Investigation on Numerical Modeling of Water Vapour Condensation from a Flue Gas with High COR2R Content

doi:10.4236/epe.2011.32023

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Energy and Power En gi neering, 2011, 3, 181-189

doi:10.4236/epe.2011.32023 Published Online May 2011 (http://www.SciRP.org/journal/epe)

Investigation on Numerical Modeling of Water Vapour

Condensation from a Flue Gas with High CO2 Content

Hamid Nabati

School of Sustainable Development of Society and Technology, Mälardalen University,

Västerås, Sweden

E-mail: [email protected]

Received February 2, 2011; revised April 7, 2011; accepted April 11, 2011

Abstract

In this paper, condensation of water vapor from a mixture of CO2/H2O is studied numerically. To simplify

the study and focus on the physical model, a simple vertical plate was chosen. Two condensation models are

developed and numerical approach is considered to implement these models. The main objective in the cur-

rent paper was to study the capability of numerical modeling in prediction of complex process. Results

showed that developed condensation models in combination with numerical approach can predict the trends

in condensation behavior of binary mixture very well. Results from this study can be developed further to be

used in design of condensers which are suitable for oxy-fuel power plants.

Keywords: Condensation, Two Phase Flow, Numerical Modeling, CO2 Capturing, Oxy-Fuel Power Plants

1. Introduction

Oxy-fuel power plants are one of the recently promising

processes for clean energy production with CO2 captur-

ing and condensers for separation of water vapour from

flue gas are essential components in these new proposed

power plants.

As the combustion process in the oxy-fuel power

plants is performed with fuel and pure oxygen, the re-

sulting flue gas consists mostly of H2O and CO2.The

water vapour separation process from the flue gas im-

pacts the thermal efficiency of the plant and the opera-

tional cost; thus the precise design of such CO2/H2O

condenser systems is a vital demand in the industry [1-3].

Currently there are commercialized condensers in the

market that most of them are installed for separation of

air and water or for condensation of nearly pure water

steams and thus far the condensers for separation of wa-

ter vapor and CO2 are not off the shelf products yet.

Considering the specific characteristics of such condens-

ers for application in the CO2/H2O separation, makes

them as special process heat exchangers in the industry.

Despite of wide referring to this kind of condenser in

different proposed Oxy fuel cycles, the required design

data are not available yet and there is a demand for more

studies to achieve the desired efficiency for CO2 captur-

ing and steam separation and basic studies on condensa-

tion of water vapour from a flue gas with high CO2 con-

centration could provide such useful technical data for

designers.

Naturally, condensation happens whenever the vapour

temperature is decreased by cooling until it reaches the

saturation temperature Tsat at the operational pressure and

usually this takes place when vapor is brought into con-

tact with a solid surface whose temperature Ts is less than

saturation temperature of the vapor. However condensa-

tion can also occur in a gas or on the interface of a liquid

and a gas. When condensation occurs in a gas, the liquid

droplets usually suspend in the gas. As it is more com-

mon in the industry to operate and control surface con-

tacts condensers and also based on what is shown for

oxy-fuel CO2 capturing, in the current study the conden-

sation on solid surfaces with focus on film condensation

is considered.

It is a common practice in the literatures to accept film

condensation in heat exchanger design [4-6]. In this

process, the condensate forms a liquid film is formed on

the solid surface. This liquid film slips down under the

influence of gravity. The thickness of the liquid film in-

creases gradually with more vapors condensation on the

film in the flow direction. Liquid covers the surface and

eventually a liquid film takes place between vapour and

solid surface. This liquid film resists against heat transfer

flow. It means that released heat from vapour condensa-

182 H. NABATI

tion at the vapour-liquid interface must transmit through

this layer before it approaches the cooling solid surface.

Moreover, the presence of even a small quantity of

non-condensable gas significantly affects the heat trans-

fer resistance in the region of the vapour-liquid interface.

Experimental studies show that the non-condensable

gases existence in the mixture has an unfavorable con-

sequence on condensation process [7,8]. As an example,

the presence of less than 1 percent (by mass) of air in

steam decreases the condensation heat transfer coefficient

more than half [4]. Vapour carries the non-condensable

gas towards the vapour-liquid interface and it accumu-

lates there. Thus special consideration should be applied

when condensation from CO2/H2O mixture is studied. In

this case a large portion of gas stream is occupied by

CO2 which is a non-condensable gas in the normal con-

dition. This is one of the cases that are referred as

multi-component (n > 2) mixture condensation where n

represents the number of components. The CO2/H2O flue

gas is a binary (n = 2) mixture that its phase equilibrium

characteristics is important in flue gas condenser design

and operation.

Condensation of water vapor from CO2/H2O flue gas

mixture on a vertical smooth surface is shown schemati-

cally in the Figure 1.

Referring to this figure, the condensation behavior can

be explained as the following:

When water vapor starts to condense, only the non-

condensable gas part in the mixture remains in the vicin-

ity of the interface surface. This gas layer acts as an ob-

stacle between vapor and surface, and makes it difficult

for the vapor to penetrate and reach the surface. Conse-

quently, the efficiency of the condensation process is

reduced.

There are limited works about surface condensation

which have been performed either experimentally or

numerically. Recently, some studies have been per-

formed on vapour condensation from mixture of non-

condensable gases and steam. However, nowadays, the

main idea is to use available tools (like numerical meth-

ods) to develop robust and reliable methods which can be

used to simulate the heat and mass transfer in condensa-

tion process.

Three main categories of condensation models are

available: models with experimental correlations, models

using Nusselt theory (based on heat and mass transfer

analogy) and mechanistic models based on the boundary

layer equations [9]. The main advantage of the first

group models is their simplicity and therefore they can

be easily adopted for numerical modeling. Also they can

be used for initial verification of CFD condensation

modeling. However they are bounded to a very limited

set of data. The second group models are also suitable to

Figure 1. Water vapour condensation on a smooth vertical

plane from a mixture containing non-condensa ble gas.

be implemented in the numerical simulation and they

give more realistic results. The last group models are

consistent with the numerical treatment of governing

equations. But at the moment, implementation of such

models requires high expenses because of their high de-

gree of complexity. These are the best models that

probably will be implemented in high accuracy CFD

codes in future. This work evaluates two condensation

models through a numerical simulation and propose the

optimum model based on available two-phase flow mod-

els and apply it in a numerical scheme with the help of

Fluent© software to study water vapour condensation

from a mixture containing mainly H2O and CO2. Also

effect of fin installation on the condensation surface is

partly studied.

2. Physical Model for Binary System

Condensation

As it was stated before, if any quantity of non-condensable

gas exists in the mixture, there would be major effects on

the heat and mass transfer resistance in the liquid-vapour

interface. The non-condensable gas is carried towards the

interface and accumulates there. This accumulation

causes the partial pressure of gas at the interface became

greater than its partial pressure in the binary mixture.

This effect produces a driving force for non-condensable

gas to diffuse again toward the bulk. This diffusive mo-

tion is contrary to water vapour diffusion toward liq-

uid-vapour interface. Also when the vapor which is

mixed with a non-condensable gas is condensing, only

the non-condensable gas remains in the vicinity of the

liquid-gas surface. This gas layer forms a resisting wall

H. NABATI

183

between condensing liquid and vapor and makes it hard

for the vapor to contact with the surface. Therefore vapor

should diffuse through this layer first before reaching the

surface. These effects reduce the condensation process

efficiency. These conditions are worse for the CO2/H2O

condenser, where the concentration of CO2 is much

higher than water vapour. The accumulation of CO2 in

the vicinity of condensation layer makes a barrier to re-

maining water vapour in the flue gas stream and de-

creases the effectiveness of condensation process.

In the mixture of CO2/H2O, the partial pressure PA, of

component H2O is that pressure which would be exerted

by H2O alone in the mixture appropriate to the concen-

tration of H2O in the flue gas at the same temperature.

Here A represents H2O to simplify the proceeding equa-

tions. Since P = ΣPA, then the partial pressure, PA, is

proportional to the mole fraction of H2O in the vapour

phase.

AAA

PxP (1)

The most known correlation that is used to relate the

partial pressure in the vapour phase to the concentration

of component A in the liquid phase is the Rault’s law.

This law states that the partial pressure PA is related to

the mole fraction xA and saturation pressure of pure

component A at the same temperature:

AAA

PxP



 (2)

where xA represents mole fraction of component A in the

liquid phase. During mass transfer process, water vapour

travels from high concentration region to region where it

has a low concentration. Just as thermal energy diffuses

from region of high temperature to low temperature re-

gion (following the temperature gradient), the mass

transfer follows the concentration gradient. Fick’s law of

diffusion represents that diffusion mass flux of any spe-

cies in a multi-components stream has direct proportion

to the species concentration gradient. This law can be

expressed as:

iim





 

J (3)

is the total mass flux of species i and gets the unit

of (kg/m2·s). Effective diffusivity, Dim is an indicator of

diffusion intensity of species i into a mixture. Diffusivity

coefficient is a function of composition, temperature and

pressure and for gases is typically on the order of 10–5

near the room temperature. The Chapman-Enskog corre-

lation is one of the known formulas which is based on

kinetic theory and takes into account all the molecular

effects precisely [9]. The correlation is:



1.8583 1011

AB D













(4)



AB is the average molecular diameter in Å and is

equal to (



A +



B)/2. MA and MB are molecular

components A and B. D can be obtained from

cor l

weight of

available

reations in literatures [4,10,11]. Table 1 shows the

required data for diffusivity calculation.

