Determining of the mathematical model heater systems of the stem of shaft, страница 2

Schematic model of calorifer setting

Determine of mathematical model of calorifer plant

In a basis of the mathematical description of the processes running in setting lays the generalized equation of thermal balance:                                         ,

where ΔF - fluctuation of an entalpy; Q - thermal streams.

Let's break calorifer setting into three interconnected sectors with specific character of the physical phenomena in them: area of calorifer plant with established heat exchange devices; the chamber of mixture; section of a shaft from its mating to mixing chamber up to a setting point of temperature detectors (.-50 m).

The first section three tandem sections divide in thermal reservoirs is representable: a) the internal cavity of the fixed calorifers, the filled the overheated water; b) metal wall of calorifers; c) air space of calorifers.

The differential equation of a heat balance for the first reservoir looks like:

where T_sr(t) - average value of overheated water temperature, defined by the formula:

С₁ – specific heat capacity of superheated water, kcal / kg *⁰С; G₁(t) – the flowof overheated water through calorifers, kg / hour; D₁ – quantity of overheated water in an internal cavity of calorifers, kg; Т₁(t), Т₂(t) – temperature of overheated water on an input and an output from heaters, accordingly, ⁰С; Т_С(t) – temperature of a metal wall of calorifers, ⁰С; К_С – heat-transfer coefficient from water to walls of tubes of calorifers, kcal /m²*hour*⁰С; F_С – internal surface of all tubes of fixed calorifers, м²; t – current time, sec.

The differential equation of a heat balance for the second reservoir looks like:

where К_V – heat-transfer coefficient from walls of a calorifer to heated air, kcal /m²*hour*⁰С; F_V – heat-exchange surface of the fixed heaters, m²; С_С – specific heat capacity of a wall, kcal /kg*⁰С; D_С – weight of a metal wall, kg; Т^'_V, T^''_V – temperature of cold and heated air, ⁰С;

The differential equation of a heat balance for the third reservoir looks like:

where G_V(t) – the flow of air through calorifers, kg/hour; С_V  - specific heat capacity of air, kcal/kg*⁰С;  D_V – quantity of heated air in volume of calorifers, kg.

The heat-transfer coefficient from water to a wall of calorifer К_С is definable from following expression:

where λ – heat conduction coefficient of walls of a calorifer, kcal/m*hour*⁰С; d – internal diameter of tubes of calorifers, m;  - Reynolds's criterion;  - Prandtles's criterion;

where ω – rate of overheated water in tubes of calorifers, m/hour; α – thermal diffusivity of superheated water, m²/hour; ν – kinematic factor of viscosity of superheated water, m²/hour.

Heat-transfer coefficient from a wall to air:

where δ – thickness of walls of tubes of a calorifer, m; К_О – factor of fining, representing the ratio of an internal surface of tubes of a calorifer to a surface of all ribs; К – heat-transfer coefficient of a calorifer:

where a, n, m – empirical factors; ν, ρ – rate of movement of air in effective cross-section of a calorifer and its density, m/hour; кг/м³; ω – rate of movement of superheated water in section of a calorifer, m/hour.

The second section, or mixing chamber, joint with shaft, it is possible to consider practically as an instantaneous element:

where Т_sm(t) – temperature of air after the mixing chamber, ⁰С; G_max(t) – the maximal flow of air through the mixing chamber, kg/hour;

The third section - a section of shaft up to a setting point of temperature detectors can be taken into account by introduction of transportation lag: