1 Introduction

There are building constructions where many detectors and sprinklers are installed on ceilings with beams or other obstructions. Ceiling obstructions make ceiling jet flow complicated, and temperature and velocity profiles under these conditions are not fully characterized. Recent works in CFD simulation have made a lot of progress in understanding the complex ceiling jet flows [1, 2]. However, it is difficult to fully validate these models and they are still too complex to be used for engineering design and assessment.

Current practices and the NFPA standards [3–5] do not provide access to the necessary engineering information to address obstructed ceiling jets. This work attempts to provide some of the basic information needed to enhance the current practices. Previously, other researchers have also addressed these flows [6–11]. The complexity of this problem has restricted the applicability of the results to a limited number of ceiling configurations. Normally, multiple beams exist and construct multiple channels on the ceiling, and detectors or sprinklers may be installed at positions within a bay not containing the initial fire plume. In this work, the beamed ceiling jet temperature profile was investigated for multiple beam spacing in the midline of the secondary bay and an empirical correlation has been developed for the maximum temperature variations. The temperature and velocity profiles for beamed ceiling jet flows are functions of the fire size and geometric variables associated with beams and ceilings. These geometric variables include the beam width, w, beam depth, d, beam spacing, 2l b, and ceiling height, H (see Figure 1). It is important to define key regions of the flow in this arrangement. The primary bay or channel is the region between two ceiling beams where the fire plume impinges. The secondary bay is the bay adjacent to the primary bay. Both the primary and secondary bays are illustrated in Figure 2.

Figure 1
figure 1

Schematic of the beamed ceiling jet simulation apparatus

Full size image
Figure 2
figure 2

Plane top view and side vies (bottom) of the experimental configuration defining terms and measurement points

Full size image

This work expands on the previous research performed by Koslowski and Motevalli [12], who reported an empirical correlation validated for the flow within the secondary bay at locations perpendicular to the beam. No correlations for the beamed ceiling jet temperature and velocity on the midline within the secondary channel have been developed previously. For design purposes, the parameters of the experimental research are limited. Heat detector and sprinkler placement is typically at center of bays or under the beams. For design considerations, placement of detectors & sprinklers below the beam normally does not pose a problem as actuation would often happen more readily below the beam regardless of fire location being directly below the beam or not.

2 Theory and Approach

The temperature correlations for the smooth ceiling jet can be expressed as a function of ceiling configuration, i.e., height, distance from plume axis and convective heat release rate. Similarly, the temperature correlation for beamed ceiling jet can also be expressed in terms of the geometric variables of a beamed ceiling, such as beam depth, width and spacing, (shown via dimensional analysis by Koslowski and Motevalli [12]) these geometric variables of the beamed ceiling are shown in Figure 1. The ceiling jet temperature for a smooth, horizontal and unconfined ceiling jet is a function of the radial position, r, the ceiling height, H, and the non-dimensional fire size, \(\dot{Q}^\ast\). The scaling laws referred to here are based on Froude modeling and their accuracy in representing large-scale fire-induced flows have been well-established with very good results when implemented carefully and with noted limitations. Quintiere has described the applicability of small-scale experiments and the validity of scaling laws in a comprehensive review paper [13].

Based on the scaling laws, for steady fires, the ceiling jet maximum temperature can be expressed by:

$$ \Updelta T_{\max,s}^{\ast} =F\left(\frac{r}{H},\dot{Q}^\ast\right) $$
(1)

where subscript “max, s” refers to a maximum value for a smooth ceiling condition.

In general, the temperature difference \(\Updelta T\) for a ceiling jet can be non-dimensionalized by the following form:

$$ \Updelta T^{\ast}=\frac{\Delta T}{\dot{Q}^{\ast2/3}T_\infty} $$
(2)

where, T ∞ is the absolute ambient temperature.

