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Experimental investigation of spray formation as affected by sprinkler geometry
Fire Safety Journal 46 (2011) 140–150
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Experimental investigation of spray formation as affected by sprinkler geometry Xiangyang Zhou n, Hong-Zeng Yu FM Global, 1151 Boston-Providence Turnpike, Norwood, MA 02062, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 October 2009 Received in revised form 27 October 2010 Accepted 10 January 2011 Available online 5 February 2011
A preliminary study at low water discharge pressures was conducted to investigate the fire sprinkler spray formation as affected by sprinkler geometry using a laser-based shadow imaging system. A sprinkler typically consists of the following components: a deflector with tines and slots, a boss above the deflector and sprinkler frame arms. Prototypes of these components were fabricated to evaluate their effects on the spray formation process. The water sheet thickness formed on non-slotted deflectors, slot spray discharge angle, sheet breakup distance from the deflector perimeter, water flux and drop size distributions were measured. It was found that deflector diameter and boss structure have little impact on drop size and sheet breakup distance. However, wider slots form larger drops. At constant operating pressure, the slot spray discharge angle is insensitive to the slot width, but sensitive to the boss that helps directing the slot spray toward the sprinkler centerline. The frame arm tends to produce a vertical spray sheet downstream of the frame arm, which increases the complexity of overall spray formation. An integral model was developed to calculate the development of water sheet thickness and speed on the deflector for different degrees of viscous effect exerted by the deflector. An empirical correlation was also established to estimate the spray flux fraction discharging from a deflector slot. The above measurements and observations are useful for the development of a spray formation model for fire sprinklers. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Fire sprinkler Spray formation Atomization Shadow imaging system
1. Introduction A fire sprinkler is designed to deliver water to the burning area to reduce the burning rate, to wet the surrounding combustibles to stop or reduce the flame spread and at the same time to cool the fire environment. To achieve the above design objective, the sprinkler spray must have sufficient momentum to penetrate the fire plume to deliver sufficient water fluxes to the fire area, and have sufficient heat absorption capability to lower the temperature of the fire environment. One of the existing research findings is that larger drops are more capable in penetrating the fire plume to reach the burning materials, and smaller drops are more efficient in cooling the fire plume [1]. Effective sprinkler protection requires optimal water flux and drop size distributions for a targeted fire hazard. To date, there are a large number of sprinkler types commercially available to meet the fire challenges resulting from changes in building construction and occupancies. A typical pendent sprinkler is shown in Fig. 1, which includes frame arms, boss, deflector tines
n
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Zhou). 0379-7112/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2011.01.003
and slots between the tines. When a sprinkler is actuated, water is discharged from the orifice to form a water jet. The water jet impinges on the deflector that redirects the water flow radially and forms thin sheets on the tines and through the slots. These water sheets eventually break up into drops with different sizes and velocities. Different sprinkler spray patterns can be obtained by adjusting the configurations of these key components. The initial drop size, drop velocity, water flux and their distributions are determined by the initial spray formation processes which in turn are affected by the specific sprinkler configuration. However, because of the processes’ complexity and stochastic behavior of the drop formation breakup process, the spray formation as affected by sprinkler geometry has not been fully modeled. As a result, sprinkler design has relied on the triedand-true approach to optimize the sprinkler’s fire protection performance for different applications. With the continuing advances in the computer technology and in mathematical techniques for solving fluid flow equations, computer modeling of sprinkler fire protection is becoming more feasible. One of the advantages of performing numerical simulations is that these simulations are cost-effective in screening candidate sprinkler protections before any fire tests are conducted, thus greatly reducing the number of full-scale fire tests required in the development. The delivery of water from the sprinkler to the burning material is
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typically modeled using the Lagrangian tracking approach [2,3]. The spray transport calculation using this approach is quite sensitive to the initial spray specification. Currently, the initial spray condition is prescribed based on tedious near-field spray measurements due to the lack of a validated, comprehensive spray formation model for fire sprinklers. Spray measurements have been performed by Yu [4] to characterize the spray patterns of three upright sprinklers with K-factors ranging from 81 to 162 lpm/bar1/2 using a laser-based shadowgraph system. The gross droplet-size distributions of the tested sprinklers were found to be representative of a composite of log-normal and Rosin–Rammler distributions. Using phase Doppler interferometry (PDI) and particle image velocimetry (PIV), Widmann et al. [5] measured the droplet size and velocity of residential sprinklers with K-factors ranging from 43 to 81 lpm/bar1/2. They reaffirmed the previous finding that droplet size inversely varies with 1/3-power of sprinkler operating pressure. Sheppard [6] used the PDI technique to measure the droplet size distribution at a radial distance of 0.38 m from the sprinkler. He also measured the droplet velocities using the PIV technique in a spherical coordinate system with the sprinkler at the origin. He showed that the droplet velocity varied with the elevation and azimuthal angles in the spherical coordinate.
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To predict the initial spray characteristics, Wu et al. [7] have developed a physics-based atomization model for sprinklers by integrating a film formation sub-model proposed by Watson [8] with a sheet disintegration sub-model proposed by Dombrowski and Johns [9]. The sprinkler was modeled as an axisymmetric impinging jet and the effects of the frame arms and tines were not incorporated into the model. For an ‘ideal’ sprinkler with simplified configuration, the volume median diameters predicted from the model agreed with the actual drop size measurements. Based on the measurements of a uniformly distributed spray pattern, Ren et al. [10] developed compact analytical formulations to predict the sprinkler spray. The high order curve-fit polynomials were derived using the Fourier series, and the generated spray distributions agreed reasonably well with the measurements. In this investigation, an experimental study was conducted at low operating pressures to investigate the sprinkler spray formation as affected by sprinkler geometry using a laser-based shadow imaging system. Prototypes of typical sprinkler components: a deflector with tines and slots, a boss above the deflector and simulated frame arms, were fabricated to evaluate their effects on the spray formation process. Measured in this investigation were the water sheet thickness formed on different non-slotted deflectors, spray discharge angle, sheet breakup distance from the deflector perimeter, water flux and drop size distributions. An integral model was developed to calculate the development of water sheet thickness and speed on the deflector for different degrees of viscous effect exerted by the deflector. An empirical correlation was also established to estimate the spray flux fraction discharging from the deflector slot. The objective of this study was to identify and evaluate the pertinent parameters that govern spray formation so that a model can be eventually developed to provide the initial spray information for spray transport calculations in performing numerical fire suppression simulations. The obtained results will also be useful in developing a protocol by which the initial spray characteristics of fire sprinklers can be measured efficiently at strategic locations in the vicinity of the sprinkler.
2. Experimental setup
Fig. 1. A typical pendant sprinkler design.
As shown in Fig. 1, a sprinkler consists of these key components: nozzle, deflector tine, deflector slot, boss and frame arm. To understand the effects of these components on the sprinkler spray development, four basic sprinkler configurations were investigated in this work. They are designated as the disk-sprinkler, slotsprinkler, boss-sprinkler and arm-sprinkler in the following discussion. Fig. 2 shows three disk-sprinklers with diameters of 25.4, 38.1 and 50.8 mm, respectively. A nozzle with an orifice of 9.5 mm was placed 20 mm above the disk. These disk-sprinklers have no tines,
Fig. 2. Three disk-sprinklers with diameters of (a) 25.4 mm, (b) 38.1 mm and (c) 50.8 mm.
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slots, boss or frame arms. This basic deflector configuration provides a baseline for evaluating the impact of additional deflector features on the discharge characteristics. Fig. 3 shows three slot-sprinklers with the same disk diameter (25.4 mm) and the same slot length (7.9 mm) but with different slot widths of 1.59, 3.18 and 4.76 mm. The slot sprinklers were used to evaluate the effect of slot width on the initial slot spray characteristics. Fig. 4 shows two boss-sprinklers with the same disk diameter (25.4 mm) and slot width (1.59 mm). Both bosses are of conical shape, and have the same base radius of 4.8 mm but different angles of 1271 and 901. As opposed to the slotsprinkler (no boss), the boss-sprinkler was used to evaluate the effect of cone angle on the initial sprinkler spray orientation. The fourth investigated sprinkler component, the frame arm, was simulated by attaching a cylindrical metal bar close to the disk as shown in Fig. 5, which displays a picture from the shadow imaging system. The bar diameter was 3.2 mm. The distance between the edge of the disk and the bar was 6.2 mm. For all the experiments conducted in this investigation, a nozzle with an orifice diameter of 9.5 mm was used. The shadow imaging system (LaVision) was used to record the spray images and measure drop sizes. This technique is based on high-magnification shadow imaging with pulsed backlight illumination. The measurement volume is defined by the field-of-view area and the depth of field of the imaging system. The light source is a double-cavity-frequency Nd:YAG laser (NewWave), which provides two light pulses of 532 nm wavelength, each with a pulse energy of about 120 mJ at 15 Hz double-pulse rate. The detection system consists of a 14-bit dual-frame CCD camera with 2048 2048 pixel resolution and a 12X zoom system that can achieve a field-of-view down to 400 mm. Using a short laser pulse (less than 16 ns) as illumination, it is possible to ‘‘freeze’’ drop motions faster than 100 m/s. A double-pulse laser combining with a double-frame camera allows the investigation of size-dependent drop velocities. This technique is independent of the drop shape and opacity and allows the measurement of drop sizes down to 5 mm. The spray measurements were conducted in the Water Mist Laboratory at the FM Global Norwood Campus. At a low pressure of 0.034 bar, Fig. 6 shows the drop formation from a breaking up water sheet when the jet was impinging on a disk-sprinkler. A 4 m
Fig. 5. The arm-sprinkler simulated by attaching a metal bar close to the disk.
Fig. 6. Experimental setup for water spray measurements illustrated at pressure 0.034 bar.
