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The control of energy in the industrial sector and the reduction of consumption in the building sector will be key elements in the energy transition. In order to achieve these objectives it is necessary to use materials with energy performance adapted to their use as well as insulators or super-insulators. In both cases, a thorough knowledge of their thermal properties will be required for optimal success.
This revised and updated 2nd edition of Thermal Properties Measurement of Materials enables the reader to choose the measurement method best suited to the material they are characterizing and provides all of the information required in order to implement it with maximum precision.
This work is intended to be accessible to anyone who needs to measure the thermal properties of a material, whether or not they are a thermal engineer.
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Veröffentlichungsjahr: 2024
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Revised and Updated 2nd Edition
Thermal Properties Measurement of Materials
Yves Jannot
Alain Degiovanni
First edition published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc., © ISTE Ltd 2018.
This edition published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2024The rights of Yves Jannot and Alain Degiovanni to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2024941895
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-883-2
Preface
Why write a book on thermal metrology and more specifically on the measurement of the thermal properties of materials?
The answer to this question is twofold:
to know and improve materials;
to have an input for calculating the modeling of complex systems.
Specialists in materials engineering are capable of manufacturing new materials with specific properties on demand, making it imperative to be able to characterize them as precisely as possible, and thus to guide new research.
The second challenge is to have an input for the calculation codes under good conditions. Indeed, the latter, by becoming more and more efficient, demands more and more precise parameter values; at each improvement of the codes, measurements need to be improved.
We could add a more societal reason to these two technical reasons: the control of energy, which goes through the use of efficient insulating materials in particular. The latter is of the utmost importance in highly energy-intensive sectors, in particular:
industries that are very energy-intensive, such as glass furnaces, cement factories or the steel industry;
buildings that need to develop more efficient thermal insulation;
transport with reduced structures.
This book is therefore accessible to all, heat transfer specialists and non-specialists, who need to measure the thermal properties of a material. The objective is to allow them to choose the measurement method best suited to the material to be characterized and to transmit to them all the information allowing the implementation of the measurement method with maximum precision.
The first chapter presents the different modes of heat transfer and the thermal properties of materials. Heat transfer equations are solved for a number of cases, which will appear later in the presented characterization methods.
The second chapter describes the tools and methods of thermal metrology, beginning with the often underestimated problem of temperature measurement. Theoretical tools such as the study of reduced sensitivities and methods of parameter estimation are then presented.
The following three chapters describe the main thermal characterization methods classified into steady-state type, transient flow/temperature type and transient temperature/temperature type, each method being presented at two levels:
The first part describes the theoretical aspects of the method with a modeling of the heat transfer in the measuring device. This theoretical study most often makes it possible to determine the limits of the method’s application in terms of values of the thermal properties and dimensions of the materials to be characterized on the one hand and measurement duration in the case of a transient method on the other hand.
The second part describes the experimental device, the measurement procedure and the mode of exploitation of the results in detail. This second part, being practical in nature, is intended for heat transfer non-specialists and can be approached independently from the first part. The application conditions set by the theoretical study will be systematically recalled in a simple manner.
This second edition was enriched with the presentation of five recently developed methods enabling characterization in extreme conditions: for superinsulating materials, for highly anisotropic materials and measurements at very high temperatures.
We present a synthesis chapter summarizing the fields of application of each of the methods presented and proposing the choice of the most adapted methods for different materials and dimensions.
In the case of a porous medium, the calculation of a temperature field and the heat flows passing through it does not only depend on the medium’s thermal properties. It is also influenced by mass transfers (water vapor in particular) and by any air flow that can pass through the medium. Complete resolution of the equations requires knowing, in additional to thermal properties, the following quantities: the mass diffusion coefficient of water vapor, porosity and permeability. After having shown the strong analogy between heat transfer equations and equations governing mass transfer and flow of a fluid in a porous medium, a new chapter has been dedicated to the most commonly used methods to measure these parameters.
Additional documents in appendix form providing numerous additional data are available for download.
Happy measuring!
July 2024
List of Notations
a
Thermal diffusivity, m
2
s
−1
.
b
Klinkenberg coefficient, Pa.
Bi
Biot number.
c
Specific heat capacity, J kg
−1
K
−1
.
D
Diameter, m; mass diffusivity, m
2
s
−1
.
E
Thermal effusivity, J m
−2
K
−1
s
−1/2
.
e
Thickness, m.
F(t)
Transfer function in real space.
f
Frequency, Hz.
Fo
Fourier number.
g
Acceleration of gravity, m s
−2
.
Gr
Grashof number.
Δ
H
Latent heat of phase change, J kg
−1
.
h
Convective heat transfer coefficient, W m
−2
K
−1
.
h
r
Radiative heat transfer coefficient, W m
−2
K
−1
.
H(p)
Transfer function in Laplace space.
I
0
,
I
1
Modified Bessel functions of the first kind.
I
Intensity, A.
K
0
,
K
1
Modified Bessel functions of the second kind.
k
Permeability, m
2
.
L, ℓ
Length, m.
Laplace transform operator.
Mass flow rate, kg s
−1
.
m
Mass, kg.
Nu
Nusselt number.
Pr
Prandtl number.
P
Pressure, Pa.
P
e
Perimeter, m.
P
Laplace variable, s
−1
.
q
Surface energy density, J m
−2
.
Power density, W m
−3
.
Q
Heat quantity, J.
r,R
Radius, m.
R
c
Thermal contact resistance, K W
−1
.
R
el
Electrical resistance, Ω.
Re
Reynolds number.
S
Surface area, m
2
.
T
Temperature, K.
t
Time, s.
U
Tension, V.
u
Velocity, m s
−1
.
V
Volume, m
3
.
X
Dry basis water content, kg
w
kg
dm
−1
.
x,y,z
Space variables.
Z
Impedance, K W
−1
.
Greek letters
α
Radiation absorption coefficient.
β
Coefficient of volume expansion.
δ
Thickness, m.
ε
Emissivity, porosity.
η
Yield or efficiency.
θ
Temperature Laplace transform.
λ
Thermal conductivity, W m
−1
K
−1
; wavelength, m.
μ
Dynamic viscosity, Pa s.
ν
Kinematic viscosity, m
2
s
−1
.
ρ
Density kg m
−3
, radiation reflection coefficient.
σ
Stefan–Boltzmann constant, W m
−2
K
4
.
τ
Time constant, s.
ϕ
Heat flux, W m
−2
.
φ
Heat flow rate, W.
Φ
Heat flow rate Laplace transform.
Ω
Solid angle sr.
ω
Pulsation, rd.
1Modeling of Heat Transfer
This chapter presents a reminder of courses on heat transfer limited to what is necessary to understand and master the methods of measuring the thermal properties of materials, which will be described in the rest of this book.
1.1. The different modes of heat transfer
1.1.1. Introduction and definitions
We will first define the main quantities involved in solving a heat transfer problem.
1.1.1.1. Temperature field
Energy transfers are determined from the evolution of the temperature in space and time: T = f(x, y, z, t). The instantaneous value of the temperature at any point of space is a scalar quantity called a temperature field. We will distinguish two cases:
time-independent temperature field: the regime is called steady state or stationary;
evolution of the temperature field over time: the regime is called variable, unsteady or transient.
1.1.1.2. Temperature gradient
If all the points of space which have the same temperature are combined, an isothermal surface is obtained. The temperature variation per unit length is maximal in the direction normal to the isothermal surface. This variation is characterized by the temperature gradient:
where:
: the normal unit vector;
: the derivative of the temperature along the normal direction.
Figure 1.1.Isothermal surface and thermal gradient
1.1.1.3. Heat flux
Heat flows under the influence of a temperature gradient from high to low temperatures. The quantity of heat transmitted per unit time and per unit area of the isothermal surface is called the heat flux ϕ (W m−2):
where S is the surface area (m2).
Note that φ(W) is called the heat flow rate and is the quantity of heat transmitted to the surface S per unit time:
1.1.1.4. Energy balance
The determination of the temperature field involves the writing of one or more energy balances. First, a system (S) must be defined by its limits in space and the different heat flow rates that influence the state of the system must be established and can be seen in Figure 1.2.
