A Phenomenological Theory of Polycomponent Interactions in Non-ideal Mixtures Application to NH3H2O...

7/29/2019 A Phenomenological Theory of Polycomponent Interactions in Non-ideal Mixtures Application to NH3H2O and Other

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A phenomenological theory of polycomponent interactions

in non-ideal mixtures. Application to NH3/H2O and

other working pairs

M.D. Staicovici

Thermopower Equipment Research and Design Institute (SC ICPET- CERCETARE SA) 236 Vitan Road, 74369, Bucharest, Romania

Received 23 October 1998; received in revised form 23 June 1999; accepted 24 June 1999

Abstract

The paper proposes an original linear phenomenological theory (Ph T) of evolution physical mono-, bi- and

particular polycomponent gasliquid interactions with non-ideal mixture. The expressions of the phenomenological

factors (entropy source, force, coecient and coupled heat and mass transfer currents) are deduced. The theory is

particularized to the NH3/H2O and other gasliquid systems used in the thermal absorption technology. The work's

conclusions are listed next. The paper raises the problem of ammonia bubble absorption which is dicult to answer with

current theory of interface mass transfer and absorption as a surface phenomenon. The heat and mass transfer at the

gasliquid interface is governed by the thermodynamic force, which applies also to solidliquid, solidgas, or liquid

liquid, gasgas type interactions and continuous or discontinuous media. The paper mentions a postulate referring to

the force behavior approaching an ideal point, previously formulated by the author. According to its consequence, the

mass and heat currents suer an ideal point approaching (i.p.a.) eect, not mentioned so far in the specialized literature,consisting in a continuous increase of their absolute value by several percent (for a pure component), to several hun-

dred times (for a binary system) when the interacting system approaches an ideal state, as compared to the values of

states which are far from the same ideal point. In this way, ``far from equilibrium'' becomes synonymous to ``low

interaction''. The classic assessment of the interface mass transfer by analogy with heat transfer lacks basic physics.

The (Ph T) satisfactorily explains the problem of ammonia bubble absorption. Absorption is a mass phenomenon, not

a surface one. An intensive way of improving absorption is emphasized, which seeks to promote the i.p.a. eect

appearance rather than the extensive way currently used, based on increasing gasliquid interaction area. To this

extent, the bubble absorber is hereby proposed for ecient absorption. The i.p.a. eect existence oers an additional

chance for a satisfactory explanation of the Marangoni eect.# 2000 Elsevier Science Ltd and IIR. All rights reserved.

Keywords: Refrigerating system; Absorption system; Ammonia/water; Absorption; Eciency

Interactions entre les composants de me langes non-ide aux :

the orie et application au uide actif NH3 / H2O et a d'autres

uides actifs

Re sume

Cet article propose une approche lineaire selon l'evolution des interactions gazliquide dans un melange non-ideal ou ce

melange comporte 2, 3 ou davantage de composants, selon les phenomenes presents. L'auteur donne les expressions

couvrant facteurs impliques (source d'entropie, force, coecient de transferts de chaleur et de masse). Le uide actif NH3 /

H2O est utilise comme exemple principal et d'autres systemes gazliquide sont utilises pour illustrer la technologie

0140-7007/00/$20.00 # 2000 Elsevier Science Ltd and IIR. All rights reserved.

P I I : S 0 1 4 0 - 7 0 0 7 ( 9 9 ) 0 0 0 3 4 - 1

International Journal of Refrigeration 23 (2000) 153167

www.elsevier.com/locate/ijrefrig


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d'absorption thermique. Les conclusions sont fournies. Un probleme souleve dans cette communication est celui de l'ab-

sorption de la bulle d'ammoniac : il est dicile a resoudre ce probleme dans l'etat actuel de la theorie sur l'interface entre le

transfert de masse et l'absorption. Le transfert de chaleur et de massea l'interface gaz-liquide est gouverne par la force

thermodynamique qui s'applique egalement aux interactions solidesolide, solidegaz, liquideliquide ou gazgaz en mili-

eux continu ou discontinu. La communication mentionne egalement un postulat sur le comportement des forces a proximite

d'un point ideal, deja expose par l'auteur. Selon ce postulat, les ux de chaleur et de masse subissent un certain eet en

s'approchant du point ideal (i.p.a.) ; ce point n'a pas ete evoque dans la litterature. Ce point implique une augmentationconstante des valeurs absolues, variant d'un faible pourcentage (pour un composant unique)a plusieurs centaines de fois

pour un systeme binaire dans lequel le systeme est proche d'un etat ideal, ce qui n'est pas le cas des systemes loin de ce

point ideal. De cette maniere, l'expression loin de l'equilibre devient synonyme de faible interaction . Les principes

de physique de base ne tiennent pas compte de l'evaluation classique du transfert de masse a l'interface par rapport au

transfert de chaleur. L'approche phenomenologique fournit une explication pour l'absorption de la bulle d'ammoniac.

L'absorption est un phenomene de masse, pas un phenomene de surface. Une fac on d'ameliorer l'absorption de fac on

marquee est soulignee ; cette approche est destinee a promouvoir l'eet i.p.a. et non d'utiliser la methode actuellement

utilisee qui est fondee sur l'augmentation de la supercie d'interaction entre le gaz et le liquide. Un absorbeur de bulle est

propose comme methode d'absorption ecace. L'eet i.p.a. donne une explication supplementaire pour l'eet de Mar-

angoni. # 2000 Elsevier Science Ltd and IIR. All rights reserved.

