Fault Detection and Diagnosis for Refrigeration Systems

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Abstract

The theme of cooling is recurrent in the world around us. Air conditioning systems in residential environments are the most common examples of refrigeration systems. However, also in the treatment, storage, transport and distribution of food products, as well as in the health and tertiary sectors, refrigeration plays a central role. The main purpose of this research is the analysis of some techniques for the detection and diagnosis of faults in this type of systems, also called chillers.

In the work, the chiller is analyzed in all its components, for which the operating principle and the signicant variables for the fault detection task are derived.

The research proceeds to the analysis of static methodologies based on the data for the detection of anomalies. Each of them builds a model of the system.

This model is then used in the monitoring stage. The static nature of the methods proposed in the thesis refers to the use, in the model identication phase, of data relating to steady state of the system instead of the entire time evolution of the signals. In this way, the system is monitored in conditions of thermodynamic stationarity and sudden transients, dicult to characterize mathematically, are eliminated from the nal database. The choice of data- driven methods is consistent with the direction of the current literature, mainly focused on those approaches that do not require a detailed physical description of the system. The ability to ne-tune the model from the data makes these techniques easily applicable to dierent plants.

In particular, the thesis considers three techniques for the detection of anomalies. Two of them, the multiple linear regression and the Principal Components Analysis (PCA), identify a model for the data in the form, respectively, of a surface and a regression hyper-plane, while the third, the Mahalanobis's distance, takes into account the probabilistic characteristics of the dataset.

These techniques are generally used for the prediction or for the dimensional reduction. In the thesis their eectiveness is tested in the context of the detection of anomalies. The dierent philosophies from which they take inspiration and the advantages and disadvantages of each approach are considered. The comparison is proposed for some faulty dataset generated with software and on a real case.

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Sommario

Il tema del rarescamento è ricorrente nel mondo che ci circonda: i sistemi di climatizzazione negli ambienti residenziali sono gli esempi più comuni di sistemi di refrigerazione. Tuttavia anche nel trattamento, stoccaggio, trasporto e distribuzione di prodotti alimentari, così come nel settore sanitario e terziario, la refrigerazione svolge un ruolo centrale. Lo scopo principale della ricerca è l'analisi di alcune tecniche per l'individuazione e la diagnosi di guasti in questa tipologia di sistemi, anche detti chillers.

All'interno del lavoro, il chiller è analizzato in tutti i suoi componenti, per i quali vengono dedotti il principio di funzionamento e le variabili signicative per la rilevazione dei guasti.

La ricerca procede all'analisi di metodologie statiche basate sui dati per il ril- evamento di anomalie. Ognuna di esse prevede la costruzione di un modello del sistema; tale rappresentazione viene poi utilizzata nella fase di monitorag- gio. La natura statica dei metodi proposti nella tesi riferisce all'uso, nella fase di identicazione del modello, di dati relativi a stati stazionari del sistema in- vece dell'intera evoluzione temporale dei segnali. In questo modo, il sistema è monitorato in condizioni di stazionarietà termodinamica e transitori improvvisi, dicili da caratterizzare matematicamente, sono eliminati dal database nale.

La scelta di metodi basati sui dati è coerente con la direzione della letteratura corrente focalizzata su quegli approcci che non richiedono una descrizione sica dettagliata del sistema monitorato. La possibilità di mettere a punto il modello dai dati rende tali tecniche facilmente applicabili a dierenti impianti.

In particolare, la tesi considera tre tecniche per la rilevazione di anomalie. Due di esse, la regressione lineare multipla e l'Analisi delle Componenti Principali (PCA), identicano un modello per i dati nella forma, rispettivamente, di una supercie e di un iper-piano di regressione, mentre la terza, la distanza di Maha- lanobis, prende in considerazione le caratteristiche probabilistiche dell'insieme di dati.

Queste tecniche sono generalmente utilizzate a scopo previsionale o per la riduzione dimensionale: nella tesi ne viene testata l'ecacia nel contesto della rilevazione di anomalie, illustrando le diverse losoe dalle quali esse prendono spunto e commisurandone vantaggi e svantaggi. Il confronto viene proposto per degli insiemi di guasti simulati via software e per un caso reale.

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Ringraziamenti

Scorrono i titoli di coda di un Dottorato che è stato capace non solo di coinvolgermi, ma di rapirmi.

Ringrazio il mio supervisore, il Prof. Alessandro Beghi, sempre disposto ad ascoltarmi, a consigliarmi, ad assecondare le mie passioni.