According to non-dimensionalization methods and as described by Motevalli [14], the maximum temperature for smooth unconfined ceiling can be depicted as follows:

$$ \Updelta T(r)_{\max,s} =\Updelta T_{\max,s}^\ast \dot{Q}^{\ast\frac{2}{3}}T_\infty $$
(3)

where

$$ \dot{Q}^\ast =\frac{\dot{Q}_c}{(c_p \rho _\infty T_\infty \sqrt{g} H^{2.5})} $$
(4)

and correlation reported by Motevalli and Marks [15] is noted below:

$$ \Updelta T_{\max,s}^\ast =0.166 \left(\frac{r}{H} \right)^{-2} +1.2 \left(\frac{r}{H} \right)^{-1} +2.0\quad\quad\quad\quad0.26\le \frac{r}{H}\le 1.0 $$
(5)

Note that \(\dot {Q}_c \) is the convective heat release rate, i.e., \(\dot {Q}_c =\dot{Q}_{tot} -\dot {Q}_r \), where \(\dot {Q}_r \) is the radiative loss.

Motevalli and Marks [15] further defined the ceiling jet temperature profile by the following:

$$ \frac{\Updelta T(r)_s }{\Updelta T(r)_{\max,s} }=4.24 \left(\frac{z}{l_T}+0.094 \right)^{0.755} e^{[-2.57(\frac{z}{l_T})]}\quad\quad\quad\quad 0.26\le \frac{r}{H}\le 2.0 $$
(6)

where the non-dimensional ceiling jet thermal Gaussian thickness, l T, is defined by:

$$ \frac{l_T}{H}=0.112 \left(1-e^{[-2.24(\frac{r}{H})]} \right) $$
(7)

Although correlations 5–7 are based on small-scale experiments and for weak plume range, Motevalli and Marks [15] compared them with large-scale correlations and showed that if one can quantify the convective heat release rate of large-scale fire experiments accurately, then reasonably good agreement is obtained. Additionally, small-scale correlation of Motevalli and Marks offer a correlation for temperature profile which large-scale experiments lack. Using dimensional analysis, the maximum temperature of the unconfined beamed ceiling jet flow within a secondary channel can be described by:

$$ \Updelta T(r,\alpha)_{\max,b} =F \left(\frac{X}{H},\frac{Y}{H},\frac{w}{H},\frac{d}{H}, Q^\ast \right) $$
(8)

where subscript, b, refers to the beamed ceiling.

Since X = 2l b and Y = 2l b/tan(α), Equation (8) can also be rewritten as follow:

$$ \Updelta T(r,\alpha)_{\max,b} =F \left(\frac{l_b}{H},\frac{w}{H},\frac{d}{H},\frac{\alpha }{90}, Q^\ast \right) $$
(9)

Here, α is being used as a convenient way to identify the location of interest within a bay.

Following the approach taken by Koslowski and Motevalli [12, 17], the beamed ceiling jet temperature can be related to the smooth ceiling jet in the following normalization:

$$ \theta =\frac{\Updelta T^\ast_b}{\Updelta T^\ast_s} $$
(10)

Equation (10), along with the previous relations, allows the analysis of the scaled experiments. The non-dimensional temperature, θ, is in general form and can be expressed as θmax for the maximum temperature, then \(\theta _{\rm max} =\Updelta T^{\ast}_{b,{\rm max}} /\Updelta T^{\ast}_{\rm max}\). The range of θ is between horizontal smooth ceiling (no beam, θ = 1.0) to some upper limit of beam depth (d), which would make the flow approach a corridor flow. For a corridor flow, θ would approach zero when there is no spillage from the primary channel into the secondary channel.

3 Experiment

The Beamed Ceiling Jet Simulation Apparatus and other related equipment were developed previously [12] and experiments were performed at the Fire Science and Combustion Laboratory at Worcester Polytechnic Institute (WPI) [16] and are fully described including burner size, etc. in these references. The experiments, listed in Table 1, were designed to maximize the range of \(\dot {Q}^{\ast}\), w/d, l b/H, d/H and α/90 being tested. The non-dimensional heat release rate, \(\dot {Q}^{\ast}\), ranging from 0.0024 to 0.0351 corresponds to fire sizes ranging from 38 kW to 560 kW in a 3.0 m high room. This provides a large range that covers plume strength beyond weak plume range. In addition, use of premixed methane-air flame ensured accurate measurements of convective heat release rate to within ±3%. In each experiment, steady state temperature data were collected at five different positions on the midline of the secondary bay corresponding to angles of 90, 75, 60, 45, and 35 degrees measured with respect to the centerline of the primary bay (see Figure 2). These points correspond to positions Y along the midline of the secondary bay such that tan(α) = X/Y. Locating of the probes horizontally were done within an accuracy of 0.01 m and vertically to within 0.001 m. In order to examine the effect of geometric variables on the ceiling jet flow according to Equation (9), and the heat release rate, a matrix of experiments shown in Table 1 was developed. Temperature data were collected over 10 s internal at 15 vertical positions below the ceiling corresponding to: z = 0, 5, 10, 15, 20, 30, 40, 50, 60, 75, 90, 105, 120, 140, 160 mm (±1 mm). The experiments were all started at t = 0 with ambient laboratory conditions when flame was ignited. The ceiling was unconfined and insulated on top (simulating adiabatic conditions) and was painted black to an emissivity of approximately one.

