Horizontal air--water flow in a square cross-section channel: pseudo-slug flow regime.

Ribeiro, A.M. ; Ferreira, V. ; Mayor, T.S. 等

INTRODUCTION

The study of simultaneous gas and liquid horizontal flow has been reported mainly for circular tubes with diameters larger than 10 mm (Lee and Lee, 2001). However, there are several industrial applications where gas-liquid flow in rectangular channels is used, as for example, in plate heat exchangers. Troniewski and Ulbrich (1984) pointed out that, in some types of thin film evaporators, the use of rectangular channels increases the area of heat transfer when compared with the use of bundles of circular tubes, which also reduces the volume of the apparatus.

Different flow patterns are observed when gas and liquid flow simultaneously through a pipe or channel. These patterns are governed by the physical properties of the fluids, the ratio of gas/liquid flow rates and the system geometry. In horizontal flow, gravity introduces an asymmetry into the system: the density difference between the two-phases causes the liquid to travel preferentially along the bottom of the channel. According to Hewitt et al. (1994), the following flow regimes can be identified: bubbly, plug, stratified (smooth and wavy), slug, pseudo-slug, and annular flow.

In the bubbly flow regime, the gas phase is distributed as discrete bubbles within a continuous liquid phase, and due to gravity, these bubbles tend to congregate near the top of the tube.

The plug flow occurs at low gas velocities and consists of elongated gas bubbles that move along the top part of the tube.

Stratified flow is characterized by the liquid flowing at the bottom of the channel while the gas passes over it. At low gas and liquid flow rates the gas-liquid interface is smooth (smooth stratified flow), but as gas flow rate increases the interface becomes wavy, with the waves travelling in the flow direction (wavy-stratified flow). For even higher gas flow rates, drops are torn from the surface of these waves giving drop entrainment into the gas.

The slug flow is characterized by the intermittent appearance of frothy slugs of liquid which bridge the entire pipe section and propagate along the pipe at a high velocity. Pressure fluctuations typify this flow pattern.

Pseudo-slug flow occurs near the annular/slug, stratified/slug, and stratified/annular flow transitions. As studied by Lin and Hanratty (1987), this flow regime is characterized by the presence of liquid flow patterns that have the appearance of slugs, but which do not give the identifying pressure pattern associated with a liquid slug. The liquid may touch the top of the channel momentarily, but it does not block the entire cross-section.

In annular flow the gas travels in the centre of the tube. The liquid flows as a slower film on the pipe wall and also as entrained drops in the gas core. In horizontal flow, gravity causes an asymmetry in the film, where the liquid film and drop concentration are greatest at the bottom of the tube. At extreme gas velocities this asymmetry becomes less pronounced.

Several authors have carried out studies of gas-liquid flow in rectangular channels, including Troniewski and Ulbrich (1984), Ide and Matsumura (1990), Wambsganss et al. (1991), Coleman and Garimella (1999), Lee and Lee (2001) and Ide et al. (2007), as shown in Table 1.

Troniewski and Ulbrich (1984) performed an experimental investigation on flow patterns and pressure drop for air-water vertical and horizontal flow in rectangular channels. The hydraulic diameter of the channels varied between 7.45 and 13.1 mm. The determination of flow patterns was conducted for air-water and for air-aqueous solutions of sugar, with pressure drop measurements taken only for air-water systems. The air mass flux was varied between 0.6 and 43 kg/([m.sup.2] s) and the water mass flux ranged from 200 to 1600 kg/([m.sup.2] s). Flow patterns maps were developed for vertical and horizontal flows. A new method for calculating pressure drops in rectangular channels was proposed, based on the Lockhart-Martinelli correlation, and which includes a correction for the difference between the flow in rectangular channels and that in circular pipes.

Ide and Matsumura (1990) proposed a correlation to estimate pressure drop in rectangular channels based on experiments carried out for air-water flow and for various inclinations of the channels (from horizontal to vertical flow). The superficial gas velocity varied from 0.1 to 6.9 m/s and the superficial liquid velocity ranged from 0.3 to 4.2 m/s. The geometry of the channels had the following characteristics: ratio of sides from 1 to 40 and hydraulic diameters from 7.3 to 21.4 mm. The new correlation, based on a separated flow model, included the effects of the ratio of sides of the rectangular channel, its hydraulic diameter, the inclination of the channel and the Reynolds number in the liquid. The correlation developed was shown to be accurate when both the gas and the liquid phases are in turbulent flow.

Wambsganss et al. (1991) studied the flow of air-water in a horizontal channel with a rectangular cross-section (19.05 mm x 3.18 mm), using the short and long sides as the base of the channel. Pressure drop measurements and flow pattern determination were carried out in adiabatic conditions for a total mass flux between 50 and 2000 kg/([m.sup.2] s).

Coleman and Garimella (1999) presented a flow pattern map for air-water flowing in a horizontal rectangular channel with a hydraulic diameter of 5.36 mm and a ratio of sides of 0.725. The air and water superficial velocities varied in the range 0.1-100 and 0.01-10.0 m/s, respectively. Flow patterns were determined from the analysis of high-speed videos. The flow pattern transitions in this rectangular channel were compared with the ones occurring in a circular tube with a diameter of 5.5 mm.

