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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Heat Transfer 2 deals with radiation, heat exchangers and flat plate solar collectors. It presents the treatment of radiation in semi-transparent media to be taken into account for insulation or recovery of high temperature waste heat (energy saving in industry), as well as in certain solar applications (energy transition). The numerous solved exercises allow the reader to grasp the whole range of applications, whether in the field of building, transport, materials or the environment. The appendices contain all the data needed to solve the exercises and will be a valuable source of information. This book is designed for masters and engineering students who are interested in all aspects of heat transfer, but also for engineers who will find the bases needed to understand similar phenomena (conduction-convection-radiation), but which require a different form of reflection and approach.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright Page
Preface
Nomenclature
Chapter 1. Heat Transfer by Radiation Between Surfaces
1.1. General: definitions
1.2. Laws of radiation
1.3. Reciprocal radiation of several surfaces
1.4. Emission and absorption of gases
1.5. Solar radiation
1.6. Corrected exercises
Chapter 2. Radiation Heat Transfer in Semi-transparent Media
2.1. General
2.2. Definitions
2.3. Radiation balance
2.4. Special cases
2.5. Conditions at the boundaries
2.6. Example of an approximate solution: two-flux or Schuster–Schwarzschild approximation
2.7. Conduction–radiation coupling
2.8. Corrected exercises
Chapter 3. Introduction to Heat Exchangers
3.1. Double-pipe heat exchangers
3.2. Complex bundle heat exchangers
3.3. Corrected exercises
Chapter 4. Flat Plate Solar Collectors
4.1. Principle
4.2. Global heat balance
4.3. Thermal balances of the different constituents
4.4. Heat flow rate gained by the fluid as a function of the temperatures
4.5. Other characteristic quantities
4.6. Calculation method for a solar collector
4.7. Corrected exercises
Appendices
References
Index
Summary of Volume 1
Other titles from ISTE in Energy
List of Tables
Chapter 1
Table 1.1. Solar radiation on a horizontal plane: notations used
Table 1.2. Average monthly values of daily irradiation (in kWh m
-2
j
-1
) in different locations
Table 1.3. Value of G (kJ m
-2
j
-1
) in Ouagadougou in August 1987
Chapter 3
Table 3.1. Fouling resistance values R
en
for some fluids
Table 3.2. Order of magnitude of the global transfer coefficient h (W m
−2
K
−1
) of various types of exchangers
Table 3.3. Order of magnitude of the global exchange coefficient for various types of refrigeration exchangers (according to IIF (1976))
List of Illustrations
Chapter 1
Figure 1.1. Principle of William Herschel’s experiment. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.2. Spectrum of electromagnetic waves (λ in m). For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.3. Diagram of the solid angle
Figure 1.4. Diagram of definition of angles
Figure 1.5. Diagram of definition of angles in Bouguer’s formula. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.6. Diagram of the distribution of an incident radiation flux on a wall
Figure 1.7. Simplified representation of the monochromatic absorption coefficient of white paint
Figure 1.8. Schematization of energy intensity
Figure 1.9. Schematization of the luminance and the energy intensity of an isotropic source
Figure 1.10. Monochromatic emittance of a black body at two different temperatures. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.11. Schematization of radiation fluxes on a surface. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.12. Equivalent electrical diagram of the radiative flux lost by a surface
Figure 1.13. Equivalent electrical diagram of the radiative heat flow rate exchanged between two surfaces
Figure 1.14. Equivalent electrical diagram of the net radiative heat flow rate exchanged between two surfaces
Figure 1.15. Spectral distribution of solar radiation outside the atmosphere. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.16. Diagram of the motion of the Earth around the Sun
Figure 1.17. Apparent motion of the Sun observed from a point of latitude L
Figure 1.18. Pinpointing the position of the Sun
Figure 1.19. Time difference from the Greenwich meridian
Figure 1.20. Equation of time ET (—) and declination δ(—) depending on the day of the year. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.21. Spectral distribution of solar radiation at ground level
Figure 1.22. Atmospheric radiation spectrum
Figure 1.23. Average global irradiation in kWh m
-2
j
-1
. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.24. Typical variation of solar irradiance (W m
-2
) during an undisturbed day. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Chapter 2
Figure 2.1. Schematization of radiative fluxes in a semi-transparent medium. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.2. Schematization of luminance. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.3. Diagram of notations used. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.4. Diagram of the studied medium. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.5. Diagram of the studied medium. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.6. Schematic representation of the considered system. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.7. Diagram of the semi-transparent wall. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.8. Diagram of the 1D non-diffusing medium. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.9. Diagram of the 1D non-scattering medium with isotropic luminance. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.10. Diagram of an optically thick medium with parallel faces. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 2.11. Diagram of a medium with parallel faces and transparent boundaries
