Eindhoven University of Technology MASTER Measurement of an interfacial temperature jump during steady state evaporation Betsema, R.

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Eindhoven University of Technology

MASTER

Measurement of an interfacial temperature jump during steady state evaporation

Betsema, R.

Award date:

2019

Link to publication

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Measurement of an interfacial temperature jump during steady state evaporation

Ruben Betsema Student ID: 0959507

Graduation committee:

dr.ir. C.C.M. Rindt dr.ir. A.J.H. Frijns dr.ir. Y.B. van de Burgt

Project supervisors:

dr.ir. A.J.H. Frijns dr.ir. S.V. Gaastra-Nedea

Eindhoven, Friday 12

^th

July, 2019

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Evaporation is an important phenomena that occurs in a wide range of natural and industrial processes. Although this phenomena has been a subject of research for many years, it is still not fully understood. Experimental results of the last few decades seem to contradict with each other, and with the theory which describes this process (which are kinetic theory of gasses (KTG) and non-equilibrium thermodynamics (NET)). A remarkable example is the measurement of a temperature jump of 15^◦C at the interface of a steady state evaporating water droplet at a pressure of 200P a. In order to determine whether this temperature jump exists and what influences this jump, a new experimental setup has been developed. This setup had to measure the temperature profile at the interface of this evaporating droplet in the range of 200 to 1000P a. The development of this experimental setup was the main goal of this project.

The developed setup consisted of an evaporating spherical droplet in a large vacuum chamber.

Purified, degassed and temperature controlled water (by a heat exchanger) was supplied to an evaporating geometry by a mass flow controlled syringe. The temperature profile at the interface of this droplet was measured by a K-type thermocouple made of two 25.4µm wires, with a precision of 2µm.

An attempt has been made to conduct the aimed measurement. However, due to the occurrence of bubbles inside the syringe and the channel between the syringe and the droplet geometry, it was not possible to do an experiment lower than 780P a and to measure the evaporating mass flow.

One experiment has been conducted to determine whether to other parts of the experimental setup are working properly. The measured temperature profile has been analyzed by a theoretical model. The shape of the temperature profile agrees well with the shape of the measurement but the calculated mass flux seems to be too low compared with other experiments and the theory (KTG and NET). It is not clear what the cause is of these differences. However, without the measurement of mass flow it is difficult to draw any conclusion.

In order to investigate the evaporation phenomena further, to determine whether this temperature jump exists and what may influence this temperature jump. One has to determine what the cause is of the bubbles and how these can be prevented. Without the bubbles, the mass flow can be measured and the experiments can be conducted at lower pressure.

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List of abbreviations

KTG Kinetic Theory of Gasses

NET Non-Equillibrium Thermodynamics MD molecular dynamics

DF Distribution function

MDF Maxwellian distribution function HK Hertz-Knudsen

HKS Hertz-Knudsen-Schage emf Electromotive force

RTD Resistance temperature detector EBL Electron-beam lithography NTC Negative temperature coefficient PTC Positive temperature coefficient LIF laser induced fluorecence PI Polyimide

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kb Boltzmann constant ^m_s2²K^kg

N0 Avogadro constant _mol¹

R Universal gas constant _molK^j

M Molar mass of water _mol^kg

v_p Velocity of a particle ^m_s

f Probability distribution −

m_p Mass of a particle kg

T Temperature K

Tl Temperature of the liquid at the interface K T_v Temperature of the vapor at the interface K

p Pressure _ms^kg2

p_l Pressure in the liquid phase _ms^kg₂

pv Pressure in the vapor phase _ms^kg2

n Number density _m¹3

nl Number density of the evaporating stream of gas _m¹3

ng Number density of the condensing stream of gas _m¹3

j Mass flux per unit area _m^kg₂

jtotal Net evaporating mass flux per unit area _m^kg2

Q_tot Overall energy tranport over the interface _m^j₂

h Enthalpy _kg^j

µ Chemical potential _kg^j

q_l Conductive heat flow through liquid _m^j₂ qv Conductive heat flow through vapor _m^j2

k_h Proportionality factor of crossterms −

M a Marangoni number −

γ Surface tension ^kg_s2

η Dynamic viscosity _ms^kg

αth Thermal diffusivity ^M_s²

κ_v Thermal conductivity vapor _msk^J

κl Thermal conductivity liquid _msk^J

α Evaporation coefficient _msk^J

β Condensation coefficient _msk^J

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Introduction

1.1 Background

Evaporation is an important phenomena which occur in a wide range of natural and industrial processes. An example of these are, the generation of power by a steam turbine, the formation of clouds and the cooling of electronics. Although this process has been a subject of research for more than hundred years, the physics is still not fully understood, as the two fundamental classical theories, the kinetic theory of gasses (KTG) and the theory of non-equilibrium thermodynamics (NET) cannot explain the experimental findings of the last decades [30][13][1][14][8]. A remarkable example of this is the measurement of a large temperature jump of 15^◦C at the interface of evaporating water at a pressure of about 200P a, this temperature jump is about 20 times as large as predicted by kinetic theory [17][31]. Although these experiments have been conducted very carefully, large differences can be found between the different experiments. Further analyses suggest that the experimental results are very sensitive to design features of the experimental setup, such as the geometry of the interface and unaccounted heat fluxes towards the interface[24][23] [10].

