Heat transfer characteristics of in-tube steam condensation process under stratified flow

International Journal of Heat and Mass Transfer 145 (2019) 118798

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Heat transfer characteristics of in-tube steam condensation process under stratified flow Luyuan Gong, Zihao Zhong, Yali Guo, Shengqiang Shen ⇑ National Joint Engineering Research Center for Thermal Energy Integration, Liaoning Laboratory of Desalination, Dalian University of Technology, Dalian 116024, China

a r t i c l e

i n f o

Article history: Received 8 May 2019 Received in revised form 7 September 2019 Accepted 27 September 2019

Keywords: Falling film evaporator In-tube condensation Effective tube length Partition map

a b s t r a c t For stratified flow condensation process, the heat transfer mechanisms in the film condensation zone above the vapour-liquid interface and the condensate zone below the interface shows significant differences. In the multi-effect evaporation device, due to the high sensitivity of the thermal performance to the variation of the heat transfer parameters, the heat transfer parameters distributions in the film condensation zone and in the condensate zone are worth exploring. In this paper, a validated distributed parameter model was established to simulate the heat transfer and flow process in a single tube of the falling film evaporator. The variation of the heat flux in the condensate zone was presented under various operating conditions. The concept of the effective tube length and the partition map of the in-tube condensation heat transfer process were proposed for the better understanding of the flow and heat transfer process. Results show that the heat transfer in the condensate zone to the total heat transfer can accounts for more than 10% of the total heat transfer, thus cannot be neglected. For the design of falling film evaporator, the condensation heat transfer process in the tube should be located in the high-efficiency heat transfer section. The steam mass flow rate, heat transfer temperature difference and the length of the heat transfer tube should matched to each other to guarantee the condensation heat transfer process be located in the high-efficiency heat transfer zone. Ó 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The water shortage problem has become the bottleneck for the development of society. Multi-effect evaporation (MEE) desalination, owing the advantages of high heat transfer coefficient, high quality of fresh water, capable of utilize low-grade energy, has become one of the most popular desalination technologies [1]. Horizontal-tube falling film evaporator is the main heat transfer component in MEE desalination device (as shown in Fig. 1). It has the operating characteristics of ‘‘small temperature difference, low flow resistance, saturated state and high sensitivity” [2]. The device is sensitive to the variance of the thermodynamic parameters. Relatively small design deviation can leads to huge economic loss [3]. For the in-tube condensation process in the MEE desalination device, the in-tube steam mass flow rate is normally less than 10 kg/(m2
s). Under such operating condition, according to the flow pattern map of EI Hajal [4], the stratified flow is the most common flow pattern [5]. Under such flow pattern, the heat transfer coefficient in the condensate zone below the liquid-vapor inter face ⇑ Corresponding author. E-mail address: [email protected] (S. Shen).

show much lower values compared with the film condensation zone above the interface. Ursenbache et al. [6,7] developed a new non-intrusive computerized image analysis and optical observation method to accurately capturing the wetted angle and void fraction for stratified two-phase flow. Chato [8] assumed that the heat transfer at the bottom of the tube can be neglected. The heat transfer mechanism at the top of the tube is similar to the analysis of the condensation process on the vertical plate by Nusselt [9]. Later, Dobson and Chato [10] proposed new heat transfer model for stratified flow. In the model, the heat transfer at the bottom of the tube was not neglected. The liquid angle was introduced to calculate the average Nusselt number of the wavy stratified flow. The similar model was also proposed by Singh et al. [11]. Jaster and Kosky [12] added the parameter of void fraction (calculated by the model of Zivi et al. [13]) to Chato’s model [8], considering the impact of the liquid angle on the heat transfer performance. However, this correlation is not suitable for the larger tube diameter and high vapor density situations [14]. Ren et al. [15] considered the impact of the non-condensable gas on heat transfer performance and present the heat transfer parameters along the tube axial direction. Tandon et al. [16] proposed a new correlation for calculation condensation heat transfer coefficient for wavy flow with Reynolds number larger than 3  104.

https://doi.org/10.1016/j.ijheatmasstransfer.2019.118798 0017-9310/Ó 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

Nomenclature A d F G g h K k L N Nu p Pr Q Re r S T Xtt x

cross section area, m2 diameter, m heat transfer area, m2 mass flow rate, kg/(m2
s) gravity, m/s2 local heat transfer coefficient, W/(m2
K) overall heat transfer coefficient, W/(m2
K) conductivity, W/(m
K) length, m number of nodes Nusselt number pressure, Pa Prandtl number heat transfer, W Reynolds number latent heat, J/kg salinity, % temperature, °C Lockhart-Martinelli number vapor quality

h

C

angle, ° dynamic viscosity, Pa
s density, kg/m3 surface tension, N/m spray density, kg/(m
s)

Subscript bot e eff in l out st sw tt top tp vo w wet

lower part of the tube evaporation effective inlet; inside the tube liquid outlet; outside the tube steam seawater total upper part of the tube two phase vapor only wall wet

l q r

Greek symbol a void fraction

Fig. 1. Falling film evaporation evaporator in MEE desalination system.

Feng et al. [17] found that the inertia force and gravity force are the major parameters that affect the flow pattern and condensation heat transfer inside the tube. A new correlation was proposed using the non-dimensional parameter of Froude number. In experimental study of Shen et al. [18], it was found that at the bottom of the tube, although the heat transfer coefficient has much lower values than the top, the local temperature difference between the condensate and the tube wall has much larger values at the bottom than the top. Rosson and Meyers [19] proposed the correlation for calculating the heat transfer coefficient in the film condensation zone and convection condensation zone. The vapor Reynolds number was added to Chato’s model for consideration of the effect of the vapor velocity on film condensation heat transfer. Thome et al. [20] defined the liquid level angle as the ratio of the length of wall covered by the film condensation zone to the entire circumferential length. The average heat transfer coefficient of condensation heat transfer is calculated as the weighted average

of the film condensation heat transfer coefficient above the liquidvapor interface and the convective condensation heat transfer coefficient below. Zhang [21] conducted 2-dimensional simulation of the condensation heat transfer process, the distributions of heat transfer coefficient and film thickness along the tube axial direction were shown under different steam velocities, surface tensions and tube diameters. Kouhikamali [22] numerically simulated 2dimensional force convective condensation in 4 mm circular cylindrical channel. It was found that the condensation heat transfer increases with the steam velocities but decreases with the tube diameter and the temperature difference. Due to the characteristics of ‘‘high sensitivity” of MEE desalination system, the operating parameters of the system are normally limited to a narrow range. Under specific operating conditions, the local temperature difference might be zero and the reverse heat transfer phenomenon may occur which severely deteriorates the local heat transfer performance [3]. The detailed variation of