Ringrazio gli Ing. Luca Cecchinato, Pierluigi Ceccolin, Marco Corradi e William Sausa, per aver creduto in questo Dottorato industriale, cogliendone opportunità e profondità. In ogni momento hanno mostrato disponibilità al dialogo.

Desidero ringraziare in particolar modo l'Ing. Andrea Cervato per avermi guidato in questi anni di lavoro anco a anco e per avermi concesso di carpire i segreti del mestiere. Esempio di totale dedizione e competenza, non potevo sperare in una gura tanto calzante per la mia crescita.

All'Ing. Mirco Rampazzo devo molto, forse tutto. Gli devo il percorso che ho compiuto, che grazie a lui ha preso, prima vita, poi forma e colore.

Ringrazio l'Ing. Francesco Simmini per essersi mostrato attento al metodo scientico, giudicando severamente e costruttivamente ogni fase del lavoro.

Dedico questo lavoro a tutti loro e alla mia famiglia, sostegno e rifugio per i miei giorni.

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Introduction

1.1 Refrigeration: why and where?

The purpose of the refrigeration is to maintain the temperature of a substance (air, food, medical products, ...) at a value lower than the surroundings. De- pending on the entity of this desired value, refrigeration systems are distinguished by chilled and frozen storage. In chilled storage the substance is maintained to a particular temperature in the range of about 0°C up to 8°C while frozen storage requires lower temperatures, typically between -18°C and -25°C.

The choice between chilled or frozen refrigeration is clearly based on the application: for the climate control in residential areas and in the tertiary sector (hotels, oces, banks, community centers, data centers, ...) a chilled refrigeration is used while, for example, in chemical and in the food industry frozen systems are needed to avoid deterioration in products.

In order to illustrate how dierent industrial sectors are aected by cooling de- mand, we refer to the pie chart of gure (1.1) where the percentage of chilled and frozen systems installed in the last ten years have been taken into account.

Data refers to the company Geoclima S.r.l.¹. Geoclima designs and develops liquid refrigerators for dierent applications that can be divided into the following categories:

chemical and petrochemical manufacturing: coal and potassium mining, processing of plastics and semiconductors;

food industry: treatment and distribution of vegetables, cheese and curds;

hospitals and pharmaceutical industry;

tertiary sector: banks and data centers, oces, universities, hotels, community and sport centers;

1Geoclima S.r.l. Via Dell'industria, 12, 34077 Ronchi dei Legionari (Gorizia), Italy

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often respond by adjusting thermostat settings or other set points and the root cause of the problem is often not diagnosed. For this reason, typically problems reoccur, and the operator responds again by making an adjustment [25]. Such adjustments are often based on rudimentary or incorrect physical reasoning and rules-of-thumb built on personal experience. Moreover, some problems do not manifest themselves in conditions that directly aect occupants of the building (residential applications) or the quality of the products (industrial applications) and, as a result, go undetected. However these undetected problems may aect energy costs and may reduce the life of the cooling system. On the other hand, unwarranted maintenance leads to excessive maintenance costs. Such considerations led to the interest in applying automated fault detection and diagnosis methods for refrigeration systems, since they would allow the improvement of system operation by automatically detecting performance problems and bring- ing them to the attention of the operators.

In the next section a brief literature review of the fault detection approaches is given. The dierent approaches are introduced and pros and cons of each of them are reported.

1.3 State of the art

Fault Detection and Diagnosis (FDD) is an area of investigation concerned with automating the processes of detecting faults in physical systems and diagnosing their cause. For many years, FDD has been an active area of research and development in the aerospace, automotive, manufacturing, nuclear, and national defense elds and continues to be today. Over the last decade, eorts have been addressed to the automated fault detection in the Heating, Ventilation and Air Conditioning (HVAC) eld. While FDD is well established in other industries, it is still in its infancy in HVAC. In fact FDD research for HVAC&R systems did not begin until the late 1980s and early 1990s [12]. In the late 1980s, the authors in [29] and [44] explored automated FDD for vapor-compression-based refrigeration. In the 1990s, several FDD applications for building systems were developed and tested in laboratories. Most of these investigations focused on FDD for vapor compression equipment (air conditioners, heat pumps, and refrigeration systems). In general, these applications of FDD used measured temperature and/or pressure at various locations in the system and thermodynamic relationships to detect and diagnose common faults. Much of the research and development in HVAC context is still performed in universities and laboratories, and commercial tools using these techniques are only beginning to emerge in this eld, while in other elds such as the automotive industry, an automatic FDD has been incorporated in products for more than two decades.

The goal of an FDD system is the early detection of faults and diagnosis of their causes, enabling correction before additional damage or before loss of service occurs. This is typically accomplished by continuously monitoring the operations of the system. The rst step is to monitor the physical system and detect any

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CHAPTER 1. INTRODUCTION 16 abnormal conditions (faults). This step is generally referred as fault detection.

