Maximum temperature of thermal plume beneath an unconfined ceiling with different inclination angles induced by rectangular fire sources

Applied Thermal Engineering 120 (2017) 239–246

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Research Paper

Maximum temperature of thermal plume beneath an unconfined ceiling with different inclination angles induced by rectangular fire sources Xiaochun Zhang, Zunmeng Guo, Haowen Tao, Jingyong Liu, Yufang Chen, Aihua Liu, Wenbin Xu ⇑, Xiaozhou Liu ⇑ School of Environmental Science and Engineering, Guangdong University of Technology, Guangzhou, Guangdong 510006, China

h i g h l i g h t s
Experiments are carried out to measure the maximum temperature of inclined ceiling jet.
The maximum temperature increase along with the increasing of ceiling inclination angle.
The maximum temperature is proportional to that of free thermal plumes at ceiling level.
A new global correlation for the maximum temperature of weak impinging flow is established.

a r t i c l e

i n f o

Article history: Received 4 November 2016 Revised 9 March 2017 Accepted 12 March 2017 Available online 28 March 2017 Keywords: Fire thermal plume Inclined ceiling Maximum temperature rise Unified correlation Rectangular fire source

a b s t r a c t The maximum temperature of thermal plume beneath an unconfined ceiling with different inclination angles induced by rectangular fire sources were investigated experimentally and theoretically. The experimental results show that the maximum temperature rise at the ceiling level with different ceiling inclination angles is proportional to that of a free fire thermal plume; and the maximum temperature rise increases according to the ceiling inclination angle when other experiment conditions remain unchanged for a given fire source; A new global non-dimensional correlation combined the effects of source-ceiling height, heat release rate, source aspect ratio and especially the ceiling inclination angle are proposed to predict the maximum temperature rise. The new proposed correlation can predict the maximum temperature rise beneath the ceiling induced by axisymmetric, rectangular and line fire sources uniformly. This work provides supplementary results over previous knowledge of temperature distribution beneath the inclined ceilings. It can also provide guidance for thermal risk assessment and fire safety design of openstyled spaces. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Thermal ceiling jet flow is an essential scientific problem and there is a lot of engineering application about it [1–4]. The knowledge of maximum temperature rise beneath the ceilings is important for the building fire risk assessment and fire protection design [5–7]. The thermal flow is not only hazardous to construction roof, but it also imposes heat feedback to other inside occupants and combustible objects. These two effects can significantly affect the stability of building ceiling and may increase the flashover potential of the building [8,9]. The maximum temperature rise beneath the

ceiling is directly related to the property damage and casualties. For example, Khoury’s work shows that most concretes will encounter a strength reduction if the temperature researches 300 °C [10]. Furthermore, Connolly’s experiments results show that the concretes will spall if their surface temperature reaches about 250–420 °C [11]. So, the maximum temperature rise beneath the ceiling has attracted much attention from thermal engineers and fire researchers [12–15], Alpert [12] established a formula to predict the maximum thermal plume temperature which has become the earliest well-known work about ceiling jet. This classic model has an empirical form as following:

DT m ðHÞ ¼ 16:9 ⇑ Corresponding authors. E-mail addresses: [email protected] (W. Xu), [email protected] (X. Liu). http://dx.doi.org/10.1016/j.applthermaleng.2017.03.121 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.

Q_ 2=3 H5=3

ð1Þ

where Q_ is the heat release rate; H is the source-ceiling height. Li et al. studied the maximum temperature rise beneath the tunnel

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

Nomenclature a, b C C2 cp ds Frs g H Hmod L q Q_ Q_ 0L Q_ 

mod

T1 DT m ðHÞ

coefficient in Eq. (8) constant related to experimental scenarios entrainment constant specific heat of air at constant pressure (kJ/(kg  K)) equal-perimeter diameters(m) source Froude number gravitational acceleration (kg  m/s2) source-ceiling height (m) modified non-dimensional height length of rectangular-source burner (m) flow rate (m3/h) heat release rate of the fire source (kW) heat release rate per unit length (kW/m) _ non-dimensional heat release rate (Q_ mod ¼ q c T QpffiffigLW 3=2 ) 1 p 1 ambient air temperature (K) maximum temperature rise of the impingement region (K)

ceiling with longitudinal ventilation and they found that the maximum temperature can be predicted by a familiar former as Eq. (1) while the dimensionless longitudinal ventilation velocity less than 0.19 [13]. The only difference is that the coefficient of Eq. (1) turned to be 17.5. Another famous and widely used model about the maximum temperature rise comes from Heskestad and Delichatsios [14]:

DT m ðHÞ ¼ 6:3Q_ 2=3 T1

ð2aÞ

where DT m ðHÞ is the maximum temperature rise at the impingement region; T 1 is the ambient air temperature; Q_  is the nondimensional heat release rate defined as:

Q_  ¼

Q_ pffiffiffi q1 T 1 C P g H5=2

ð2bÞ

where q1 is the ambient air density; C P is the specific heat of air at constant pressure; g is the gravitational acceleration; H is the source-ceiling height; Q_ is the heat release rate of the fire source; T 1 is the ambient air temperature. Heskestad and Delichatsios’ model has a wider application scope in fire engineering and the calculated value by using Eqs. (2a) and (2b) is larger than that of Eq. (1). However, both of them only considered the flat ceiling thermal plumes driven by axisymmetric fire source, which could not applied to the inclined ceiling jet induced by rectangular fire sources. For the flat ceiling jet flow, another comparatively famous work comes from Hu et al. [2,6] and Kurioka et al. [15]. They all established semi-empirical formulas for maximum temperature of smoke layer based on scaling analysis. Along with the above investigations about horizontal ceiling jet flow. Oka et al. carried out a series of work to research the temperature, velocity and entrainment properties of thermal flow beneath the inclined ceiling [16–19]. For the maximum temperature rise, they focused on the temperature attenuation law for the thermal ceiling flow while the maximum temperature rise at impingement point was not studied and their values were directly calculated by Eqs. (2a) and (2b) [16]. In addition, it is well known that the source geometry should be an essential factor which can seriously affect combustion behavior [20–22] and inevitably influences the maximum temperature rise beneath the ceiling [6,23–25]. However, the correlations mentioned above only considered the thermal plumes driven by

ws W

nozzle exit velocity (m/s) width of the rectangular -source burner (m)

Greek symbols h inclined angle of ceiling (°) a coefficient in Eq. (3) q1 ambient air density (kg/m3) D difference between variables U dimensionless maximum temperature rise k constant Subscript mod m p s 1

modify maximum pressure fire source ambient

axisymmetric fire sources while the rectangular fire sources are actually more common in the real fires. Many previous studies have been focused on axisymmetric sources [e.g. 2,7,16,20–22]. Instead, relatively small amount works were carried out for rectangular or line fire sources [e.g. 22,25,26]. We have conducted experiments to investigate the flat ceiling temperature profiles and flame extension length under the ceiling for ceiling jets driven by line fire plumes [25]. It is remarkable that the literatures mentioned above are mainly focused on horizontal thermal ceiling flow. The few works involved ceiling inclination effects have not given any correlation for the temperature rise at the impingement zone either. So, these previous correlations cannot be directly used to predict the maximum temperature rise in an inclined ceiling flow while such inclined roofs are extensively existing in our buildings. It is needed to discover the influences of inclination angle and source geometry on the maximum temperature rise. Wood is also widely used as ceiling in our buildings. The roof will be heated and began to thermal decomposition, cracked [27–29], even burning [30,31] when a fire occurred. These physical processes are all in the scope of fire science. It is easy to see that the thermal flow beneath a wooden ceiling will be more complicated than that of an unburned ceiling. These physical processes are all researched respectively and there is no comprehensive literature report can be found about a flammable ceiling jet. This will be a significant and challenge work to investigate the thermal flow beneath a flammable ceiling in the future. So, we only consider a non-flammable ceiling in this work. Experiments were conducted by employing three rectangularsource gas burners with different aspect ratios. Five sourceceiling heights and four angles of the unconfined plate are considered to measure the maximum thermal plume temperature rises beneath the ceiling. The fundamental purpose of this paper is to establish a global correlation to predict the maximum thermal plume temperature rises beneath an inclined non-flammable ceiling. 2. Experimental investigation A series of sub-scale experiments are carried out in this work. An unconfined heat-proof plate with dimension of 1.5 m (W)  2 m (L) is used to simulate the building ceiling which formed a 1:4 scale model. This suspended ceiling was made of good fire resistance performance and low thermal conductivity of

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

Fig. 1. Outline of the experimental setup.

0.035 W/(mK) and had a smooth surface[thickness of 0.02 m]. The unconfined ceiling could be adjusted to an arbitrary inclination angle and height during the process of experiments. Three rectangular-source gaseous burners made of stainless steel with exit size of 2 mm  142.5 mm, 4 mm  71.25 mm, and 6 mm  47.5 mm were employed. The fire sources have a same exit area with a homolographic diameter 19 mm compared to a circular fire source. High purity propane is adopted as fuel with supply rate monitored and controlled by a gas flow rate meter with minimum scale of 0.02 m3/h. The heat release rates are regulated in a range of 2.40–9.60 kW in order to have turbulent flames. Experimental scenarios that flame height lower than sourceceiling heights are selected for further research. Five source-ceiling heights (0.475 m, 0.57 m, 0.665 m, 0.76 m and 0.855 m) and four inclination angles (5°, 8°, 20°, 30°) are considered in our experiments. The maximum distance H (0.855 m) is 45 times of the burner effective diameter and the corresponding slope ranged in 8.7–57.7%. Such experimental design could cover most of the building fire scenarios in China [32]. Each case is repeated 3 times. The average value is taken as the final data and the uncertainty is taken as their error. The whole experimental setup is shown in Fig. 1. The temperatures of the thermal impinge flow were measured using 17 K-type thermocouples with a wire diameter of 0.50 mm. These thermocouples are positioned 1.5 cm (longitudinal direction: along the inclination angle direction) or 1 cm (transversal direction: perpendicular to the inclination angle direction) below the ceiling. This position is selected based on the results reported by Oka [16] and our previous experimental results [25]. The horizontal intervals between thermocouples are 0.025 m and the arrangement of thermocouples can be seen in Fig. 2. The temperature data were collected every second and then stored in a PC for further analysis. The summary of experimental scenarios are listed in Table 1. 3. Results and discussion We have conducted experiments to investigate the flat ceiling temperature profiles and flame extension lengths under a nonflammable ceiling for ceiling jets driven by line fire plumes [25]. The maximum temperature rise can be calculated by the following correlation for a line fire:

Fig. 2. Temperature measurements.

Table 1 Experimental conditions. W (mm)

L (mm)

ds (m)

q (m3/h)

Q_ (kW)

ws (m/s)

Q_ mod

Frs

2.00

142.50

0.09 0.09 0.09 0.09 0.09 0.09

0.09 0.14 0.19 0.22 0.26 0.29

2.40 3.59 4.83 5.80 6.72 7.68

0.09 0.13 0.18 0.22 0.25 0.29

169.15 252.80 340.74 408.89 473.44 541.27

0.09 0.14 0.19 0.23 0.26 0.30

4.00

71.25

0.05 0.05 0.05 0.05 0.05 0.05

0.09 0.14 0.19 0.22 0.29 0.33

2.40 3.59 4.83 5.80 7.68 8.64

0.09 0.13 0.18 0.22 0.29 0.32

119.61 178.76 240.94 289.13 382.74 430.58

0.13 0.20 0.26 0.32 0.42 0.47

6.00

47.50

0.03 0.03 0.03 0.03 0.03 0.03 0.03

0.09 0.14 0.19 0.22 0.26 0.29 0.37

2.40 3.59 4.83 5.80 6.72 7.68 9.60

0.09 0.13 0.18 0.22 0.25 0.29 0.36

97.66 145.96 196.73 236.07 273.44 312.51 390.63

0.15 0.23 0.31 0.38 0.43 0.50 0.62

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

 2=3 DT m ðHÞ 1 Q_ 02=3 ¼ 0:586 H1 pffiffiffi L T1 aq1 C p T 1 g

ð3Þ

where a is the coefficient; Q_ 0L is the heat release rate per unit length; the other symbols have the same physical mean as Eqs. (2a) and (2b). The flat ceiling jet experiment data could be exacted from Ref. [25]. Fig. 3 shows the maximum temperature rise at impingement zone against fire source heat release rates with the source-ceiling height H = 0.38 m for three burner dimensions (4 mm  71.25 mm, 6 mm  47.5 mm, 2 mm  142.5 mm), along with a comparison with the correlation suggested by Alpert [12] and Heskestad and Delichatsios [14]. It clearly shows that the experiment data sets are scattered around the lines calculated by Heskestad and Delichatsio’s [14] and Alpert’s models [12]. Much of this discrepancy is to a reason that both of Heskestad and Delichatsio’s [14] and Alpert’s models did not consider the effects of fire source shapes which can affect the air entrainment of the thermal plume. Fig. 4 shows the dimensionless maximum temperature profile against non-dimensional heat release rates which shows the comparison of experiment data with Heskestad and Delichatsio’s [14] model. It can be seen that the dimensionless form of Heskestad and Delichatsios’ model can give very crude predicted values.

360

1:71 1:18 1:8

300

ΔTm(H)

240

(a) The measured values increase with the increasing of heat release rates for each inclination angle and fire burner respectively while the measured values turned larger by comparing the three fire burner experiment data for each inclination angle. This result is consistent with the former results of free thermal plume theory [20]. (b) As shows in Fig. 5, the correlations proposed by Alpert and Heskestad and Delichatsios cannot well predict the measured values. This result maybe come from three reasons. For the first reason, our experimental data are measured by thermocouples with diameter 0.5 mm which have a less thermal radiation loss than Alpert and Heskestad and Delichatsios’ measurements. For the second reason, the dimension of fire sources in our experiments are smaller than that of Alpert [12] Heskestad and Delichatsios [14], which lead to less air entrainment for the thermal plume. At last, their correlations did not contain the effects of inclination angle while the inclination angel can increase the maximum temperature rise [16]. Based on the above observation, we need to establish a correlation which can reflect the influences of inclination angle and fire burner dimensions. Fortunately, numerous studies have shown that the maximum temperature for a flat ceiling jet above the fire source is proportional to the centerline temperature of a free fire plume at the same height [e.g. 6,12–14]. Thus, we would like to establish a unified correlation to predict the maximum temperature based on the classical free fire plume model proposed by Quintiere and Grove [33,34]:

180 120 60 Heskestad and Delichatsios [14] Alpert [12]

0

The experimental data are still scattered and this implicit that a correlation is needed to reflect the influence of source aspect ratio. Fig. 5(a)–(c) shows the maximum temperature rise at impingement zone against fire source heat release rates at different inclination angles 5°, 8°, 20° and 30° with the source-ceiling height remain unchanged H = 0.665 m for each fire burner, along with a comparison with the correlation suggested by Heskestad and Delichatsios [14] and Alpert [12]. It clearly shows that:

-60 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Q_ mod ¼

Q Fig. 3. Maximum temperature rises at impingement zone against fire source heat release rates for flat ceiling jets (H = 0.38 m): comparison with Alpert, Heskestad and Delichatsios’ model.

1.2

ΔTm(H) T∞

0.9

Q_ pffiffiffi q1 cp T 1 g LW 3=2

ð4bÞ

and

1:71 1:18 1:8

Q_ mod

Heskestad and Delichatsios [14]

0.00

ð4cÞ

where C 2 is almost equal to 0.1 which reflects buoyant entrainment, C is a constant related to experimental scenarios which can be derived by the measured experimental data; k is a constant; W is the width of the rectangular-source burner; L is the length of rectangular-source burner. Eq. (4a) can be rewritten as the following form through a simple transformation:

0.3 0.0

DT m ðHÞ T1

U¼

0.6

-0.3 -0.04

ð4aÞ

where

Q_ mod ¼

1.5

     C p 3=2 H 1=2 H U  1 þ 2C 2 ðk þ 1Þ 4 W W     W H  1 þ 2C 2 L W

0.04

0.08

0.12

0.16

U3=2



¼A

Fig. 4. Dimensionless maximum temperature rises against non-dimensional heat release rates for flat ceiling jets (H = 0.285 m, 0.38 m, 0.475 m).

1=2

       H W H  1 þ 2C 2  1 þ 2C 2 W L W

ð5aÞ

0.20

Q *2 / 3

H W

where

A¼

pC 4ðk þ 1Þ

ð5bÞ

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

300

300

200 150

200 150 100

100 50 0

θ=5° θ=8° θ=20° θ=30°

1:18

250

ΔTm(H)

1:71

250

ΔTm(H)

θ=5° θ=8° θ=20° θ=30°

50

Heskestad and Delichatsios [14] Alpert [12]

0

1

2

3

4

5

6

7

8

9

0

10

Heskestad and Delichatsios [14] Alpert [12]

0

1

2

3

4

5

6

7

8

9

10

Q

Q (a) 2mm*142.5mm

(b) 4mm*71.25mm

300 θ=5° θ=8° θ=20° θ=30°

1:8

250 200

ΔTm(H) 150 100 50 0

Heskestad and Delichatsios [14] Alpert [12]

0

1

2

3

4

5

6

7

8

9

10

Q (c) 6mm*47.5mm Fig. 5. Maximum temperature rise at impingement zone against fire source heat release rates at different inclination angles (5°, 8°, 20° and 30°) with the source-ceiling height remain unchanged (H = 0.665 m) for each fire burner.

Therefore, we can plot

Q_ mod

U3=2

H against W and give the best fitted value of

A for each inclination angle as shown in Fig. 6(a)–(d). It shows that the experimental data can be well described by Eq. (5a). Apparently, the value of A decreases with the increasing of inclination angle and this result is consistent with the result obtained from Fig. 5. This monotone decreasing trend of ‘‘A” implicit that the maximum temperature rises of the impinging flow are proportional to the free thermal plume at the ceiling level for a given ceiling inclination angle. Fig. 7 shows the variation of constant against source Froude number for all tested inclination angles. Here, the source Froude number defined as follow [20]:

ws Fr s ¼ pffiffiffiffiffiffiffi gds

ð6Þ

where ws is the nozzle mean exit velocity, ds is the characteristic length of the nozzle which taken as the equal-perimeter diameter ðds ¼ 2ðW þ LÞ=pÞ of the nozzle. The following correlations can be obtained based on the experimental data for each inclination angle:

8 0:46 > C > ¼ 0:684  0:25ðFr 2s Þ > kþ1 > > > > < C ¼ 0:648  0:23ðFr 2 Þ0:46 kþ1

s

> C > ¼ 0:621  > kþ1 > > > > : C ¼ 0:592  0:19ðFr 2 Þ0:46 s kþ1 0:46 0:21ðFr 2s Þ

h ¼ 5 h ¼ 8 h ¼ 20 h ¼ 30

It can be seen that the value of lowing form:

C kþ1

can be predicted by the fol-

0:46 C ¼ a  bðFr2s Þ kþ1

ð8Þ

Fig. 8 plots the variation of parameters a and b against the sinusoidal values of inclination angles and the data for flat ceiling comes from Ref. [25]. The corresponding correlations for a and b can be obtained based on data fitting:

(

a ¼ 0:744  0:221ðsinhÞ

1=2

ð9Þ

1=2

b ¼ 0:282  0:127ðsinhÞ

So, we can obtain correlation for

C kþ1

finally:

C 1=2 ¼ 0:744  0:221ðsinhÞ kþ1 1=2

 ½0:282  0:127ðsinhÞ

ðFr2s Þ

0:46

ð10Þ

In order to give the explicit correlation for maximum temperature rise, we further rewritten the Eq. (4a) as the following form:

("  1=2    DT m ðHÞ pC H H ¼  1 þ 0:2 T1 4ðk þ 1Þ W W 2=3 )1     W H  _ Q mod  1 þ 0:2 L W

U¼ ð7Þ

ð11Þ

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

2500

2500 θ=5° fitted line

1500 1000 500 0

1500 1000 500

Q∗mod H H 2 H )( ) = 0.521× ( )1 2 × 1 + 0.2( ) × 1 + 0.2( Φ3 2 142.5 W W W

0

θ=8° fitted line

2000

Q ∗m o d Φ3 2

Q ∗m o d Φ3 2

2000

0

100 200 300 400 500 600 700 800

Q∗mod 2 H H H )( ) = 0.509 × ( )1 2 × 1 + 0.2( ) × 1 + 0.2( 142.5 W Φ3 2 W W

0

HW (b) nozzle aspect ratio 1:71 and θ = 8°

HW

(a) nozzle aspect ratio 1:71 and θ = 5° 2500

2500

θ=30° fitted line

θ=20°

fitted line

2000 1500

1500

Q ∗m o d Φ3 2

Q ∗m o d Φ3 2

2000

1000

1000

500 0

0

100

200

300

400

500

600

Q_ mod

U3=2

700

θ=8° θ=20°

H W

100 200 300 400 500 600 700 800

HW (d) nozzle aspect ratio 1:71 and θ = 30° at four inclinations (burner: 2 mm  142.5 mm; source-ceiling height (H): 0.475 m).

U ¼ H1 mod

C λ +1

ð13Þ

As we can see form Eq. (12), the correlation can be approximately used to predict corresponding temperature rise induced by an axisymmetric fire source if W:L equal to 1 and Eq. (13) can be simplified in the following form:

0.01

0.1

Frs Fig. 7. Variation of constant inclination angles.

