Team:FCB-UANL/Model
One of the most important features of firefighting foams to measure their quality is their stability, as effectiveness relies on it. Foam stability depends on many factors, such as the environment, components, usage, etc. However, for us as producers, it is important to ensure our product’s quality by regulating the concentrations of the components to be optimal for both foam formation and fire combat. To achieve that, we made up a mathematical model able to predict the optimal concentrations of each component, thus allowing us to get to know better the parameters required for obtaining the highest efficiency while carrying out the experimental process, the foam formulation and industrial upscaling.
GENERAL PROCESS
Our model is based on a draining flow simulation model of a soapy foam1, which allows us to model a single bubble and analyze how its stability changes depending on different parameters; thus, we can identify the most important ones to control in order to optimize the bubble’s draining flow. These results provide valuable information for both laboratory and industrial production, as we can determine the necessary amount of the components to produce the foam.
The model will help us to know the optimum amount of necessary surfactants, Ranaspumin-2 and surfactin to maximize the foam’s stability.
HOW DOES THE MODEL WORK?
For the modeling, we used the following parameters that were already defined1:
- Draining flow (it refers to the quantity of liquid a bubble loses over time and speed at which this happens)
- Initial Bubble Thickness
- Initial surfactant concentration
- Vertical length of the bubble
- Component densities
- Surface tension
Before describing the Partial Differential Equations (PDE), we must first define the three main topics on which our mathematical model is built: The Navier Stokes Equations, The Continuity Equation for Fluids and The Theory of Lubrication.
We now proceed to the physical analysis of the model. The system consists of a 2-dimensional bubble where the film is held by two wires of thickness h0, which is the initial thickness of the system. The wires are separated by a length L.
The system to be analyzed will evolve as time passes, and draining flow, a phenomenon whose causal agent is gravity, will occur. We must outline that the content of the thin film will be drained in a vertical way, as it is shown in the image.
The initial system is static with the following border conditions and assumptions:
- Constant atmospheric pressure,
- Surfactant is insoluble,
- The edges are exposed to the atmosphere,
- Symmetry of the film is assumed,
- No external forces
We decided to take into account these considerations in order to have a less complex simulation and have more control in the situation we are approaching. Hence, after a deep analysis we obtained 3 coupled partial differential equations, which together are able to explain the draining flow. The mathematical development from which these equations are derived is shown in Schwartz and Roy’s (1999) work1. Each of these 3 equations has a physical meaning within the bubble analysis.
These 3 equations were previously non-dimensioned in order to be able to program the equations in a simpler way, thus simplifying the process.
The model was non-dimensionalized as follows:
This is the general model of the draining flow in a soapy film. To make it more specific to the project, it is necessary to introduce input parameters which will differentiate the different types of foams.
CODING
Solving the model’s equations analytically is not considered practical because they are of a very large order. Hence, it was decided to make a program that solves them numerically.
Attempts were made to solve using software such as Matlab, FreeFem + +, Wolfram Mathematica, as well as using the Python language since it contains various libraries for solving PDEs, but due to their complexity it was not possible in either way.
Therefore, we decided to use finite difference methods along with the Thomas algorithm to solve the equations because these methods are the most used to solve High order PDE’s. With these methods, along with the discretization of the domain, the results obtained will be more accurate. In particular, the Thomas algorithm allows us solve the slip velocity equation. Its structure, which is shown below, helps us to calculate the coefficients used in the finite differences method.
Then, the corresponding transformation of the equations was done through finite differences in the following way:
The code was written in the FORTRAN programming language and was run on a Linux machine with a 4-core processor at 3GHz each. The calculations can take from a few seconds to a few minutes.
Sample of the code:
https://github.com/iGEM-UANL/2020RESULTS
It is important to remember that the results will be in their dimensionless form, so this consideration must be taken into account in their interpretation. To model our type of foam, we used some parameters that correspond to the main components, which are Ranaspumin-2 and surfactin, which are the following:
With the following initial conditions:
Now that we know that G must be at the same magnitude as B, we are now going to use factors of 1000 on G. In order to obtain results that we can measure in the laboratory, we have the relation:
This is an equation of the slope point form obtained through experimental data of surface tension and surfactant concentration, where K is a constant that represents the slope of the data. This has to be obtained experimentally in order to complete the dimension of the amount of surfactant mentioned in the graphs and in the final part of the results. Despite not having the experimental values, we can continue working with the model’s dimensionless form to approximate a result for its further usage when we were able to obtain that parameters.
We introduced the parameters of Ranaspumin-2 into the model, and the results we obtained for different concentration profiles are shown in the following graphs, where the green line represents an initial surfactant concentration in 2000 and the purple line in 8000:
We can see that there is a notable difference depending on the surfactant concentration because, as shown in the graph above, they do not follow the same behavior. What the lines indicate is how the surfactant is distributed through the bubble, which is directly related to slip velocity; the longer the distance (X axis), the greater the slip velocity.
Here we can corroborate the inference we did from the first graphic, since it shows that for a lower amount of initial surfactant the slip velocity is higher (green line). Hence, we can say that this factor will impact the draining flow significantly, as the content inside the bubble will slide down the surface more quickly.
Here, we can observe that for a higher surfactant concentration, the thickness increases. Also, a greater growth in the thickness of the black film (region of the foam where the thickness decreases) of the bubble with higher surfactant concentrations can be noticed in comparison with one with smaller concentrations. Therefore, being a thicker film, the bubble has better resistance to forces that oppose it, in addition to a considerable delay in the draining flow.
Before we mention the conclusions generated from the graphs obtained with the model, we must have in mind some considerations. The graphs are the result of the simulation of the draining flow at different surfactant concentrations. In the case of surfactin, the results obtained for different concentration profiles are the same as the results given with the Ranaspumin-2, so we decided to not include it because there will be no difference in the process of drainage.
Another important point that has to be taken into account for the correct understanding of the values shown is that, due to the nature of the model, the drainage of the foam is very fast. This means that adjusting the time 1 second in the simulation corresponds to approximately 1000 seconds in the experiments.
Another aspect to outline is that a bubble with a length of 1 cm is being analyzed, so, even though the variations that are present in the experiments are apparently small, any change in the observed magnitude is still considerable and affects its stability.
Now, several conclusions can be established by taking into account the obtained results. These concluding remarks are also very convenient to consider in future experimental tests. First, an easy result to observe is the fact that the draining flow is a function of time and the amount of surfactant, so this aspect will be taken as a key factor for the development of the foam.
Another outcome that can be observed when analyzing the model is that the greater the amount of surfactant, the more stable the bubble is and, consequently, the foam in the whole set. It is easy to appreciate that the slip velocity does not play such an important role within the draining flow model.
In the same way, it is observed that there is not a considerable difference for values higher than 1000 in the initial concentration of surfactant (remember it is its dimensionless form), but rather for values lower than this. Hence, we can say that 1000 is the minimum concentration at which surfactant provides greater support to the external forces that can get to break the bubble or accelerate the flow.
Although we have a robust approximation with this model, there are still certain aspects of it to be improved in the future, such as the consideration of various environmental factors as well as the addition of other compounds that can modify the physical characteristics of our foam. To further complement our model, future experiments will be carried out to corroborate the results along with establishing other parameters necessary to increase its accuracy.
REFERENCES
[1]. Schwartz L.W.; Roy R.V. (1999). Modeling draining flow in mobile and immobile soap films. Journal of Colloid and interface Science 218 (1):309-323.