Figure 1.2.System and energy balance
The first principle of thermodynamics is then applied to establish the energy balance of the system (S):
After having replaced each of the terms by its expression as a function of the temperature, we obtain a differential equation whose resolution, taking into account the boundary conditions of the system, makes it possible to establish the temperature field. We will first give the possible expressions of the heat flow rates that can enter or exit a system by conduction, convection or radiation before giving an expression of the flux stored by sensible heat.
NOTE.– In steady state: φst = 0.
1.1.2. Conduction
Conduction is the transfer of heat within an opaque medium, without displacement of matter, under the influence of a temperature difference. The transfer of heat via conduction within a body takes place according to two distinct mechanisms: transmission via atomic or molecular vibrations and transmission via free electrons.
Figure 1.3.Conductive heat transfer scheme
The theory of conduction is based on the Fourier hypothesis: the heat flux ϕ is proportional to the temperature gradient:
The heat flow rate transmitted by conduction in the direction Ox can therefore be written in algebraic form:
where:
ϕ
: the conductive heat flux (W m
−2
);
φ
: the conductive heat flow rate (W);
λ
: the thermal conductivity of the medium (W m
−1
K
−1
);
x
: the space variable in the heat flow’s direction (m);
S
: the surface area of the passage of the heat flux (m
2
).
The values of the thermal conductivity λ of some of the most common materials are given in Table 1.1. A more complete table is given in Appendices 1 and 2.
Table 1.1.Thermal conductivity of certain materials at room temperature
Material
λ
(W m
−1
K
−1
)
Silver
419
Copper
386
Aluminum
204
Mild steel
45
Stainless steel
15
Ice
1.9
Concrete
1.4
Clay brick
1.1
Glass
1.0
Water
0.60
Plaster
0.48
Asbestos
0.16
Wood (hard, soft wood)
0.12–0.23
Cork
0.044–0.049
Stone wool
0.038–0.041
Glass wool
0.035–0.051
Expanded polystyrene
0.036–0.047
Polyurethane (foam)
0.030–0.045
Extruded polystyrene
0.028
Air
0.026
1.1.3. Convection
Figure 1.4.Convective heat transfer scheme
Here, we will only consider the heat transfer between a solid and a fluid, the energy being transmitted by the fluid’s displacement. A good representation of this transfer mechanism is given by Newton’s law:
where:
φ
: the heat flow rate transmitted by convection (W);
h
c
: the convective heat transfer coefficient (W m
−2
K
−1
);
T
p
: the solid’s surface temperature (K);
T
∞
: the temperature of fluid away from solid surface (K);
S
: the area of solid/fluid contact surface (m
2
).
The value of the convective heat transfer coefficient hc is a function of the fluid’s nature, temperature, velocity or the temperature difference and the geometrical characteristics of the solid/fluid contact surface. The correlations in the most common cases of natural convection are given in Appendix 3, that is, when the fluid’s movement is due to temperature differences (no pump or fan).
Thermal characterization aims to measure the conductive and diffusing properties of a material. Convection most often occurs as a mode of “parasitic” transfer on the boundaries of the system by influencing the internal temperature field. We therefore have to take this into account. The correlations presented in Appendix 3 show that the coefficient of heat transfer by natural convection depends on the temperature difference between the surface and the surrounding fluid. Most often this difference is not perfectly uniform on surfaces and varies over time. It is therefore not possible to calculate it precisely and it will most often have to be estimated.
In natural convection, its value is generally between 2 and 5 W m−2 K−1. The radiation heat transfer coefficient that will be defined below is of the same order of magnitude. It will therefore be noted that placing a device under vacuum makes it possible to reduce losses by decreasing convective transfers but not canceling them, because radiation transfer is not affected by pressure.
1.1.4. Radiation
Radiation is a transfer of energy by electromagnetic waves (it does not need material support and even exists in a vacuum). We will only focus here on the transfer between two surfaces. In conduction problems, we take into account the radiation between a solid (whose surface is assumed to be gray and diffusing) and the surrounding environment (of large dimensions). In this case, we have the following equation:
where:
φ
: the radiation heat flow rate (W);
σ
: Stefan’s constant (5.67 × 10
−8
W m
−2
K
−4
);
ε
p
: the surface emission factor;
T
p
: the surface temperature (K);
T
∞
: the temperature of the medium surrounding the surface (K);
S
: the area of surface (m
2
).
