Entropy generation analysis of Rayleigh convection in gas–liquid mass transfer process

Accepted Manuscript Title: Entropy generation analysis of Rayleigh convection in gas-liquid mass transfer process Authors: Dong Li, Man Chen, Song Zhao, Aiwu Zeng PII: DOI: Reference:

S0263-8762(18)30185-0 https://doi.org/10.1016/j.cherd.2018.04.011 CHERD 3128

To appear in: Received date: Revised date: Accepted date:

15-8-2017 3-4-2018 10-4-2018

Please cite this article as: Li, Dong, Chen, Man, Zhao, Song, Zeng, Aiwu, Entropy generation analysis of Rayleigh convection in gasliquid mass transfer process.Chemical Engineering Research and Design https://doi.org/10.1016/j.cherd.2018.04.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Entropy generation analysis of Rayleigh convection in gas-liquid mass transfer process

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Dong Li, Man Chen, Song Zhao, Aiwu Zeng*

State Key Laboratory of Chemical Engineering, School of Chemical Engineering and Technology, Tianjin University,

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Tianjin 300072, China

Correspondence: Prof. Aiwu Zeng ([email protected]). State Key Laboratory of Chemical

Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072,

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China.

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Graphical abstract

Highlights

Entropy generation is first applied to explore an unsteady mass transfer process.

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The critical value offers a quantified thermodynamic criterion for the transient state.

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Rayleigh convection enhances mass transfer by reducing the local irreversibility.

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Rayleigh convection changes the irreversible route with the ordered plume structures.

Abstract The exploration of the transient state of Rayleigh convection is implemented via the entropy 1

generation analysis, which is always used to identify the irreversibility with the available energy losses for the non-equilibrium thermodynamic processes of CO2 absorbed into some solvents. The calculation of local entropy generation rate for the convective mass transfer is worked out with the quantitative Schlieren technology, and its profiles perpendicular to the gas-liquid interface are applied to analyze the evolution of Rayleigh convection. The results show that entropy generation reaches approximately 2.50 W·K-1·m-3 for the CO2-ethanol system until interfacial convection

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occurs to reduce local irreversibility with the synergistic effects of velocity vector and concentration gradient, as a result of providing a quantified thermodynamic criterion for predicting the onset of

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Rayleigh convection. Moreover, the critical value caused by mass transfer is determined by its

density difference and boundary condition for different solvents. The variation of global entropy generation shows that Rayleigh convection changes the irreversible route from a linear to non-linear

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thermodynamic branch with the emergence of the ordered plume-like structures. The detailed

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entropy generation analysis for the onset and development of convective cells is discussed to

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investigate Rayleigh convection and deepen the comprehension for the internal mechanism of

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convection enhancing mass transfer from the viewpoint of irreversible thermodynamics.

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Keywords

Entropy generation; Irreversible thermodynamics; Rayleigh convection; Transient; Quantitative;

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Nomenclature

light variation of blade shading [m]

C

mass concentration of solute [kg·m-3]

ΔC

difference between the concentration of pure solvent and saturated solution [kg·m-3]

C*

saturated molar concentration of solute [kg·m-3]

c

mole concentration of solute [mol·m-3]

D

diffusion coefficient of solute in the solvent [m2·s-1]

d

vertical distance from the interface [mm]

F

external force [N]

f2

focal length of the second concave mirror [m]

g

the acceleration of gravity [m·s-2]

HA

Henry’s constant of solute [atm]

ji

diffusive mass transfer flux [mol·m-2 s-1]

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∆a

2

conductive heat transfer flux [J·m-2 s-1]

jk

chemical reaction rate flux [mol·m-2 s-1]

k

the thermal conductivity [W·K-1·m-1]

L

the width between two optical glasses [m]

l

the characteristic length [m]

M

molecular weight of species [kg·mol-1]

n

refractive index of liquid

P

global entropy generation rate [W·K-1]

p

pressure [kPa]

p*

saturated vapor pressure of solvent [kPa]

R

the molar gas constant [J·mol-1·K-1]

Ra

Rayleigh number

T

temperature [K]

t

time [s]

tc

experimental critical time observed by the Schlieren images [s]

t'c

critical time predicted by the critical entropy generation [s]

u

velocity vector [m·s-1]

V

liquid volume [m3]

v

gas flow rate [mL·min-1]

xA

mole fraction of solute

x, y, z

coordinate [mm]

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N A M

Greek letters

thermal diffusion coefficient [m2·s-1]

γ

surface tension [N·m-1]

Δγ

difference between the surface tensions of pure solvent and saturated solution [N·m-1]

η

dynamic viscosity [Pa·s]

μ

chemical potential [J·mol-1]

ρ

density [kg·m-3]

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α

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jq

Δρ

difference between the densities of pure solvent and saturated solution [kg·m-3]

σ

entropy generation rate per unit volume [W·K-1·m-3]



mass fraction

τ

stress tensor

Λ

chemical reaction driving force

Subscripts 0

reference

A

carbon dioxide, solute

B

solvent

c

critical transient state

g

gas phase 3

i

species

k

chemical reaction

surf

at the interface

1. Introduction

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Rayleigh convection (Rayleigh, 1916) is an interfacial turbulent phenomenon occurring in mass/heat transfer processes due to a density gradient caused by differences of concentration or

temperature across the interface in a multiphase fluid system. Such a convection will pull the liquid

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stream downward into a plume-like convective pattern and promote the renewal of interfacial liquid

and exchange between the interfacial vicinity and liquid bulk. The effects turn out to be significant for enhancing mass/heat transfer in many engineering applications such as absorption/desorption,

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distillation, extraction, crystallization and so on (Sun and Yu, 2006; Yu and Yuan, 2014).