When an abnormal condition is detected, fault diagnosis is used to determine its causes. These two steps constitute the FDD process. Generally, the detection is a relatively easier stage than the diagnosis because the latter typically requires a deepest knowledge of the system. Moreover, when the components of the systems inuence each other, it is very dicult to diagnose which component represents the origin of the fault.

In the refrigeration context, the aim of the FDD techniques is to implement the monitoring of refrigeration systems in an automated way, in order to identify the presence of malfunctions and provide information on the possible causes of anomalous behavior to operators. In particular, the attention is directed not only to the catastrophic malfunction of the system, easy to detect because it interrupts any operation (such as the breaking of the compressor or other problems to the mechanical and electrical parts of the machine), but also to the faults associated with the degradation of the components that can cause an increase in energy consumption and a decrease in performance over the time.

For example, faults of this type are the loss of refrigerant gas in the pipe joints, the heat exchangers fouling, the increase or reduction in the water ow rate etc. The detection of such faults is usually based on the identication of a model from good-operating data collected on the plant. The model identied on past observations is used to understand if the new recorded data from the system are in accordance (fault-free) or not (faulty) with the model prediction.

In particular, an anomaly is detected when the dierence, or residual, between model prediction and actual data exceeds a predened threshold. Basically, fault detection and diagnosis methods dier from one to another for the type of model used. In the next subsection we investigate the dierent model types proposed in the literature.

1.3.1 Fault Detection and Diagnosis methods

As we have already pointed out, FDD techniques dier for the model used to monitor the system, or rather, they dier for the a priori physical knowledge required in the model formulation. In this sense three categories are known in the open literature:

quantitative models: a high level of a priori physical knowledge is needed to write the equations that dene the behavior of the system;

rule-based models: an intermediate level of physical or expert knowledge is used to dene the expected behavior of the system;

black box models or process history based models: the model identication is based on the data collected from the plant. No a priori knowledge on the physics of the involved processes is required.

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1.3.1.1 Quantitative models: white and gray-box representations A model based entirely on rst principles (such as conservation of energy and thermodynamic equations that govern the behavior of the system) requires a high level of a priori physical knowledge. This type of modeling refers to the so called quantitative model-based fault detection. In a physical model-based FDD approach, the behavior of the system is modeled (predicted or estimated) for a given set of measured inputs (temperatures, pressures, ow rates) and model parameters (heat transfer geometry and coecients, type of refrigerant used in the circuit).

Quantitative model-based approaches can be further subdivided into detailed and simplied physical models.

Detailed physical models are based on coupled space-time partial dierential equations in mass, momentum, and energy balances and detailed knowledge of the characteristics of all components in the system (compressors, valves, heat exchangers, pipes) is needed.

In contrast to the detailed physical modeling techniques, simplied physical modeling generally employs a lumped parameter approach, which is computationally simpler because partial dierential equations are transformed into ordinary dierential and algebraic equations. Clearly, simplied models are less accurate than detailed ones in the prediction and estimation, and cannot characterize the value of the output variables or states in each point of the space of interest, but the computational complexity and the time required to their derivation and implementation is reduced.

Quantitative models can also be divided into dynamic and steady-states models.

Dynamic physical models are particularly important for capturing faults during transient operation. Steady-states quantitative model-based fault detection is typically based on the check of the energy balances and other thermodynamic relationships when the thermodynamic stationarity is reached inside the system, that is when the measured variables (temperatures, pressures, mass ow rates, ...) are approximately constant in time.

When the development of quantitative models does not require any data collection from the plant, the physical model is called a white-box representation. In this case the physics is sucient to completely write the model. Measurements are only used during the monitoring in order to compare the model prediction with the actual data and to alert in presence of a mismatch between the two values. Bendapudi and J.E. Braun in [6] provide a detailed list of available dynamic models for vapor compression equipment and a dynamic centrifugal chiller model from rst principles.

Conversely, when the data collection is required to experimentally tune some unknowns parameters of the physical model (heat transfer coecients, thermal resistances, ...) the quantitative model is said to be a gray-box representation.

Examples of studies of parameter estimation applied to building systems include the contributions of Sonderegger [43], Subbarao [45], Rabl [37], Reddy [38], Braun [11], Gordon and Ng [17], and Guyon and Palomo [19]. In par-

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CHAPTER 1. INTRODUCTION 18 ticular, Gordon and Ng in [17] developed a model based on rst principles of thermodynamics and linearized heat losses. The Gordon and Ng's model has been also employed in [1] to characterize the coecient of performance of a refrigeration system. Other examples of gray-box quantitative descriptions applied in the chiller modeling are proposed in [21]. In the latter a set of models has been derived using the rst principles of thermodynamics; each model represents one of the components of the system (compressor, condenser, evaporator, and expansion valve).