The controversy in this topic started by the discovery of Pao in 1971 that KTG predicts a temperature jump in the range of 0 to 1^◦C at the interface of an evaporating liquid at low pressure (200 to 1000Pa) at the scale of a few mean free path[31][17]. This led to some controversy in this topic because this temperature jump implies that in a certain situation, the overall heat transfer in a system might be, at a local point, in the opposite direction of the temperature gradient. Which is for example a gas between a hot evaporating liquid, and a cold condensing liquid [21][20].

The first successful attempt to measure this jump directly has been conducted by Fang and Ward [13]. In order to be able to measure in the range of a mean free path, they measured the temperature of evaporating water at low pressure (200 to 1000P a) with a thermocouple made of two 25.4µm wires. Surprisingly, they measured a temperature jump up to 7.8^◦C for a pressure of 195P a. This is much larger than the temperature jump of 0.2^◦C predicted by KTG.

In order to explain these unexpected results, Bond and Sturchtrup [23] made a model of the mass flux and heat flux towards an evaporating interface for a 1D spherical and 1D planar model.

According to their model, the evaporating mass flux is twice as high as what would be possible from the conservation of energy. They suggest that the disagreement between their model and the experiments is due to extra heat fluxes towards the evaporating interface. In addition, they discovered that the evaporating mass flux is strongly limited by the heat supply through the vapor and liquid layer towards the interface and, as a consequence, it depends strongly on the geometry of the interface. The hypotheses that the evaporation is limited by heat transfer toward the interface is also supported by a molecular dynamics (MD) simulations and experiments on evaporating nano droplets conducted by Holyst et al. [27][26].

As a consequence of these experimental results and the findings of Bond et al., Baman et al conducted a new series of experiments [30][29]. Similar to the experiment of Fang and Ward

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but with some improvements. They suggested that the extra heat supply towards the interface, suggested by Bond et al., is due to the conductance of heat through the glass funnel, which holds the water droplet so they used a PVC slot as the liquid-droplet container. In addition, they measured the influence of heat flux in the vapor phase on the mass flux and the temperature jump with a resistance wire located 3mm above the interface. The maximum measured temperature jump was up to 15^◦C for a heater with a temperature of 90^◦C. Which is even more peculiar than the results of Fang and Ward.

A third remarkable experiment is the experiment conducted by Phillips et al.[8][7] in which they tried to measure the heat flux from the interface with inverse temperature profile directly.

They constructed a cell, consisting of two plates. One cold plate from which the liquid is evaporating and a hotter plate at which the liquid condensates. They claimed to have measured a cold to warm transport of energy. However, Struchtrup and Bond[23] analyzed the result with the use of non-equilibrium thermodynamics (NET) and determined that the resistivities are up to an order of 5 times as high as that one should expect from kinetic theory.

Some researchers claim that they have found an explanation for the unexpected high temperature jump. Which are, for example, non-equilibrium, rarefaction or conduction effects. Although these explain to some degree the difference between the experiments and the theory, they do not explain why the different experiments differ so much between each other.

1.2 Problem description

It is not clear why the experimentally measured temperature jump is many times higher than what would be expected from the KTG and NET, and why this jump differs so much between the different experiments. However the model of Bond et al., [10] indicates that the temperature jump strongly depends on the evaporating mass flow, and that this mass flow strongly depends on the heat fluxes towards the interface. This hypotheses is supported by other research[27][26].

The heat fluxes towards the interface are not only sensitive to the shape of the interface but also to heat leakages towards the interface.

The goal of this project is to develop a setup and to conduct experiments similar to the experiments of Fang et al.[13] and Badam et al.[30]. This to determine whether this temperature jump exists and how the results can be compared with other experiments with the use of KTG and NET. This goal had led to the research question of this project:

What is the temperature profile and temperature jump of a slowly evaporating water droplet water in the pressure range of 200 to 1000P a, in the region of a few mean free path, and how does it compare to KTG, NET and previously conducted experiments.

This question will be answered with the use of the following sub questions:

1. What are the classical evaporation theories and how can this be compared with measurement results?

2. What is the current state of research and what are important design features that can be used in the current setup?

3. What is the most suitable temperature measurement method and how does calibration affect the measurement results?

4. What is the best design to conduct the measurement stated in the main research question?

5. How does the measurement results compare with the theory and with the previously conducted research?

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CHAPTER 1. INTRODUCTION

1.3 Outline

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.

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Overview of theory, experiments and simulations

This chapter provides an overview of the relevant research in the field of the evaporation phenomena. 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:

f_M(v_p) = r m_p

2πkbTexp(−mpv²_p 2T kb

) (2.1)

Here f is the probability that a particle will travel with a certain velocity v_p, m_pis the mass of a single particle, k_b 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

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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

Z ∞

−∞

vpfM(Tl, vp)dvp− nvmp

Z ∞

−∞

vpfM(Tv, vp)dvp (2.3)

jtot= 1 4mpnv

s 8Tlkb

πmp

−1 4mpnl

s 8Tvkb

πmp

(2.4) With the use of the ideal gas law (n = _{T k}^p

b) this expression can be rewritten to:

jtot= pl

r mp

2πTlKb

− pv

r mp

2πTvKb

(2.5) However, experimental observations suggest that the real mass flow is lower than this theoretical value. That is why commonly an evaporation (α) and condensation coefficient (β) is introduced:

j_tot= αj_evap− βj_cond= r m_p

2πkb

(α p_l

√T_l − β p_v

√T_v) (2.6)

This relation is known as the general Hertz-Knudsen equation (HK-equation) and it is commonly simplified with the assumption of α = β to:

jtot= α r mp

2πkb

( pv

√Tv

− pl

√Tl

) (2.7)

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) = f_M(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:

j_tot= α r mp

2πk_b( pl

√T_l − Ψ( ˜V ) pv

√T_v) (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 K_b (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:

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jtot= α r mp

2πkb

( pl

√Tl

− (1 − j mpnv

r mpπ 2T kb

) pv

√Tv

) (2.13)

jtot= 2α 2 − α

r mp

2πk_b( pv

√T_v − pl

√T_l) (2.14)

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 situations.

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 (σ^s), is the sum of thermodynamic fluxes (Ji) times its thermodynamic force (Xi):

σ^s=X

i

J_iX_i≥ 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]:

J_i=X

j

ˆ

r_ijX_j (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 gradient 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]:

Q_tot= j_toth_l+ q_l= j_toth_v+ q_v (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

1 T_s− 1

T_l

+ qv

1 T_v − 1

T_s

− J 1

T_s(µv− µl) ≥ 0 (2.18) Where µ_v, µ_l and T_s are the chemical potential of the vapor and the chemical potential of the liquid and the inter facial temperature respectively. The heat flux, q_l, in the liquid phase and the

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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

1 T_v − 1

T_l

− jtot

1

T_l(µv− µl) (2.20)

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

T_l(µ_v− µ_l), 1 T_g − 1

T_l] (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:

"

J qv

#

=

"

ˆ r11 rˆ12

ˆ r21 rˆ22

# "₁

T_l(µv− µl)

1 Tv −_T¹

l

#

(2.23)

With description of chemical potential for a vapor in equillibrium with a liquid per mol:

µl= µ⁰_v+ RTlln(Psat(Tl)) (2.24)

µ_v= µ⁰_v+ RT_lln(P_v) (2.25)

Equation2.23can be further simplified to:

"

J q_v

#

=

"

ˆ r11 rˆ12

ˆ r₂₁ rˆ₂₂

# "_RT

l

M (µv− µl)

1 T_v −_T¹

l

#

(2.26)

The interfacial resistivities of experimental results can be calculated with the use of this equation. However, the resistivities can also be calculated with the use of kinetic theory. According to [10] and [17] these coefficients are:

ˆ

r11= 0.79 1 − 0.40σ 1 − 0.46σ

!

nvRT^vv (2.27)

ˆ

r21= ˆr12= − 0.10σ 1 − 0.46σ

!

nvM v (2.28)

ˆ

r₂₂= 0.23 1 − 0.46σ

!n_vvM²

RT^v (2.29)

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.

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2.2 Important experiments and simulations of the past dec- ades

The discovery that kinetic theory predicts a temperature jump at a liquid-vapour by Pao [31], has led to great interest and controversy in this subject matter by many researches [21],[20],[13][10].

According to some of them, an inverted temperature profile has been expected for a vapor phase between a hot evaporating liquid and a cold condensing liquid. Here, due to the temperature jump, the temperature in the vapor phase close to the evaporating hot liquid is colder than the temperature of the vapor close to the condensing cold liquid. This implies that temperature gradient is in the opposite direction of the overall heat flux of the system. This has sometimes been referred to be anomalous. The existence of this phenomena has been supported by molecular dynamic simulation [28][2].

A first attempt to measure this anomalous temperature distribution experimentally was by Shankar and Deshpande[25]. They measured the temperature distribution between a evaporating liquid surface and cooler condensing surface by a 0.3mm K-type thermocouple. They measured a slightly inverted temperature gradient. However, they had difficulties with controlling the pressure inside the chamber. Which might be caused by the contamination of the liquid and which they estimated to be around 133 to 266P a. In addition, the thermocouples they used are relatively large which cause an extra flow of heat to the system.

2.2.1 Fang et al.

To overcome the problems of Shankar et al., Fang et al.[13] constructed a different experimental setup, in which they evaporate purified degassed water from a glass funnel. This to reduce the effect of contamination. The water was cooled down to a temperature of 15, 26 and 35^◦C. The Funnel was placed in a double walled vacuum chamber to reduce the effect of heat conduction.

The temperature profiles at the interface was measured by a 25.4µm K-type thermocouple for pressures ranging from 194.7 to 596P a with intervals of 10µm. The small thermocouple allows them to measure the temperature jump within the Knudsen layer. The results of these experiments are shown is table2.2. Unexpectedly, they measured a temperature discontinuity at the interface up to 7.8^◦C. This is much larger than what would be expected from KTG and NET.