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To acquire the comprehensive evolution-information of Rayleigh convection and explore the

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mechanism of mass transfer, many researchers have used qualitative or quantitative experiments

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(Okhotsimskii and Hozawa, 1998; Arendt et al., 2004; Liu et al., 2008; Sha et al., 2010), numerical simulations (Sha et al., 2002; Fu et al., 2011, 2013), theoretical analyses (Kurenkova et al., 1970;

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Tan and Thorpe, 1992, 1999; Hyun and Min, 2003; Ennisking et al., 2005; Kim et al. 2006; Sun, 2012) and other methods to study this transport phenomenon. Moreover, it is acknowledged that its

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structure could be described as a plume convection pattern. Considering the practical importance of the stability criterion for the critical onset of Rayleigh convection, Blair and Quinn (1969) identified

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the occurrence of disturbances that would grow to some discernable sizes or structures with the onset of convection, resulting in possibly not being able to properly determine a critical time due to the poor sensitivity of the naked eyes compared with optical instruments. Tan and Thorpe (1992,

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1999) utilized the maximum-Rayleigh-number criterion to define the transient state and theoretical onset time of Rayleigh convection for soluble and sparingly soluble gases diffusing in water with the boundary conditions of constant mass flux (CMF) and fixed surface concentration (FSC). However, it is limited to exactly predict the critical parameters of Rayleigh convection due to the lack of influences of the time-dependent nonlinear base concentration profile on the disturbances of convective velocity and concentration. Then, Sun (2012) developed a spatial base-profile influenced 4

frozen-time marginal state analysis for the buoyancy-driven convection and the calculated critical parameters were in good agreement with the published experimental results. Apart from these methods, Fu et al. (2013) and Guo et al. (2015) obtained the critical point for the transient state of Rayleigh convection through the variation of instantaneous mass flux across the gas-liquid interface and mass transfer coefficient compared with penetration theory, which is also an approximate time for the transient state of convection. Up till now, there are few researchers who have investigated

the critical onset time of Rayleigh convection in the processes of mass transfer.

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the non-equilibrium thermodynamic property to provide a precise quantified criterion for predicting

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From the principle of Prigogine’s Dissipative Structure and Self-Organization Theory (Glansdorf and Prigogine, 1971; Prigogine and Nicolis, 1985), interfacial convection, such as Bénard convection occurs in the region of non-linear and non-equilibrium thermodynamics. To

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characterize the convective mass or heat transport phenomena through thermodynamics, the theories

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based on the Second Law of Thermodynamics have been extended to irreversible thermodynamics

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for an open system, and it could be concluded that entropy generation has gradually drawn much more attention for exploring the irreversibility caused by the available energy losses. Bejan (1982,

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1996) demonstrated the usefulness of entropy generation analysis for the identification and

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reduction of thermodynamic irreversibility of the heat and fluid flow processes through a new thermodynamic engineering approach. Magherbi et al. (2003) calculated the entropy generation rate numerically for natural convection in a heat transfer process. The results showed that the total

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entropy generation had a maximum when the transient state commenced, and it increased with the Rayleigh number and irreversibility distribution ratio, and the result was further verified by Oliveski

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et al. (2009). Adeyinka and Naterer (2005a, b) introduced entropy generation to analyze the flow and free convection in a square enclosure with the profiles of velocity and temperature obtained by particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF) and claimed that

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entropy generation could be considered as a practical tool when identifying the local available energy losses. Furthermore, Sciacovelli (2015) summarized the advantages of global and local entropy generation rates at different levels when they were respectively used to model the irreversibility of operating conditions or designing parameters and to allow identifying the areas where the irreversibility occurs. Nevertheless, those studies on entropy generation analysis mainly focused on numerically simulated Rayleigh convection in the process of heat transfer or 5

optimization of the performance of convective heat and mass transfer with the minimum entropy generation. An experimental study on entropy generation for the transient state of buoyancy-driven Rayleigh convection has not yet been carried out based on non-equilibrium thermodynamics. The objective of the present work is to introduce entropy generation to quantitatively explore the thermodynamic irreversibility of the non-stationary mass transport phenomenon with Rayleigh convection through an experimental method. It is concluded that Rayleigh convection changes the

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irreversible route with a critical entropy generation rate of approximately 2.50 W·K-1·m-3 for the

transient state, which is different from penetration theory when CO2 is absorbed into quiescent

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ethanol. The critical entropy generation provides an easier and more precise quantified criterion to elaborate the critical non-equilibrium thermodynamic properties and predict the onset of Rayleigh convection for different gas-liquid systems. Moreover, considering the usefulness of entropy

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generation analysis for the internal mechanism of convection enhancing mass transfer, it is

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instructive for improving the mass transfer performance in the condition of the synergetic function

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between the velocity vector and concentration fields with less available energy dissipated.

2. Experiment of mass transfer process

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2.1 Experimental device and operation

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The different solvents with a declared purity of 99.7% (mass content) were provided by Guangfu Technology Development Company (Tianjin, China). The purities of the absorbed gas CO2 and the shielding gas N2 used for the experiments were 99.99% (mass content). The physical

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properties of CO2-solvents system are presented in Table 1. All the experiments are operated at 298.15 K and 101.325 kPa.

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It is found that these procedures are all Rayleigh-unstable and Marangoni-stable processes

(Okhotsimskii and Hozawa, 1998) by considering these physicochemical properties with ∆ρ>0 and ∆γ<0, which are the differences between the liquid densities and surface tensions of the solvent

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saturated with CO2 on the surface and of the liquid bulk, respectively, in the absorption process. The Z-type Schlieren and mass transfer analysis device is similar to that in Yu and Zeng (2014)

and Chen et al. (2016), as shown in Fig. 1 and Fig. 2 with some different gas flow rates. Before the experiment, the pure ethanol, taken as an example, was degassed by an ultrasonic equipment and then was emitted into the device by a micro-flow velocity injection pump. The height of the initially quiescent liquid layer was 60 mm. The shielding gas N2 and the high-purity CO2 pre-saturated by 6

the ethanol vapor prevented the volatilization of liquid phase ethanol. The CO2 gas, measured by a rotameter, passed into the mass transfer apparatus by five inlets on the top side (Fig. 2a). A clean sponge was fixed on the position under the gas inlets to distribute the gas flow, and a PID (Proportion-Integration-Differentiation) temperature controller ensured that the temperature was identical between the sweep gas and liquid layer to avoid the emergence of natural convection triggered by the temperature differences. Once CO2 gas contacts with the static liquid bulk, it

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triggers the absorption process.