Fig. 3. Three slot-sprinklers with the same disk diameter (25.4 mm), the same slot length (7.9 mm) but different slot widths of (a) 1.59 mm, (b) 3.18 mm and (c) 4.76 mm.
diameter water collection pool was positioned below the disksprinkler to collect water. A pump was used to circulate the water between the pool and the discharging nozzle. The camera and the laser light diffuser were mounted on a 3-D traverse, which could move in the X, Y and Z directions with 1 mm accuracy. The laser light was transmitted via an optical fiber. Both the camera and the light diffuser were situated in air-purged protection housings. The discharge pressure was monitored during the measurement. The pressure fluctuation at 1.38 bar was about 1%.
3. Calculation of spray sheet thickness
Fig. 4. Two conical boss-sprinklers with the same base radius (4.8 mm), disk diameter (25.4 mm) and slot width (1.59 mm), but different angles of (a) 1271 and (b) 901.
The atomization model developed by Wu et al. [7] showed that the most important parameters for drop formation are the sheet thickness and average sheet speed U. The sheet speed governs the wave growth rate and the sheet thickness determines the diameter of the drop. As shown in Fig. 6, a radially expanding sheet is
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formed after the jet impinges on the disk. Based on mass conservation, the sheet thickness d at the edge of the disk can be calculated as follows:
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thickness d and speed U were expressed as
d
dU dd d þU þ U ¼ 0, dr dr r
ð4Þ
2
D U0 , 8r U
ð1Þ
where D is the diameter of the nozzle orifice, r is the disk radius, U0 is the average jet speed from the nozzle and U is the average sheet speed along the disk. For constant values of D, r and U0, the production of sheet thickness and the sheet speed are also constant as dU¼D2U0/8r. The average value of U can be decided by the measurement of d. If there is no viscous loss, U ¼U0, the inviscid sheet thickness d0 can be calculated as
d0 ¼
D2 : 8r
ð2Þ
At a constant nozzle size, Eq. (2) shows that d0 decreases only with the disk radius but is not related to the sheet speed (or discharge pressure). When the nozzle size is larger, the spray sheet is thicker and the atomization model of Wu et al. [7] shows that larger drops will be produced. Because of the skin friction arising from the interaction between the water and the disk surface, the sheet speed U decreases along the disk radius and the sheet thickness becomes thicker than that calculated from Eq. (2). Assuming the sheet flow on the disk is turbulent and applying the Blasius similarity solution, the sheet thickness at the edge of the disk can be expressed by Watson’s sheet development model [8] as 1=5 D2 7nl þ 0:0166 d¼ r 4=5 , ð3Þ 8r U0
water flow rate (lpm)
where nl is the water kinematic viscosity. Eq. (3) shows that the sheet thickness is increased by the viscous effect, and this viscous effect is related to the disk radius and inversely related to the initial jet speed (or discharge pressure). In order to show the importance of the viscous effect, a non-dimensional thickening factor b ¼ d/d0 is defined, i.e. the ratio of the actual thickness to the inviscid sheet thickness. An integral model was formulated in the present study so that the sheet thickness can be calculated for different degrees of viscous effect (see the Appendix). To simplify the analysis, a tophat velocity profile in the water sheet and a given friction coefficient were assumed. The radial development of the sheet
2U dr
dU dd 1 þ U 2 d ¼ rg d Cd rU 2 , dr dr 2
where Cd is the average friction coefficient on the disk surface and g is the gravitational acceleration. The thickness and speed of the water sheet at the deflector perimeter can be obtained by integrating Eqs. (4) and (5) numerically with unknown starting thickness and speed of the water sheet. Assuming that the sheet starts at a radial distance twice the impinging jet radius r ¼D and equating the starting sheet speed U to the jet speed U0 (i.e. inviscid assumption), the starting thickness could be calculated with Eq. (2).
4. Measurements and discussions In this exploratory study, the shadow imaging system was used to investigate the initial spray characteristics, including spray sheet thickness, slot spray discharge angle, sheet breakup distance, water flux and drop size, for selected sprinkler geometric features. 4.1. K-factor and slot spray flux fraction pffiffiffi For the nozzle with 9.5 mm orifice, the K-factor, K ¼ q= p, was firstly determined to calculate the water flow rate at the discharge pressure, where q is the flow rate in liters per minute (lpm), p is the pressure at the nozzle in bar and K is the K-factor in units of lpm/bar1/2. It was measured using a stopwatch, a water barrel and a weight scale for discharge pressures up to 0.83 bar. Fig. 7 shows the measured water discharge rates and the corresponding K-factors in a range of discharge pressures. Fig. 8(a) shows the flow stream from the 3.18 mm wide slotsprinkler at a pressure 0.034 bar. The water sheet was divided into two distinct sections by the slot: a separate stream from the slot and the remaining mostly undisturbed water sheet. A 38 mm diameter pipe was used to collect the stream originating from the slot. Using a stopwatch, a water barrel and a weight scale, the slot stream flow rate was measured. Fig. 8(b) shows the slot spray
60
6
50
5
40
4
30
3 flow rate
20
K-factor
10 0 0.0
0.2
ð5Þ
0.4 0.6 discharge pressure (bar)
0.8
2
K-factor (lpm/bar1/2)
d¼
1 0 1.0
Fig. 7. Measured water flow rate (lpm) and calculated K-factors (lpm/bar1/2) for the nozzle with 9.5 mm orifice.
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4.2. Water sheet thickness
0.12
slot spray flow rate fraction
0.1 0.08 0.06
1.59 mm slot
0.04
3.18 mm slot 0.02
4.76 mm slot 0 0
0.05
0.1
0.15
0.2
0.25
0.3
discharge pressure (bar) Fig. 8. (a) Slot stream from a 3.18 mm wide slot illustrated at pressure 0.034 bar and (b) slot spray flow rate fractions measured for three slot widths.
flow rate fraction measured for slot widths of 1.59, 3.18 and 4.76 mm. For each slot width, the slot spray flow rate fraction was almost constant in the range of test pressures from 0.03 to 0.28 bar. At higher pressures, the slot stream became difficult to discern visually from the remaining water sheet, and was not measured in the current exploratory study. The average slot spray flow rate fraction was about 11.2% for the 4.76 mm slot, 7.8% for the 3.18 mm slot and 5.6% for the 1.59 mm slot. For the two larger slots, the slot spray flow rate fraction was almost linearly proportional to the slot width. For the 1.59 mm slot, however, the slot spray flow rate fraction was relatively high for its width. This is because the slot was too narrow to form a clearly defined slot spray jet. As a result, a portion of the water sheet from the other part of the disk was also collected by the 38 mm pipe. Considering the effect of the slot geometrical size on the water flow, the slot spray flow rate fraction, V_ sf , was correlated for the 4.76 and 3.18 mm slots as follows: V_ sf ¼
Ws , 2pð0:75Rin þ0:25Rout Þ
ð6Þ
where Ws is the slot width, Rin is the radius at the base of the slot and Rout is the disk radius. The calculated slot spray flow rate fraction from Eq. (6) is 11.23% for the 4.76 mm slot and 7.5% for the 3.18 mm slot, reasonably close to the measurements.
In this study, two methods were used to measure the sheet thickness at the disk edge. The first method used particle image velocimetry (PIV) to cast a vertical laser sheet at the edge of the disk. The illuminated images with and without water discharge were recorded by the high resolution CCD-camera. The sheet thickness was determined by taking the difference between the interface heights. However, it was quickly found that the results obtained from this method were not repeatable. This was due to the fact that the recorded water-and-air interface might not have been the actual interface. The water is semi-transparent to the light. The wavy water surface reflected and refracted the light in various directions. The image of the water-and-air interface could therefore be shifted upward or downward. Consequently, this method was abandoned. The second method used back-shadow imaging. The disksprinkler was located between the camera and the light source. For no water discharge, Fig. 9(a) shows that the edge of the 25.4 mm diameter disk is near the right side in the image. For water discharging at a pressure of 0.034 bar, Fig. 9(b) shows the shadow image of the water layer on the disk. The sheet thickness at the edge of the disk was estimated to be 1.095 mm based on the two images. The shadow imaging method gives reproducible sheet thickness data. Table 1 presents the measured sheet thicknesses at the edges of 25.4 and 50.8 mm diameter disks for two operating pressures. This method was applied only to low pressure conditions. At high pressures, the far-reaching horizontal spray tended to wet the camera lens quickly and obscure the images for the current setup. Table 1 shows that the thickening factor and, therefore, the viscous effect increase with the disk diameter and the discharge pressure. The sheet thickness measurements were compared to the model predictions. Fig. 10 shows the development of sheet thickness calculated with Watson’s model (Eq. (3)) and with the present integral model (Eqs. (4) and (5)) along the radial distance. The sheet thicknesses measured at 0.034 and 0.068 bar are also presented for comparison. Based on the measured K-factor, the average speed of the impinging jet was U0 ¼3.25 m/s at 0.034 bar and U0 ¼3.98 m/s at 0.069 bar. Two friction coefficients (Cd ¼0.01 and 0.04) were assumed for the present model. Fig. 10 shows that the sheet thickness decreases with the radial distance, indicated by both the model and data. Watson’s model in general under-predicted the water sheet thickness. For the 25.4 mm diameter disk, the calculated thickening factor from Eq. (3) was b ¼1.042 at 0.034 bar and b ¼1.040 at 0.069 bar, which are less than the measurements of b ¼ 1.226 and 1.349 shown in Table 1. By increasing the discharge pressure from 0.034 to 0.069 bar, Watson’s model shows that the thickening factor reduced by 0.2%, however the measured thickening factor increased by 10%. A possible reason is that the water layer shown in Fig. 9 was still weakly turbulent but Watson’s model was developed based on the assumption of turbulent flow. To further examine Watson’s model, it is needed to improve the current measuring technology and to conduct measurements at higher operating pressures. After accounting of different degrees of viscous effect exerted by the disk, Fig. 10 shows that the predictions of the present integral model are better, especially at a radius of 25.4 mm with Cd ¼0.04. However, at a given constant friction coefficient, calculations show that the thickness predictions made with the present model are insensitive to the impinging jet speed (or discharge pressure). It is expected that the friction coefficient would vary toward the disk perimeter as the speed decreases. The discrepancy could be resolved by implementing a speed dependent friction coefficient in the present model.