Lee and Lee (2001) reported studies on pressure drop for adiabatic air-water flow in rectangular channels with a fixed width of 20 mm and heights of 0.4, 1, 2, and 4 mm. Overall, the superficial gas and liquid velocities varied in the range 0.05-18.7 and 0.03-2.39 m/s, respectively. A new way to calculate the parameter C of the Lockhart-Martinelli pressure drop correlation was proposed for rectangular channels with small heights.

Ide et al. (2007) investigated the characteristics of air-water flow in rectangular channels with cross-section dimensions of: 1 mm x 1 mm; 2 mm x 1 mm, 5 mm x 1 mm, and 9.9 mm x 1.1 mm. The direction of the flow was vertical upward, vertical downward, and horizontal. Studies were carried out to determine the influence of the aspect ratio on flow patterns, hold-up and frictional pressure drop. During the tests the liquid and gas superficial velocities were varied between 0.03-2.3 and 0.1-30 m/s, respectively.

As may be noted from the above review of published articles on gas-liquid flow, most reported works studied flow patterns and measurements of pressure drop and hold-up using rectangular channels with several different dimensions and orientations. In particular these studies used channels whose cross-sections included one dimension which was much larger than the other. Only two articles referred to work with square-section channels, Troniewski and Ulbrich (1984) and Ide et al. (2007), in which channels with hydraulic diameters of 13.1 and 1 mm, respectively, were used.

The present article reports new data on liquid hold-up, gas and liquid velocities, and frequency and velocity of the waves, obtained in a square cross-section horizontal channel with a hydraulic diameter of 24.25 mm, for both wavy-stratified and pseudo-slug flows. From video observations the flow characteristics were analyzed and the flow regime transitions identified. Empirical correlations for the liquid hold-up, gas and liquid velocities and roll-wave frequency, were developed for a range of gas and liquid flow rates occurring in the pseudo-slug flow.

EXPERIMENTAL DETAILS

Experimental Facility

The flow apparatus used in these experiments is illustrated schematically in Figure 1. It consisted of a 32 mm ID circular horizontal tube (the entering section) followed by a horizontal square cross-section channel with 24.25 mm sides (the test section). This apparatus is fully described by Ferreira (2004). The filtered air, delivered from a rotative compressor, was controlled by a pressure regulator (PR) and a valve (V5), and measured using a calibrated rotameter (R3). The water was stored in a 100 L tank (T) and circulated through the system by means of a centrifugal pump (PP), being metered by a set of calibrated rotameters (R1 and R2).

The air and the water entered the circular tube by a tee and were allowed to travel for 5.2 m, in order to reach developed conditions, before entering the square test section. The test section was constructed in Perspex having a square cross-section (H=24.25 mm), and a length of 2.3 m. Both ends included converging pieces (CP) that were constructed to provide a very smooth transition from the circular to the square geometry and from the square-section back to the circular tube. The length of the square channel, L, was chosen in order to be longer than 50 [D.sub.h], as specified for rectangular channels by Troniewski and Ulbrich (1984), where [D.sub.h] is the hydraulic diameter of the channel (for the channel used in this study [D.sub.h] = H and L = 95[D.sub.h]. Before entering the square-section channel, the air/water mixture travelled for 5.2 m through the circular tube, which was sufficiently long to attain fully developed conditions, before passing through the converging piece that made the transition from circular to the square geometry. Under these conditions, and given the length of the square test section, it is assumed that fully developed flow was reached at the section were experimental data was measured. The air/water mixture was returned to the stock tank through a section of PVC tube, where the air was vented and the water was re-circulated. The horizontal circular tube and the square-section channel were carefully levelled in order to avoid interference in flow patterns by pipe misalignment.

[FIGURE 1 OMITTED]

Visualization studies using still photography and a high-speed video camera allowed the identification of flow patterns (wavy-stratified and pseudo-slug flows). In addition data on liquid hold-up, gas and liquid velocities, and roll-wave frequency were obtained from the high-speed videos. The velocity of roll-waves was determined from analysis of the signal from two pressure transducers coupled with films from the video camera. The average pressure in the test-section was measured with a manometer.

Identification of Flow Patterns

The identification of flow patterns in the square-section channel was conducted using a digital video camera (Canon XM1) operating at 25 Hz and a digital still camera (Nikon Coolpix 3500). The digital camera was positioned at the end of the test section, perpendicular to the flow direction. A 500 W light source, positioned opposite to the lens of the camera provided illumination. A white sheet of paper was used to reflect and diffuse light onto the test section. The photographs were recorded using an equivalent film speed setting of ISO 100, and shutter speeds ranging from 1/1500 to 1/3000 s at F3.5.

The procedure for operating the high-speed video camera, which was also used to collect data for liquid hold-up, is described in the following section.

