Heat and mass transfer of ammonia-water in falling film evaporator

Abstract

To investigate the performance of heat and mass transfer of ammonia-water during the process of falling film evaporation in vertical tube evaporator, a mathematical model of evaporation process was presented, the solution of which that needed a coordinate transformation was based on stream function. The computational results from the mathematical model were validated with experimental data. Subsequently, a series of parameters, such as velocity, film thickness and concentration, etc., were obtained from the mathematical model. Calculated results show that the average velocity and the film thickness change dramatically at the entrance region when x<100 mm, while they vary slightly with the tube length in the fully developed region when x>100 mm. The average concentration of the solution reduces along the tube length because of evaporation, but the reducing tendency becomes slow. It can be concluded that there is an optimalβrelationship between the tube length and the electricity generated. The reason for the bigger concentration gradient in the y direction is that the smooth tube is chosen in the calculation. It is suggested that the roll-worked enhanced tube or other enhanced tube can reduce the concentration gradient in the film thickness direction and enhance the heat and mass transfer rate.

Keywords

falling film evaporation / ammonia-water / heat and mass transfer

Introduction

Falling films are widely employed in the heat and mass transfer processes of industrial equipment, such as vertical condensers, film evaporators, absorption towers and falling film coolers. The inherent advantages of falling thin film flow are short contact time between the process fluid and heated surface, high heat flux, minimal pressure drop, minimal static head and small process fluid holdup. Since the mechanism of the falling film is closely related to the transfer efficiency, there have been a number of attempts to investigate the liquid film flow by both practical and theoretical approaches in the past few years [1-6].

Yang and Shen [7] carried out an experimental study of falling film heat transfer in order to show how the heat transfer coefficient is affected by different parameters. Zhang et al. [8] performed an experimental study of falling film evaporation of solar desalination system. They concluded that the thermal performance of the system is greatly improved because of the falling film evaporation technology used. Liu and Yi [9] conducted an experimental study on enhancement of falling film evaporation heat transfer of pure water and water/salt mixtures. Their experimental results showed that the low-cost roll-worked tube can greatly enhance the evaporation heat transfer performance of the falling film. Johansson et al. [10] studied the heat transfer in evaporating black liquor falling film and presented experimental heat transfer data from black liquor evaporation. Souza et al. [11] investigated the performance of a solar energy powered falling film evaporator with film promoter. Shi et al. [12] investigated the heat transfer performance of lithium bromide solution in falling film generator, and obtained an experimental correlation of falling film heat transfer coefficient.

Raisul et al. [13] developed a coupled heat and mass transfer model to extract the transfer coefficients for falling-films. Xu et al. [14] researched the mass transfer across the falling film by numerical simulation. Kim and Ferreira [15] proposed a new method to determine heat and mass transfer coefficients from experimental data. Morison et al. [16] researched the minimum wetting and distribution rates in falling film evaporators. Gommed et al. [17] developed a numerical tool to predict the heat and mass transfer during the absorption process of ammonia-water vapor into a liquid layer falling inside a vertical tube. Niu et al. [18] built a mathematical absorption model for ammonia-water falling film absorption in magnetic field to study the influence of magnetic field on the absorption process. Du et al. [19] made energy analysis of evaporating thin falling film instability in vertical tube. The analysis indicated that the main reason for film breakup by increasing tube wall heat flux is that the stability effect of capillary adsorbability on tube wall is weakened as surface tension waving is enhanced by improving tube wall temperature. Assad and Lampinen [20] proposed a mathematical model of evaporation process from a laminar falling liquid film on a vertical plate of constant temperature. The results showed that lower liquid mass flow produces higher cooling rate. Feddaoui et al. [21] analyzed the evaporative cooling of liquid film falling inside a vertical insulated tube in turbulent gas stream.

Although the evaporation and absorption phenomena in the ammonia/water falling-film evaporators and absorbers have been widely investigated, little research has been conducted on the heat and mass transfer during falling film process by simulated and experimental approaches simultaneously, and the mathematical models has not yet been validated with experimental results.