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Table 1 Sheet thicknesses measured at the disk edge using the shadow imaging method. 0.034 bar
Disk (25.4 mm) Disk (50.8 mm)
0.069 bar
Sheet thickness (mm)
Thickening factor
Sheet thickness (mm)
Thickening factor
1.094 70.065 0.691 70.055
1.226 1.548
1.205 7 0.105 0.877 7 0.092
1.349 1.964
1.4
spray sheet thickness (mm)
1.2 1.0 0.8
measurements at 0.034 bar
0.6
measurements at 0.069 bar 0.4
present model, Cd=0.01 present model, Cd=0.04
0.2
model of Watson 0.0
0
5
10
15
20
25
30
radial distance (mm) Fig. 10. Comparison between calculated and measured sheet thicknesses along the radial distance.
Fig. 9. Shadow images recorded at (a) no water discharge and (b) with water discharging at 0.034 bar for the 25.4 mm diameter disk.
4.3. Sheet breakup distance Once the water sheet leaves the deflector, it transforms into an unconfined expanding sheet. The sheet expands radially outwards from the deflector, becoming increasingly unstable and creating aerodynamic waves. These sinuous waves grow until the sheet begins to break up at a critical wave amplitude. The sheet finally disintegrates into ring-like ligaments and drops. Fig. 11 shows two sheet breakup images at discharge pressures of 0.14 and
0.55 bar, respectively. The 25.4 mm diameter disk is partly shown at the bottom of the image. The camera and the light source were all placed above the nozzle to photograph the water sheet. Using a short laser pulse (less than 16 ns) as illumination, the motion of the water sheet was ‘‘frozen’’ in the image. The breakup is defined as the location where the sheet is completely broken up into ligaments. The breakup distance, Rb, was recorded between the edge of the disk and the first breakup location. As shown in Fig. 11, the breakup distance is about 210 mm at 0.14 bar, and about 170 mm at 0.55 bar. The sheet breakup distance was shortened as the discharge pressure was increased. For the three slot-sprinklers used in this study, the slot divided the sheet into two sections (see Fig. 8(a)). Compared to the water sheet formed by a solid disk, the stream along the slot broke up earlier and the ligaments of the remaining sheet were disconnected. As the slot width was increased, more water was directed into the slot, the stream along the slot broke up faster and the angle of the disrupted portion of the expanding sheet became larger. For each experimental condition, at least 21 sheet breakup images were recorded. Table 2 tabulates the average sheet breakup distance measured for all the three disk-sprinklers and the 901 boss-sprinkler in a pressure range of 0.14–0.83 bar. The bosssprinkler had a disk diameter of 25.4 mm. As shown, the sheet breakup distance decreased with the increase of the discharge pressure. Table 2 shows that the disk diameter and the boss had little effect on the sheet breakup distance. As shown in Fig. 12, the dimensionless sheet breakup distance (2Rb/D) is correlated with the Weber number, We, which is defined as We ¼
rl U02 D , s
ð7Þ
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55 50
2Rb/D
45 40 35 30 25 20 1000
10000 Weber number
100000
Fig. 12. Dimensionless sheet breakup distance (2Rb/D) correlated with the Weber number.
can be correlated with the following equation: 2Rb ¼ 370We1=4 : D
ð8Þ
4.4. Slot spray discharge angle
Fig. 11. Characteristic sheet breakup images taken from above the spray operating at (a) 0.14 bar and (b) 0.55 bar for the 25.4 mm diameter disk-sprinkler. Table 2 Sheet breakup distance (mm) measured for the three disk-sprinklers and the 901 boss-sprinkler.
0.14 bar 0.28 bar 0.55 bar 0.83 bar
Disk (25.4 mm)
Disk (38.1 mm)
Disk (50.8 mm)
Boss (aperture 901)
223 713 198 711 166 711 153 711
228 718 197 716 167 712 154 710
2357 16 2077 10 1737 10 1527 9
235 7 13 192 7 12 171 7 13 154 7 11
where rl is the water density, U0 is the jet speed, D is the diameter of the orifice and s is the water surface tension. The data obtained in this study for all the disk-sprinklers and the 901 boss-sprinkler
For the slot-sprinkler with 1.59 mm wide slot, Fig. 13 shows the slot sprays at two discharge pressures of 0.14 and 0.83 bar. The images were recorded by pointing the camera at 901 to the slot spray sheet. The slot spray discharge angle is defined as the angle between the vertical centerline of the impinging jet and the lower edge of the spray. In Fig. 13, the slot spray angle was measured to be 581 at 0.14 bar, and 511 at 0.83 bar. The angle decreased with the increase of the pressure but the relative change was small. For the other two slot-sprinklers with slot widths of 3.18 and 4.76 mm, the spray angle at 0.14 bar was measured to be 591 for the 3.18 mm wide slot, and 601 for the 4.76 mm slot. Although the slot spray flux increased as the slot width was increased, the spray angle changed little with the increase of slot width. For the two boss-sprinklers introduced in Fig. 4, the spray angle at 0.14 bar was measured to be 481 for the 1271 boss, and 441 for the 901 boss. Compared to the slotsprinkler without the boss (see Fig. 13), the conical boss tended to direct the slot spray further toward the sprinkler centerline. Fig. 14 shows the measured slot spray angles for the three slotsprinklers and two boss-sprinklers operating from 0.034 to 0.83 bar. In general, the slot spray angle tended to decrease with the increase of discharge pressure. However, for the slot-sprinkler with the 4.76 mm slot, the slot spray angle was nearly constant in the range of the investigated pressures. While the effect of slot width on the slot spray angle was small, the boss had a strong impact on the angle. For the 901 boss, the slot spray angle decreased from 561 to 121 as the discharge pressure was increased from 0.034 to 0.83 bar. Reducing the slot spray angle is useful to improve the water flux distribution directly below the sprinkler. 4.5. Effect of frame arm As shown in Fig. 5, the effect of the frame arm was examined by positioning a 3.2 mm diameter metal bar close to a disksprinkler. After the spray sheet leaves the disk, the sheet impinges on the bar. Fig. 15 shows the effect of the bar on the spray sheet at
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147
70
slot spray angle
60 50 40 30 slot width 1.59 mm
20
slot width 3.18 mm slot width 4.76 mm
10
boss aperture 127 deg boss aperture 90 deg
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
discharge pressure (bar) Fig. 14. Measured slot spray angles for three slot-sprinklers and two bosssprinklers.
slot. Therefore, the presence of the frame arm increases the complexity of the overall spray formation. 4.6. Drop size The shadow imaging technique is able to determine the sizes and shapes of individual drops. Measuring the sizes and velocities of drops detected in a control volume for sufficient time duration, statistical spray properties can be derived. One of the characteristic drop diameters describing the statistical spray information is the measuring-volume-weighted median drop diameter, Dv50, which denotes that 50% of the cumulated drop volume is from drops with diameters smaller than Dv50. The Dv50 provides a picture of the overall drops present in a control volume. If individual drop velocities are considered, a flux-weighted median diameter, Df50, can be determined. The water flux, Vi (m3/m2 s), of a drop can be expressed as Vi ¼
Fig. 13. Slot sprays recorded at discharge pressures of (a) 0.14 bar and (b) 0.83 bar for the slot-sprinkler with a slot width of 1.59 mm.
two discharge pressures. At the lower pressure of 0.014 bar, Fig. 15(a) shows that the water climbed up along the bar to a height of 15 mm above the disk. At the higher pressure of 0.034 bar, Fig. 15(b) shows that water reached a height of 20 mm. It was observed that as the discharge pressure increased further, the water could reach even higher along the bar. Both images of Fig. 15 show that a vertical water sheet was formed behind the bar. At the lower pressure of 0.014 bar, the vertical sheet eventually re-joined the sheet on the disk. After increasing the discharge pressure to 0.034 bar, it was observed that the water sheet behind the bar did not re-join the sheet on the disk, and eventually disintegrated into drops independently of the breakup processes of the sheets originating from the disk and
ð1=6Þpd3i ui , A dof
ð9Þ
where di is the drop diameter, ui is the drop velocity, A is the area of the field view of the camera and dof is the depth of the field. The dof value is linearly proportional to the drop size. Because the drop size is much smaller than the field view area, the A value is assumed to be constant. The measuring volume is the product of P A and dof. The total water flux is N i ¼ 1 Vi , where N is the number of all detected drops. To illustrate the drop formation process, Fig. 16 shows the side view of a horizontal spray sheet at a discharge pressure of 0.34 bar from a 25.4 mm diameter disk-sprinkler. The horizontal distance between the disk edge and the left end of the photograph is 170 mm. The physical viewing area of each image record was 31.6 31.6 mm2. The image shown in Fig. 16 is a composite of four such images recorded at different times and horizontal locations (124.4 31.6 mm2). As shown in Fig. 16 from left to right, the spray sheet first broke up at a critical wave amplitude and then disintegrated into ligaments and drops. Fig. 17 shows the shadowgraph of a vertical slot spray sheet discharging from the slot of the 901 boss-sprinkler at 0.34 bar. The image is a composite of nineteen smaller images. Each smaller image’s dimension is 31.6 31.6 mm2. The top-left region is not shown because the image was obscured by the water spray. The boss-sprinkler was located 20 mm from the left side of the composite image. It shows that the slot spray sheet follows the same breakup process as that of the water sheet originating from the disk edge.
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To illustrate the effect of disk diameter of the disk-sprinkler on drop size, Table 3 shows the measuring-volume-weighted Dv50 and the flux-weighted Df50 for discharge pressures of 0.34 and 0.69 bar. The measurements were conducted at 0.3 and 0.6 m horizontally from the edge of the disk. At each location, the camera was traversed vertically across the entire spray to measure the drop size and velocity distributions. The diameters shown in Table 3 are the overall result at each radial location. The drop size decreased for all the three disk-sprinklers as the discharge pressure was increased. Because the dof value of Eq. (9) increases with the drop size, the flux-weighted Df50 was smaller than the volume-weighted Dv50 based on these near-field measurements. Table 3 also shows that the disk diameter has little impact on the drop size. As recalled from the data in Table 1, the larger disks result in thinner sheets at the disk edge but greater thickening factors. This implies that, for the same discharge pressure, the sheet speed at the edge of the larger disk is lower than that of the smaller disk. The atomization model [7] indicates that both a higher sheet speed and a thinner sheet produce smaller drop sizes. As the disk diameter increases, the sheet at the disk edge is thinner but the speed is also lower. The countering result is that the drop size is little affected by the disk diameter. Table 2 also shows that the sheet breakup distances are comparable for all the three disk-sprinklers. The result is consistent with the disk diameter’s small effect on the drop size as discussed above. To evaluate the effect of slot width on drop size, measurements were performed at 0.6 m from the disk edge for all the three slot-sprinklers. Table 4 shows the median drop diameters of the slot sprays. Based on these data, it is apparent that wider slots produce larger drops. This is because the initial slot spray is thicker for a wider slot.
5. Conclusions
Fig. 15. Vertical water sheet formed behind the sprinkler arm at discharge pressures of (a) 0.014 bar and (b) 0.034 bar.
Using a shadow imaging system, an exploratory study at low water discharge pressures was performed to gain insight into the spray formation of fire sprinklers as affected by the key sprinkler components. The findings from this investigation will be helpful in developing a spray formation model for fire sprinklers operating at typical discharge conditions, which can be used to provide the initial spray conditions for spray transport calculations using a numerical simulation. The understanding is also useful in developing a protocol by which sprinkler sprays can be efficiently quantified by conducting as few measurements as possible at strategic locations in the spray. Prototypes representing key sprinkler components were fabricated to evaluate their effects on the spray formation processes. These prototypes were designated as: disk-sprinkler, slot-sprinkler, boss-sprinkler and arm-sprinkler, with which the effects of deflector diameter, deflector slot width, boss configuration and sprinkler frame arms could be evaluated separately. The water
Fig. 16. Side view of a horizontal spray sheet from a 25.4 mm diameter disk-sprinkler discharging at 0.34 bar.
X. Zhou, H.-Z. Yu / Fire Safety Journal 46 (2011) 140–150
149
Fig. 17. Side view of a vertical slot spray sheet discharging at 0.34 bar from the slot of the 901 boss-sprinkler.
Table 3 Measured drop sizes of the three disk-sprinklers. Disk diameter (mm)
Pressure (bar)
Horizontal distance from the edge of the disk 0.3 (m)
25.4 38.1 50.8 25.4 38.1 50.8
0.34 0.34 0.34 0.68 0.68 0.68
0.6 (m)
Dv50 (mm)
Df50 (mm)
Dv50 (mm)
Df50 (mm)
1.273 1.238 1.332 1.028 0.980 1.056
1.054 0.996 1.124 0.811 0.788 0.866
1.069 1.428 1.224 1.001 0.983 1.214
0.931 1.251 1.028 0.870 0.802 0.966
Table 4 Median drop diameters of the slot sprays of the three slot-sprinklers. Slot width (mm)
Pressure (bar)
Dv50 (mm)
Df50 (mm)
1.59 3.18 4.76 1.59 3.18 4.76
0.34 0.34 0.34 0.68 0.68 0.68
1.697 2.863 3.080 1.366 2.186 2.733
1.421 2.691 2.767 1.139 1.705 1.811
sheet thickness, slot spray discharge angle, sheet breakup distance, water flux, drop size and velocity were measured. It was found that the deflector diameter and the deflector boss had little impact on the drop size and sheet breakup distance. Wider deflector slots formed larger drops. At the same operating pressure, the slot width had little effect on the slot spray angle, but the boss helps directing the slot spray toward the sprinkler centerline. The slot spray angle tended to decrease with the
increase of discharge pressure, but the relative change was small. The frame arm tended to produce an up-lifted vertical sheet downstream of the arm, thus increasing the complexity of overall spray formation. An integral model was developed so that the sheet thickness at the disk edge could be calculated for different viscous effects exerted by the deflector. The model provides reasonable predictions of water sheet thickness and speed, which are required by existing spray atomization models. An empirical correlation was also established in this study to estimate the fraction of water discharging from a single slot of a sprinkler deflector. The fraction of water discharging from all the slots of a deflector is another piece of essential information required by spray atomization models. While representative with regard to the overall phenomenon, the findings from this investigation may not be directly applicable to fire sprinklers operating at typical (higher pressure) discharge conditions. As a first step in a systematic approach to investigate the mechanism of spray formation, experimental study will be conducted on higher pressure with the improvement of the measuring technology.
Acknowledgements This study was funded by FM Global in its sprinkler technology research program. We are grateful for Stephen D’Aniello’s assistance in the preparation of the measurement setup.
Appendix A. A formulation for the calculation of water sheet thickness and speed on a cylindrical disk The global atomization model currently being developed at the University of Maryland (UM) for a water jet impinging vertically
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on the center of a cylindrical disk requires the information of water sheet thickness and speed at the disk perimeter [7]. Currently, UM estimates the required information from the semi-empirical correlation developed by Watson [8]. The correlation gives a thinner water sheet at the disk perimeter than the laser-based back-shadow measurements recently made in this exploratory investigation. It is speculated that Watson’s Blasius boundary layer approximation does not adequately account for the viscous effect on the water sheet development on the disk. Therefore, an alternate formulation was developed so that calculations of water sheet development on the disk can be performed for different degrees of viscous effect exerted by the disk. A.1. Physical problem
z,v
g r,u
A.2. Formulation With the above coordinate and speed component designations, the continuity equation and the r- and y-momentum equations for an axisymmetrical boundary layer on a horizontal surface are @ @ ðr ruÞ þ ðr rvÞ ¼ 0, @r @z @u @u @p 1 @ @u þ rv ¼ þ rm , @r @z @r r @z @z
d
0
Eq. (A.6) can be further consolidated to Z Z d @ @ d @p @u ðr ru2 Þ @zud @zr m ðr ruÞ @z ¼ r : @r @r 0 @z z ¼ 0 0 @r ðA:7Þ
0
d
Applying Leibnitz’s rule for differentiation of integral, Z @ d d dd ðr ru2 Þ @z ¼ ðr ru2 Þ @zr ru2d @r dr 0 dr
and Z Z d @p d d dd @z ¼ : p @zpd dr 0 dr 0 @r
ðA:8Þ
ðA:9Þ
Substituting Eqs. (A.8) and (A.9) into Eq. (A.7) and approximating top-hat velocity profile in the water sheet, we have Z d dd d d ðr ru2 dÞr ru2 ¼ r ðppd Þ @zr tw dr dr 0 dr Z d d ¼ r rgðdzÞ dzrtw dr 0 d 2 ¼ r ð12rg d Þr tw dr rg d 1 2 f 2 ð rd Þ ru , ¼ ðA:10Þ 2 dr 2 2
V
ðA:1Þ
ðA:2Þ
and @p ¼ rg: @z
Z
Z
When a water jet impinges on the center of a cylindrical disk, what are the thickness and speed of the water sheet at the disk perimeter?
ru
Substituting Eq. (A.5) into Eq. (A.2) and integrating across the sheet layer, we have Z d Z Z @u u @ d 1 d @ ru @z d ðr ruÞ @z þ u ðr ruÞ @z @r r 0 @r r @r 0 0 Z d @p @u @zm ¼ : ðA:6Þ @z z ¼ 0 0 @r
ðA:3Þ
The formulation below follows the traditional procedure for performing flow integration across the boundary layer. From Eq. (A.1), we obtain Z 1 @ z v¼ r ru @z: ðA:4Þ r r @r 0 Multiplying r@u=@z on both sides of Eq. (A.4), we have Z z Z 1 @ @ z @ ðr ruÞ @z ¼ ðr ruÞ @z u ðr ruÞ : u r @z @r 0 @r 0
@u 1 @u @ rv ¼ @z r @z @r
ðA:5Þ
where tw ¼ mð@u=@zÞz ¼ 0 ¼ f u2 =2. Note that Eq. (A.7) can also be simplified with other presumed speed profiles. References [1] C. Yao, Extinguishing role of water sprinklers, in: Proceedings of the Engineering Applications of Fire Technology Workshop, National Bureau of Standards, Gaithersburg, MD, April 16–18, 1980, pp. 51–88. [2] K.B. McGrattan, G.P. Forney, Fire Dynamics Simulator (Version 4), Technical Reference Guide, NIST Special Publication 1018, National Institute of Standards and Technology, Gaithersburg, Maryland, March 12, 2006. [3] S. Nam, Development of a computational model simulating the interaction between a fire plume and a sprinkler spray, Fire Safety Journal 26 (1996) 1–33. [4] H.Z. Yu, Investigation of spray patterns of selected sprinklers with the FMRC drop size measuring system, in: Proceedings of the First International Symposium on Fire Safety Science, New York, 1986, pp. 1165–1176. [5] J.F. Widmann, D.T. Sheppard, R.M. Lueptow, Non-intrusive measurements in fire sprinkler sprays, Fire Technology 37 (2001) 297–315. [6] D.T. Sheppard, Spray Characteristic of Fire Sprinkler, NIST GCR 02-838, National Institute of Standards and Technology, Gaithersburg, MD, 2002. [7] D. Wu, D. Guillemin, A.W. Marshall, A modeling basis for predicting the initial sprinkler spray, Fire Safety Journal 42 (2007) 283–294. [8] E.J. Watson, The radial spread of a liquid jet over a horizontal plane, Journal of Fluid Mechanics 20 (1964) 481–499. [9] N. Dombrowski, W.R. Johns, The aerodynamics instability and disintegration of viscous liquid sheets, Chemical Engineering Science 18 (1963) 203–214. [10] N. Ren, A.W. Marshall, H. Baum, A comprehensive methodology for characterizing sprinkler sprays, Presented at the 33rd International Symposium on Combustion, Beijing, China, 2010.