Bedeaux et al [10] calculated the Onsager resistivities of the results with the use of Equation 2.23to examine the results with NET. Because only two of the three resistivities can determined directly from the measurement data they calculated the resistivity for the case of no coupling ( ˆr₁₂= 0) and for a coupling which is proportional to the theoretical values. This proportionality factor, defined as kh is derived by dividing the theoretical value mass flux resistivity by the theoretical value of the coupling resistivity (Equation 2.29 by Equation 2.28). Which results in the following:

ˆ r₁₂ ˆ r11

= 0.44RT_v= 0.18∆_vapH = k_h∆_vapH (2.30) Here ∆_vapH is the enthalpy of evaporation approximated by 2.5RT_v. The calculated resistivities for the measurement data have been presented in Table2.2. The values of the resistivities calculated by Equation2.27, Equation2.28and Equation2.29for T_v= 273.9K have been presented in Table2.1. For both values of k_h, the measured resistivities are up to 30 times larger than the theoretical values.

Bond et al. [23], developed a mathematical model to determine the influence of heat fluxes towards the interface through both liquid and vapor for a planar-spherical and mixed planar- spherical geometries. An interesting feature of this model is that it shows that the evaporation rate is strongly dependent on the heat fluxes towards the interface and as a consequence, the geometry of the interface. As a result, they conclude that the measurement is very sensitive to heat leakages and deviations in the evaporation geometry. They cannot explain the observed

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σ = 1 σ = 0.1 dimension ˆ

r₁₁ 3.5x10⁵ 3.1x10⁵ J/m²s ˆ

r21 −32.5 −1.84 mol/m²s ˆ

r22 3.3x10⁻² 1.9x10⁻³ mol²/jm²s

Table 2.1: Transfer coefficients for evaporation of water using kinetic theory for different values of the accumulation coefficient.

ˆ

rqqx10³(J/m²s) ˆrwwx10⁸(mol²/10⁴jm²s)

Pa µlh⁻¹ gm⁻²s⁻¹ Tv Tl ∆T kh= 0 kh =

0.18

kh= 0 kh = 0.18

596.0 70 0.2799 3.3 -0.2 3.5 10.6 8.1 4.6 1.4

493.3 75 0.2544 0.9 -2.8 3.7 9.9 7.9 4.4 1.00

426.6 85 0.3049 -0.5 -4.7 4.2 9.1 7.1 4.6 1.05

413.3 90 0.4166 -0.9 -5.1 4.2 9.7 6.9 6.2 1.43

310.6 100 0.3703 -3.6 -8.7 5.1 8.1 5.6 3.3 0.95

342.6 100 0.3480 -2.3 -7.6 5.3 8.3 7.1 7.6 1.05

333.3 100 0.3970 -1.5 -7.7 6.2 7.2 4.9 2.8 0.84

269.3 110 0.4081 -4.2 -10.5 6.3 7.2 5.4 4.0 0.92

277.3 110 0.4347 -4.1 -10.2 6.1 6.7 5.0 5.2 1.05

264.0 120 0.4097 -4.8 -11.0 6.2 7.7 6.8 9.3 1.08

269.3 120 0.4860 -4.0 -10.5 6.5 7.4 5.4 4.8 1.07

245.3 130 0.4166 -5.8 -11.8 6 8.5 6.9 5.3 1.03

233.3 140 0.4938 -4.9 -12.3 7.4 6.9 5.1 4.3 0.95

213.3 150 0.5086 -5.9 -13.4 7.5 7.0 4.6 2.9 0.87

194.7 160 0.5386 -6.5 -14.5 7.8 6.7 5.0 4.6 0.97

Table 2.2: Measurement results Fang and ward [13]

evaporation rate by their model because the energy required for this is twice as high as that could be conducted through the liquid and vapor towards the interface. The hypotheses that the evaporation is limited by heat transfer toward, the interface is also supported by a molecular dynamics (MD) simulation and an experiment on evaporating nano droplets conducted by Holyst et al. [27][26].

2.2.2 Badam et al.

As a possible explanation for the extra heat supplied towards the interface, Duan et al. suggest that the heat is supplied by thermocapillary convection just below the interface in the liquid layer[12].

This idea is supported by the fact that Fang and Ward measured the temperature to be constant for a layer of a thickness of 100µm in the direction perpendicular to the interface, just below the interface. If the energy was supplied purely by diffusion, one would expect a linear temperature profile up to the interface. According to them, this is the result of a parabolic temperature profile at the interface caused by heat leakages through the funnel wall. The temperature gradient at the interface will cause a gradient in surface tension because this is a function of temperature. The net tangential force caused by this gradient will cause a convective flow. The dominance of this effect can be expressed by the Marangoni number:

M a =dγ(T ) dT

∆T L ηαth

(2.31) In which, γ(T ) is the temperature dependent surface tension, ∆T the temperature difference over a characteristic length scale , L the characteristic length between these temperature (which