Measurement of Liquid Hold-Up

The video camera was mounted perpendicular to the frontal surface of the channel, close to the end of the test section. Two light stops were placed behind, making angles of 60[degrees] with the posterior surface, illuminating a white sheet of paper used to reflect and diffuse light onto the test section. The flow was recorded for periods of 5 min which provided a reasonable quantity of video data, without exceeding the processing capacity of the computer (a 30 min video segment, converted to Microsoft AVI format, requires approximately 4 GB of disk space). The raw videos were stored in their original format (Microsoft DV) and subsequently processed using a commercial video editing software package (Adobe Premiere), in order to crop the area of interest, and permit conversion to the AVI format.

A Matlab program was developed to analyze each of the resulting videos (Mayor et al., 2007). Each individual frame, corresponding to a snapshot of the flow every 0.04 s, was converted from the original colour format to greyscale (256 grey levels, where 0 represents black and 255 represents white) with the associated pixel intensity value being used as an indication of luminosity. The greyscale frames were subsequently transformed to binary mode by reducing the number of grey levels to two (where 0 represents black and 1 represents white). This transformation was achieved by defining a threshold value, that is a pixel value defining the transition point from black to white. All pixels with a value lower than the threshold were then considered to be black, and all pixel values above the threshold were considered to be white. This procedure allowed the definition of the interface height (at the same flow section) for all the recorded video frames.

Millimetric rulers placed on the front and rear faces provided the conversion factor (pixels into millimetres) for calibration purposes. The liquid height distribution was obtained based on large samples (more than 3000 readings) to assure statistical significance. The statistical parameters of this distribution (mean, mode, median, and standard deviation) were calculated.

Hold-up data were obtained for liquid flow rates between 2.8 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s and gas flow rates in the range from 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s. The experiments were performed at ambient temperature with the pressure in the test section varying between 1.2 and 2.2 bar, depending on flow conditions.

Determination of Liquid and Gas Velocities

The mean velocity of the liquid phase was determined from the mean liquid height. The cross-sectional area of the channel occupied by the liquid was calculated taking the mean liquid height and the transverse dimension of the channel. By dividing the volumetric liquid flow rate by this cross-section, the mean velocity of the liquid in the channel was obtained. A similar procedure was adopted to calculate the mean gas velocity, taking into account that the gas volumetric flow rate was calculated using the average temperature and pressure conditions in the test section. Liquid and gas velocity data were obtained for liquid flow rates between 2.8 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s and gas flow rates in the range from 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s.

Measurement of Roll-Wave Frequency

The frequency of roll-waves was obtained by counting the number of waves that occurred per minute of film, the procedure being repeated for the entire duration of the each film segment (5 min). The frequency was then calculated as the average of the five counts. Data were obtained for liquid flow rates between 7.5 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s and gas flow rates from 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s and the flow regime observed was always pseudo-slug.

Measurement of Wave Velocity

The velocity of the roll-waves was determined from analysis of the signal of two differential pressure transducers coupled with films from the video camera. When a roll-wave reached the first tap of a transducer there was a sudden and distinct increase in the pressure drop signal. The taps of the first and second pressure transducers were 0.121 and 0.097 m apart, respectively, and the distance between the first pressure taps of the transducers was 0.558 m.

The video camera was necessary in order to identify which waves touched the top of the channel. It was positioned in front of the first pressure tap of the first transducer in order to freeze the moment that the front of the wave reached this location. The connection between the timing of the video camera and the timing of the pressure drop signal was determined with a chronometer synchronized with the pressure drop acquisition time (an image of the synchronized chronometer was grabbed in video at the beginning of each experiment). The velocity of the wave was calculated as the ratio of the distance between the first taps of the pressure transducers and of the time taken by the wave to travel this distance.

Wave velocity was determined for gas flow rates of 7.7 x [10.sup.-3] and 9.9 x [10.sup.-3] kg/s with liquid flow rate varying between 7.5 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s and for the gas flow rate of 1.4 x [10.sup.-2] kg/s with liquid flow rate varying between 9.9 x [10.sup.-2] and 1.7 x [10.sup.-1] kg/s.

Experimental Uncertainties

The uncertainties associated with the channel dimensions, the pressure and the gas and liquid flow rates are shown in Table 2. Following calibration of the flow meters, liquid and gas mass flow rates uncertainties were calculated using propagation analysis (Holman, 2001).

RESULTS AND DISCUSSION

Flow Regimes

For the characterization of the flow regimes, the liquid flow was maintained constant while the gas flow was increased in steps. To assist in the identification of flow patterns, both a digital still camera and a high speed video camera were used to capture images of the flow under various conditions, following the procedures described in Identification of Flow Patterns and Measurement of Liquid Hold-Up Sections.

[FIGURE 2 OMITTED]

Wavy-stratified flow was observed for liquid flow rates equal to 2.8 x [10.sup.-2] and 5.1 x [10.sup.-2] kg/s and all gas flow rates (7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s). In this regime the liquid flows at the bottom of the tube and the gas above it. The gas-liquid interface has waves of small amplitude, which do not reach the top of the channel. Also, an increase in the liquid flow rate produced increases in both the level of the liquid and the amplitude of the waves. Figure 2 shows the wavy-stratified flow regime for a gas and a liquid mass flow rate of 7.7 x [10.sup.-3] and 5.1 x [10.sup.-2] kg/s, respectively.