Figure 2.12. Diagram of a medium with parallel faces and opaque boundaries
Figure 2.13. Diagram of a medium with opaque parallel faces with diffuse emission and specular reflection
Figure 2.14. Schematization of a transparent interface
Figure 2.15. Quadripolar diagram for the purely scattering medium
Chapter 3
Figure 3.1. Diagram of a double-pipe heat exchanger
Figure 3.2. Diagram of parallel flow and counter flow heat exchangers
Figure 3.3. Diagram of elementary flows in a double-pipe heat exchanger
Figure 3.4. Temperature evolution in a parallel flow heat exchanger
Figure 3.5. Evolution of temperatures in a counter flow heat exchanger
Figure 3.6. Simplified representation of the benefit generated by a heat recovery unit
Figure 3.7. Diagram of a 1–2 heat exchanger
Figure 3.8. Diagram of a 2–4 heat exchanger
Figure 3.9. Diagram of two types of cross flow exchangers
Figure 3.10. Evolution of temperatures in a condenser
Figure 3.11. Evolution of temperatures in an embedded evaporator
Figure 3.12. Evolution of temperatures in a dry expansion evaporator
Chapter 4
Figure 4.1. Block diagram of a flat plate solar collector
Figure 4.2. Diagram of convective heat transfers in a type 1 covered collector. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 4.3. Equivalent electrical diagram of heat transfers in a type 1 solar collector
Figure 4.4. Diagram of convective flows in a type 2 covered solar collector. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 4.5. Equivalent electrical diagram of heat transfers in a type 2 solar collector
Figure 4.6. Diagram of convective flows in an uncovered type 3 solar collector. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 4.7. Equivalent electrical diagram of heat transfers in a type 3 solar collector
Figure 4.8. Diagram of convective flows in an uncovered type 4 solar collector. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 4.9. Equivalent electrical diagram of heat transfers in a type 4 solar collector
Figure 4.10. Diagram of the elementary thermal balance on the air
Figure 4.11. Sectional diagram of the absorber
Figure 4.12. Instantaneous efficiency of a solar collector as a function of . For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Guide
Cover Page
Title Page
Copyright Page
Preface
Nomenclature
Table of Contents
Begin Reading
Appendices
References
Index
Summary of Volume 1
Other titles from ISTE in Energy
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Series Editor
Alain Dollet
Heat Transfer 2
Radiation and Exchangers
Yves Jannot
Christian Moyne
Alain Degiovanni
First published 2023 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 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2023The rights of Yves Jannot, Christian Moyne 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: 2023932777
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-928-0
Preface
In addition to its obvious interest, which the current energy transition will certainly not contradict, thermal energy has a special place among the engineering sciences. Often taught at the beginning of engineering courses, it teaches how to analyze the phenomena involved in a real situation, identify the important aspects and develop a description that is likely to provide, before any more sophisticated approach, some solution elements, or even if possible, a first approximate solution that is easy to calculate.
Each of the three modes of heat transfer – conduction, convection and radiation – are associated with three different ways of thinking. Conduction, due to the apparent simplicity of Fourier’s law and the heat equation, will allow different modeling and resolution methods depending on the simplifications used. The complexity of flowing fluids will generally make such an approach impractical in thermal convection without using numerical calculation: to remain relatively simple, we will resort to dimensional analysis approaches without, however, giving some elementary notions about essential approaches, especially the concept of boundary layer. Radiation opens onto the vast universe of physics, requiring simplifying assumptions that must be taken into account by the thermal engineer.
In all cases, the understanding of heat transfer processes requires the resolution and, above all, the meditation of a few exercises ranging from elementary cases (to begin with) to more complex situations, ultimately requiring a simultaneous approach to the different transfer modes that are often coupled.
This heat transfer course is intended for graduate students in universities and engineering schools, as well as practicing engineers. It presents the main modes of heat transfer: conduction, convection (Volume 1) and radiation (Volume 2). Volume 2 also includes two chapters on systems in which several modes of heat transfer occur simultaneously: heat exchangers and solar collectors. The main methods for solving the heat equation are presented and illustrated by using the Laplace transform, the separation of variables, the integral transformation, the quadrupole method and the numerical methods.
