Advanced Analytic and Control Techniques for Thermal Systems with Heat Exchangers

Advanced Analytic Control Techniques for Thermal Systems with Heat Exchangers presents the latest research on sophisticated analytic and control techniques specific for Heat Exchangers (HXs) and heat Exchanger Networks (HXNs), such as Stability Analysis, Efficiency of HXs, Fouling Effect, Delay Phenomenon, Robust Control, Algebraic Control, Geometric Control, Optimal Control, Fuzzy Control and Artificial Intelligence techniques. Editor Libor Pekar and his team of global expert contributors combine their knowledge and experience of investigated and applied systems and processes in this thorough review of the most advanced networks, analyzing their dynamics, efficiency, transient features, physical properties, performance, feasibility, flexibility and controllability. The structural and dynamic analyses and control approaches of HXNs, as well as energy efficient manipulation techniques are discussed, in addition to the design of the control systems through the full life cycle. This equips the reader with an understanding of the relevant theory in a variety of settings and scenarios and the confidence to apply that knowledge to solve problems in an academic or professional setting.Graduate students and early-mid career professionals require a robust understanding of how to suitably design thermal systems with HXs and HXNs to achieve required performance levels, which this book offers in one consolidated reference. All examples and solved problems included have been tried and tested, and these combined with the research driven theory provides professionals, researchers and students with the most recent techniques to maximize the energy efficiency and sustainability of existing and new thermal power systems.

• Analyses several advanced techniques, the theoretical background of these techniques and includes models, examples and results throughout
• Focusses on advanced analytic and control techniques which have been investigated or applied to thermal systems with HXs and HXNs
• Includes practical applications and advanced ideas from leading experts in the field, as well as case studies and tested problems and solutions

• Language: English
• Publisher: Academic Press
• Release date: Jul 10, 2020
• ISBN: 9780128194232

Book preview

development.

Section 1

Fundamentals of heat exchangers

Chapter 1

Introduction to heat exchangers

Libor Pekař    Faculty of Applied Informatics, Department of Automation and Control Engineering, Tomas Bata University in Zlín, Zlín, Czech Republic

Abstract

This concise introductory chapter is aimed at providing an unacquainted reader with very basic essentials about heat exchangers. There are two main parts of this chapter. Namely, the definition of heat exchangers and several possible classifications are given first. Afterward, the reader is acquainted with some basic mathematical laws that model the behavior and heat transfer of a general heat exchanger. References to various literature resources are provided so the reader can find more detail therein.

Keywords

Classification; Fourier's law; Heat exchangers; Heat exchanger construction; Heat exchanger types; Heat transfer rate; Logarithmic mean temperature difference; Newton's cooling law

1 Introduction

This concise introductory chapter is aimed at providing an unacquainted reader with very basic essentials about heat exchangers (HXs). It is definitely not intended to be exhaustive. There are two main parts of this chapter. Namely, the definition of HXs and several possible classifications are given first. Afterward, the reader is acquainted with some basic mathematical laws that model the behavior and heat transfer of a general HX. References to various literature resources are provided so the reader can find more detail therein.

A HX is a heat transfer device that exchanges heat between two or more process fluids, at different temperatures, and in thermal contact [1–3]. In HXs, heat transfer takes place from a high-temperature fluid to a low-temperature fluid. Usually, the phase of the fluids is not changed during the heat transfer. Besides heat conduction, thermal convection and radiation apply, which indicates that it is a complex process. HXs have many domestic and industrial applications, such as in building heat and air conditioning systems, refrigeration units, steam power plants, chemical processing plants, transportation power systems, etc. [3].

HXs can be classified according to their transfer processes, flow arrangement, fluid phase, and geometric construction [2]. Several possible classifications are concisely given to the reader in the following section. As HXs transfer thermal energy between fluids, a very basic mathematical and physical background of the heat transfer is provided as well [4–6].

2 Heat exchangers classification

As mentioned before, there are several ways to classify HXs. In addition, several authors use diverse points of view to which the XHs are classified. Therefore, the presented classification cannot be unique, and the reader may find many other approaches to HX categorization.

