THE MODELING OF THE HARDENING VELOCITY By Ph.D. Igor Bobin

The hardening is an important physical process at the industrial production. The most famous examples are pelletizing of iron ore or briquetting of coal and peat. We spoke about the hardening kinetics in the articles [1-3].
Analytical curves of the hardening kinetics for the ore briquetting are shown in Fig.1.

Fig 1. The hardening kinetic curves obtained at the modeling [1-3] by formulas                
(1) - artificial desiccation and (2) - native desiccation Time is presented in seconds.

Now we shall deal with hardening velocity or hardening rate. What is that?

Hardening velocity (rate) vS is ratio of the compressive strength of specimen Rs to unit of time t. Hardening velocity is derivative of time vS = dRs /dt  (Pa / sec).

Our modeling approach to the hardening kinetics allows carry direct calculation of the hardening velocity, pioneering move. Formulas (1) and (2) of the hardening kinetics are presented in articles [1-3].

I have proposed the following my own formula with delay for the direct analytical calculations of the hardening velocity vS (the differential equation in the operational form of Laplace).

The parameters Rs max, rs0, τ and T are determined from an experimental data of the hardening process.

The mathematical model (3) of the hardening velocity vS of the first order with delay is Transfer Function W(s), which is convenient for modeling using MATLAB. The mathematical model (3) has enough accuracy for engineering calculations.

The parameters Rs max, rs0, τ, τtr and T2 are determined from an experimental data of the hardening process.

The mathematical model (4) of the hardening velocity vc of the second order with delay is Transfer Function W(s), which is convenient for modeling using MATLAB. The mathematical model (4) provides very high accuracy of scientific calculations and engineering.

Analytical curves of the hardening velocity are shown in Fig. 2 and Fig. 3.

Fig 2. The hardening velocity curves obtained at the modeling by formulas                
(3) - artificial desiccation and (4) - native desiccation. Time is presented in seconds.

Fig 3. The hardening velocity curves obtained at the modeling by formulas                
(3) - artificial desiccation and (4) - native desiccation. Time is presented in hours.

The use of I. Bobin's operational formulas of the hardening kinetics and velocity allows:
1. Reduce the cost of obtaining products with the required strength.
2. Effectively manage the process of hardening of products.
3. Optimize the process of hardening finished products and semi-finished products.
4. Reduce the number of production areas.
5. Reduce the cost of drying and hardening products.

In this manner the modeling of hardening kinetics is an indispensable tool for analisis of the mineral technology and other. The immediate analytical description and a visual representation of the hardening velocity in time are very important for solving optimization problem of production processing. The formulas of the hardening velocity can be used everywhere with success where a product (or semi-product) acquires strength over time.
MINERAL MODELING 

© Ph.D. Igor Bobin
September 24, 2017
bobin.igor@yahoo.com