Energy Transfers by Convection

By Abdelhanine Benallou

Whether in a solar thermal power plant or at the heart of a nuclear reactor, convection is an important mode of energy transfer. This mode is unique; it obeys specific rules and correlations that constitute one of the bases of equipment-sizing equations.

In addition to standard aspects of convention, this book examines transfers at very high temperatures where, in order to ensure the efficient transfer of energy for industrial applications, it is becoming necessary to use particular heat carriers, such as molten salts, liquid metals or nanofluids. With modern technologies, these situations are becoming more frequent, requiring appropriate consideration in design calculations.

Energy Transfers by Convection also studies the sizing of electronic heat sinks used to ensure the dissipation of heat and thus the optimal operation of circuit boards used in telecommunications, audio equipment, avionics and computers.

• Language: English
• Publisher: Wiley
• Release date: Jan 3, 2019
• ISBN: 9781119579175

Book preview

Preface

The secret of change is to focus all your energy not on fighting the old, but on building the new.

Dan Millman (1946–), artist, author, record-holder and sportsman

For several years, I have cherished the wish of devoting enough time to the writing of a series of books on energy engineering. The reason is simple: for having practiced for years teaching as well as consulting in different areas ranging from energy planning to rational use of energy and renewable energies, I have always noted the lack of formal documentation in these fields to constitute a complete and coherent source of reference, both as a tool for teaching to be used by engineering professors and as a source of information summarizing, for engineering students and practicing engineers, the basic principles and the founding mechanisms of energy and mass transfers leading to calculation methods and design techniques.

But between the teaching and research tasks (first as a teaching assistant at the University of California and later as a professor at the École des mines de Rabat, Morocco) and the consulting and management endeavors conducted in the private and in the public sectors, this wish remained for more than twenty years in my long list of priorities, without having the possibility to make its way up to the top. Only providence was able to unleash the constraints and provide enough time to achieve a lifetime objective.

This led to a series consisting of nine volumes:

– Volume 1: Energy and Mass Transfers;

– Volume 2: Energy Transfers by Conduction;

– Volume 3: Energy Transfers by Convection;

– Volume 4: Energy Transfers by Radiation;

– Volume 5: Mass Transfers and Physical Data Estimation;

– Volume 6: Design and Calculation of Heat Exchangers;

– Volume 7: Solar Thermal Engineering;

– Volume 8: Solar Photovoltaic Energy Engineering;

– Volume 9: Rational Energy Use Engineering.

The present book is the third volume of this series. It concerns the study of convection heat transfer.

As we will see, the calculation methods established in this book present multiple applications in engineering: heat exchanger calculation, demand-side energy, improvement of heat dissipation in electronic circuits, etc.

A series of exercises is presented at the end of the book, aimed at enabling students to implement new concepts as rapidly as possible. These exercises are designed to correspond as closely as possible to real-life situations occurring in industrial practice or everyday life.

Abdelhanine BENALLOU

October 2018

Introduction

Whilst conduction is of great importance in the transmission of thermal energy through continuous media, particularly solid media, heat transfer in industrial equipment (heat exchangers, evaporators, etc.) very often involves exchanges between fluids (gas or liquid) and solid walls (Leontiev, 1985). Indeed, in most situations, we heat a liquid or gas by putting it in contact with a hot surface, or cool a fluid through contact with a cold surface. To resolve such problems, conduction equations are no longer applicable, given that there is a large difference: in convection, the fluid concerned is in motion.

In this document we will therefore be focusing on convective heat transfer between solids and fluids in motion.

Generally, the motion of the fluid is one of two types:

– It is induced by the temperature gradients existing in the system, without the intervention of any external device (pump, compressor or other). In this case we speak of natural or free convection.

– It is created by an external device that sets the fluid in motion (pump, fan, etc.). In this case we speak of forced convection.

Yet whether the motion is natural or forced, we are often faced with a problem where the equations describing the fluid dynamics are coupled with those reflecting the heat balance. It will therefore be necessary to simultaneously solve the equations reflecting the energy and momentum balances (Navier-Stokes equation: see Bird, Stewart and Lightfoot, 1975; Kays and Crawford, 1993; Giovannini and Bédat, 2012). The problem is therefore far from straightforward and this difficulty has driven various researchers to propose approximation methods leading to practical results.

Since the approximation proposed by Boussinesq (1901) to solve the natural convection equations, several analyses have been developed to provide answers to classic problems of thermal engineering, such as the transfer of heat to a fluid in forced circulation within a tube (the Graetz problem), the heat transfer between a fluid and a flat plate (the Blasius problem), or the heat transfer generated by Archimedes forces in natural convection (the Boussinesq problem).