Mots cles: Syste me frigorique ; Syste me a absorption ; Ammoniac / eau ; Absorption ; Ecacite

Nomenclature

A reduced excess heat

cp specic heat at constant pressure (kJ/kmol K)

C local thermal capacity (kJ/kmol)

f gure of merit quantifying the increase of a

phenomenological function approaching an

ideal state, as compared to its far equilibrium

states

h specic enthalpy (kJ/kmol)

j interface current (mass, heat, etc.) (mol/s, kg/

s, W)

k global mass transfer coecient (kg/m2s)

m bubble feeding mass ow rate (kg/s)

M match function intervening in the generalized

thermodynamic force equation

Li,j phenomenological coecient associated with

the coupled currents i and j (mol2K/Js, kgKs/

m2)

N mass ux (kg/m2s)

Nu Nusselt number

p partial pressure (kPa, bar, ata)

Pr Prandtl numberR gas constant (kJ/kmolK)

Re Reynolds number

s specic entropy (kJ/kmolK)

Sc Schmidt number

Sh Sherwood number

S.

entropy source (W/K)

t time (s)

T,t temperature (K, Celsius)

U total internal energy (J)

V volume, molar volume (m3, m3/kmol)

x liquid phase molar fraction (mol/mol)

x liquid phase molar fraction in equilibrium

with the gas phase molar fraction y (mol/mol)

X generalized thermodynamic force (kJ/kmolK)

y gas phase molar fraction (mol/mol)

y gas phase molar fraction in equilibrium with

the liquid phase molar fraction x (mol/mol)

Greek symbols

d nite dierence (variation)

driving force (according to classical inter-

pretation)

X

exergy destruction velocity (W)

" chemical potential (kJ/kmol)

' supercial tension (N/m)

Subscripts

abs absorption

c condensation

e equilibrium

g gas, gaseous phase

gen generation

i mass

q heatl liquid, liquid phase

max maximum

p constant pressure, pure

ph phenomenological

r reduced

T constant temperature

v vaporizationH before interactionHH after interaction

0 environmental

1,2 mixture species

154 M.D. Staicovici / International Journal of Refrigeration 23 (2000) 153167


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1. Introduction

The NH3/H2O system is the most common working

combination in the thermal absorption technology.

Although there is quite a comprehensive theoretical and

practical experience in the use of such combination, the

gasliquid interaction in this medium is, most surpris-ingly, still dealt with empirically. The analogy with heat

transfer is commonly used for the calculation of dis-

continuous media (interface) mass transfer,

N k I

In Eq.(1), N is the mass ow transferred as a result of

the action of the motive force . The motive force is

equal to the deviation from the equilibrium conditions.

Proportionality factor k is the mass transfer coecient.

According to the known concept ``of the two lms'' [1],

N is proportional to the dierence of the concentrationsof the gas or liquid phases and to the global mass

transfer coecients of the phases

N Kg y y Kl x x

P

In Eq.(2), x* and y* are the concentrations of the liquid

and gas phases in equilibrium with the gas phase of

concentration y and the liquid phase of concentration x,

respectively. Coecients Kg and Kl are expressed as

functions of the global resistance to mass transfer,

which have additive properties to the partial resistance

of the phases, similarly to the heat transfer. More often

than not it is considered that mass transfer is controlled

by the phase with the greatest partial resistance. In

order to decrease global resistance, the dispersion of the

process controlling phase is usually used. That is why,

according to the classic interpretation, it is implicitly

admitted that the NH3/H2O gasliquid interaction is a

surface phenomenon.

The literature referring to mass transfer has mainly

dealt with the dependence of transfer coecients on

molecular diusivity. This approach includes ``the two

lms'' concept referred to above, as well as the theory of

penetration [2], and renewal [3]. Specialized literature

has paid less attention to the relation between absorp-tion rate and hydrodynamic conditions. Kropholler and

Carr [4], Chilton and Colburn [4] and Ciborowski and

Richlicki [5], have proved experimentally that in certain

given conditions there is a Reynolds analogy in the case

of mass transfer

Sh amRebm Sccm Q

and

Nu ahRebh Prch R

in the case of heat transfer, in two-phase gas, liquid ow

in pipes. Several authors have studied heat and mass

transfer in the NH3/H2O system, [511]. Most of them

express their results by means of correlations similar to

Eqs. (3) and (4). A detailed study was conducted by

Keizer [12], for two-phase NH3/H2O ow of the slug

type in vertical pipe absorbers. For the purpose of eval-

uating mass transfer, he made use of prediction correla-

tions of partial coecient kl, which are due to Banerjeeet al. [13], Gregory and Scott [14], Jepsen [15], Jagota

[16], Kulic and Rhodes [17], Kasturi and Stepanek [18]

and others. Comparing his experimental data with the

ones predicted and nding that they sensibly dier,

sometimes even by an order of magnitude, Keizer places

under a question mark the applicability of the correla-

tions of the authors mentioned above to the prediction

of mass transfer in the case of the NH3/H2O system and

comes to the conclusion that it cannot be precisely

determined which partial resistance, of the liquid phase

or of the gas phase, respectively, controls the NH3/H2O

absorption process. Finally his correlations are satisfac-tory by adapting Eq. (3).

In an attempt to nd a more accurate method of pre-

diction of NH3/H2O gasliquid interactions, this paper

will take into consideration the non-equilibrium ther-

modynamics [19]. Although it seems to be a natural

direction of investigation, it has not been used so far to

analyse neither the NH3/H2O system nor other working

combinations.

Herein below a simple experiment made by the

author, which strengthens the conviction that the

method of approaching the interactions which form the

object of this study should be reviewed, will be descri-

bed. The experiment consists of injecting a single gas-

eous ammonia bubble into the NH3/H2O solution at

normal pressure, ambient temperature and small molar

fractions. Based upon the numerous observations

recorded, it has been found out that subject to the gas

phase feeding rate the evolution of the bubble may be

characterized as follows: (i) at low and moderate feeding

rates (108107 kg/s), the dynamics of the bubble takes

place in two distinct stages: (a) growth, when the bubble

progressively expands its volume up to a maximum

value, concurrently with gas absorption, followed by (b)

collapse, when the bubble is rapidly absorbed, reducing

its volume up to zero, although it continues to be fedwith gas through the injection nozzle; throughout its

evolution the bubble does not detach itself from the

nozzle and has an almost spherical shape; in the case of

a continuous gas feeding, the phenomenon becomes

quasi-periodical; the frequency of complete evolution

increases with the increase of the ow rate; the duration

of the increase stage is much longer than the one of the

collapse stage; (ii) at high ow rates, collapse no longer

occurs, but a high frequency phenomenon of oscillation

of the bubble volume between two values, a maximum

and minimum one, is noticed; the bubble is no longer

spherical but elongates in the direction of the nozzle axis

M.D. Staicovici / International Journal of Refrigeration 23 (2000) 153167 155


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and has an irregular contour. The study of the bubble in

its complete growthcollapse evolution gives rise to two

questions which are dicult to answer by means of the

current theory of interface mass transfer and absorption

as a surface phenomenon, synthesized by Eqs. (1) and

(2) (the problem of the ammonia bubble absorption, [20]):