Pseudo-slug flow was observed for liquid mass flow rates between 7.5 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s, and all gas flow rates (7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s). This flow regime has roll-waves with an amplitude that reaches the top of the tube. After the passage of these waves, the gas-liquid interface was very similar to the one observed in the wavy-stratified flow. The passage of a roll-wave in pseudo-slug flow is illustrated in Figure 3, for a gas and a liquid mass flow rate of 7.7 x [10.sup.-3] and 1.5 x [10.sup.-1] kg/s, respectively. For the lower liquid flow rates, the top of the roll-waves was formed by a liquid mist which was carried by the gas. An increase in the gas flow rate (for a constant liquid flow) produced an increase in the length of these misty zones and they appeared to be more diluted. As the liquid flow rate was increased, the top of these waves became thicker and for the highest liquid flow rates, the top of the waves was formed by compact liquid that almost blocked the gas from flowing. For the conditions under study in the pseudo-slug regime, the length and velocity of the roll-waves increased by increasing the liquid flow rate.

Liquid Hold-Up

Information on liquid hold-up, [[epsilon].sub.L], was obtained from measurements of the liquid height inside the tube, as described in Measurement of Liquid Hold-Up Section. The gas mass flow rate [m.sub.G], was varied in the range of 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s, and the liquid mass flow rate, [m.sub.L], in the range of 2.8 x [10.sup.-2] to 2.7 x [10.sup.-1] kg/s. For these flow conditions hold-up varied between 0.11 and 0.38 and the associated uncertainties from 0.2% to 2.4%. The uncertainties were calculated according to the method developed by Mayor et al. (2007).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Figure 4 shows a graph where the liquid hold-up is plotted against the ratio between the gas and the liquid mass flow rate [m.sub.G]/[m.sub.L], for several values of the liquid mass flow rate, and with flow conditions including wavy-stratified and pseudo-slug flows. In general, for a given liquid flow, liquid hold-up decreases when the ratio [m.sub.G]/[m.sub.L] increases. Also, for the data in the pseudo-slug flow regime ([m.sub.L] in the range of 7.5 x [10.sup.-2] to 2.7 x [10.sup.-1] kg/s) hold-up is a function of [m.sub.G]/[m.sub.L] and is independent of the liquid flow rate. For wavy-stratified flow conditions, the number of data points taken was quite small (corresponding to [m.sub.L] = 2.8 x [10.sup.-2] and 5.1 x [10.sup.-2] kg/s), however the resulting analysis shows that hold-up appears to be dependent on both [m.sub.G]/[m.sub.L] and on the liquid flow rate. In order to support this statement, more experiments with wavy-stratified flow would have to be carried out. In the graph shown in Figure 4, the data points corresponding to [m.sub.L] = 5.1 x [10.sup.-2] kg/s (wavy-stratified flow) are located close to the data for the pseudo-slug flow regime. It is possible that for this flow rate, the system is already approaching the transition to the pseudo-slug flow.

An attempt was made to establish the relationship between liquid hold-up and the ratio between gas and liquid mass flow rates, for the experimental points in the pseudo-slug regime. A polynomial expression of second order was chosen to fit the experimental points and the parameters were determined by using an optimization process (available through the Solver add-in in Microsoft[TM] Excel) in order to minimize the sum of the squares of the differences between the experimental and the calculated values from the chosen function. The final expression obtained was:

[[epsilon].sub.L] = 2.1 x [10.sup.-3] [([m.sub.G/[m.sub.L]).sup.2] + 1.0 [m.sub.G]/ [m.sub.L] + 1.5 x [10.sup.-4] (1)

To evaluate the ability of Equation (1) to fit the experimental data, the mean deviation Dm, defined as:

[D.sub.m] = [square root of [N.summation over (i=1)] [([[gamma].sub.cal,i] -[gamma].sub.exp,i]/[[gamma].sub.exp,i]).sup.2] (2)

was determined (Wen and Chen, 1982). In Equation (2) [[gamma].sub.cal,i] and [[gamma].sub.exp,i] represent, respectively, the calculated and the experimental values of the variable, in this case the liquid hold-up. The average deviation obtained was [D.sub.m] = 0.2%, confirming that Equation (1) is suitable for representing the hold-up for the flow conditions under study.

Liquid Velocity

Mean liquid velocities, [v.sub.L], were determined as indicated in Determination of Liquid and Gas Velocities Section, for gas mass flow rates in the range of 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s and for liquid mass flow rates between 2.8 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s. For these flow conditions, liquid velocity varied between 0.27 and 1.4 m/s, with associated uncertainties in the range from 0.4 to 1.3%.