Contents lists available at ScienceDirect
Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Experimental investigation of spray formation as affected by sprinkler geometry Xiangyang Zhou n, Hong-Zeng Yu FM Global, 1151 Boston-Providence Turnpike, Norwood, MA 02062, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 October 2009 Received in revised form 27 October 2010 Accepted 10 January 2011 Available online 5 February 2011
A preliminary study at low water discharge pressures was conducted to investigate the fire sprinkler spray formation as affected by sprinkler geometry using a laser-based shadow imaging system. A sprinkler typically consists of the following components: a deflector with tines and slots, a boss above the deflector and sprinkler frame arms. Prototypes of these components were fabricated to evaluate their effects on the spray formation process. The water sheet thickness formed on non-slotted deflectors, slot spray discharge angle, sheet breakup distance from the deflector perimeter, water flux and drop size distributions were measured. It was found that deflector diameter and boss structure have little impact on drop size and sheet breakup distance. However, wider slots form larger drops. At constant operating pressure, the slot spray discharge angle is insensitive to the slot width, but sensitive to the boss that helps directing the slot spray toward the sprinkler centerline. The frame arm tends to produce a vertical spray sheet downstream of the frame arm, which increases the complexity of overall spray formation. An integral model was developed to calculate the development of water sheet thickness and speed on the deflector for different degrees of viscous effect exerted by the deflector. An empirical correlation was also established to estimate the spray flux fraction discharging from a deflector slot. The above measurements and observations are useful for the development of a spray formation model for fire sprinklers. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Fire sprinkler Spray formation Atomization Shadow imaging system
1. Introduction A fire sprinkler is designed to deliver water to the burning area to reduce the burning rate, to wet the surrounding combustibles to stop or reduce the flame spread and at the same time to cool the fire environment. To achieve the above design objective, the sprinkler spray must have sufficient momentum to penetrate the fire plume to deliver sufficient water fluxes to the fire area, and have sufficient heat absorption capability to lower the temperature of the fire environment. One of the existing research findings is that larger drops are more capable in penetrating the fire plume to reach the burning materials, and smaller drops are more efficient in cooling the fire plume [1]. Effective sprinkler protection requires optimal water flux and drop size distributions for a targeted fire hazard. To date, there are a large number of sprinkler types commercially available to meet the fire challenges resulting from changes in building construction and occupancies. A typical pendent sprinkler is shown in Fig. 1, which includes frame arms, boss, deflector tines
n
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Zhou). 0379-7112/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2011.01.003
and slots between the tines. When a sprinkler is actuated, water is discharged from the orifice to form a water jet. The water jet impinges on the deflector that redirects the water flow radially and forms thin sheets on the tines and through the slots. These water sheets eventually break up into drops with different sizes and velocities. Different sprinkler spray patterns can be obtained by adjusting the configurations of these key components. The initial drop size, drop velocity, water flux and their distributions are determined by the initial spray formation processes which in turn are affected by the specific sprinkler configuration. However, because of the processes’ complexity and stochastic behavior of the drop formation breakup process, the spray formation as affected by sprinkler geometry has not been fully modeled. As a result, sprinkler design has relied on the triedand-true approach to optimize the sprinkler’s fire protection performance for different applications. With the continuing advances in the computer technology and in mathematical techniques for solving fluid flow equations, computer modeling of sprinkler fire protection is becoming more feasible. One of the advantages of performing numerical simulations is that these simulations are cost-effective in screening candidate sprinkler protections before any fire tests are conducted, thus greatly reducing the number of full-scale fire tests required in the development. The delivery of water from the sprinkler to the burning material is
X. Zhou, H.-Z. Yu / Fire Safety Journal 46 (2011) 140–150
typically modeled using the Lagrangian tracking approach [2,3]. The spray transport calculation using this approach is quite sensitive to the initial spray specification. Currently, the initial spray condition is prescribed based on tedious near-field spray measurements due to the lack of a validated, comprehensive spray formation model for fire sprinklers. Spray measurements have been performed by Yu [4] to characterize the spray patterns of three upright sprinklers with K-factors ranging from 81 to 162 lpm/bar1/2 using a laser-based shadowgraph system. The gross droplet-size distributions of the tested sprinklers were found to be representative of a composite of log-normal and Rosin–Rammler distributions. Using phase Doppler interferometry (PDI) and particle image velocimetry (PIV), Widmann et al. [5] measured the droplet size and velocity of residential sprinklers with K-factors ranging from 43 to 81 lpm/bar1/2. They reaffirmed the previous finding that droplet size inversely varies with 1/3-power of sprinkler operating pressure. Sheppard [6] used the PDI technique to measure the droplet size distribution at a radial distance of 0.38 m from the sprinkler. He also measured the droplet velocities using the PIV technique in a spherical coordinate system with the sprinkler at the origin. He showed that the droplet velocity varied with the elevation and azimuthal angles in the spherical coordinate.
141
To predict the initial spray characteristics, Wu et al. [7] have developed a physics-based atomization model for sprinklers by integrating a film formation sub-model proposed by Watson [8] with a sheet disintegration sub-model proposed by Dombrowski and Johns [9]. The sprinkler was modeled as an axisymmetric impinging jet and the effects of the frame arms and tines were not incorporated into the model. For an ‘ideal’ sprinkler with simplified configuration, the volume median diameters predicted from the model agreed with the actual drop size measurements. Based on the measurements of a uniformly distributed spray pattern, Ren et al. [10] developed compact analytical formulations to predict the sprinkler spray. The high order curve-fit polynomials were derived using the Fourier series, and the generated spray distributions agreed reasonably well with the measurements. In this investigation, an experimental study was conducted at low operating pressures to investigate the sprinkler spray formation as affected by sprinkler geometry using a laser-based shadow imaging system. Prototypes of typical sprinkler components: a deflector with tines and slots, a boss above the deflector and simulated frame arms, were fabricated to evaluate their effects on the spray formation process. Measured in this investigation were the water sheet thickness formed on different non-slotted deflectors, spray discharge angle, sheet breakup distance from the deflector perimeter, water flux and drop size distributions. An integral model was developed to calculate the development of water sheet thickness and speed on the deflector for different degrees of viscous effect exerted by the deflector. An empirical correlation was also established to estimate the spray flux fraction discharging from the deflector slot. The objective of this study was to identify and evaluate the pertinent parameters that govern spray formation so that a model can be eventually developed to provide the initial spray information for spray transport calculations in performing numerical fire suppression simulations. The obtained results will also be useful in developing a protocol by which the initial spray characteristics of fire sprinklers can be measured efficiently at strategic locations in the vicinity of the sprinkler.
2. Experimental setup
Fig. 1. A typical pendant sprinkler design.
As shown in Fig. 1, a sprinkler consists of these key components: nozzle, deflector tine, deflector slot, boss and frame arm. To understand the effects of these components on the sprinkler spray development, four basic sprinkler configurations were investigated in this work. They are designated as the disk-sprinkler, slotsprinkler, boss-sprinkler and arm-sprinkler in the following discussion. Fig. 2 shows three disk-sprinklers with diameters of 25.4, 38.1 and 50.8 mm, respectively. A nozzle with an orifice of 9.5 mm was placed 20 mm above the disk. These disk-sprinklers have no tines,
Fig. 2. Three disk-sprinklers with diameters of (a) 25.4 mm, (b) 38.1 mm and (c) 50.8 mm.
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slots, boss or frame arms. This basic deflector configuration provides a baseline for evaluating the impact of additional deflector features on the discharge characteristics. Fig. 3 shows three slot-sprinklers with the same disk diameter (25.4 mm) and the same slot length (7.9 mm) but with different slot widths of 1.59, 3.18 and 4.76 mm. The slot sprinklers were used to evaluate the effect of slot width on the initial slot spray characteristics. Fig. 4 shows two boss-sprinklers with the same disk diameter (25.4 mm) and slot width (1.59 mm). Both bosses are of conical shape, and have the same base radius of 4.8 mm but different angles of 1271 and 901. As opposed to the slotsprinkler (no boss), the boss-sprinkler was used to evaluate the effect of cone angle on the initial sprinkler spray orientation. The fourth investigated sprinkler component, the frame arm, was simulated by attaching a cylindrical metal bar close to the disk as shown in Fig. 5, which displays a picture from the shadow imaging system. The bar diameter was 3.2 mm. The distance between the edge of the disk and the bar was 6.2 mm. For all the experiments conducted in this investigation, a nozzle with an orifice diameter of 9.5 mm was used. The shadow imaging system (LaVision) was used to record the spray images and measure drop sizes. This technique is based on high-magnification shadow imaging with pulsed backlight illumination. The measurement volume is defined by the field-of-view area and the depth of field of the imaging system. The light source is a double-cavity-frequency Nd:YAG laser (NewWave), which provides two light pulses of 532 nm wavelength, each with a pulse energy of about 120 mJ at 15 Hz double-pulse rate. The detection system consists of a 14-bit dual-frame CCD camera with 2048 2048 pixel resolution and a 12X zoom system that can achieve a field-of-view down to 400 mm. Using a short laser pulse (less than 16 ns) as illumination, it is possible to ‘‘freeze’’ drop motions faster than 100 m/s. A double-pulse laser combining with a double-frame camera allows the investigation of size-dependent drop velocities. This technique is independent of the drop shape and opacity and allows the measurement of drop sizes down to 5 mm. The spray measurements were conducted in the Water Mist Laboratory at the FM Global Norwood Campus. At a low pressure of 0.034 bar, Fig. 6 shows the drop formation from a breaking up water sheet when the jet was impinging on a disk-sprinkler. A 4 m
Fig. 5. The arm-sprinkler simulated by attaching a metal bar close to the disk.
Fig. 6. Experimental setup for water spray measurements illustrated at pressure 0.034 bar.