1–light source; 2–spike filter; 3–lens; 4–slit; 5–flat mirror; 6–concave mirror; 7–blade; 8–CCD camera;

9–gas tank; 10–rotameter; 11–pre-saturator; 12–PID temperature controller; 13–gas-liquid mass transfer

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apparatus

To acquire more legible images, the position of the flat mirror and CCD camera ensured that the optical axis was at the center of the CCD sensor, and the resolution of the CCD camera was set

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up as 1628×1236 pixels, so the actual length was 0.0384 mm/pixel. The recorded Schlieren images

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with a time step of 1 second were obtained with 1/100 second of exposure time. The mass-transfer

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region of 23-45 mm in the abscissa and 45-60 mm in the ordinate was screened and ranked to obtain

much as possible.

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2.2 Concentration profile

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the relative precise information, avoiding the perturbations of contact angle and vertical walls as

The refractive index profile of liquid ethanol vs. different CO2 concentration is obtained by

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analyzing the gray variation of Schlieren images on the basis of the sketch of light passing through an experimental cell and quantitative Schlieren technology (Liu et al., 2008; Panigrahi and

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Muralidhar, 2012), and the corresponding equation is shown in Eq. (1): n( x0 , y, t ) x  n( x0 , y0 , t ) 

1 Lf 2



y

y0

ady

(1)

where n represents the refraction index and ∆a [m] is the light variation of blade shading when the

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parallel light passes two concave mirrors and is shaded by the horizontal blade. To obtain the concentration field, it is necessary to identify the relationship between the

concentration and refractive index. The non-linear relational expression of n and concentration of CO2 (CA) can be obtained by a quadratic polynomial fitting. The result is shown in Eq. (2): (Yu and Zeng, 2014) n  1.552 105 CA2  2.006 104 CA  1.3595

(2) 7

In the liquid bulk near the bottom, where the influence of the dissolved gaseous CO2 does not exist, the relevant refractive index can be regarded as the same as that of pure ethanol. Therefore, the corresponding refractive index distribution and concentration fields are worked out with the aid of Eq. (1) and Eq. (2), respectively.

3. Entropy generation rate Since Rayleigh convection occurs in the non-linear and non-equilibrium thermodynamic state,

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the local equilibrium hypothesis (Prigogine and Nicolis, 1985) that each infinitesimal volume of a system can be considered to be in a state of thermodynamic equilibrium, though the overall system

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operates irreversibly at a given time, is introduced to this non-stationary mass transfer process. Thus, the formulation of the Second Law of Thermodynamics and the entropy balance equation within the

framework of continuum theory is allowed, and the entropy generation rate can be derived by

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combining the conservation equations of mass, composition and energy.

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For a general transfer phenomenon with the external force field and internal viscous flow, the

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detailed expression of the entropy generation per unit volume σ [W·K-1·m-3] (Li, 1986; Pope et al., 2010; Sciacovelli et al., 2015) is shown in Eq. (3):

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  i  Fi  1 k 1 jk    ji         : u   T T  i k  T  T T

   h   m   f   r  jq  

(3)

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which shows that it is consisted of four terms with the summation of the product of the “flux” and “force”: The first term is the vector product of heat flux and the gradient of inverse temperature,

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which is generated by heat transfer; the second one is the vector product between mass flux and the subtraction of the body force for the chemical potential gradient due to mass transfer; the third term

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is the tension product of viscous force and velocity gradient with the production of the fluid friction, while the last is the irreversibility caused by the factors of driving force and chemical reaction rate. In the process of CO2 absorbing into ethanol, the solution is satisfied by Raoult’s law at 298.15

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K and 101.325 kPa according to the small differences between the experimental and ideal data (Dalmolin et al., 2006) in Fig. 3 and could be considered as an ideal solution for the range from 0 to 0.0062, i.e., the maximum mole fraction of CO2 in the saturated ethanol solution, since the CO2 gas is sparingly soluble in pure ethanol without the existence of chemical reactions. Therefore, the Gibbs excess energy, excess volume, excess entropy and excess enthalpy are not considered. Buoyancy-driven Rayleigh convection, whether it is caused by concentration or thermal 8

boundary conditions, could all be characterized by the dimensionless Rayleigh (Ra) number: Ra 

d 3 g  , where the characteristic length d is defined as the penetration depth measured by the D

gray level; g is the acceleration of gravity; Δρ is the density difference between the fluid on the surface and in the bulk, including the effects of mass and heat transfer. If the Ra number is positive and higher than a critical value, interfacial convection will occur resulting in acceleration of the

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mass transfer rate. From the data reported by Okhotsimskii and Hozawa (1998) and Postigo and Katz (1987), the effects of temperature caused by mass transfer have been compared with the mass

change, which could be calculated as:

Rac  C   c  77 RaT T T D

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transport and the ratio of Rayleigh convection strength due to the concentration and temperature

, where ∆C, ∆T are the maximum

concentration and temperature difference between the surface and liquid body; βc, βT are the

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variation rate of the solution density with the influences of concentration and temperature, respectively. It could be found that the strength of Rayleigh convection caused by mass transfer is

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almost 76 times larger than the thermal effects, so that the effects of absorption heat could be

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neglected. By considering the fact that Rayleigh convection in the gas-liquid mass transfer process

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would be complicated in influencing the mass transfer rate referring to the vaporization effects and the trigger of Marangoni convection caused by the thermal boundary conditions, it is imperative to

