By Nils Johansson

Several theoretical relationships and engineering methods are available for studying conditions in the room of fire origin. However, there is a lack of simple engineering methods that can be used to describe the conditions in rooms adjacent the room of fire origin. This means that studies of the conditions in such spaces have to be conducted with two-zone models or Computational Fluid Dynamics (CFD) models.

Such advanced computer models are generally excellent tools for fire engineering purposes. Even so, there is still a need for simple hand-calculation methods for several reasons. Johansson, Svensson, and van Hees ^[1] have developed and evaluated two simple methods that can be used to estimate both temperatures and hot gas layer height in adjacent rooms. This article gives a brief insight into their work.

Simple and transparent methods are considered to be valuable tools for understanding the fundamentals of complex fire dynamics problems, since such methods include the variables that govern the studied phenomenon. Simple hand-calculation methods can also increase the possibility for engineers to estimate calculations in a fire safety design. Such simpler methods might have a lower accuracy compared to more-advanced methods, but simpler methods can provide estimates that can be satisfactory for the studied problem. Rough estimates can also be used to determine whether a more-detailed and time-consuming calculation using a more-advanced model is needed.

Furthermore, a simple method can be executed quickly, while a CFD model might require hours or days to provide a result. This means that simple methods can provide possibilities for conducting fire risk analyses, such as Monte Carlo analyses, where hundreds or thousands of scenarios are studied, and this is currently not feasible with a CFD model.

Fires in rooms are usually described as starting in furnishings and then developing by spreading to other items in the room. However, a fire can involve several rooms. There are a few examples of engineering methods to study multi-room post-flashover fires, but pre-flashover conditions in multi-room compartments seem to have been studied even less, and no hand-calculation methods, comparable to something like the MQH-correlation, existed before the work of Johansson, Svensson, and van Hees.^[1]

A Simple Energy Balance

A general energy balance, Equation 1, can be formulated for a multi-room compartment where each room is connected to the next room by an opening and the final room is connected to the outside by an opening (see Figure 1). 

where Q̇ w,i is the heat loss due to conduction through the exterior boundaries in room i and Q̇_L is the convective losses through the opening in the final room. The radiative loss (Q̇_R) and the change in energy storage in the hot-gas-layer (Q̇_B) are disregarded in Equation 1 because the two included components are considered to be dominating. For slow-growing or constant fires, Q̇_B can be neglected because the temperature increase in the hot-gas-layer will be small. Q̇_R is considered to be larger than Q̇_B, but it is still small in comparison to Q̇_W and Q̇_L. 

Figure 1: A simple energy balance in a multi-room compartment with n-number of rooms and with an opening to the outside.

The general energy balance in Equation 1 can be reformulated for a two-room case (see Figure 2) and the specific case as follows:

Figure: Illustration of an energy balance in a two-room compartment.

The heat losses to the boundaries in contact with hot gases in each room are expressed using the terms, q̇ _loss,1 and q̇ _loss,2, which in turn can be expressed as being a function of the temperature difference between the boundary surface and the surroundings, the area in contact with hot gases, and a heat transfer coefficient (see Equation 3).

Several heat transfer phenomena are lumped into the heat transfer coefficient, h, . It is not trivial to estimate, but it can be done with simplified relationships for semi-infinite or thermally thin materials.

An alternative method to estimate the heat transfer coefficient is to use empirically determined values or correlations. A_w is the area in contact with hot gases; this does not taken into account the fact that the two rooms have a common wall.

Two Methods to Predict Conditions in Adjacent Rooms

It is not reasonable to derive simple mathematical expressions for estimating temperatures in an adjacent room directly from theory, due to the complexity of the fire phenomena. Two alternative approaches have been used to develop two hand-calculation methods. The methods are based on the same simplified energy balance as presented in Equation 1.

The first method consists of a single empirical correlation (Equation 4). The correlation was found with the help of a numerical experiment and the data from the numerical experiment were used in a multiple regression analysis to find a correlation for the hot-gas-layer temperature in the adjacent room^[2]. The room dimensions were randomly selected within certain intervals, with the restriction that the resulting geometry was within the limits of the two-zone model assumption. The width and height of the openings varied between 0.5-2.2 and 1-3.7 m, respectively.

The fire source was placed in the centre of the fire room and the HRR varied between 320 and 2000 kW. The properties of the enclosure boundaries were similar to concrete, lightweight concrete, or brick. The resulting correlation (Equation 4) had a coefficient of determination, R²-value, of 0.93.

