Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS
March 26-30, 2017, Jeju Island, Korea
ACTS-P00051
ANALYSIS ON INFLUENCING FACTORS OF EVAPORATION PROCESS
BASED ON STATISTICAL RATE THEORY
Li Zhang, Chun-Mei Wu, You-Rong Li*
College of Power Engineering, Chongqing University, 174 Shazhengjie, Shapingba, Chongqing 400044, China

Presenting Author: [email protected]
Corresponding Author: [email protected]
*
ABSTRACT
In order to understand the influence of various factors on the evaporation rate on gas-liquid interface, the
evaporation process of water in pure steam environment was calculated based on the statistical rate theory (SRT),
and the results were compared with those from the traditional Hertz-Knudsen equation. It is found that: (1) the
evaporation rate on vapor-liquid interface increases with the increase of evaporation temperature and evaporation
temperature difference and the decrease of vapor pressure. (2) When the steam is in a superheated state, even if the
temperature of the liquid phase is lower than that of the vapor phase, the evaporation may also occur on the gasliquid interface; at this time, the absolute value of the critical temperature difference for occurring evaporation
decreases with the increase of vapor pressure. (3) When the evaporation temperature difference is smaller, the
theoretical calculation results based on SRT are basically the same as the predicated results from the Hertz-Knudsen
equation; but the deviation between them increases with the increase of temperature difference.
KEYWORDS: Statistical rate theory, Evaporation rate, Gas-liquid interface, Theoretical analysis
1. INTRODUCTION
The evaporation phenomenon exists widely in daily life and industrial production process, such as, spray drying,
electronic cooling, spray printing, DNA detection and protein chip technology and so on, in which the calculation
and control of evaporation rate has a very important influence, and has already attracted the attention of many
scholars [1-3]. Many factors will affect the interfacial evaporation rate, including gas composition, temperature,
pressure and flow velocity and so on; when the evaporation of liquid is in the static pure steam environment, the
evaporation rate jm can be calculated according to the theory of classical kinetic theory (CKT) [4-7], in which HertzKnudsen-Schrage equation or Hertz-Knudsen equation is used commonly:
M  Ps Tl  P V 
jm  

(1)


2 R  Tl
Tv 
In the formula, M is the medium molar mass, R the universal gas constant, R=8.314 J/(mol·K), Ps the saturation
pressure corresponding to interfacial liquid-phase temperature Tl, Pv and Tv as the vapor pressure and the temperature
respectively, and the evaporation adaptation coefficient determined by the experiment.
Eq. (1) shows that the relationship of the evaporation flux and the temperature difference between two sides of gasliquid interface is nonlinear, if the vapor and the liquid are in equilibrium state, Tl=Tv, then the evaporation flux jm=0,
that is, no evaporation process occurs. The Hertz-Knudsen equation (1) is linearized around equilibrium, and
becomes the following form [8-9]:
M
M
jm   v hfg
(Tl  Tv )= v hfg
T
(2)
3
2 RTv
2 RTv3
in which, v is vapor density, hfg latent heat of vaporization, T temperature difference between the liquid and vapor
phases, namely,T=TlTv.
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It should be noted that both Eqs. (1) and (2) are the calculation formulas of vapor-liquid interfacial evaporation flux
on basis of the experiment, in which the evaporation coefficient  is constant and is determined by the experiment.
Therefore, Ward and Fang [10] derived the evaporation mass flux formula according to the statistical rate theory
(SRT), which was certified by a series of experimental results [11-14]. Compared with the classical Hertz-Knudsen
equation, there is no empirical constant in this expression, which not only can calculate the interfacial evaporation
rate more accurately, but also make up for the deficiencies of the classical kinetic theory that cannot reflect the
interfacial non-equilibrium effect.
This paper analyzes the influences of various non-equilibrium thermodynamic effects of the vapor-liquid interface
on the interfacial evaporation mass flux, which is based on the interfacial evaporating statistical rate theory, with
water as the working medium; the results are compared with those of the classical Hertz-Knudsen equation.
2. EVAPORATION STATISTICAL RATE THEORY
Consider the single component steady evaporation system with liquid phase, vapor phase and the interface. The
temperatures of vapor phase and liquid phase in the vapor-liquid interface were TIV and TIL, the vapor pressure PV,
the saturation pressure Ps (TIL) corresponding to the liquid phase temperature. To maintain steady state evaporation,
the continuous liquid is supplied from the bottom part, and the continuous steam that is generated by evaporation is
removed from the upper part. The liquid supplement rate should be equal to the evaporation rate to maintain the
vapor-liquid interfacial position. Simultaneously, the liquid phase is touched with the constant temperature heat
source TL, and all the heat for vaporing comes from the heat source.
For simplicity, assume that the interfacial thickness is zero, so there is no adsorption process at the interface; namely,
a liquid-phase molecule enters the vapor phase through the interface and does not stay at the interface, and vice versa.
Meanwhile, each phase of vapor and liquid is homogeneous and all the intensive parameters are constant. According
to the statistical rate theory, the evaporation mass rate at the gas-liquid interface can be deduced [10-14]:
 s 
jm  2mw Ke sinh  LV 
(3)
 kb 
in which,
Ps TIL 
 vf TIL   P L
 