Considering the steady state behavior at the all loca-

tion in the flow field and referring to Figure 1, the en-

ergy balance implies that the heat transfer from the flue

gas to the condensation film plus the latent heat of con-

densation should be equal to the heat which is passing

through the condensation film. As the temperature level

is the place that condenser is going to be used is some-

thing about 100˚C (more or less), the radiation from flue

gas to the film can be neglected.

The first attempt to analyze the film condensation

process was done by Nusselt with some simplifying as-

sumptions like laminar film flow, stationary vapour and

conduction heat transfer trough the film. Based on the

analysis, he proposed the following correlation for the

mean value of heat transfer coefficient over the whole

vertical surface [12]:



0.943 ff gfgf

film

ghk

 

f filmS























A number of attempts have made to improve the Nus-

selt theory and modify it to suit the real pro

condensate in a real condensation process always is

(5)

cess. The

oled down further to a temperature which is less than

saturation temperature and higher than the cooling wall

temperature. Rohsenow [13] showed that this effect can

be accounted by using modified latent heat defined as:

For the water vapour content in the flue gas, there is

superheat state and the vapor should be cooled down to

Tsat before the condensation occurs. In this case the

modified latent heat can be written as:



0.63

film filmS

fg fg

CTT

hh h

 









CTT

 









(7)

Considering the definition for modified late

convective heat transfer coefficient through the con-

densing film (hfilm) can be written as [4]:

M (kg/kmol)

nt heat, the

Table 1. Required data for diffusivity calculation [10].

Species



/kB (K)



(Å)

Air 78.6 3.711 28.96

HO 363 2.655 18.02

CO2 195.2 3.941 44.01

184 H. NABATI



0.94 ff fgf

ilm

TTL















(8)

All the liquid properties should be calculated at aver-

age temperature (Tsat + Ts)/2.

The heat transfer from flue gas to interface

tween flue gas and condensing film consists of two parts:

fir

3gh





at S









layer be-

stly, heat which passes the diffusion layer directly and

reach to the interface (sensible heat) and secondly the

latent heat of the vapour which reaches the interface and

condenses. With an assumption that the liquid-vapor

interface has a temperature between saturation and sur-

face temperatures and also considering the constant

temperature for the liquid film interface, the energy bal-

ance equation is written as following:



totfilmconv condens

totsff s

 

cond f

T hTT

 

  

qqq q

hTThTT





 (9)

And condensation film flow rate is given by:





condfgfsfs

mhhATT



 

 (10)

sfer coef-

fic der-

ing the mass transfer and is calculated

automatically using wall functions. So it is

2] and is simple to be implemented in the numerical

our atmos-

here using the saturation condition is calculated and

h∞ in (9) represents the convective heat tran

ient between flue gas and condensation film consi

in the CFD code

not explained

re and details can be found in the user guide [13]. The

last coefficient in (9) is the most challenging part in

condensation modeling. In the current paper, two differ-

ent condensation models are implemented and compared

together which are explained in the following sections.

2.1. Model I: Based on Nusselt Theory

The first condensation model is based on Nusselt theory

code Firstly, the heat transfer in a pure vap

then effect of non-condensable gas is introduced by a

degradation factor. The following formula is derived

with this methodology:



 

ff gfg

condncsat f

ghk

qf TT

TTL

 









 







(11)

1 0.9464.9894.135

1 15.48

ncncnc nc





 















(12)

The condensation flow rate is calculated using the

modified latent heat,



:

cond fgcond

mhq





 (13)

2.2. Model II: Based on Diffusion Boundary

Layer Theory

nsfoefficient (h) is given

by following correlation [4]:

This model is originally developed by Peterson [15]. The

condensation heat traer ccondens

,ABfgp gv

cond

hLR

DhC M







 













(14)

ShL is the average Sherwood number for the whole

condensing plate which is given by [4]:

0.664Sh R

e Sc (15)



issteam and non-condensable gas conc

the vicinity of interface and defined as [4]:

entration in

ln 1

nc mean

nc int

nc mean















,ln

erface

steam meannc







(16)

3.1. Governing Equations

The governing equations for conversation of mass, mo-

mentum and energy given by [17] can be formulated by

sing the tensor notation. The flue gas mixture and con-

Newtonian fluids. The

ployed to solve the proc-

, momentum and conti-

ies are solved in a struc-

,mean 



nc interface



3. Numerical Approach

densation layer are considered as

luent© CFD code has been emF

ess governing equations. Energy

uity equations for each specn

atf 

The degradation factor is calculated based on experi-

mental works and here the following correlation is im-

plemented [15]:

tured or unstructured mesh using finite volume method.

Most of the following explanations on governing equa-

tions have been derived from available literatures [14],

[17].

3.1.1. Continuity Equa ti o n

If conservation equation is applied to a species and then

rearranged in a general form, the following equation is

obtained:

H. NABATI

185

 

iiii

yy S





J (17)

Si represents mass source term for species i and calcu-

volume. Diffusive flux for lated based on mixture unit

each species i

was previously presented by (3).



qqqqqpq

 





u (18)

p

where αq represents volume fraction of qth phase:

Volume of the phase in a celldomain

Volume of the celldomain



 (19)

3.1.2. Moment um Equation

This equation is based on second law of N

represents that for each fluid particle, rate of momentum

Transient term + Convection term = Pressure force +

Body force + Shear force + Inter-phase forces

mentum exchange + other external forces

Or in the mathematical form:

ewton and

change is equal to sum of all forces on that particle

and mo-

 



q qqqqqq

qqq pqpqqqqq

pRm

 

 

 







uuu

guF

(20)

1p



 is the mass transfer rate from phase q to phase p.

The momentum equation can be simplified for different

flons.

3.1.3. Multiphase Species Transport Equation

General multiphase species transport equation for s

i which belongs to mixture of the qth phase is expressed

as following:

ws based on the flow conditio

pecies

 



qqq qqqq

 

 

u

(21)

1pq q p

p

ij ji

qq qi



 



3.1.4. Energy Equation

The energy conservation equation is based on enthalpy

equations for all phases. This equation is representing in

where the i

y represents the mass fraction of the spe-

cies i in the qth phase and Si is the volumetric rate of

mass increase (could be also a negative value) for com-

ponent i.

the followinform [14]:

 



qqpqpqpqqpqp

mhmh





 



(22)

qqqqq q

 







qq q

Here hq is the specific enthalpy of the qth phase and



is the phase stress-strain tensor. Explanations on

other terms in energy equ

ation can be found in [14].

3.1.5. Mass Transfer Consid er ations

In the Fluent©, contributions due to mass transfer are

added only to the momentum, species and energy equa-

tions and no source term is added to other scalars

in-

ut the appropriate model for condensation and sink of

tion.

The later model has been used to primary study how

model output. Fins are con-

idered as a simple rectangles attached to the cooling

dary layer section. Two adiabatic inlet and

rbulence [14]. More detail can be found in the Fluent

ebsite or user manuals. No general model exist for

ss transfer and it depends on the case like evaporation,

boiling or condensation, The UDF should be used to

mass is imposed into the continuity equa

3.2. Simulation Model

Two 2D models were used to study the described water

vapour condensation model, when mixture contains high

CO2 concentration (Figure 2). First model is a simple

vertical plate and the second one is a vertical plate with

same dimension as the first one which is equipped with

some fins.

fins affect on condensation

wall. Copper was selected as the material for fins. The

generated meshes for both models are structured map

element meshes. Meshes were generated several times to

assure a mesh dependent solution. Also meshes were

refined in the areas close to walls to get accurate results

in the boun

tlet sections were considered to ensure that the correct

flow condition in the condensation zone was achieved.

Figure 2. 2D models that are used for condensation study.

H. NABATI

186

For simulation start up, first the interface temperature

is approximated and condensing film properties and heat

transfer coefficient has been determined in first cells

adjacent to the cooling wall. Condensation mass flow

rate and heat flux are calculated then based on the equa-

tion that described in the former sections. Then the heat

balance calculation is performed and new film interface

temperature is calculated. This procedure continuous

until the good agreement between old and new film in-

terface is achieved.

