1. Introduction
There is continuous progress in the reflection of the fire phenomenon in computer models [
1]. There is also a growing number of applications of computer software to complex fire problems [
2]. Computer models are continually gaining accuracy, as is the scope of the modeled fire phenomena. However, due to limitations in representation of many fire-related phenomena, computational costs, and uncertainties related to numerical investigation, physical experimentation in reduced-scale modeling is still a popular tool. Since the first studies on the smoke control of factories performed by Thomas et al. [
3], reduced-scale research allowed for the development of better techniques and solutions to prevent fires and limit their consequences to occupants, buildings, and the environment. Such studies could also be used in forensic applications [
4,
5] and building design.
There are two governing non-dimensional numbers in low-Mach–fire-related flows, namely the Reynolds (
Re) and Froude (
Fr) numbers. However, the Fr scaling conflicts with the
Re preservation by providing two different results for velocity in the scale models. There are analyses [
6] that prove that in fire-related flows, one should preserve the Froude-number, not the Reynolds number when scaling is applied. In this case, so-called partial scaling is adopted that favors the
Fr over the
Re [
7]. Quintiere et al. mentioned that maintaining sufficient height of the scale model compartment (>0.3 m) may be adequate to maintain flow turbulence [
5].
The
Fr scaling assumes that two fires are similar to each other, if the Froude-number characterizing both fires is equal, and there is geometrical and hydraulic similarity of the systems in which the fires take place. An illustration of the Froude-number scale fire modeling concept is shown in
Figure 1. Variable
x describes dimensions of the model in full- (index “
f”) and reduced- (index “
m”) scale. If the Heat Release Rate (HRR,
Q) of the fire is scaled accordingly and the
Fr is preserved, the temperatures (
T) measured in the reduced- and full-scale fires should be equal.
The approach of the
Fr scaling must be used with care, despite its popularity. Spalding emphasized that partial scaling is “an art” and not a science, due to problems in properly relating the behavior of the full-scale system to the one of the model [
9]. Among these problems, the most common are with the scaling the chemistry of combustion and the flow turbulence [
9]. Nevertheless, Froude-number scale modeling of fires has proven useful in understanding various fire phenomena, and their consequences [
3,
10,
11,
12].
In this research, we focus on some issues identified with the practical use of the Froude-number scale modeling of a small fire (HRR of approximately 150 kW) in a small-sized compartment (approximately 10 × 10 × 4 m³). We have observed a discrepancy in the measurements of the temperature between the full- and reduced-scale experiments that match observations previously reported in the literature. Furthermore, the experiment was recreated with the use of CFD modeling in larger number of scales as an attempt to confirm the experimental observations, investigate the turbulent flow structure in more details, and form generalised conclusions. The temperature of smoke measured in the reduced-scale analyses was lower than in a full-scale experiment, despite conforming to the Froude-number scaling principles. The discrepancy was bigger when the scale used wassmaller. Moreover, in very small scales we have observed problems with maintaining a sufficiently high Reynolds number, and the fire behavior has changed from turbulent to laminar, leading to significant error in the modeled smoke temperatures and invalidating the scientific value of these experiments.
2. Froude-Number Based Fire Modeling
In the early 1960s, mass fires were considered to be an extremely complex phenomena that did present formidable scaling problems. A body of literature did exist (reviewed by Williams [
9]) related to the scaling of specific, simple types of fires. It was in the review paper by Williams [
9] where the foundation for future scaling of fires was laid, based on the combination of several different models based on the principal laws, mathematical operations, and empirical observations. The paper introduced 29 dimensionless groups that describe the scaling principles for fire phenomena. In this modeling technique, some of the phenomena are simplified, and others are omitted on purpose. However, the overall image of the reduced-scale fire may be useful and provide a reasonably good picture of the characteristics of the full-scale system that is of interest. The concept of reduced-scale modeling was summarized by Quintiere in reference [
13] and more recently in references [
5,
14].
In the technique of scale modeling using dimensionless numbers of similarity, the starting point is the Buckingham theorem, also known as the “Pi Theorem” [
14]. The theorem states that:
“Each function involving a certain number n of physical variables ai, and these variables are expressible in terms of k independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n – k dimensionless parameters constructed from the original variables. If all dimensionless parameters are identical, the phenomenon will be identical despite the different parameters of the ai type.”
