A mathematical model for a thermally coupled ... - ScienceDirect

Desalination 196 (2006) 177–187

A mathematical model for a thermally coupled humidification– dehumidification desalination process Rihua Xiong, Shichang Wang*, Zhi Wang Chemical Engineering Research Center, State Key Laboratory of Chemical Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072, China Tel. +86 (22) 2789-2155; Fax +86 (22) 2740-4757; email: [email protected]

Received 7 November 2005; accepted 9 January 2006

Abstract The humidification–dehumidification process is an interesting technique that has been adapted for water desalination. Previous works experimentally investigated desalination processes in the shell and tube columns, where the humidification and dehumidification were thermally coupled and simultaneously performed at the tube and shell sides, respectively. In this work, a comprehensive steady-state mathematical model was developed for such a humidification–dehumidification desalination process by taking into account the heat and mass balances on both sides of the desalting column, the mass transfer rate at the humidification side, and the heat transfer rate between the dehumidification side and humidification side. Meanwhile, the mass transfer coefficient at the humidification side and the total heat transfer coefficient between the dehumidification side and humidification side were discussed and correlated. The correlations could represent the experimental data very well. Keywords: Desalination; Humidification; Dehumidification; Mathematical model; Mass transfer coefficient; Heat transfer coefficient

1. Introduction In many arid zones, coastal or inlands, seawater or brackish water desalination may be the only solution to the shortage of fresh water [1]. Standard desalination processes such as multistage flash (MSF), multi-effect distillation (MED) and reverse osmosis (RO) have gained much success *Corresponding author.

in the last half-century. However, their requirements of high initial capital costs and permanent qualified maintenance workers have limited their access to many developing countries [2]. Further, it is not economical to employ them when the capacity is small, especially when there is a need to utilize renewable energy such as solar energy. As a result, small- or medium-scale water desalination units with good flexibility in capacity,

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R. Xiong et al. / Desalination 196 (2006) 177–187

moderate capital cost, and the possibility of using renewable energy, are of great interest in regions like the northwest zone and many islands of China as well as many other remote arid area across the world where infrastructure is fairly low and the water demand is decentralized. The humidification–dehumidification process is an interesting technique, which has been adapted for water desalination, where air is used as a carrier gas to evaporate water from the saline feed and to form fresh water by subsequent condensation. It is likely to be the most promising process of solar desalination [3]. Most researchers [4–8] have performed the humidification–dehumidification desalination process in two separate columns, one for humidification and another for dehumidification, with the columns constructed in different structures with various materials. However, separate structure increases both the complexity of the system and the capital cost. Furthermore, in a separate humidification column the latent heat of evaporation for humidification can only come from the sensitive heat of saline water fed to humidify the carrier gas, which limits the amount of the water that can be vaporized, resulting in a limited humidification effect of the carrier gas. At the same time, large amounts of condensation latent heat in another column are nearly lost or partially reused to preheat the saline feed water with low efficiency due to a large temperature difference in such heat transfer processes. In recent years, Beckman et al. [9] proposed an interesting desalination process known as dewvaporation, in which the humidification and dehumidification process were simultaneously performed in one continuous contact tower. They also developed a simplified mathematical model for such a desalination process, with the assumption that the liquid film temperature is the average temperature of the gas phases [10], which limited the application of the model [11]. In previous works [12,13], thermally coupled desalination processes in the shell and tube columns were experimentally investigated, where the

humidification and dehumidification were simultaneously performed at the tube and shell sides, respectively. In this work, a comprehensive mathematical model was developed for such a thermally coupled humidification-dehumidification desalination process. Meanwhile, the corresponding mass and heat transfer coefficients in such processes were correlated and discussed.

