A more elaborated literature review is given in the second chapter of this thesis. This chapter firstly describes the fundamental theories, KTG and NET, which will be used later on to analyze the measurement results. In the second part of the chapter the design of the relevant previous experimental setups are discussed, this information is used for the design of the setup for this project. In the third chapter different temperature measurement techniques are compared to each other and a selection of the most suitable has been made. This measurement technique is tested by a simple setup and is discussed in Chapter 4. The design of the evaporation setup is discussed in Chapter 5. In Chapter 6, the experimental results are discussed and compared with the results of the previous research. Finally, conclusions are drawn and recommendations are given for further research.
Overview of theory, experiments and simulations
This chapter provides an overview of the relevant research in the field of the evaporation phenom-ena. The first part reviews the classical theories briefly, which will later be used to analyze and compare the measurement results with other relevant experimental and numerical findings. These experiments are discussed in the second part of the chapter, as critical design features will be a guide for the design of the experimental setup of this project.
2.1 Classical evaporation theory
The two well established techniques to analyze the evaporation process are the kinetic theory of gasses (KTG) and non equilibrium thermodynamics (NET). Those two theories are widely used to compare experiments and simulations with each other, which is typically done by the calculation of an evaporation coefficient and the Onsager resistivities. The derivation of this coefficients with assumptions are briefly presented in this chapter.
2.1.1 The evaporation coefficient
The kinetic theory of gasses (KTG) describes a gas as a large number of moving particles in-teracting with each other by collisions. This theory can be used to find an expression for the evaporating mass flux. Which can be done by a closer analyses of the kinetics just above the liquid-gas interface. In general, the probability that a particle will travel with a certain velocity is given by a distribution function (DF). In the case of a gas in equilibrium, the DF is Maxwellian (MDF)[14]. In one direction this function is:
fM(vp) = r mp
2πkbTexp(−mpv2p 2T kb
) (2.1)
Here f is the probability that a particle will travel with a certain velocity vp, mpis the mass of a single particle, kb is the Boltzmann constant and T is the absolute temperature. The simplest approach is to assume that the velocity of the particles that evaporate follow a MDF with the temperature of the liquid and the condensing flux follows a MDF with the temperature of the vapor. The mass flux can be determined from the distribution function in the following way:
j = nmp
Z ∞
−∞
vpf (vp)dvp (2.2)
Where j is the mass flux through a unit area, n is the number of particles per unit volume of the gas. This equation can be applied to the evaporation problem. The net evaporating mass flux
CHAPTER 2. OVERVIEW OF THEORY, EXPERIMENTS AND SIMULATIONS
is the mass flux of the stream of gas emitted from the liquid phase and the mass flux of the stream of gas which will be condense:
jtot= jevap− jcond= nlmp
b) this expression can be rewritten to:
jtot= pl However, experimental observations suggest that the real mass flow is lower than this the-oretical value. That is why commonly an evaporation (α) and condensation coefficient (β) is introduced:
This relation is known as the general Hertz-Knudsen equation (HK-equation) and it is com-monly simplified with the assumption of α = β to:
jtot= α
However, the assumption of a MDF for the condensing particles in the gas may not be valid for a rapidly evaporating process, as the gas above the interface is not in complete equilibrium. This effect can by taken into account if one assumes that an evaporating stream can be represented by a shifted MDF: f (v) = fM(v − V ) in which V is the average bulk velocity of the gas moving away from the interface. The derivation of the total mass flux can be done in the same way as has been done for the Hertz-Knudsen equation. In this way, the Hertz-Knudsen equation becomes:
jtot= α r mp
2πkb( pl
√Tl − Ψ( ˜V ) pv
√Tv) (2.8)
where the function Ψ is defined as:
Ψ( ˜V ) = exp(− ˜V2) + ˜V Γ(1
2, ˜V2) (2.9)
Where Γ is the gamma function and ˜V is defined as:
V = V˜
r m
2T Kb (2.10)
Note that in the case V ,→ 0 , the expression reduces to the standard Hertz-Knudsen equation.