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eliminate the effects of temperature and its thermodynamic irreversibility generated by the heat transfer and the coupling effects of mass and heat transfer through the isothermal operations of the

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PID temperature controller and the pre-saturated absorbed gas. Apart from these parameters, the gravity is the exclusive external body force per unit mass in

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the process which can be eliminated. As a result, the simplification of the entropy generation rate for a two-dimensional mass-transfer system is demonstrated in Eq. (4) (Carrington and Sun, 1991;

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Pope et al., 2010): 

         2 RD A A B      M A M B AB c  x   y   T 2

2

2 2   u 2  u y   ux u y     2  x   2      x     x   y   y 

(4)

where ωA is the mass fraction of CO2 in the solvent; ux and uy are the convective velocity in x or y dimension, respectively, and thus the entropy generation rate is mainly caused by mass transfer and viscous dissipation for the isothermal mass transfer process in an unsteady gas-liquid system. According to the measurement of velocity by the PIV technique in the literature (Fu et al., 9

2011), the maximum of the convective velocity and velocity gradient of ethanol with the dissolved gas CO2 is approximately 10-3 m·s-1 and 10 s-1, separately; thus, the estimation of the maximal magnitude of entropy generation caused by viscous friction is approximately 10-3 W·K-1·m-3. Compared with the entropy generation caused by mass transfer, in which the magnitude is at least 2 W·K-1·m-3 for the convective stage, the irreversibility produced by viscous friction is negligible for some liquid flows with low viscosity such as ethanol, methanol or water during the absorption

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processes, but it is uncertain whether it is universal and applicative for other high-viscosity liquids.

Consequently, the entropy generation rate caused by the existence of concentration gradient for

       2 RD A A      M A M B AB c  x   y  2

2

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these absorption processes is given by:   

(5)

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Additionally, the global entropy generation rate P in the gas-liquid mass transfer process is the

V

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V

  2   2   2 RD  A    A  dV M A M B AB c  x   y  

4. Results and discussions

(6)

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P    dV  

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integral over the system volume of the local entropy generation σ, which is shown in Eq. (6):

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With the precondition that the local disturbances emerged in the interphase surface are only dependent on the absorbed concentration perturbations, all experiments are conducted as mentioned

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above at different gas flow rates of 50, 100, 130, 150 and 200 mL·min-1, in order to investigate Rayleigh convection within mass transfer. Due to the similarities of these absorption processes, an

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analysis of the entropy generation rate and thermodynamic irreversibility in the energy conversion caused by Rayleigh convection is carried out only for CO2 absorbed into ethanol at 130 mL·min-1 gas flow rate.

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4.1 Distributions of local entropy generation rate According to the Schlieren images, there are two stages during the absorption of CO2 into pure

ethanol, which shows close agreement with the benchmark data (Fu et al., 2013; Guo et al., 2015; Chen et al., 2016). The first one is molecular diffusion described by penetration theory, while the second is Rayleigh convection generated by the density gradient in the process of mass transfer. Rayleigh convection affects the maps of carbon dioxide mass fraction as well as the variations of 10

convective velocity after the occurrence of a plume-like structure, which eventually influences the liquid-phase mass transfer rate, as a result of changing the profiles of local entropy generation; therefore, the corresponding thermodynamic properties of Rayleigh convection can be extracted and obtained. Different colors represent different local mass fractions of the absorbed gas CO2 and different entropy generation rates distributed in Fig. 4. The local entropy generation maps are examined to

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note possible parts of the system where thermodynamic irreversibility is unusually high with more dissipated available energy (Sciacovelli et al., 2015). In the initial dominated molecular diffusion

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stage, the profiles of entropy generation rate caused by the diffusive resistance are almost horizontal,

and its maximum occurs in the vicinity of the gas-liquid interface, which demonstrates that CO2 is penetrated into ethanol with nearly homogeneous mass-transfer resistance before 9 s and its

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concentration distribution is flat near the surface. When the random perturbed regions emerge at the

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interface, the increasing potential energy is accumulated sufficiently to trigger Rayleigh convection,

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and local high entropy generation is generated beneath the interface (Fig. 4b/B). The reason is that density increases and surface tension decreases with the absorption process, and at the time that the

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influence of surrounding higher surface tension and viscous friction is overcome, the corresponding

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regions have the tendency to evoke convection. Therefore, it produces local high entropy generation owing to the mass-transfer resistance near the gas-liquid interface. As more disturbances occur on the surface, convection becomes increasingly significant and gradually grows into plume-like

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structures downward into the liquid bulk with the local convective velocity generated. Due to the conversion between the potential energy and kinetic energy, the less available energy is dissipated,

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which means that Rayleigh convection reduces the local irreversibility caused by the mass-transfer resistance to enhance mass transfer rate. Compared with the concentration profiles in Fig. 4 (a-e), it can be observed that the

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distribution of local low entropy generation rate is almost identical to the area where Rayleigh convection occurs, while the local high entropy generation in the bulk of liquid mainly exists on both sides of Rayleigh convective cells, showing a nearly symmetrical distribution in one convective cell. According to the profiles of velocity and streamlines obtained through the PIV method (Fu et al., 2011), a similar tendency appears that the concentration and velocity profiles are maximum, but their gradients are minimum in the center of one convective cell. Therefore, the stronger convection 11

would result in lower entropy generation profiles since the velocity vector cooperates with the concentration gradient to reduce the mass transfer irreversibility for the denser fluids moving from near the surface layer to the deep liquid bulk. However, on both sides of the plume stream where low concentration flows are raised up to refresh the original denser liquid located in the vicinity of the surface, the diffusive resistance caused by the concentration gradient is identical to the direction of flow velocity, which accelerates the losses of mechanical energy and produces higher local

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entropy generation rate due to the shear stress. Moreover, while the velocity vector is normal to the concentration distributions, the conversion of energy generated by the interfacial circular convection

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does not make any contribution to reduce the local irreversibility due to the mass-transfer resistance occurring in the direction of concentration gradient. Given the above phenomena, it can be concluded that Rayleigh convection reduces the local entropy generation in the condition of the

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synergetic function of the velocity vector and concentration gradient; on the other hand, it increases

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inversely in regions where the concentration gradient degrades convective velocity.

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In the diffusive regions, besides these plume structures, Fig. 4 (D/E) shows that there is a thin liquid layer with lower entropy generation at approximately 0.5-1 mm below the surface. Based on

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the literature (Okhotsimskii and Hozawa, 1998), Rayleigh convection includes not only the plume-

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like structures downward into the bulk liquid but also various kinds of interfacial turbulence beneath the gas-liquid interface. As a specific characteristic of the experimental phenomenon, small local vortexes exist near the interface, which accelerate the conversion of the mechanical energy with less

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dissipation of the available energy in order to generate local low irreversibility.

4.2 Entropy generation of the liquid bulk

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Fig. 5 shows the distribution of average entropy generation of the horizontal cross-sections

along the y-direction in the bulk liquid during the absorption process. Before the occurrence of local convection, entropy generation has the maximum in the vicinity of the gas-liquid surface and then

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directly decreases to approximately zero at the edge of the diffusion thickness along the vertical distance from the interface due to the diffusive resistance, which is similar to the variation of local Ra number (Tan and Thorpe, 1992) with the exclusive difference that its maximum value of entropy generation rate gradually decreases with the contact time. The result can also be verified in Table 2 by the phenomenon that before 10 s, the depth of zero value of entropy generation in the y-direction is identical to the diffusion thickness, which is defined as a certain depth that CO2 is penetrated into 12

ethanol near the gas-liquid interface. However, once Rayleigh convection occurs, the diffusion thickness and the depth of the zero value is no longer consistent, since the distribution of entropy generation protrudes downward into the bulk liquid (Fig. 4B-E). Similarly, the local minimum values at approximately 0.5-1 mm depth from the interface also demonstrate the existence of local vortexes. With the increasing strength of Rayleigh convection, the local mass transfer is reinforced with the aid of these local vortexes, so that lower entropy generation and deeper depth in the bulk

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liquid are obtained. In addition, it is interesting to note that besides the interfacial local maximum

value after the commencement of Rayleigh convection, entropy generation increases to a depth of

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the peak value in the bulk liquid, which is lower than the gas diffusion thickness and shows a good agreement with the results obtained from previous literature (Chermiti et al., 2011). In particular, the peak values overpass the maximum near the surface with the dominant effect of circulation flow

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caused by Rayleigh convection reinforcing mass transfer in the bulk liquid over molecular diffusion,

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and the stochastic disturbances and intermittent renewal of the gas-liquid interface reduce the

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thermodynamic irreversibility due to the conversion between different forms of energy. With the plume convection downward to the deeper liquid along the vertical distance from the

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interface, the peak values at different times increase in the first 30 s, then decrease and become more

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flattened gradually until the convection weakens and reaches a relative quasi-equilibrium state at approximately 35 s. The phenomenon also indicates that local mass-transfer resistance accompanied with the available energy losses results in the production of dissipative structures, such as ordered

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plume convections. After achieving its peak value, entropy generation with small oscillations decreases to zero in the deeper liquid because the plume-like structures with the denser fluids extend

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into different depths of the liquid body, but the general tendency is becoming deeper eventually with

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the development of Rayleigh convection due to the effect of gravity.

4.3 Critical onset of Rayleigh convection To study the critical thermodynamic properties for the transient state with the quantitative

Schlieren technology, the variation of local entropy generation rate beneath the interface is used to mark the onset of Rayleigh convection (Fig. 6). The entropy generation rate increases within the diffusive time and is almost constant along the horizontal direction before 8 s. After 9 s, the local 13

small disturbance emerges at approximately 34-38 mm of the x-direction, with the local entropy generation reaching 2.45 W·K-1·m-3, and then evolves into Rayleigh convection, which can be observed by the naked eye at t=10 s. From the theory of irreversible thermodynamics, the onset of Rayleigh convection should be an exact state for the absorption of CO2 into pure ethanol, and thus the critical entropy generation should be a determinate value independent of gas flow rates whenever Rayleigh convection initiates. The data in Table 3 verify this inference well and it can be concluded

IP T

that the critical entropy generation is approximately 2.50 W·K-1·m-3 as a quantitative unstability criterion for the onset of Rayleigh convection, which is equivalent to the available energy dissipated

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per unit volume per second at a specific centigrade degree.

With the precondition that the local disturbances emerged on the surface are only dependent on the absorbed concentration perturbation, increasing gas flow rate reduces the gaseous phase

U

mass-transfer resistance, resulting in shortening the onset time of convection and accelerating the

N

proceeding of mass transfer. Furthermore, the critical transient Ra number for the experiments is

A

found to be 850 (Fig. 7), which is 17% lower than the theoretical value of 1028 due to the actual boundary condition that the interfacial concentration is smaller than the equilibrium value on the

M

condition of a static absorption in the CO2-ethanol system at 298.15 K and 101.325 kPa (Fig. 8).

ED

With the same transient Ra number independent of gas flow rates, all experiments have a similar trend of interfacial average concentration and obtain a consistent transient value of the critical entropy generation (Table 3). This demonstrates the fact that it is controlled by the liquid phase

PT

mass-transfer resistance for the sparingly soluble gas diffusing in solvents, and thus the increasing gas flow rate does not influence the mass transfer rate. Compared with the simulated concentration

CC E

profiles of CO2 dissolved in ethanol through the LBM method (Fu et al., 2013; Guo et al., 2015), the local entropy generation could be carried out with Eq. (5) in the vicinity of the surface and the estimated value is respectively approximately 2.77 W·K-1·m-3 from Fu et al. (2013) and 2.59 W·K-

A

1·m-3

from the study of Guo et al. (2015), which proves the experimental result to be valid.