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

the heat transfer parameters in the falling film evaporator is worth of detailed study. Coupling the heat transfer process of the falling film evaporation outside the tube and the condensation inside the tube is interesting because the heat transfer parameters are non-uniformly distributed on both sides. Especially under stratified flow pattern, the inner tube is divided into different zones by the liquid-vapor interface. In this paper, a validated model was established to simulate the heat transfer and flow process under stratified flow. The heat transfer and flow process were coupled for both the inside and outside of the tube. The effective tube length and the partition map of in-tube condensation heat transfer process are proposed for the better understanding and design of the condensation process in the falling film evaporator for desalination. 2. Physical and mathematical model 2.1. Physical model The falling film evaporator in the low-temperature multi-effect evaporation seawater desalination (LT-MED) system contains a large number of heat transfer tubes. In the evaporator, the steam flows inside the tubes and condenses, releasing its latent heat to the seawater outside the tube. The most commonly in-tube flow in LT-MED device is the stratified flow. On the shell side, the seawater flows from the upper tubes to the lower ones by gravity, forming falling films on the outer surfaces of the tubes. Part of the seawater evaporates when heated by the in-tube steam and the rest continues to flow downward. The physical model of the heat transfer tube is shown in Fig. 2. 2.2. Mathematical model The distributed parameter model is adopted to calculate the heat transfer and flow characteristics in a horizontal tube.

3

Uniform grids throughout the calculation domain are adopted. As shown in Fig. 2(b). Along the tube axial and circumferential directions, the tube is divided into a two-dimensional network with identical elements. Along the tube axial direction, the tube wall is divided into Ni calculation nodes. Each node represents a small section of tube with the length of dL. i = 1 represents that the node is at the steam inlet and i = Ni reprents that the node is at the steam outlet. Considering that the flow and heat transfer characteristics inside and outside the horizontal tube show horizontally symmetric distribution, half of the tube wall is selected as the calculation area for reducing the calculation load. For each selected tube with the length of dL, the inner tube is divided into Nj nodes from the bottom to top of the tube. When Nj = 1, it means that the calculation node is at the bottom of the tube and j = Nj the top. For each divided grid point, the local pressure drop, the wetted angle, the steam heat transfer coefficients in different zones are calculated. The mathematical model established in this paper is based on the following assumptions: (1) The temperature, spray density and salinity of the seawater outside the tube remain unchanged along the tube axial direction; (2) The heat released by the steam in the tube is completely absorbed by the seawater outside the tube, ignoring the heat loss; (3) The effects of gravity on the distribution of both the in-tube condensation film thickness and the local condensation heat transfer coefficient in the film condensation zone are not considered; (4) The fouling resistance of the heat transfer tube is ignored. The heat conduction in the circumferential direction and axial direction of the tube wall are ignored. For each element, the governing equations of mass, momentum and energy and be expressed as follows:

(a) Physical model

(b) Mesh division Fig. 2. Schematic diagram of the physical model and mesh division of the heat transfer tube.

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L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

Gin ¼ Gout

ð1Þ

pst;out ¼ pst;in  Dptotal

ð2Þ

Q ¼ hin F in ðT st  T w Þ ¼ hout F out ðT w  T sw Þ

ð3Þ

For the calculation of the local pressure drop inside a horizontal tube, several models are proposed. For homogeneous model, liquid and vapor flow are assumed to have the same velocty. For the stperated model, the vapor and liquid phases flow at different velocities. Most of the previously proposed models were based on the experimental results with the working fluids of refrigerant [5,23–26]. Recently, Shen et al. [5] experimentally measured the local pressure drop of steam in a horizontal tube. The uncertainty of the obtained pressure drop is less than 0.5%. Based on the experimental data, a new correlation for calculating the steam pressure drop was proposed as shown in Eq. (4).

Dp G2 1:03 x ¼ 0:03 DL 2dqtp




ll ls

0:02 ð4Þ

For the calculation of the local heat transfer coefficient under the gravity-driven flow regim, most of the correlations [8,12] were based on the Nusselt’s model [9]. This model is based on the analys of pure vapor condensation on a vertical wall. Latter scholars improved the model by calculating the condensation heat transfer coefficient above liquid-gas inter-face based on the Nusselt’s model but using different model below the liquid-gas interface [10,11]. The calculation of void fraction and wetted angle became the key parameters for calculating the local condensation heat transfer coefficiet. Rouhani and Axelsson [27] proposed a correlation for calculating the void fraction based on the drift flux flow model as shown in Eq. (5). The drift flux model was firstly proposed by Zuber and Findlay [28] in which the drift velocity was used to represent the difference between the average velocities of the liquid phase and the gas phase. For the various models that calculates the void fraction, the model of Rouhani and Axelsson is among the most widely used. Biberg [29] proposed a correlation for calculating the wetted angle of the stratified flow at a given void fraction, as shown in Eq. (6). The void fraction model proposed by Rouhani and Axessoncan well predict the wetted angle with relative low steam flow rate as shown in Fig. 3.

a¼

x

(




½1 þ 0:2ð1  xÞ

qs

x

qs

þ

1x

ql

 þ

 0:25 )1 1:18 g rðql  qs Þ G q2l ð5Þ


1=3 h i 3p 1  2ð1  aÞ þ ð1  aÞ1=3  a1=3 2 h  i 1  ð1  aÞa½1  2ð1  aÞ 1 þ 4 ð1  aÞ2 þ a2 200

hwet ¼ pð1  aÞ þ

ð6Þ

Shen et al. [30] compare the previous correlations for calculating the condensation heat transfer coefficient with their experimental data and found remarkable deviations between them. In their experimental study, the uncertainty of the obtained condensation heat transfer coefficient is less than 15.7%. Based on the experimental date, a new set of correlations for calculating the local condensation heat transfer coefficient in both the film condensation zone and the condensate zone inside a circular tube were proposed as shown in Eqs. (7) and (8) [30]. For the calculation of the heat transfer coefficient in the film condensation zone, the mean relative erro of the model is 1.3%. For the calculation of the heat transfer coefficient in the condensate zone, the mean relative error are 4.4% and 0.9% at 0° and 20°. But it is up to 25% at 40°.