Fig. 3. Three slot-sprinklers with the same disk diameter (25.4 mm), the same slot length (7.9 mm) but different slot widths of (a) 1.59 mm, (b) 3.18 mm and (c) 4.76 mm.
diameter water collection pool was positioned below the disksprinkler to collect water. A pump was used to circulate the water between the pool and the discharging nozzle. The camera and the laser light diffuser were mounted on a 3-D traverse, which could move in the X, Y and Z directions with 1 mm accuracy. The laser light was transmitted via an optical fiber. Both the camera and the light diffuser were situated in air-purged protection housings. The discharge pressure was monitored during the measurement. The pressure fluctuation at 1.38 bar was about 1%.
3. Calculation of spray sheet thickness
Fig. 4. Two conical boss-sprinklers with the same base radius (4.8 mm), disk diameter (25.4 mm) and slot width (1.59 mm), but different angles of (a) 1271 and (b) 901.
The atomization model developed by Wu et al. [7] showed that the most important parameters for drop formation are the sheet thickness and average sheet speed U. The sheet speed governs the wave growth rate and the sheet thickness determines the diameter of the drop. As shown in Fig. 6, a radially expanding sheet is
X. Zhou, H.-Z. Yu / Fire Safety Journal 46 (2011) 140–150
formed after the jet impinges on the disk. Based on mass conservation, the sheet thickness d at the edge of the disk can be calculated as follows:
143
thickness d and speed U were expressed as
d
dU dd d þU þ U ¼ 0, dr dr r
ð4Þ
2
D U0 , 8r U
ð1Þ
where D is the diameter of the nozzle orifice, r is the disk radius, U0 is the average jet speed from the nozzle and U is the average sheet speed along the disk. For constant values of D, r and U0, the production of sheet thickness and the sheet speed are also constant as dU¼D2U0/8r. The average value of U can be decided by the measurement of d. If there is no viscous loss, U ¼U0, the inviscid sheet thickness d0 can be calculated as
d0 ¼
D2 : 8r
ð2Þ
At a constant nozzle size, Eq. (2) shows that d0 decreases only with the disk radius but is not related to the sheet speed (or discharge pressure). When the nozzle size is larger, the spray sheet is thicker and the atomization model of Wu et al. [7] shows that larger drops will be produced. Because of the skin friction arising from the interaction between the water and the disk surface, the sheet speed U decreases along the disk radius and the sheet thickness becomes thicker than that calculated from Eq. (2). Assuming the sheet flow on the disk is turbulent and applying the Blasius similarity solution, the sheet thickness at the edge of the disk can be expressed by Watson’s sheet development model [8] as 1=5 D2 7nl þ 0:0166 d¼ r 4=5 , ð3Þ 8r U0
water flow rate (lpm)
where nl is the water kinematic viscosity. Eq. (3) shows that the sheet thickness is increased by the viscous effect, and this viscous effect is related to the disk radius and inversely related to the initial jet speed (or discharge pressure). In order to show the importance of the viscous effect, a non-dimensional thickening factor b ¼ d/d0 is defined, i.e. the ratio of the actual thickness to the inviscid sheet thickness. An integral model was formulated in the present study so that the sheet thickness can be calculated for different degrees of viscous effect (see the Appendix). To simplify the analysis, a tophat velocity profile in the water sheet and a given friction coefficient were assumed. The radial development of the sheet
2U dr
dU dd 1 þ U 2 d ¼ rg d Cd rU 2 , dr dr 2
where Cd is the average friction coefficient on the disk surface and g is the gravitational acceleration. The thickness and speed of the water sheet at the deflector perimeter can be obtained by integrating Eqs. (4) and (5) numerically with unknown starting thickness and speed of the water sheet. Assuming that the sheet starts at a radial distance twice the impinging jet radius r ¼D and equating the starting sheet speed U to the jet speed U0 (i.e. inviscid assumption), the starting thickness could be calculated with Eq. (2).
4. Measurements and discussions In this exploratory study, the shadow imaging system was used to investigate the initial spray characteristics, including spray sheet thickness, slot spray discharge angle, sheet breakup distance, water flux and drop size, for selected sprinkler geometric features. 4.1. K-factor and slot spray flux fraction pffiffiffi For the nozzle with 9.5 mm orifice, the K-factor, K ¼ q= p, was firstly determined to calculate the water flow rate at the discharge pressure, where q is the flow rate in liters per minute (lpm), p is the pressure at the nozzle in bar and K is the K-factor in units of lpm/bar1/2. It was measured using a stopwatch, a water barrel and a weight scale for discharge pressures up to 0.83 bar. Fig. 7 shows the measured water discharge rates and the corresponding K-factors in a range of discharge pressures. Fig. 8(a) shows the flow stream from the 3.18 mm wide slotsprinkler at a pressure 0.034 bar. The water sheet was divided into two distinct sections by the slot: a separate stream from the slot and the remaining mostly undisturbed water sheet. A 38 mm diameter pipe was used to collect the stream originating from the slot. Using a stopwatch, a water barrel and a weight scale, the slot stream flow rate was measured. Fig. 8(b) shows the slot spray
60
6
50
5
40
4
30
3 flow rate
20
K-factor
10 0 0.0
0.2
ð5Þ
0.4 0.6 discharge pressure (bar)
0.8
2
K-factor (lpm/bar1/2)
d¼
1 0 1.0
Fig. 7. Measured water flow rate (lpm) and calculated K-factors (lpm/bar1/2) for the nozzle with 9.5 mm orifice.
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4.2. Water sheet thickness
0.12
slot spray flow rate fraction
0.1 0.08 0.06
1.59 mm slot
0.04
3.18 mm slot 0.02
4.76 mm slot 0 0
0.05
0.1
0.15
0.2
0.25
0.3
discharge pressure (bar) Fig. 8. (a) Slot stream from a 3.18 mm wide slot illustrated at pressure 0.034 bar and (b) slot spray flow rate fractions measured for three slot widths.
flow rate fraction measured for slot widths of 1.59, 3.18 and 4.76 mm. For each slot width, the slot spray flow rate fraction was almost constant in the range of test pressures from 0.03 to 0.28 bar. At higher pressures, the slot stream became difficult to discern visually from the remaining water sheet, and was not measured in the current exploratory study. The average slot spray flow rate fraction was about 11.2% for the 4.76 mm slot, 7.8% for the 3.18 mm slot and 5.6% for the 1.59 mm slot. For the two larger slots, the slot spray flow rate fraction was almost linearly proportional to the slot width. For the 1.59 mm slot, however, the slot spray flow rate fraction was relatively high for its width. This is because the slot was too narrow to form a clearly defined slot spray jet. As a result, a portion of the water sheet from the other part of the disk was also collected by the 38 mm pipe. Considering the effect of the slot geometrical size on the water flow, the slot spray flow rate fraction, V_ sf , was correlated for the 4.76 and 3.18 mm slots as follows: V_ sf ¼
Ws , 2pð0:75Rin þ0:25Rout Þ
ð6Þ
where Ws is the slot width, Rin is the radius at the base of the slot and Rout is the disk radius. The calculated slot spray flow rate fraction from Eq. (6) is 11.23% for the 4.76 mm slot and 7.5% for the 3.18 mm slot, reasonably close to the measurements.
In this study, two methods were used to measure the sheet thickness at the disk edge. The first method used particle image velocimetry (PIV) to cast a vertical laser sheet at the edge of the disk. The illuminated images with and without water discharge were recorded by the high resolution CCD-camera. The sheet thickness was determined by taking the difference between the interface heights. However, it was quickly found that the results obtained from this method were not repeatable. This was due to the fact that the recorded water-and-air interface might not have been the actual interface. The water is semi-transparent to the light. The wavy water surface reflected and refracted the light in various directions. The image of the water-and-air interface could therefore be shifted upward or downward. Consequently, this method was abandoned. The second method used back-shadow imaging. The disksprinkler was located between the camera and the light source. For no water discharge, Fig. 9(a) shows that the edge of the 25.4 mm diameter disk is near the right side in the image. For water discharging at a pressure of 0.034 bar, Fig. 9(b) shows the shadow image of the water layer on the disk. The sheet thickness at the edge of the disk was estimated to be 1.095 mm based on the two images. The shadow imaging method gives reproducible sheet thickness data. Table 1 presents the measured sheet thicknesses at the edges of 25.4 and 50.8 mm diameter disks for two operating pressures. This method was applied only to low pressure conditions. At high pressures, the far-reaching horizontal spray tended to wet the camera lens quickly and obscure the images for the current setup. Table 1 shows that the thickening factor and, therefore, the viscous effect increase with the disk diameter and the discharge pressure. The sheet thickness measurements were compared to the model predictions. Fig. 10 shows the development of sheet thickness calculated with Watson’s model (Eq. (3)) and with the present integral model (Eqs. (4) and (5)) along the radial distance. The sheet thicknesses measured at 0.034 and 0.068 bar are also presented for comparison. Based on the measured K-factor, the average speed of the impinging jet was U0 ¼3.25 m/s at 0.034 bar and U0 ¼3.98 m/s at 0.069 bar. Two friction coefficients (Cd ¼0.01 and 0.04) were assumed for the present model. Fig. 10 shows that the sheet thickness decreases with the radial distance, indicated by both the model and data. Watson’s model in general under-predicted the water sheet thickness. For the 25.4 mm diameter disk, the calculated thickening factor from Eq. (3) was b ¼1.042 at 0.034 bar and b ¼1.040 at 0.069 bar, which are less than the measurements of b ¼ 1.226 and 1.349 shown in Table 1. By increasing the discharge pressure from 0.034 to 0.069 bar, Watson’s model shows that the thickening factor reduced by 0.2%, however the measured thickening factor increased by 10%. A possible reason is that the water layer shown in Fig. 9 was still weakly turbulent but Watson’s model was developed based on the assumption of turbulent flow. To further examine Watson’s model, it is needed to improve the current measuring technology and to conduct measurements at higher operating pressures. After accounting of different degrees of viscous effect exerted by the disk, Fig. 10 shows that the predictions of the present integral model are better, especially at a radius of 25.4 mm with Cd ¼0.04. However, at a given constant friction coefficient, calculations show that the thickness predictions made with the present model are insensitive to the impinging jet speed (or discharge pressure). It is expected that the friction coefficient would vary toward the disk perimeter as the speed decreases. The discrepancy could be resolved by implementing a speed dependent friction coefficient in the present model.