Ke 
exp 

1


V
L
2 mkbTIL
 vg TI   Ps TI   
(4)
L
 T V 4 Ps TIL  
 qvib TIV    1 1  3n 6  
 TIV 
i  vf TI   L
sLV
I
i
 4  1  L   ln  L 
P  Ps TIL 
  ln 
   V  L      /T V  
V
L 
L
i
I
kb
 TI  PI 
 qvib TI    TI TI  i 1  2 e  1  kbTI
 TI 
ei /2T
hi
, i 
 i / T
2 kb
i 1 1  e
(5)
3n  6
qvib T   
(6)
In Eq. (4), the PL is the liquid phase pressure under equilibrium state, and should satisfy the following relations:
2 LV TIL 
L
V
(7)
P  PI 
rl
Above all, Δ sLV indicates the entropy change that is caused by liquid molecules into the vapor space, kb is
Boltzmann constant, kb=1.38×10-23 J/K, and h is Planck constant, h=6.626×10-34 J·s; mw is for the molecular mass of
evaporation, n for atomic number, vf and vg for the specific volume of saturated liquid molecules and gas
molecular,LV as the surface tension of gas-liquid interface, l as the characteristic temperature,i as the phonon
frequency, and qvib vibrational partition function. If the evaporation interface is flat, the curvature radius of the
interface is rl=0, and the type (7) becomes P L  PIV .
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3. RESULTS AND ANALYSIS
The working medium is water, the number of atoms is n=3, and the molar mass is M=0.018015kg/mol. The
thermodynamic properties of water and steam are calculated according to Ref. [15]. Vibrational frequency of
covalent bond of water molecules includes symmetric stretching vibration frequency, anti-symmetric stretching
vibration frequency and flexural vibration frequency, and stretching vibration frequency is usually about 2 times of
flexural vibration frequency. The values for three frequencies i were respectively [14]: 1=3.651×106×2.9979×108
=1.0945×1015 Hz, 2=3.756×106×2.9979×108=1.1260×1015 Hz, and 3=1.590×106×2.9979×108=0.4767×1015 Hz.
It was found that when the vapor is saturated, if the liquid-phase temperature is higher than the vapor temperature,
right now the liquid is in the over-saturated state, and the saturated pressure corresponding to the liquid-phase
temperature must be higher than the vapor pressure. Therefore, the evaporation process will occur at the vapor-liquid
interface. When both the vapor and the liquid are in the saturated state, the change of pressure with temperature is
described by the Clausius-Clapeyron equation, that is dp/dT=vhfg/T. Thus we can see that under the given
evaporation temperature difference, the vapor density v increases rapidly with the increase of temperature, which
makes the difference between the saturation pressure of the liquid-phase temperature and the vapor pressure increase,
so the interfacial evaporation rate increases rapidly. At a given liquid-phase temperature, the interfacial evaporation
flux increases linearly with the increase of the temperature difference. Special attention should be paid to that when
the vapor is saturated, if the liquid-phase temperature is lower than the vapor temperature T<0, the liquid phase
must be in a supercooled state, and not the evaporation process but the condensation process occurs at the vaporliquid interface, so interfacial evaporation flux is negative.
In the current experimental studies on vapor-liquid interfacial evaporation phenomenon [11-14], continuous
pumping on the vapor chamber is to maintain a constant pressure; at this time, the vapor is usually in a superheat
state. When the vapor is superheated, a new parameter  is introduced to describe the superheat degree and is
defined as the ratio of vapor pressure and saturation pressure corresponding to vapor temperature, i.e. =PIV/PsV(TIV).
Obviously, is smaller, the superheat degree is higher; =1 means the saturated vapor. In superheated vapor
environment, the change rule of evaporation mass flux in the superheated vapor is overall similar to that in the
saturated vapor. However, the difference has three points: (1) because the vapor is superheat, pressure drops, vaporphase chemical potential decreases, the chemical potential difference between liquid phase and vapor phase as the
vaporing driving force increases, so the interfacial evaporation mass flux increases. (2) The chemical potential of the
pure substance is not only related to the pressure, but also changes with the temperature; the lower the pressure is