Table 1 Experimental Parameters [16]
Full size table

A 0.0254 mm (0.001′′) diameter E type (chromel-constantan) thermocouple was employed to collect the temperature data. The probe could be moved both horizontally and vertically using a sliding motor-driven mechanism. The probe was connected to the data acquisition system with a pair of insulated and shielded cable to reduce the noise level. A program written in the BASIC language was used with a Metro byte data acquisition system to collect the data.

4 Results and Analysis

4.1 Temperature Profiles

Temperature profiles at the midline within the secondary channel were measured with a variety of multiple beam spacings, beam widths, beam depths, ceiling heights, and fire sizes as described in Table 1. Previous studies reviewed by Koslowski and Motevalli [12, 17] and their experimental results indicates that the width of the obstruction, w, (when of the same order of magnitude or less, as the beam depth, d) is not a significant parameter. In order to further examine this hypothesis, the temperature profiles at two locations along the secondary bay centerline. (α = 90° & α = 45°) are shown in Figures 3 and 4, while all fire and ceiling variables are kept the same except w/d ratio, which is varied by a factor of 2.

Figure 3
figure 3

Non-dimensional temperature profiles at 90-degree positions, Q = 5.0 kW, H = 1.00 m, d/H = 0.09

Full size image
Figure 4
figure 4

Non-dimensional temperature profiles at 45-degree positions, effect of beam width; Q = 5.0 kW, H = 1.00 m, d/H = 0.09

Full size image

Clearly, the effect is not observable within our experimental resolution. In all figures, the smooth-unconfined horizontal ceiling jet non-dimensional temperature profile [15] is plotted for reference.

A similar comparison examining the effect of beam spacings, l b, on the ceiling jet temperature profile is shown in Figures 5 and 6. Effect of variation in l b/H is less than 10%. Although, the variation of l b/H in these cases is not large, these spacing fall within most practical ranges. The beam spacing effect may become significant in special cases. It is also interesting to note that for the beam depth of Figures 5 and 6 (i.e., d/H = 0.12) a clear 1-dimensional flow within the bay confinement has developed where the temperature profiles are initially flat, for z less than the beam depth.

Figure 5
figure 5

Non-dimensional temperature profiles at 90-degree positions, effect of beam spacing; Q = 5.0 kW, H = 0.75 m, d/H = 0.12

Full size image
Figure 6
figure 6

Non-dimensional temperature profiles at 45-degree positions, effect of beam spacing, Q = 5.0 kW, H = 0.75 m, d/H = 0.12

Full size image

As previous researchers have determined [10–12], the beams certainly reduce the temperature of the ceiling jet within the secondary channel. Comparison of temperature profiles for different d/H values are shown in Figures 7 and 8, where only α is varied in the two figures. Clearly, with decreasing α, each non-dimensional temperature profile of beamed ceiling jet at a given d/H becomes steeper and is closer to that of the smooth unconfined ceiling. Therefore, temperature profiles are mainly affected by d/H and α/90. The gradient of ceiling jet temperature profile reduces significantly as d/H is increased. The uniform temperature near the ceiling, at α/90 ≈ 1, is caused by large vortices being shed and recirculation regions when d/H is increased. This is slightly more evident for α = 90° (Figure 7) since ceiling jet flow is perpendicular to the beam. Along the secondary bay, flow develops and temperature profile shape becomes more pronounced. It is observed that regardless of the variation in geometry, ceiling height, and fire size, the secondary bay ceiling jet temperature is significantly lower than the smooth ceiling jet temperature for d/H greater than 0.03. This has an important effect on actuation time of heat detectors and sprinklers for minor obstructions.