The chapters on steady-state conduction, convection, radiation between surfaces and on heat exchangers, as well as their application exercises can be approached by undergraduate students.
The proposed exercises are all corrected in detail; they present practical applications covering all the theoretical aspects of the course. Some long exercises are real case studies, and show that the assimilation of this course makes it possible to solve concrete problems in many fields of interest: for example, calculation of the evolution of the temperature and thermal losses of a building, calculation of losses in double glazing, measurement of the thermal properties of solids, sizing of a heat exchanger and sizing of a solar collector.
The appendices contain all the physical data and correlations needed to solve the exercises and problems presented and will be a valuable source of information for the engineer.
January 2023
Nomenclature
a
thermal diffusivity (m
2
s
-1
)
Bi
Biot number
c
specific heat (J kg
-1
K
-1
)
C
f
friction coefficient
D
diameter (m)
D
h
hydraulic diameter (m)
e
thickness (m)
E
thermal effusivity (W m
-2
K
-1
s
1/2
)
F
conduction form coefficient (m)
Fo
Fourier number
g
acceleration of gravity (m s
-2
)
Gr
Grashof number
h
convection heat transfer coefficient (W m
-2
K
-1
)
H
enthalpy (J)
ΔH
latent heat of phase change (J kg
-1
)
I
electrical energy intensity (A)
I
n
modified Bessel function of the first kind of order
n
J
n
unmodified Bessel function of the first kind of order
n
K
n
modified Bessel function of the second kind of order
n
ℓ
width (m)
L
length (m)
mass flow rate (kg s
-1
)
unit vector normal to a surface
Nu
Nusselt number
p
Laplace variable (s
-1
)
p
e
perimeter (m)
Pr
Prandtl number
energy volume density (W m
-3
)
Q
heat quantity (J)
q
c
heat flow (W K
-1
)
r, R
radius (m), resistance (Ω)
Ra
Rayleigh number
R
c
contact resistance (K W
-1
)
Re
Reynolds number
s
curvilinear abscissa (m)
S
area (m
2
)
t
time (s)
T
temperature (K)
average temperature (K)
u
velocity (m s
-1
)
V
volume (m
3
)
x, y, z
space variables (m)
Y
n
unmodified Bessel function of the second kind of order
n
Greek letters
α
absorption coefficient of a wall
β
cubic expansion coefficient (K
-1
)
δ
declination (°)
ε
emissivity
ϕ
heat flux (W m
-2
)
Φ
Laplace transform of the heat flow rate (W s)
φ
heat flow rate (W)
λ
thermal conductivity (W m
-1
K
-1
), wavelength (m)
μ
dynamic viscosity (Pa s
-1
)
ν
kinematic viscosity (m
2
s
-1
), frequency (Hz)
η
yield or efficiency (%), emission coefficient of a medium (W m
-3
sr
-1
)
κ
absorption coefficient of a medium (m
-1
)
Ω
solid angle
τ
transmission coefficient of a wall
θ
Laplace transform of the temperature (K s)
1Heat Transfer by Radiation Between Surfaces
1.1. General: definitions
1.1.1. Type of radiation
All bodies, whatever their state, solid, liquid or gaseous, emit electromagnetic radiations. This emission of energy is done at the expense of the internal energy of the emitting body.
The radiation propagates in a straight line at the velocity of light; it consists of radiation of different wavelengths λ as demonstrated by William Herschel’s experiment shown in Figure 1.1.
Figure 1.1.Principle of William Herschel’s experiment. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Figure 1.2.Spectrum of electromagnetic waves (λ in m). For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
By passing through a prism, the radiation is more or less deflected depending on its wavelength. The radiation emitted by a source at temperature T0 is sent to a prism and the deflected beam is projected onto an absorbing (blackened) screen, thus obtaining the decomposition of the total incident radiation into a spectrum of monochromatic radiation.
If a thermometer is moved along the screen, the temperature Te is measured, characterizing the energy received by the screen in each wavelength. By constructing the curve Te = f(λ), we obtain the spectral distribution of the radiated energy for the temperature T0 of the source. We then see that:
– the energy emitted is maximum for a certain wavelength
λ
m
variable with
T
0
;
– the energy is only emitted over an interval [
λ
1
,
λ
2
] of wavelength characterizing thermal radiation.