Nutop ¼

0:02Re0:318 vo 1 þ 1:11X 0:755 tt

"

ql ðql  qs Þgrd3 kl ll ðT s  T w Þ "

Nubot;h ¼

0:33 0:033Re0:8 l Pr l




ql 1þ qs

#0:25

0:5 

ð7Þ #0:8
1:31 x  coshwet 1x cosh ð8Þ

Shen et al. [31] conducted an experimental study on the falling film evaporation of seawater outside the horizontal tube with an outer diameter of 19 mm and obtained the distribution of the local falling film heat transfer coefficient hout along the circumferential direction. Experimental results show that the local out-tube falling film evaporation heat transfer coefficient is independent of the heat flux. Based on assumption (1), for any angular position on the wall, the local falling film evaporation heat transfer coefficient remains unchanged along the tube axial direction. Combining Eqs. (4)–(8) with the experimental data of the seawater evaporation heat transfer process outside the tube, the mathematical model of coupling the flow and heat transfer process in and outside a single heat transfer tube in the falling film evaporator is established. The in-tube and out-tube heat transfer coefficients of the nodes (i,j) on the tube wall are hin,(i,j) and hout(i,j), respectively. The overall heat transfer coefficient Ki,j of the nodes (i,j) on the tube wall can be calculated by the following equation:

K i;j ¼

1 1 hin;ði;jÞ

din þ 2k ln ddout þ h w in

din

ð9Þ

out;ði;jÞ dout

In Eq. (9), din and dout are the inner and outer diameters of the heat transfer tube, respectively. kw is the thermal conductivity of the tube. The boundary conditions are as follows: (1) the mass flow rate of seawater outside the tube and steam inside the tube are given; (2) the saturation temperature of the seawater and steam are specified; (3) the steam enters the tube with the initial vapor quality of 1. 2.3. Outline of solution process

Fig. 3. Comparison between experimental value and predicted value of wetted angle.

The algorithm of the model is shown in Fig. 4. The initiated parameters required in the calculation process are the total tube length L, the steam inlet temperature Tin, the steam mass flow rate

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

Fig. 4. . The flow chart of numerical calculation.

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L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

G, the temperature of seawater outside the tube Tsw, the spray density C and the salinity of seawater S. Vapor quality at the inlet of the tube is 1. The calculation procedure is explained in the following steps: (1) The heat transfer tube is divided into N  181 network, and the calculation starts at i = 1. (2) In the ith section of the tube axial direction, Eq. (5) is used to calculate the two-phase flow void faction ai. Eq. (6) is used to calculate the stratified flow wetted angle hwet,i. Eq. (4) is sued to calculate the local pressure drop Dpi of the twophase flow. (3) For each crosssection of the tube, the tube is divided into two parts by the wetted angle hwet,i. Starting from the position of j = 1 at the bottom of the tube, if the position angle is greater than the wetted angle hwet,i, the node is in the filmlike condensation zone and the local in-tube heat transfer coefficient is calculated by Eq. (7). If the position angle of the node is less than or equal to the wetted angle hwet,i, the node is in the condensate zone, and the local in-tube heat transfer coefficient is calculated by Eq. (8). The overall heat transfer coefficient Ki,j of the node (i,j) on the tubewall is calculated using Eq. (9) . (4) For each cross section of the tube, repeat step (3) until j = 181. (5) The calculation of the overall heat transfer Qi of the ith node along the axial direction. Taking into account the symmetry characteristics of the flow and heat transfer parameters in the cross-section, the calculation method is shown in Eq. (10). The variation of the Dxi of the steam is calculated by Eq. (11).

Q i ¼ Q i;1 þ 2 

180 X

Q i;j þ Q i;181

ð10Þ

Table 1 The geometrical parameters and operation conditions of heat transfer tube. Parameter

Description

tube inner diameter wall thickness tube length steam inlet temperature steam mass flow rate seawater temperature seawater spray density seawater salinity

18 mm 0.5 mm 6m 63 °C 6 kg/(m2
s) 60 °C 0.043 kg/(m
s) 3%

Fig. 5. The grid independence verification.

j¼2

Dxi ¼

Qi rGA

ð11Þ

where r is the latent heat of the steam, G is the inlet mass flow rate of steam, A is the cross-sectional area of the tube. (6) The vapor quality xi+1 and local pressure pi+1 at the (i + 1)th node can be calculated by Eqs. (12) and (13). Then the steam temperature Ts,i+1 is determined according to the water vapor property table.

xiþ1 ¼ xi  Dxi

ð12Þ

piþ1 ¼ pi  ðDp=DLÞi dL

ð13Þ

(7) Repeating the step 2–5 until either xi+1 or dTi+1 equals to 0, or i equals to N. 2.4. Grid independence The geometric parameters of the tube as well as the operating conditions are shown in Table 1. Along the circumferential direction, half of the tube is divided into 181, 201, and 226 discrete nodes respectively. The calculation of the total heat transfer Q varies less that 0.03%. Thus, in the mesh generation process, the tube is divided at an interval of 1°along the circumferential direction. In the tube axial direction, four types of tube length increment dL are selected that are 0.1 m, 0.05 m, 0.01 m and 0.005 m, respectively. The calculation results of the total heat transfer Q under above four

dL are shown in Fig. 5. When dL is 0.01 m and 0.005 m, the total heat transfer is 2647 W and 2645 W, respectively, with the latter 0.05% less than the former. It is considered that when dL is less than or equal to 0.01 m, the calculation result of the model is independent of the value of dL. Therefore, dL is chose to be 0.01 m in the subsequent calculations in this paper. The impact of the condensate zone on the heat transfer performance under various operating conditions were analyzed in detail. A new partition map of in-tube condensation heat transfer process is proposed for the better understanding and design of the falling film evaporator. 3. Results and discussions 3.1. Distribution of heat transfer at different areas on the tube wall For the stratified condensation heat transfer process in the horizontal tube, the tube wall is divided into two zones by the stratified vapor-liquid interface: the film condensation zone above the interface and the condensate zone below. The heat flux in the condensate zone is smaller than that in the film condensation zone. The proportion of the heat transfer in the condensation zone to the overall heat transfer is shown in Fig. 6 with the steam inlet temperature in the tube Tst of 62 °C, the steam mass flow rate Gst varying between 2 and 10 kg/(m2
s), and the temperature of the seawater outside the tube Tsw of 60 °C. It can be seen that the proportion of the heat transfer of the condensate zone to that of the entire cross-section of the tube (hereafter referred as ‘‘the proportion”) increases first and then decreases. In the front part of the tube, the condensate at the bottom of the tube has relatively low liquid level. However, near the steam inlet, the area of the tube

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

Fig. 6. Variation of the proportion of heat transfer per unit length of the condensate zone along the tube axial direction under different mass flow rates.

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the proportion exhibit lower values compared with 6 and 8 kg/(m2
s). It is because under such high Gst, the area of the wall surface covered by the condensate is remarkably reduced which result in the decrease of the heat transfer in the condensate zone. Fig. 7 shows the proportion along the tube axial direction when maintaining Gst of 5 kg/(m2
s), the total temperature difference, DTtt, of 2 °C, and setting the evaporation temperature of the seawater outside the tube, Te, of 50 °C, 60 °C and 70 °C, respectively. It can be seen that the proportion firstly increases, reaching a maximum near the tube length of 2.2 m, and then decreases slowly along the tube axial direction. In the first 1 m of the tube, the impact of the saturation temperature on the proportion is not obvious. In the region longer than 1 m away from the steam inlet, the proportion decreases with the increase of steam saturation temperature. This is due to the difference between the heat transfer mechanisms in the film condensation zone and the condensate zone. In the film condensation zone, the heat transfer is dominated by heat conduction between the tube wall and the thin liquid film. As the temperature increases, the decrease of the water viscosity leads to thinner film thickness. Thus, the heat transfer coefficient in this area has higher values. For the condensate zone, the heat transfer is dominated by the convection heat transfer between the condensate and the tube wall. Under the same inlet flow mass rate, the increase of the temperature reduces the density of the steam. The shear force of the steam to the condensate layer at the bottom is weakened which leads to smaller heat flux in the condensate zone. In the condensate zone, the proportion decreases with the increment of the saturate temperature. Fig. 8 shows the variation of the proportion along the tube axial direction when setting DTtt of 2 °C, 3 °C and 4 °C, respectively, maintaining Tsw of 60 °C, the spray density Csw of 0.043 kg/(m
s), the salinity Ssw of 3%, and Gst of 6 kg/(m2
s). In the first 2 m of the tube, the proportion increases with the increment of DTtt. With the increase of DTtt, the film thickness is increased in the film condensation zone which leads to the decrease of the heat transfer coefficient. On the other hand, in the condensate zone, the axial flow velocity of the condensate remains almost unchanged with the increase of the temperature. The heat transfer coefficient in the condensate zone remains almost unchanged. Thus the proportion has larger values under larger temperature difference. As the steam flows towards the outlet, the vapor quality gradually dominates the heat transfer. At the same position of the tube, the larger

Fig. 7. The variation of the proportion of heat transfer per unit length of the condensate zone along the tube axial direction at different steam saturation temperatures.

wall covered by the condensate increases more rapidly along the tube axial direction compared with the latter part of the tube due to the larger ratio of the arcuate area to the condensate height. Thus, the proportion per unit length shows increasing trend along the tube axial direction. As the liquid level of condensate continuously increases, the heat transfer in the condensate zone is weakened because of the increase of the thickness of the condensate layer. The proportion is then reduced. When Gst is 2 kg/(m2
s), the proportion reaches its maximum at 1 m away from the steam inlet, which is 7.1%, followed by a decreasing trend in the latter part of the tube. When Gst is increased, both the shear force between the vapor and liquid interface and the velocity of condensate at the bottom of the tube increase. The proportion tends to have higher values at lager Gst. Moreover, under higher Gst, the steam has relatively higher velocity at the steam outlet, which leads to smaller decreasing amplitude of the proportion. For example, when Gst is 6 kg/(m2
s), the proportion reaches 10% at the tube length of 2.2 m, but for 2.2 m after, the proportion remains almost unchanged. It is interesting that when Gst is 10 kg/(m2
s),

Fig. 8. The variation of the proportion of heat transfer per unit length of the condensate zone along the tube axial direction under different total temperature differences.

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L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

DTtt, the lower the vapor quality and the thicker of the condensate layer at the bottom of the tube. Therefore, in the region longer than 3.3 m away from the steam inlet, the proportion exhibits smaller values at larger DTtt. The flow resistance of the vapor-liquid two-phase flow along the tube axial direction causes the decreases of the steam saturation temperature. The steam saturation temperature is more sensitive to the change of the saturation pressure under vacuum conditions. In this paper, the difference between the temperature of the steam at the tube inlet and the temperature of the seawater outside the tube is defined as the total heat transfer temperature difference. Then, the film condensation zone introduced above is further divided into two zones. The area where the local condensation temperature difference is less than 30% of DTtt is defined as the low temperature difference zone, and the rest of the film condensation zone(where the local temperature difference is no less than 30% of DTtt) as the high temperature difference zone. Based on such division, the tube wall is divided into the condensation zone where the tube is covered by the condensate at the bottom of the tube, the low and the high temperature difference zones depending on the value of the local temperature difference. Fig. 9 demonstrates the area of the three zones and the heat transfer proportion of each zone to the total heat transfer under the operating condition that the tube length of 6 m, the steam inlet temperature of Tst 62 °C, the sea water temperature outside the tube of Tsw 60 °C, and Gst varying between 6 and 10 kg/(m2
s). When Gst is lower than 6 kg/(m2
s), the steam temperature has relatively smaller decrease along the tube axial direction, so the area of the low temperature difference zone is small which leads to the unobtrusive impact of the insufficient temperature difference section on the heat transfer. With the increment of Gst, the height of the condensate layer decreases due to the increase of the shear force of the steam to the condensate layer. The ratio of the area of the condensate zone decreases from 21.4% to 14.8% when Gst is increased from 6 to 10 kg/(m2
s). The heat flux in the condensate zone is smaller compared with in the film condensation zone, so the heat transfer proportion of the condensate zone is less than the area ratio as shown in Fig. 9(2). The proportion of heat transfer slightly decreases from 9.3% to 8.0% when Gst is increased from 6 to 10 kg/(m2
s). The increase of Gst leads to the increase of the velocity of the condensate layer and reduces the condensate height, enhancing the heat transfer in the condensate zone. Therefore, the heat transfer proportion decrease at a slower speed than the area ratio with the increment of Gst. Under larger Gst, due to the larger pressure drop, the steam saturation temperature decreases more remarkably along the tube

(1) Ratio of area

axial direction. For a certain location of the tube, the local condensation temperature tends to have smaller values under larger Gst which leads to the remarkable increase of the area of the low temperature difference zone. The ratio of the low temperature difference area is increased from 10.6% to 52.7% when Gst is increased from 6 to 10 kg/(m2
s).When Gst is chosen to have too large values, for part of the tube wall, the utilization of DTtt is greatly reduced due to significant decrease of the local temperature difference. The proportion in this part also increases with the increase of Gst. But the proportion is smaller than the area ratio, and the difference between the two (the heat transfer ratio and the area ratio) increases with the increment of Gst. Due to the decrease in steam saturation temperature under larger Gst, the tube wall area of the high temperature difference zone is significantly reduced, and the area ratio is reduced from 68% to 32.6%. In the three different tube wall zones, the heat flux in the high temperature difference zone is the largest, and the smallest is in the condensate zone. Therefore, the heat transfer proportion in the high temperature difference zone is larger than the area ratio. For the high temperature difference zone, the proportion decreases with the increase of Gst. But the decrease is less than that of the area ratio. The proportion decreases from 80.6% to 49.8% when Gst is increased from 6 to 10 kg/(m2
s). Fig. 10 demonstrates the ratio of wall area and heat transfer in the three zones under different DTtt with the total tube length Ltt of 6 m, Tsw of 60 °C, and Gst of 8 kg/(m2
s). Under larger DTtt, more steam is condensed into liquid. The height of the liquid layer at the lower part of the tube increases which result in the increase of the area of the condensate zone. The ratio of the area of the condensate zone is increased from 11.8% to 24.3% when DTtt is increased from 1 °C to 4 °C. Although the area of the condensate zone is greatly increased, the increase of the height of the condensate layer greatly reduce the heat transfer coefficient. The proportion of heat transfer in the condensate zone increases only by 4%. The effect of steam flow resistance on the condensation heat transfer is significant if DTtt has relatively small values. When DTtt is only 1 °C, the ratio of the area of the low temperature difference zone is as high as 65.8%, and the proportion of heat transfer is 58.8%. As DTtt increases, the sensitivity of the heat transfer to the flow resistance gradually decreases, and the proportion of the area of the low temperature difference zone is significantly reduced. When DTtt is 2 °C and 3 °C, the ratio of the area of the low temperature difference zone is 35.1% and 9.3%, respectively. The proportion of heat transfer is also slightly reduced. When DTtt reaches 4 °C, the low temperature difference zone disappears. The ratio of the area and heat transfer of the high temperature difference area

(2) Ratio of heat transfer

Fig. 9. Ratio of wall area and heat transfer in three zones at different mass flow rates.

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

(1) Ratio of area

9

(2) Ratio of heat transfer

Fig. 10. Proportion of area and heat transfer in three zones under different total temperature differences.

(1) Effective tube length

(2) Total heat transfer of the tube

Fig. 11. Effect of steam mass flow rate and temperature difference of the total heat transfer performance of tubes.

both increase significantly with the increase of DTtt. In the high temperature difference area, the proportion of heat transfer is slightly higher than the ratio of the area. For the condensation heat transfer process in the horizontal tube, the condensate accumulates at the bottom of the tube, and the steam pressure decreases which leads to the decrease of the local steam temperature difference. Above analysis shows that when Gst is greater than 6 kg/(m2
s), as Gst increases, the influence of the condensate at the lower part of the tube on the heat transfer is weakened, while the effect of the steam saturation temperature on heat transfer is enhanced. The area of the high temperature difference zone (which has the best heat transfer performance among the three zones) is significantly reduced, resulting in a decrease in the total heat transfer of the entire tube. With the increment of DTtt, the influence of condensate zone on heat transfer increases, and the effect of the steam saturation temperature is weakened. The area of the high temperature difference zone increases, directly leading to the increase of the total heat transfer of the entire tube.

3.2. Variation of the effective heat transfer length and total heat transfer Fig. 11 demonstrates the variation of the effective length of a tube with Ltt of 6 m, Tsw of 60 °C, Tst varying from 60.5 °C to 64 °C, and Gst increasing from 1 to 10 kg/(m2
s). The effective tube length Leff is defined as the maximum tube length within which the local temperature difference is larger than 0 °C or the vapor quality is larger than 0. When DTtt is only 0.5 °C and Gst is between 1 and 3 kg/(m2
s), Leff equals to Ltt, which is 6 m. With the increase of Gst, the steam temperature at the steam outlet gradually decreases to 0 °C. Leff decreases remarkably. When Gst reaches 10 kg/(m2
s), the Leff is only 0.55 m. When DTtt is between 1 and 2.5 °C, Leff increases to 6 m, reaching a plateau, and followed by a decreasing trend as the further increase of the steam flow mass rate. When Gst is relatively small, Leff is small due to the complete condensation of the steam. On the other hand, when Gst is large, Leff is small because the temperature difference is reduced to 0 °C. Under the condition

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that DTtt is larger than 3 °C (and the total tube length of 6 m), it is the complete condensation of steam rather than the local temperature difference reducing to 0 °C that leads to the decrease of Leff. Therefore, Leff increases firstly with the increase of Gst, and then remains unchanged to 6 m. It is indicated that selecting an appropriate Gst is beneficial for preventing the steam from completely condensed before the steam outlet and guaranteeing the local temperature difference larger that 0 °C. The maximum allowed Gst increases with the increase of the DTtt for guaranteeing Leff equals to Ltt. When Gst is smaller than 3 kg/(m2
s), the increase of DTtt leads to the decrease in the heat transfer length required for the complete condensation of the steam at the steam outlet. Thus Leff monotonously decreases with the increment of Ttt. When Gst is between 4 and 6 kg/(m2
s), the local heat transfer temperature is reduced to 0 °C under the small temperature difference. Therefore, with the increase of DTtt, Leff firstly increases due to that the location where the local temperature difference is 0 °C gradually approaches the steam outlet. Then Leff reaches plateau which is 6 m. As the further increase of DTtt, Leff is reduced due to the complete condensation of steam. It is noted that when Gst is greater than 7 kg/(m2
s), the steam cannot be completely condensed in the tube. Leff increases with DTtt until it reaches the total tube length. The total heat transfer of the tube has different values with the change of Gst and DTtt. With the increase of Gst, the total heat transfer shows an increasing trend followed by a decreasing trend. The increase of Gst is beneficial for the enhancement of the condensation heat transfer. However, when Gst is too large, the steam saturation temperature in the tube decreases rapidly along the tube axial direction which leads to the remarkable decrease of the local temperature difference. Under the operating condition that DTtt is 0.5 °C, the total heat transfer amount reaches a maximum value of 553 W when Gst is 2 kg/(m2
s). As Gst further increases to 10 kg/(m2
s), the total heat transfer is reduced to only 49 W. Under larger DTtt, more steam is condense in the tube which leads to the reduce of the local steam velocity as well as the local temperature difference drop. Thus the maximum heat transfer and the corresponding steam mass flow rate also increase. When DTtt is 4 °C, the total heat transfer of the tube reaches a maximum value of 4249 W, which is 7.8 times the maximum heat transfer when DTtt is 0.5 °C. When Gst is smaller than 7 kg/(m2
s), as the increase of DTtt, the maximum heat transfer shows an increasing trend before reaching a plateau. This is because that under large DTtt, the steam is completely condensed. The maximum heat transfer amount is equal to the latent heat released by the completely condensing of the steam. When Gst is greater than or equal to 7 kg/(m2
s), the steam cannot be completely condense even DTtt reaches 4 °C. The maximum heat transfer increases monotonously with the increase of DTtt. At a given tube length, when the operating conditions outside the tube, such as the temperature, salinity and spray density of the seawater are determined, the parameters that affect the heat transfer performance of the tube are only the in-tube temperature and mass flow rate at the steam inlet. The partition map of in-tube condensation heat transfer process is proposed according to whether the effective tube length equals to the total tube length. The in-tube condensation heat transfer process is divided into three sections: the high-efficiency heat transfer section, the insufficient temperature difference section and the insufficient mass flow rate section. The partition map is shown in Fig. 12 with Ltt of 6 m, Tsw of 60 °C, Csw of 0.043 kg/(m s) and Ssw of 3%. The insufficient mass flow rate section is located at the lower right part of Fig. 12. In this section, Gst has relatively lower values while DTtt has relatively larger values. The vapor quality decreases rapidly along the tube axial direction, and the steam is condensed

Fig. 12. Three areas of condensation heat transfer in the tube.

completely before reaching the steam outlet. It makes Leff shorter than Ltt. It indicates the insufficiency of the steam mass flow rate, and the insufficient utilization of the heat transfer area. In Fig. 12, the insufficient mass flow rate section accounts for 37.6% of the sum of the three sections. In the middle part of Fig. 12, it is the high-efficiency heat transfer section in which both the vapor quality and the steam saturation temperature have relatively slower drop along the tube axial direction. In this section, Leff remains 6 m which is Ltt. The upper and lower limits of the steam mass flow rate in the high-efficiency heat transfer section are the corresponding steam mass flow rates when the vapor quality reaches 0 and the local temperature difference reaches 0 °C at the steam outlet, respectively. When DTtt is 0.5 °C, the upper and lower limits of the steam mass flow rate in the high efficiency heat transfer section are 3.5 and 0.8 kg/(m2
s), respectively. As DTtt increases, the upper and lower limits of the steam mass flow rate in the high efficiency heat transfer section also increase. It can be seen from Fig. 7 that the lower limit of mass flow rate (corresponding to the appearance that the steam is completely condensed at steam outlet) increases almost linearly with the increase of DTtt. On the other hand, the upper limit of the steam mass flow rate (corresponding to the appearance that the local temperature difference reaches 0 °C at the steam outlet) increases with the increase of DTtt, but the increase rate gradually decreases. In this paper, the steam mass flow rate is set to be less than or equal to 10 kg/(m2
s). When DTtt is greater than 2.6 °C, even if the steam mass flow rate reaches 10 kg/(m2
s), the temperature difference at the steam outlet is larger than 0 °C. Therefore, only the insufficient mass flow rate section and the high-efficiency heat transfer section exist when DTtt is greater than 2.6 °C with the disappearance of the insufficient temperature difference section. Among the three sections, the high-efficiency heat transfer section accounts for the largest part which is 41.9% of the entire section. The insufficient temperature section is located in the upper left part of Fig. 12. In this section, Gst is relatively higher while DTtt is relatively smaller. Due to the larger vapor flow pressure drop, the steam saturation temperature decreases rapidly along the tube axial direction. The local temperature difference is reduced to 0 °C which greatly reduce the effective heat transfer area as well as the total heat transfer. The insufficient temperature difference section accounts for the smallest proportion, which is only 20.5% of the entire section. The total heat transfer of the tube has a maximum value in the high-efficiency heat transfer section at a given DTtt. The steam mass flow rate corresponding to the maximum heat transfer is