Figure 1.5.Radiation heat transfer scheme
NOTE.– In equations [1.6] and [1.7], temperatures can be expressed either in °C or K because they appear only in the form of differences. On the contrary, in equation [1.8], the temperature must be expressed in K.
1.1.4.1. Linearization of the radiation flux
In the case where the fluid in contact with the surface is a gas and where the convection is natural, the radiation heat transfer with the walls (at the average temperature Tp) surrounding the surface can become of the same order of magnitude as the convective heat transfer with the gas (at temperature T∞) at the contact with the surface and can no longer be neglected. The heat flow rate transferred by radiation is written according to equation [1.8]:
It can take the form:
where hr is called the radiation transfer coefficient:
The radiation transfer coefficient hr varies very slightly for limited variations of the temperatures Tp and T∞ and can be regarded as constant for a first simplified calculation. For example, with ε = 0.9, Tp = 60°C and T∞ = 20°C, the exact value is hr = 6.28 W m−2 K−1. If Tp becomes equal to 50°C (instead of 60°C), the value of hr becomes equal to 5.98 W m−2 K−1, and we get a variation of only 5%. When Tp is close to T∞, we can consider: hr ≈ 4σεT∞3. It is also noted that the calculated values are of the order of magnitude of a natural convection coefficient in air.
It is to be remembered that when the convection exchange of a surface with its environment takes place by natural convection, we write the global heat flow rate (convection + radiation) exchanged by the surface in the form of: φ = hS(Tp − T∞), where h = hc + hr.
1.1.4.2. Case of a high-temperature source
The radiation transfer also occurs in the exchange of a wall with temperature Tp with a high-temperature heat source at Ts, for example, the Sun (≈ black body at 5,760 K). In this case, equation [1.8] becomes:
where K is a constant taking into account the surface and source emissivities as well as the geometric shape factor between the surface and the source.
If: Ts ≫ Tp, then equation [1.10] becomes: φ = −KSTs4 = −φ0 and the radiation source is modeled by a constant heat flow rate φ0 imposed on the wall (this is, for example, the case of a thermal sensor exposed to the Sun).
1.1.5. Heat storage
The storage of energy in sensitive form in a body corresponds to an increase of its enthalpy in the course of time from which (at constant pressure and in the absence of change of state):
where:
φ
st
: the stored heat flow rate (W);
ρ
: density (kg m
-3
);
V
: volume (m
3
);
c
: specific heat (J kg
−1
K
−1
);
T
: temperature (K);
t
: time (s).
The product ρVc (J K−1) is called the thermal capacitance of the body.
1.2. Modeling heat transfer by conduction
1.2.1. The heat equation
In its mono-dimensional form, this equation describes the one-directional transfer of heat through a flat wall (see Figure 1.6). Consider a system of thickness dx in the direction x and section of area S normal to direction Ox. The energy balance of this system is written as:
where:
Figure 1.6.Thermal balance of an elementary system
By plotting in the energy balance and dividing by dx, we obtain:
or:
In the three-dimensional case, we obtain the heat equation in the most general case:
This equation can be simplified in a number of cases:
If the medium is isotropic:
λ
x
=
λ
y
=
λ
z
=
λ.
If there is no generation of energy inside the system: .
If the medium is homogeneous,
λ
is only a function of
T
.
The hypotheses (a) + (b) + (c) make it possible to write:
If
λ
is constant (moderate temperature deviation), we obtain the Poisson equation:
The ratio is called thermal diffusivity (m2 s−1); it characterizes the propagation velocity of a heat flux through a material. Values can be found in Appendix 1.