Pros and cons of FDD based on quantitative models are summarized below.

Pros:

models are based on physical or engineering principles, so they can provide a complete characterization of the system;

both normal and faulty operation can be modeled.

Cons:

they may be complex and computationally intensive: the eort required to develop a model is signicant, especially when a detailed model is derived;

these models generally require many inputs to describe the system, and many physical coecients need to be available or identiable. Therefore white box representations are rare and some measurements are required to tune the model;

dierent systems require dierent models: in other words, models gener- alization is not possible.

1.3.1.2 Rule-based models

Rule-based modeling techniques use a priori knowledge to derive a set of if-then- else rules and an inference mechanism that searches through the rule-space to draw conclusions. Rule-based systems can be based on expert knowledge in- ferred from experience. This category also includes FDD methods that use simple thresholds (to trigger alarms). In developing expert systems, the knowledge of domain experts is usually obtained through interviews with a knowledge engineer, who later enters the collected information into a database (often referred to as a knowledge base). Expert systems have been developed to diagnose and analyze problems in many elds. Systems for medical diagnosis are among the most widely recognized. Authors of [23, 24] describe the development of an embedded expert system for monitoring packaged HVAC equipment while the report [22] describes an expert system for FDD in building systems. Expert systems are easy to develop, transparent, and the uncertainty can be taken into account as observed in [46]. The main weaknesses are that they are very specic to a system, are dicult to update or change and these models depend on the expertise of the developer.

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1.3.1.3 Black-box models

When the knowledge of the physical laws that govern the system is not available (or not required) and the model is identied in a blindly way simply using the data collected on the plant, we have the so called black-box representation.

Black-box models can be distinguished into parametric and non parametric models and also into dynamic and steady-state representations. Parametric models include linear, non linear and multiple regression. Polynomial and logis- tic regression are examples of linear parametric models. Non linear regression category is dened by the tting functions which are non-linear with respect to the model parameters (rational functions with at least one parameter at the de- nominator, Articial Neural Networks,...). In the multiple regression case more than one input signal is used in the model to explain the output variable(s).

Authors of [40] and [13] developed and evaluated a complete FDD system for packaged air-conditioning equipment. Seven black-box steady-state models are used to describe the relationship between the driving conditions and the expected output states in a normally operating system. In a normally operating all of the output states in the system are assumed to be functions of only three driving conditions that aect the operating states of the unit: the ambient air temperature (Tair), the temperature of the return air into the evaporator coil (Tra), and wet-bulb temperature (Twb) of the return air entering the evaporator coil. The black-box models take the polynomial form:

yi= a1+ a2Twb+ a3Tra+ a4Tair+ a5T_wb² + a6T_ra²+

a7T_air² + a8TwbTra+ a9TraTair+ a10TwbTair+ a11T_wb³ + ...

where yi is the i-th output variable prediction and the vector of parameters

a,





 a1

...

a11

...







is determined using the least square algorithm. The polynomial is t to steady- state training data obtained in the laboratory and compared with a separate set of steady-state test data for validation. This is an example of linear (the model is linear with respect to the polynomial coecients) multiple regression (the model inputs are Twb, Tra, Tair, T_wb² , T_ra², ...).

Researchers in [35, 27, 2, 32, 48, 39] also used black-box models based on Arti-

cial Neural Networks (ANN) for FDD. ANNs are so named because they were

rst proposed as a model of neuron-biological processes. ANNs can be viewed as sets of interconnected nodes usually arranged in multiple layers (input, hid- den, and output): inside the node, the input signal is rstly multiplied by a

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CHAPTER 1. INTRODUCTION 20 parameter and an oset coecient is added, then this result is elaborated in a non linear way (by a sigmoidal or radial function) to t the output.

Examples of non parametric black-box modeling are the so called correlation or covariance representations. In this case the collected data are not used to estimate the coecients of a model to obtain the best t, but to discover the mutual relation (correlation) between dierent plant variables, in order to measure the entity of this relationships and to alert when some characteristics in the correlation model change (fault). In recent years, the research on fault detection is mainly addressed on the static and non-parametric black box models:

an example is the Principal Component Analysis, central in the papers [3, 5, 42].

