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Heat Mass Transfer (2007) 43:677–686
DOI 10.1007/s00231-006-0148-0
ORIGINAL
Effect of vapor condensation on forced convection heat transfer
of moistened gas
Yongbin Liang Æ Defu Che Æ Yanbin Kang
Received: 22 November 2005 / Accepted: 16 May 2006 / Published online: 14 June 2006
Springer-Verlag 2006
Abstract The forced convection heat transfer with
water vapor condensation is studied both theoretically
and experimentally when wet flue gas passes downwards through a bank of horizontal tubes. Extraordinarily, discussions are concentrated on the effect of
water vapor condensation on forced convection heat
transfer. In the experiments, the air–steam mixture is
used to simulate the flue gas of a natural gas fired
boiler, and the vapor mass fraction ranges from 3.2 to
12.8%. By theoretical analysis, a new dimensionless
number defined as augmentation factor is derived to
account for the effect of condensation of relatively
small amount of water vapor on convection heat
transfer, and a consequent correlation is proposed
based on the experimental data to describe the combined convection–condensation heat transfer. Good
agreement can be found between the values of the
Nusselt number obtained from the experiments and
calculated by the correlation. The maximum deviation
is within ±6%. The experimental results also shows
that the convection–condensation heat transfer coefficient increases with Reynolds number and bulk vapor
mass fraction, and is 1~3.5 times that of the forced
convection without condensation.
List of symbols
A
heat transfer area (m2)
cp
specific heat at constant pressure (J/(kg K))
Y. Liang Æ D. Che (&) Æ Y. Kang
School of Energy and Power Engineering,
Xi’an Jiaotong University, Xian 710049, China
e-mail: [email protected]
d
D
gm
h
Hfg
m
m¢ ¢
Nu
P
Pr
q
Q
Re
Sc
Sh
T
W
q
tube outer diameter (m)
diffusion coefficient (m2/s)
mass transfer coefficient (kg/(m2 s)
heat transfer coefficient (W/(m2 K))
latent heat of condensation (kJ/kg)
mass flow rate (kg/s)
interfacial condensate flux (kg/(m2 s)
Nusselt number hd/k
pressure (Pa)
Prandtl number
heat flux (W/m2)
heat transferred (W)
Reynolds number (ud/v)
Schmidt number
Sherwood number (gm d/q D)
temperature (C)
mass fraction
density (kg/m3)
Subscripts
b
bulk
c
cooling water
conv convection
cond condensation
g
bulk vapor–gas mixture
i
interface
nc
non-condensable gas
pc
pure component
sat
saturation
tot
total
w
outer tube wall
v
vapor
123
678
Heat Mass Transfer (2007) 43:677–686
1 Introduction
The energy loss in a boiler is mainly due to the heat
released by the exhaust flue gas to the atmosphere. For
a conventional boiler, in order to avoid low temperature corrosion, the exit flue gas temperature is usually
higher than 150C, sometimes as high as 200C. At
such temperatures, the water vapor entrained in the
exhaust flue gases does not condense, so that the released heat consists of both sensible heat and latent
heat. Recently, for the purpose of environmental protection, the Chinese government has enforced the energy policy that will lead to a larger market share of gas
fired boilers in China. As fuel gas, natural gas in particular, contains more hydrogen atoms instead of carbon, compared to anthracite or bituminous coal, larger
fraction of water vapor in combustion products results.
Generally the volumetric fraction of the water vapor in
flue gases can be in the range of 10–20%, large amount
of latent heat of vaporization is lost to the atmosphere.
Figure 1 shows the theoretical relationship of boiler
efficiency based on lower heating value versus exit flue
gas temperature. It can easily be seen that after a relatively gradual change in the temperature range of 60–
180C, the boiler efficiency rises sharply at the dew
point (about 60C) and a value larger than 100% can
be attained at very low exhaust gas temperatures [1].
As such, it is profitable to recover the latent heat by
condensing as much water vapor as possible in the flue
gases from a gas fired boiler.
There is a large amount of literature available on
condensation, but the literature directly related to the
content in this paper is few. Literature survey shows
that the condensation of vapor in vapor–gas mixture
was studied long time ago, but all the investigations
hM
1:17
¼ 0:11 ;
hNu C
ð1Þ
where C is the air mass fraction and ranges from 0.002
to 0.04.
Later, Hampson [3] studied the condensation of
steam on a vertical plate, using nitrogen as the noncondensable gas instead of air, and suggested that the
heat transfer ratio should vary as:
hM
¼ 1:2 20:0X;
hNu
ð2Þ
where the weight ratio X of nitrogen to steam ranges
from 0 to 0.02. In terms of the data obtained for steam–
air and toluene–nitrogen mixtures condensing on a
horizontal pipe, Henderson and Marchello [4] proposed the ratio in the following form:
hM
1
¼H¼
:
1 þ C 1 y þ C 2 y2
hNu
108
Boiler efficiency, %
(Based on lower heating value)
conducted were on the condensation in the presence of
a relatively small amount of non-condensable gases
(approximately £20%), with the aim being at clarifying
the effect of non-condensable gases during condensation.
In the investigations of vapor condensation in the
presence of non-condensable gases, it is a common
practice to modify the classical Nusselt equation or
other simplified equation by introducing a correction
factor that takes into account the effect of non-condensable gases. Additionally, several empirical correlations and mechanistic models were derived [2–6].
Meisenburg [2], who investigated the effect of noncondensable gases on the condensation of steam on a
vertical tube, found that the ratio of observed heat
transfer coefficient, hM, to the heat transfer coefficient
predicted by the Nusselt equation, hNu, could be described by the following equation:
ð3Þ
106
Excess air ratio =1.05
104
102
100
Dewpoint
98
96
94
92
90
88
0
20
40
60 80 100 120 140 160 180 200
Exit flue gas temperature,°C
Fig. 1 Relationship of boiler efficiency versus exit flue gas
temperature
123
It can be seen that the treatment result in a polynomial expression for H in terms of the molar percentage of gas y. The experimental data were fitted by
least square method to obtain the constants C1 and C2.
Abdullah et al. [5] studied the condensation of
steam and R113 on a bank of horizontal tubes in the
presence of a non-condensable gas. They concluded
that the detrimental effects on heat transfer were
due to the reduction in vapor velocity and the
presence of air (non-condensable gas). To take into
account these adverse effects, an expression based
on Nusselt equation for vapor side was derived as
follows:
Heat Mass Transfer (2007) 43:677–686
Nu ¼ a
q2L ghfg d3
lL kL DT
679
1=4
ð4Þ
where a was a constant to modify the Nusselt equation
and was determined by experiments.
Groff et al. [6] developed a correlation for the
Nusselt number during laminar, forced-convection film
condensation from steam–air and steam–hydrogen
mixtures on horizontal isothermal plates, which is:
Nux
pffiffiffiffiffiffiffiffiffiffiffi ¼
Rex;L
Nux
pffiffiffiffiffiffiffiffiffiffiffi
Rex;L
!
Gi ;
ð5Þ
0
pffiffiffiffiffiffiffiffiffiffiffi
where Nux = Rex;L 0 was the value for pure vapor at
the same T¥ and DT. Gi, which accounts for the heat
transfer reduction due to the presence of the noncondensable gas, is a function of T¥, W¥ and DT.
For the wet flue gas from a gas fired boiler, the
water vapor volume fraction is only about 10–20%
and the non-condensable gas mass fraction is lager
than 80%. When latent heat recovery is performed,
the correlations mentioned above are inappropriate.
According to previous studies [7–12], the heat transfer
coefficient in this situation is higher than that of the
forced convection without condensation, but on the
same order, and much lower than that of the condensation of pure steam. Therefore, the effect of
water vapor condensation in the wet flue gas on
convection heat transfer, rather than the effect of
non-condensable gases during condensation, should
be investigated. Unfortunately, there is a great
shortage of investigations on this issue.
By introducing the Colburn-Hougen method, Che
et al. [8] analyzed the heat and mass transfer process
of high moisture flue gases flowing downward in a
bank of horizontal carbon steel tubes. They indicated
that the influential factors of convective-condensation
coefficient include the partial pressure of the vapor
pv, the temperature of the outer tube wall Tw, the
mixture temperature Tg, Re and Pr. Based on
dimensional analysis, a dimensionless number Ch
called condensation heat transfer factor was obtained
to describe the effect of vapor condensation, which is
expressed as:
Ch ¼
TsatðpvÞ Tw
;
Tg Tw
ð6Þ
where Tsat(pv) is the saturation temperature corresponding to the partial pressure of vapor pv. Finally, a
normalized correlation of convection–condensation
heat transfer was derived and expressed as:
Nuh ¼ aReb Pr1=3 Chc :
ð7Þ
But there is a limitation of the factor Ch, i.e., a
contradictory physical meaning will be induced for Ch,
if the vapor fraction is extremely small. Whenever the
vapor fraction is vanishing, the value of Tsat(pv) will be
approaching even less than zero, and the value of
Tsat(pv) – Tw can be minus. Therefore, Ch is only
applicable to relatively high vapor fraction.