These different analyses lead to solutions that may be implemented using numerical tools (Landau and Lifshitz 1989; Kays and Crawford 1993; Giovannini and Bédat, 2012). This is done using certain approximations, notably regarding the incompressibility of Newtonian fluids in forced convection. However, the validity of these approximations is sometimes disputed (Boussinesq approximation: see Lagrée, 2015), and their real justification is, above all, related to the simplifications necessary in order to soften the rebel equations, as Boussinesq himself calls them:

Thanks to the simplifications then obtained, the question, still very difficult and almost always rebellious to integration, is no longer unaffordable. (Boussinesq, 1901)

The results of the analytical developments are available in various works (Knudsen and Katz, 1958; Bird, Stewart and Lightfoot, 1975; Landau and Lifshitz, 1989; Kays and Crawford, 1993; Giovannini and Bédat, 2012). Yet, the complexity of the developments and the tools to be implemented for their application makes them unsuited to our perspective, which is to provide fast orders of magnitude for engineering calculations. Thus, where the analytical demonstration allows, and particularly to give an example of the complexity that might be encountered, the equations used to determine the transfer coefficient are established; this is the case for forced convection inside a pipe in laminar flow (see Chapter 2).

In the opposite case, which represents the majority of situations, dimensional analysis is applied to determine the general expressions of the transfer coefficients in forced and natural convections, then the correlations available for calculating these coefficients are presented.

It is for this reason that in Chapter 1 of this volume, we propose to proceed with a phenomenological analysis of convection, based on a representation of all the complexities by a convection heat transfer coefficient, h, such that the heat flux transferred by convection is given by (see Volume 1, Chapter 2):

where:

A is the transfer area

h is the convection heat transfer coefficient, also known as the convective heat transfer coefficient

Δθ = θ2 – θ1 is the heat transfer potential difference. This is the temperature gradient between the fluid and the solid being considered.

In this way, all the analytical complexity of the problem is grouped together in the coefficient h, for which dimensional analysis reveals the expression structure.

Chapters 2 and 3 present the correlations which make it possible to calculate the coefficient h for a fluid circulating in forced convection inside cylindrical pipes or pipes of other shapes. Forced convection on the outside of pipes or around objects is dealt with in Chapter 4.

Yet, while forced convection is the most common case in industrial installations where the fluid is set in motion by means of fans, pumps or compressors, the fact remains that several natural convection situations are also encountered: central-heating radiators, cold rooms, electronic heat sinks, etc. For this reason, Chapter 5 is dedicated to this type of convection.

Moreover, for specific circuits operating at very high temperatures (around 700°C), the usual heat-transfer fluids are no longer suitable. This is the case for concentrated solar power plants or fast neutron nuclear reactors where liquid metals, nanofluids and, sometimes, molten salts are used to ensure the transfer of the energy produced. Of course, the correlations established for ordinary fluids are no longer valid. Correlations specific to this type of situation are developed in Chapter 6, whilst the physical data required for the calculations are grouped together in Appendix 1 (database).

1

Methods for Determining Convection Heat Transfer Coefficients

1.1. Introduction

In convection, heat transfer occurs between a fluid in motion and a neighboring surface. In reality, the fact that the fluid is in motion is the most specific aspect of convection. It logically follows that the convection heat transfer coefficient will depend intrinsically on the nature of the flow established. In fact, intuitively, we do not expect to have the same heat transfer in a fluid flowing in a laminar regime and a fluid in turbulent flow.

As such, when determining convection heat transfer, the first step is always to characterize the motion of the fluid.

1.2. Characterizing the motion of a fluid

Let us recall here that the flow of a fluid is characterized by a dimensionless number, known as the Reynolds number, Re (Bird, Stewart and Lightfoot, 1975; Knudsen and Katz, 1958; Landau and Lifshitz, 1989), defined by:

math

where:

d is the diameter of the pipe where the flow occurs

v is the flow velocity of the fluid

ρ is the density of the fluid

μ is the viscosity of the fluid

Two structurally different flow regimes can be encountered, depending on the Reynolds number (see Figure 1.1):

– a laminar flow for Re ≤ 2,300: in this regime the fluid flow is orderly, moving in parallel layers;

– a turbulent flow for Re > 4,000: in this regime the fluid flow is disorderly, but generally results in the fluid traveling in a clearly defined direction.

The interval corresponding to Reynolds numbers whereby 2,300 < Re < 4,000 corresponds to a transition zone between these two regimes.