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

shown by the solid line connected by the red dots in Fig. 12. The maximum heat transfer and its corresponding steam mass flow rate increase with the increase of DTtt, but the increasing rate gradually decreases. When DTtt is increased from 0.5 to 4 °C, the maximum heat transfer increases from 553 to 4252 W and the steam mass flow rate corresponding to the maximum heat transfer increases from 1.9 to 8.2 kg/(m2
s). The steam mass flow rate corresponding to the vapor quality of 0.2 at the outlet of the tube is represented by the dotted square points in the high efficiency heat transfer section in Fig. 12. As DTtt increases, the difference between the steam mass flow rates corresponding to the outlet mass vapor quality of 0.2 and of 0 gradually increases. When DTtt is 0.5 °C, such difference is 0.2 kg/(m2
s), and when DTtt is 4 °C, the difference is 1.8 kg/(m2
s). When the local temperature difference of the tube is close to 0 °C, the heat transfer is very low, Increasing DTtt at the outlet of the tube helps to improve the local heat transfer performance. The dotted line connected by triangular points in the high efficiency heat transfer section in Fig. 12 represents the steam mass flow rate corresponding to DTtt at the steam outlet of 0.3 °C. This mass flow rate is slightly lower than the steam mass flow rate corresponding to the temperature difference at the outlet of 0 °C, and the difference between the two decreases with the increase of DTtt. The insufficient mass flow rate section and the low temperature difference section can be categorized as the inefficient heat transfer section. When the heat transfer process is located in the inefficient heat transfer section, due to the low local vapor quality or the small local temperature difference, the heat transfer area of the tube is utilized at lower efficiency, which leads to smaller total heat transfer of the entire tube. For the design of the falling film evaporator in multi-effect evaporation desalination system, the condensation heat transfer process in the tube should be working in the high efficiency heat transfer section. By optimizing the matching of the steam mass flow rate, the temperature difference and the tube length, the heat transfer area of the evaporator can be better utilized, and the larger total heat transfer can be obtained. Fig. 13 show the upper and lower limits of the steam mass flow rate in the high efficiency heat transfer section when Tsw is 50 °C, 60 °C and 70 °C, respectively. As mentioned above, with other operating conditions of the same, the total heat transfer of the tube increases slightly with the increase of the saturation temperature; On the other hand, the latent heat of the steam slightly decreases

Fig. 13. The upper and lower limits of steam mass flow rate in high efficiency heat transfer section at different saturation temperatures.

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with the increase of the saturation temperature. Therefore, under high saturation temperature, more steam is needed for the complete condensation at the steam outlet. That is, the lower limit of the steam mass flow rate in the high efficiency heat transfer section increases slightly with the increase of the saturation temperature. It is indicated in Fig. 13 that there is no significant impact of the temperature on the lower limit of the high efficiency heat transfer section. When DTtt is smaller than 2.0 °C, the lower limit of the mass flow rate at the three saturation temperatures are almost of the same. When DTtt reaches 4 °C, the difference of the lower limit of the steam mass flow rate in the high efficiency heat transfer section between 50 °C and 70 °C is only 0.8 kg/(m2
s). The proportions of the areas of the insufficient mass flow rate at the three saturation temperatures are 35.6%, 37.6% and 38.4%, respectively. Under the same steam inlet mass flow rate, the local steam pressure drop increases significantly with the decrease of saturation temperature due to the lareger specific volume. Moreover, the variation of the steam saturation temperature is more sensitive to the pressure drop at lower saturation temperatures. Therefore, under the same inlet temperature difference, the local temperature difference decreases more rapidly along the tube axial direction at lower saturation temperatures. The upper limit of the mass flow rate in the high-efficiency heat transfer section decreases remarkably with the decrease of the saturated temperature. It can be seen from Fig. 13 that when Te is 50 °C and DTtt reaches 4.0 °C, the insufficient temperature difference section still exist. With the increase of saturation temperature, the upper limit of steam mass flow rate increases rapidly to the maximum value of steam mass flow rate selected in this paper which is 10 kg/(m2
s). The range of insufficient temperature difference section decreases remarkablely. When Te is 70 °C, the insufficient temperature difference section exists only when DTtt is less than 1.6 °C. Under three saturation temperatures, the proportions of high-efficiency heat transfer sections are 25.6%, 41.9% and 52.2%, respectively and the proportions of the insufficient temperature difference sections are 38.8%, 20.5% and 9.4% respectively. Therefore, as the increase of the saturation temperature of the seawater outside the tube, the area of the insufficient mass flow rate section slightly increases. The lower limit of the mass flow rate of the high-efficiency heat transfer section increases slightly. The range of high-efficient heat transfer sections is greatly increased and the range of the insufficient temperature difference is significantly reduced. When Te is 60 °C and 70 °C, meanwhile, DTtt is greater than 2.6 °C and 1.6 °C correspondently, the condensation heat transfer will not be ceased by the decrease of the local temperature difference. Therefore the insufficient temperature difference does not exist under such operating conditions. Fig. 14 shows the variance of the maximum heat transfer and the corresponding steam mass flow rate under different DTtt. As described above, under the same operating conditions, the total heat transfer of the tube increases slightly with the increment of the saturation temperature. Therefore, the maximum heat transfer and the corresponding steam mass flow rate slightly increase with the increment of the saturation temperature. Both increase almost linearly with the increase of DTtt. The higher the saturation temperature, the greater the increase. When DTtt is only 0.5 °C, the maximum heat transfer of the tube at three saturation temperatures (50 °C, 60 °C and 70 °C) are 518 W, 553 W and 587 W, respectively, with the maximum difference of 13.3%. The corresponding steam mass flow rates are 1.4 kg/(m2
s), 1.9 kg/(m2
s) and 2.6 kg/(m2
s). When DTtt is 4 °C, the maximum heat transfer at three saturation temperatures are 3857 W, 4252 W and 4574 W, respectively, with the maximum difference of 18.6%. When Te are 50 °C and 60 °C, the steam mass flow rates corresponding to the maximum heat transfer the are 6.6 kg/(m2
s) and 8.2 kg/(m2
s). When Te is 70 °C, the steam mass flow rate

