For more than 100 years, automatic sprinkler systems have been considered an effective method to control fires in enclosed areas and to provide property and life protection. An automatic sprinkler is one of the most important components in the sprinkler system.
Automatic sprinklers must be well designed so that, when flames appear, they can successfully detect the fire’s heat, activate the water flow and begin suppression. In order to handle fire hazards in a variety of applications, different automatic fire sprinklers of various types and sizes are available. One way to categorize automatic fire sprinklers is by discharge coefficient, also known as K-factor. K-factor of a fire sprinkler is defined as: K= Q/√p, where Q is flow rate, and p is static pressure. For instance, several automatic sprinklers with different nominal K-factor values are shown in Figure 1. The nominal K-factor value is also used to determine the sprinkler’s flow rate or pressure drop when calculating hydraulically designed systems.
It is required by the listing agencies that a variety of performance tests be passed before the developed automatic sprinkler can be approved or listed (for instance: FM Approvals, 2018; UL, 2008; and others). One of these tests is the test of discharge coefficient (K-factor). For typical automatic fire sprinklers, it is required that individual K-factor values at each pressure be within ±5% of the calculated mean discharge coefficient for the entire range of pressures that are tested (FM Approvals, 2018; UL, 2008). Measurements of the K-factor are performed in the test equipment as specified by the authorization agencies. A schematic drawing of the test equipment for measuring an automatic fire sprinkler’s K-factor is shown in Figure 2.
Although fire sprinklers have been in use for over 100 years, they are developed mainly on the basis of experimental testing. There has been little progress toward developing analytical or numerical methods of calculating their effectiveness (Yao, 1997; Wess and Fleming, 2020). Engineering calculations are restricted within a limited number of tasks, such as: hydraulic calculations of water flow through piping (Fleming, 2016). With rapid progress in computer hardware and numerical algorithms over the past decades, Computational Fluid Dynamics (CFD) has increasingly received attention. Computer simulations allow designers and engineers to design ever more challenging structures, components and processes with minimal use of expensive experimental testing. CFD has been widely used to determine flow field and pressure distribution in nozzles, orifices, valves and many others. In the present work, the theoretical formulation for sizing sprinkler orifices is first presented. Then, CFD modeling of sprinkler nozzles with different waterway profiles is shown and discussed.
The theoretical K-factor value, which is calculated based on Equation (7), represents an ideal or the maximum discharge coefficient for a given sprinkler nozzle exit diameter. This also can be used to quickly estimate the minimum required nozzle size for a specific nominal K-factor value. In practice, the actual K-factor value of a sprinkler nozzle is affected by many factors, such as: wall friction loss, abrupt change in flow areas, flow separation and recirculation, and so on. In some cases, the coefficient of velocity (cv) and the coefficient of contraction (cc) are utilized to account for effects of flow separation and recirculation (Linder, 2008). The coefficients of velocity and/or contraction can be estimated based on empirical values. However, such estimation would lead to relatively high errors in comparison with the tolerance which is allowed by the listing agencies for fire sprinklers. Alternately, such parametric study can be accomplished with verified and validated CFD models. An example is described in the present manuscript to demonstrate the process.
The studied sprinkler nozzle has an exit diameter of 0.404in (0.0103m), and a length of 0.866in (0.022m). From Equation (7), the theoretical K-factor value of this nozzle (Kmax) is 4.87 gpm/psi0.5 (70.1 lpm/bar0.5). Two different waterway profiles are numerically studied: (a) sharp-edged, and (b) linear. The sharp-edged profile is essentially a straight hole. Their cross-sectional views are shown in Figure 4.
For improved accuracy, the ‘standard’ test geometry is selected as the computational domain. Due to symmetry, an axisymmetric domain is used, as displayed in Figure 5. The shear-stress transport (SST) k–w model is selected for modelling turbulent separated flow. Compared with the standard k–w model, the SST k–w model includes the term accounting for the transport of the turbulence shear stress in the definition of the turbulent viscosity (µt). This allows the SST k–w model to achieve more accurate and reliable results for flows with adverse pressure gradient than the standard k–w model (Ansys Inc., 2018). The CFD package, Fluent v19.3, is used for solving the flow and pressure fields. During the numerical calculations, the convergence criterion required that the maximum relative mass residual based on the inlet mass be smaller than 5 × 10-6. More details about the present CFD model and its validations can be found in (Nie and Cutting, 2020). Meshed computational domains for the sharp-edge and the linear waterway profiles are displayed in Figure 6. Mesh refinement is employed near solid walls or in the region where velocity gradient is expected to be high.
Pressure distributions in the nozzle at 100psi (0.689MPa) are shown in Figure 7. For the sharp-edged waterway profile, most of the pressure drop is located at the entry of the nozzle. Sudden change in the flow area leads to flow separation at the sharp edge and then a subsequent recirculation region, where adverse pressure gradient exists. Separated flow is much alleviated with the linear waterway profile, where pressure gradually drops along the nozzle’s length.
The computed K-factor value for the sharp-edge waterway profile is 3.86 gpm/psi0.5 (55.6 lpm/bar0.5). In comparison, for the linear waterway profile, it is 4.74 gpm/psi0.5 (68.2 lpm/bar0.5). To evaluate hydraulic performance of the nozzle waterway profiles, a K-factor efficiency is introduced, which is defined as nk = K/Kmax. The K-factor efficiencies for the sharp-edged and the linear waterway profiles are 0.793 and 0.973, respectively. It can be observed that the waterway profile directly impacts the K-factor value of the sprinkler nozzle. Essentially, the K-factor efficiency (nk) is equivalent to the general discharge coefficient or factor for flow through orifices or nozzles, as used in Heald (2010) and Fleming (2008). As stated by the above authors, these coefficients or factors are usually determined by laboratory tests. Different design iterations will require extensive time and cost to make prototypes, and to accomplish the laboratory tests. By contrast, the present CFD model requires about 15 to 30 minutes to complete one parametric modelling on a standard multi-core desktop computer, which provides a promising alternate for quick evaluations and conceptual designs.
Different automatic fire sprinklers of various types and sizes are available. The K-factor value is crucial to determine the sprinkler’s flow rate or pressure drop when calculating hydraulically designed systems. Two methods are included in the present manuscript: (1) the quadratic equation for estimating the theoretical values; and (2) the CFD model with detailed waterway profiles considered for high-fidelity values.
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We gratefully thank Carson Coelho for his valuable discussions and suggestions.
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