Figure 1.1. Flow regimes

It should be noted that the limits between the different flow areas defined in Figure 1.1 are not as fixed, as they depend to a certain extent on the situations involved and the fluids used: whether the pipes are smooth or not, whether the fluids are viscous are not, etc. Consequently, it is not unusual to find that different authors consider different limits for the switching of the flow regimes.

Indeed, depending on the situation considered, the switch from the laminar zone to the transition zone can spread out from Re = 2,1000 to Re = 2,400. Likewise, the switch from the transition zone to the turbulent zone can span from Re = 2,400 to Re = 10,000. For the purposes of the engineering calculations of interest to us, however, we will retain the limits shown in Figure 1.1.

The flow regime of a fluid can be determined systematically by following the flowchart presented in Figure 1.2.

Figure 1.2. Determining flow regime

1.3. Transfer coefficients and flow regimes

In the various studies of interest to the engineer, we are called upon to determine the flux of energy exchanged by convection between a fluid (gas or liquid) in motion and a solid wall (pipe, furnace wall, etc.) that is in contact with said fluid.

It should be recalled that, whatever the flow regime (laminar or turbulent), the flux exchanged between the fluid and the wall is given by ϕ = h A (Δθ) where:

A is the transfer area between the fluid and the wall

Δθ is the difference between the average temperature of the fluid and that of the wall

h is the convection heat transfer coefficient

Intuitively, we can make the following observations regarding the influence of the flow regime on the convective flux (see Figure 1.3):

– in laminar flow, the orderly movement of the fluid in parallel layers results in limited renewal of the layers of the fluid that are in contact with the solid wall;

– in contrast, the disorderly motion of the fluid in turbulent flow results in higher rates of contact surface renewal.

The heat flux corresponding to turbulent flow, due to higher surface renewal rates, will be the larger in the laminar case:

math

These intuitive observations reflect the fact that energy transfer will be more effective in turbulent flow than in laminar flow, and also that the convection heat transfer coefficients resulting from turbulent flows will be much greater than those obtained for laminar flows.

Figure 1.3. Laminar and turbulent convective fluxes

Thus, determining the transfer coefficients is of great importance when calculating the fluxes transferred by convection. These coefficients are determined from correlations established based on experimentation and dimensional analysis.

1.4. Using dimensional analysis

As we have seen (Volume 1, Chapter 4), dimensional analysis is generally used when we wish to establish one or more equations (correlations) describing a given phenomenon or physical parameter, based on a series of experiments. It makes it possible to describe the evolution of a system through the variations of dimensionless numbers. The use of such numbers makes it possible to render the use of correlations that are general and independent of the system of units considered.

1.4.1. Dimensionless numbers used in convection

The different dimensionless numbers used in engineering calculations were defined in Chapter 4 of the first volume in this set. Let us recall in this section that the dimensionless groups (or numbers without dimensions) usually used to describe heat transfer by convection are:

– The Nusselt number: math where:

h is the convection heat transfer coefficient, also known as the specific heat transfer coefficient

d is the inner or outer diameter of the pipe (depending on whether we are interested in the heat transfer between a fluid and the inner or outer wall of the pipe)

λ is the heat conductivity of the fluid

– The Prandtl number: math where:

Cp is the specific heat of the fluid

μ is the viscosity of the fluid

λ is the heat conductivity of the fluid

It should be recalled that we can also write: math .

I.e. math , where: math and math .

– The Péclet number: math where:

d is the inner or outer diameter of the pipe

v is the average velocity of the fluid

ρ is the density of the fluid

Cp is the specific heat of the fluid

λ is the heat conductivity of the fluid

We recall that: Pe = Re Pr.

– The Grashof number: math , where:

d is the inner or outer diameter of the pipe

ρ is the density of the fluid

g is the acceleration of gravity

β is the volumetric expansion coefficient of the constant-pressure fluid

Δθ is the difference between the wall temperature and the average fluid temperature, or vice versa

– The Stanton number: math , where:

h is the convection heat transfer coefficient

Cp is the specific heat of the fluid

v is the average velocity of the fluid

ρ is the density of the fluid

We also recall here that: math .

– The Graetz number: math , where:

W is the fluid mass flow rate

Cp is the specific heat of the fluid

λ is the heat conductivity of the fluid

L is the characteristic dimension of the area considered: length or width

– The Elenbaas number: math where:

L is the length of the base comprising fins

Z is the distance between two fins

ρ is the density of the fluid

β is the volumetric expansion coefficient of the constant-pressure fluid

g is the acceleration of gravity

Δθ is the difference between the wall temperature of the fins and the temperature of the fluid at large

Cp is the specific heat of the fluid

μ is the viscosity of the fluid

1.4.2. Dimensional analysis applications in convection

1.4.2.1. Application in forced convection inside a tube

Consider a fluid flowing within a cylindrical pipe of diameter d and length L.