In steady state, we obtain Laplace’s equation:
Moreover, hypotheses (a), (c) and (d) make it possible to write:
heat equation in cylindrical coordinates (
r,θ,z
):
In the case of a cylindrical symmetry problem where the temperature depends only on r and t, equation [1.16] can be written in simplified form: ;
heat equation in spherical coordinates (
r,θ,φ
):
1.2.2. Steady-state conduction
1.2.2.1. Simple wall
We will place ourselves in the situation where the heat transfer is one directional and where there is no energy generation or storage.
We consider a wall of thickness e, thermal conductivity λ and large transverse dimensions whose extreme faces are at temperatures T1 and T2 (see Figure 1.7).
Figure 1.7.Basic thermal balance on a simple wall
By carrying out a thermal balance on the system (S) constituted by the wall slice comprised between the abscissae x and x + dx, we obtain:
where: and T(x) = A x + B
with boundary conditions: T(x = 0) = T1 and T(x = e) = T2.
Thus:
The temperature profile is therefore linear. The heat flux passing through the wall is deduced by equation: .
Thus:
Equation [1.19] can also be written as: ; this equation is analogous to Ohm’s law in electricity that defines the intensity of the current as the ratio of the electrical potential difference on the electrical resistance. The temperature T thus appears as a thermal potential and the term appears as the thermal resistance of a plane wall of thickness e, thermal conductivity λ and lateral surface S. We thus get the equivalent network represented in Figure 1.8.
Figure 1.8.Equivalent electrical network of a single wall
NOTE.– The heat flow rate is constant; it is a general result for any tube of flux in steady state (system with conservative heat flow rate).
1.2.2.2. Multilayer wall
This is the case for real walls (described in Figure 1.9) made up of several layers of different materials and where only the temperatures Tf1 and Tf2 of the fluids that are in contact with the two faces of the lateral surface wall S are known.
In steady state, the heat flow rate is preserved when the wall is crossed and is written as:
where:
It was considered that the contacts between the layers of different natures were perfect and that there was no temperature discontinuity at the interfaces. In reality, given the roughness of the surfaces, a micro-layer of air exists between the surface hollows contributing to the creation of a thermal resistance (the air is an insulator) called thermal contact resistance. The previous formula is then written as:
NOTE.–
A thermal resistance can only be defined on a flux tube.
The thermal contact resistance between two layers is neglected if one of them is a thermal insulator or if the layers are joined by welding.
Figure 1.9.Schematic representation of heat flow and temperatures in a multilayer wall
The equivalent electrical diagram is shown in Figure 1.10.
Figure 1.10.Equivalent electrical network of a multilayer wall
1.2.2.3. Composite wall
This is the case most commonly encountered in reality where the walls are not homogeneous. Let us consider, by way of example, a wall of width L consisting of hollow agglomerates (Figure 1.11).
Considering the symmetries, the calculation of the wall’s thermal resistance can be reduced to that of a unit cell defined by the diagram in Figure 2.6. This unit cell is a flux tube (isotherm at x = 0 and x = e1 + e2 + e3, and adiabatic at y = 0 and y = ℓ1 + ℓ2 + ℓ3) and can therefore be represented by a resistance R.
The exact calculation of this resistance is complex because each medium does not constitute a flux tube. If there is no need for high accuracy, several approximate calculations are possible by making assumptions on isothermal and adiabatic surfaces.
For example, assuming the surfaces at x = e1 and x = e1 + e2 to be isothermal and the surfaces at y = ℓ3 and y = ℓ2 + ℓ3 adiabatic and by using the laws of association of the resistors in series and in parallel, we obtain:
It will be noted that this solution is only approximate since the real transfer is 2D because of the differences between the thermal conductivities λ1 and λ2.
Here:
which can be represented by the equivalent electrical network shown in Figure 1.12.
Figure 1.11.Diagram of a composite wall
Figure 1.12.Electrical equivalent network of a composite wall
1.2.2.4. Long hollow cylinder (tube)
We consider a hollow cylinder with a thermal conductivity λ, internal radius r1, external radius r2, length L and with internal and external face temperatures being T1 and T2, respectively (see Figure 1.13). It is assumed that the longitudinal temperature gradient is negligible compared to the radial gradient.
Let us carry out the thermal balance of the system constituted by the part of the cylinder comprised between radii r and r + dr : φr = φr+dr.
where: and .