Figure 7
figure 7

Non-dimensional temperature profiles along the secondary bay demonstrating corridor flow effect, α = 90°

Full size image
Figure 8
figure 8

Non-dimensional temperature profiles along the secondary bay demonstrating corridor flow effect, α = 45°

Full size image

In Figure 9, the locations of crossing points between the smooth ceiling jet temperature profiles (d/H = 0) and beamed ceiling jet temperature profiles also points to the development of corridor type flows. This crossing is pronounced for results bracketed by, 0.07 ≤ d/H ≤ 0.09, and they seem to be at vertical distances, z ≈ d. For example, in Figures 3, 4, 7 and 8, the crossing points are located within 0.8 ≤ z/l T ≤ 1.2. The distance z can be determined and are approximately equal to the beam depth for each condition. For α = 90 (perpendicular flow to the beam), r/H, = 2l b/H, applying this condition to Equation (7) shows that for 0.2 ≤ l b/H ≤ 0.5, l T/H varies between 0.07 to 0.1. Therefore, z/l T = 1.0 (ratio of thermal Gaussian thickness, l T, to the ceiling height, H) is approximately close to d/H = 0.1. This means that the ceiling jet flow with temperature less than 37% of the maximum value is flowing to the secondary bay. This percentage is further reduced at lower α as also demonstrated in Figures 9 and 10. Examination of our experimental data showed that the temperature profile becomes increasingly uniform for d/H > 0.03 since the ceiling jet impinges on the beam and the flow is directed downward. This effect on the temperature profile can be ascribed to the redirection of the ceiling jet flow under the beam. Koslowski and Motevalli addressed this phenomenon and showed the transition to be evident for 0.06 ≤ d/H ≤ 0.09 [17]. For lower d/H values, the corridor flow does not fully develop and the temperature profile retains a form similar to the smooth ceiling jet case. For larger d/H values, development of the corridor flow makes the temperature profiles more uniform along with the reduction of temperature and prevents the crossing of the smooth ceiling jet temperature profile (see Figures 7 and 8).

Figure 9
figure 9

Non-dimensional temperature profiles, H = 1.0 m, Q = 5 kW, d/H = 0.09 and l b/H = 0.25

Full size image
Figure 10
figure 10

Non-dimensional temperature profiles, H = 0.75 m, Q = 10 kW, d/H = 0.2 and l b/H = 0.4

Full size image

4.2 Maximum Temperatures

The data from experiments 1–15 involve a wide range of non-dimensional fire sizes, beam widths, beam spacing, beam depths and ceiling heights. Despite these variations, non-dimensional maximum temperature can be reasonably considered as a function of d/H and α/90. The normalized data were analyzed to determine a polynomial fit shown in Equation (11).

$$ \theta_{\max} =[a_0 + a_1 \left(\frac{d}{H} \right)+ a_2 \left(\frac{d}{H}\right)^2 \ldots][ b_0 + b_1 \left(\frac{\alpha}{90} \right)+ b_2 \left(\frac{\alpha}{90} \right)^2 +\ldots] $$
(11)

The constants in the above two sub-polynomials have been obtained by using a modified Levenbery-Marquardt algorithm and a finite-difference Jacobia [18] to solve for this nonlinear least square fit. The constants obtained are:

$$ a_{0}=0.417, a_{1}=-4.148, a_{2}=23.493, a_{3}=-51.718, b_{0}=3.899, b_{1}=-3.021, b_{2}=1.381 $$

The goodness of this fit is determined by the correlation coefficient defined in a standard fashion [18]. This correlation has a 0.995 coefficient and is plotted in Figure 11 and seems to fit the experimental data very well for the 0.03 ≤ d/H  ≤ 0.25. In Figure 11, f(α) was determined by calculation of θmax using Equation (11) at different (α) resulting in the following relation:

$$ f(\alpha)=3.90-3.02 \left(\frac{\alpha }{90} \right)+1.38 \left(\frac{\alpha }{90} \right)^2 $$
(12)
Figure 11
figure 11

