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The energy balance within a bubble column evaporator
Article in Heat and Mass Transfer · May 2018
DOI: 10.1007/s00231-017-2234-x
3 authors, including:
Chao Fan
Muhammad Shahid
UNSW Sydney
Australian Defence Force Academy
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The energy balance within a bubble column
Chao Fan, Muhammad Shahid &
Richard M. Pashley
Heat and Mass Transfer
Wärme- und Stoffübertragung
ISSN 0947-7411
Heat Mass Transfer
DOI 10.1007/s00231-017-2234-x
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Author's personal copy
Heat and Mass Transfer
The energy balance within a bubble column evaporator
Chao Fan 1,2 & Muhammad Shahid 1 & Richard M. Pashley 1
Received: 26 January 2017 / Accepted: 14 November 2017
# Springer-Verlag GmbH Germany, part of Springer Nature 2017
Bubble column evaporator (BCE) systems have been studied and developed for many applications, such as thermal desalination,
sterilization, evaporative cooling and controlled precipitation. The heat supplied from warm/hot dry bubbles is to vaporize the
water in various salt solutions until the solution temperature reaches steady state, which was derived into the energy balance of the
BCE. The energy balance and utilization involved in each BCE process form the fundamental theory of these applications. More
importantly, it opened a new field for the thermodynamics study in the form of heat and vapor transfer in the bubbles. In this
paper, the originally derived energy balance was reviewed on the basis of its physics in the BCE process and compared with new
proposed energy balance equations in terms of obtained the enthalpy of vaporization (ΔHvap) values of salt solutions from BCE
experiments. Based on the analysis of derivation and ΔHvap values comparison, it is demonstrated that the original balance
equation has high accuracy and precision, within 2% over 19–55 °C using improved systems. Also, the experimental and
theoretical techniques used for determining ΔHvap values of salt solutions were reviewed for the operation conditions and their
accuracies compared to the literature data. The BCE method, as one of the most simple and accurate techniques, offers a novel
way to determine ΔHvap values of salt solutions based on its energy balance equation, which had error less than 3%. The thermal
energy required to heat the inlet gas, the energy used for water evaporation in the BCE and the energy conserved from water vapor
condensation were estimated in an overall energy balance analysis. The good agreement observed between input and potential
vapor condensation energy illustrates the efficiency of the BCE system. Typical energy consumption levels for thermal desalination for producing pure water using the BCE process was also analyzed for different inlet air temperatures, and indicated the
better energy efficiency, of 7.55 kW·h per m3 of pure water, compared to traditional thermal desalination techniques.
C gp
W Taire
Enthalpy of vaporization
Specific heat capacity of air in units of J·m−3 K−1 under constant
Steady state temperature near the top of the column in the units
of K
water vapor density
Temperature difference between the gas entering and leaving the
column in the units of K
Differential pressure, between the gas inlet into the sinter and
atmospheric pressure at the top of the column
Specific heat capacity of air in units of J·g−1 K−1 under constant
Energy used for the evaporation of the water in the solution
Volume of the bubble at temperature T
Energy supplied by the bubble from Tin to Te
Mass of air (or gas) in gram per cubic meter
Moles of air in the bubble
Moles of water vapor in the bubble
Moles of water vaporized into the bubble
Universal gas constant
Absolute temperature
Atmospheric pressure
Water vapor pressure of the solution
Bubble column evaporator
Reverse osmosis
Work done the air pump through the colum
Initial (dry) bubble volume, Vf Final (wet) bubble volume
1 Introduction
* Chao Fan
School of Physical, Environmental and Mathematical Sciences,
University of New South Wales, Canberra 2600, Australia
School of Environmental Science and Engineering, Sun Yat-sen
University, Guangzhou 510275, People’s Republic of China
The bubble column evaporator (BCE) system is a simple and
novel process in which warm/hot gas is pumped through a
multi-porous sinter into a solution-filled column. A high density of uniform bubbles can be produced throughout the column. This BCE system, due to highly efficient heat and rapid
vapor transfer through bubbles contacting the column
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Heat Mass Transfer
solution, has been studied and developed for a wide range of
useful applications [1], such as, evaporative cooling [2], thermal desalination [3–5], thermal sterilization [6–8], solutes
thermolysis in aqueous solution [9] and precise determination
of thermophysical properties of concentrated salt solutions [2,
3, 10]. Although the potential mechanism of each application
has been introduced and studied, the energy balance involved
in the BCE process, as a fundamental aspect for the BCE
application, is important for analysis. Also, the mass and heat
transfer of bubbles are studied in the typical areas of nucleate
boiling [11, 12] and flow dynamics [13, 14], and sub-boiling
system regarding the use of warm/hot bubbles, say the BCE,
also has significant potentials for developing new applications
based on the fundamental theory study.