The main purpose of this study is to investigate the heat and mass transfer and the distributions of parametersβduring falling film evaporation process. First, an experimental study of falling film evaporation of ammonia water in vertical tube evaporator was conducted. Then, a mathematical model for describing heat and mass transfer during falling film evaporation process was developed and solved by numerical method. After that, the mathematical model was validated with experimental results. Lastly, parametric studies were performed and the distributions of the parameters of concentration, velocity and film thickness were analyzed in order to make the evaporator much more compact, low-cost and efficient.

Experimental systems

Experimental setup

The experimental setup for testing ammonia-water evaporation outside vertical tube and absorption inside vertical tube, as illustrated in Fig. 1, consists of six subsystems, the ammonia-water evaporation subsystem, namely the ammonia vapor generator subsystem; the ammonia-water absorption subsystem; the heating subsystem; the cooling water subsystem; the circulation pipeline subsystem; and the measurement and controlling subsystem. The aim of this experimental setup is to investigate the performance of the heat and mass transfer of ammonia-water in falling film evaporation and absorption process.

The ammonia-water evaporation subsystem is mainly composed of an evaporation tube and a solution distributor. The evaporation tube is made of stainless steel with an outer diameter of 25 mm, an inner diameter of 21 mm and a length of 5000 mm. The solution distributor lies at the top of the evaporator. The width of the annular spaceβof the solution distributor is fixed at 2 mm, and the film flows downward along the wall outside the evaporation tube. The perpendicularity of the tube is calibrated to ensure uniform film spreading along the circumference. There are two small sight glasses at the top and bottom of the evaporation tube through which the flow pattern can be observed. The ammonia-water absorption subsystem composes of an absorption tube and a solution distributor. The solution will be distributed uniformly at the internal surface of the absorption tube by the solution distributor. The cooling water enters from the bottom of the absorber to remove the absorption heat. The absorption tube has an outside diameter of 32 mm and an inner diameter of 28 mm, and the length of the absorption tube is 6000 mm. The heating subsystem includes a hot water pump, a hot water tank, a rotor flowmeter and an electric heater.βBecause there is no correspondingly miniaturized turbine which can make a good match with so small an experimental setup, a reducing valve is used as a substitute for the turbine in the experimental system.

The measurement and controlling subsystem is composed of thermocouples, pressure measurement devices, flow measurement devices, temperature controllers and an adjustable electric heater. The temperatures are measured with thermocouples; the system pressure is measured with two calibrated pressure transducers. The concentration of the ammonia-water solution is a critical variable. Two absolute pressure transducers are placed at the upper end and lower end of the vertical tube connected toβthe inlet of evaporator respectively. The distance between the upper and lower pressure transducers is fixed. The density of the ammonia-water solution can be estimated by measuring the pressure difference between the two pressure transducers [22,23]. The concentration of strong solution is determined from the temperature and the density measured with the pressure transducers. The volume flow of the solution and ammonia vapor is measured by the electromagnetic flowmeter and the vortex flowmeter respectively. All the signals of the thermocouples, pressures and volume flow are recorded by the data acquisition system.

Experimental principle and procedures

Discharging the non-condensable gas by the vacuum pump was required before the experiment. Then liquid ammonia and distilled waterβwere charged in the experiment system. The initial concentration of the solution was dependent on the mass of liquid ammonia and distilled water charged.

The strong solution from the ammonia-water storage tank was pumped to high pressure by dosing pump, whose volume flow was measured by the electromagnetic flowmeter. Then the strong solution was heated by the weak solution from the evaporator when it passed through the plate type heat exchanger. Afterwards, the strong solution entered the evaporator at the upper entrance. In the evaporator, liquid film was formed by the solution distributor, and it flowed downwards along the external surfaces of the evaporation tube. Meanwhile, the hot water which was heated by the electric heater flowed upwards inside the evaporation tube. The ammonia vapor was evaporated when the strong solution was heated by the hot water. As a result, the strong solution became weak solution due to evaporation, and the weak solution exited the evaporator via the bottom exit. At the same time, the ammonia vapor exited the evaporator through the upper exit. The weak solution was cooled by the strong solution when it passed through the plate type heat exchanger. Subsequently, the weak solution reached the absorber and entered the solution distributor at the top of the absorber after passing through the reducing valve 6. The ammonia vapor from the evaporator passed through the reducing valve 13 and entered the absorber at the upper entrance. In the absorber, absorption occurred when the ammonia vapor encountered the weak solution. Consequently, the weak solution became strong solution. The absorption heat was removed by coolant, which flowed from the bottom to the top of the absorber. Finally the strong solution returned to the ammonia-water storage tank. During the experimental process, the temperature, pressure and flow data were recorded in the data acquisition system.