The specific heat transfer coefficient at the surface of the tube, h, is a function of:

– the physical properties of the fluid, ρ, μ, Cp, λ;

– its average velocity, v;

– the characteristics of the pipe, d, L.

We can therefore write that the convection heat transfer coefficient is a function, f, of this set of parameters: h = f (ρ, μ, Cp, λ, v, d, L).

We assume that the function, f, can develop in a series of the form:

math

Thus, each term in the series must have the dimension of h, namely:

math

Yet:

mathmath

Under these conditions, the dimensional equation of the series gives:

math

or:

math

This dimensional equation can be broken down as follows:

– for dimension M: 1 = αi + βi + δi;

– for dimension L: 0 = 2 γi + δi + εi + σi + ηi –3 αi – βi;

– for dimension T: – 3 = – βi – 2 γi – 3 δi – εi;

– for dimension θ: – 1 = – γi – δi.

We thus obtain the system of equations: math .

The series giving h can thus be written in the form:

math

Or alternatively by grouping together the terms of power αi:

math

By proceeding in the same way with the terms of power γi, we obtain:

math

Lastly, by grouping together the terms of power σi, we arrive at:

math

Thus, dimensional analysis shows that the Nusselt number can be expressed as a function of the powers of Re, Pr and math .

This makes it possible to orient the experiments to be conducted in order to determine the correlations that give the Nusselt number. Indeed, the correlations sought need to be of the form:

math

Parameters a, m, n and σ are to be determined from experiments during which we successively vary:

– the Reynolds number, whilst keeping the other parameters constant: for example, for a given fluid and a set tube diameter, we can vary the fluid’s velocity by varying its flow rate;

– the Prandtl number, whilst keeping the other parameters constant by changing the nature of the fluid;

– the ratio math , whilst keeping the other parameters constant by taking, for example, different tube lengths.

Several researchers have worked on these types of experiments to determine the different parameters for flow situations in forced convection. This has led to correlations that often bear the names of the researchers who developed them. These correlations generally enable the calculation of the Nusselt number, and therefore the convection heat transfer coefficient, h, as a function of the Reynolds and Prandtl numbers.

This chapter presents the most important of these correlations, which are of great use when performing calculations in forced convection inside or outside of pipes.

1.4.2.2. Application in natural convection along a tube

We know that the phenomenon of natural convection must depend not only on the physical properties of the fluid considered, but also on the expansion coefficient, β, as well as of the acceleration of gravity, g, and the temperature difference, Δθ. We can thus write: h = f (β, g, Δθ, d, ρ, μ, Cp, λ).

If we proceed as above, we can admit that the function, f, can develop in a series of the form:

math

Thus, each term in the series must have the dimension of h, namely:

math

Yet:

math

Hence the dimensional equation:

math

This dimensional equation can be broken down as follows:

– for dimension M: 1 = γi + δi + εi;

– for dimension L: 0 = αi + βi – γi + δi – 3 εi + 2 ηi;

– for dimension T: – 3 = – 2 αi – γi – 3 δi – 2 ηi;

– for dimension θ: – 1 = – δi – ηi.

We thus obtain the system of equations: math .

Thus, by grouping together the terms of power εi and those of power ηi, the series giving h can be written in the form:

math

Since math , we can conclude that the Nusselt number will be a function of the Prandtl number and the Grashof number.

I.e. Nu = C Pra Grb.

Thus, by applying dimensional analysis to natural-convection problems, we are able to show that the Nusselt number can be expressed as a function of the Grashof and Prandtl numbers. Parameters C, a and b and are to be determined from experiments during which we successively vary:

– the Prandtl number, whilst keeping the other parameters constant by changing the nature of the fluid;

– the Grashof number, whilst keeping the other parameters constant, for example by varying the temperature difference, Δθ.

Several experiments of this type have been conducted and have enabled appropriate parameters to be determined. Chapter 5 of this volume presents the most significant of these correlations, which make it possible to calculate, in cases of natural convections, the Nusselt number, and therefore the convection heat transfer coefficient, h, as a function of the Grashof and Prandtl numbers.

1.5. Using correlations to calculate h

Let us recall that, in order to calculate convective fluxes, it is essential to know the convection heat transfer coefficient, h. This coefficient is calculated based on the physical data relating to the problem being considered: the nature of the fluid (ρ, Cp, λ, μp, etc.), the flow type (forced or natural), flow rate or velocity, pipe type (characteristic length: diameter or radius). Knowledge of these different parameters enables the dimensionless numbers to be calculated: Pr, then Re (in the case of forced convection) or