(i) if during the growth phase, as the bubble increases itsvolume, the molar fraction x of the interface approaches

the saturation point so that the dierence in the con-

centration of phases (x x) and implicitly, the currentabsorbed decrease, then how could the occurrence of the

collapse phase, which, quite to the contrary, assumes a

continuous increase of the current absorbed in order to

exceed the relatively constant value of the feeding ow

rate, be explained?; (ii) if absorption is considered to be a

surface phenomenon, then, yet again, how couldcollapse,

during which the current absorbed, proportional to the

gasliquid contact area, decreases by the radius second

power, instead of increasing as noted in the rst ques-tion above, be explained? As will be demonstrated fur-

ther, these questions may be accurately answered by

applying the non-equilibrium thermodynamics (i.e.

phenomenological equations) to the interface of the

subject interactions. Its application brings to light an

important feature of the mass and heat currents beha-

vior near an ideal (equilibrium) point, which the specia-

lized literature does not point out, namely the ideal

point approaching eect. Previous author's works use

the phenomenological theory to model absorption of

the ammonia bubble absorption, having as a tool a non-

empirical linear Phenomenological Hydro-GasoDyna-

mical (PhHGD) approach. The nal part of the paper

presents some results of this modeling, in order to elu-

cidate from both qualitative and quantitative points of

view the problem of the ammonia bubble absorption,

and shows the conclusions deriving from it.

2. Phenomenological equations of evolution physical

mono-, bi- and particular polycomponent interactions

with non-ideal mixture

In the general case, interacting binary two-phase evo-

lution phenomena, of gasliquid, solidliquid, solidgasor gasgas, liquidliquid type, may be considered. For

the sake of clarity, here we shall use the notation for the

physical interactions of gasliquid type. The liquid

phase may additionally contain other dissolved non-

volatile components. The state parameters characteriz-

ing the interface, i.e. temperature, pressure and molar

fraction, marked as TgYpg and y for gas and as TlYpl and

x for liquid, are time functions. At interface, surface

tension forces, given by Laplace equation, written in a

new wording [21], by equality

p div '3

S

where '3

is a normal positive vector to the surface and

equal to the surface tension of the gasliquid, are gen-

erated. The theoretical approach holds for both open or

closed systems. It evolves towards the state of equili-

brium through a nite sequence of stationary states. It is

equally assumed that the total energy of the system is

approximately equal to its internal energy, that is itsmacroscopic kinetic energy, the external elds of any

nature which act on it as well as its internal tensions are

negligible. Under these circumstances, when the

mechanical work exchanged by the system with the

exterior is negligible, the heat exchanged between the

system and its environment and between its various

areas is sensibly equal to the total enthalpy exchange.

The gasliquid interaction implies the simultaneous

existence of coupled mass and heat currents `j'', noted

by indices ``i'' and ``q'', respectively, of gas and liquid,

marked by indices ` g'' and, ``l'', respectively, which get

in and out of the system, marked as prime (`) and sec-ondary (``), respectively.

The entropy source, SX

, shows the contribution of the

currents to the global entropy yield. In a stationary

regime this is written [22]

SX

To

ji

HsiH

ji

HHsiHH

jqHTH

jHHq

THH

T

where X

is the exergy dissipation velocity and To is the

temperature of the innite reservoir. Dimensionally jiH

and jqH are expressed in mol/s or kg/s and in W, respec-

tively. Eq.(6) is particularized to two opposed elemen-

tary processes which are permanently encountered in the

heat absorption technology: (i) the gas interacts with the

liquid, resulting in a liquid with new properties

(absorption); (ii) the liquid decomposes in liquid and gas

with new properties (generation). Taking the above into

account as well as the common conventional signs in

thermodynamics, Eq. (6) can be rewritten for case (i) as

SX

jiYgH sg

H jiYlH sl

H jiYlHH sl

HH

jqYgH

1

THgjqYl

H 1

TlH jqYl

HH jq

1

TlHH

4 5U

where jq is a heat current, considered as an excess ele-

ment resulting from the deviation from the ideal state of

the mixture. The following currents balances are con-

sidered in the system

process heat

jqYgH jqYl

H jqYlHH jq V

total mass

jiYgH jiYl

H jiYlHH W



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mass for one of the species

yjiYgH xjiYl

H x x jiYlHH IH

In Eq. (10) x is the nite variation of molar fraction,

suered by the liquid as a result of its interaction with

the gas phase. The heat ows are extensive elements.Taking into account the previous observation concern-

ing the calculation of the heat exchange by means of

total enthalpy, they can be written as

jqYgH jiYg

H hHg

jqYlH jiYl

H hHl

jqYlHH jiYl

HH hHHl II

Taking into consideration Eqs (9) and (10), the rstbracket of Eq. (7) can be written as

jiYg sH

g jiYlH sHl jiYl

HH sHHl

jiYgH sHg s

Hl

y x

sHHl sHl

x

! IP

It is noted that the rst bracket of Eq. (7) represents the

contribution of mass currents to the entropy source

only, therefore the ratio in the right side of Eq. (12)

becomes

sHHl sHl

x

dsl

dx

pYT

IQ

Temperature TlHH is related to the nite variation of

liquid temperature due to the deviation from the ideal

state T TlHH Tl

H by

1

TlHH

1

Tl 1 T

TlH

1TlH

1 T

TlH

T

TlH

2 F F F

4 5IR

Following simple calculations by means of Eqs. (14) and(8)(11) the second bracket of Eq.(7) may be written as

jqYgH 1

TgHjqYl

HH 1

TlH jqYl

HH jq 1

TlHH

!