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ˆ

rqqx10⁴(J/m²s) ˆrwwx10⁸(kg²/jm²s) Pa 10⁻⁴kg/(mT²vs) Tl ∆Tmsd ∆TKT G kh= 0 kh =

0.18

kh= 0 kh = 0.18 No heating

561.0 4.70 -1.10 0.73 1.83 0.06 1.33 1.05 162 15.7

490.0 4.92 -2.96 -0.93 2.03 0.08 1.26 1.03 171 14.8

389.1 5.49 -6.02 -3.76 2.27 0.10 1.23 0.66 58.7 12.6

336.5 5.79 -7.90 -5.29 2.60 0.12 1.09 0.41 37.7 10.5

292.4 6.15 -9.61 -6.83 2.78 0.15 1.07 0.16 23.4 8.96

245.3 6.59 -11.67 -8.42 3.25 0.18 0.95 0.03 17.9 7.60

Heating at 30^oC

736.0 5.78 2.65 6.64 3.99 0.15 1.62 1.62 3610 9.93

569.5 6.07 -0.91 2.93 3.84 0.17 1.52 1.49 546 10.5

483.0 6.36 -3.18 1.04 4.22 0.20 1.36 1.34 833 10.0

391.2 6.87 -6.02 -1.26 4.76 0.26 1.27 1.19 202 9.20

295.2 7.37 -9.51 -4.01 5.50 0.33 1.06 0.74 31.3 6.91

240.3 7.68 -11.85 -6.09 5.76 0.43 1.10 0.62 17.8 5.87

Heating at 40^oC

736 6.71 2.65 7.97 5.33 0.20 1.61 1.60 2860 8.67

567 7.16 -0.97 4.82 5.79 0.27 1.50 1.49 642 8.34

485 7.34 -3.12 3.02 6.14 0.31 1.41 1.40 713 8.02

392 7.80 -5.95 0.57 6.52 0.39 1.34 1.26 123 7.56

288.5 8.32 -9.80 -2.22 7.59 0.53 1.17 0.96 34.9 6.04

236.6 8.53 -12.18 -4.00 8.18 0.66 1.10 0.89 30.0 5.61

Table 2.3: Measurement results Badam et al.[30]

is between the wall and the middle of the droplet in the case of the experiment), η the surface tension and α the thermal diffusivity. There are three regimes which can be described by this number [12]. For M a < 100, the flow can be neglected. In the range M a = 100 − 22000, the flow becomes more dominant. And M a > 22000, this is a turbulent regime in which this effect is highly dominant.

To avoid this effect of thermocapillar convection. Badam et al. constructed a new setup from which they evaporated a droplet from a PMMA because of the low thermal conductivity [30],[29]

(see Figure 2.1). In addition, because Bond et al. indicated the dependency of heat flux on the evaporation rate, they equipped the setup with a heating element just above the interface to determine this dependence experimentally. The results of this experiment are presented in Table 2.3. The measurement results were surprising as they measured a temperature discontinuity up to 14^◦C for a temperature of the heating element of 80^◦C The evaporating mass flux, and so the temperature jump increased as pressure decreased or the heat flux at the interface increased. In addition, the temperature jump was not the same under the same conditions as in the case of the experiment by Fang and Ward. An explanation for this is that the difference in geometry reduces the heat flux to the interface and so the evaporating mass flux and so the temperature jump.

The evaporation coefficient was calculated for the different experiments with the use of the Hertz-Knudsen-Schage equation (Equation 2.14) for the different experiments. The calculated coefficients are scattered in the range of 0.03 to 0.2. In addition, they calculated the Onsager resistivities of the evaporation in the same way as Bedeaux et al. did for the results of Fang and Ward. The calculated resistivity values are much higher than the predicted value of kinetic theory (see Table2.1).

Kazemi et al. [24] have investigated the influence of conduction through the wires of the thermocouple on the measurement results by a CFD simulation. They assumed that the experiment can be simulated as a continuum as they calculated a Knudsen number of maximum 0.8 if one uses

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(a) (b)

Figure 2.1: The thermocouple used in the experiments of Badam et al.(a) and a schematic overview of the experimental setup (b) [29].

the diameter of the thermocouple junction as the characteristic diameter is used in the following way:

Kn = λ Lchar

= µ ρdp

r π

2RT (2.32)

Where,λ is the mean free path, Lcharis a characteristic length scale, µ is the viscosity, ρ the density, dp the characteristic diameter, R the universal gas constant and T the temperature. The for the value of the temperature jump they used the Smoluchowski temperature jump condition as boundary condition.

They found that the temperature is overestimated by a factor of 2 due to the combination of heat conduction and the temperature jump due to the Smoluchowski boundary condition.

However, as they indicate, the assumption of that the flow can be simulated as a continuum is doubt able. As the normally limit is about Kn = 0.1

2.2.3 Phillips-Onsager Cell

In this series of experiments Phillips and coworkers claimed that they have measured the cold to warm transport, as was predicted by [31], [21]. The experimental setup consisted of chamber partly filled with purified and degassed liquid water and partly with water vapor. This chamber was made of two large plates placed closed to each other in a way that the system could be analyzed as a 1-D system [8]. During the experiments, the lower plate was kept at a constant temperature and the temperature of the upper plate was varied between −8^◦C and 10^◦C. By measuring the pressure inside the chamber (in the range of 520 to 1300P a), and measuring the temperature of the upper and lower plate, the heat of transport is described by the following relation:

Q∗ = − T_L psat(TL)

dp(T_H) dTH

(2.33) Sturchup et al. [15], have made a mathematical model of the setup and calculated the Onsager resistivity at the evaporating liquid vapor inter phase. With the use of the measurement results of Phillips et al. [8], they calculated the resistivity of heat transport at the liquid/vapor interface to be: ˆr11= 61500. This value is about five times as low as the value predicted by kinetic theory.