2. Mathematical model 2.1. Process description Briefly, the desalination process focused on here is a carrier gas desalination process where humidification and dehumidification take place in one column. As an example, the shell and tube column was used to perform such a desalination process, which was described in detail in our previous work [13]. In this case, room temperature air is brought into the tube side of the column from the bottom and then flows upward in the tube side. The inside wall of the tubes are wet by preheated feed water, which is fed into each tube at the top of the column through overfall holes in the wall of the tube. As the air moves from the bottom to the top in the tubes, heat is transferred from the shell side to the tube side through the tube wall, allowing the air to rise in temperature and evaporate water from the water coating the inside tube wall. Concentrated water leaves from the bottom of the column, while hot and nearly saturated humid air leaves the column from the top. This hot and humid air exit from the tube side is further externally heated and humidified a little. This hotter saturated air is then sent directly into the shell side of the column from the upper inlet. The shell side of the column, being slightly hotter than the tube side, allows the humid air to cool and condensate, while the condensation latent heat is transferred to the tube side. Finally, water condensate and the dehumidified air leave the shell side of the column from the lower outlet.


2.2. Model development As mentioned above, a key point of the present desalination process is that the latent heat of the dehumidification side should be effectively transferred to the humidification side and reused, by which the process is expected to get an improved energy efficiency. Focusing on a differential element of the desalting column, a sketch of the heat and mass transfer model for the present desalination process is presented in Fig. 1, in which the balance and transfer of heat and mass are indicated. On the humidification side, the carrier gas is flowing upward and humidified by downward hot saline water. On the dehumidification side, the humidified and hot gas is flowing downward and dehumidified on the wall while the condensation latent heat is transferred to the dehumidification side. As indicated in Fig. 1, the energy balances at the tube side and the shell side of the column can be described as follows, respectively: d ( G ⋅ We ) = d ( L ⋅ CP ⋅ Tw ) + dQ

(1)

d ( G ⋅ Wd ) + d ( F ⋅ CP ⋅ Td ) = dQ + dQLoss

(2)

At the same time the mass balances of water

179

at the tube side and the shell side lead to the following two equations: dL = G ⋅ dH e

(3)

dF = −G ⋅ dH d

(4)

The humidification process of the carrier gas by the falling film of the feed water at the tube side is treated as a mass transfer process with the humidity difference between the water surface and the bulk carrier gas as driving force [14], which give rise to the following mass transfer equation: G ⋅ dH e = kH ⋅ ( H w − H e ) ⋅ dAH

(5)

where kH is the mass transfer coefficient introduced to describe the humidification process and AH is the corresponding surface area where vaporization takes place. As for the heat transfer process between the dehumidification and humidification side of the column, the driving force is taken as the difference of the saturated temperatures of the humid air at the humidification side and the temperature of the saline water coating on the inside wall of the humidification side. As a result, such a heat transfer process can be expresed as follows:

Fig. 1. Sketch of heat and mass transfer in a differential element.

180


dQ = K ⋅ ( Td − Tw ) ⋅ dA

(6)

where K is the corresponding total heat transfer coefficient and A is the heat transfer area. Besides this, the heat loss through the outside wall of the column is evaluated with a heat loss coefficient KLoss, namely: dQLoss = K Loss ⋅ ( Td − Tamb ) ⋅ dALoss

(7)

where Tamb is the temperature of the ambient and ALoss is the heat loss area. The specific heat capacity of the feed water and fresh water is assumed as equal and constant, then: d ( L ⋅ CP ⋅ Tw ) = L ⋅ CP ⋅ dTw + CP ⋅ Tw ⋅ dL

(8)

d ( F ⋅ CP ⋅ Td ) = F ⋅ CP ⋅ dTd + CP ⋅ Td ⋅ dF

(9)

Combining Eqs. (1)–(9) and carrying out transformation, the following five equations can be derived:

⎛ dWe ⎞ G ⋅⎜ ⎟ ⋅ dTe = L ⋅ CP ⋅ dTw + CP ⋅ Tw ⋅ dL ⎝ dTe ⎠ ⎛ dA ⎞ + K ⋅⎜ ⎟ ⋅ (Td − Tw ) ⋅ dZ ⎝ dZ ⎠

(10)

⎛ dWd ⎞ G ⋅⎜ ⎟ ⋅ dTd + F ⋅ CP ⋅ dTd + CP ⋅ Td ⋅ dF ⎝ dTd ⎠ ⎛ dALoss ⎞ ⎛ dA ⎞ = K ⋅⎜ ⎟ ⋅ (Td − Tw ) ⋅ dZ + K Loss ⋅ ⎜ ⎟ (11) ⎝ dZ ⎠ ⎝ dZ ⎠

⋅ (Td − Tamb ) ⋅ dZ

⎛ dH ⎞ ⎛ dA G ⋅ ⎜ e ⎟ ⋅ dTe = kH ⋅ ⎜ H ⎝ dZ ⎝ dTe ⎠

⎞ ⎟ ⋅ ( H w − H e ) ⋅ dZ ⎠ (12)

⎛ dH ⎞ dL = G ⋅ ⎜ e ⎟ ⋅ dTe ⎝ dTe ⎠

(13)

⎛ dH dF = −G ⋅ ⎜ d ⎝ dTd

(14)

⎞ ⎟ ⋅ dTd ⎠

In order to simplify the developed equations, seven model parameters were introduced and defined as follows:

m1 =

dWe dWd dH e dH d , m2 = , m3 = , m4 = dTe dTd dTe dTd

n1 =

dA dA dA , n2 = H , n3 = Loss dZ dZ dZ

Substituting the above parameters in Eqs. (10)– (14) gives: G ⋅ m1 ⋅

dTe dT dL = L ⋅ CP ⋅ w + Cp ⋅ Tw ⋅ dZ dZ dZ + K ⋅ n1 ⋅ ( Td − Tw )

(15)

dTd dT dF + F ⋅ CP ⋅ d + CP ⋅ Td ⋅ dZ dZ dZ = K ⋅ n1 ⋅ (Td − Tw ) + K Loss ⋅ n3 ⋅ (Td − Tamb )

(16)

dTe = kH ⋅ n2 ⋅ ( H w − H e ) dZ

(17)

G ⋅ m2 ⋅

G ⋅ m3 ⋅

dT dL = G ⋅ m3 ⋅ e dZ dZ

(18)

dT dF = −G ⋅ m4 ⋅ d dZ dZ

(19)


By rearranging the above five equations, the mathematical model of the present desalination process is finally presented as follows:

181

dTd K ⋅ n1 ⋅ ( Td − Tw ) + K Loss ⋅ n3 ⋅ (Td − Tamb ) = dZ G ⋅ m2 + F ⋅ CP − G ⋅ m4 ⋅ CP ⋅ Td

(20) dTe kH ⋅ n2 ⋅ ( H w − H e ) = dZ G ⋅ m3

(21)

dTw kH ⋅ n2 ⋅ ( m1 − m3 ⋅ CP ⋅ Tw ) ⋅ ( H w − H e ) − m3 ⋅ K ⋅ n1 ⋅ ( Td − Tw ) = dZ m3 ⋅ L ⋅ CP

(22)

dL = kH ⋅ n2 ⋅ ( H w − H e ) dZ

(23)

dF −G ⋅ m4 ⋅ ⎡⎣ K ⋅ n1 ⋅ (Td − Tw ) + K Loss ⋅ n3 ⋅ (Td − Tamb )⎤⎦ = dZ G ⋅ m2 + F ⋅ CP − G ⋅ m4 ⋅ CP ⋅ Td

(24)

Thus, the above set of ordinary differential equations has been developed to describe the thermally coupled humidification–dehumidification desalination process. 2.3. Model parameters There are three classes of parameters in the developed model. The first class of parameters is related to the column structure, including n1, n2 and n3. Those parameters can be easily determined by the structure information of the column used. In our previous work [13], a baffled shell and tube desalination column was described in detail. This column is made up of 73 copper tubes with an effective height of 2.1 m. The effective heat transfer area, mass transfer area and heat loss area of the column are 2.75 m2, 2.52 m2 and 0.53 m2, respectively. In this case, the structure parameters can be easily calculated according to their definitions mentioned before, namely n 1=1.38 m, n2=1.26 m and n3=0.25 m. The second class of parameters is intermediate parameters m1, m2, m3 and m4, which, by definition,

are the temperature derivatives of enthalpy or humidity of the humid air at the humidification side or dehumidification side. These derivatives can be numerically calculated by finite difference method as follows:

m1 =

dWe We = dTe

m2 =

dWd Wd = dTd

m3 =

dH e H e = dTe

Te +∆t

−We

Te −∆t

2 ⋅ ∆t

Td +∆t

−Wd

Td −∆t

2 ⋅ ∆t

Te +∆t

−He

2 ⋅ ∆t

Te −∆t

(25)

(26)

(27)

182


dH d H d = m4 = dTd

Td +∆t

−H d

Td −∆t

2 ⋅ ∆t

(28)

where W and H are the enthalpy and humidity as functions of temperature, while the subscripts ‘e’ and ‘d’ refer to the humidification side and dehumidification side, respectively. ∆t is a finite temperature difference with a value of 0.1ºC applied in the numerical process of this work. The third class of parameters is heat and mass transfer coefficients, including mass transfer coefficient kH describing the humidification process at the humidification side, total heat transfer coefficient K describing the heat transfer process between the dehumidification and humidification sides, and heat loss coefficient KLoss describing the heat loss to the ambient from the column. As is known, the determination of heat and mass transfer coefficients are critical to describe the present thermally coupled desalination process, which will be discussed in detail in the following two sections. 3. Mass transfer coefficients 3.1. Calculation of mass transfer coefficient As mentioned before, the mass transfer coefficient kH here describes the humidification rate of the carrier gas by saline feed water. From Eq. (5) one can find that the definition of kH is:

dH (Te ) G kH = ⋅ H (Tw ) − H (Te ) dAH

G ⋅ ⎡⎣ H (Te, out ) − H (Te, in ) ⎤⎦ AH ⋅ ∆H m

∆H m =

⎡ H (Tw,in ) − H (Te,out ) ⎤ − ⎡ H (Tw,out ) − H (Te,in ) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ H (Tw,in ) − H (Te,out ) ⎤ ln ⎢ ⎥ ⎢⎣ H (Tw,out ) − H (Te,in ) ⎥⎦

(31) Based on Eqs. (29)–(31), the mass transfer can be calculated from experimental results under certain operating conditions. By this means, mass transfer coefficients under various operating conditions were determined with the experimental results of two shell and tube desalination columns [12,13]. 3.2. Correlation of mass transfer coefficient It is well known that the humidification effect is greatly dependent on both the gas and liquid flow rates. Therefore, it is reasonable to correlate mass transfer coefficients in present process with the mass velocity of carrier gas U, which is a specific gas flow rate, and film flow rate Γ, which is a specific liquid flow rate. Accordingly, a multiple regression analysis was conducted on such a correlation, which yields the following relationship:

kH = 0.0021 ⋅ U 1.27 ⋅ Γ −0.29 (29)

In this work kH is simplified as a mean value for the whole column, namely: kH =

mean humidity difference between the water/air interface and the bulk air, which is:

(30)

where the mass transfer driving force is the log

(32)

The above correlation can be used when U is in the range of 0.06–3.0 kg/(m2 s) and Γ in the range of 0.0004–0.006 kg/(m s). The predicted mass transfer coefficients by Eq. (32) could reproduce the experimental results [12,13] fairly well, as shown in Fig. 2. The dependency of mass transfer coefficients on specific gas flow rate and specific liquid flow rate is further shown in Fig. 3. It is worth noting that the exponent of the specific liquid flow rate

R. Xiong et al. / Desalination 196 (2006) 177–187 0.08

0.05

0.07

Γ, kg m-1 s-1 0.0006 0.001 0.002 0.003 0.004

0.06 -1

0.04

-2

kH / kg m s

-2

kH, pred. / kg m s

-1

0.06

0.03 0.02 0.01

183

0.05 0.04 0.03 0.02 0.01

0.00 0.00

0.01

0.02

0.03

0.04 -2

0.05

0.06

0.00 0.0

-1

0.5

1.0

1.5

2.0 -2

kH, exp. / kg m s

U / kg m s

2.5

3.0

3.5

-1

Fig. 2. Comparison of predicted mass transfer coefficients with experimental results.

Fig.3. Mass transfer coefficients by developed correlation.

is negative in Eq. (32). This is mainly because an increase of water flow rate will greatly decrease the heat transfer driving force between the dehumidification side and the humidification side, which was discussed in detail in previous work [13].

which is based on the assumption that the ratio of the condensation coefficient and the convection coefficient represents the ratio of their corresponding heat transfer loads. Meanwhile, the total heat transfer resistance between the dehumidification side and the humidification side is dominated by the gas film resistance of condensation due to the presence of large amounts of non-condensable gases. Therefore, the local total heat transfer coefficient can be treated as follows:

4. Heat transfer coefficients 4.1. Calculation of local total heat transfer coefficient In the present desalination process, the desalting column was typically placed vertically, in which the local heat transfer coefficients were always changing from the bottom to the top due to the change of the local situations in both the dehumidification and humidification sides. Therefore, it is necessary to deal with the heat transfer coefficients with point to point consideration. The approximate relationship between the condensation heat transfer coefficient of humid air and convection heat transfer coefficient of air was introduced by Nawayseh et al. [15], namely: hc dW = ha CP,air ⋅ dT

(33)

K = K0 ⋅

dWd m = K0 ⋅ 2 CP,air ⋅ dTd CP,air

(34)

where K0 was assumed to be constant everywhere in the desalination column under certain operating conditions. So it is critical to determine the value of K0 in order to get the local value of K. In this work, the mathematical model developed in Section 2 was used to determine the value of K0 under each operating condition by trial and error method. An initial value of K0 was applied to determine the value of K at different height of the column via Eq. (34). Then the mathematical model was numerically solved using the standard fourth order


Runge–Kutta method. The estimation of K0 was gradually adjusted and finally determined according to the comparison of the model prediction and the corresponding experimental result. By this means, values of K0 under various operating conditions were determined with the experimental results of two shell and tube desalination columns [12,13]. 4.2. Correlation of total heat transfer coefficient For the correlation of K0 with operating conditions, the following standard relationship was used as the basis:

Nu = C1 ⋅ Rem ⋅ Pr n

(35)

where Nu is the Nusselt number with the definition that:

Nu =

K0 ⋅ d t λ mix

(36)

where dt and λmix are the outside diameter of the heat transfer tube and the thermal conductance of the mixed humid air, respectively. The mean temperature of the humid air at the dehumidification side was used as the characteristic temperature, namely: T=

Td,in + Td,out

K = 0.902 ⋅ ⋅ (U

However, a preliminary analysis showed that Pr numbers were almost constant with values in the range of 0.72–0.74. Meanwhile, in spite of Re, Nu was also found to be greatly dependent on the film flow rate of the water at the humidification side. Such a dependency was then taken into consideration by additional terms in the correlation. By multiple regression analysis and careful comparison studies, the relationship between the total heat transfer coefficient and operating conditions was finally obtained, namely:

− 150Γ )

(38)

2.04

3.0 2.5 2.0

(37)

2

0.1

λ mix m2 ⋅ ⋅ Re0.62 ⋅ Γ 0.58 d t CP,air

The correlation Eq. (38) can be used with Re in the range of 15–2700 while U and in the same ranges with Eq. (32). The predicted heat transfer coefficient by Eq. (38) could represent the experimental results [12,13] fairly well as shown in Fig. 4, which shows the comparison of correlation and experimental results in terms of Nu. The heat transfer gas boundary layer at higher Re tends to be thinner, while the vapor fraction at higher saturation temperature tends to be higher. Therefore, the heat transfer coefficient increases with the increase of the saturation temperature and Re, as is shown in Fig. 5. On the other hand, the effect of the water film flow rate on the heat transfer coefficient is more complicated, which is shown in Fig. 6. However, the tendency in Fig. 6 is consistent with what was discussed in the experimental investigation in terms of the effect of film flow rate on the water productivity [13].

Nu pred.

184

1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Nu exp. Fig. 4. Comparison of predicted Nu with experimental results.