jiYgH 1

TgH

1

TlH

hHg

1

TlHhHg h

Hl

hHljr

T

TlH 1 T

TlH

VbbbbX

WbbabbY

IS

where

jr jiYg

H

jiYlHHIT

is the reduced mass current of gas (mol mol1), involved

in the interaction. The ratio TTlH is calculated by means ofthe liquid specic heat equation at constant pressure

cpYl dhl

dT

pYx

IU

resulting in

T

TlH

hHlTlH cpYl

Ajr IV

where the non-dimensional factor A in Eq. (18), herein

called ``reduced excess heat'' is obtained with Eqs. (9),(10) and (16) and has the expression

A y x dhHldx

pYT

1

TlH cpYlIW

Eqs. (12)(19) are introduced in Eq. (7) and the entropy

source is written as

SX

jiYgH X PH

where

X

@sHg s

Hl

y x

dsHldx

pYT

4 5

hHg1

TgH

1

TlH

A

TlH 1 Ajr hHg h

Hl

jr h

Hl

h iA

PI

Eq. (20), written as above, points out the proportion-

ality of the entropy source with the coupling gas mass

current jiYgH and with factor X , which is the thermo-dynamic (generalized) force of the irreversible process at

hand [22,23]. The force diers of forces governing the

coupled heat and mass transfer in continuous media and

originating from Fourier's and Fick's laws, respectively.

It only depends on the entry values of the system, for

which reason the marking primary (H) is hereinafter

given up to. Because it depends only of the interface p,

T, x and y state parameters, force is a way independent

function, therefore a state function. According to Eqs.

(8)(11) the coupled heat current jq has the expression

jq jiYgH C PP



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where Cis the local thermal capacity (availability) of the

binary mixture [24]

C hHg hHl

y x

dhHldx

pYT

PQ

This paper is conned to considering that the currentsinvolved are linear functions of the thermodynamic for-

ces. Later on, this consideration will be justied. Hence,

the phenomenological equations are reduced to the

simplest linear expression

jiYgH L11 X PR

where L11 is the strictly positive phenomenological

coecient of proportionality, having the dimension of

mol2 KJ1 s1. In a rst approximation L11 is con-

sidered to be constant, independent ofX. With Eq. (24),

the entropy source, [Eq. (20)], acquires the denitepositive form

SX

L11X2 PS

for all interface p, T, x and y values, except the ideal

points, given by

pl pg pe

Tl Tg Te

"1 2 Yl "1 2 Yg

where, according to current opinion it vanishes [23].

Later, the paper will refute this last statement. It is use-

ful to further proceed to non-dimensioning the equa-

tions referred to above. In order to do this, the

following reduced elements are considered

L11Yr L11

jiYlHH

R

YXr XRYSX

r SX

RjiYlHH

hr h

TbRY cpYr cpaRY sr saR

pr p

pbYTr

T

TbPT

In Eq. (26) R, pb and Tb are the gas constant, a reduc-

tion pressure and, respectively, an absolute reduction

temperature. In view of the above, the previously

deduced equations can be re-written in a reduced form

(index ``r'') as follows

SX

r jr X L11Yr X2r PU

jr L11Yr Xr PV

Xr sgYr slYr

y x dslYr

dx

prTr

4 5

hgYr1

TgYr

1

TlYr

Ar

TlYr 1 Ar jr

hgYr hlYr

jr hlYr

PW

Ar dhlYr

dx

pr Tr

y x

TlYr cpYlYrQH

jqYr jr Cr QI

where

Cr hgYr hlYr

y x

dhlYr

dx

pr Tr

QP

The partial reduced thermodynamic force of a purely

mass nature, given by the rst square bracket of the

right side of Eq. (29) is marked as XiYr

XiYr sgYr slYr

y x dslYr

dx

pr Tr

QQ

We have to remark here the similitude of Eqs. (32) and

(33).

Bearing in mind that L11Yr must be in a rst approx-

imation independent ofXr, from Eqs. (28) and (29) the

reduced thermodynamic force is expressed in an implicit

form as [20,25,26]

Xr

XiYr hgYr1

TgYr

1

TlYr

ArhlYr

TlYr 1 ArL11Yr Xr

1 Ar hgYr hlYr

TlYr 1 ArL11Yr Xr M prYeY xe

QR

where M prYeYxe is a new function discussed later inSection 3. It may be derived rather solving Eq. (34) with

respect to Xr by means of a simple iterative method[27] (which the author preferred), but also as a second

degree algebraic equation [26]. Force is useful inapproaching interaction mass and heat currents of

mono-, bi- and particular poly-component gasliquid

interactions with non-ideal mixture, coupling it with

Eqs. (28) and (31) and the classic hydro-gasodynamics.

For instance, the two last types of interactions are

encountered in the binary systems NH3/H2O, H2O/

LiBr, NH3/LiNO3, NH3/NaSCN. CH3NH2/H2O etc.,

or in the ternary systems such as NH3/H2O-LiBr,

CH3NH2/H2O-LiBr etc. The resulting Phenomen-

ological Hydro-GasoDynamical (PhHGD) method was

already successfully used to model a few ammonia/

water absorption/generation applications [20,25,28,29].



7/15

The thermodynamic force enjoys several important

properties [26]. First, as already mentioned, force

applies also to interactions of solidliquid, solidgas or

gasgas, liquidliquid type, provided that appropriate

indexes in Eq. (34) be used (e.g. for a liquidsolid inter-

action with a solid nal phase, the indicesg and lchange

in l and s, respectively). Force is valid for both con-tinuous or discontinuous media. Obviously, the currents

assessment needs to analytically determine the right

phenomenological coecients, which often is a dicult

task. Second, force must be not dened in the ideal

points, because otherwise it should take at the same time

and with the same probability two opposite values, cor-

responding to the elementary processes at hand, with

opposite signs of the homologue currents, which is

impossible. This property extends to the currents and to

the entropy source, not dened in an ideal point as well.