2.1 Classification according to the functional point of view

Kothandaranman [7] divides HXs into the following three groups according to their functioning.

(i)Recuperative HXs

(ii)Regenerative HXs

(iii)HXs with direct mixing

Note that some authors subsume these classifications under the constructional point of view (see, e.g., [3, 8]). The arrangement of HXs in groups according to their geometric construction features is given in Section 2.4.

2.1.1 Recuperative HXs

Recuperative HXs (or recuperators) have separated flow paths for each fluid, which flow simultaneously through the HX and across the wall separating the paths. The heat is transferred through the separating wall. They are more common compared with regenerative HXs. Sometimes they are grouped into indirect-contact, direct-contact, and special HXs [3]. However, we herein use these categories in Section 2.2.

2.1.2 Regenerative HXs

Regenerative HXs, also called accumulative or capacitive [9], are characterized by the process during which the heat transfers from the hot fluid to a material (so-called matrix) that accumulates the heat energy. Later, the cold fluid flowing through the HX removes the energy from the matrix, and it is heated. The matrix becomes colder, which recovers its initial state, and the cycle repeats. Some authors [8] divide this family of HXs into:

(a)Static HXs

(b)Dynamic HXs

This subclassification, which can also be subsumed under a constructional point of view, is characterized by a matrix movement. In the case of static HXs, the matrix is stationary (fixed), and it is exposed to a single fluid stream. Once the heat is accumulated, another fluid stream enters to remove the energy. On the contrary, the dynamic ones incorporate a moving matrix exposed to both the streams at once. One-half of the rotor is faced with the hot fluid and the second half with the cold one. It yields a movement of the rotor, in which the initial state is recovered within a single turn.

Accumulative HXs can be found in houses equipped with a solar heating system, in regenerative units with exhaust gases energy recovery, and in some other energy-intensive industries. In the future, the use of regenerators is likely to increase as attempts are made to recover lower grade heat and improve energy efficiency.

2.1.3 HXs with direct mixing

The direct-mixing-type HXs are characterized by fluid flows that are not separated at all. Therefore, the fluids are mixed during the heat transfer, and corresponding temperatures become equal. This type of HXs can, however, be rarely found in practice.

2.2 Classification according to transfer processes

HXs can be grouped according to transfer processes into:

(i)Direct contact HXs

(ii)Indirect contact HXs

(see, e.g., [2]).

2.2.1 Direct contact HXs

For this type of HX, there is no dividing wall, so heat transfers directly from the hot fluid to the cold fluid. According to Shah & Sekulić [10], direct HXs can be subclassified with respect to fluid types as follows:

(i)Two immiscible liquids

(ii)Liquid-gas

(iii)Liquid-steam

However, other possibilities may also exist, such as gas-solid, gas-liquid, liquid-solid, or solid-solid streams [3]. The only condition is that the two (or more) streams must be immiscible (e.g., water and oil). The advantages of utilizing direct contact HXs include the lack of surfaces to corrode or foul, which yields to heat transfer performance degradation. On the contrary, the streams must be at the same pressure, which leads to extra costs. The mixing (even if the fluids are immiscible) is inevitable, which results in the streams’ contamination. This family of HXs can be used for many systems; typical representatives are open-evaporative cooling towers of power stations, jet condensers for water vapor, open-feed water heaters in power plants, or barometric condensers used in the petroleum industry [11].

2.2.2 Indirect contact HXs

Two (or more) fluid streams are separated by the use of an impervious surface, such as tubes, plates, etc. It means that the fluids are not mixed. A commonly known representative of this group is the automobile radiator [3].

2.3 Classification according to flow arrangement

The usual classifications of HXs based on flow path configuration are [4]:

(i)Co-current flow HXs

(ii)Counter-current flow HXs

(iii)Crossflow HXs

2.3.1 Co-current flow HXs

The co-current (or parallel) flow is characterized by the two fluid streams entering together at one end of the HX, flowing through in the same direction (i.e., parallel to each other), and leaving together at the opposite end of the HX. A schematic sketch is shown in Fig. 1.