In this paper, an in-depth research is conducted both
theoretically and experimentally to investigate the
forced convection condensation on a bank of horizontal tubes with vertical downflow of steam–air mixture. The effect of vapor condensation is considered.
Firstly, by theoretical analysis, a new dimensionless
number defined as augmentation factor is obtained to
take into account the effect of condensation of relatively small amount of water vapor on convection heat
transfer, and a consequent correlation is derived to
describe the combined convection–condensation heat
transfer. Secondly, experiments have been conducted
to determine the unknown constants in the correlation,
and to investigate the effects of various influential
factors. The parametric study is conducted by varying
the moistened air flow rate (Re) and temperature, the
vapor mass fraction in the free mixture stream, the
cooling water mass flow rate and temperature and
the tube wall temperature. During the experiments, the
air–steam mixture is used to simulate the flue gas of a
natural gas fired boiler, the vapor mass fraction ranges
from 3.2 to 12.8%.
2 Theoretical analysis
2.1 Fundamentals of heat and mass transfer
There is a great difference between the condensation
process in the presence of a non-condensable gas and
that of pure vapor. In the analysis on the condensation
in the presence of non-condensable gases, Colburn and
Hougen [13] proposed that condensation mass transfer
is controlled by diffusion across a thin film at which the
accumulation of non-condensable gas occurs (noncondensable gas boundary layer). The temperature and
pressure distributions of both situations are illustrated
in Fig. 2. The solid and dashed lines represent the vapor–liquid interfaces in the cases of the condensation
of pure vapor and the condensation of the vapor with
non-condensables, respectively.
When non-condensable gases are absent, the bulk
vapor temperature Tb,pc and the liquid–vapor interface
123
680
Heat Mass Transfer (2007) 43:677–686
Non-condensable
Interface
gas boundary layer (pure component)
Pure component
Tb,pc= Tsat
Ti,pc
Coolant
side
Tw,pc
Ti
Tg
With non-condensables
Ptot
Tc
Tw
Qtotal
Tube
wall
Pi,v
Pi,nc
Pb,v
In Eq. 9, hcv is the forced convection heat transfer
coefficient due to sensible heat; in Eq. 10, m¢ ¢cond is the
interfacial condensate flux, Hfg is the latent heat of
vaporization of the vapor under the corresponding
partial pressure.
According to Newton’s law of cooling, the overall
heat transfer coefficient hoverall can be determined by
the following.
hoverall ðTg Ti Þ ¼ hcv ðTg Ti Þ þ m00 cond Hfg :
Pb,nc
Interface
(with non-condensables)
Qconv
Qcond
As we know, forced convection heat transfer and
mass transfer can be described by the Nusselt number
Nu and the Sherwood number Sh as:
Nu ¼ CRem Pr1=3
Condensate film
ð11Þ
0
ð12Þ
Fig. 2 Boundary layer temperature and pressure distributions
Sh ¼ C0 Rem Sc1=3 ;
temperature Ti,pc are identical and equal to the saturation temperature of pure vapor Tsat; when non-condensable gases exist, there is a drop between the vapor
saturation temperature in the bulk and the temperature at the liquid–vapor interface, which is due to the
reduction in vapor partial pressure (Pb,v > Pi,v) and can
be considered as a penalty in the condensation process.
It can be imagined that the vapor must diffuse through
the non-condensable gas boundary layer for it to attain
the liquid–vapor interface, so that the vapor concentration difference between the bulk and the interface is
the driving force of mass transfer. In addition, there is a
drop from bulk gas temperature Tg to the interfacial
temperature Ti, which is the driving force of convection
heat transfer.
Based on this model, a heat balance at the liquid–gas
interface between the heat transfer through the noncondensable gas boundary layer and the heat transfer
through the condensate film can be obtained. As the
heat transfer though non-condensable gas boundary
layer consists of two components: the sensible heat flux
qconv due to temperature difference between bulk vapor–gas mixture and interface, and the latent heat flux
qcond given up by the condensing vapor, therefore
where Re=ud/v is the Reynolds number.