(see Table 2.1). In addition, they determined that the calculated resistivity is very sensitive to the temperature of the plates in the setup and that disagreement between the measured value and KTG may be a consequence of a measurement error. A measurement error 0.5^◦C can explain the difference between the theoretical and experimentally resistivities.

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Pressure T_l T_v [T_v− T_l] 266 -10.83 10.46 0.36 303.2 -9.15 -8.82 0.33 369.4 -6.69 -6.40 0.29 435.7 -4.52 -4.28 0.24 544.6 -1.61 -1.40 0.21

672.5 1.39 1.56 0.17

815.5 4.08 4.22 0.14

Table 2.4: Temperature jump of a concave evaporating liquid- vapor interface. Conducted by Kazemi et al. [1]

2.2.4 Kazemi et al.

The experiment conducted by Kazemi et al. [1] are similar to that of Fang and Ward and Badam et al. [13][30], but instead of a convex spherical or cylindrical geometry they used a concave spherical geometry by evaporating the water from a vertical tube with an inner diameter of 13mm. They used the same size and type and size of thermocouples as in the other experiments. In contrast with the other experiments they found a temperature jumps in the range of 0.14to0.40^◦C (See Table 2.4) they could not compare their results with the temperature jump predicted by KTG because they did not measured the overall mass flow. However they did make a CFD simulation of the process to determine the influence of heat fluxes towards the interface. They found that the evaporation process is dominated by the heat transfer towards the interface. Which was concluded after imposing different theoretical expressions for the evaporation rate in the simulation. In this way they confirmed the hypotheses of [23], which concluded the same results by a mathematical model instead of a CFD simulation.

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Chapter 3

Selection of a temperature sensor

The first step in the design of the experimental setup was the selection of a temperature sensor to measure the temperature at the interface, as this is leading for the overall design of the setup. As an example, an optically measurement method requires a different type of setup than a setup with an measurement tip. The different considered temperature sensors are discussed in this chapter and used to select the most suitable sensor.

It is not clear to what extend the use of a thermocouple as a temperature sensor has influence on the measurement results of the experiments conducted by Fang et al[13]. and Badam et al.[30].

A way to do this is to conduct a measurement with a different type of sensor. However, as will be argued in the end of this chapter, a thermocouple is the only type of sensor which can be made small and accurate enough to conduct the desired measurement.

The requirements for the temperature measurement, used to make a selection, are:

• The sensor should be small enough to measure the temperature at the scale of a mean free path above the interface. Which is estimated to be 50 µm at 200 pascal and 0^◦C with the following formula from KTG[9]: l = ^√^k^b^T

2πd²_pp. Where l is the mean free path ,dpis the kinetic diameter of the gas particle.

• The sensor should be able to measure temperatures in the range between −20 and 100^◦C.

The temperature of the evaporating water cools down during the evaporation process and may become super cooled. The upper limit has been set to be able to calibrate the sensor boiling water.

• The temperature sensor should be able to conduct a measurement with an accuracy of

+

−0.1^◦C.

• The act of measurement should influence the evaporation process as less as possible. In the way that it should not supply an extra energy flux to it.

The working principle and the significant properties of different temperature sensors are presented in the first section of this chapter and how this can be implemented to conduct the desired measurement. The properties of the thermocouple are discussed more elaborately as this type of sensor is used in the setup.

3.1 Overview of considered temperature sensors

3.1.1 Thermocouple

A thermocouple is an electronic measuring device, which generates a temperature dependent potential difference between two wires, of a different material, as a consequence of the Seebeck

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effect. At one end, the two wires are joint together to a measurement tip. A thermocouple can only measure a temperature difference, which is between the object or fluid to be measured and a reference temperature (see Figure3.1for a schematic overview).

The Seebeck effect is the generation of an electric potential as a consequence of a temperature gradient in a material. As described by the theory of non-equilibrium thermodynamics, a temperature gradient will not only cause a flow of heat but also a electric current. In this case the generated flux is the flow of heat through the wire and the force is the potential difference. This can be written as:

E = SdT

dx (3.1)

The value S, is the Seebeck coefficient, which relates the electric field with the temperature gradient. This coefficient is material and temperature dependent.

In terms of a measurable potential:

V = Z T₂

T₁

S(x, T )dT (3.2)

This potential difference can be measured with a voltmeter and it can be used to calculate the temperature difference between both sides or to determine the temperature at one side if the temperature at the other side is at a known constant value. A schematic example is shown in Figure3.1

It is important to stress that the net electromotive force (emf) is not generated in the junction but in the wires.

Figure 3.1: Schematic principle of a thermocouple. A temperature gradient inside generates an emf in the wire. If the wires A-B and A-C are made from a different material, the generated potential difference does not cancel each other out and a net potential between B and C exists.

To be able to conduct a precise measurement, it is important that the generated electromotive force (emf) is predictable. The thermocouple can be calibrated by measuring the generated voltage at different known temperatures. After this, the generated emf should only be a function of the temperature in the junction. In practice this can be quite difficult, because the Seebeck coefficient is a function of the temperature, temperature gradient and microscopic material structure. The first two dependencies can be controlled by a proper thermocouple design. Large temperature gradients should be avoided because second order effects can influence the measurement, and the wires should be well isolated. An example of a poorly design thermocouple is depicted in Figure 3.2. Here, the emf is generated in a small part of the wire, which is the consequence of a bad insulation.