Fig. 1 Co-current flow configuration (the hot stream is red/thick , the cold stream is blue/thin ).

2.3.2 Counter-current flow HXs

In a counter-current HX, the two fluids flow parallel to each other yet in opposite directions; see the schematic sketch in Fig. 2.

Fig. 2 Counter-current flow configuration (the hot stream is red/thick , the cold stream is blue/thin ).

This type of HX provides less-uniform wall temperatures; however, efficiency is higher than for a co-current HX, and it requires less relative heat transfer surface area [12]. The reader is referred to Section 3.3 for more detail about efficiency.

2.3.3 Crossflow HXs

In a crossflow HX, the two fluids flow at right angles to each other. This type of HX is usually used for the combination of liquid-gas or gas-gas streams [13]. The following two families of crossflow HXs can be raised:

(a)HXs with mixed streams

(b)HXs with unmixed streams

In mixed-stream HXs, the main fluid flow is perpendicular to another flow; however, it is partially diverged in the parallel direction to another flow; see Fig. 3. Contrariwise, HXs with unmixed streams use partitions, plates, or other barriers to avoid mixing of the fluid streams, as shown in Fig. 4.

Fig. 3 Crossflow configuration with mixed streams (the hot stream is red/thick , the cold stream is blue/thin ).

Fig. 4 Crossflow configuration with unmixed streams (the hot stream is red/thick , the cold stream is blue/thin ).

Another possible subclassification of crossflow HXs is based on the number of moves through the heat transfer matrix. We can define the following two classes:

(a)Single-pass crossflow HX

(b)Multipass crossflow HX

In single-pass crossflow units, the fluids move through the heat transfer matrix perpendicularly to each other yet still in one (fixed) direction, as displayed in Figs. 3 and 4. Contrariwise, one fluid stream shuttles back and forth across the flow path of the other fluid stream for multipass HXs [4, 12]. Usually, a combination of cross and counter-current flows is then approximately obtained, as seen in Fig. 5.

Fig. 5 Multipass crossflow configuration (the hot stream is red/thick , the cold stream is blue/thin ).

Crossflow HXs are used, for example, in gas turbines. They are intermediate in efficiency and in the required relative heat transfer surface area between counter-current flow and parallel flow exchangers [3]. For efficiency, HXs are designed to maximize the surface area of the wall between the fluids while minimizing resistance to fluid flow through the HX [14]. An aspect affecting counter-current versus crossflow choice is that of pressure drop [11].

In industrial practice, hybrids of all three flow types are often used. Usually, a combination of counter-current and crossflows can be found.

2.4 Classification according to geometric construction

HXs can be arranged in groups according to their construction features. A possible classification of HXs based on their geometric construction can be as follows:

(i)Tubular HXs

(ii)Plate HXs

(iii)Compact HXs

(iv)Regenerative HXs

Items (i) to (iii) can also be included in families of recuperative and indirect-contact HXs (see Sections 2.1.1 and 2.2.2). Note that regenerative HXs (iv) have already been introduced in Section 2.1.2. We do concisely provide the reader with a description and a possible further subclassification.

2.4.1 Tubular HXs

This type of HX is widely used in engineering applications. Tubular HXs are manufactured in many types, sizes, and flow arrangements. The advantage and popularity of tubular HXs lie in their applicability for a wide range of operating temperatures and pressures. They can be subdivided into a number of categories and according to various points of view [11, 15], for example, as:

(a)Shell-and-tube HXs

(b)Tube-in-tube HXs

(c)Tube-in-plate HXs

(d)Coiled-tube HXs

(e)Furnaces

(f)Air-cooled HXs

(g)Special tubular HXs

Shell-and-tube HXs are likely the most used HXs in industry. They consist of a number of round tubes mounted inside a cylindrical shell and has the following five major parts: the front and rear headers where the fluid enters and leaves the tube side, respectively; the tube bundle; the shell; and the baffles. The baffles are used to support the tubes, to direct the fluid flow approximately transversely to the tubes (which increases heat transfer intensity), and to increase the turbulence of the shell fluid [11]. One fluid flows over the outside of the tubes while the second fluid flows through the tubes, which yields heat exchange. The differences between shell-and-tube HX variations lie in the arrangement of flow configurations (co-current, counter-current) and in the details of construction [16, 17]. The following main three types of a shell-and-tube HX can be found [4]: (1) one-shell-pass and one-tube-pass (see Fig. 6), (2) one-shell-pass and two-tube-pass (see Fig. 7), and (3) two-shell-pass and four-tube-pass (see Fig. 8). Fig. 7 may also represent a double-pipe HX (consisting of one or more tubes contained within a larger pipe).

Fig. 6 One-shell-pass and one-tube-pass HX (the hot stream is red/thick , the cold stream is blue/thin ).

Fig. 7 One-shell-pass and two-tube-pass HX (the hot stream is red/thick , the cold stream is blue/thin ).

Fig. 8 Two-shell-pass and four-tube-pass HX (the hot stream is red/thick , the cold stream is blue/thin ).

This family of HXs may use liquid-to-liquid, liquid-to-gas, or gas-to-gas fluids [5], and it can be found through a number of industrial application, e.g., in the petrochemical or pharmaceutical industry.

The simplest tubular configuration is represented by tube-in-tube type HXs. They are composed of two concentric tubes where the one with the smaller diameter is placed inside the tube with the larger diameter. Different fluids flow inside each of the tubes. This configuration is suitable to be used for low heating power application and it is, hence, not widely used in the industry.

Regarding tube-in-plate HXs, the tubes are mounted in some form of pipe, whereas the plates act as supports and provide extra surface area in the form of fins. This family of HXs can be found in air conditioning applications and heat recovery [11, 15].

A coiled-tube HX is composed of tubes or pipes that are shaped to spirals, which enhances heat transfer [18]. This shape, however, suffers from a tendency to fouling [19]. Fouling can be defined as the deposit of unwanted material (the so-called foulant) on heat transfer surfaces during service. The foulant can be formed by dirt, sand, rust, salt crystals, etc., and it reduces the HX efficiency. Moreover, coiled tubes almost disable their cleaning.

In furnaces, the process fluid passes through them in helically wound or straight tubes, and either burners or electric heaters provide the heating. Furnaces are widely used in steam- and gas-turbine power plants [4].

An air-cooled HX consists of a bundle of tubes, a fan system, and a supporting structure [11]. An additional surface area for better heat transport is provided by fins mounted to tubes. Fans are either placed below the bundle to blow air through the tubes or above tubes to suck up air via them. This family of HXs can be found in applications where cooling water cannot be used.

Last but not least, special tubular HXs include, e.g., heat pipes that consist of a pipe, an operating fluid, and a wicking material. The fluid absorbs heat first. Then, it is evaporated and passed to the other end of the pipe where it condenses and releases heat. It is, finally, returned by capillary action to the hot side of the pipe, which yields evaporating again [20].

Note that another subclassification of tubular HXs can also be found in the literature [21]:

(a)HXs with direct tubes

(b)HXs with coiled tubes

(c)HX with Fields’ tubes

All these types of tubular HXs (except for coiled-tube HXs) are included in the family of HXs with direct tubes.

2.4.2 Plate HXs

These types of HXs consist of a number of welded or bolted plates that serve for heat exchange. They are usually designed for moderate temperature and pressure differentials because of the plate geometry. A relatively high surface area-to-volume ratio is their advantage that enables their usability in various industrial applications, such as food processing, cryogenic, or chemical industries [22].

