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2009 | Book

Prologue on ideal gases and incompressible fluids Read chapter Read first chapter

Fundamentals of Thermodynamics and Applications

With Historical Annotations and Many Citations from Avogadro to Zermelo

Authors: Ingo Müller, Wolfgang H. Müller

Publisher: Springer Berlin Heidelberg

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

Thermodynamics is the much abused slave of many masters • physicists who love the totally impractical Carnot process, • mechanical engineers who design power stations and refrigerators, • chemists who are successfully synthesizing ammonia and are puzzled by photosynthesis, • meteorologists who calculate cloud bases and predict föhn, boraccia and scirocco, • physico-chemists who vulcanize rubber and build fuel cells, • chemical engineers who rectify natural gas and distil f- mented potato juice, • metallurgists who improve steels and harden surfaces, • - trition counselors who recommend a proper intake of calories, • mechanics who adjust heat exchangers, • architects who construe – and often misconstrue – ch- neys, • biologists who marvel at the height of trees, • air conditioning engineers who design saunas and the ventilation of air plane cabins, • rocket engineers who create supersonic flows, et cetera. Not all of these professional groups need the full depth and breadth of ther- dynamics. For some it is enough to consider a well-stirred tank, for others a s- tionary nozzle flow is essential, and yet others are well-served with the partial d- ferential equation of heat conduction. It is therefore natural that thermodynamics is prone to mutilation; different group-specific meta-thermodynamics’ have emerged which serve the interest of the groups under most circumstances and leave out aspects that are not often needed in their fields.