Absolutely, the critical entropy generation of 2.50 W·K-1·m-3 is an approximate experimental value because the calculation of entropy generation exclusively includes the irreversibility caused by mass transfer, although the effects of viscous friction and heat transfer are small and negligible. Nevertheless, to some extent, the critical value could be proposed to be a quantified criterion for predicting the transient state of Rayleigh convection from a general view of irreversible 14

thermodynamics. As for the system of CO2 absorbed into some solvents on the condition that the density of liquid bulk changes sparingly with the dissolved CO2 while that of the gas-liquid interface keeps constant, it is reasonable to assume that the concentration gradient could be predicted by penetration theory until Rayleigh convection commences. Thus, the local entropy generation rate of Eq. (5) can be simplified into a form of Eq. (7). 1

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 RC *  2dDt d )  e  erfc( M A t  2 Dt  2



(7)

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According to Tan and Thorpe (1992), the maximum transient Ra number for the onset of natural convection is found by setting   Ra   0 to gain the position as d  2 2Dt for the FSC  d  boundary condition. The transient time could be calculated with the known critical Rayleigh (Rac)

U

number for the onset of Rayleigh convection. With a similar variation between local Ra number and

N

entropy generation before the occurrence of Rayleigh convection, the method can be introduced to

A

explore the critical thermodynamic property for the same condition. Therefore, the onset times

M

predicted by the critical entropy generation with different gas flow rates indicate a close agreement with the onset time of Rayleigh convection observed by the naked eye through Schlieren technology

ED

(Fig. 9). Although the critical times have great differences between the values predicted by the critical entropy generation rate and penetration theory (Tan and Thorpe, 1992), it is still reasonable

PT

to be attributed to two reasons induced first by the variable non-equilibrium concentration at the gas-liquid interface (Fig. 8) being different from the FSC boundary condition and second by the

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high probability and magnitude of the concentration perturbation for the experimental operations. From the results in previous study of Fu et al. (2013), increasing the probability and magnitude of the concentration perturbation would reduce the corresponding onset time of Rayleigh convection

A

for the unsteady mass transfer process. Moreover, the critical entropy generation rate can also be acquired, and the approximate value

is 6.94 W·K-1·m-3 at d= 0.39 mm based on penetration theory for the CO2-ethanol system. Similarly, the experimental critical entropy generation and its vertical distance from the interface are compared with the predicted values by penetration theory for CO2 absorbed into several solvents in Table 4. It can be found that the critical entropy generation and its depth in the bulk liquid are different for 15

various solvents and mainly dependent on the physiochemical properties of different gas-liquid systems (Table 1). The general variation tendency shows that the higher density difference between the interfacial fluid and the bulk liquid with less onset time would result in producing the larger critical entropy generation rate at the shallower liquid layer. Moreover, the reason could be inferred by the fact that more mechanical energy is dissipated to trigger Rayleigh convection for the actual boundary condition with these properties. Additionally, there are some differences between the

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theoretical and experimental values because the theoretical values are limited to the conditions that assume no convection and validated for the FSC boundary condition, whereas the absorption

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experiments are operated with the variable and non-equilibrium interfacial concentration at 298.15 K and 101.325 kPa. However, according to the experimental critical entropy generation to predict

the onset time of Rayleigh convection, the relative error can be acceptable compared with the ones

U

observed by the Schlieren images (Table 4). As a consequence, the critical entropy generation rate

N

can be regarded as a new useful method to predict the onset of Rayleigh convection and the critical

4.4 Global entropy generation rate

A

value is determined by its density difference and boundary condition for a Rayleigh-unstable system.

M

For the whole mass-transfer process with Rayleigh convection, the global entropy generation

ED

rate and its variation rate of the liquid system during 0-60 s are calculated (Fig. 10). A comparison between Fig. 10 (a) and (b) shows that the global entropy generation is small, and its variation rate nearly decreases to zero before 10 s, which fits the minimum entropy generation of Prigogine’s

PT

theorem for the diffusion stage, since the mass transfer system is in the linear branch of irreversible thermodynamics, where Fick’s diffusion dominates. When the concentration perturbation gives rise

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to Rayleigh convection, the route of the irreversibility changes and is different from that in the pure diffusion stage; thus, the global entropy generation rate increases rapidly because the convection promotes more CO2 being dissolved in the solvents with more mechanical energy dissipated until

A

approximately 35 s, and the whole variation tendency is identical to that of the corresponding area under the curve of average entropy generation in the horizontal cross-section with the coordinate axes at different times in Fig. 5. The variation rate of entropy generation also suggests that it changes from the linear branch to the non-linear one of the irreversible thermodynamics with the emergence of macroscopic ordered plume-like structures. After 35 s, the global entropy generation rate fluctuates in the vicinity of 1.4×10-6 W·K-1 for a long time with the inherent disturbances, and its 16

variation rate gradually reaches a relative quasi-stationary state after some oscillations near zero. However, the relative quasi-stationary state, which is only a temporary stage due to the development and dissipation of the plume convection, could develop into chaos without any influence of external force and eventually reach a state of saturated ethanol solution. This result means that Rayleigh convection accelerates the mass transfer process through changing the irreversible route.

5. Conclusions

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An innovative analysis of entropy generation on Rayleigh convection was developed for the absorption of CO2 into some solvents on the basis of irreversible thermodynamic theory. The

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quantitative experimental exploration of the thermodynamic property of Rayleigh convection for

non-stationary mass transfer processes had not yet been encountered. The results indicated that the critical entropy generation rate for the CO2-ethanol system was approximately 2.50 W·K-1·m-3 when

U

Rayleigh convection commenced, where the key point was in providing a new and more precise

N

quantified thermodynamic criterion for predicting the onset of plume convection. Moreover, the

A

critical values, being dependent on its density difference and boundary condition for different systems, were proved to be valid in comparison with the ones predicted by penetration theory.