There are several types of plate HXs in practice; for example:

(a)Plate-and-frame HXs

(b)Plate-fin HXs

(c)Spiral HXs

(d)Lamella HXs

Plate-and-frame HXs are constructed of a large number of relatively thin embossed plates mounted between two rectangular end members that serve as a frame. The plates have holes for fluid flows, and they are separated by a gasket. One side of each of the plates is exposed to the heat fluid flow, whereas the opposite side to the cold fluid flow. The advantage of this type of plate HX lies in the possibility to add or remove some plates if heat power should be changed. Moreover, they can easily be taken apart to clean [3, 23]. On the contrary, they may suffer from leakage, which can be solved by welding the plates or blazing them together followed by welding on the inlet and outlet elements.

Plate-fin HXs can also be classified within the group of compact HXs [2] (see Section 2.4.3), as they are characterized by a very high compactness factor (up to 6000 m²/m³). The compactness factor is expressed by the ratio of the total amount of heat transfer surface area and the volume of the HX [5]. These HXs consist of fins placed between parallel plates. The fins allow co-current, counter-current, and crossflows, or their combinations. This type of HX is usually used for gas-to-gas, gas liquefaction, or cryogenic applications [20] and in low-pressure conditions.

Spiral HXs consist of two long flat parallel plates formed by a roll mandrel into a coil shape. The distance between the two surfaces is adjusted by using a spacer pin. The ends are welded or sealed with gaskets so that fluids can flow through the tunnel. The advantage of this type of HX lies in a better heat transfer; moreover, there is less tendency to fouling. They are mainly used for fouling and viscous fluids or those containing particles; however, they are quite expensive due to the used manufacturing technology.

The term lamella expresses a plate tube. Lamella HXs are mounted either as a bundle of welded lamellae or in a lamella-fin configuration (which is close to a tube-in-plate HX; see Section 2.4.1). Their compactness factor is a bit less than that of plate-fin HXs [22]. Typically, they apply when an extended surface or high operating pressure are required.

2.4.3 Compact HXs

Regardless of their structural design, HXs with a compactness factor greater than about 700 m²/m³ are (quite arbitrarily) referred to as compact HXs. For example, human lungs have a factor of about 20,000 m²/m³ and radiators in cars of about 1000 m²/m³ [11]. Contrariwise, despite their volumes, shell-and-tube or tube-in-tube HXs are not considered compact due to their compactness factor ranging from 70 to 500 m²/m³ [5].

Fins or a corrugation are used to form a larger heat exchange surface. The fins are mounted by welding, brazing, adhesive bonding, or by a mechanical joint [2]. Two types of compact HXs are distinguished:

(a)Plate-fin compact HXs

(b)Tube-fin compact HXs

Both types were shortly introduced earlier in Sections 2.4.1 and 2.4.2. In the latter case, fins can be mounted to either the inner or outer tube surface. They can be usually found for gas-liquid applications where fins are placed on the gas side.

2.5 Classification according to fluid phase

When the fluid temperature reaches a particular critical value, the fluid phase (state) changes. This phase change is done for a constant pressure value. The following three basic categories can be determined:

(i)Condenser

(ii)Evaporator

(iii)Crystallizer

In condensers, the heat is released from the fluid to the surrounding environment. They are used for various applications in steam power plants, chemical processing plants, and nuclear electric plants for space vehicles. The major types include surface condensers, jet condensers, and evaporative condensers [24]. A different classification includes families of direct and indirect condensers with further subclassification [3].

An evaporator is a device in which the fluid turns the liquid into a gas. If heat is transferred from the whole fluid volume (not only from its surface), a boiler rather than an evaporator is obtained. In the case of gas-to-gas HXs, it is required that the surface area is up to 10 times larger on the condenser side or evaporator side compared with liquid-liquid HXs [25].

Finally, a crystallizer represents an HX responsible for the fluid change from the liquid state to a solid one. This is done by heat release to the surrounding environment.

2.6 Other HX grouping

It is worth noting that HXs can also be classified in accordance with the application for which they are intended. The specialized requirements have led to the development of many construction types. Some of them are unique to particular applications [4]. One can find radiators (e.g., in cars) and cooling towers (e.g., in nuclear power plants) [3], etc.