Figure 5 shows the ratio between the liquid velocity and the superficial liquid velocity [v.sub.L]/[v.sub.SL], as a function of the ratio between the gas and the liquid mass flow rates [m.sub.G]/[m.sub.L], for flow conditions that include the wavy-stratified and pseudo-slug flows. This graph shows that [v.sub.L]/[v.sub.SL] increases linearly with [m.sub.G]/[m.sub.L], for a constant liquid mass flow rate. Also, for the data in the pseudo-slug flow regime ([m.sub.L] in the range of 7.5 x [10.sup.-2] to 2.7 x [10.sup.-1] kg/s), [v.sub.L]/[v.sub.SL] is a function of [m.sub.G]/[m.sub.L] and is independent of the liquid flow rate. As already seen in the previous section (Liquid Hold-Up Section), for the wavy-stratified flow conditions, the number of data points taken was quite small (corresponding to [m.sub.L] = 2.8 x [10.sup.-2] and 5.1 x [10.sup.-2] kg/s), but [v.sub.L]/[v.sub.SL] appears to be dependent on both [m.sub.G]/[m.sub.L] and on the liquid flow rate. The data points corresponding to [m.sub.L] = 5.1 x [10.sup.-2] kg/s are placed close to the data for the pseudo-slug flow regime.

Based on the information provided by Figure 5, a linear relationship was determined between [v.sub.L]/[v.sub.SL] and [m.sub.G]/[m.sub.L] for the data in the pseudo-slug flow regime. The optimization process, previously mentioned in liquid Hold-Up Section, was used to determine the parameters of the equation from the experimental points. The expression obtained was:

[v.sub.L]/[v.sub.SL] = 18 [m.sub.G]/[m.sub.L] + 2.3 (3)

[FIGURE 5 OMITTED]

To evaluate the performance of Equation (3) when fitting the experimental points, the mean deviation was calculated using Equation (2). The value obtained was [D.sub.m] = 5%, showing that Equation (3) provides a good representation of the data under study.

Gas Velocity

Mean gas velocities [v.sub.G], were calculated following the procedure described in Determination of Liquid and Gas Velocities Section, for gas mass flow rates in the range of 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s and for liquid mass flow rates between 2.8 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s. For these flow conditions, gas velocity varied between 10 and 17 m/s and the associated uncertainties were in the range from 5 to 10%.

The graph in Figure G shows the ratio of gas velocity to the superficial gas velocity [v.sub.G]/[v.sub.SG], versus the ratio between gas and liquid mass flow rates, [m.sub.G]/[m.sub.L], for flow conditions that include wavy-stratified and pseudo-slug flow regimes. It was observed that, for a constant value of the liquid flow rate, [v.sub.G]/[v.sub.SG] decreases with increasing [m.sub.G]/[m.sub.L]. For the data in the pseudo-slug condition ([m.sub.L] in the range of 7.5 x [10.sup.-2] to 2.7 x [10.sup.-1] kg/s), [v.sub.G]/[v.sub.SG] varies with [m.sub.G]/[m.sub.L] and is independent of the liquid flow rate. On the other hand, for the wavy-stratified flow conditions, the number of data points taken was quite small (corresponding to [m.sub.L] = 2.8 x [10.sup.-2] and 5.1 x [10.sup.-2] kg/s), however it would appear that [v.sub.G]/[v.sub.SG] is both dependent on [m.sub.G]/[m.sub.L] and on the liquid flow rate. The data points corresponding to [m.sub.L] = 5.1 x [10.sup.-2] kg/s are placed close to the data for the pseudo-slug flow regime. As already mentioned, a possible interpretation is that for this flow condition, the gas-liquid flow system is close to the transition to the pseudo-slug flow regime.

An equation was developed to establish the relationship between the non-dimensional gas velocity [v.sub.G]/[v.sub.SG] and the ratio [m.sub.G]/[m.sub.L], for the data in the pseudo-slug flow regime. A polynomial expression of second order was chosen to fit the experimental points and the parameters were determined by using the optimization process mentioned earlier. The final expression obtained was:

[v.sub.G/[v.sub.SG] = 18 [([m.sub.G]/ [m.sub.L]).sup.2] - 5.9 [m.sub.G]/[m.sub.L] + 1.7 (4)

[FIGURE 6 OMITTED]

To establish the performance of Equation (4) in representing the experimental data, the mean deviation given by Equation (2) was calculated. The resulting value, [D.sub.m] = 2%, shows that Equation (4) provides a good fit for the pseudo-slug flow conditions under study.

Roll-Wave Frequency

Data on the frequency of roll-waves was only obtained for conditions of pseudo-slug flow, where the liquid flow rates varied between 7.5 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s and gas flow rates ranged from 7.7 x [10.sup.-3] to 1.5 x [10.sup.-2] kg/s. For these flow conditions the roll-wave frequency varied from 1.3 to 4.3 [s.sup.-1], with uncertainties in the range of 2-22 %.

Figure 7 shows the wave frequency, [f.sub.w], as a function of the ratio between the gas and the liquid mass flow rates, [m.sub.G]/[m.sub.L], for several values of the gas mass flow rate. It may be noted that [f.sub.w] decreases with the ratio [m.sub.G]/[m.sub.L] for a given value of the gas flow rate and increases with gas mass flow rate.

Figure 8 provides a representation of the data as a three-dimensional graph, where the wave frequency is plotted against the gas and the liquid mass flow rates.