The second method, presented in Equations 5–9, is more tiresome to apply than Equation 4, but at the same time, more transparent than the correlation. The second method is also more flexible because there might be alternatives to the different supporting models—that is, different types of plume models. In such a case, the best supporting model for the problem should be used. The second method also offers the possibility of taking the different enclosure materials in different rooms into account.

The second method includes these five steps.

1. The height to the hot-gas-layer interface in the first room, z_int,1, is calculated by using these three equations.
2. When z_int,1 is known from step 1, it is possible to calculate ṁ_g using Equation 6.
3. Since no mass disappears in the compartment (see Figure 2), ṁ_g at opening B will be the same as at opening A and the hot-gas-layer interface height in the adjacent room, z_int,2, can be calculated using Equation 6.
4. The temperature increase in the room of fire origin can now be calculated using a simple energy balance (Equation 8), that accounts for the heat losses due to convective flow thorough the opening and heat transfer to the boundaries.
5. Finally, it is possible to solve the energy balance (Equation 9) and to calculate the excess temperature in the adjacent room.

Both methods apply to simple two-room geometries similar to that in Figure 2. Even so, they illustrate which variables will be of importance in a multi-room geometry. The second method can be modified and applied in more complex situations, such as more rooms or openings.

Evaluation of the Methods

Predictions with the two methods have been compared to data from a small-scale compartment fire experiment. Using two different-sized rooms (one small and one larger) in the experiment created two different room configurations. The small room corresponded to a 1/4^th-scale ISO room, and the large room was 1.2 × 1.2 × 0.8 m. The two rooms were connected with an opening and there also was an opening from one of the rooms to the outside. The size of the openings varied between a small (0.2 × 0.5 m) and a larger opening (0.3 × 0.5 m). The fire was placed in the centre of the inner room (see Figure 2).

The two room configurations and the two opening sizes were combined into five different geometrical scenarios. Varying the size of the fire and the type of fuel resulted in a total of 16 unique fire test setups. Tests were performed three or four times for each setup, and a total of 52 tests were conducted in the experimental test series.

Thermocouple trees were placed in both rooms to approximate the hot-gas-layer temperature and height. The experimental uncertainty for the hot-gas-layer temperature and height was estimated and concluded to be 12% and 13%, respectively. It also was possible to approximate the model uncertainty with the results from the tests and the estimated experimental uncertainties.

Both methods have been used to predict the temperature (see Figure 3) in the experimental test series. Step 1 in the second method was also used to predict the hot-gas-layer height (see Figure 4)

.

Figure 3: Predicted values plotted against experimental values of the hot-gas-layer temperature for both methods in the adjacent room. The dashed lines are the estimated experimental uncertainty

Figure 4: Predicted values plotted against experimental values of the hot-gas-layer height with the second method in the adjacent room. The dashed lines are the estimated experimental uncertainty

Both methods predict the hot-gas-layer temperature within 10–15% of the experimental values, and that is, in many cases, within the estimated experimental uncertainty.

There are benefits and drawbacks to both methods. The first method consists of two simple-to-use and, with regard to the result of the evaluation study, reasonably accurate mathematical expressions. The second method is slightly less accurate than the first method, but it is more transparent and has a stronger theoretical basis.

The paper by Johansson, Svensson, and van Hees^[1] contains two major research contributions. Firstly, the experimental data are rather comprehensive, with a total of 52 performed tests, and the experimental setup, is described in detail in the paper, as is the experimental uncertainty. Secondly, they present an evaluation study of two novel methods that can be used to calculate the hot-gas-layer temperature and height in a two-room compartment, as well as the precision and bias of the two methods.

If you would like to know more about the research presented in this article, have a look at the paper by Johansson, Svensson, and van Hees in Fire Safety Journal^[1].

Nils Johansson with the Division of Fire Safety Engineering, Lund University, Sweden

1. Johansson, N., Svensson, S., van Hees, P. “An evaluation of two methods to predict temperatures in multi-room compartment fires”, Fire Safety Journal, 77, 2015, pp. 46–58. DOI: 10.1016/j.firesaf.2015.07.006.
2. Johansson, N., van Hees, P., “A correlation for predicting smoke layer temperature in a room adjacent to a room involved in a pre-flashover fire”, Fire and Materials 38, 2014, pp.182–193. DOI: 10.1002/fam.2172.