Another possible classification reflects flow arrangements (single-pass vs multipass HXs) and heat transfer mechanics (single-phase/double-phase convection on one side/two sides) [24].

3 Basic heat exchange formulae

This section is aimed at providing the reader with very basic laws, formulae, and conditions regarding heat exchange, its mechanism, heat balance, and HX efficiency.

3.1 Heat transfer mechanism

Whenever two systems with different temperatures interact, heat energy transfers between each of them. This process is spontaneous. The following three heat transfer mechanisms are distinguished [6, 7]:

(i)Conduction

(ii)Convection

(iii)Radiation

3.1.1 Heat conduction

Disorganized microscopically colliding particles, such as atoms, molecules, and electrons, transfer their potential and kinetic energy. Heat (thermal) conduction means the transfer of heat energy due to these collisions. The energy transfers from the particle with higher energy to that of the lower one. The temperature difference determines the heat transfer rate. Conduction applies mainly to solid materials; however, it can be observed in liquids and gases as well.

Heat conduction can be expressed via Fourier's law

   (1)

where q(t) (W/m²) stands for flow rate density, λ (W/(m K)) is the material heat conductivity, and gradTx(t, x) means the temperature gradient (K/m) of space coordinate x (m). The value of λ is mostly considered as a constant in practice, yet it holds true for isotropic (homogenous) materials only; it depends on temperature, generally.

If one assumes a homogenous one-dimensional material and two close points with constant temperature, Eq. (1) becomes the known difference form

   (2)

where ΔQ (J) is the amount of heat transferred per time interval Δt, k (W/(m² K)) is the heat transfer coefficient, A (m²) expresses the surface area, and ΔT (K) represents the temperature difference between the points at the distance Δx (m).

3.1.2 Thermal convection

Convection is the heat transfer process that combines heat conduction and a macroscopic (bulk) transport and mixing of the fluid substance (i.e., advection) [26]. In addition, diffusion due to a concentration gradient may apply. Thermal convection takes place in liquids and gases. It may also take place between solids and liquids (gases), which is the most common type of convection in HXs; this process gives rise to the notion of heat transmission.

Some authors present the following Newton's law of cooling as the basic heat convention relationship

   (3)

(W) is the heat transfer rate, Ts(t) means the temperature of the solid body, and Tf expresses fluid temperature (considered as constant). The law states that the rate of heat loss of solid material is proportional to the difference in temperatures between the solid body and its environment (fluid) [27]. In fact, Eq. (3) is a special version of Eq. (1). Moreover, it applies when the heat transfer coefficient is almost independent of the temperature difference. But, in reality, it depends on the temperature values and physical properties of the fluid. Therefore, Eq. (3) represents an approximation of real conditions suitable for practice, and k is set experimentally [26].

3.1.3 Radiation

The surface of matter with a temperature above absolute zero emits electromagnetic radiation due to the thermal motion of particles. This total radiated power is described by the Stefan-Boltzmann law [26]

   (4)

where σ = 5.67 ⋅ 10− 8 W/(m² K) is the Stefan-Boltzmann constant and ɛ represents the emissivity. For black-body surfaces, it holds that ɛ = 1; otherwise (for the so-called gray bodies), ɛ ∈ (0, 1).

The radiation has a particular energy that can be transferred to another surface. The total heat transfer caused by radiation is given by the difference between radiation leaving one surface (body 1) and that coming from another (body 2). For black bodies, the rate of energy transfer reads

   (5)

where subscript 1 holds for body 1, whereas subscript 2 is related to body 2. F is the so-called angle factor that expresses the proportion of the radiation leaving surface 1 and striking surface 2 [28]. The heat transfer rate for gray bodies is

   (6)

where

   (7)

In practice, these three heat transfer mechanics apply simultaneously. The combination of conduction and convection can usually be found, for example, when heat transfers through a wall from a hot fluid to the cold one.