4. Investigated Cases

Two main cases are considered: Simple flue gas channel

and flue gas channel fitted with pin fins internally. In the

first case, two condensation models are implemented and

compared. The first model is based on Nusselt theor

available correlation from literature

8,18]). Following correlation is proposed by Dehby [8]:

nd the latter is based on diffusion boundary layer theory

described in section 2. Then based on this study, the

selected condensation model is selected and imple-

mented in the second case to study the effect of fins in

thermo-hydraulic behavior of condensation process.

5. Validity of Models

First model was run with a binary mixture of air and wa-

ter vapor. This is done first to validate the model and

compare results with

([



2438 458.3log



 



(23)

Correlation is valid within these ranges:

0.3 m < L < 3.5 m; 1.5 atm. < Ptot < 4.5 atm.;

10˚C < (T∞ – TS) < 50˚C

Heat transfer coefficient dependency on mass fraction

of air at inlet is illustrated in Figure 3. As it is observed,

the trend of both numerical condensation models is simi-

ar to the experimental correlation result. Even thoug

0.05

L3.7 28.7

condens

hP



eling results differ slightly,

firms that presented models

1) Errors produced by measuring system (categorized

as experimental errors).

2) Intrinsic numerical errors caused by computational

procedures such as truncations (modeling errors).

fication and

assumptions (modeling errors).

experimental data and mod

he presented diagram cont

are relatively capable to predict the condensation behav-

ior of such condensers.

Some reasons for slight discrepancies between experi-

mental and modeling results can be identified as follow-

ings:

3) Errors resulted from modeling simpli

Figure 3. Verification of model validity (air/water vapour

mixture).

To minimize the CFD modeling errors, mesh depend-

ency of the solution was examined by solving the flow

and temperature fields for different mesh configurations

made of different cells. These profiles were compared in

several sections for all configurations to be sure that the

maximum difference in the flow field properties between

the coarser and finer meshes are less than 1% and the

final mesh lead to mesh-independent solutions. Based on

these results, the flue gas mixture then changed from

Air/H2O to CO2/H2O and appropriate material properties

supplied to the model to make it more suitable for

Oxy-fuel process.

6. Results and Discussion

Presented results in the current section include outpu

h a validation of reference case to examine the

ondensation models and then it followed by a different

wever the predicted results from

e first condensation model (based on Nusselt theory (or

)) stand higher. Both

odel show that total heat transfer decreases rapidly

data obtained from different simulations. The modeling

started wit

se containing pin-fins. Results obtained for this model

are presented in Figures 4 to 8. Several checks were

performed in order to verify accuracy of the generated

results. The contour plots for velocity, temperature and

pressure were observed separately to confirm that the

results satisfy the boundary conditions and also they are

independent of grid size.

A comparison of two condensation model is presented

in Figure 4. The relation between total heat transfer co-

efficient and CO2 mass fraction are illustrated here.

As the CO2 mass fraction increases, both models intend

to give closer results. Ho

heat and mass transfer analogy

with any increase in CO2 concentration. A rough conclu-

sion from figure is that 1% increase in CO2 mass fraction

decrease the heat transfer coefficient about 1%.

Figure 5 illustrates prediction of condensation rate.

The trend of condensation rate is consistent with theory.

As the CO2 mass fraction increases, the condensation

H. NABATI

187

Figure 4. Comparison of two condensing models in mean

heat transfer coefficient prediction (CO2/H2O mixture).

Figure 5. Comparison of condensation rate for two con-

densation models.

rate decreases more sharply. The reason is that firstl

ecame more difficult.

Results also showed that average heat transfer coeffi-

cient is more sensitive to inlet velocity at low speeds.

Figure 6 shows the results that are obtained with model

based on diffusion boundary layer theory. It can be seen

that at velocities more than 1.5 m/s, the velocity effects

is negligible. However, when the CO2 mass fraction is

lower at the inlet, velocity can be an affecting parameter

as well. The trend was found to be same for lower CO2

fractions.

Effect of flue gas inlet temperature on mean heat

transfer coefficient is depicted in Figure 7. Also here for

higher inlet temperatures, rate of heat transfer coefficien

ature with less heat removal. Subse-

uently the rate of condensation would be higher at

mparison of total heat transfer

there is less water content in the flow gas and secondly

the diffusion of water vapour toward the cooling surface

change is less. It is natural, as water vapour reach to

saturation temper

lower inlet temperature.

Second case that was considered in this study was a

vertical plate which was equipped with some pin fins.

Figure 8 shows the co

efficient in the simple condensing plate and surface

Figure 6. Average heat transfer coefficient vs inlet flue gas.