and the higher the temperature is, the lower the chemical potential is. Therefore, when the vapor is in the
superheated state, even if its temperature is higher than the liquid-phase temperature T=Tl-Tv<0, it also ensures that
the chemical potential of the vapor phase is lower than that of the liquid phase. Therefore, the evaporation process
can still be carried out and interfacial evaporation mass flux is still positive, which have been confirmed by a large
number of experimental results [11-14]. (3) Due to the superheated vapor, the Clausius-Clapeyron equation is no
longer applicable, and the evaporation mass flux no longer increases linearly with the increase of evaporation
temperature difference.
In order to reveal the influence of vapor superheat degree on the interfacial evaporation mass flux, the variation law
of the interfacial evaporation mass flux with vapor pressure under different conditions was determined. It was found
that the interfacial evaporation mass flux increases gradually with the decrease of vapor pressure; the lower the
pressure is, the greater the increasing rate of the evaporation mass flux is.
From above, when the vapor is saturated, if the liquid-phase temperature is lower than the vapor temperature T<0,
not the evaporation but the condensation occurs on the interface; when the vapor is superheat, whether the
evaporation process occurs on the interface or not depends on the two phase temperature difference and the vaporphase pressure. Obviously, when the liquid-phase temperature is low enough, the condensation process will occur on
the interface. In this case, the critical conditions whether the evaporation process can occur on the interface are
calculated. The absolute value of the critical temperature difference Tcri in the evaporation process decreases with
the increase of the vapor-phase pressure. At =1, vapor is saturated state and the vapor and the liquid are in
equilibrium state, therefore, Tcri=0.
Compared with the traditional Hertz-Knudsen equation, the greatest merit of the SRT theory for calculating the
evaporation rate on the vapor-liquid interface is that it does not require any empirical constant or coefficient, and is a
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pure theoretical calculation formula. The comparison between the two calculation results showed that when the
vapor is saturated and the evaporation temperature difference is small, the calculated results of SRT theory are very
close to that of the Hertz-Knudsen equation, which is in the vicinity of the equilibrium state. As long as the suitable
evaporation coefficient is selected, both calculation results are basically the same. However, when the evaporation
temperature difference is larger, the difference between the calculated results is increased, and the increasing speed
of the interfacial evaporation rate predicted by SRT theory with the increase of the liquid-phase temperature is
slower than that of the Hertz-Knudsen equation. On the other hand, when the vapor is in a superheat state, the
difference of interfacial evaporation rate between the two indicating results increases.
4. CONCLUSIONS
The evaporation process of water in pure steam environment was calculated by adopting the statistical rate theory,
and the main conclusions are as follows: (1) the evaporation rate of vapor-liquid interface increases with the increase
of the temperature difference and evaporation temperature and the decrease of vapor pressure; (2) when the vapor is
in superheated state, even if the liquid-phase temperature is lower than the gas-phase temperature, the evaporation
process may also occur on the vapor-liquid interface. At this point, the absolute value of the critical temperature
difference decreases with the increase of vapor pressure; (3) when the evaporation temperature difference is small,
the calculation results for the SRT theory and the Hertz-Knudsen equation are basically the same, but the deviation
increases with the increase of temperature difference.
ACKNOWLEDGMENT
This work is supported by National Natural Science Foundation of China (Grant No. 11532015).
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