The use of dynamic, parametric black box modeling to solve a fault detection problem in the automotive cooling system is discussed in [36] where the model is represented by a dierence equation which coecients are identied using the process history data. A similar approach is used by Beghi et al. in [4] for an application which involve a refrigeration system.

Pros and cons of FDD based on black-box models are summarized below.

Pros:

these models are suited to problems for which theoretical models of behavior are poorly developed or inadequate to explain observed performance;

they are suited where training data are plentiful or inexpensive to collect;

black-box models identication does not require an understanding of the physics of the system being modeled;

computational requirements are generally manageable.

Cons:

black-box accurate modeling requires an in-depth knowledge in statistics and probability theory;

the stage of data collection needs to be carefully designed and a large amount of training data is needed, specially in presence of high order or non linear models;

most models cannot be used to extrapolate beyond the range of the training data;

the coecients of the identied models are specic to the system for which they are trained: for new systems the models need to be re-identied.

1.4 Objective of the thesis

Due to the complexity of the physical processes involved in the refrigeration systems, for which many parameters are unknown (heat exchanger coecients,

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thermal resistances, ...), and due to the complexity of the dynamics that characterize the phenomena, steady-state black box FDD methods may represent a good compromise among performance, complexity, and robustness. As high- lighted in subsection 1.3.1.3, black box representations allow to maintain a low computational eort (both the implementation and the storage eorts are op- timized) and they can be tuned from the collected data: this implies that a deepest physical characterization of the system is not required. However, black- box techniques need some preliminary critical analysis: which are the hypothesis at the basis and which the approaches eectively useful for the application? How the nal fault detection system can be tuned in a robust way? The answers for these questions are often in the data and the suitability of a particular technique is related to the satisfaction of the hypothesis at the basis. For these reasons, techniques with less restrictive hypothesis are devoted to perform well in a wide range of situations. The thesis emphasizes this fact by comparing dierent methods, each one with a dierent hypothesis set at the basis. Three approaches are compared: the multiple linear regression (equipped with the interquartile range statistics), the Principal Component Analysis and the Mahalanobis's distance.

The comparison is made on both simulated and real data. Numerical data are obtained through a software that simulates the chiller in steady state conditions (thermodynamic equilibrium) in both normal and faulty situations. Real data refer to a refrigeration system installed for residential application. In this case some control signals have been electronically deteriorated via a Programmable Logic Controller to generate the fault; a steady state detector is necessary to

lter out the transients contained in the real dynamics.

1.5 Manuscript outline

The thesis is outlined as follows: in chapter 2 the working principle of the refrigeration system is introduced and its components are characterized from a thermodynamic point of view in order to highlight the variables involved in the fault detection and diagnosis task. For each failure class, the symptomatic variables for the fault diagnosis are given. The static black box fault detection approaches are derived in a mathematical way in chapter 3: in the same chapter the hypothesis at the basis of each method and the fault detection indexes are discussed. The three approaches are used to detect and diagnose dierent type of faults on both simulated and real data in chapter 4, where their performances are compared. Some concluding remarks are given in chapter 5.

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CHAPTER 1. INTRODUCTION 22

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Chapter 2

Refrigeration systems

The refrigeration system, or chiller, represents the subject of the manuscript.

Therefore in this chapter its physical description is taken into account. Some thermodynamic aspects are initially considered in order to derive the parameters that characterize the cooling system. Then, the operation principle of the system is explained and a focus on its components is given in order to emphasize which signals are fundamental in the fault detection and diagnosis task.

2.1 Some thermodynamic aspects

2.1.1 State and exchange quantities: the concepts of steady state, heat and work

In each time instant a uid is characterized by some variables such as the mass, volume, pressure, temperature and may be subject to chemical reactions. When all the variables are constant in time and there are no chemical reactions inside the substance, we say that the system is in a steady state condition. Steady state is less restrictive than the concept of physical equilibrium in which the mechanical equilibrium (constant volume and uniformity of the pressure inside the substance), the chemical equilibrium (no chemical reactions or diusion phenomena inside the uid) and the thermal equilibrium (constant value for the temperature in each part of the substance) are contemporary required.

The transition from a steady state condition to a new one is denoted with the term thermodynamic process and is characterized by the absorption or the deliver of energy with the external environment or with other substances. This energy can be exchanged in the form of heat or work and its eect is to modify the internal energy¹ of the substance. Therefore heat and work are exchange variables (no state quantities) that characterize the transients in a time-analysis of the system.

1A denition of internal energy is given in the appendix.