In the investigation on forced convection heat
transfer inside tube or over tube bundles, it is a
common practice to define the Reynolds number by
taking the tube diameter or equivalent diameter as
the characteristic dimension, regardless of whether or
not the fluid flow is one-dimensional or two-dimensional. This definition has been used by many
scholars. For example, Zukauskas [14] and Grimson
[15] used the definition of the Reynolds number
based on the tube diameter when they conducted
their study on forced convection heat transfer on
horizontal tube bundles. For a given process, if the
boundary conditions are same, there is an analogy
between the heat transfer and the mass transfer [16,
17]. Thus the coefficients and the exponents in
Eqs. 12 and 13 should be identical, i.e., C=C¢, m=m¢.
Therefore
qtot ¼ qconv þ qcond ;
where h and gm are the heat and mass transfer coefficients, respectively, Le=Pr/Sc=a/D is known as the
Lewis number [18].
ð8Þ
where qconv and qcond can be expressed, respectively, as
follows:
qconv ¼ hcv ðTg Ti Þ
qcond ¼ m00 cond Hfg :
123
ð9Þ
ð10Þ
ð13Þ
Nu
¼
Sh
1=3
Pr
Sc
ð14Þ
h
¼
gm
1=3
a 2=3
D
k
¼ qcp
¼ qcp Le2=3 ;
a
D
D
ð15Þ
2.2 Convection–condensation heat transfer
As mentioned above, the vapor must diffuse through
the non-condensable gas boundary layer to reach the
Heat Mass Transfer (2007) 43:677–686
681
liquid–vapor interface, so that a mass balance at the
interface yields the following equations:
m00 cond
@Wv
¼ qD
þ Wv;i ðm00 tot Þi
@y i
ð16Þ
m00 nc ¼
@Wnc
qD
þ Wnc;i ðm00 tot Þi ;
@y i
ð17Þ
where (m¢ ¢tot)i is an imaginary compensatory mixture
flow at the interface to account for the situation that
the diffusion process is in one direction only, Wv,i and
Wnc,i are the vapor and gas mass fraction at the interface, respectively.
As the condensate surface is impermeable to the
non-condensable gases, the value of m¢ ¢nc should be
zero, and Eq. 17 can be simplified as
@Wnc
qD
¼ Wnc;i ðm00 tot Þi :
@y i
ð18Þ
In this study, the mixture is composed of two components: condensing vapor and non-condensables, thus
the following correlation can be obtained
Wnc þ Wv ¼ 1:
ð19Þ
Therefore
@Wnc @Wv
þ
¼ 0:
@y
@y
m
cond
hcd ¼
m00 cond Hfg
1
¼ gm B Hfg :
ðTg Ti Þ
ðTg Ti Þ
By using the definition of the Nusselt number,
Nucd=hcd d/k, Eq. 24 can be rewritten as
Nucd ¼ gm B Hfg d
:
ðTg Ti Þk
By using the definition of the Sherwood number,
Sh=gm d/q D, Eq. 25 can be written as the following
dimensionless form
Nucd ¼ Sh B Hfg
Pr
;
Sc Cp ðTg Ti Þ
m00cond
¼
1 Wv;i
ð26Þ
where Pr/Sc is the Lewis number, represented by Le,
the group Hfg /Cp (Tg – Ti) is the ratio of the phase
change energy to the sensible heat transfer, and is
known as the Jakob number, represented by Ja [18].
According to the work of Osakabe [12], under the
present situation, the boundary conditions are same,
and the heat and mass transfer are analogical, therefore the expressions are identical, i.e., the relevant
coefficients and exponents in Eqs. 12 and 13 should be
the same. Substituting Eq. 13 into the above equation
yields the following correlation
ð21Þ
Nuoverall ¼
hoverall d ðhcv þ hcd Þd
¼
¼ Nucv þ Nucd :
k
k
ð28Þ
i
ð27Þ
The Nusselt number for the overall heat transfer
coefficient of both sensible and latent components can
be obtained as
Simplifying Eq. 21 gives
v
qD @W
@y
ð25Þ
Nucd ¼ CRem Pr1=3 Le2=3 B Ja:
@Wv
qD @Wv
¼ qD
þ Wv;i :
Wnc;i @y i
@y i
ð24Þ
ð20Þ
Combining Eqs. 16, 18 and 20 and rearranging yield
the equation of condensate flux at the interface as
follows:
00
Therefore
From Eqs. 12 and 27, Eq. 28 can be rewritten as
ðWv;b Wv;i Þ
¼ gm B
¼ gm
ð1 Wv;i Þ
ð22Þ
where gm is the mass transfer coefficient, and
B=(Wv,b – Wv,i)/(1 – Wv,i) is the driving force of mass
transfer.