Concerning the entropy source, our conclusion contra-

dicts the current opinion, previously mentioned in thisparagraph. The set of ideal points separates the strictly

positive thermodynamic forces ( Xr b 0), character-ized by absorption currents, jr b 0, from the strictly

negative forces, ( Xr ` 0), which fact gives rise togenerating currents, jr ` 0. The third property is related

to the force behavior near an ideal point. So far, the

specialized literature does not point out this aspect and

refers only to the total entropy increase to maximum

values when the system approaches such a point, caus-

ing the entropy source strictly positive denition. In this

work it is considered that real systems naturally making

for equilibrium obey the following postulate [26]: The

generalized thermodynamic forces of closed and open

systems continuously increase in absolute value to max-

imal nite values approaching an ideal point. Here are

included the systems with physical or chemical interac-

tions in the rst place. The phenomenological factors,

proportional to the force, must have the same force

feature.

The case of pure component interactions is important

and worth to paying a little bit more attention to. The

Eq. (34) holds here too, simply considering that x and y

take close values to 0, or 1. However, the reduced excess

heat Ar and the term y x dslYrdx

prYTr

in Eq. (33) are

vanishing when x and y approach 0 or 1, so that Eq.(34) simplies in case of pure components, namely

XrYp sgYr slYr

hgYr1

TgYr

1

TlYr

QS

In Eq. (35), the two terms of the right member have

dierent signs, but XrYp is strictly negative and holdsfor vaporization

XrYpY) XrYp

jrY) L11Yr XrYpY)

TgYr

b TlYr

QT

In case of condensation, the currents are changing the

sign, therefore

XrYpYc XrYp

jrYc L11Yr XrYpYc

TgYr ` TlYr QU

When the system approaches an ideal point, the second

term in the right side of Eq. (35) vanishes and force

takes maximum absolute values, in accordance with the

postulate. The next paragraph includes case studies for

binary and pure component interactions.

The phenomenological coecient may be theoreti-

cally estimated making a simple imaginary experiment.

Let us consider an isolated gaseous mixture bubble, of

volume Vg t , at pressure pg t and temperature Tg,which interacts with the liquid phase surrounding it. For

convenience, it is assumed that the bubble has an iso-

therm adiabatic evolution in the liquid. In accordancewith the rst principle of thermodynamics, the variation

of its internal energy within the time unit is equal to the

power exchanged between the gas and its environment,

namely

UX

g pgVX

g

or,

jr&ljVX

gjug pgVX

g QV

where ug cvYgTg is the gas specic internal energy.Upon absorption VXg ` 0, and upon generation VXg b 0

and Eq. (38) is simplied to the expression

jr pg

&lc)YgTgQW

Eq. (39) shows that positive reduced currents are

obtained upon absorption, and negative reduced cur-

rents are obtained upon generation, as previously men-

tioned herein. This equation is valid when the

homobaricity condition in the bubble is met [30]

RX aC 2

`` 1

where RX

and Care the interface velocity and the velocity

of sound in gas, respectively. Physically, the above con-

dition is met in the case of mass currents with low

values, that is, anticipating the conclusions hereof, mass

currents which are far from the state of equilibrium,

when the pressure from the wall is sensibly equal to the

pressure in the rest of the gas volume and has a known

value. The determination of the phenomenological

coecient is not aected by the position of the system

relative to the point of equilibrium in respect of which it



8/15

has been assumed that L11Yr is independent ofXr. L11Yr is

calculated by means of Eq. (28)

L11Yr jr

XrRH

where jr and Xr are determined through an iterativeprocess from Eqs. (39) and respectively (34) [20]. A

similar result is obtained if, instead of the energy equa-

tion, the mass balance in the bubble is written

jr&ljVX

j &gVX

RI

resulting in

jr &g

&lRP

For an ideal gas

&g pg

RTgRQ

and the reduced mass ow acquires the form

jr pg

&lRTgRR

which is likewise valid and useful in homobaricity con-

ditions.

3. Application of phenomenological equations to the

gasliquid systems. The ideal point approaching

(I.P.A.) eect

The thermal factors involved in the calculation of the

thermodynamic force have been rst analytically

expressed for the NH3/H2O system [20], and its pure

components [26,29], by means of Ziegler and Trepp

state equation [31]. In a second step, the applications

were extended also to other known working combina-

tions, namely NH3/LiNO3 and NH3/NaSCN, using

thermodynamic and physical property data alreadypublished for them [32]. From the experience, the gas

liquid interactions of these systems are nonlinear and

the assumption of a linear set of phenomenological

equations needs to be justied. Indeed, the currents are

linear with respect to the thermodynamic force, only. In

fact, they are strong nonlinear with respect to all inter-

face variables through the thermal factors srY hrY Ar and

Cr intervening in the calculation of the thermodynamic

force [31,32].

For the beginning we shall present the results of the

ammonia/water system. The phenomenological coe-

cient has been estimated by means of Eqs. (40), (39) and

(34), resulting in L11Yr 103jRj1 1a8314. Prior to

starting the application, we paid special attention to the

consistency of the thermodynamic force equation with

respect to the equilibrium thermodynamics, regarding

the ideal points calculation of a gasliquid system and

particularly of the ammoniawater mixtures. As we

already mentioned, the force is not dened in an idealpoint and never cancels. Practically, such a point is

determined by progressively restricting the interval of

denition throughout which the function has nite

values, of equal absolute value and contrary signs, up to

the desired level of accuracy (here usually obtained up

to the sixth decimal). In principle, the two approaches,

equilibrium and phenomenological, are equivalent, as

far as the determination of the ideal point is concerned.