Thus: where:
with boundary conditions: T(r1) = T1 and T(r2) = T2.
Here:
Figure 1.13.Diagram of transfers in a hollow cylinder
By applying equation , we obtain:
This equation can also be written as: with and can be represented by the equivalent electrical network of Figure 1.14.
Figure 1.14.Equivalent electrical network of a hollow cylinder
1.2.2.5. Multilayer hollow cylinder
This is the practical case of a tube covered with one or more layers of different materials and where only temperatures Tf1 and Tf2 of the fluids in contact with the inner and outer faces of the cylinder are known; h1 and h2 are the global (convection + radiation) heat transfer coefficients between the fluids and the internal and external faces (see Figure 1.15).
Figure 1.15.Diagram of heat transfers in a multilayer hollow cylinder
In steady-state conditions, the heat flow rate φ is conserved when passing through the different layers and is written as:
Hence:
which can be represented by the equivalent electrical network of Figure 1.16.
Figure 1.16.Equivalent electrical network of a multilayer hollow cylinder
1.2.2.6. Fins
We consider a cylindrical bar of section S (in the Oyz plane) and length L (following x). The bar is made up of a homogeneous and isotropic material (thermal conductivity λ). The boundaries along x (in x = 0 and x = L) are subject to uniform conditions. The lateral boundary is subject to heat exchange through an exchange coefficient h with the constant and uniform exterior temperature T∞. We call pe the perimeter of the bar.
Heat transfer is 3D. The so-called fin hypothesis simplifies the problem by assuming that the straight sections of the bar are isothermal.
Consider the example of a cylindrical part of a circular section (of radius R) exchanging with the surrounding environment on its lateral surface as represented in Figure 1.17.
Figure 1.17.Representation of a problem in cylindrical geometry.
The fin hypothesis is justified if (internal thermal resistance ≪ external thermal resistance; see equation [1.32]), where is the Biot number corresponding to heat transfer in a radial direction. We pass from a 2D axisymmetric problem in T(x, r) to a 1D problem in T(x).
This involves establishing the energy conservation equation; it can be established without it being necessary to specify the shape of the cross section provided that it has a constant area.
We carry out a heat balance on a slice of length dx (see Figure 1.18) and obtain:
in
x
: conductive flow ;
in
x
+
dx
: conductive flow ;
where S is the straight section of the bar.
At the solid–fluid interface:
ϕ
c
=
hp
e
dx
(
T
−
T
∞
).
Here,
p
e
dx
is the lateral exchange surface.
Figure 1.18.Diagram of a heat balance
The balance is written as:
The so-called “fin” equation is therefore:
It is valid whatever pe and S, respectively, the perimeter and the straight section of the fin. By posing: and θ = T − T∞, it can still be written as: .
It is a second-order differential equation with constant coefficients whose general solution is of the form:
The solution of the fin equation depends on the boundary conditions in x = 0 and in x = L.
Case 1: The fin is isolated at its end, and the boundary conditions are:
The solution is written as:
Case 2: We take into account the heat exchanges in x = L. The boundary conditions are given as:
The solution is written as:
The solution to the fin equation (equation [1.25]) can be put in matrix form by introducing the flow in x = 0 and x = L, that is:
thus:
φ
0
,
φ
L
: heat flow rates in
x
= 0 and ;
θ
0
,
θ
L
: temperature differences
T
(0) −
T
∞
and
T
(
L
) −
T
∞
.
where:
The equivalent electrical diagram is given in Figure 1.19.
Figure 1.19.Equivalent electrical diagram of a fin
Here, R1, R2 and R3 are thermal resistances given by: , and .