The condensation heat transfer coefficient hcd
accounting for the latent heat transfer can be defined
as follows:
hcd ðTg Ti Þ ¼ m00 cond Hfg
ð23Þ
Nuoverall ¼ CRem Pr1=3 ð1 þ Le2=3 BJaÞ:
ð29Þ
Equation 29 is the normalized correlation of convection–condensation heat transfer, in which C and
m are constants to be determined by experimental
data.
The Prandtl number Pr and the Schmidt number Sc
in the investigated range can be considered to be
invariant, and calculations have shown that Pr 0.696
and Sc 0.670, therefore, Le=Pr/Sc 1.04. On the
123
682
Heat Mass Transfer (2007) 43:677–686
other hand, as the non-condensable gas mass fraction is
above 80% in the range investigated in this paper,
according to previous studies [11], the thickness of the
condensate film is very thin and the heat resistance of
the condensate film is negligible, the tube wall temperature can be considered to be equal to the temperature at the interface, i.e., Ti=Tw.
The work of Osakabe [12] shows that under the
present situation the thickness of the condensate film is
within the range of 0.04–0.17 mm, and it is also indicated that the temperature difference across the film is
less than 2.5 K, the treatment to neglect the condensate film will lead to very small deviation. Some other
researchers, such as Jia et al. [9], Cao [7], used this
treatment as well, and good performance could be
found.
A qualitative analysis on Eq. 29 shows that if the
vapor mass fraction is zero, i.e., Wv,b=Wv,i=0,
B=(Wv,b – Wv,i)/(1 – Wv,i) will be equal to zero, and
(1+Le2/3 BJa) is reduced to unit. Consequently,
Eq. 29 is reduced to the form of Nuoverall=CRem
Pr1/3, which is the formula for single-phase convection heat transfer (without condensation). By an
in-depth analysis, (1+Le2/3 BJa) can be considered as
an augmentation factor to account for the enhancement effect induced by the condensation of water
vapor on forced convection heat transfer. In the
condensation process, the mass fraction Wv,i at the
interface is maintained to be equal to the value at
the saturated gas state corresponding to the interface
temperature Ti.
10 T
I-8
5
T
I-7
6
E-20
E-8
3
E-21
E-12
TI-9
12
Figure 3 is the schematic of the experimental apparatus.
As shown in Fig. 3, air is blown by forced draft fan
into the channel in which electric heaters are located.
After preheated in the primary heater (right half of the
channel), the air meets the steam supplied from an
electrically heated boiler at the middle of the channel.
The air–steam mixture flows over the secondary heater
(left half of the channel) where it is heated to the
temperature as required, and then flow downwards
though the heat exchanger where condensation occurs.
After the steam entrained in the gas is condensed, latent heat is released and reclaimed. The condensed
water is collected by a tray and is weighed. The cooling
water flows into the heat exchanger from bottom, in a
counter-flow arrangement with the air–steam mixture
stream. The flow rate of air, steam and cooling water
are changed by regulating valves and is measured by
pitot tube flow meter, vortex shedding flow meter and
rotameter, respectively. The room temperature, the
moistened gas stream temperature, the condensing
heat exchanger tube wall temperature and the cooling
water temperature are measured by nickel chromium–
nickel silicon thermocouples for its high sensitivity and
good thermo-electrical performance. The thermocouple signals are transferred to a personal computer for
analysis. For safety, electrical power is adjusted by
voltage regulator during the operation.
In the experiments, carbon steel tubes are employed. Show in Fig. 3 is the arrangement of test tubes.
The heat exchanger consists of an in-line tube bank of
3 rows and 60 stages in total. The detailed geometrical
parameters are listed in Table 1.
2
The parametric study is conducted by varying the
moistened air flow rate (Re) and temperature, the vapor mass fraction in the free mixture stream, the
cooling water mass flow rate and temperature and the
E-7
FI
8
3.1 Test rig
3.2 Measurement and data reduction
4
E-9
3 Experimental
9
11
1
Table 1 Geometrical parameters of heat exchanger
7
Fig. 3 Experimental apparatus. 1 Forced draft fan; 2 Pitot tube
flow meter; 3 Electric boiler; 4 Vortex shedding flow meter; 5
Heating up section; 6 Condensing heat exchanger; 7 Water
pump; 8 Rotameter; 9 Condensed water collecting pan; 10
Thermocouples; 11 Transformer; 12 Electrical heater
123
Out side diameter
Wall thickness
Cross-section
Transverse pitch
Longitudinal pitch
Number of tubes
20 mm
2 mm
500 mm · 120 mm
40 mm
40 mm
180
Heat Mass Transfer (2007) 43:677–686
683
Table 2 Experimental ranges and uncertainties
4.6
Uncertainty
Range
Coolant flow rate
Moistened air velocity
Free stream vapor mass fraction
Inlet coolant temperature
Outlet coolant temperature
Inlet moistened air stream
temperature
Tube wall temperature
0.5 m3/h
0.4 m/s
0.3%
0.1C
0.1C
0.1C
1–4.5 m3/h
4–7 m/s
3.2–12.8%
10.7–15.1C
17.7–29.7C
106.2–154.9C
0.1C
12.3–42.1C
(1+Le2/3BJa)=1.67
4.4
lnNuoverall
Parameter
4.2
4.0
3.8
tube wall temperature. The ranges and uncertainties of
the measurements are given in Table 2.
During steady operations, the total heat Qtot transferred from the steam–air mixture is equal to the heat
absorbed by the cooling water and can be calculated by
Qtot ¼ mc cp;c ðT 0 c T 00 c Þ;
ð30Þ
where mc is the mass flow rate of the cooling water, cp,c
is the specific heat of cooling water, T¢c and T¢ ¢c are the
outlet and inlet temperatures of the cooling water,
respectively.
The overall heat transfer coefficient for convection–
condensation heat transfer is obtained by:
hoverall ¼
Qtot
w Þ:
AðTg T
ð31Þ
w are the mean temperatures of steam–
g and T
where T
air mixture and outer tube wall, respectively.
Wilke method [19] and Lindsay method [20] are
used to calculate the viscosity and heat conductivity of
the mixture, respectively. Other physical properties
such as density can be obtained in terms of the law of
mixed gases. The method of Kline and McClintock [21]
is employed to evaluate the uncertainties of the
experimental results. The maximum uncertainty in the
Reynolds number was estimated to be 8% and the
uncertainty in the Nusselt number was 8.5% and the
maximum uncertainty in the total heat transfer was
8.3%.
4 Results and discussions
Experiments have been performed to examine the effects of various factors, including mixture flow rate
(Reynolds number Re), mass fraction of water vapor
Wv,b etc, on the convection–condensation heat transfer
coefficient.
Based on the experimental data, the effects of the
Reynolds number Re and the augmentation factor are
discussed. The relationship of the Nusselt number
3.6
8.0
8.2
8.4
lnRe
8.6
8.8
Fig. 4 Relationship between Nuoverall and Re
Nuoverall versus Re and (1+Le2/3 BJa) are shown in
Figs. 4 and 5. It can be seen that Nuoverall is increased
with Re and (1+Le2/3 BJa).
If Re is raised, the sensible heat transfer is enhanced.
From Eq. 29, it can be found that for a given value of
(1+Le2/3 BJa), the increase of sensible heat transfer
leads to the increase of total convection–condensation
heat transfer.
In the expression of the augmentation factor
(1+Le2/3 BJa), Ja=Hfg /Cp (Tb – Ti) is the ratio of the
phase change energy to the sensible heat transfer;
B=(Wv,b – Wv,i)/(1 – Wv,i) is the driving force of mass
transfer, and its value increases as the vapor mass
fraction increases. In the case of higher vapor mass
fraction (i.e., higher value of B), more water vapor can
be condensed and more latent heat can be released, the
overall heat transfer coefficient is higher. Therefore,
higher value of (1+Le2/3 BJa) means higher value of
Nuoverall. Figure 6 shows the relationship between
Nuoverall and Wv,b. It can be found that Nuoverall is
increased with increased Wv,b, which agrees well to the
above discussions.