There are many different types of thermocouples. Among them, eight are standardized and their emf-temperature relationship is internationally recognized. Those eight thermocouples are named with a letter and are listed in Table3.1.

Platinum-based thermocouples(B,R,S) are mostly suitable for precision laboratory measure- ments and for industrial applications above 1000^◦C but they are quite expensive. The second group, which are the nickel based thermocouples(N and K), are used in application who do not require high temperature measurement.

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CHAPTER 3. SELECTION OF A TEMPERATURE SENSOR

Figure 3.2: An example of the emf generated in the thermocouple wires. Due to a bad isolation, the emf is generated locally and equally over the wire [4].

Type Temperature range S at 0^◦ C S at 100^◦ C (^◦ C) (µV K⁻¹) (µV K⁻¹)

B 0 - 1600 -0.2 0.9

R -50 - 1600 5.3 7.5

S -50 - 1600 5.4 7.3

K -200 - 1200 39.5 41.4

N -200 - 1200 25.9 29.6

E -200 - 1000 58.7 67.5

J -200 - 700 50.4 54.4

T -200 - 400 38.7 46.8

Table 3.1: Properties of standard thermocouple types [4].

The third group, (E,J and T) has one constantan wire. The Seebeck coefficient for constantan is higher than other material, this makes the signal stronger. The latter two groups are typically used for temperatures below the 1000^◦C.

The thermocouple can be fabricated in the required size by welding the thin wires together by a spot weld caused by a spark or laser. An alternative is to fabricate a thermocouple by electron-beam lithography (EBL) which can be used to manufacture smaller sensors. Balcytis et al. [3] managed to fabricate a Au-Ni 2, 5µm thermocouple at a 30nm thick membrane. This technique can be used to fabricate smaller thermocouples but the production process is much more complicated.

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3.1.2 Resistance temperature detector

A Resistance temperature detector(RTD) is a device of which the electrical resistance of the measurement tip is sensitive to the temperature. By applying an electrical current through a calibrated tip, one can measure its electrical resistance by Kelvin or Wheatstone bridge. If the temperature-resistance relation of the sensor is known, one can deduce te temperature from the resistance measurement.

Most commonly used metals for a RTD are copper, nickel and platinum[19]. Copper is relatively cheap but has a small temperature range as it may oxidate above the 100^◦C, nickel has a larger temperature range but the resistance temperature dependence is nonlinear, which makes it more complicated to use. Platinum can be used up to 1000^◦C and has a linear relation between the resistance and temperature. The minimum temperature at which these sensors can be used is close to absolute zero.

Typically the metal wire is wrapped around a ceramic cylinder to increase the length of the wire. The typical uncertainty is about 0.01 to 0.1degreeCelsius but that depends strongly on the relative length of the temperature dependent part.

It will be difficult to fabricate a RTD as small as 25µm because the wire has to made even smaller than this to be able to wrap it around a ceramic insulator. Another possibility is to make a wire element with the use of etching. But this as well needs to have some support to make the film element flat enough to make precise measurement. So the design of an RTD will be a trade-off between accuracy(long wire) and size(short wire)

[6].

3.1.3 Semiconductor based resistors

An alternative to the metal RTD are the thermistors, which are semiconductors whose resistance is a function of the temperature. Most thermoresistors are made of ceramic, which limits the possible shape of the resistor to the shape of beads rods or disks. The minimum size of an negative temperature coefficient (NTC) or positive temperature coefficient (PTC) resistor is about 400µm.

and the uncertainty is about 0.01 to 0.1^◦C [6] at the size of 1 to 3mm. Thermisistors are compared to RTDs cheaper to fabricate, but less sensitive at smaller size. [19]. So the themorisitor does not meet the requires size that is needed for the experiment.

3.1.4 Micro-fiber thermometers

A micro-fiber sensor is a sensor which consist of one or multiple optical wires [11]. The optical properties of this fiber or the properties of a coating, are sensitive to the temperature and can be used to measure the temperature. Which is done by sending a light beam trough the fibre to the optical sensitive part of the sensor, and measure the reflected or transmitted light. In the case of transmission, the light source is at one end of the fibre, temperature sensitive part is in the middle and the sensor at the other end. In the case of reflection, the light source end sensor are at one end of the wire and the temperature sensitive part is at the other end.

The measurement part of the optical fibre, which is a lose end or the middle of a closed loop depending on whether the reflection or transmission is measured, can be made temperature sensitive by a coating. This can be a coating for which refraction index is a function of the temperature, and one can deduce the temperature by the spectrum of the transmitted light through the wire, or it can be a fluorescence coating. For the latter, the coating will emit light after exciting by a laser pulse. The characteristic time of the decay of the emitted light is a function of temperature.

An alternative is to apply a grating to the measurement part of the fiber. In this way the reflective index is periodically manipulated along the length of the wire. Due to thermal expansion of the material, the shape and so the reflective index will change at the grating as function of the temperature. This will only work if the diameter of the fibre is in the range of multiple times the wavelength of the light, which is 3 to 5µm for red light.