Figure 7. Relation between inlet temperature and total heat

transfer coefficient.

Figure 8. Comparison of total heat transfer coefficient be-

tween simple case and pin-finn e d surface

ible. It means that models that are

eveloped here are not really accurate to predict the ex-

act heat transfer coefficient in more complicate geome-

tries. The reason is that the model just look into the cell

adjacent to wall and the geometry is not accounted at all.

Especially the correlations that were implemented are

developed for vertical case and some surfaces of pins are

with pin fins. Figure shows that the difference in coeffi-

cient values is neglig

H. NABATI

188

horizontal.

7. Conclusions

Simulating results have been presented for two conden-

sation models and two different geometries. The physics

of the problem and the heat transfer characteristics have

been discussed for these models. The aim was to evalu-

ate numerical modeling capabilities to predict water va-

pour condensation from a flue gas that contains high

concentration of CO2. The results are summarized as

followings:

e to experimen-

ture. However at higher inlet temperatures

nd velocities the sensitivity to these parameters de-

oefficient was estimated by calculat-

ppreciated

1) Both models are capable to predict the trends i

condensation process. However, the model based on

oundary layer theory shows closer valu

tal correlation. The effect of the CO2 presence in the flue

gas as a non-condensable gas was predicted correctly by

both models.

2) Heat transfer coefficient decreases as a consequence

of the increase in CO2 mass fraction for constant wall

temperature as a result of the higher resistance to diffuse

from the flue gas bulk to the boundary layer.

3) The total heat transfer rate depends on inlet velocity

and tempera

creases.

4) Heat transfer c

ing the interface temperature. However, it was found that

it is possible to get approximately same results by as-

suming this temperature equal to wall temperature. This

assumption facilitates the numerical efforts.

5) A brief description of the technical approach that

was implemented for current study is:

quires the Eulerian model. This Eulerian multiphase

model is an advanced model of Fluent and requires quite

a bit of experience to handle. In addition, modification of

these model to suit condensation process, which itself is

a very complex process, would require both, good under-

standing of the physical process and good knowledge of

model inside the Fluent. The accurateness of the nu-

merical modeling results is determined by the empirical

correlations specified to model the condensation process.

In the industry, there is a practice to model the process

with some correlations available in the open literature

and then tweak various parameters to results which are

close to the experimental results. Such a tuning is neces-

sary in numerical modeling as well for most of the cases,

as the general correlations may not yield accurate results

for a specific set up. It is advisable that designing a con-

denser just based on Numerical results may be a difficult

and expensive task.

8. Acknowledgements

Fluent Inc.’S solver capabilities are highly a

d I hereby knowledge use of it in the current paper.

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omenclature

- surface area [m2]

p - specific heat [JkgK–1]

- diameter [m]

- effective mass diffusivity [m2·s–1]

- external force vector

- acceleration of gravity [m.s–2]

- heat transfer coefficient [W·m2·K–1]

- specific enthalpy in energy equation [j·kg–1]

- evaporation latent heat [J·kg–1]

Computational Fluid Dynamics: The Finite Volume

Method,” 2nd Edition, Pearson Educat

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Transfer, Vol. 87, No. 283, 1991, pp. 19-28.

hfg



h - modified latent heat of evaporation (sub-

ffect) [J·kg–1] cooling e

h - modified latent heat of evaporation (sub-

g effect) [J·kg–1]

·s–1]

–1·K–1]

Greek symbols

α - volume fraction



- steam and non-condensable gas concentration



- dynamicviscosity [Pa·s]



- density [kg·m–3]



- mean molecular diameter [Å]



cooling-superheatin

J - total mass flux [kg·m–2

[W.mK - heat conductivity

L - length [m]

·mol–1] M - molecular weight [kg

–1

 - mass flow rate [kg·s]

P - pressure [Pa]

Q - intensity of heat exchange between phases

–1

[j·s ]

R - reaction term in momentum equation

- source term in governing equations

- Schmidt number (Sc =





.d))

- Sherwood number

T - temperature [K]

u - velocity vector

se molar fraction

x - vapour pha

x' - liquid phase molar fraction

y - mass fraction

- phase stress-strain tensor

Subscripts

conv. - convection

g ase

ies

erent phases

s bulk

cond. - condensation

f - liquid phase, film

- gas ph

i - species i

L - dimensional length

Mean - average

nc - non-condensable spec

p,q - representative of diff

s - wall condition

sa - saturation condition

v - vapour

tot - total

∞ - flue ga

Superscripts

ave - averaged

i - species