A comparison can be made between the condensation factor Ch=(Tsat – Tw)/(Tg – Tw) proposed in [8]
and (1+Le2/3 BJa). In Ch, the numerator Tsat – Tw is
the deference between the saturation temperature
corresponding to the partial pressure of water vapor
and the tube wall temperature, i.e., the tube wall
subcooling, whose physical meaning is the driving force
of mass transfer. It has the same meaning as B in
(1+Le2/3 BJa). The values of both Hfg and Cp in the
experimental range vary little and can be considered
to be constant, thus Ja is a function of Tg – Tw only,
which can be reduced to the denominator in Ch.
Therefore, generally, it can be concluded that Ch and
(1+Le2/3 BJa) has the same physical meaning.
123
684
Heat Mass Transfer (2007) 43:677–686
140
2.6
2.4
Re=5600
Re=5600
100
2.2
(1+Le2/3BJa)
Nusselt number Nuoverall
120
80
60
40
2.0
1.8
1.6
1.4
20
1.2
0
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
0.02
0.04
(1+Le2/3BJa)
Fig. 5 Relationship between Nuoverall and (1+Le2/3 BJa)
ð32Þ
Nusselt number Nuoverall
Re=5600
120
100
80
60
40
0.02
0.04
0.06
0.08
0.10
0.12
Water vapor mass fraction Wv,b
Fig. 6 Relationship between Nuoverall and Wv,b
123
Equation 32 is applicable to the parametric range as
follows:
0.032 < Wv,b < 0.128
3,900 < Re < 7,300
1.21 < (1+Le2/3 BJa) < 2.57.
This paper mainly investigates the heat and mass
transfer of high moistened gas in the condensing heat
exchanger of a condensing boiler. Therefore, the
velocity corresponding to the chosen Re falls in the
range recommended in boiler design, and the correlation can be applied to the design practice of condensing
boiler. Of course, more experimental data will be
needed to apply the correlation in a wider range of Re.
Figure 8 shows the comparison of the Nusselt
numbers calculated by Eq. 32 with those obtained by
Nusselt number Nuoverall (Experimental)
160
140
0.14
Fig. 7 Relationship of (1+Le2/3 BJa) versus Wv,b
But there is a disadvantage for the condensation
factor Ch. As mentioned above, when the vapor mass
fraction is extremely small, the value of Ch will be zero
or be minus. An improvement of the augmentation
factor lies in that it overcomes the disadvantage of Ch.
Shown in Fig. 7 is the relationship of (1+Le2/3 BJa) and
Wv,b, it can be seen that (1+Le2/3 BJa) decreases continuously as Wv,b decreases and approaches 1.0 at very
low Wv,b.
By employing the multivariate regression method,
the unknown constants in Eq. 29 can be determined by
experimental data, and the final correlation is obtained
as follows
Nuoverall ¼ 0:0641Re0:772 Pr1=3 ð1 þ Le2=3 BJaÞ:
0.06
0.08
0.10
0.12
Water vapor mass fraction Wv,b
0.14
140
120
100
80
60
40
40
60
80
100
120
Nusselt number Nuoverall (Calculated)
Fig. 8 Calculated and experimental Nuoverall
140
Heat transfer coefficient hcv, hoverall (W/m2K)
Heat Mass Transfer (2007) 43:677–686
685
180
160
3.
hoverall (Vapor-air Mixture)
hcv (Air)
140
120
100
80
60
40
20
0
0.02
0.04
0.06
0.08
0.10
0.12
Water vapor mass fraction Wv,b
0.14
condensable gases on the condensation of vapor,
should be emphasized.
By theoretical analysis, a new dimensionless number defined as augmentation factor (1+Le2/3 BJa) is
obtained to take into account the effect of condensation of relatively small amount of water vapor on convection heat transfer. Based on the
experimental data, a simple correlation to describe
the combined convection–condensation heat
transfer is put forward, which is expressed as
Nuoverall=0.0641Re0.772 Pr1/3 (1+Le2/3 BJa) in the
experimental range. Satisfactory agreement has
been obtained, and the maximum deviation is
within ±6%.
Fig. 9 Comparisons of heat transfer characteristic of air and
vapor–air mixture
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± 6%, which shows satisfactory agreement.
The experiments on the heat transfer characteristics
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heat transfer characteristic of vapor–air mixture, as
shown in Fig. 9. The experimental conditions are
Re=5,600, Pr=0.696, (1+Le2/3 BJa)=1.59.
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5 Conclusions
1.
2.
The convection–condensation heat transfer coefficient increases as the gas–vapor mixture flow rate
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of vapor condensation on the convection heat
transfer, instead of the detrimental effect of non-
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