3.2 Heat transfer between two fluids

If one intends to design an HX or to predict its performance, it is essential to relate the total heat transfer rate to measurable quantities such as the total surface area of an HX, the inlet and outlet fluid temperatures, the overall heat transfer coefficient, etc. The following two relations can be obtained by applying overall energy balances to the hot and cold fluids [29]:

   (8)

   (9)

Eq. (8) holds for the hot fluid (subscript H) whereas Eq. (9) is related to the cold fluid (subscript Cis the mass flow rate, h⋅(t) means the particular fluid enthalpy, and subscripts I and O stand for the inlet and the outlet conditions, respectively. These formulae hold under certain constraints, such as [29]:

(i)No energy loss appears to the surroundings.

(ii)The HX is at steady-state.

(iii)There are no negligible potential and kinetic energy changes.

If, in addition, specific heat capacities (cpH, cpC) of the fluids are independent of temperature and there is no phase change in the fluids, Eqs. (8) and (9) become

   (10)

   (11)

respectively, in which average fluid temperatures (at the particular location) are considered.

The fluid temperature is dependent qualitatively on the position inside an HX. Let us consider a single-pass flow HX, which is the most used flow arrangement [5]. Then, particular temperature profiles in co-current and counter-current flows are displayed in Figs. 9 and 10, respectively.

Fig. 9 The temperature profile in co-current-flow single-pass HX (the hot fluid is red/thick , the cold stream is blue/thin ).

Fig. 10 The temperature profile in counter-current-flow single-pass HX (the hot fluid is red/thick , the cold stream is blue/thin ).

3.2.1 Logarithmic mean temperature difference

Because temperature difference ΔT(x) is changing during the heat transfer process (see Figs. 9 and 10), there appears the question of how to assess the temperature difference in the sense of Eq. (2) or, (3) here. A mean temperature ought to be found.

If the heat transfer at a particular point x on the heat transfer surface is considered, then by the analysis of heat transfer from the surface to the fluid on cold and heat sides and by the combination of Eqs. (1), (3), (10), (11) and the analysis of the heat flux crossing the wall between the two fluids, followed by the integration, one can obtain the logarithmic mean temperature difference as

   (12)

where

   (13)

holds for the co-current flow, whereas

   (14)

applies for the counter-current flow configuration. Note that it may hold that THO < TCO [29].

Then, the basic heat transfer equation reads

   (15)

where kt is the total (overall) heat transfer coefficient.

Eqs. (12)–(14) are applicable to single-pass HXs only; however, they can be extended to multipass and crossflow configurations [10]. It holds that ΔT(x) = ΔT = const. only if the counter-current configuration is used and both heat capacity rates (for the cold and hot fluids) are equal.

3.2.2 Resistance to the heat transfer

When analyzing the overall heat transfer coefficient kt in Eq. (15), one has to consider the convective heat transfer from the hot fluid to the partition between fluids, the conductive heat transfer through the partition, and the convective heat transfer from the partition to the cold fluid. The process can also be characterized by the total heat (thermal) resistance Rt (K/W), which is the inverse of kt scaled by the surface area A, i.e.,

   (16)

where

   (17)

In Eq. (17), RH, RC are convective heat transfer resistances for the hot and cold fluids, respectively, and Rp means the conductive resistance for the partition.

The value of RH depends on the flow rate, its geometry, and fluid properties. This dependence is characterized by some dimensionless numbers, such as the Reynolds, Nusselt, and Prandtl numbers [30]. Their mutual relationship is affected by the fact whether the flow is laminar or turbulent. Analogously, the same statements hold for RC. The system geometry decides the value of Rp [29].

In fact, fouling causes additional resistance. The foulant yields an insulation layer that worsens the heat transfer from the hot fluid to the cold one. Moreover, the HX efficiency (see Section 3.3) decreases whereas the pressure drop increases [16].

In the case of a planar wall (see Fig. 11), it holds that

   (18)

where αC, αH (W/K) is the (heat) convection coefficient, λp (W/(m K)) represents the thermal conductivity of the wall, and dp (m) expresses its thickness. For other types of geometries (spherical, cylindrical), the reader is referred to Bergman et al. [4].