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CHAPTER 2. REFRIGERATION SYSTEMS 24 2.1.1.1 Heat

Nature has the tendency to balance each disequilibrium. From an electric point of view, when two objects are charged with opposite signs, their particles (typically electrons) tend to move among the two materials in order to restore the neutral condition. In chemistry, similarly, matter tends to ow from a high density side to a low density one. The same considerations hold in the thermal context; after some time, two objects at dierent temperatures have the natural tendency to assume the same temperature (intermediate to the initial ones).

In the rst example (electrical charges), the equilibrium restoring is associated to a passage of particles (current). In the second example (concentration of matter), the ow of the matter compensates the density dierences. In the third case, the thermal equilibrium is reached thanks to the so-called heat ow.

Dierently from the rst two examples, the thermal re-balance is not always attributable to a ow of particles or matter; for this reason the heat ow is de-

ned as an energy ow and it can be explained from a molecular point of view.

Due to the thermal stirring, the molecules inside the object with higher temperature have a kinetic energy higher than the molecules in the lower temperature one. When the two objects are placed in contact, the collisions between the molecules of the two substances create an increase in the speed of the molecules in the lower temperature object (initially only on the contact surface and, subse- quently, in the rest of the object thanks to a domino eect). In contemporaneity, the internal kinetic energy of the higher temperature object decreases. After some time all the molecules inside the two objects will have the same speed, that is the same kinetic internal energy and, therefore, the same temperature.

This is, intuitively, the denition of the heat ow. The heat ow required to increase/decrease for 1°C the temperature of 1Kg of a substance mainly depends on the substance itself and it is called specic heat (cspec). The unit of measure of the specic heat is^J/^KgKand the heat Q needed to change the temperature of mKg of the substance from the initial value (T1) to the nal one (T2) is dened as

Q = mcspec(T2− T1) (2.1)

and it is expressed in Joule (J). The heat is positive if the substance increases its temperature (T2> T1), negative if the temperature decreases (T2< T1).

Another useful denition is the heat per unit mass (q,^J/^Kg) which can be simply derived from the previous formula by dividing both sides with the mass of the substance.

When the substance, for example a uid, is in motion inside a pipe with constant mass ow rate ˙m (^Kg/s), the denition (2.1) can be rewritten as:

Q = ( ˙m∆t) cspec(T2− T1) (2.2) where ˙m∆t is the total mass owed, for which the temperature is changed from T1 to T2 in the considered time interval ∆t. By assuming only an innitesimal

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time interval (dt in place of ∆t) and by dividing both sides of (2.2) for the interval time (let's denote the variation in the temperature with dT in place of (T2− T1)), we obtain:

d ˜Q = ˙mcspecdT. (2.3)

The quantity (2.3) is called elementary heat ow. When the uid increases its temperature thanks to the energy exchange, the convention imposes to assign a positive sign for the elementary heat ow (the heat is absorbed). If the temperature decreases the sign is negative (the heat is lost). These two cases lead to formulas

d ˜Q = ˙mcspecdT (2.4)

for the heat absorption, and

d ˜Q = − ˙mcspecdT (2.5)

for the heat loss.

The detailed analysis of the processes through which the heat takes place is the purpose of a discipline called heat transmission. Based on the fundamental concepts of thermodynamics, it studies the evolution over the time of the events generated by thermal non-equilibrium or non-steady states. Even this branch of physics has a strong experimental character and its own from the experimental observation of thermal phenomena, it has been found that the thermal exchange between bodies, having dierent temperatures, can take place in three ways [8]:

by conduction: the bodies are placed in contact and there is no relative motion between them;

by convection: the bodies (at least one is a uid) are placed in contact and there is relative motion between them;

by radiation: there is no direct contact between the objects which are separated by a space or vacuum.

In all these cases the experimental observation of the phenomenon shows that the temperatures of the two bodies change over time until both reach the same temperature, whose value is intermediate between their initial temperatures.

Thermal conduction: consider a system consisting of a cylindrical rod of ho- mogeneous material (uniform material structure at each point) and isotropic (thermal properties independent of the direction). The bar extremities consist of two parallel at surfaces, which section is A, at a distance ∆x, initially at dierent and uniform temperatures T1 and T2 (in Kelvin, [K]), with T1 > T2. Let also assume that the system is thermally insulated. The observed heat ow Q, assumed taking place only along the cylindrical axis, is:˙

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CHAPTER 2. REFRIGERATION SYSTEMS 26

Q = λ˙ A

∆x(T1− T2) (2.6)

where λ is the conductivity coecient that depends on the material and is measured in_W

mK

. Therefore the heat ow is proportional to the temperature dierence ∆T , T1− T2 and to the sectional ow area, while it is inversely proportional to the distance between the considered extremities. The unit of measure for the heat ow is the Watt [W]. From (2.6) it is possible to dene the total heat Q owed in the bar during the time interval ∆t as

Q = λ A

∆x(T1− T2) ∆t (2.7)

which is measured in Joule [J].