Mathematical model

In the evaporator under consideration, a film of liquid ammonia-water solution flowed downwards along the outside wall of a vertical evaporation tube, and hot water flowed upwards inside the vertical evaporation tube. The ammonia vapor which evaporated from the liquid film exited the evaporator at the top.

The falling-film evaporation was a complicated process in which the mass transfer and heat transfer influenced each other. The coupled heat and mass transfer which depended on fluid properties, geometry of the heat exchanger, and various operating parameters made it difficult to analyze the falling film evaporation process. The driving force of the evaporation was the differential pressure between the ammonia vapor evaporated (p) and the partial pressure of ammonia vapor at solution surface (p*) [18,24-27]. In the process of evaporation, p* decreased with the decrease ofsolution concentration and temperature, and the evaporation driving force of the solution, and therefore, would be reduced, thus the solution should be heated effectively so that the evaporation could run further.

Governing equations

The following assumptions were made in formulating the model:

1) The heat and mass transfer in the direction of circle of the falling film tube were neglected;

2) The flow was in a steady state;

3) There were no molecular diffusion and heat conduction in the direction of falling; and

4) The pressure gradients were negligible.

Under the above assumptions, the combined heat and mass transfer process in the binary system at steady-state conditions was governed by the following equations:

Continuity equation

(1){\displaystyle \frac{\partial \rho u}{\partial x}}+{\displaystyle \frac{\partial \rho v}{\partial y}}=\mathrm{0.}

Momentum equation

(2)\rho u{\displaystyle \frac{\partial u}{\partial x}}+\rho v{\displaystyle \frac{\partial u}{\partial y}}={\displaystyle \frac{\partial }{\partial y}}\left(\mu {\displaystyle \frac{\partial u}{\partial y}}\right)+\rho g.

Energy equation

(3)\rho C_{p}u{\displaystyle \frac{\partial T}{\partial x}}+\rho C_{p}v{\displaystyle \frac{\partial T}{\partial y}}={\displaystyle \frac{\partial }{\partial y}}\left(\lambda {\displaystyle \frac{\partial T}{\partial y}}\right).

Quality equation

(4)\rho u{\displaystyle \frac{\partial \xi }{\partial x}}+\rho v{\displaystyle \frac{\partial \xi }{\partial y}}={\displaystyle \frac{\partial }{\partial y}}\left(\rho D_{\mathrm{m}}{\displaystyle \frac{\partial \xi }{\partial y}}\right).

The x axis was set along the falling direction, and the y axis along the film thickness direction.

Initial and boundary conditions

The initial and boundary conditions were expressed as follows:

1) At the inlet of the solution

{\delta |}_{x=0}={\delta }_{\mathrm{i}\mathrm{n}},{v|}_{x=0}=0,{T|}_{x=0}=T_{\mathrm{i}\mathrm{n}},{\xi |}_{x=0}={\xi }_{\mathrm{i}\mathrm{n}},{u|}_{x=0}=u_{\mathrm{i}\mathrm{n}}={\displaystyle \frac{\mathrm{\Gamma }}{\rho {\delta }_0}},{\delta }_{\mathrm{i}\mathrm{n}}=2\mathrm{m}\mathrm{m}.

2) At the outside wall of the evaporation tube

{v|}_{y=0}=0,{u|}_{y=0}=0,{{\displaystyle \frac{\partial \xi }{\partial y}}|}_{y=0}=0,{T|}_{y=0}=T_{\mathrm{w}}.

3) At the vapor-liquid interface

\lambda {{\displaystyle \frac{\partial T}{\partial y}}|}_{y=\delta }=\mathrm{\Delta }H{\displaystyle \frac{\rho D_{\mathrm{m}}}{1-\xi }}{{\displaystyle \frac{\partial \xi }{\partial y}}|}_{y=\delta },{{\displaystyle \frac{\partial u}{\partial y}}|}_{y=\delta }=0.

