12.5.5. 1 mathematical model of shelf life and commodity quality
There is a basic assumption that there is a set of parameters (Q) (survival rate in the experiment) that affect the quality of products and can be measured. Further quantitative analysis requires the relationship between quality change time and some factors, which is usually expressed by the following equation:
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Each factor changes with time. For example, temperature is generally the main factor for the stability of biological samples; According to the water activity, the influence of humidity and other important parameters affecting the reaction rate is explained. Now, this method is well applied to predict the stability of food and medicine (Labuza et al., 1983). Because the quality change of samples during storage is completely derived from chemical reactions, the simplest and most commonly used empirical model is as follows:
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In the formula (12. 1), k is the rate constant; N is the reaction order (food or medicine is usually 0 or 1). This formula completely ignores the participation of machinery. In actual production, in an appropriate coordinate system (such as the semi-logarithmic diagram of the first-order reaction), the parameter Q is designed to change with time, and whether the device is good (r2 is high) is evaluated by ordinary statistical methods. Generally speaking, when the data generated by the reaction is more than 50% complete, a definite value of n can be given (Labuza et al., 1983).
Assume that the rate constant k is affected by humidity and temperature. The most common arrhenius formula uses a mathematical model to describe the dependence of K on T:
k=k0exp(-Ea/RT) ( 12.3)
In this formula, k0 refers to the pre-factor; R is the ideal gas constant; Ea is the activation energy. If the spectrum of ln(k) versus T- 1 is a straight line, then arrhenius formula can be applied, and the activation energy is constant outside the temperature range. At least four temperature data are needed to determine the linear relationship between them. When the temperature exceeds a certain reaction temperature, the straight line shifts. Under the condition of low humidity, it is impossible for the product to appear glassy, so it is more appropriate to use the Williams-Randall-Ferry (WLF) equation to describe the change with temperature:
log(kref/k)=[-c 1(T-Tref)]/[C2+(T-Tref)]( 12.4)
Where C 1 and C2 are constants, depending on raw materials; Tref is the reference temperature, which generally refers to the glass transition temperature (glass transition temperature). According to the concept of water activity, the influence of humidity on product stability has been well confirmed (Karel, 1975). The first step is to detect the linear relationship between rate constant and water activity:
lnk=aaw ( 12.5)
It is necessary to further understand the relationship between water content m (water content per kilogram of dried spores) and water activity (under constant t), which is called adsorption isotherm. In addition, the most commonly used formula for this isotherm is as follows:
m =(moKCVaw)/{( 1-Kaw)[ 1+K(C- 1)aw]}( 12.6)
Combining the formulas (12.3) and (12.5), the relationship between rate constant k and temperature and water activity is obtained:
lnk = a 1+β/T+γaw+δaw/T( 12.7)
In this formula, a, β, γ and δ are all constants, which can be determined by statistical methods such as nonlinear regression.
The application of 12.5.5.2 mathematical model
Pedreschi et al. (1997) prepared two types of Trichoderma harzianum T. P 1 spores for shelf-life experiments: M 1 was cultured at 28℃ for 60h, and M2 was heat-shock treated at 40℃ for 90min under the same culture conditions. Two samples were filtered with glass wool to remove hyphae, and then filtered with Sartorius nitrocellulose membrane (pore size 1.2 μm) of SaiDolis to obtain spore mud. The solid was dried in a dryer containing silica gel desiccant (AW = 0.03) for 3 d, and the absolute survival rate was detected. The absolute survival rate (AV) was determined by comparing the number of colony forming units (CFU) with the total spore number (ts).
Absolute survival rate was measured. The absolute survival rate (AV) was determined by comparing the number of colony forming units (CFU) with the total spore number (ts).
The relative survival rate (v) is defined as:
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AVt in the formula is the absolute survival rate of dried spores after time t; AV0 is the absolute survival rate of dried spores.
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Where a = [- 1], β = [1-], ε =; Kb and c are equation constants; M0 is a single layer value.
The first-order kinetic equation used to describe the relationship between relative survival rate (V) and time is as follows:
lnV = 4.6 1-kt( 12. 10)
At the beginning of storage period, the relative survival rate V0 was 100, and LN (100) = 4.6 1.
The dependence of K on T and aw accords with Arrhenius semi-empirical model of areni, and the dynamic equation is as follows:
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The experimental results showed that M 1 and M2 showed high survival rate (55%) and similar trehalose content (4.0% and 5.4% respectively) after slow test. After storage 10d at different temperatures T(8℃, 33℃ and 42℃) and water activity aw(0.03, 0.33 and 0.75), there was no significant difference among the groups. When AW = 0.03, the spore survival rates at 8℃ and 33℃ were 65438 000% and 70%, respectively. When AW = 0.75 and 42℃, trehalose content and spore survival rate decreased the fastest. The spore survival rate was 100% after 52 days of heat shock at 8℃, and the trehalose content was slightly higher than that of M 1.