The function can be simplified with the use of mass conservation to relate the mass flux and ˜V : jtot= nvmpV = ˜V np2mpT kb (2.11) For relatively low velocities the function Ψ can be approximated by:
Ψ( ˜V ) = 1 + ˜V√
π + .... (2.12)
With this first order approximation and the relation between the mass flow and ˜V . The total heat flux can be expressed as:
jtot= α
This equation is known as the Hertz-Knudsen-Schage HKS-equation. Both Equation2.8, and Equation 2.14are commonly used to calculate evaporation coefficient. Where the HK-equation is generally used in a equilibrium situations and the HSK-equation more in non-equilibrium situ-ations.
2.1.2 Onsager resistivities
Another way to analyze the evaporation problem is with non-equilibrium thermodynamics. In this theory it is assumed that the total entropy (σs), is the sum of thermodynamic fluxes (Ji) times its thermodynamic force (Xi):
σs=X
i
JiXi≥ 0 (2.15)
An example of such a flux-force pair is the flow of heat, and a temperature gradient respectively.
For many relatively slowly irreversible processes, it has been observed that every flux is linear dependent on all the other thermodynamic forces[5]:
Ji=X
j
ˆ
rijXj (2.16)
These flux force relations are called the phenomenological relations. The values ˆrij, are called the Onsager resistivities and give the relation between the thermodynamic fluxes and forces.
Onsager discovered that these relations can be simplified by the fact that the matrix ˆr,consisting of the elements ˆrij, is a positive semi-definite matrix. This is a consequence of the second law of thermodynamics as entropy production has to be greater than zero. Another property is that the matrix is symmetric, which implies that the cross coefficients,ˆrij and ˆrjiare the same.
The flux-force relations imply that a thermodynamic force can give rise to a thermodynamic flux which is not based on the same thermodynamic variable. For example, a temperature gradi-ent in an electrically conducting wire can give not only rise to a heat flux, but also to a electrical current. And vice versa, a electrical potential difference can give rice to an electrical current and a heat flux. The dependencies are not limited to two flux/force pairs but the amount of relevant thermodynamic fluxes and forces in a system.
In the case of an evaporating liquid, it is assumed that the two main thermodynamic fluxes towards and from the interface are the conductive heat and mass flux. The Onsager relations for this flux-force pairs can be found by combining Equation2.15and Equation2.16with the energy balance and the generation of entropy at the interface[10]:
Qtot= jtothl+ ql= jtothv+ qv (2.17) In which, Qtot is the overall heat flux, jtot is the overall mass flux, hl and hv is the enthalpy of the liquid and of the vapor respectively, ql and qv are the heat fluxes in the vapor and in the liquid. The entropy production at the interface:
σ = ql Where µv, µl and Ts are the chemical potential of the vapor and the chemical potential of the liquid and the inter facial temperature respectively. The heat flux, ql, in the liquid phase and the
CHAPTER 2. OVERVIEW OF THEORY, EXPERIMENTS AND SIMULATIONS
inter-facial temperature can be eliminated in the equation of entropy production with the use of the energy balance (Equation2.17) and the following thermodynamic identity:
∂(µ/T )
∂(1/T ) = h (2.19)
Equation2.18can be re-written to:
σ = qv
In this latter equation, one can identify the form of Equation2.15. Where the two thermodynamic fluxes are defined as:
J = [qv, jtot] (2.21)
And the two thermodynamic forces are defined as:
X = [1
Tl(µv− µl), 1 Tg − 1
Tl] (2.22)
In combination with the phenomenological relations (Equation2.16), These fluxes and forces can be related to each other with the following system of equations:
"
With description of chemical potential for a vapor in equillibrium with a liquid per mol:
µl= µ0v+ RTlln(Psat(Tl)) (2.24)
µv= µ0v+ RTlln(Pv) (2.25)
Equation2.23can be further simplified to:
"
The interfacial resistivities of experimental results can be calculated with the use of this equa-tion. However, the resistivities can also be calculated with the use of kinetic theory. According to [10] and [17] these coefficients are:
ˆ
Here, σ is the accumulation coefficient, v the mean thermal velocity ( q3RT
M ) and M the molar mass. The accumulation coefficient gives the fraction of particles which reflect from the vapor on the interface diffusively. The other amount of particles reflect spatially, which means that the particles bounce back elastically. The Onsager resistivities for the evaporation for water are presented in Table2.1 for two different values of the accumulation coefficient.