When the heat transfer is completed, both the extremities, as well as the internal sections of the bar, are at the same temperature. However, during the heat exchange the temperature in the bar changes in time and in relation to the position along the axis. At a specied position along the axis, all the points in a section are at the same temperature: for this reason this types of sections are called isotherm. If we consider two arbitrarily closest sections inside the bar with innitesimal distance dx and characterized by the temperature dierence dT, equation (2.6) assumes the dierential form

Q = λA˙ dT

dx. (2.8)

The thermal ow is always orthogonal to the isotherm sections and, in the general case of ow along the space direction ~n (instead of the x-axis), equation (2.8) takes the form:

Q = λA˙ ∂T

∂~n (2.9)

which is equivalent to:

Q = λA˙

∂T

∂x~i + ∂T

∂y~j +∂T

∂z~k

(2.10)

where ~i, ~j and ~z are the versors of the three Cartesian axis. Equation (2.10) is called Fourier's law [8].

Convection: the thermal convection takes place when at least one of the two bodies exchanging heat is a uid. Therefore the convection may take place between a solid and a liquid, between a solid and a gas, between a liquid and a gas, but also between two liquids. A necessary condition for the convection to happen is that the uid is placed in relative motion with respect to the second body. The relative motion of the uid can have dierent causes. It may be

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due to mechanical devices (fans, pumps, etc.) or to natural phenomena (wind, sea currents, etc.) which impose a certain speed on the uid. In the rst case we have the so-called forced convection. On the contrary, when the motion is naturally generated, the convection is called natural; this occurs also when the heat ow causes dierences in the density of the uid form which originates a mass displacement. Fluid volumes with lower density tend to rise and volumes of uid with greater density are recalled in their place. The distinction between the two types of convection is not strict and often in real situations they coexist.

Moreover, in the convection the microscopic modes of energy transmission are the same as in the case of conduction. The substantial dierence lies in the fact that, in addition to the transport of energy due to molecular interactions, it ap- pears also the motion of matter that conveys this energy. In absence of motion, the mechanism of heat transmission would be the same of the conduction case and the energy would be transmitted among the particles without macroscopic motion of matter. The equation that denes the heat ow in the convection case is the following:

Q = αA (T˙ 1− T²) . (2.11)

T1, T2are respectively the body and the uid temperature (in Kelvin; we assume that the uid is used to produce a cooling eect, so T1> T2), A is the contact surface while α is the convection thermal exchange coecient. This value is dicult to estimate: it is related not only to the uid properties but also to the motion characteristics and to the geometry of the body-uid system. α is expressed in _W

m²K

.

Thermal radiation: conversely to the conduction and convection, the radiation does not need direct contact between the bodies and does not require a propa- gation medium. It is a phenomenon that occurs in every material state (solid, liquid or gaseous) and also takes place in a vacuum. Radiation consists in the emission of electromagnetic waves generated by atoms and molecules excited by thermal stirring. Protons and electrons vibrations generate variable electric currents and therefore magnetic elds. This phenomenon occurs at any temperature, but only at quite high temperatures the contribution to the total heat exchange exceeds the conduction and convection ones. The heat exchanged by radiation is mainly transmitted from the body at a higher temperature than that at a lower temperature. In reality, radiation energy propagates in both directions, but with less intensity from the cold element to the hot one.

2.1.1.2 Conduction and convection in the heat exchangers characterization

A heat exchanger is a device used to transfer the heat between two or more uids.

The uids may be separated by a solid wall to prevent mixing or they may be in direct contact. In typical refrigeration applications the two uids (refrigerant

uid and water/air) are maintained separately by a tube. Therefore, for our

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is the tube surface, Th denotes the hot uid temperature, and Ttube,int the temperature of the internal pipe wall.

In the second stage, the heat ows inside the tube thickness, so there is a conduction process

Q˙2= λA

s (Ttube,int− Ttube,ext) (2.13) where λ is the conductivity coecient, s is the tube thickness, and Ttube,extthe temperature at the external pipe wall.

Finally the heat ows from the external pipe wall to the cold uid by convection:

Q˙3= αextA (Ttube,ext− Tc) (2.14) where αext is the convection thermal exchange coecient on the shell and Tc

the temperature of the cold uid.

Since no heat dissipation is assumed, it holds ˙Q = ˙Q1 = ˙Q2 = ˙Q3 (we denote with ˙Q the heat ow).