R. Xiong et al. / Desalination 196 (2006) 177–187 550

1000

Γ=0.002kgm s

-1 -1

600

450 400 -1

700

Re=1500 o Td , C 85 80 70 60

500

C 85 80 70 60

500

K/Wm

-2 o

C

-1

800

o

C

Td ,

-2 o

900

K/Wm

185

400 300 200

350 300 250 200 150 100

100

50

0 0

500

1000

1500

2000

2500

3000

0 0.000

0.001

Re

0.002

0.003

0.004

0.005

Γ / kg m s -1

-1

Fig.5. The variation of predicted K with Re and saturated temperature of mixed gas.

Fig. 6. The variation of predicted K with film flow rate of water.

4.3. Heat transfer coefficient profiles in desalting columns

4.4. Heat loss coefficient

As mentioned above, the local heat transfer coefficients along the height were always changing from the bottom to the top in a vertically placed desalting column. As examples, Fig. 7(a) and (b) show profiles of the heat transfer coefficient along the height in the unbaffled and baffled shell and tube desalting columns under typical operating conditions, respectively. In both cases, the local heat transfer coefficient increases from the bottom to the top due to the increase of the operating temperature as well as vapor fraction of the humid air in the columns. The heat transfer coefficient in the unbaffled column was found between 10– 30 W/(m2 ºC), which happens to be in the same range with the work of Nawayseh [15]. However, the heat transfer coefficient in the baffled column was in the range of 50–200 W/(m2 ºC), which is much higher than that of in the unbaffled column. The different ranges of the heat transfer coefficients for the baffled and unbaffled columns give a quantitative reason why the water productivity of the baffled column was 3–6 times that of the unbaffled one [13].

Heat loss coefficients were calculated based on heat balance. Since the errors in experimental measurements were cumulated in the heat balance calculation, the obtained heat loss coefficients were not correlated with the operating conditions. Instead, an average heat loss coefficient based on hundreds of experimental runs was obtained and found to be 1.57 W/(m2 ºC), which was very close to the estimation in a humidification desalting unit by Al-Hallaj [6]. 5. Conclusion Based on previous experimental investigations, a comprehensive mathematical model has been developed for a thermally coupled humidificationdehumidification desalination process. The model was presented as a set of five-element first order ordinary differential equations, which can be solved numerically by standard fourth order Runge–Kutta method. The mass transfer coefficients at the humidification side were studied and correlated based on experimental results. A calculation method for the

186

R. Xiong et al. / Desalination 196 (2006) 177–187 2.0

2.0

(a)

1.6

1.6

1.4

1.4

1.2

1.2

1.0 0.8 0.6 0.4

1.0 0.8 0.6 0.4

0.2

0.2

0.0

0.0

12

(b)

1.8

Height / m

Height / m

1.8

14

16

18

20

K/Wm

22 -2 o

24

26

28

80

100

120

140

-1

C

K/Wm

160 -2 o

180

200

-1

C

Fig. 7. Typical heat transfer coefficient profile in shell and tube desalting columns: (a) unbaffled column; (b) baffled column.

local total heat transfer coefficients between the dehumidification side and humidification side was developed based on the developed mathematical model. Meanwhile, a correlation for the local total heat transfer coefficients was developed based on experimental results. These correlations could represent the experimental data very well. The heat transfer coefficients in the baffled column were found to be in the range of 50– 200 W/(m2 ºC), while similar heat transfer coefficients in the unbaffled column were in the range of 10–30 W/(m2 ºC). The heat transfer process in the shell and tube desalting column could be greatly enhanced by baffle plates. Symbols A — AH — ALoss — CP — CP, air — dt — F —

Heat transfer area, m2 Mass transfer area, m2 Heat losss area, m2 Heat capacity of water, J/(kg ºC) Heat capacity of humid air, J/(kg ºC) Diameter of the heat transfer tube, m Fresh water flow rate, kg/s

