**2. Materials and Methods**

*2.1. Theoretical Basis of the Conducted Research*

2.1.1. Generalization of Air Distribution in Grate Furnaces

The course of the combustion process depends mainly on the initial composition of the fuel, its initial amount, the time fuel stays on the grate (measured from the beginning of the combustion process), and the distribution of air fed to the furnace. In the case of a furnace with a movable grate, the time of fuel residence on the grate, at a constant grate move speed, determines the location (position) of the fuel on the grate:

$$l = w \ t\_\prime \tag{1}$$

where:

*l*—fuel location on the grate, distance from the point of fuel supply onto the grate, m; *w*—grate movement speed, m/min;

*t*—fuel-stay time on the grate, min.

In the literature [3,25,29], information can be found on the applied experimental distribution of combustion air supplied to the grate. The authors of the present study in [24] defined the following generalized function of air distribution in grate furnaces:

$$\dot{V}(\mathbf{x}) = a \left( 1 - \frac{\mathbf{x}}{\mathbf{X}} \right) \mathbf{x} \exp \left( b \frac{\mathbf{x}}{\mathbf{X}} \right) \tag{2}$$

where:

*a*—empirical coefficients, m3/min2 or m3/m2;

*b*—empirical coefficients, dimensionless quantity;

*<sup>x</sup>*, *<sup>X</sup>*—variable specifying the fuel condition on the grate and its maximum value; .

*V*(*x*)—air stream corresponding to the variable *x*;

with:

• in furnaces with a fixed grate, the following should be assumed: *x* = *t*; *X* = *T*, min,

• in a furnace with a movable grate: *x* = *l*; *X* = *L*, m,

where:

*T*—maximum fuel-stay time on the grate, min.,

*L*—active length of the movable grate, m.

The function . *<sup>V</sup>*(*x*) should satisfy the following conditions: . *<sup>V</sup>*(*<sup>x</sup>* <sup>=</sup> <sup>0</sup>) <sup>=</sup> 0; . *V*(*x* = *X*) = 0. The form of the function is characterized by empirical coefficients. The numerical value of the factor b follows from the condition:

$$\frac{d\dot{V}(\mathbf{x})}{d\mathbf{x}} / \iota\_{\mathbf{x}=\mathbf{R}\mathbf{X}} = \mathbf{0},\tag{3}$$

where:

*R*—relative value of the variable *x* which defines the maximum stream of supplied air, referenced to the value of *X*; a dimensionless quantity.

In turn, the numerical value of the coefficient "*a*" determines the relationship:

$$V\_{ad} = \int\_{\mathbf{x}=0}^{\mathbf{x}=X} \dot{V}(\mathbf{x})d\mathbf{x} \tag{4}$$

where:

*Vad*—total demand for combustion air resulting from the initial composition and amount of fuel and from the ratio of excess air, m3.

Using Equations (3) and (4) we obtain:

$$b = \frac{1 - 2R}{R(R - 1)},\tag{5}$$

$$a = \frac{V\_{ad}}{X^2 \cdot (Y\_1 - Y\_2)} \,\text{}\tag{6}$$

where:

$$Y\_1 = \frac{1}{b^2} [(b-1)\exp(b) + 1] \tag{7}$$

$$Y\_2 = \left(\frac{1}{b} - \frac{2}{b^2} + \frac{2}{b^3}\right) \exp(b) - \frac{2}{b^3},\tag{8}$$

When analyzing the system of Equations (2), (5) and (6), it can be seen that the . *V*(*x*) curves of air distribution in grate furnaces are explicitly determined by the value of the variable *x* = *R* and by the total combustion air demand *Vad*.