C kþ1

8" #2=3 91  1=2   2 = DT m ðHÞ < pC H H  _ U¼ ¼  1 þ 0:2 =Q mod : 4ðk þ 1Þ W ; T1 W

1

ð14aÞ

2

against source Froude number for all tested

Substituting Eq. (10) into Eq. (11), a unified correlation which combined effects of heat release rate, burner aspect ratio, ceiling-source height and especially the ceiling inclination angle for maximum temperature rise can be established. Fig. 9 plots the maximum dimensionless temperature rise against the modified non-dimensional height which defined as:

p

0

All of the experiment data, tighter with the experiment data sets extracted from Ref. [25] and Heskestad and Delichatsios’ work, can be well predicated by the following correlation:

θ=30°



1=2





C H H  1 þ 0:2 W 4 ðk þ 1Þ W 2=3     W H  1 þ 0:2 Q_ mod L W

Hmod ¼

Q∗mod H H H 2 )( ) = 0.461× ( )1 2 × 1 + 0.2( ) × 1 + 0.2( Φ3 2 W W 142.5 W

θ = 20°

against non-dimensional source-ceiling height

1

0

800

θ=5°

"

500

Q∗mod 2 H H H )( ) = 0.479 × ( )1 2 × 1 + 0.2( ) × 1 + 0.2( 142.5 W W W Φ3 2

HW (c) nozzle aspect ratio 1:71 and Fig. 6. Combined physical term

100 200 300 400 500 600 700 800



ð12Þ

The correlation can be used to predict the temperature rise induced by a line fire source if W:L is so small that the third part can be considered as equal to 1. Then Eq. (13) can be simplified in the following form:

DT m ðHÞ T1 8" #2=3 91  1=2   , < = pC H H  _ ¼  1 þ 0:2 Q mod : 4ðk þ 1Þ W ; W

U¼

ð14bÞ

So, The correlation established in this work can predict the maximum temperature rises of weak thermal impinging flow induced by axisymmetric, rectangular and line sources uniformly. The experiment data sets are collected from sub-scale experiments and the results do need to be re-examined by the fullscale experiments. The results and conclusion obtained in this work can only suitable to a non-flammable ceiling. It cannot be

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X. Zhang et al. / Applied Thermal Engineering 120 (2017) 239–246

1.2

0.5

b = 0.282 − 0.127 ( sinθ )

12

a = 0.744 − 0.221 ( sinθ )

12

1.0

0.4

0.8

a

0.3

b

0.6 0.4

0.2 0.1

0.2 -0.1

0.0

0.1

0.2

0.3

0.4

sinθ

0.5

0.6

0.0 -0.1

0.0

0.1

0.2

sinθ

0.3

0.4

0.5

0.6

Fig. 8. Variation of parameters a and b against the sinusoidal values of inclination angles and the corresponding correlations.

Acknowledgement θ=5° θ=8° θ=20° θ=30°

1

Φ

−1 Φ = H mod

This work was supported by National Natural Foundation of China under Grant No. 51506032, Natural Science Foundation of Guangdong Province under Grant No. 2014A030310190, Natural Science Foundation of Guangdong Province under Grant No. 2016A030310341 and National Natural Foundation of China under Grant No. 51508110.

References

θ=0° [25] Heskestad and Delichatsios [14]

0.1 1

H m od

10

Fig. 9. Dimensionless maximum temperature profile against modified non-dimenC sional height including kþ1 dependency on source Froude number of rectangular jet fires.

directly used to predict the temperature rise in thermal flow beneath a flammable ceiling. 4. Concluding remarks The maximum thermal plume temperature beneath the inclined ceiling was investigated both experimentally and theoretically, focusing on weak thermal plume impingement. Major findings can be addressed: (1) The ceiling inclination angles have a significant influence on the maximum temperature rises which increase along with the increasing of ceiling inclination angle in our experimental range (Figs. 5 and 6). (2) The maximum temperature rises of the impinging flow are proportional to that of free thermal plumes at the ceiling level for a given ceiling inclination angle up to 30 degree (Eqs. (7)–(10); Figs. 7 and 8). (3) A new global correlation which can reflect the influence of source-ceiling height, heat release rate, source aspect ratios and especially the ceiling inclination angles are proposed to predict the maximum temperature rises beneath the inclined ceiling. This correlation can predict the maximum temperature rises of weak thermal impinging flow induced by rectangular, axisymmetric and line sources uniformly (Eqs. (13), (14a) and (14b); Fig. 9).

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