M

Beyond the transient state, Rayleigh convection changed the irreversible route to decrease the local

ED

entropy generation rate, since the conversion between the potential energy and kinetic energy reduced the available energy losses in order to reinforce mass transfer when the velocity vector cooperated with the concentration gradient but increased inversely in the regions where the

PT

concentration gradient degraded convective velocity. Therefore, entropy generation could be applied to analyze the dominated thermodynamic

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property and internal mechanism of Rayleigh convection enhancing mass/heat transfer rate, which would be an easier and more precise method for exploring the onset of Rayleigh convection fundamentally rather than the traditional approaches of optimizing the mass/heat transfer

A

performance, even providing the potential possibility to control the convection as required.

Acknowledgements This work was supported by the National Key Technology Research and Development Program of the Ministry of Science and Technology of China [2007BAB24B05].

17

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Arce, A., Jr., A.A., Rodil, E., Soto, A., 2000. Density, Refractive Index, and Speed of Sound for 2Ethoxy-2-Methylbutane + Ethanol + Water at 298.15 K. Journal of Chemical & Engineering Data

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45, 536-539.

Arendt, B., Dittmar, D., Eggers, R., 2004. Interaction of interfacial convection and mass transfer effects in the system CO2 –water. International Journal of Heat & Mass Transfer 47, 3649-3657.

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Bejan, A., 1982. Entropy Generation Through Heat and Fluid Flow. Wiley New York.

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Bejan, A., 1996. Entropy generation minimization: The new thermodynamics of finite-size devices

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Carrington, C.G., Sun, Z.F., 1991. Second law analysis of combined heat and mass transfer phenomena. International Journal of Heat & Mass Transfer 34, 2767-2773. Chen, M., Zhao, S., Zeng, A., Yu, H., 2016. Quantitative analysis of Rayleigh convection in

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interfacial mass transfer process. CIESC Journal 67, 4566-4573 (in Chinese with English abstract) . Chen, S., Lei, Q., Fang, W., 2002. Density and refractive index at 298.15 K and vapor–liquid

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equilibria at 101.3 kPa for four binary systems of methanol, n-propanol, n-butanol, or isobutanol with N-methylpiperazine. Journal of Chemical & Engineering Data 47, 811–815. Chermiti, I., Hidouri, N., Brahim, A.B., 2011. Entropy generation in gas absorption into a falling

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liquid film. Mechanics Research Communications 38, 586-593. Dalmolin, I., Skovroinski, E., Biasi, A., Corazza, M.L., Dariva, C., Oliveira, J.V., 2006. Solubility of carbon dioxide in binary and ternary mixtures with ethanol and water. Fluid Phase Equilibria 245, 193–200. Dittmar, D., Fredenhagen, A., Oei, S.B., Eggers, R., 2003. Interfacial tensions of ethanol–carbon dioxide and ethanol–nitrogen. Dependence of the interfacial tension on the fluid density— 18

prerequisites and physical reasoning. Chemical Engineering Science 58, 1223-1233. Ennisking, J., Preston, I., Paterson, L., 2005. Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Physics of Fluids 17, 239-248. Frank, M.J.W., Kuipers, J.A.M., van Swaaij, W.P.M., 1996. Diffusion coefficients and viscosities of CO2 + H2O, CO2 + CH3OH, NH3 + H2O, and NH3 + CH3OH liquid mixtures. Journal of Chemical & Engineering Data 41, 297–302.

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Fu, B., Liu, B., Yuan, X., Chen, S., Yu, K.T., 2013. Modeling of Rayleigh convection in gas–liquid interfacial mass transfer using lattice Boltzmann method. Chemical Engineering Research & Design

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91, 437-447.

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Experimental Verification. Chinese Journal of Chemical Engineering 19, 845–854.

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Glansdorf, F., Prigogine, I., 1971. Thermodynamic theory of structure, stability and fluctuations.

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Wiley-Interscience, a division of John Wiley & Sons, Ltd. London –New York –Sydney -Toronto. Guo, K., Liu, C., Chen, S., Liu, B., 2015. Modeling with statistical hydrodynamic quantities of mass

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Li, R., 1986. Non-equilibrium thermodynamics and dissipative structure. Tsinghua University Press, Beijing, China (in Chinese). Liu, C., Zeng, A., Yuan, X., Yu, G., 2008. Experimental study on mass transfer near gas–liquid interface through quantitative Schlieren method. Chemical Engineering Research & Design 86, 201207. Magherbi, M., Abbassi, H., Ben, B.A., 2003. Entropy generation at the onset of natural convection. 19

International Journal of Heat & Mass Transfer 46, 3441-3450. Okhotsimskii, A., Hozawa, M., 1998. Schlieren visualization of natural convection in binary gas– liquid systems. Chemical Engineering Science 53, 2547-2573. Oliveski, R.D.C., Macagnan, M.H., Copetti, J.B., 2009. Entropy generation and natural convection in rectangular cavities. Applied Thermal Engineering 29, 1417-1425. Panigrahi, P.K., Muralidhar, K., 2012. Schlieren and Shadowgraph Methods in Heat and Mass

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Transfer. Springer New York.

Pope, D.N., Raghavan, V., Gogos, G., 2010. Gas-phase entropy generation during transient

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methanol droplet combustion. International Journal of Thermal Sciences 49, 1288-1302.

Postigo, M.A., Katz, M., 1987. Solubility and thermodynamics of carbon dioxide in aqueous ethanol solutions. Journal of Solution Chemistry 16, 1015-1024.

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Prigogine, I., Nicolis, G., 1985. Self-Organisation in Nonequilibrium Systems: Towards A

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Dynamics of Complexity. D. Reidel Publishing Company, Springer Netherlands.

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Rayleigh, L., 1916. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philosophical Magazine 32, 529-546.