An attempt was made to establish a non-linear relationship between wave frequency and the gas and liquid mass flow rates, using an equation of the form:

z = [am.sub.L] + [bm.sup.2.sub.L] + [cm.sub.G] + [dm.sup.2.sub.G] + [em.sub.L][m.sub.G] + f (5)

where z represents the dependent variable (in this case the wave frequency, [f.sub.w]), and a to f are coefficients to be determined. The relevance of each coefficient was assessed by determining the corresponding standard errors (for a 95% confidence level). Non-significant parameters were not considered in the fit. Table 3 shows estimated coefficients and the corresponding standard errors for the wave frequency.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The fitted surface represented in Figure 8 was obtained from Equation (5) with the estimated coefficients in Table 3. Overall, the graph in Figure 8 and the fitting equation show an increase in wave frequency, [f.sub.w], with increasing gas (first order) and liquid (second order) mass flow rates. The increasing rate of [f.sub.w] with [m.sub.G] is independent of [m.sub.L], while the increasing rate of [f.sub.w] with [m.sub.L] is independent of [m.sub.G].

Velocity of the Roll-Waves

From the video films and the pressure transducers (see Measurement of Wave Velocity Section), it was possible to determine the velocity of roll-waves for gas flow rates of 7.7 x [10.sup.-3] and 9.9 x [10.sup.-3] kg/s and for liquid flows between 7.5 x [10.sup.-2] and 2.7 x [10.sup.-1] kg/s, as well as for a gas flow of 1.4 x [10.sup.-2] kg/s and liquid flow in the range of 9.9 x [10.sup.-2] to 1.7 x [10.sup.-1] kg/s. It was impossible to determine the roll-wave velocity for other flow conditions since the top of the waves were quite diluted. For the flow conditions under study, the wave velocity varied between 2.7 and 8.5 m/s and the associated uncertainties were in the range from 4 to 27%.

Figure 9 shows the variation of wave velocity, [v.sub.w] with the liquid flow rate, for three gas mass flow rates. As the liquid flow rate increases, the wave velocity increases, until it seems to reach a plateau. For high liquid flow rates, the wave velocity appears to be almost independent of the gas flow rate.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

The wave velocity assumes higher values than the liquid velocity but lower than the gas velocity. Figure 10 presents the ratio between the wave velocity and the gas velocity versus the liquid mass flow rate, for constant values of the gas flow rate. It can be seen that for the conditions studied, [v.sub.w]/[v.sub.G] always has values less than 1 and it is almost independent of the gas mass flow rate.

The ratio between the wave velocity and the liquid velocity, [v.sub.w]/[v.sub.L], varied in the range 5.3-7.3 and shows, overall, a smooth increase with the liquid mass flow rate.

COMPARISON WITH PUBLISHED DATA

In this section the data on liquid hold-up, wave frequency and wave velocity for the pseudo-slug flow conditions in the square-section channel are compared with experimental results obtained by Soleimani and Hanratty (2003). These authors studied pressure drop, liquid hold-up, wave frequency and wave velocity in a horizontal circular tube with an internal diameter of 25.4 mm, mainly for pseudo-slug flow. For liquid hold-up these authors presented data for superficial gas velocities, [v.sub.SG], of 3, 5, 6, and 8 m/s and for superficial liquid velocities, [v.sub.SL], in the range of 0-0.3 m/s. Results for wave frequency were shown for [v.sub.SG] equal to 5, 8, and 10 m/s and for [v.sub.SL] in the range 0.05-0.3 m/s. Wave velocity was measured for [v.sub.SG] equal to 5, 6, and 8 m/s and superficial liquid velocities in the range of 0.05-0.3 m/s.

In Figures 11 to 13, open and full symbols represent the data from the present study (pseudo-slug flow) and that from Soleimani and Hanratty, respectively. Only data obtained for similar superficial gas velocities were used in the comparison.

Figure 11 shows the liquid hold-up versus the superficial liquid velocity, [v.sub.SL], for constant values of the gas superficial velocity, [v.sub.SG]. A striking feature in this graph is that for the square-section channel, the pseudo-slug regime initiates and occurs at higher values of the superficial liquid velocity than for the circular tube (for the same superficial gas velocity). As a consequence, liquid hold-up results are higher for the square-section channel than for the circular tube. However the behaviour of the channel hold-up values resembles that of the values in the circular tube: they follow similar trends and present a smooth transition. For the present study hold-up data shows an initial increase with superficial liquid velocity, [v.sub.SL], followed by the attainment of a plateau (for a constant [v.sub.SG]), and a small dependence on the gas superficial velocity, [v.sub.SG]. Overall, the results by Soleimani and Hanratty also show an initial increase in hold-up followed by the attainment of a plateau, and a more evident dependence on the superficial gas velocity.