Comparison of experimental data for all measured positions within the secondary bay with the formulated correlation

Full size image

Finally combining Equations (11) and (12) the following equation is obtained:

$$ \frac{\theta_{\max}}{f(\alpha)}=0.42-4.15 \left(\frac{d}{H} \right)+23.49 \left(\frac{d}{H} \right)^2 -51.72 \left(\frac{d}{H} \right)^3 $$
(13)

The significance of the analysis and correlation developed (Equation 13), which is shown in Figure 11 is that all the data are collapsed for the entire range of d/H, various l b/H and \(\dot{Q}^{\ast}\). The correlation behaves physically correct. As d/H approaches zero the smooth-unconfined horizontal ceiling jet is reached asymptotically θmax approaches 1. As d/H increases beyond 0.25, θmax approaches zero which is a no-spillage or corridor flow condition. Furthermore, it has been demonstrated that a relatively flat profile is obtained in the secondary bay as d/H reaches about 10% value. Thus, determining θmax may be sufficient to characterize the ceiling jet temperature. Although the flow velocity was not measured in this work, previous work [12] indicates a large reduction in velocity in the secondary bay.

Work of Vettori [19] comes closest to large-scale validation test. However, his detailed measurements were limited to 4 locations in a beamed ceiling configuration with 16′′ beam spacing, plume impinged on several bays and beams and the measurement was at least 4 bays away from the plume impingement. The data graphed in report by Vettori shows general agreement with the predictions using Equation (13). The ceiling height for these experiments was 2.4 m. This type of data can only provide qualitative comparisons and the measurement details can not match small-scale conditions.

5 Application Examples

The failure or delay in heat detector or sprinkler actuation jeopardizes the detection and suppression system reliability. It is very important to be able to determine if the minimum heat release rate required for detector or sprinkler operation exceeds a design threshold fire size for sprinklers or heat detectors positioned in every other bay.

The maximum temperature of ceiling jet within the secondary bay can be used to determine the minimum fire size required for the actuation of each detector or sprinkler based on the developed correlation regardless of l b/H. For example, if ceiling height H is 3.5 m and temperature rating of detector or sprinkler is 57.2°C (135°F), the minimum fire size for different d/H and r/H can be calculated. The results of this calculation are plotted in Figure 12. It is shown that with increasing d/H, the minimum fire size \(\dot{Q}_m \) vs. d/H rise drastically. The larger r/H, the steeper is the slope of \(\dot{Q}_m\) vs. d/H. Furthermore, the minimum fire size under the condition of 0.03 < d/H < 0.25 and 0.25 < r/H < 1.0 can be predicated on the basis of correlation established in this work, detector or sprinkler’s temperature rating and configuration of beamed ceiling.

Figure 12
figure 12

Calculated minimum fire size needed to actuate a heat detector or sprinkler head placed in the secondary bay

Full size image

6 Conclusions

This paper established an empirical equation for beamed ceiling jet maximum temperature within the secondary bay (or channel) as a function of the ratio of the beam depth to ceiling height and the position along the secondary bay axis. It is obvious that for d/H  > 0.20, the non-dimensional maximum temperature decreases dramatically, and the flow within the bay approaches a corridor flow. There is, on the average, 15–90% reductions in the temperature of ceiling jet within secondary channels for the beam depths of 3–25% of ceiling height, respectively, when compared with the smooth horizontal unconfined case. This equation was demonstrated for its practical application to determine the minimum fire size detected by a heat detector placed in a secondary channel. The experimental data in this work may be used to validate the numerical results generated by field models. The data and correlation developed here can also be used to perform parametric studies of ceiling geometry effects on smoke and heat detector performance. The results reported here are limited to those cases when the fire plume impinges the ceiling within the primary bay and the flame height is considerably less than the ceiling height. The \(\dot {Q}^{\ast}\) range is large enough to include practical fire sizes, however, the relation to the smooth horizontal ceiling correlation limits its validity to r/H = 1.0 with the added caveat that the this correlation was developed using small-scale experiment within a weak-plume range. Although, there is an advantage as early detection of growing fires often falls within very small fires and weak plume region.