Solution method

The film thickness of the liquid was variable along the tube length because of the evaporation, so, the calculation domain of the heat and mass transfer problem to be solved was nonuniform. The nonuniformity and the singularity at the evaporation tube required a coordinate transformation that was based on stream function normalization, to make the calculation domain uniform. The stream function \psi below was adopted:

(5)\psi ={\displaystyle {\int }_{\mathrm{0}}^y\rho u\mathrm{d}y}.

Differentiating Eq. (5) yields

(6){\displaystyle \frac{\partial \psi }{\partial y}}={\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle {\int }_{\mathrm{0}}^y\rho u\mathrm{d}y}\right)=\rho u.

Substituting Eq. (6) into Eq. (1) gives

(7){\displaystyle \frac{\partial \psi }{\partial x}}={\displaystyle {\int }_{\mathrm{0}}^y{\displaystyle \frac{\partial \rho u}{\partial x}}\mathrm{d}y}={\displaystyle {\int }_{\mathrm{0}}^y-{\displaystyle \frac{\partial \rho v}{\partial y}}\mathrm{d}y}=-\rho v.

Three dimensionless numbers are defined as

(8)\omega ={\displaystyle \frac{\psi -{\psi }_{\mathrm{e}}}{{\psi }_{\mathrm{i}}-{\psi }_{\mathrm{e}}}},\theta ={\displaystyle \frac{x}{L}},\overline{u}={\displaystyle \frac{u}{u_{\mathrm{i}\mathrm{n}}}},

where {\psi }_{\mathrm{e}} is the stream function at the outside wall of the evaporation tube, and {\psi }_{\mathrm{i}} is that at the vapor liquid interface.

Using Eqs. (5)–(8), the irregular coordinate system x-y were changed into the regular θ-ω coordinate system [17,18].

The above partial differential equations were discretizedβby finite volume method [28,29], and then solved by tri-diagonal matrix algorithm (TDMA) [30,31]. A grid of 5001 points was selected in the x direction, and the grid in film thickness direction contained 200 points.

Validation of mathematical model

The above mathematical model was thoroughly validated with experimental results obtained from the above experimental setup, as shown in Fig. 1.

Fourteen different experimental conditions were chosen to perform the experimental test, and the heat and mass transfer were calculated by the mathematical model under like conditions. The experimental and calculated results are presented in Table 1.

Table 1 shows the input conditions and output values of the different experimental tests and calculation. The values in each column are the sequence number of experiment, inlet hot water temperature, volumetric flow of hot water, inlet solution concentration, inlet solution mass flow, experimental outlet concentration of solution, calculated outlet concentration of solution, experimental outlet mass flow of solution, calculated outlet mass flow of solution, experimental heat exchange capacity and calculated heat exchange capacity, respectively.

Several comparisons were made between the experimental results and the calculated ones with the mathematical model developed. As can be seen from the data in Table 1, there is a good agreement between the experimental and calculated results, the biggest relative error of heat exchange capacity between experimental results and calculated ones being 8.6%, while the biggest relative error of the outlet solution concentration being not more thanβ3.34%. It is interesting to note that the calculated heat exchange capacity is always less than experimental heat exchange capacity due to heat lossβto the environment. The reason for the discrepancies between the experimental and calculated results mainly results from two factors. One is the errors in the measurement of temperature, pressure and flow. The other is the errors in the calculation of thermal propertiesβof ammonia water. The formulas used for calculating the thermal propertiesβof ammonia water are also not accurate enough.

Results and discussion

By comparing the calculated results with the experimental data, it can be concluded that the mathematical model describing the falling film evaporation of vertical tube is valid. So, the model can be used for calculating the film thickness, velocity, concentration and other parameters in the process of evaporation. The numerical results from the mathematical models are analyzed in this section. The data from the sixth row in Table 1 is chosen as input conditions and output values.

