 4)m, )Vv +lv (4)m, (I)In, lv )1)1 * d

pidi'

From which an expression for the evaporating mass flow is deduced:

33 (A.30) (A.31)

. 0

(A.33)

dv,

d v in V 111 ) 0 (A.34) dp dp (A.35)

-This poses a potential problem for the simulation since equation (A.35) uses which in is determined

d t with equation (A.28) to determine where 0,., itself is used.

This algebraic loop has been solved as shown in the next section.

(4)1/1,VO -.4)7?1,IR) vV + (4)711. 40ii) v1 +)1. dP / dv,, m +_v rut dv1 m 1 cut di' \ dp dp tr?!!4.4

Interim Report Evaporator Model

TUDe ft

dv dv

dP sot ??1

v

d p _satM

'van fluuren-van Swaay

V

dp

+

van 8uuren-van Swaay

A.2.2 Solving the algebraic loop in the equations for the two phase vessel

It is possible to substitute equation (A.35) in equation (A.28) in order to eliminate the algebraic loop Equation (A.28) is rewritten as

ICMOS - CRP

TUDelft

dp

A -

B4)..1

(A.36)

=

dt

And similarly for equations (A.35):

= C D- dp (A.37)

dt

Which results in:

dp

A - BC

(A.38) art 1

BD

With:

.4)(h -u,)

- - 4),H.10(hi-ui)

A

-v-I

Br

dv in sat

dui

In' sat (A.39) dp dp

du,

dui in1 sat (A.40) dp In + v dp sat (4) - _)vv C _ (A.41) dvy V_v dvi

-v

dp

Dr

M v. dp sat 1 sat (A.42) -V I

A.2.3 One phase flow -gas- region

For a control volume within an open system as

illustrated in Figure A.3, with one phase flow, the following conservation laws are applicable

dm -'V,N. - (I) c,

di

dU

dQ h h

di

di

d(um)

dt

4)h.eu-r 41m,i Ih - _{4)m. h} 0

For the left hand term follows:

d(um)

du

_m+u

dm

dt

di

di

R

= 1

Combining equations (A.47) and (A.48), and rewriting it using equation (A.49), leadsto: V

mu =

PVcT-RT

v x - 1

An 'idealized' gas is an ideal gas with constant specific heats, and subsequently with a constant ic (A.43)

35

Control

volume V

Conservation of mass

The change in stored mass equals the difference between out and in flowing mass:

Figure A.3 Control volume in an open system

Conservation of energy

Starting from the general description of the first law of thermodynamics, given in equation (A.22), the first assumption is that the work performed on the surroundings is zero, because the size of the control volume does not change. The second assumption is that differences in height and velocity are negligible. Thus the first law then becomes:

Assuming ideally mixed conditions within the vessel, and assuming the heat flux the only supplied heat, this can be rewritten as:

The left hand term -change of internal energy- can be rewritten using the following equations for an idealized gas:

u = c

_(A.47) = mRT (A.48) (A.44) (A.49) 4014.$

Interim Report Evaporator Model

TUDelft

a

van Buurenvan Swaay

-4

+ o (A.48)

(A.46)

T p V

Differentiating this equation with time gives:

d(m u ) V dp

d t K-1 cl t

The enthalpy contained in the inflowing gas can be rewritten using:

h = u

pv

and combined with equations (A.47) and (A.49):

1.

R

CC

lc

Ii .c.Ttpv

. p v -R PV - PV V R 1-K C v

Resulting in the following description of the change pressure in the vessel:

dp - 1 d t (P P ov 04) m. 0) V tu-r Rots>

!CMOS - CRP

T U Delft

-an Buurenv-an Svvaay =

-van Buuren-van Swaay

4.3

Equations for resistance parts in refrigerant segment

A.3.1 Two phase flow region

The pressure drop - mass flow relations found in 'literature can, be divided into two categories: those describing local pressure drop, and those describing overall pressure drop. Local pressure drop is.

commonly described with the Lockhardt Martinelli correlation [Hsu c.s., 1976]. The second

variable - the velocity ratio - is an integral part of the equations. These correlations, however, are very complex and require the use of small elements if applied in a finite element model, or prior integration. of the equations, which seems mathematically impossible [Wang, 1991]. A further drawback of the Lockhardt Martinelli correlation is the calculation order: the correlation are developed to be applied when the mass flow is known and pressure drop is unknown, whereas in the model developed here, the pressure drop is known and the mass flow is unknown.

In order to describe the mass flow as a function of the 'pressure drop, the following equation

{Pierre, d 9641 is. used:

I'

i

x -x

Apr

* o

I+

.it

i

_i fr x_av From which is derived:

It'In. With: .0 25 Bo Re 2 V 1 av X V_{aV V} + (1 - X )v._QV

Where:

f

= 0.0185 for oil-free conditions

This equation is also used by Wang [1994] to 'describe the overall _{pressure drop over the evaporator.}

37 X

av

Interim Report Evaporator Model

=-= tfroi

TU'Delft

n b

i

CI, 2 av

A!

_till 0%.521 Pi-Po fr2 b 1(A.53)1 r 1 2 V

e

van Buuren-van Swaay

-I,-.47

ICMOS - CRP

The distribution of vapour and liquid along the length of the evaporator is mainly influenced by the velocity ratio between the vapour and liquid. This ratio is influenced by the local quality and by the friction force between the two phases. Several descriptions are knownvf from literature, but they all use

complex non-linear equations, which are unsuited for this model. Wang [1991] has used a

flow simulation program to determine the velocity ratios for three different interfacial friction factors

f. He

uses 10th order polynomials of the void fraction to describe the results within his model!:

A plot of the

resulting

ration shows a rather

starightforward shape. Here, the values for fly= 30 are

approximated with a linear interpolation according

Table A. The choice of the value for the interfacial friction factor has to be confirmed when the model is

being validated

'Velocity raiiavs., void fraction,

-f_lv = 300 --f _Iv = 3 --f_lv = 3 -f_Itcalc 1

Figure A.4 Velocity ratios

Table A Coordinates for approximation of velocity ratio acc. Wang 1199191

tv,= 30.

TUDelft

l(A:56) , velocity ratio,

'1

0.3 3.5 0.85 3.5 10 E 11366 WI i.0

Which results in:

fly= 300 fh, = 30 fb, = 3

A 9.79E-01

8.31E-01 8.32E-01

A 1.61E+00

1.92E+01 11.92E+01

A 4.66E+01 -2.54E+02 3.95E+00

A -7.39E+02

ig4 5.67E+03 3.90E+03, -2.84E+04 2.03E+03 -2.64E+04

/1's -2.37E+04 1.11E+05 1.35E+05

A 5.77E+04 -2.58E+05 -3.69E+051

137 -8.40E+04 3.66E+05 5.90E+05.

A 7.26E+04 -3.12E+05 -5.53E+05

A -3.43E+04

1.48E+05 2.83E+05 filo 6.84E+03 -2.98E+04 -6.08E+04

0.2 01 0.4 0.6 0.8 1 void fraction 6 5 4 1.13 0