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

The frequency of roll-waves, [f.sub.w], is plotted in Figure 12 against the superficial liquid velocity. Overall, the wave frequency increases with VSL and with superficial gas velocity. As mentioned previously, in the channel the pseudo-slug regime initiates and occurs at higher values of the superficial liquid velocity (for the same superficial gas velocity). Principally for this reason, the frequency data in the channel are higher than for the circular tube, under similar superficial gas velocities. Again, the frequency values in the channel resemble those in the circular tube (similar trends and a smooth transition), particularly the values of the lower superficial gas velocity studied. The variation of the frequency with the superficial liquid velocity (for a constant VSG) is linear in both the circular tube (Soleimani and Hanratty, 2003) and the square-section channel. However, the frequency has a more evident dependence on the gas flow rate in the channel than in the circular tube.

The velocity of the roll-waves [v.sub.w], is presented in Figure 13 as a function of the superficial liquid velocity. For both sets of data (square-section channel and circular tube), the wave velocity increases with the superficial liquid velocity and shows a small dependence on the superficial gas velocity. However, the data for the circular tube shows a sudden increase in the wave velocity around [v.sub.SL] = 0.2 m/s, for [v.sub.SG] = 8 m/s. This behaviour was not observed in the data obtained in the present study.

CONCLUSIONS

In the present investigation new experimental data on liquid hold-up, gas and liquid velocities, wave velocity and wave frequency were obtained for air-water flow in a horizontal square cross-section channel. The following conclusions can be drawn for the flow conditions under study:

* Liquid hold-up, [[epsilon].sub.L], decreases with the ratio [m.sub.G]/[m.sub.L].

* The dimensionless liquid velocity [v.sub.L]/[v.sub.SL], increases linearly with the ratio [m.sub.G]/[m.sub.L].

* The dimensionless gas velocity [v.sub.G]/[v.sub.SG], decreases with the ratio [m.sub.G]/[m.sub.L].

* Data on the frequency of roll-waves shows that there is an increase in wave frequency with increasing gas and liquid mass flow rates.

* For the pseudo-slug regime, new empirical correlations were obtained for the liquid hold-up, the velocities of the gas and of the liquid phases and the frequency of the roll-waves.

* For a constant value of the gas flow, the velocity of the roll-waves, [v.sub.w], increases with liquid flow rate and, for the higher liquid flow rates, [v.sub.w] is almost independent of the gas flow rate.

* Comparison of the data obtained in the present work with the results by Soleimani and Hanratty (2003), for a circular tube of similar dimensions, shows that the pseudo-slug regime initiates and occurs in the square-section channel at higher values of the superficial liquid velocity than for the circular tube (for the same superficial gas velocity). It should be noted, however, that the behaviour of the channel data resembles that of the circular tube, showing similar trends and a smooth transition.

NOMENCLATURE

a coefficient in Equation (5)
b coefficient in Equation (5)
c coefficient in Equation (5)
d coefficient in Equation (5)
[D.sub.h] hydraulic diameter (mm)
[D.sub.m] mean deviation
e coefficient in Equation (5)
f coefficient in Equation (5)
[f.sub.w] wave frequency ([s.sup.-1])
G total mass flux (kg/([m.sup.2] s))
[G.sub.G] gas mass flux (kg/([m.sup.2] s))
[G.sub.L] liquid mass flux (kg/([m.sup.2] s))
H height of the channel (mm)
L length of the test section (m)
[m.sub.G] gas mass flow rate (kg/s)
[m.sub.L] liquid mass flow rate (kg/s)
N number of data points in Equation (2)
[v.sub.G] gas velocity (m/s)
[v.sub.L] liquid velocity (m/s)
[v.sub.SG] superficial gas velocity (m/s)
[v.sub.SL] superficial liquid velocity (m/s)
[v.sub.w] wave velocity (m/s)
[y.sub.cal,i] calculated value of the variable y, in Equation (2)
[y.sub.exp,i] experimental value of the variable y, in Equation (2)
z dependent variable in Equation (5)

Greek Symbols

[[epsilon]
.sub.L] liquid hold-up

Manuscript received April 4, 2007; revised Manuscript received November 5, 2007; accepted for publication November 28, 2007.

REFERENCES

Coleman J. W. and S. Garimella, "Characterization of Two-Phase Flow Patterns in Small Diameter Round and Rectangular Tubes," Int. J. Heat Mass Transfer 42, 2869-2881 (1999).

Ferreira, V. C. F., "Estudos Hidrodinamicos e de Sujamento em Condutas Horizontais de Seccao Recta Quadrada," MSc. Thesis, Faculty of Engineering, University of Oporto (2004).

Hewitt, G. F., G. L. Shires and T. R. Bott, "Process Heat Transfer," CRC Press LLC, BocaRaton (1994), pp. 391-421.

Holman, J. P., "Experimental Methods for Engineers," McGraw-Hill, New York (2001).

Ide, H. and H. Matsumura, "Frictional Pressure Drops of Two-Phase Gas-Liquid Flow in Rectangular Channels," Exp. Thermal Fluid Sci. 3, 362-372 (1990).

Ide, H., A. Kariyasaki and T. Fukano, "Fundamental Data on the Gas-Liquid Two-Phase Flow in Minichannels," Int. J. Thermal Sci. 46, 519-530 (2007).

Lee, H. J. and S. Y. Lee, "Pressure Drop Correlations for Two-Phase Flow Within Horizontal Rectangular Channels With Small Heights," Int. J. Multiphase Flow 27, 783-796 (2001).