Figure 2 describes the profiles of average velocity of solution as a function of tube length. The average velocity denotes that velocity is averaged in the y direction for fixed x. As is evident, the average velocity increases quite rapidly at the entrance region, after a short distance along the tube it reaches the maximum value, and then decreases gradually along the tube length. At the entrance, the average velocity is 0.21 m/s, and it reaches 0.86 m/s rapidly at x=100 mm. The maximum velocity, 0.94 m/s, occurs at x=300 mm. The reason for the rapid increase of average velocity at the entrance is that the liquid film near the entrance region is thicker, as demonstrated in Fig. 3. This means that the gravity of the liquid film is greater than the viscous stressesβat the outside wall of the tube. As a result, the average velocity increases with the tube length. The liquid film becomes thin with increasing velocity, and the gravity of liquid film also becomes small, while the viscous stress increases with the increase of velocity. The average velocity reaches the maximum value when liquid gravity is equal to viscous stresses. Subsequently, the average velocity decreases gradually along the tube length because the gravity of the liquid is slightly less thanββthe viscous stressesβwhen x>300 mm. This is due to ammonia vapor evaporation, resulting in the decrease of solution mass flow and gravity along the tube length. The average velocity is 0.886 m/s and 0.85 m/s at x=2500 mm and x=5000 mm, respectively. This indicates that the velocity change from x=300 mm to x=5000 mm is very small, which is very useful for designing practical evaporators.

The film thickness of ammonia-water solution along the tube length are displayed in Figs. 3(a) and 3(b), respectively. It is found from Fig. 3(a) that the film thickness of the solution decreases dramatically at the entrance region, while it is almost invariable far away from the entrance region. The reason for the sharp decrease of the film thickness at the entrance region is that the gravity of the liquid film is greater than the viscous stresses, resulting in the increase of the velocity and the decrease of the film thickness, which can be explained by the average velocity, as exhibited in Fig. 2. The initial film thickness is 2 mm at the entrance, and it drops to 0.47 mm quickly at x=50 mm. The decreasing tendency of the film thickness becomes slow when x>50 mm, and the film thickness is 0.39 mm at x = 100 mm. The film thickness decreases gradually with the increasing length of the tube when x>300 mm, as shown in Fig. 3(b). The film thickness is 0.346 mm, 0.3444 mm and 0.344 mm at x=1000 mm, 3000 mm, and 5000 mm, respectively. As expected, the film thickness at x=5000 mm is somewhat lower than that at x=1000 mm due to the ammonia vapor which is evaporated from the ammonia-water solution, resulting in the decrease of mass flow along the tube length. Therefore, it can be concluded that the average velocity and the film thickness change drastically at the entrance region when x<100 mm, while they vary slightly with the tube length in the fully developed region when x>100 mm, as illustrated in Figs. 2, 3(a) and 3(b).

Figure 4 shows the distribution of average concentration of the solution in the x axis direction. As can be seen in Fig. 4, the average concentration of the solution reduces along the tube length because of the evaporation. The reducing tendency of the solution concentration becomes slow along the tube length, as evident from Fig. 4.

Figure 5 describes the amount of evaporated ammonia vapor and percentage as a function of the tube length. The amount of evaporated ammonia vapor, mE, at x section means the sum of evaporated ammonia vapor from 0 to x. PE means the proportion of mE to the amount of evaporated ammonia vapor at the whole tube length. mE increases with the increase of the tube length, but the increasing tendency becomes slow, as shown in Fig. 5. PE is 66.6% at x=2000 mm, while it reaches 80.64% at x=3000 mm. This could be explained by the fact that the solution concentration is higher at the upper side of the tube where the partial pressure of the ammonia vapor at the solution surface is also higher, so, the driving force of the evaporation, namely the differential pressure between the partial pressure of ammonia vapor at solution surface and the ammonia vapor evaporated, is also bigger, resulting in a faster evaporation rate (namely generation rate of vapor). Consequently, the solution concentration at the upper side of the tube reduces rapidly due to the faster evaporation rate, as can be seen in Fig. 4. While the partial pressure of the ammonia vapor at the solution surface and the driving force of the evaporation are smaller at the bottom of the tube because of the lower solution concentration, as a result, both the evaporation rate and the solution concentration vary slowly. The concentration difference between the entrance and exit of the evaporation tube is 6.75%, and the evaporation pressure is 1105.9 kPa.