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Table 1 Sheet thicknesses measured at the disk edge using the shadow imaging method. 0.034 bar
Disk (25.4 mm) Disk (50.8 mm)
0.069 bar
Sheet thickness (mm)
Thickening factor
Sheet thickness (mm)
Thickening factor
1.094 70.065 0.691 70.055
1.226 1.548
1.205 7 0.105 0.877 7 0.092
1.349 1.964
1.4
spray sheet thickness (mm)
1.2 1.0 0.8
measurements at 0.034 bar
0.6
measurements at 0.069 bar 0.4
present model, Cd=0.01 present model, Cd=0.04
0.2
model of Watson 0.0
0
5
10
15
20
25
30
radial distance (mm) Fig. 10. Comparison between calculated and measured sheet thicknesses along the radial distance.
Fig. 9. Shadow images recorded at (a) no water discharge and (b) with water discharging at 0.034 bar for the 25.4 mm diameter disk.
4.3. Sheet breakup distance Once the water sheet leaves the deflector, it transforms into an unconfined expanding sheet. The sheet expands radially outwards from the deflector, becoming increasingly unstable and creating aerodynamic waves. These sinuous waves grow until the sheet begins to break up at a critical wave amplitude. The sheet finally disintegrates into ring-like ligaments and drops. Fig. 11 shows two sheet breakup images at discharge pressures of 0.14 and
0.55 bar, respectively. The 25.4 mm diameter disk is partly shown at the bottom of the image. The camera and the light source were all placed above the nozzle to photograph the water sheet. Using a short laser pulse (less than 16 ns) as illumination, the motion of the water sheet was ‘‘frozen’’ in the image. The breakup is defined as the location where the sheet is completely broken up into ligaments. The breakup distance, Rb, was recorded between the edge of the disk and the first breakup location. As shown in Fig. 11, the breakup distance is about 210 mm at 0.14 bar, and about 170 mm at 0.55 bar. The sheet breakup distance was shortened as the discharge pressure was increased. For the three slot-sprinklers used in this study, the slot divided the sheet into two sections (see Fig. 8(a)). Compared to the water sheet formed by a solid disk, the stream along the slot broke up earlier and the ligaments of the remaining sheet were disconnected. As the slot width was increased, more water was directed into the slot, the stream along the slot broke up faster and the angle of the disrupted portion of the expanding sheet became larger. For each experimental condition, at least 21 sheet breakup images were recorded. Table 2 tabulates the average sheet breakup distance measured for all the three disk-sprinklers and the 901 boss-sprinkler in a pressure range of 0.14–0.83 bar. The bosssprinkler had a disk diameter of 25.4 mm. As shown, the sheet breakup distance decreased with the increase of the discharge pressure. Table 2 shows that the disk diameter and the boss had little effect on the sheet breakup distance. As shown in Fig. 12, the dimensionless sheet breakup distance (2Rb/D) is correlated with the Weber number, We, which is defined as We ¼
rl U02 D , s
ð7Þ
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55 50
2Rb/D
45 40 35 30 25 20 1000
10000 Weber number
100000
Fig. 12. Dimensionless sheet breakup distance (2Rb/D) correlated with the Weber number.
can be correlated with the following equation: 2Rb ¼ 370We1=4 : D
ð8Þ
4.4. Slot spray discharge angle
Fig. 11. Characteristic sheet breakup images taken from above the spray operating at (a) 0.14 bar and (b) 0.55 bar for the 25.4 mm diameter disk-sprinkler. Table 2 Sheet breakup distance (mm) measured for the three disk-sprinklers and the 901 boss-sprinkler.
0.14 bar 0.28 bar 0.55 bar 0.83 bar
Disk (25.4 mm)
Disk (38.1 mm)
Disk (50.8 mm)
Boss (aperture 901)
223 713 198 711 166 711 153 711
228 718 197 716 167 712 154 710
2357 16 2077 10 1737 10 1527 9
235 7 13 192 7 12 171 7 13 154 7 11
where rl is the water density, U0 is the jet speed, D is the diameter of the orifice and s is the water surface tension. The data obtained in this study for all the disk-sprinklers and the 901 boss-sprinkler
For the slot-sprinkler with 1.59 mm wide slot, Fig. 13 shows the slot sprays at two discharge pressures of 0.14 and 0.83 bar. The images were recorded by pointing the camera at 901 to the slot spray sheet. The slot spray discharge angle is defined as the angle between the vertical centerline of the impinging jet and the lower edge of the spray. In Fig. 13, the slot spray angle was measured to be 581 at 0.14 bar, and 511 at 0.83 bar. The angle decreased with the increase of the pressure but the relative change was small. For the other two slot-sprinklers with slot widths of 3.18 and 4.76 mm, the spray angle at 0.14 bar was measured to be 591 for the 3.18 mm wide slot, and 601 for the 4.76 mm slot. Although the slot spray flux increased as the slot width was increased, the spray angle changed little with the increase of slot width. For the two boss-sprinklers introduced in Fig. 4, the spray angle at 0.14 bar was measured to be 481 for the 1271 boss, and 441 for the 901 boss. Compared to the slotsprinkler without the boss (see Fig. 13), the conical boss tended to direct the slot spray further toward the sprinkler centerline. Fig. 14 shows the measured slot spray angles for the three slotsprinklers and two boss-sprinklers operating from 0.034 to 0.83 bar. In general, the slot spray angle tended to decrease with the increase of discharge pressure. However, for the slot-sprinkler with the 4.76 mm slot, the slot spray angle was nearly constant in the range of the investigated pressures. While the effect of slot width on the slot spray angle was small, the boss had a strong impact on the angle. For the 901 boss, the slot spray angle decreased from 561 to 121 as the discharge pressure was increased from 0.034 to 0.83 bar. Reducing the slot spray angle is useful to improve the water flux distribution directly below the sprinkler. 4.5. Effect of frame arm As shown in Fig. 5, the effect of the frame arm was examined by positioning a 3.2 mm diameter metal bar close to a disksprinkler. After the spray sheet leaves the disk, the sheet impinges on the bar. Fig. 15 shows the effect of the bar on the spray sheet at
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70
slot spray angle
60 50 40 30 slot width 1.59 mm
20
slot width 3.18 mm slot width 4.76 mm
10
boss aperture 127 deg boss aperture 90 deg
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
discharge pressure (bar) Fig. 14. Measured slot spray angles for three slot-sprinklers and two bosssprinklers.
slot. Therefore, the presence of the frame arm increases the complexity of the overall spray formation. 4.6. Drop size The shadow imaging technique is able to determine the sizes and shapes of individual drops. Measuring the sizes and velocities of drops detected in a control volume for sufficient time duration, statistical spray properties can be derived. One of the characteristic drop diameters describing the statistical spray information is the measuring-volume-weighted median drop diameter, Dv50, which denotes that 50% of the cumulated drop volume is from drops with diameters smaller than Dv50. The Dv50 provides a picture of the overall drops present in a control volume. If individual drop velocities are considered, a flux-weighted median diameter, Df50, can be determined. The water flux, Vi (m3/m2 s), of a drop can be expressed as Vi ¼
Fig. 13. Slot sprays recorded at discharge pressures of (a) 0.14 bar and (b) 0.83 bar for the slot-sprinkler with a slot width of 1.59 mm.
two discharge pressures. At the lower pressure of 0.014 bar, Fig. 15(a) shows that the water climbed up along the bar to a height of 15 mm above the disk. At the higher pressure of 0.034 bar, Fig. 15(b) shows that water reached a height of 20 mm. It was observed that as the discharge pressure increased further, the water could reach even higher along the bar. Both images of Fig. 15 show that a vertical water sheet was formed behind the bar. At the lower pressure of 0.014 bar, the vertical sheet eventually re-joined the sheet on the disk. After increasing the discharge pressure to 0.034 bar, it was observed that the water sheet behind the bar did not re-join the sheet on the disk, and eventually disintegrated into drops independently of the breakup processes of the sheets originating from the disk and
ð1=6Þpd3i ui , A dof
ð9Þ
where di is the drop diameter, ui is the drop velocity, A is the area of the field view of the camera and dof is the depth of the field. The dof value is linearly proportional to the drop size. Because the drop size is much smaller than the field view area, the A value is assumed to be constant. The measuring volume is the product of P A and dof. The total water flux is N i ¼ 1 Vi , where N is the number of all detected drops. To illustrate the drop formation process, Fig. 16 shows the side view of a horizontal spray sheet at a discharge pressure of 0.34 bar from a 25.4 mm diameter disk-sprinkler. The horizontal distance between the disk edge and the left end of the photograph is 170 mm. The physical viewing area of each image record was 31.6 31.6 mm2. The image shown in Fig. 16 is a composite of four such images recorded at different times and horizontal locations (124.4 31.6 mm2). As shown in Fig. 16 from left to right, the spray sheet first broke up at a critical wave amplitude and then disintegrated into ligaments and drops. Fig. 17 shows the shadowgraph of a vertical slot spray sheet discharging from the slot of the 901 boss-sprinkler at 0.34 bar. The image is a composite of nineteen smaller images. Each smaller image’s dimension is 31.6 31.6 mm2. The top-left region is not shown because the image was obscured by the water spray. The boss-sprinkler was located 20 mm from the left side of the composite image. It shows that the slot spray sheet follows the same breakup process as that of the water sheet originating from the disk edge.