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CHAPTER 3. SELECTION OF A TEMPERATURE SENSOR

One of the advantages of the use of an optical wire for the measurement at the liquid-vapor interface is that it has a very low thermal conductivity compared to a metal based thermometer which minimizes heat flow towards the interface. However, these types of sensors are not as accurate as required. And in addition, the fabrication is relatively complex.

3.1.5 Bimetals

The thermal expansion of a metal can be used to measure a temperature. In a bimetal-thermometer, two metals with a different expansion coefficient are glued together. If the temperature is increased the different expansion will cause a stress in the material and, as a consequence, a net rotation.

The bending of the material can be used to read a temperature.The typical operating range is between -20 and 400^◦C. The uncertainty is about 1 to 10^◦C

A possibility to use this in in the setup is to measure the bending of a two thin films glued on top of each other by a camera or a laser. This thin film has then to be supported by a guidance.

however, It will be difficult to make this small and accurate enough to meet the requirements.

3.1.6 Liquid in glass

The thermal expansion coefficient of a liquid can be used to measure a temperature. This is typically done by a bulb on which a capillary has been attached. The bulb acts as a measurement tip and a reservoir for the expandable liquid. the interface of the liquid will be inside the capillary part of the thermometer and will act as an indication of the temperature, as the interface will move up if the fluid expand, the position of the interface can be measured in the same way as in the case of the bimetal sensor. To use this type of sensor in the setup, one has to fabricate a liquid container smaller than 25µm and an even smaller capillary part for reading the temperature.

3.1.7 Infrared

Every material emits photons who’s spectrum is given by the Planck law, and is it known as black-body radiation:

Eλ,b= C1

λ⁵[exp(C₂/λT ) − 1 (3.3)

The temperature of an object can be determined by measuring the emitted spectrum of an object by for instance a camera. A Planck curve has to be fitted to the measured spectrum by adjusting the temperature in the above equation. At room temperature, the peak of the curve is at the wavelength of infrared light 800 to 1000 nm. The advantage of Infrared is that it does not disturb the evaporation process by an extra supply of heat or the flow of vapor. however, the intensity of the infrared is very low at low temperatures. Which makes it difficult to use this technique at the requirement of the measurement around zero degree.

[6].

3.1.8 Raman spectroscopy

The raman scattering of some molecule bands is sensitive to the temperature. A molecule can be exited to a higher energy level by the absorption of a photon. In the case of raman spectroscopy, this is a virtual state. After excitation, the molecule falls back into a ’normal’ quantum state.

This state is the same, one high or one lower than the original energy state, known as Rayleigh, Stokes and anti-Stokes scattering respectively.

Raman spectroscopy has been used in [16] and [18] to measure the temperature of a stream of evaporating water droplets in a vacuum, in such a way that the water molecule evaporated balistically (without collisions). The temperature was used to calculate the evaporation coefficient. Raman spectroscopy was used because the OH spectrum of liquid water is very sensitive to temperature. The accuracy of the measurement was about 2^◦C.

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The minimum size of the measurement spot is typically 1 − 10 µm and it is a function of the wavelength. If the measurement point becomes too small, the point starts to diffract and is known as the Airy gab (which is a consequence of Heisenberg uncertainty principle, one cannot define momentum and position of a light particle with complete certainty).

Typically in Raman spectroscopy experiments, a laser is used as a light source and a CCD camera in combination with a spectrograph is used as a detector.

The disadvantge of raman spectroscopy is that it is not accurate enough in the required range of o^◦C and at low pressure.

3.1.9 Laser induced fluoresence

In laser induced fluorescence (LIF), the molecule is exited to a quantum state with a higher energy level by a photon. After the excitation, the molecules fall one by one back to their ground state by emitting a photon, this process is known as fluorescence. The timescale of the fluorescence depends on the temperature of the molecule, which can be used to derive the temperature. Mostly, one has to add a special dye to the fluid, as most molecules, such as water, are not suitable enough to apply LIF for temperature measurement. The equipment used for these measurement are the same as in the case of Raman spectroscopy. Namely, a laser as light source and a CCD camera and spectrograph as detector. The advantage of LIF is that it is a non invasive measurement, as it does not disturb the flow at the interface of causes an extra supply of heat towards it.

3.2 Conclusion

Although the wide variety of temperature measurement methods, the thermocouple is still the best option. The combination of small size and high accurate around 0^◦C cannot be met by other techniques. The alternatives invasive measurement methods can not be made small enough and the non-invasive are not precise enough at the working temperature. The trade-off that has been paid is that one cannot determine whether the temperature jump is due to the use of a thermocouple.

Among the thermocouples, it has been decided to use a K-type thermocouple. The only available thermocouple wires in the size of 25.4µm was a K-type and a E-type thermocouple (Omega). Although the E-type thermocouple has a higher Seebeck coefficient, it has been decided to use the same K-type thermocouple as has been used in the experiments of Fang et al.[13], Badam et al.[30] and Kazemi et al. [1]. In this way, the experimental results of the current setup can be better compared with the previous experiments.

Eindhoven University of Technology MASTER Measurement of an interfacial temperature jump during steady state evaporation Betsema, R.