Some types of concentrated salt solutions have unexpected
effects on the coalescence inhibition of bubbles [15, 16], which
has facilitated the BCE process with the controlled production of a
high density of fine bubbles produced by pumping warm or hot
dry gases through the sinter into a column containing a solute
which inhibits bubble coalescence. These 1–3 mm diameter bubbles, due to coalescence inhibition behavior, become rapidly saturated by water vapor within a few tenths of a second [17], which
cannot be estimated by the Fick’s law [2]. At the same time, the
heat from the bubbles is also transferred for water evaporation,
rather than heating of the column solution after reaching the
steady state condition. The efficient heat and vapor transfer have
been used for the basis of each BCE application, and they encompass the basic physics of bubble rise rates [17–19], vapor collection rate [4] and interactions within the aqueous solution [20, 21].
In this paper, the methodology regarding the BCE experiment procedure and the derived equations was illustrated. The
procedure and derived balances were used to determine
ΔHvap values of various salt solutions, which were analyzed
for the accuracy of the ΔHvap determination and compared
with that from different techniques. Based on the physics analysis and ΔHvap determination of balances in the BCE, it is
established that originally proposed balance has a wide temperature range for determining ΔHvap values of salt solutions
with high accuracy and precision. In addition, the energy consumption using the BCE for desalination was estimated.
2 Methodology
The BCE used in the experiment was supplied with dry air
following passage through a large amount of fresh silica gel to
dehumidify the normal inlet air and an electrical gas heater
was then used to obtain the desired inlet temperature. As warm
or hot dry air was pumped by a HIBLOW air pump into the
column, fine bubbles (around 1–3 mm in diameter) were produced through a glass sinter disc (porosity No.2) and a steady
state column temperature (near the column top) can be
reached when the heat supplied from the temperature change
of the inlet bubbles was balanced by the thermal energy required for evaporation of water vapor from the salt solutions,
assuming no heat transfer with the environment and no heat
loss. The BCE system is shown schematically in Fig. 1 (left
side). The details of the column and a photograph of bubble
column are also given in Fig. 1. In the BCE experiment, the
inlet gas, steady-state column top temperature and the differential pressure between the inlet and outlet gas were measured
to determine ΔHvap values.
Earlier works [2, 3, 10] described how the bubble column
can be maintained at a steady state condition, where the stable
column top temperature is obtained when the heat supplied
from bubbles (per unit volume of dry gas) exactly equals the
enthalpy of vaporization required to reach the equilibrium
water vapor pressure within the bubbles. This method is based
on the steady state volumetric balance in the bubble column,
which has been used for the determination of ΔHvap values of
concentrated salt solutions [3, 10], and is described by the
following Eq. (1):
ΔT C p ðT e Þ þ ΔP ¼ ρv ðT e Þ ΔH vap ðT e Þ
(in units of J·m−3)
where Cp(Te) is the specific heat of the gas flowing into
the bubble column at constant pressure, in units of J·
m−3 K−1; Te is the steady state temperature near the top of
the column in the units of K; ρv is the water vapor density
at Te, in units of mol·m−3, can be calculated from the water
vapor pressure of salt solutions at the steady state temperature, using the ideal gas equation; ΔT is the temperature
difference between the gas entering and leaving the column, in units of K; ΔP is the differential pressure between
the air enters and leaves column, in units of J·m−3, which
represents the work done by the gas flowing into the base
of the column until it is released from the solution.
This equation was first time reported by Francis and
Pashley [2] for low vapor pressures, e.g. low column top temperatures of about 283 K. The equation’s accuracy and precision was further tested in later studies [3, 10], summarized in a
review of BCE applications [1] and used for undergraduate
teaching of ΔHvap [22]. However, apart from the use of this
equation, it is still worth, in details, analyzing the physics
behind this thermodynamic process between the bubble and
solution and comparing alternatively proposed balances.
3 Analysis and discussion
3.1 Analysis of original balance in the BCE
The original equation Eq. (1) was found to be accurate for
determination of ΔHvap values of salt solutions over a wide
range of temperature. Although Eq. (1) is on the basis of per
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Heat Mass Transfer
Fig. 1 Schematic diagram of the BCE system (left side), details of the bubble column (right side, upper) and a photograph of a bubble column for
0.5 mol·kg−1 NaCl (right side, lower)
volume of gas in the bubble column, the temperature change
from the cooling of bubbles does not have effects on the balance. Any given bubble released into the column at high temperature (Tin) will contract until it reaches the column solution
steady state temperature Te. The heat supplied by the bubble
(of volume Vb at temperature T) to the column, as it contracts,
in a very rapid way, is given by the sum of Cp(T) × Vb(T) over
the temperature range from Tin to Te, i.e. over temperature
range of ΔT. As Cp(T) and Vb(T) are both functions of temperature T, the heat Hb, supplied by the bubble to the column,
is given by:
H b ¼ ∫T in C p ðT Þ V b ðT ÞdT
Importantly, Cp(T) is linearly proportional to 1/T (see Fig. 2)
and Vb(T), volume of the bubble, is linearly proportional to T,
according to the ideal gas equation; so the product of Cp(T) ×
Vb(T) is constant, and therefore at steady state:
H b ¼ C p ðT Þ V b ðT Þ ΔT
Throughout the column, there is pressure decrement from
the air enters and leaves the column, which corresponds to the
mechanical work done on the column by the pump. It is assumed that this work is converted completely into heat, which
is absorbed by the column. This component can be directly
related to the original work-heat conversion in Joule’s original
paddle wheel experiments in 1843. The bubbles mixed well
with the solution and there is no hot or cool region in the
column. The dry bubbles are saturated rapidly [17] and it is
assumed that bubble in the column is always Vb(Te) when it
reaches the steady state. So the work is:
W Taire ¼ ΔPV b ðT e Þ
It is easy and direct to obtain the energy used for the
vaporization of nwater mole water in the solution (Qwater).
This is:
Qwater ¼ nwater ΔH vap
Fig. 2 The linear relationship between specific heat capacity values Cp of
air and 1/T. Cp values are from CRC Handbook [23]
This analysis regarding the supply and consumption of
energy in the bubble column presented the basic physics
process between the bubble and solution. Volumetric basis, in the form of per volume of the bubble gas, makes
the BCE experiment applicable for the ΔHvap determination through the measurements of inlet air temperature,
column top temperature and the differential pressure.
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Heat Mass Transfer
3.2 Bubble volume expansion during water vapor
The physical conditions of a bubble in a solution before and
after being saturated by water vapor, at the same temperature
and under atmospheric pressure, are described in Fig. 3.
Based on Fig. 3, dry and wet bubble volume Vi and Vf
can be easily obtained using the ideal gas equation and
Dalton’s law:
Vi ¼
nair RT
Vf ¼
ðnair þ nw ÞRT
where Vi, Vf are the initial (dry) and final (wet) bubble volumes; nair and nw are the moles of air and water vapor in the
bubble; R is the universal gas constant; T is absolute temperature and Patm is atmospheric pressure. Then the mole of gases
in saturated bubble is given by:
nair þ nw ¼ nair þ ρv V f ¼ nair
where ρv is the water vapor density, which is equal to nw/Vf.
Then the final bubble volume is given below after rearranging
Eq. (8) and using the ideal gas equation:
nair RT
Vf ¼
Patm −Pv
where Pv is the water vapor pressure of the solution. Then the
bubble expansion percent before and after being saturated is
given by:
V f −V i
Percentexpansion ¼
100 ¼
−1 100 ð10Þ
Patm −Pv
expansion over a range of temperatures is shown in Fig. 4
(vapor pressure data of pure water, NaCl and CaCl2 are
from CRC book [23], Clarke et al. [24] and Patil et al.