1.2.2.7. General case
We have just given the value of the thermal resistance for some particular cases of practical interest. More generally, the thermal resistance of a flux tube (see Figure 1.20) is written as:
Figure 1.20.Diagram of a flux tube
1.2.3. Conduction in unsteady state
1.2.3.1. Medium at uniform temperature
We will study the transfer of heat to a medium at a uniform temperature (“small body” hypothesis), which is a priori contradictory because it is necessary that there is a thermal gradient for heat transfer to occur. This approximation of the medium at uniform temperature may nevertheless be justified in certain cases, which will be specified. For example, the quenching of a metal ball that consists of immersing a ball initially at the temperature Ti in a bath of constant temperature T0. Assuming that the temperature inside the ball is uniform, which will be all the more true as its size is small and its thermal conductivity is high, we can write the thermal balance of this ball between two points in time as t and t + dt:
Hence:
This solution is shown in Figure 1.21. We note that the grouping is homogeneous at a time which will be called τ and is the system’s time constant. Hence:
This quantity is fundamental insofar as it gives the order of magnitude of the physical phenomenon at that point in time; we have in fact: .
Figure 1.21.Evolution of the temperature of a medium at uniform temperature
It is always interesting in physics to present the results in dimensionless form. Two dimensionless numbers are particularly important in variable regime:
The Biot number: ,
ℓ
is the medium’s characteristic dimension,
ℓ
=
R
/3 for a sphere:
The Fourier number:
The Fourier number characterizes heat penetration in a variable regime.
The definition of these two dimensionless numbers makes it possible to write the equation of the temperature of the ball in the form:
Knowing the product of the Biot and Fourier numbers makes it possible to determine the evolution of the sphere’s temperature. It is generally considered that a system such as Bi < 0.1 can be considered to be at a uniform temperature; the criterion Bi < 0.1 is called the criterion of “thermal accommodation”.
1.2.3.2. Semi-infinite medium
A semi-infinite medium is a wall of sufficiently large thickness, so that the perturbation applied to one face is not felt by the other face. Such a system represents the evolution of a wall of finite thickness for a sufficiently short time so that the disturbance created on one face has not reached the other side (true for the point in time that the temperature of the other face does not vary). We will consider, for example, a layer of thickness 1 μm as a semi-infinite medium in the 3ω method!
1.2.3.2.1. Imposed surface temperature
Method: Integral Laplace transform in time and inversion by tables.
The semi-infinite medium is initially at the uniform temperature Ti (see Figure 1.22). The surface temperature is suddenly maintained at a temperature T0; this boundary condition is called the Dirichlet condition.
Figure 1.22.Diagram of semi-infinite medium with imposed surface temperature
The heat equation is written as:
with boundary conditions:
The following variable change is made: .
Hence: and
Equation [1.35] can thus be written as:
The boundary conditions become:
The Laplace transform of with respect to time is written as (see Appendix 4 on integral transformations): .
The Laplace transform of equation [1.39] leads to: . Here, , which justifies the change of variable.
This equation is therefore of the form: with .
Hence: θ(x, p) = A exp(−qx) + B exp(qx).
The temperature keeps a finite value when x → ∞, thus B = 0 and θ(x, p) = A exp(−qx).
The Laplace transform of equation [1.41] leads to: , hence: and: .
Using the inverse Laplace transform tables presented in Appendix 5 leads to the following result:
where: .
The function erf is called the error function (see values in Appendix 6).
1.2.3.2.2. Imposed heat flux at the surface
Method: Integral Laplace transform in time and inversion by tables.
Considering the same configuration but instead by brutally imposing a heat flux density on the surface of the semi-infinite medium (see Figure 1.23), this boundary condition is called the Neumann condition.
Figure 1.23.Diagram of the semi-infinite medium with imposed surface flux
The heat equation is written as:
with the boundary conditions:
Condition [1.46] expresses the conservation of heat flux at the surface of the semi-infinite medium.
The following change of variable is made: . Equation [1.44] can then be written as:
The boundary conditions become:
The Laplace transform of equation [1.48] leads to: with .
Hence: θ(x, p) = A exp(−qx) + B exp(qx).
The temperature keeps a finite value when x → ∞, thus B = 0 and θ(x, p) = A exp(−qx).
The Laplace transform of equation [1.50] leads to: .
Hence: and .
Using the inverse Laplace transform tables presented in Appendix 5 leads to the following result:
where: , this function is tabulated in Appendix 6.
1.2.3.2.3. Imposed heat transfer coefficient
Method: Integral Laplace transform in time and inversion by tables.