The statement above (Pi Theorem) is based on the principle of dimensionless unity. Moreover, it states that if the equation expresses a proper relation between variables in physics, the process shall be dimensionally homogeneous, and its parameters will have the same dimensions [
15].
Thus, a dimensionally homogeneous equation containing n mathematical variables:
can be recorded in the dimensionless form:
It was already identified by Williams [
9] that creating a model that conserves all 29 dimensionless groups would be formidable. Thus, subsets were created, of which the basic subset is presented below. The geometrical scale of the model is the ratio of the characteristic dimensions of the scaled-down model (index “m”) to the corresponding dimensions in the full scale (index “f”).
Froude-number similarity (4), convection and radiation groups, and gas-phase heat release were grouped together with fuel gasification and loading groups, and ambient atmospheric condition groups [
9].
These were foundations for future scaling principle developments. The heat release rate in small scale is determined by the scaling of Zukoski number (5), also known as dimensionless group Π9.
In modern fire modeling, it may be assumed that two fires can are similar if the following requirements are met [
8]:
The Froude-numbers of both of the fires are equal;
All geometrical features related to the fire and the environment are scaled with the same scale;
The fires are occurring at well-ventilated conditions, i.e. the combustion is not significantly influenced by the reduced-scale, and the combustion efficiency in full and reduced-scale is similar;
The flow in the buoyant plume is turbulent.
If the Froude similarity criterion is met, and Reynolds criterion is satisfied, the other relevant parameters that describe the flow of mass and heat in the compartment will scale as summarized in reference [
16] and listed below.
The heat release rate of the fire [kW]
For a more thorough introduction to
Fr scaling of fires, the reader is kindly referred to reference [
14].
The
Fr scaling approach can be potentially used in an investigation of other phenomena relevant to fire safety, where buoyancy is the dominant force in the model. An example can be flammable gas dispersion, and a recent example of full-scale research can be found in reference [
17]. However, in scaling of such phenomena other set of relations may apply, as the change of density in Equation (4) will not be related to heat release, as in Equations (5) and (6).
The abovementioned set of similarities was used widely in fire safety science, in a vast array of scales. Fire sizes in literature review range from 0.31 kW to 100 MW, and the scales from full scale (1:1, [
18]) to 1:48 [
19]. A summary of investigated research performed in reduced-scale is shown in
Table 1.
6. Conclusions
The paper has shown observed discrepancies and problems with the application of Froude-number scaling for modeling compartment fires. The experiments were performed with a wide range of Reynolds numbers, showing the essential role that turbulent flow has on the temperatures in the plume and the compartment. The previously mentioned rule of thumb value of Re = 10,000 was confirmed, to sufficiently minimize the error of the method related to the flow turbulence.
In case of small scales (1:2 and 1:4) the average temperatures measured were up to 30% lower than in the full-scale experiment, however in most of the experiment duration this difference was up to 10% (which in the opinion of the authors can be considered as an acceptable value). The temporal change in the temperature was well represented in small scale. These results indicate that the scaling method can be useful for investigation of the flow of smoke in buildings. For smaller scales (1:10 and smaller) the differences in the temperatures measured were significant, and in case of very small scales (1:20 and 1:50), the results have no scientific value due to change of the flow from turbulent to laminar.
CFD modeling with FDS software did sufficiently represent the full- and reduced-scale experiments and was used to analyze a wide array of scaled fires. A similar approach can be used before future experiments, to verify the sensitivity of the experiment to the scale, and estimate the Reynolds number of the flow. Furthermore, the numerical modeling may help with investigating the effects of materials used in the reduced-scale model on the heat transfer to the model boundaries.
As first discussed by Spalding, and mentioned by Williams [
9] and Quintiere [
14], the reduced-scale modeling is an art that requires the user to choose which dimensionless relations are conserved, and which are omitted. To maintain the high scientific value of scaled-down experiments, the user should make informed decisions, and use modern tools (such as CFD modeling) to assess the model sensibility to the changes introduced in the reduced-scale. As shown in this paper, just following the basic scaling theory is not sufficient to guarantee that the results of reduced-scale experiment are valid. Also, the temperature should not be considered as the only variable to be assessed. Other parameters such as flow velocity, mass concentration of pollutants, or heat fluxes may be useful for evaluating the validity of the model.