This expansion may affect the proposed balance Eq. (1), as
the water vapor is included in the original balance for bubble
heat supply calculation. Hence, a factor θ can be introduced
into the Eq. (1) to reduce the contribution from the gas specific
heat Cp, in order to account for the presence of saturated water
vapor in the bubble:
ΔT C p ðT e Þ θ þ ΔP ¼ ρv ðT e Þ ΔH vap ðT e Þ
ρ ðT e Þ RT e
θ ¼ 1− v
The factor θ is actually the percent that dry air occupied in
the bubbles, excluding the component of water vapor, which
indicates the heat contribution is all from the dry gas in the
BCE. In the same principle of Eq. (11), instead of using the
heat capacity per unit volume, Cp, the weight-based unit, i.e.
heat capacity per unit weight of gas C gp , is used. This gives
another version [1] of the balance equation which produces
same results as Eq. (11):
ΔT C gp ðT e Þ ρa þ ΔP ¼ ρv ðT e Þ ΔH vap ðT e Þ ð13Þ
where ρa is the density of air (or gas) in a bubble. This was
obtained from the ideal gas equation using the molar mass of
air (28.96 g·mol−1) and by subtracting the number of moles of
the absorbed water vapor from the total number of moles of
gas in the bubble per unit volume at Te and 1 atm pressure. It
should be noted that the mass of air or gas in a bubble remains
constant as the bubble passes through the column. This new
balance equation removes the water vapor in the bubbles from
the heat-supply calculation; compared with Eq. (1).
This percent can easily be found using above relation if
the vapor pressure data are available. For example, in pure
water, 0.5 mol·kg−1 NaCl and 5 mol·kg−1 CaCl2, the %
Fig. 3 Two situations of the bubble before and after being saturated by
water vapor
Fig. 4 Individual bubble expansion percent by capturing water vapor in
pure water, 0.5 mol·kg−1 NaCl and 5 mol·kg−1 CaCl2 with temperature
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Heat Mass Transfer
Table 1 Different error percent of
ΔHvap measurements using BCE
process for various common salt
solutions based on different
developed energy balances
Temperature (°C)
Salt solutions
Difference percent from different energy balance (%)
NaCl (1.0 m)
CaCl2 (1.0 m)
CaCl2 (5.0 m)
KCl (4.0 m)
NaCl (0.5 m)
All literature values were calculated from published vapor pressure values using the Clausius-Clapeyron equation,
which was demonstrated [3] to give authentic literature ΔHvap values. NaCl vapor pressure values from [24];
CaCl2 and KCl vapor pressure values from [25]
However, ΔHvap term in Eq. (11) and Eq. (13) includes the
work done by bubble-vapor expansion in the solution due to
water vapor absorption, although the left side of both equations consider removing the effect of the vapor expansion in
the bubble. Hence, the balance becomes:
ΔT C gp ðT e Þ ρa þ ΔP ¼ ρv ðT e Þ ΔH vap ðT e Þ−PΔV ðT e Þ ð14Þ
cases, the bubble expansion had a significant effect on the
results and the original equation gives more accurate ΔHvap
values. It is established that the original equation Eq. (1) has
high accuracy and precision based the above analysis and
previous works [3, 4, 10].
3.3 The accuracy of the original balance of the BCE
where PΔV term is the work done by the bubble-vapor
expansion. Based on the energy balance summary [1]
and previous works data [3, 4, 10], the difference percent
of ΔHvap results from Eq. (1), Eq. (11), Eq. (13) and Eq.
(14) with the literature ΔHvap values was calculated on
average, as given in the Table 1.
As can be seen from Table 1, all balance equations give the
similar difference percent in the relatively low temperature
cases, i.e. 19–26 °C, due to the low percent of vapor expansion
in the bubbles (see Fig. 3); while in the high temperature
When the BCE reaches steady state, the column top solution
temperature can be accurately predicted based on the original
equation Eq. (1) through a fixed inlet gas temperature. The
experiment BCE temperature data in previous works were
found to well fit with this prediction, as shown in Fig. 5.
The potential application in terms of the evaporative cooling
effect is also presented in this figure.
Figure 5 also indicated that a different ΔP value would
be revealed as a slight change in the steady-state column
Fig. 5 The relationship between inlet gas temperature and column
solution temperature at steady state based on Eq. (1) for 0.5 m NaCl in
BCE of different ΔP. Experimental temperature data based on BCE from
Francis and Pashley [2], Fan et al. [3, 10] and Shahid et al. [1, 4] are on the
predicted curve over the wide range of inlet gas temperature. Arrows
show the slightly different column top-solution temperatures at a constant
inlet-gas temperature, due to the work done by pumped inlet gas to the
Fig. 6 The linear trend of literature ΔHvap of NaCl over 0.5–1.2 m with
temperature. Literature ΔHvap values are from Clarke and Glew [24],
Hubert [26] based on the calculation using Clausius-Clapeyron equation
and Chou [27] calculated from osmotic coefficient, specific heats and
solubilities. Experimental data based on BCE are from Francis and
Pashley [2] with accuracy of 3% for 0.5 M NaCl at around 10 °C, Fan
et al. [3, 10] with accuracy of within 1% for 1 m NaCl over around 20–
25 °C and Shahid et al. [4] with accuracy of 2% for 0.5 m NaCl over
around 40–55 °C
1.0–3.8 m, 30 °C b
0–14.2 m, boiling point c
0–1.6 m, boiling point c
0–1.6 m, boiling point c
0–2.7 m, boiling point c
7.9 m (saturated), 109 °C
0–25.7 m, 100–115 °C a
Boiling point a
Chou [27]
Shahid et al. [1, 4]
Fan et al. [3, 10]
Francis and Pashley [2]
Topor et al. [31]
Gurovich et al. [30]
Hunter and Bliss [29]
Lunnon [28]
Wagner and Pruss [32]
Patil et al. [25]
Clarke and Glew [24]
Accuracy on average Less than 10%
4.2 m (saturated), 106.8 °C a
3.3 m (saturated), 104.8 °C a
1.1–2.6 m, 30 °C b
0–33.5 m, 100–116 °C a
2.6–15.4 m, 30 °C b
6.8 m (saturated), 110 °C a
Theoretical technique
Less than 5%
1.4–23.2 m, room
temperature-boiling point d
Less than 3%
11–12 °C e
4.0 m, 20 °C f
1.0–3.0 m, 25 °C f
3.0 and 3.5 m, 25 °C f
0.5 and 5 M, 10 °C e
1.0 m, 20–25 °C f
0.5 m, 40–55 °C g
1.0 and 5.0 m, 20–25 °C f
1.0–4.0 m, less than 70 °C j
1.0–6.0 m, less than 70 °C j
0.5–6.0 m, 0–110 °C i
Reasonable agreement with each other f
0.01–373.95 °C k. calculated from the equation of state.
0.9–5.7 m, 0–176 °C h
Calorimetric measurement (CM) thermogravimetric analysis (TGA) Bubble column evaporator (BCE) Calculated from specific Calculated from vapor pressures at different
heats and solubilities
temperatures (Clausius-Clapeyron equation)
Experimental technique
Comparison of operation temperature and concentration of salt solutions using different experimental and theoretical technique
Salt Solutions
Table 2
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Heat Mass Transfer
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Heat Mass Transfer
temperature in the BCE if the inlet-gas temperature was
kept constant. That is, as a necessary term (work to the
column) for the energy balance in the BCE (Eq. (1)), the
increase of this work leads to a slight increase of the
solution temperature, as indicated from arrows for the
same inlet gas temperature.
Also, a summary of determined ΔHvap values using the
BCE supported the high accuracy of the original equation
with a wide range of temperature, as shown in Fig. 6. It
should be noted that literature ΔHvap of 0.5 m and 1.0 m
NaCl at same temperature (5–55 °C) has similar values,
with less than 0.04%.
There are also other techniques to determine ΔHvap of salt
solutions, but the BCE method offers a novel and relatively
more accurate way for the determination over a wide range of
temperature. As can be seen in a summary of different techniques of Table 2, BCE approach opened a new technique for
ΔHvap determination with high accuracy compared to the conventional ways.
3.4 Energy utilization analysis of the BCE process
In all of the experiments with high inlet-air temperatures, the
solution-top temperatures always reached steady state after
about 5 mins, as shown in Fig. 7 [4]. It is also worth noting
that the observed column-top temperature was unaffected by
the presence or absence of C12EO8 surfactant, added to capture more water vapor when using the BCE for desalination
[4], and that these steady-state temperatures were very close to
those estimated in Fig. 7, based on Eq. (1).