Table of Contents

Frontmatter
Prologue on ideal gases and incompressible fluids
Thermal and caloric equations of state
The systematic development of the thermodynamic theory and its applications begins in Chap. 1 of this book. Some of the applications concern ideal gases, notably air. Others concern nearly incompressible fluids, notably water. Therefore it is appropriate to have the equations for ideal gases and incompressible fluids available at the outset. There are two of them, the thermal equation of state and the caloric one.
The equations of state of ideal gases and of incompressible liquids are the results of the earliest researches in the field of thermodynamics. Their best-known pioneers are Robert Boyle (1627-1691), Edmé Mariotte (1620-1684), Joseph Louis Gay-Lussac (1778-1850), James Prescott Joule (1818-1889) and William THOMSON (Lord Kelvin, 1824-1907) and their work is nowadays a popular subject of the physics curricula in high schools.
Therefore in this somewhat advanced - or intermediate - book on thermodynamic we feel that we may assume those equations as known. We just list them in order to introduce notation and for future reference.
Ingo Müller, Wolfgang H. Müller
Objectives of thermodynamics and its equations of balance
Fields of mechanics and thermodynamics
Mass density, velocity, and temperature
During a process the mass density, the velocity, and the temperature of a fluid are, in general, not homogeneous in space, nor are they constant in time. Therefore mass density, velocity, and temperature are called time-dependent fields.
Fluid mechanics proposes to calculate the fields of mass density ρ(x i ,t), and velocity w j (x i ,t) in a fluid. Thermodynamics proposes to calculate the fields of mass density ρ(x i ,t), velocity w j (x i ,t), and temperature T(x i , t) in the fluid. Therefore thermodynamics is more accurate than fluid mechanics: In addition to the motion of the fluid and its inertia, it takes into consideration how “hot” the fluid is.
On the interface between two bodies - the fluid and the wall of the container (say) - the temperature is continuous. This property defines temperature, it is the basis of all measurements of temperature by contact thermometers, and it is often referred to as the Zeroth Law of Thermodynamics.
Most thermometers rely on the thermal expansion of the thermometric substance, often mercury. In this book we shall usually employ the Celsius scale - or centigrade scale - but often also the absolute or Kelvin scale. Both scales use the same degree of temperature such that melting ice and boiling water at normal pressure differ by 100 degrees. The values of temperature of these fix points are 0°C and 100°C, or 273.15K and 373.15K, respectively.
Ingo Müller, Wolfgang H. Müller
Constitutive equations
On measuring constitutive functions
The need for constitutive equations
We recall the objective of thermodynamics which is the determination of the five fields
mass density ρ(x i , t), velocity w j (x i ,t), temperature T(x i , t). (2.1)
For this purpose we need five equations and we choose the five equations of balance of mass, momentum, and (internal) energy
\({{\partial\rho} \over {\partial{t}} } + {{\partial\rho{w}_i} \over {\partial{x}_i} } = 0,\)
\({{\partial\rho{w_j}} \over {\partial{t}} } + {{\partial(\rho{w_j}{w_i}-t_{ji})} \over {\partial{x}_i} } = \rho f_j, \)
\({{\partial\rho{u}} \over {\partial{t}} } + {{\partial} \over {\partial{x}_i} } (\rho u w_i + q_i) = t_{ji} {{\partial{w}_j} \over {\partial{x}_i} } + \rho{z}.\) (2.2)
We shall assume that f j and z are given functions of x i and t. In our applications f j is the gravitational force f j =(0,0,-g), and z is usually set equal to zero. Even then, the equations (2.2) are not sufficient for the determination of ρ, w j , and T, because they contain additional fields, namely the stress tensor t ji , the specific internal energy u, and the heat flux q i ; and the temperature does not occur at all.
Therefore we need additional equations which relate t ji , u, and q i to the basic fields ρ, w j , and T and these are called the constitutive equations of thermodynamics. Indeed, experience with the thermodynamic substances - usually fluids, vapors, and gases - has taught us that stress, internal energy, and heat flux depend on the fields ρ, w j , and T in a materially dependent manner.
Ingo Müller, Wolfgang H. Müller
Reversible processes and cycles. “p dV thermodynamics” for the calculation of thermodynamic engines
Work and heat for reversible processes
For a heuristically important and qualitatively correct treatment of thermodynamic processes one usually ignores shear stresses, heat conduction, and temperature and pressure gradients. We have discussed the working \(\dot{W}\) of such idealized - reversible -processes in Paragraph 1.5.6. The stress work -or internal work* - done in the time dt is given by
\(\dot{W}_{\rm stress}{\rm d}t = - p{\rm d}V. (3.1)\)
Ingo Müller, Wolfgang H. Müller
Entropy
The Second Law of thermodynamics
Formulation and exploitation
Formulation
Rudolf Julius Clausius (1822-1888) has drawn conclusions from the following experience
Heat cannot pass by itself pass from a colder to a warmer body
or, in a later version
Heat cannot pass from a colder to a warmer body without compensation.
These are two formulations of the Second Law of thermodynamics.
We have already mentioned – as an assumption – that there are always positive and negative parts of the heat exchanged with a heat engine. Indeed, this is a consequence of the Second law. If it were different, we should be able to convert the heat of a part of the cold sea fully into work, and then convert the work back into heat by stirring a hot liquid. In this manner, in effect, heat would have passed from the cold sea to the hot liquid and this contradicts the Second Law. William Thomson (Lord Kelvin) has used this argument to express the Second Law in an alternative form, viz.
It is impossible to gain work in a heat engine by just cooling a body.
All of these statements are open to the criticism that they are verbally expressed and lack the stringency of mathematical formulae. It is a somewhat idle effort, however, to try and make these suggestive formulations strict, because in the end, when we have gone through Clausius’s argument, there is a mathematical formula, an inequality, and this is the proper mathematical form of the Second Law - and that is strict.
Ingo Müller, Wolfgang H. Müller
Entropy as S=k lnW
Molecular interpretation of entropy
Mass and momentum of a gas or a fluid are easy to interpret in terms of atoms and molecules: They are simply the sums of the masses and momenta of the constituent molecules. Pressure and temperature may also convincingly be interpreted by molecular processes and properties, e.g. see Prologue 4. Energy is more subtle: It is true that the kinetic energy of a gas is the sum of the kinetic energies of the molecules, but energy also contains the potential energy of the molecular interaction between at least pairs of two particles; often clusters of more than two.
Much more difficult is the molecular interpretation of entropy. Entropy represents a property of the ensemble of all molecules and depends on their spatial distribution and on the distribution of momenta in the gas. This interpretation was found by Ludwig Eduard Boltzmann (1844-1906) who - along with James Clerk Maxwell (1831-1879) - developed the kinetic theory of gases. Here we cannot go into that theory in any detail, and therefore we cannot explain the derivation of the molecular interpretation of entropy.
Ingo Müller, Wolfgang H. Müller
Steam engines and refrigerators
The history of the steam engine
It was Denis Papin – the inventor of the pressure cooker, cf. Paragraph 2.4.8 – who first condensed vapor and lifted a weight by doing so. He used a brass tube of diameter 5 cm; some water at the bottom was evaporated and the vapor pushed a piston upward which was then fixed by a latch. Afterwards the tube was taken away from the fire, the vapor condensed and a Torricelli vacuum formed in the tube. When the latch was unlocked, the air pressure pushed the piston downward and lifted a weight of 60 pounds.
Ingo Müller, Wolfgang H. Müller
Heat Transfer
Non-Stationary Heat Conduction
The heat conduction equation
In a body at rest with constant mass density and momentum the equations of balance of mass and momentum are identically satisfied, if we neglect thermal expansion, and the energy balance (1.48) – without radiation – is reduced to the form
\( \rho{{\partial{u}}\over{\partial{t}}}+{{\partial{q_i}}\over{\partial{x_i}}}=0. \) (7.1)
Ingo Müller, Wolfgang H. Müller
Mixtures, solutions, and alloys
Chemical potentials
Characterization of mixtures
We consider mixtures, or solutions, or alloys with v constituents, which are characterized by a Greek index. Thus p α is the partial pressure of constituent α, ρ α its density, and u α or s α are the specific values of the internal energy and entropy, respectively. Pressure, density, internal energy density, and entropy density of the mixture are additively composed of the corresponding partial quantities
\( p=\sum\limits_{\alpha=1}^{v}p_{\alpha}, \rho = \sum\limits_{\alpha=1}^{v}\rho_{\alpha}, \rho{u} = \sum\limits_{\alpha=1}^{v}\rho_{\alpha}u_{\alpha}, \rho{s} = \sum\limits_{\alpha=1}^{v}\rho_{\alpha}s_{\alpha}. \) (8.1)
This does not mean, however, that ρ α , u α , and s α are given by the constitutive functions of the pure constituent α. Indeed, ρ α , u α , and s α will generally depend on p α and T , and on all the other partial pressures p β .
Ingo Müller, Wolfgang H. Müller
Chemically reacting mixtures
Stoichiometry and law of mass action
Stoichiometry
We denote the mass of a constituent α in a homogeneous mixture by m α . The corresponding mass balance reads
\( {{{\rm d}m_\alpha} \over {{\rm d}t}} = \tau_{\alpha}V, \alpha = 1,2, \cdots, v, \) (9.1)
where τ α is the density of mass production of constituent α. It is clear that the sum of all the v values τ α must be equal to zero, because the total mass is conserved in a chemical reaction. However, this is not the only condition which the production densities τ α must satisfy. Further conditions follow from the fact that the number of atoms – and their masses – must be conserved in the reaction; the atoms present before the reaction are still present after the reaction, although they may have changed their arrangement in molecules. The investigations of the conditions for this kind of arrangement is the subject of stoichiometry (Greek: stoichos “order” and metron “measure”).
Ingo Müller, Wolfgang H. Müller
Moist air
Characterization of moist air
Moisture content
Unsaturated moist air is a mixture of air and water vapor. Both are considered as ideal gases. Indeed, we use the ideal gas laws for water vapor all the way down to the state of saturation, where condensation occurs. For a given temperature this occurs at the pressure p′= p(T) which may be read off from Table 2.4.
Ingo Müller, Wolfgang H. Müller
Selected problems in thermodynamics
Droplets and bubbles
Available free energy
We investigate the equilibrium of a liquid droplet in its vapor, and – at the same time – we discuss the equilibrium conditions for a vapor bubble surrounded by liquid. Fig. 11.1 shows the systems to be considered and introduces some notation. Droplets and bubbles are considered as spherical, p is the pressure on the outside and T the temperature. In some instances we divide the page along a vertical line; in such cases the formulae to the left of the line refer to droplets, those to the right refer to bubbles.
Ingo Müller, Wolfgang H. Müller
Thermodynamics of irreversible processes
Single fluids
The laws of FOURIER and NAVIER-STOKES
We recall that – at the very beginning of this book – we have defined the determination of the five fields
mass density \(\rho(\underline{x},t)\) , velocity \(w_i(\underline{x},t)\) , temperature \(T(\underline{x}, t)\) (12.1)
as the objective of thermodynamics of fluids.
Ingo Müller, Wolfgang H. Müller
Backmatter
Metadata
Title
Fundamentals of Thermodynamics and Applications
Authors
Ingo Müller
Wolfgang H. Müller
Copyright Year
2009
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-74648-5
Print ISBN
978-3-540-74645-4
DOI
https://doi.org/10.1007/978-3-540-74648-5

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