From the point of view of generating electricity, if the evaporation tube is too long, although the heat exchange capacity and the amount of evaporated ammonia vapor can increase greatly with the increase of the tube length, there will be a small increase in the evaporation pressure and the inlet pressure of the turbine due to the bigger concentration difference between the entrance and exit of the evaporation tube. On the other hand, if the evaporation tube is too short, the evaporation pressure and the inlet pressure of the turbine will be higher, while the amount of the evaporated ammonia vapor will be smaller. Therefore, a conclusion can be drawn that there is an optimalβrelationship between the tube length and the generated electricity, which will be investigated in the future.

Figure 6 depicts the concentration distribution along the film thickness direction for different x. It is evident from Fig. 6 that the solution concentration reduces with the film thickness. The solution concentration near the evaporation tube wall is 0.56176, 0.54011, and 0.51486 at x=500 mm, 2500 mm, and 4500 mm, respectively. While the solution concentration at the vapor liquid interface is 0.47288, 0.47215, and 0.47197 at x=500 mm, 2500 mm, and 4500 mm, respectively. This indicates that the ammonia vapor is rapidly evaporated from the solution at the vapor liquid interface, and that the vapor phase and liquid phase are almost in equilibrium at the vapor liquid interface. This also indicates that the mass transfer resistance is dominant in the liquid side where the concentration gradient of ammonia-water solution is very big. Moreover, it is observed from Fig. 6 that the concentration gradient of the solution in the film thickness direction varies with x. The concentration gradient near vapor liquid interface at x=500 mm is bigger due to the higher solution concentration and faster evaporation rate, while it becomes smaller at x=2500 mm and 4500 mm because of the decrease in the solution concentration. In addition, it is very interesting to note that the concentration gradient of the solution in the film thickness direction is bigger. The reason for the bigger concentration gradient in the y direction is that the evaporation tube chosen in this study is smooth tube, which results in a very low velocity in the y direction. If roll-worked enhanced tube or other enhanced tube is used, the velocity in the y direction and the turbulence near the evaporation tube wall could be enhanced. As a result, the concentration gradient in the film thickness direction would be reduced with the increase of velocity in the y direction. Consequently, the evaporation rate and the performance of the heat exchange would be also enhanced in the falling film evaporator.

Conclusions

A mathematical model was developed for describing the heat and mass transfer during falling film evaporation process and solved by coordinate transformation. Then an experimental study was conducted in order to validate the mathematical model. By comparing the calculated results with the experimental ones, it could be concluded that the mathematical model was valid and could describe the falling film evaporation process in the vertical tube evaporator. A series of parameters, such as velocity, film thickness and concentration, and etc., were obtained from the mathematical model.

The calculated results show that the average velocity and the film thickness change dramatically at the entrance region when x<100 mm, while they vary slightly with the tube length in the fully developed region when x>100 mm. The film thickness is 0.39 mm with an average velocity of 0.86 m/s at x=100 mm. The average concentration of the solution reduces along the tube length because of the evaporation, but the reducing tendency becomes slow along the tube length. It is found that the amounts of evaporated ammonia vapor increase with the increase of the tube length, but the increasing tendency becomes slow. It can be concluded that there is an optimalβrelationship between the tube length and electricity generated by analyzing the variation tendency of the solution concentration and the amount of evaporated ammonia vapor with the tube length. Moreover, the solution concentration reduces with the film thickness. The reason for the bigger concentration gradient in the y direction is that the evaporation tube used in this study is smooth tube, which results in a very low velocity in the y direction. If roll-worked enhanced tube or other enhanced tube is used, the velocity in the y direction and the turbulence near the evaporation tube wall could be enhanced. As a result, the concentration gradient in the film thickness direction would reduce with the increase of velocity in the y direction, resulting in a higher evaporation rate. This work is helpful in understanding the heat and mass-transfer mechanism during falling film evaporation process. Moreover, it can offer a support for the design of falling film evaporators. To enhance heat and mass transfer, the roll-worked enhanced tube should be applied in the further work.

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