Based on this desalination work using the BCE process, as
shown in Fig. 8, three types of energy involved in the operation were estimated with different inlet air temperature:
the dry air was heated by the heater from room temperature to different inlet-gas temperatures;
the hot air supplied heat to the solution in the BCE to
evaporate the water;
the energy produced from water-vapor condensation over
a 25-min period at steady state (see Fig. 7).
Fig. 7 Measured 0.5 mol·kg−1 NaCl solution top temperature within a
bubble column using different inlet gas temperature (data reproduced
from [4])
supply for the inlet air and there would be more
energy lost and consumed for providing hot gas at
designed inlet gas temperature.
(2) The energy supplied by the hot air to the solution for
water vaporization was calculated from the weight of
water lost during the 25-min period at steady state
and the literature ΔH vap value at the steady-state
temperature. As in [4], the water weight loss was
measured directly from the difference in the weight
of the column at the beginning and at the end of the
experiment. In the first five minutes of each experimental run (Fig. 7), before steady state was reached,
the weight loss was estimated from the water-vapor
density at the column temperature and the air flow.
The total weight loss during the 25 mins at steady
state was then obtained by subtracting the estimated
weight loss in the first 5 mins from the weight loss
over 30 mins (data from [4]).
These three types are explained and analysed as follows:
(1) The heat supplied in heating the inlet air from room
temperature to the inlet temperature was calculated
using the heat capacity of air (J·g−1·K−1), the weight
of the air flowing through the flow meter over the
25-min period and the temperature change. In practice, the temperature of inlet air before entering the
column normally has to be heated higher than desired
value because of the energy loss with room environment. Hence this is the lowest estimation of energy
Fig. 8 Comparison of the energy supplied to heat the dry air, the energy
used to vaporize water in 0.5 m NaCl and the energy from water-vapor
condensation in producing 1 mol of pure water
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Heat Mass Transfer
(3) Energy recovery rates in industrial reverse-osmosis desalination and thermal desalination can be up to 99%
under optimum conditions [33]. Assuming 99% energy
recovery here, the energy produced from condensation
was estimated using values for the enthalpy of condensation of the salt solution at the steady-state temperature
from the literature [23] and the water weight loss during
the 25-min period.
Acknowledgments We would like to thank the Australian Research
Council for funding this project (Grant Number: DP120102385).
Compliance with ethical standards
Competing interest
The authors declare no competing financial
At higher inlet-air temperatures, more pure water can
be produced from the BCE system than at lower temperatures using the same amount of electricity, due to the
increased water-vapor density and higher vapor supersaturation in the bubbles [4], as shown in Fig. 8. Industrially,
even higher inlet-air temperatures from, for example,
industrial-waste flue gases could be used, which would
save energy for pumping and heating the inlet air.
Commercial reverse-osmosis systems typically have an
energy usage in the range of 4–6 kW·h·m−3, conventional
thermal/evaporative units 14–27 kW·h·m−3 [34, 35] and
BCE systems used for thermal desalination 7.55 kW·h·
m−3 [1] at an inlet-gas temperature of 550 K. The comparison of energy consumption by different methods was
discussed in details in [1]. Higher inlet-gas temperatures
could reduce this estimate still further based on the Fig. 8,
but the solution temperature must be less than its boiling
point. The BCE system uses a non-boiling process which
is fairly independent of the concentration of the salt solution and can work effectively at salt concentrations as
high as 5 m NaCl or more.
4 Conclusions
The original balance of the BCE gives the high accurate
and precise ΔHvap values of salt solutions either in low
temperature or high temperature, even it includes the the
water-vapor expansion in the bubbles, as ΔHvap term includes the work done by bubble-vapor expansion in the
solution due to water vapor absorption. The consideration
of the water-vapor expansion in the bubble leads to different BCE balances, which affected much for the accuracy of the high temperature experiments. Based on the
original balance equation, the good prediction of the column temperature and relatively higher accuracy of ΔHvap
compared to other techniques further supported the validity of the BCE balance for the thermodynamic study.
Also, the good agreement between the energy supplied
to the inlet dry air and the energy utilized for water vaporization in the BCE demonstrate the efficiency of the
system used for thermal desalination. The process could
potentially be pursued for higher energy efficiency by
using higher inlet air temperatures.
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