However, although in both cases the same state equa-

tion for the binary mixture under study has been used,

the phenomenological calculus have led to sensibly dif-

ferent results in some parts of the solubility eld [20].First attempts to approach the results of the two meth-

ods failed when a nonlinear dependence of the mass

current with respect to the thermodynamic force was

considered. This explains why we used only a linear

dependence of the mass current with respect to the

thermodynamic force [Eq. (24)]. Best results were

obtained amplifying the second term of the denomi-

nator in Eq. (34) by a match function M prYeY xe ,already mentioned in Section 2. Several values of the

match function are given in Fig. 1 for ve reduced ideal

pressures and ve ideal solution mass fractions. These

values were good correlated by a four-order double

polynomial expansion:

Fig. 1. Several values of the match function in Eq. (34) for

ammonia/water gasliquid mixtures (pb 10rY Tb 100 K).

Fig. 1. Plusieurs valeurs correspondent a Eq. (34) pour les mel-

anges ammoniaceau (gazliquide) (pb

=10 bar ; Tb

= 100 K).



9/15

M prYeYxe

exp4iYj0

AiYjpirYex

je

2 3RS

The coecients AiYj are given below:

A00 1X55107 100Y A01 4X30114 100Y

A02 2X50709 101Y A03 5X14562 10

1Y

A04 3X53064 101Y A10 2X20458 10

1Y

A11 3X43605 102Y A12 2X15534 10

3Y

A13 5X56704 103Y A14 4X87992 10

3Y

A20 5X72617 101Y A21 9X54441 10

2Y

A22 6X00841 103Y A23 1X54652 10

4Y

A24 1X35339 104Y A30 5X53949 10

1Y

A31 9X34452 102Y A32 5X88969 10

3Y

A33 1X51448 104Y A34 1X32402 104Y

A40 1X74164 101Y A41 2X94977 10

2Y

A42 1X86140 103Y A43 4X78901 10

3Y

A44 4X18852 103

The relative dierences (%) in the approximation of

equilibrium mass fraction of the mixture solution with

Eq. (34), are plotted in Fig. 2. The dierences could

come also from the dierent units used for xe, namely

mass fraction in hx diagram, respectively molar frac-tion in the equation of state of Ziegler and Trepp, and

the use of a unique computing value for the phenomen-

ological coecient. Eq. (34) covers an usual domain for

refrigeration purposes with the NH3/H2O system,

namely of 1.04p415.0 bar, 0.14x40.45 and

0.54y40.998.

The presence of the match function in force equation

must be explained, because a less experienced reader

may think that its use was necessary to compensate for

the much simplify assumptions considered in our theo-

retical approach. In fact, M prYeYxe compensates thearbitrarily chosen zero reference state of a system gas/

liquid enthalpy and entropy, which Eq. (34) is quitesensitive to. To this extent, Eq. (45) is valid for Ziegler

and Trepp work, only.

Prior to using Eq. (34) in an evolution process

assessment, we need the ``nearest'' ideal point which the

system evolves to, in order to calculate the match func-

tion. Normally, this should result from an iterative cal-

culus, searching, according to a Prigogine theorem, the

way of minimum entropy production. However, not all

but in many cases a simpler practical method can be

considered, remarking that most systems have state

parameters with quasiconstant values all along the evo-

lution period, provening from their connection with theinnite reservoirs or other engineering purposes etc.

Depending on the system variance, these parameters

partially or completely dene the nearest ideal point. In

our case, the ammonia/water system is biphase and

bicomponent, so it suces to have two such parameters,

but the type of the parameters may change upon the

specic application. For instance, for the absorption

process of a closed system absorption refrigeration

plant, these parameters are the rich solution nal tem-

perature, close to the sink source temperature and the

absorbed gas molar fraction, but for the generation

process, they change to the nal poor solution tem-

perature, close to the warm source temperature and the

condensing pressure. In case of the bubble absorption

experience, mentioned in the introduction, the nearest

ideal point is dened by the quasiconstant absorption

pressure (pl pg) and gas molar fraction y.The phenomenological Eqs. (27), (28), (31) and (34)

have been represented for exemplication as partial

functions of the reduced state parameters xY tlYrYy and

tgYr, respectively, in Figs. 37 for the NH3/H2O system

(at interface, gas and liquid are in mechanical equili-

brium, pl pg, and no pressure partial functions exist infact). The thermodynamic force of pure components

water and ammonia is given in two study cases in Fig. 8,and that of NH3/LiNO3 and NH3/NaSCN systems in

four study cases in Figs 9 and 10, respectively. The

curves are dierent in shape, depending upon the vari-

able used in the abscissa and the nature of the system.

Partial functions which depend on the parameters of the

same phase have the same absorption or generation

monotonous variation. According to the postulate and

its consequence, the diagrams show the continuously

increase to nite values of force, entropy source, and of

the coupled mass and heat currents, approaching an

ideal state. In case of currents, an ideal point approach-

ing (i.p.a) eect is emphasized: depending on how strong

Fig. 2. The relative dierences % in the approximation of

equilibrium mass fraction of ammonia/water solutions with Eq.

(34).

Fig. 2. Dierences entre la fraction massique des solutions

ammoniac / eau et l'Eq. (34), exprime en pourcentages.



10/15

the currents coupling is, the approach to an ideal state

determines an increase eect of the function absolute

value by several percent (in case of pure components), to

several hundred times (for the binary systems), as com-

pared to the states which are far from the same ideal

point. In this way, the term of ``far from equilibrium

state'' changes its classical meaning and corresponds toits opposite, namely of ``low interaction''. The i.p.a.

eect may be quantied by a gure of merit, `f '', equal

to the ratio of two same type (absorption/condensation

or generation/vaporization) function values, corre-

sponding to the states closer, respectively, further to the

ideal state. Figs. 310 include also the maximal `f ''

values for the chosen examples. As a direct consequence

from the above it results that estimation of the interface

mass transfer by analogy with heat transfer lacks basic

physics. Force in Eq. (2) tends to zero approaching an

ideal point, which is not true, therefore is erroneous.

The specialized literature does not contain systematictheoretical or experimental accounts of the non-equili-

brium interaction at hand which the results obtained

herein might be compared to. The only conrmation of

such results is given by their compliance with a few

experimental works which are in agreement with the

graphs shown in Figs. 310, proving the increase of the

gaseous mass ow driven as a two-phase binary system

Fig. 4. The i.p.a. eect for the reduced mass (a) and heat cur-

rent (b) as functions of solution mass fraction.