The different types of electromagnetic waves and their corresponding wavelengths are represented in Figure 1.2. It should be noted that the radiation emitted by bodies and having a thermal effect is between 0.1 and 100 μm. It should also be noted that radiation is perceived by humans:
– by the eye: for 0.38 μm <
λ
< 0.78 μm, visible radiation;
– by the skin: for 0.78 μm <
λ
< 314 μm, IR radiation.
1.1.2. Definitions
1.1.2.1. Classification
The physical quantities will be distinguished according to:
– the spectral composition of the radiation:
- if a quantity
G
is relative to the entire spectrum, it is said to be total;
- if it concerns a narrow spectral interval
d
λ
around a wavelength
λ
(or a frequency
ν
), it is said to be monochromatic:
G
λ
(or
G
ν
);
– the spatial distribution of radiation:
- if the quantity is relative to all the directions in space, it is said to be hemispherical;
- if it characterizes a given direction of propagation, it is said to be directional:
G
x
or .
1.1.2.2. Definitions relative to sources
1.1.2.2.1. Heat flow rate
We call heat flow rate of a source S denoted φ, the heat radiated by S per unit time in all the space which surrounds it, on all the wavelengths. The heat flow rate φ is expressed in W.
The heat flow rate sent by an element of surface dS in an elementary solid angle dΩ along the direction between two wavelengths λ and λ + dλ is denoted .
The heat flow rate sent by an element of surface dS in an elementary solid angle dΩ along the direction is denoted .
The heat flow rate sent throughout space by an elementary surface dS is denoted dφ.
We therefore have the following equations:
1.1.2.2.2. Reminder on elementary solid angles
The solid angle Ω under which from a point O we see a surface S is by definition the area of the surface intersection of the sphere of radius unit and the cone with vertex O leaning on the contour of the surface S (see Figure 1.3).
The elementary solid angle dΩ under which is seen from a point O the contour of a small surface dS (assimilated to a flat surface) can be calculated by:
Properties:
– the value of a solid angle Ω is between 0 and 4
π
;
– for a cone with half-angle at the vertex
α
: Ω = 2
π
(1 − cos
α
).
Figure 1.3.Diagram of the solid angle
1.1.2.2.3. Energy emittance
We distinguish the following values:
– Monochromatic: an element of surface dS emits a certain heat flow rate by radiation in all directions of the ½ space. This heat flow rate is distributed over an interval of wavelengths. If we consider the heat flow rate emitted between the two wavelengths λ and λ + dλ, we define the monochromatic emittance of a source at temperature T by:
– Total: this is the heat flux emitted by radiation per dS over the entire spectrum of wavelengths. It is only a function of the temperature T and the nature of the source:
1.1.2.2.4. Energy intensity in one direction
We call energy intensity the heat flow rate per unit of solid angle emitted by a surface dS in a solid angle dΩ surrounding the direction Ox:
1.1.2.2.5. Luminance in one direction
Let α be the angle made by the normal to the emitting surface S with the direction Ox. The projection of dS on the plane perpendicular to Ox defines the apparent emitting surface dSx = dS cos α (see Figure 1.4). The elementary energy intensity in the direction Ox per unit of apparent emitting surface dSx is called the luminance, (in W m-2 sr-1).
Starting from equation [1.4], we obtain:
Figure 1.4.Diagram of definition of angles
1.1.2.2.6. Application: Bouguer’s formula
We deduce from the previous definitions the expression of the flux sent by an element dSi of luminance on another element dSk:
Figure 1.5.Diagram of definition of angles in Bouguer’s formula. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
where dΩ is the solid angle under which we see the surface dSk from the surface dSi thus:
Leading to Bouguer’s formula:
1.1.2.3. Definitions relative to a receiver
1.1.2.3.1. Irradiance
It is the counterpart of the emittance for a source. Irradiance is the heat flux (W m-2) received of receiving surface, from all directions.
1.1.2.3.2. Reception of radiation by a wall
When an incident ray of energy φλ hits a wall at temperature T, a part φλρλT of the incident energy is reflected by the surface S, another part φλαλT is absorbed by the body which heats up and the remaining part φλτλT is transmitted and continues on its way.
We obviously have: φλ = φλρλT + φλαλT + φλτλT hence: ρλT + αλT + τλT = 1.