Lin, P. Y. and T. J. Hanratty, "Effect of Pipe Diameter on Flow Patterns for Air-Water Flow in Horizontal," Int. J. Multiphase Flow 13, 549-563 (1987).

Mayor, T. S., A. M. F. R. Pinto and J. B. L. M. Campos, "An Image Analysis Technique for the Study of Gas-Liquid Slug Flow Along Vertical Pipes-Associated Uncertainty," Flow Meas. Instrum. 18, 139-147 (2007).

Soleimani, A. and T. J. Hanratty, "Critical Liquid Flows for the Transition from the Pseudo-Slug to Stratified Patterns to Slug Flow," Int. J. Multiphase Flow 29, 51-57 (2003).

Troniewski, L. and R. Ulbrich, "Two-Phase Gas-Liquid Flow in Rectangular Channels," Chem. Eng. Sci. 39, 751-765 (1984).

Wambsganss, M. W., J. A. Jendrzejczyk and D. M. France, "Two-Phase Flow Patterns and Transitions in a Small, Horizontal, Rectangular Channel," Int. J. Multiphase Flow 17, 327-342 (1991).

Wen, C. Y. and L. H. Chen, "Fluidized Bed Freeboard Phenomena: Entrainment and Elutriation," AIChE J. 28, 117-128 (1982).

A. M. Ribeiro, (1) * V. Ferreira, (2) T. S. Mayor (2) and J. B. L. M. Campos (2)

(1.) Instituto Superior de Engenharia do Porto, Rua Dr. Ant6nio Bernardino de Almeida, 431, 4200-072 Porto, Portugal

(2.) Centro de Estudos de Fenomenos de Transporte, Deportamento de Engenharia Quimica da Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias 4200-465 Porto, Portugal

* Author to whom correspondence may be addressed. E-mail address: asr@a isep.ipp.pt

Table 1. Published work on qas-liquid flow in rectanqular channels
(fluid mixtures, geometries and flow conditions)

Author Type of experiments Fluid mixture

Troniewski Flow patterns Air-water
and Ulbrich Pressure drop Air-aqueous
(1984) solutions of sugar

Ide and Pressure drop Air-water
Matsumura Void fraction
(1990)

Wambsganss Flow patterns Air-water
et al. (1991) Pressure drop

Coleman and Flow patterns Air-water
Garimella
(1999)

Lee and Lee Pressure drop Air-water
(2001)

Ide et al. Flow patterns Air-water
(2007) Pressure drop
 Hold-up

Author [D.sub.h] (mm) Aspect ratio

Troniewski 7.45-13.1 1-12
and Ulbrich (vertical)
(1984) 0, 1-10
 (horizontal)

Ide and 7.3-21.4 1-40
Matsumura
(1990)

Wambsganss 5.45 1/6 and 6
et al. (1991)

Coleman and 5.36 0.725
Garimella
(1999)

Lee and Lee 0.78-6.67 0.02-0.2
(2001)

Ide et al. 1-1.98 1-9
(2007)

Author Orientation Flow conditions

Troniewski Vertical [G.sub.G] = 0.6-43 kg/
and Ulbrich ([m.sup.2] s)
(1984) Horizontal [G.sub.L] = 200-1600 kg/
 ([m.sup.2] s)

Ide and Various inclinations, [V.sub.SG] = 0.1-6.9 m/s
Matsumura from horizontal to [V.sub.SL] = 0.3-4.2 m/s
(1990) vertical

Wambsganss Horizontal G = 50-2000 kg/
et al. (1991) ([m.sup.2] s)

Coleman and Horizontal [V.sub.SG] = 0.1-100 m/s,
Garimella [V.sub.SL] = 0.01-10 m/s
(1999)

Lee and Lee Horizontal [V.sub.SG] = 0.05-18.7 m/s,
(2001) [V.sub.SL] = 0.03-2.39 m/s

Ide et al. Vertical upflow [V.sub.SG] = 0.1-30 m/s
(2007) Vertical downflow [V.sub.SL] = 0.03-2.3 m/s
 Horizontal

Table 2. Uncertainties associated with the experimental parameters

Parameter Uncertainty
Channel side, H [+ or -] 0.05 mm
Length [+ or -] 0.5 mm
Pressure [+ or -] 0.05 bar
Liquid mass flow rate, [m.sub.L] 0.2-1.3%
Gas mass flow rate, [m.sub.G] 5-10%

Table 3. Estimated coefficients and corresponding standard errors for
wave frequency in pseudo-slug flow

 Estimated coefficient Standard error

a -- --
b 5.90 x [10.sup.0] 2.27 x [10.sup.-1]
c 1.27 x [10.sup.4] 2.59 x [10.sup.2]
d -- --
e -- --
f -- --
[R.sup.2] 0.982

[R.sup.2] is the proportion of variance that the fitting accounts
for; equation form: z = [am.sub.L] + [bm.sup.2.sub.L] +
[cm.sub.G] + [dm.sup.2.sub.G] + [em.sub.L][m.sub.G] + f.