Fig. 4. L'eet i.p.a de la masse reduite (a) et le ux de chaleur

(b) en fonction des fractions de masse de la solution.

Fig. 5. The i.p.a. eect behavior for the reduced entropy source (a)

and thermodynamic force (b) as functions of solution temperature.

Fig. 5. Comportement de l'eet i.p.a. pour la source d'entropie

diminuee (a) et la force thermodynamique (b) en fonction de la

temperature de la solution.


and thermodynamic force (b) as functions of solution mass fraction.


diminuee (a) et la force thermodynamique (b) en fonction des

fractions de masse de la solution.



11/15

is approaching a state of saturation. To this eect,

mention should be made of the experimental work of

Hui and Thome [33], who investigated the boiling of

binary azeotropic ethanolwater and ethanolbenzene

mixtures and ascertained the increase of the number of

vaporization points, until it became impossible for such

points to be individually distinguished, as the molarfraction of the mixtures approached the saturation

value.

4. The PhHGD approach of the single bubble absorption

The author used the PhHGD approach to model local

absorption/generation processes in the ammonia/water

system [20]. Because generation in an absorption plant

passes more or less through a sequence of ideal points,

but absorption, on the contrary, working usually with

rectied vapor is far from the ideal point and takes placetherefore at low gasliquid interaction, the author paid

more attention to the absorption process. However,

taking into account the ndings of Section 3, he had

additional good reasons to model the bubble absorp-

tion: (i) it was used to satisfactorily explain the ammonia

bubble absorption problem, raised in the introduction;

(ii) it contributed to a simpler experimental validation of

the PhHGD method [20,28], and (iii) it served to perform

an analytical study of absorption for the system at hand,

based on the single bubble dynamics. Here we present

only a few results referring to the rst point and some

conclusions of the analytical study.

Fig. 7. The i.p.a. eect behavior for the reduced entropy source(a) and thermodynamic force (b) as functions of gas reduced

temperature.


diminuee (a) et la force thermodynamique (b) en fonction de la

temperature reduite des gaz.


and thermodynamic force (b) as functions of gas mass fraction.


diminuee (a) et la force thermodynamique (b) en fonction des

fractions de masse des gaz.

Fig. 8. The water and ammonia i.p.a. eect behavior for the

reduced thermodynamic force as function of liquidgas reduced

temperature.

Fig. 8. Impact du comportement l'eet i.p.a. de l'ammoniac / eau

pour le force thermodynamique reduite en fonction de la tem-

perature reduite du melange liquide / gaz.



12/15

Absorption takes place between gas and the sub-

cooled absorbent, when forces and currents are positive.

The set of the PhHGD code entry data corresponds to

the absorption of the ammonia bubble (y=0.992) in

water (xI 104), at normal pressure (pI 1X001 bar)

and environment temperature (TI 293 K). The solu-

tion of the motion equation is shown in Fig. 11. Fig. 12elucidates the problem of ammonia bubble absorption.

Indeed, the reduced absorbed mass current jr and the

actual one

jHiYg jr&ljVX

j RT

continously increase during bubble growth time, causing

the bubble pressure decrease (Fig. 13). At a certain

moment, the actual current equals the bubble mass

feeding rate m and the collapse starts. Depending on the

interface parameters, it continues to increase also after

the starting of the collapse at an even higher speed, upto the total absorption of the bubble. This further

explains why the collapse duration is much shorter than

the duration of the growth. The cumulated absorbed

mass current

jHiYgYa t

t0

jHiYg z dz RU

and the corresponding coupled heat current jHHqYlYa are

plotted against the bubble growth time in Fig. 14.

Referring to the ammonia/water analytical study, the

PhHGD code proved to be a rened tool of investiga-tion of the intrinsic absorption properties of this com-

bination. According to it, a few rst conclusions have

Fig. 9. The i.p.a. eect behavior for partial thermodynamic

forces of liquid mass fraction (a) and liquid temperature (b) in

case of the NH3/LiNO3 system.

Fig. 9. Comportement de l'eet i.p.a. pour les forces thermo-

dynamiques partielles pour la fraction liquidemasse (a) et le

temperature du liquide (b) du systeme NH3/LiNO

3.

Fig. 10. The i.p.a. eect behavior for partial thermodynamic

forces of liquid mass fraction (a) and liquid temperature (b) in

case of the NH3/NaSCN system.

Fig. 10. Comportement de l'eet i.p.a. pour les forces thermo-

dynamiques partielles pour la fraction liquidemasse (a) et le

temperature du liquide (b) du systeme NH3/NaSCN.

Fig. 11. Dynamics of ammonia bubble absorption.

Fig. 11. Dynamique de l'absorption de la bulle d'ammoniac.



13/15

been drawn concerning best mass/heat transfer: (i) gas

and initial liquid temperatures should have close values

to those of the ideal point; (ii) working with as much as

low absorption pressures is desirable (absorption e-

ciency increases about 3 times when pressure decreases

from 2.9 to 0.2 bar), but this must be obviously corre-

lated with the other design parameters of a plant, and(iii) absorption eciency considerably decreases when

the gas mass fraction approaches the unity; bearing in

mind its high energy consumption, vapor rectication,

commonly used in absorption plants, has this time an

additional reason to be avoided, that of decreasing

mass/heat transfer in absorption processes; to this

extent, on the opposite side, resorption plants gain twice

eliminating rectication. More details about this analy-

tical study can be found in author's past works,

[20,28,29].