We thus define the monochromatic reflection ρλT, absorption αλT and transmission τλT coefficients, which are functions of the nature of the body, its thickness, its temperature T, the wavelength λ of the incident radiation and the angle of incidence.
Figure 1.6.Diagram of the distribution of an incident radiation flux on a wall
If we consider the incident energy over the entire spectrum of wavelengths, we obtain the total reflection ρT, absorption αT and transmission τT coefficients.
1.1.2.4. Black body, gray body
1.1.2.4.1. Black body
It is a body that absorbs all the radiation it receives regardless of its thickness, its temperature, the angle of incidence and the wavelength of the incident radiation, it is defined by: αλT = 1. A surface coated with lampblack is approximately a black body.
Black body properties:
– all black bodies radiate in the same way;
– the black body radiates more than the non-black body at the same temperature.
1.1.2.4.2. Gray body
A gray body is a body whose absorption coefficient αλT is independent of the wavelength λ of the radiation it receives. It is defined by: αλT = αT.
In many cases and in this chapter, we can consider the solid bodies as gray bodies by interval and we use an average absorption coefficient with respect to the radiation emitted for λ < 3 μm (radiation emitted by bodies at high temperature such as the sun) and an average absorption coefficient with respect to the radiation emitted for λ > 3 μm (radiation emitted by bodies at low temperature: atmosphere, solar absorber). As an example, the following values can be considered for white paint.
Figure 1.7.Simplified representation of the monochromatic absorption coefficient of white paint
The absorption coefficient values of certain materials are given in section A.7 of the Appendix.
1.2. Laws of radiation
1.2.1. Lambert’s law
A source is isotropic if the luminance is independent of the direction: .
thus: .
From the equality , we deduce Lambert’s law for an isotropic source.
Figure 1.8.Schematization of energy intensity
Thus, the emission indicator is a sphere tangent at O to the emitting surface when the latter follows Lambert’s law (see Figure 1.9).
Figure 1.9.Schematization of the luminance and the energy intensity of an isotropic source
REMARK.– For a cone with half-angle at the vertex α: Ω = 2π(1 − cos α) and dΩ = 2π sin α dα, so when a body follows Lambert’s law:
therefore:
1.2.2. Physical laws
1.2.2.1. Black body radiation
1.2.2.1.1. Monochromatic emittance
It is given by Planck’s law:
with:
–
c
, velocity of light in the medium in which it propagates: ;
–
c
0
, velocity of light in vacuum:
c
0
= 2.998 × 10
8
m s
−1
;
–
n
, refractive index of the medium;
–
h
, Planck’s constant:
h
= 6.6261 × 10
−34
J s;
–
k
, Boltzmann’s constant:
k
= 1.3806 × 10
−23
J K
−1
;
–
T
, temperature (K);
–
λ
, wavelength (m);
–
ν
, frequency: (Hz).
Planck’s law makes it possible to draw isothermal curves representing the variations of as a function of the wavelength for various temperatures (see Figure 1.10).
REMARK.–
– The monochromatic luminance of the black body at temperature T emitting in a medium of refractive index n is given by: , where is the luminance of the black body emitting in vacuum .
– The wavelength λM for which the emission is maximal varies with the temperature of the source (Wien’s laws):
and:
with T: temperature (K).
– For the sun (T ≈ 5,777 K), 90% of the energy is emitted between 0.31 and 2.5 μm, the maximum being located in the visible spectrum. On the other hand, a black body at 373 K (100°C) has its maximum emission around λ = 8 μm in the IR and zero emission below 3 μm.
1.2.2.1.2. Total emittance
The integration of Planck’s formula for all wavelengths gives the total emittance M0(T) of the black body which is only a function of the temperature T, we obtain Stefan–Boltzmann’s law:
with: σ = 5.670 × 10−8 W m−2K−4.
1.2.2.1.3. Fraction of emittance in a given interval of wavelengths [λ1, λ2]
It is the fraction of the heat flow rate emitted by the unit of surface of the black body at the temperature T between the wavelengths λ1 and λ2:
which can also be written as: .
Let us calculate F0−λT. is of the form: , with C1 = 2πhc2 and .
Figure 1.10.Monochromatic emittance of a black body at two different temperatures. For a color version of this figure, see www.iste.co.uk/jannot/heattransfer2.zip
Hence: .
We find that F0−λT only depends on the product λT. It is therefore sufficient to draw up once and for all a table with a single entry