By using (2.12), (2.13), (2.14) in the the heat conservation relation, we obtain

Q =˙ A (Th− Ttube,int)

1 αint

= A (Ttube,int− Ttube,ext)

s λ

= A (Ttube,ext− Tc)

1 αext

(2.15)

and by using a proportions property², the following result is derived:

Q =˙ 1

1

α_int +^s_λ+_α¹

ext

A (Th− Tc) = KA (Th− Tc) . (2.16) The parameter

K, 1

1

αint +^s_λ+_α¹

ext

(2.17) is said global heat transfer coecient of the heat exchanger. This coecient measures the amount of thermal power exchanged per unit of area and per unit of temperature dierence. Therefore it denes the tendency of the heat exchanger to exchange energy. The inverse of (2.17) is called thermal resistance.

(2.17) may be monitored to detect deterioration in the heat exchanger, due for example to fouling.

In reality, the temperatures of the hot and cold uids are not constant along the pipe. Therefore equation (2.16) is only a simplication but it is useful to understand what happens in each innitesimal piece of the pipe. If we denote

2if â_b= ^c_d=ê_f then â_b= ^c_d=ê_f =â+c+e_b+d+f

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CHAPTER 2. REFRIGERATION SYSTEMS 30 with x the position along the pipe, with dx the length of the innitesimal exchanging surface, and with dA the corresponding tube surface, the relationship (2.16) allows to dene the heat ow in the abscissa x as:

d ˙Q = KdA [Th(x) − T^c(x)] (2.18) where Th(x) , Tc(x) are the hot and cold uid temperatures in the point of interest (x).

By denoting with ˙mhand ˙mcrespectively the mass ow rate of the hot and cold

uid, with cspec,h and cspec,c respectively the specic heat of the hot and cold

uid (both the mass ow rate and the specic heat are assumed constant along the pipe), and with dTh and dTc the variation in the temperature for the two

uids at the point x due to the elementary heat ow, we obtain the following equations:

d ˙Q = − ˙mhcspec,hdTh (2.19)

d ˙Q = ± ˙mccspec,cdTc. (2.20) The rst expresses the heat as a loss from the hot uid, while the second expresses the heat as an absorption for the cold one (clearly the two heat ows are equals in module, because the heat losses from the hot uid is gained by the colder one). It is worth noticing that for a cocurrent ow the sign of (2.20) is positive (in fact Tc(x + dx) > Tc(x)), while in the countercurrent case the equation has to be dened with the negative sign (because Tc(x + dx) < Tc(x)).

From (2.19) and (2.20)

dTh= − d ˙Q

˙

mhcspec,h

, dTc= ± d ˙Q

˙

mccspec,c (2.21) therefore, by dening the dierence between hot and cold uid temperature as

∆, Th− Tc,one obtains

dTh− dTc= d (Th− Tc) = d∆ = −Md ˙Q (2.22) where

M = 1

˙ mccspec,c

+ 1

˙

mhcspec,h (2.23)

for the cocurrent case and

M = 1

˙

mhcspec,h + 1

˙

mccspec,c (2.24)

for the countercurrent.

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The integration of (2.22) along the overall heat exchanger gives

ˆ ∆_L

∆0

d∆ = − ˆ Q^˙

0

M d ˙Q ⇒ M = −∆L− ∆⁰

Q˙ (2.25)

where ∆0, ∆L are respectively the dierence between hot and cold uid temperatures at the point 0 and at the point L along the heat exchanger (see also

gure (2.1)).

By using (2.18) in (2.22), one obtains d∆

∆ = −MKdA (2.26)

and by integrating (2.26) along the overall heat exchanger ˆ ∆L

∆0

d∆

∆ = − ˆ A

0 M KdA ⇒ ln∆L

∆0 = −MKA (2.27)

that is (by using (2.25))

ln∆L

∆0 = −∆L− ∆0

Q˙ KA. (2.28)

Therefore the heat ow can be also expressed as

Q = KA˙ ∆L− ∆⁰

ln^∆_∆^L₀ (2.29)

which is useful to design the heat exchanger.

The quantity

∆L− ∆0

ln^∆_∆^L₀ (2.30)

is called logarithmic mean temperature dierence [9].

2.1.1.3 Latent heat: evaporation and condensation processes In the previous sections we have dened the concept of heat as a phenomenon that continuously changes the temperatures of two (or more) substances until a common temperature is reached. However, in some particular cases, really important in the refrigeration systems, the heat ows without changing the temperature inside a substance (that is, the kinetic energy of its molecules).

These special cases are the state changes. In particular, refrigeration systems involve two types of state changes: the evaporation (passage from liquid to gas) and the condensation (passage from gas to liquid).