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To illustrate the effect of disk diameter of the disk-sprinkler on drop size, Table 3 shows the measuring-volume-weighted Dv50 and the flux-weighted Df50 for discharge pressures of 0.34 and 0.69 bar. The measurements were conducted at 0.3 and 0.6 m horizontally from the edge of the disk. At each location, the camera was traversed vertically across the entire spray to measure the drop size and velocity distributions. The diameters shown in Table 3 are the overall result at each radial location. The drop size decreased for all the three disk-sprinklers as the discharge pressure was increased. Because the dof value of Eq. (9) increases with the drop size, the flux-weighted Df50 was smaller than the volume-weighted Dv50 based on these near-field measurements. Table 3 also shows that the disk diameter has little impact on the drop size. As recalled from the data in Table 1, the larger disks result in thinner sheets at the disk edge but greater thickening factors. This implies that, for the same discharge pressure, the sheet speed at the edge of the larger disk is lower than that of the smaller disk. The atomization model [7] indicates that both a higher sheet speed and a thinner sheet produce smaller drop sizes. As the disk diameter increases, the sheet at the disk edge is thinner but the speed is also lower. The countering result is that the drop size is little affected by the disk diameter. Table 2 also shows that the sheet breakup distances are comparable for all the three disk-sprinklers. The result is consistent with the disk diameter’s small effect on the drop size as discussed above. To evaluate the effect of slot width on drop size, measurements were performed at 0.6 m from the disk edge for all the three slot-sprinklers. Table 4 shows the median drop diameters of the slot sprays. Based on these data, it is apparent that wider slots produce larger drops. This is because the initial slot spray is thicker for a wider slot.
5. Conclusions
Fig. 15. Vertical water sheet formed behind the sprinkler arm at discharge pressures of (a) 0.014 bar and (b) 0.034 bar.
Using a shadow imaging system, an exploratory study at low water discharge pressures was performed to gain insight into the spray formation of fire sprinklers as affected by the key sprinkler components. The findings from this investigation will be helpful in developing a spray formation model for fire sprinklers operating at typical discharge conditions, which can be used to provide the initial spray conditions for spray transport calculations using a numerical simulation. The understanding is also useful in developing a protocol by which sprinkler sprays can be efficiently quantified by conducting as few measurements as possible at strategic locations in the spray. Prototypes representing key sprinkler components were fabricated to evaluate their effects on the spray formation processes. These prototypes were designated as: disk-sprinkler, slot-sprinkler, boss-sprinkler and arm-sprinkler, with which the effects of deflector diameter, deflector slot width, boss configuration and sprinkler frame arms could be evaluated separately. The water
Fig. 16. Side view of a horizontal spray sheet from a 25.4 mm diameter disk-sprinkler discharging at 0.34 bar.
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Fig. 17. Side view of a vertical slot spray sheet discharging at 0.34 bar from the slot of the 901 boss-sprinkler.
Table 3 Measured drop sizes of the three disk-sprinklers. Disk diameter (mm)
Pressure (bar)
Horizontal distance from the edge of the disk 0.3 (m)
25.4 38.1 50.8 25.4 38.1 50.8
0.34 0.34 0.34 0.68 0.68 0.68
0.6 (m)
Dv50 (mm)
Df50 (mm)
Dv50 (mm)
Df50 (mm)
1.273 1.238 1.332 1.028 0.980 1.056
1.054 0.996 1.124 0.811 0.788 0.866
1.069 1.428 1.224 1.001 0.983 1.214
0.931 1.251 1.028 0.870 0.802 0.966
Table 4 Median drop diameters of the slot sprays of the three slot-sprinklers. Slot width (mm)
Pressure (bar)
Dv50 (mm)
Df50 (mm)
1.59 3.18 4.76 1.59 3.18 4.76
0.34 0.34 0.34 0.68 0.68 0.68
1.697 2.863 3.080 1.366 2.186 2.733
1.421 2.691 2.767 1.139 1.705 1.811
sheet thickness, slot spray discharge angle, sheet breakup distance, water flux, drop size and velocity were measured. It was found that the deflector diameter and the deflector boss had little impact on the drop size and sheet breakup distance. Wider deflector slots formed larger drops. At the same operating pressure, the slot width had little effect on the slot spray angle, but the boss helps directing the slot spray toward the sprinkler centerline. The slot spray angle tended to decrease with the
increase of discharge pressure, but the relative change was small. The frame arm tended to produce an up-lifted vertical sheet downstream of the arm, thus increasing the complexity of overall spray formation. An integral model was developed so that the sheet thickness at the disk edge could be calculated for different viscous effects exerted by the deflector. The model provides reasonable predictions of water sheet thickness and speed, which are required by existing spray atomization models. An empirical correlation was also established in this study to estimate the fraction of water discharging from a single slot of a sprinkler deflector. The fraction of water discharging from all the slots of a deflector is another piece of essential information required by spray atomization models. While representative with regard to the overall phenomenon, the findings from this investigation may not be directly applicable to fire sprinklers operating at typical (higher pressure) discharge conditions. As a first step in a systematic approach to investigate the mechanism of spray formation, experimental study will be conducted on higher pressure with the improvement of the measuring technology.
Acknowledgements This study was funded by FM Global in its sprinkler technology research program. We are grateful for Stephen D’Aniello’s assistance in the preparation of the measurement setup.
Appendix A. A formulation for the calculation of water sheet thickness and speed on a cylindrical disk The global atomization model currently being developed at the University of Maryland (UM) for a water jet impinging vertically
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on the center of a cylindrical disk requires the information of water sheet thickness and speed at the disk perimeter [7]. Currently, UM estimates the required information from the semi-empirical correlation developed by Watson [8]. The correlation gives a thinner water sheet at the disk perimeter than the laser-based back-shadow measurements recently made in this exploratory investigation. It is speculated that Watson’s Blasius boundary layer approximation does not adequately account for the viscous effect on the water sheet development on the disk. Therefore, an alternate formulation was developed so that calculations of water sheet development on the disk can be performed for different degrees of viscous effect exerted by the disk. A.1. Physical problem
z,v
g r,u
A.2. Formulation With the above coordinate and speed component designations, the continuity equation and the r- and y-momentum equations for an axisymmetrical boundary layer on a horizontal surface are @ @ ðr ruÞ þ ðr rvÞ ¼ 0, @r @z @u @u @p 1 @ @u þ rv ¼ þ rm , @r @z @r r @z @z
d
0
Eq. (A.6) can be further consolidated to Z Z d @ @ d @p @u ðr ru2 Þ @zud @zr m ðr ruÞ @z ¼ r : @r @r 0 @z z ¼ 0 0 @r ðA:7Þ
0
d
Applying Leibnitz’s rule for differentiation of integral, Z @ d d dd ðr ru2 Þ @z ¼ ðr ru2 Þ @zr ru2d @r dr 0 dr
and Z Z d @p d d dd @z ¼ : p @zpd dr 0 dr 0 @r
ðA:8Þ
ðA:9Þ
Substituting Eqs. (A.8) and (A.9) into Eq. (A.7) and approximating top-hat velocity profile in the water sheet, we have Z d dd d d ðr ru2 dÞr ru2 ¼ r ðppd Þ @zr tw dr dr 0 dr Z d d ¼ r rgðdzÞ dzrtw dr 0 d 2 ¼ r ð12rg d Þr tw dr rg d 1 2 f 2 ð rd Þ ru , ¼ ðA:10Þ 2 dr 2 2
V
ðA:1Þ
ðA:2Þ
and @p ¼ rg: @z
Z
Z
When a water jet impinges on the center of a cylindrical disk, what are the thickness and speed of the water sheet at the disk perimeter?
ru
Substituting Eq. (A.5) into Eq. (A.2) and integrating across the sheet layer, we have Z d Z Z @u u @ d 1 d @ ru @z d ðr ruÞ @z þ u ðr ruÞ @z @r r 0 @r r @r 0 0 Z d @p @u @zm ¼ : ðA:6Þ @z z ¼ 0 0 @r
ðA:3Þ
The formulation below follows the traditional procedure for performing flow integration across the boundary layer. From Eq. (A.1), we obtain Z 1 @ z v¼ r ru @z: ðA:4Þ r r @r 0 Multiplying r@u=@z on both sides of Eq. (A.4), we have Z z Z 1 @ @ z @ ðr ruÞ @z ¼ ðr ruÞ @z u ðr ruÞ : u r @z @r 0 @r 0
@u 1 @u @ rv ¼ @z r @z @r
ðA:5Þ
where tw ¼ mð@u=@zÞz ¼ 0 ¼ f u2 =2. Note that Eq. (A.7) can also be simplified with other presumed speed profiles. References [1] C. Yao, Extinguishing role of water sprinklers, in: Proceedings of the Engineering Applications of Fire Technology Workshop, National Bureau of Standards, Gaithersburg, MD, April 16–18, 1980, pp. 51–88. [2] K.B. McGrattan, G.P. Forney, Fire Dynamics Simulator (Version 4), Technical Reference Guide, NIST Special Publication 1018, National Institute of Standards and Technology, Gaithersburg, Maryland, March 12, 2006. [3] S. Nam, Development of a computational model simulating the interaction between a fire plume and a sprinkler spray, Fire Safety Journal 26 (1996) 1–33. [4] H.Z. Yu, Investigation of spray patterns of selected sprinklers with the FMRC drop size measuring system, in: Proceedings of the First International Symposium on Fire Safety Science, New York, 1986, pp. 1165–1176. [5] J.F. Widmann, D.T. Sheppard, R.M. Lueptow, Non-intrusive measurements in fire sprinkler sprays, Fire Technology 37 (2001) 297–315. [6] D.T. Sheppard, Spray Characteristic of Fire Sprinkler, NIST GCR 02-838, National Institute of Standards and Technology, Gaithersburg, MD, 2002. [7] D. Wu, D. Guillemin, A.W. Marshall, A modeling basis for predicting the initial sprinkler spray, Fire Safety Journal 42 (2007) 283–294. [8] E.J. Watson, The radial spread of a liquid jet over a horizontal plane, Journal of Fluid Mechanics 20 (1964) 481–499. [9] N. Dombrowski, W.R. Johns, The aerodynamics instability and disintegration of viscous liquid sheets, Chemical Engineering Science 18 (1963) 203–214. [10] N. Ren, A.W. Marshall, H. Baum, A comprehensive methodology for characterizing sprinkler sprays, Presented at the 33rd International Symposium on Combustion, Beijing, China, 2010.