G — Carrier gas flow rate, kg/s H — Humidity, kg vapor/kg dry air ∆Hm — Log mean humidity difference, kg vapor/ kg dry air K — Heat transfer coefficient, W/(m2 ºC) K0 — Base heat transfer coefficient, W/(m2 ºC) kH — Mass transfer coefficient, kg/(m2 s) KLoss — Heat loss coefficient, W/(m2 ºC) L — Feed water flow rate, kg/s m — Intermediate parameter related with the humidity or entropy of the mixed gas n — Structure parameter in the model, m Nu — Nusselt number Pr — Prandtl number Q — Heat transfer rate, W Re — Reynolds number S — External steam flow rate, kg/s T — Temperature, ºC ∆t — Temperature difference, ºC U — Mass flow velocity of mixed gas, kg/(m2 s) W — Enthalpy of mixed gas, J/kg air Z — Height of the desalting column, m


Greek Γ λ

— Film flow rate of water, kg/(m s) — Thermal conductance, W/(m oC)

Subscripts 1,2,3,4— Sequence of the intermediate parameter amb — Ambient d — Condensation/dehumidification side e — Evaporation/humidification side exp — Experimental in — Inlet m — Log mean mix — Mixed gas out — Outlet pred — Predicted w — Water Acknowledgements This study is financially supported by National Natural Science Foundation of China and China Energy Conservation Investment Corporation under Contract No.20236030.

References [1] H.T. El-Dessouky, H.M. Ettouney and Y. Al-Roumi, Multi-stage flash desalination: present and future outlook. Chem. Eng. J., 73 (1999) 173–190. [2] A. Macoun, Alleviating water shortages — lessons form the Middle East. Desalination Water Reuse, 10(2) (2000) 14–20. [3] S. Paerkh, M.M. Farid, J.R. Selman and S. Al-Hallaj, Solar desalination with a humidification–dehumidification technique — a comprehensive technical review. Desalination, 160 (2004) 167–186. [4] K. Bourouni, R. Martin, L. Tadrist and H. Tadrist, Experimental investigation of evaporation performances of a desalination prototype using the aeroevapo-condensation process. Desalination, 114 (1997) 111–128.

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[5] N.K. Nawayseh, M.M. Farid, A.A. Omar, S.M. AlHallaj and A.R. Tamimi, A simulation study to improve the performance of a solar humidification– dehumidification desalination unit constructed in Jordan. Desalination, 109 (1997) 277–284. [6] S. Al-Hallaj, M.M. Farid and A.R. Tamimi, Solar desalination with a humidification–dehumidification cycle: performance of the unit. Desalination, 120 (1998) 273–280. [7] Y.J. Dai and H.F. Zhang, Experimental investigation of a solar desalination unit with humidification– dehumidification. Desalination, 130 (2000) 169– 175. [8] S.A. Hashemifard and R. Azin, New experimental aspects of the carrier gas process (CGP). Desalination, 164 (2004) 125–133. [9] B.M. Hamieh, J.R. Beckman and M.D. Ybarra, The dewvaporation tower: an experimental and theoretical study with economic analysis. Desalination Water Reuse, 10(2) (2000) 35–43. [10] B.M. Hamieh, J.R. Beckman and M.D. Ybarra, Brackish and seawater desalination using a 20 ft2 dewvaporation tower. Desalination, 140 (2001) 217– 226. [11] M.M. Farid, S. Parekh, J.R. Selman and S. Al-Hallaj, Solar desalination with a humidification–dehumidificaition cycle: mathematical modeling of the unit. Desalination, 151 (2002) 153–164. [12] R.H. Xiong, S.C. Wang, Z. Wang, L.X. Xie, P.L. Li and A.M. Zhu, Experimental investigation of a vertical tubular desalination unit using humidification–dehumidification process. Chinese J. Chem. Eng., 13 (2005) 324–328. [13] R.H. Xiong, S.C. Wang, L.X. Xie, Z. Wang and P.L. Li, Experimental investigation of a baffled shell and tube desalination column using humidification– dehumidification process. Desalination, 180 (2005) 253–261. [14] N. Milosavljevic and P. Heikkilä, A comprehensive approach to cooling tower design. Appl. Therm. Eng., 21 (2001) 899–915. [15] N.K. Nawayseh, M.M. Farid, S. Al-Hallaj and A.R. Al-Tamimi, Solar desalination based on humidification process — I. Evaluating the heat and mass transfer coefficients. Energy Conv. Manage., 40 (1999) 1423–1439.