The phenomenological modelling of the bubble

absorption may be a valuable lesson for a futureadvanced thermal absorption technology. Next, we

synthesize what is to be learned from it. First, the

absorption process of gasliquid working combinations

used in the thermal absorption technology is a mass phe-

nomenon and not a surface one, as pointed out also by

Eq. (46). From here, corroborated also by conclusion

relative to the motrice force, given in Section 3, it results

also that the use of the mass transfer coecients Kg and

Kl as surface factors is improper. Second, absorptioncan be improved in an intensive way, seeking not remo-

teness from the ideal point (see the classical point of

view), but nearness to it, when the currents naturally

increase up to very high values without any additional

technical improvements (see gure of merit f=740.7 in

Fig. 12). The tendency manifested in the construction of

NH3/H2O absorbers of increasing the gasliquid contact

area through the dispersion of one of the phases in order

to increase the absorption eciency may not result in

the anticipated eect. Quite the contrary, by extend-

ing the contact area the more rapid evolution of the

interface towards the equilibrium parameters is delayed(the mass fraction x of the liquid at interface decreases

with the increase of the contact area, at the same gas

feeding ow rate) and absorption decreases, as com-

pared to the one achieved in a less dispersed phase.

Moreover, the auxiliary pumping energy consumption is

also increased. Thus, in the light of the above, the

increase of the contact area is merely an extensive tech-

nical solution which helps to compensate the absorption

decrease caused by an articial remoteness from the

ideal point, oering in exchange a more stable func-

tioning of the absorbing device. In addition to the

above, it is noted that quasispherical bubbles with a

minimum contact area for the same volume stand the

greatest chances to achieve the best mass and heat

transfer through i.p.a. eect appearance. This gives rise

to the idea of constructing an NH3/H2O bubble ow

absorber [20]. It is at plate and mounted horizontally.

In comparison with a present NH3/H2O absorber

with vertical pipes or with liquid phase dispersion, the

bubble absorber could oer the following advantages:

Fig. 12. Phenomenological elucidation of the ammonia bubble

absorption problem.

Fig. 12. Elucidation de l'absorption de la bulle d'ammoniac.

Fig. 13. Tube and bubble pressure variation during ammonia

bubble absorption.

Fig. 13. Variation de la pression a l'interieur du tube et de la bulle

lors de l'absorption de la bulle d'ammoniac.

Fig. 14. Cumulated absorbed mass and coupled heat currents

during ammonia bubble absorption.

Fig. 14. Flux de masse et de chaleur absorbes cumules lors de

l'absorption de la bulle d'ammoniac.



14/15

(i) maximum absorption eciency; (ii) minimum pres-

sure loss on gas side; (iii) it is suited to a modern com-

pact plate construction; (iv) minimum auxiliary energy

consumption. Third, intensive theoretical and experi-

mental research work has been carried out during the

last decade aiming at improving absorption, by means

of additives (surfactants) which stimulate the Mar-angoni convection [3436]. Up to the present date there

is no satisfactory explanation of the mechanism which

generates this eect [37]. Most of the unsuccessful

attempts have been based on the eect of supercial

tension reduction in the liquid, obtained by means of

additive agents. The non-equilibrium thermodynamics

oers yet another chance to the clarication of this

problem. Without having the intention to give a quali-

tative and so much more quantitative complete expla-

nation, we will however make a primary analysis of the

cause which can lead to the occurrence of the Mar-

angoni eect from the phenomenological point of view[20]. Here, contrary to the eect obtained by the dis-

persion of phases, by means of additive agents the

interface area decreases considerably, and its parameters

approach the equilibrium values. Interface becomes

non-homogeneous, in the sense that violent absorption

centers appear at its level which alternate with weak

interaction zones, due to the presence of the surfactant

(which can be dissolved in the absorber or not). This

non-homogeneity, inuenced to a small extent by the

supercial tension, causes an accentuated perturbation

of the liquid surface, which is characteristic of Mar-

angoni instability.

5. Conclusions

The above leads to the conclusions presented herein

below. These conclusions refer to all mono-, bi- and

particular polycomponet gasliquid working combina-

tions used in the thermal absorption technology,

including in the rst place the ammoniawater system:

1. The paper raises the problem of ammonia bubble

absorption, which is dicult to answer by means

of current theory of interface mass transfer andabsorption as a surface phenomenon.

2. According to the linear phenomenological

approach, the heat and mass transfer at the gas

liquid interface is governed by the thermodynamic

force, dierent to that of forces governing the heat

and mass transfer in continuous media and origi-

nating from Fourier's and Fick's laws, respectively.

The force equation is extendable to other interac-

tions occuring in continuous or discontinuous

media, like solidliquid, solidgas, or liquid

liquid, gasgas type. The paper mentions a postu-

late referring to the force behavior approaching

an ideal point, introduced by the author in a pre-

vious work. According to its consequence, the

mass and heat currents suer an ideal point

approaching (i.p.a.) eect, not mentioned so far in

the specialized literature, consisting of a con-

tinuous increase of their absolute value by several

percent (for a pure component), to several hun-dred times (for a binary system) when the inter-

acting system approaches an ideal state, as

compared to the values of states which are far

from the same ideal point. In this way, ``far from

equilibrium'' becomes synonymous with ``low

interaction (currents)''. The entropy source, force

and coupled currents are not dened in an ideal

point.

3. Estimation of the interface mass transfer by ana-

logy with heat transfer, expressed by Eqs. (1) and

(2), lacks basic physics.

4. The phenomenological approach elucidates theproblem of ammonia bubble absorption.

5. Absorption process is a mass phenomenon and

not a surface one. The use of the mass transfer

coecients as surface factors is improper.

6. An intensive way of improving absorption is

emphasized, which seeks to promote the i.p.a.

eect appearance. This would replace the exten-

sive way currently used, based on increasing

gasliquid interaction area. To this extent, the

bubble absorber is hereby proposed for ecient

absorption.

7. The i.p.a. eect existence oers an additional chance

for a satifactory explanation of the Marangoni

eect.

Acknowledgements

The author thanks Professor M.D. Cazacu at the

Polytechnical University of Bucharest for the valuable

assistance in performing the crude material of this paper

and math. D-tru Mihai at SC ICPET-Cercetare SA for

the help in numerical solving.

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A Phenomenological Theory of Polycomponent Interactions in Non-ideal Mixtures Application to NH3H2O...

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