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Revision as of 14:59, 28 September 2018
MODEL
Introduction
This year, our team creates a mathematical representation of our concrete self-healing system. This representation, or model, constructs a judging scale with which we can utilize to regulate the four main environmental factors affecting our final concrete healing efficiency (reflected by mineralization activity).
The four factors are: 
Based on this model, we can also design the best ‘package'– the microcapsule shells with adequate nutrition combination.
This modelling process, presented below, can be seen as a feedback from the wet lab (experiment result) and the dry lab(modelling analysis).
The following page shows how we conducted modelling approaches to achieve our goals. To begin with, we make some definition.
- Variables and nomenclature:
- concentration of spores - c[Spore]
- concentration of carbon source(C3H5O3Na) - c[C3H5O3Na]
- concentration of nitrogen source(NaNO3) - c[NaNO3]
- pH of the media - pH
- Goal of model:
- Use preliminary data to guide future experiments.
1. Standardizing variables: transforming variables into the same scale. Here we utilize the z-score standardizing method, such that
where the symbols in the formula are variable, standardized variable, sample mean, andstandard deviation respectively.
2. Fitting functions of each variable with polynomial function.
3. Getting the overall relationship using linear least square method.
The graph above depicts the polynomial regression of 4 factors. In general, for each of those four factors, the mineralization activity shows the similar tendency of going up first and down later with the increase of each factor.
The fitted functions are showed as follows:
We then fitted x with y linearly to get the overall regression equation which can describe the weight of four variables respectively.
with
where ai1represents the coefficient of the highest power term of curve-fitting equation, such that the coefficient of the highest power term are of the same scale.
From the equation above, we can see that nitrogen source has the maximum weight, while pH has the minimum weight, which means nitrogen source is the most essential nutrition for B.subtilis spore. Additionally, the low weight of pH shows that spores are not sensitive to the change of pH in a relatively apt range, although there is a sharp decline in activity when pH reaches 11.
In summary, as long as we keep the environment below the boundary high-pH point, it makes no much difference how much we have improved the alkaline resistance of B.subtilis spore. In this way, this model instructs us on how to modify our formulation, such that the ingredients can be made full use of.