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Sciacovelli, A., Verda, V., Sciubba, E., 2015. Entropy generation analysis as a design tool—A review.

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Renewable & Sustainable Energy Reviews 43, 1167-1181. Sha, Y., Cheng, H., Yu, Y., 2002. The Numerical Analysis of the Gas-Liquid Absorption Process Accompanied by Rayleigh Convection. Chinese Journal of Chemical Engineering 10, 539-544.

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Sha, Y., Li, Z., Lin, F., al., e., 2010. Shadowgraph observation on interfacial turbulence phenomena in gas-liquid mass transfer. CIESC Journal 61, 844-847 (in Chinese with English abstract).

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Sun, Z.F., 2012. Onset of Rayleigh–Bénard–Marangoni convection with time-dependent nonlinear concentration profiles. Chemical Engineering Science 68, 579-594. Sun, Z.F., Yu, K.T., 2006. Rayleigh–Bénard–Marangoni Cellular Convection : Expressions for Heat

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and Mass Transfer Rates. Chemical Engineering Research & Design 84, 185-191. Takahashi, M., Kobayashi, Y., Takeuchi, H., 1982. Diffusion coefficients and solubilities of carbon dioxide in binary mixed solvents. Journal of Chemical & Engineering Data 27, 328-331. Tan, K.K., Thorpe, R.B., 1992. Gas diffusion into viscous and non-Newtonian liquids. Chemical Engineering Science 47, 3565-3572. Tan, K.K., Thorpe, R.B., 1999. The onset of convection induced by buoyancy during gas diffusion 20

in deep fluids. Chemical Engineering Science 54, 4179-4187. Yu, H., Zeng, A., 2014. Visualization and quantitative analysis for Marangoni convection in process of gas-liquid mass transfer. CIESC Journal 65, 3760-3768 (in Chinese with English abstract).

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Yu, K.T., Yuan, X., 2014. Introduction to Computational Mass Transfer. Springer Berlin Heidelberg.

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PT

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Fig. 1 Schematic diagram of experimental apparatus

22

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(a) Schematic diagram of external apparatus; (b) Schematic diagram of internal apparatus

ED

M

A

N

U

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Fig. 2 Simple block-diagram of the gas-liquid mass transfer analysis device

A

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Fig. 3 p-xA equilibrium diagram for the CO2-ethanol system

23

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Fig. 4 Distributions of local concentration (a-e) and entropy generation rate (A-E) with Rayleigh

A

convection at t=9 s, 10 s, 11 s, 15 s, 20 s during the absorption of CO2 into ethanol

24

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A

CC E

PT

ED

M

A

N

U

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Fig. 5 The average entropy generation distribution of horizontal cross-section at different time

25

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A

CC E

PT

ED

M

A

N

U

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Fig. 6 Local maximum entropy generation beneath the interface at the onset of Rayleigh convection

26

A

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PT

ED

M

A

N

U

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Fig. 7 Critical Rayleigh numbers for the condition of 130 mL·min-1 gas flow rate

27

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A

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PT

ED

M

A

N

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Fig. 8 Variation of the interfacial average concentration with times for 130 mL·min-1 gas flow rate

28

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Fig. 9 Comparison of the onset times of Rayleigh convection predicted by the experimental critical

A

CC E

PT

ED

M

A

N

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entropy generation and those observed by the Schlieren images

29

IP T SC R U N A

A

CC E

PT

ED

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Fig. 10 Variation of (a) the global entropy generation rate and (b) its variation rate of the liquid system

30

Table 1 Physical properties of CO2-Solvents system at 298.15 K, 101.325 kPa ∆γ×104

η×103

D×109

(N∙m-1)

(Pa∙s)

(m2∙s-1)

1.01c

-1.0d

1.08b

3.86b

156e

6.55b

1.45c

-0.125c

0.55b

5.13g

173h

1.57b

0.38i

-0.5c

0.89g

1.97b

1634e

ρ (kg∙m-3)

C* (kg∙m-3)

∆ρ (kg∙m-3)

Ethanol

785.22a

4.70b

Methanol 786.48f 997.04a

Water

From Arce et al. (2000);

b

From Takahashi et al. (1982);

c

From Okhotsimskii and Hozawa (1998);

d

From Dittmar et al.(2003);

e

From Miguel A. Postigo and Miguel Katz (1987);

f

From Chen et al. (2002);

g

From Frank et al. (1996);

h

From Fu et al. (2013);

i

From Tan and Thorpe (1999).

A

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PT

ED

M

A

N

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a

H (atm)

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Solvents

31

Table 2 Depth of different typical values occurring in the y-direction at different times 5s

10 s

15 s

20 s

25 s

30 s

35 s

40 s

Peak value

0.04

0.04

0.85

1.16

1.47

1.51

1.59

1.62

Diffusion thickness

0.54

0.82

1.38

1.90

2.02

2.13

2.20

2.21

Zero value

0.55

1.18

2.51

4.98

8.54

9.47

11.71

14.27

A

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ED

M

A

N

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Depth of d (mm)

32

Table 3 Critical properties at the onset of Rayleigh convection with different gas flow rates 50

100

130

150

200

tc (s)

21

12

10

9

6

Rac

905

837

850

890

744

σc (W·K-1·m-3)

2.36

2.52

2.45

2.48

2.70

A

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ED

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A

N

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vg (mL·min-1)

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Table 4 Comparison of the critical entropy generation and its vertical distance from the interface with different solvents between the predicted values and experimental results at the onset time σc (W·K-1·m-3)

dc (mm)

tc (s)

Solvents Experimental Predicted

Experimental

Predicted

Experimental

Ethanol

6.94

2.50

0.39

0.04

12.9

10

Methanol

21.51

12.88

0.30

0.04

7.7

5

Water

1.08

0.46

0.42

0.1

102.5

100

A

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PT

ED

M

A

N

U

SC R

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Predicted

34