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L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

Fig. 14. Maximum heat transfer capacity and corresponding mass flow rate at different saturation temperatures.

corresponding to the maximum heat transfer exceeds the calculation range of steam mass flow rate selected in this paper. In Fig. 15, the variance of the upper and lower limits of the steam mass flow rate in the high efficiency heat transfer section under different tube lengths are demonstrated. The tube length is selected to be 5–7 m in later discussions. With the increase of the tube length, the heat transfer area increases. More steam could condense and releases its latent heat in the tube. Therefore, the lower limit of the steam mass flow rate in the high efficient heat transfer section increases with the increase of the tube length. The greater DTtt, the larger increasing rate. When DTtt is 0.5 °C, the lower limit of the steam mass flow rate of the high heat transfer section with different tube lengths are 0.6 kg/(m2
s), 0.8 kg/(m2
s) and 0.9 kg/(m2
s), with the maximum difference of only 0.3 kg/(m2
s). When DTtt is 4 °C, the lower limit of the steam mass flow rate of the heat transfer section under different tube lengths are 5.8 kg/(m2
s), 7.0 kg/(m2
s) and 8.0 kg/(m2
s), with the maximum difference of 2.2 kg/(m2
s). The area of the insufficient mass flow rate section increases with the increase of the tube length. The proportion of the insufficient mass flow rate section

Fig. 16. Maximum heat transfer amount and corresponding mass flow rate of tubes with different tube lengths.

under three tube lengths are 30.6%, 37.6% and 43.8%, respectively. With the increase of the tube length, the flow distance of steam in the tube increases. The steam saturation temperature has larger decrease along the tube axial direction. The upper limit of the mass flow rate in the high efficiency heat transfer section decreases with the increase of the tube length. For the three tube lengths, when DTtt is greater than 2.4 °C, 2.6 °C and 2.8 °C, respectively, the local temperature difference could not be reduced to 0 °C. That is, the insufficient temperature difference area dose not exist. The area of the insufficient temperature difference increases with the increase of the tube length. The proportion of the insufficient temperature difference under three tube lengths are 18.8%, 20.5% and 22.3%, respectively. From the above analysis, it is indicated that the area of high efficiency heat transfer section of tube decreases with the increase of the tube length, and the proportion of the high efficiency heat transfer section of three tube lengths are 50.6%, 41.9% and 33.9%, respectively. Fig. 16 shows the variance of the maximum heat transfer and the corresponding steam mass flow rate when Te is 60 °C and the tube length is 5 m, 6 m and 7 m, respectively. As the tube length increases, the heat transfer area increases. The total heat transfer increases, and the increasing rate increases with DTtt. When the tube length is 6 m and 7 m, respectively, the heat transfer area is increased by 20% and 40% than that when the tube length is 5 m. Under different DTtt when the tube length is 6 m, the maximum heat transfer is 15.8–17.0% larger than that when the heat tube length is of 5 m. When the tube length is 7 m, the maximum heat transfer is 30.1–33.8% larger than that when the tube length is of 5 m. The increasing rate of the maximum heat transfer are less than the increasing rate of the tube area. The steam mass flow rate corresponding to the maximum heat transfer increases with the increase of the tube length, but the increase is not significant. When DTtt is 4 °C, the maximum increase is 0.5 kg/(m2
s). When DTtt is less than 2.5 °C, the steam mass flow rate corresponding to the maximum heat transfer under different tube lengths is almost the same. 4. Conclusions

Fig. 15. Upper and lower limits of steam mass flow rate in high efficient heat transfer section with different tube lengths.

A comprehensive distributed parameter model was established to simulate flow and heat transfer process of a horizontal tube in a falling film evaporator. Based on the simulation results, the conclusions can be summarized:

L. Gong et al. / International Journal of Heat and Mass Transfer 145 (2019) 118798

(1) The increase of the steam mass flow rate leads to larger heat transfer coefficient but smaller local temperature difference, thus the total heat transfer shows an increasing trend followed by a decreasing trend. (2) When the steam has higher saturation temperatures, the local temperature difference exhibits larger values but the local heat transfer coefficient shows lower values. Under the impact of the two, the total heat transfer increases with the increment of the saturation temperature. With the increase of total heat transfer temperature difference, despite the remarkable decrease of the heat transfer coefficient, the total heat transfer tube increase significantly. (3) With the increase of steam mass flow rate, the influence of the condensate at the lower part of the tube on heat transfer is weakened, and the influence of the saturation temperature on heat transfer is strengthened. The wall area of the high temperature difference zone decreases significantly, which leads to the decrease of the total heat transfer of the whole tube. With the increase of the total heat transfer temperature difference, the influence of the condensate on heat transfer is enhanced, and the influence of saturation temperature on heat transfer is weakened. The wall area of the high temperature difference zone increases, and the total heat transfer of the whole heat transfer tube increases. (4) According to whether the local temperature difference equals to zero or the effective heat transfer length is equal to the total tube length, the partition map of the condensation heat transfer process is proposed. The heat transfer process is divided into three sections: the high-efficiency heat transfer section, the insufficient temperature difference section and the insufficient mass flow rate section. For the design of evaporator, the steam mass flow rate, the heat transfer temperature difference and the length of the heat transfer tube should matched to guarantee that condensation heat transfer process be located in the high-efficiency heat transfer zone.

Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgement The authors are gratitude for the support of the State Key Program of National Natural Science Foundation of China (Grant No. 51936002), the Fundamental Research Funds for Central Universities of Ministry of Education of China under Grant No. DUT17RC(3) 036. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.118798. References [1] J. Wang, X. Chen, T. Lu, X. Chen, S. Shen, B. Liu, Three-dimensional film thickness distribution of horizontal tube falling film with column flow, Appl. Therm. Eng. 154 (2019) 140–149.

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