2.1.2. Determination of Total Emissions of Gaseous Combustion Products

The calculation dependencies given in the further part of the study apply to furnaces with a fixed grate (*x* = *t*). They are also correct for furnaces with a mechanical grate, by taking into account Equation (1) and by substituting the variable l instead of the variable *t*.

The emission of gaseous products is determined from the dependence:

$$
\dot{m}\_{\text{j,R}}(t) = \dot{V}\_{sp}(t)e\_{\text{j,R}}(t) \approx \dot{V}(t)e\_{\text{j,R}}(t) \tag{9}
$$

where:.

*mj*,*R*(*t*)—emission of the *j*-th gaseous product at time *t* for the *R*-variant of air separation stream, g/min or mg/min,

with: *<sup>j</sup>* <sup>∈</sup> [CO2, CO, SO2, NOx]; .

*Vsp*(*t*) —dry exhaust gas stream at time *t* for *R*-th variant of air separation, m3/min; *ej*,*R*(*t*)—concentration of the *j*-th component in the dry exhaust gas, for the *R*-th variant of air separation at time *t*, g/m3 or mg/m3.

In view of the difficulties with measuring the flue gas flow, it was assumed: . *Vsp*(*t*) = . *V*(*t*). This problem is discussed in more detail in Section 2.1.3.

The air distribution is zonal (see Section 2.4). The flow of air . *Vi* in the *i*-th zone is constant but different between specific zones. The value of the air stream in the *i*-th zone is determined using Equation (2), yielding:

$$
\dot{V}\_{i,R} = \frac{1}{t\_{i,k} - t\_{i,p}} \int\_{t\_{i,p}}^{t\_{i,k}} \dot{V}\_R(t)dt\_\prime \tag{10}
$$

where:

*ti*,*p*, *ti*,*k*—start and end value of time *<sup>t</sup>* limiting the *<sup>i</sup>*-th feed zone of air, min., *<sup>i</sup>* = (1 . . . *<sup>I</sup>*). .

*Vi*,*R*—air stream in the *i*-th delivery zone, for the *R*-th variant of its distribution, m3/min.

Air stream is a quantity that controls the combustion process. Its value results from the distribution function (2). The compliance of the above air stream with the assumptions should be controlled by measurement.

The concentrations of the *i*-th components of exhaust gas *ej*,*R*(*t*) are also measured. Since the analyzer used to measure the concentrations shows average values in a specific time step, the emission stream of exhaust gas components can be determined from the following relation: .

$$
\dot{m}\_{\text{j,R,i,n}} = \dot{V}\_{\text{i,R}} e\_{\text{j,R,i,n}} \tag{11}
$$

where:.

*mj*,*R*,*i*,*<sup>n</sup>* —average emission stream of the *j*-th product, with *R*-th air separation, in the *n*-th time step, g/min or mg/min;

*ej*,*R*,*i*,*<sup>n</sup>* —average concentration of the *j*-th exhaust component in the *n*-th time step, of the *i*-th zone and the *R*-th variant of air supply, g/m<sup>3</sup> or mg/m3;

and:

*n* = (1 . . . *N*)

where:

*N*—number of time steps in the *i*-th air supply zone.

The total emissions of gaseous combustion products can be determined from the formula:

$$m\_{\mathbf{j},\mathbf{R}} = \int\_{t=0}^{t=T} \dot{m}\_{\mathbf{j},\mathbf{R}}(t)dt = \Delta t \sum\_{n=1}^{n=N} \dot{m}\_{\mathbf{j},\mathbf{R},\mathbf{k}} = \Delta t \sum\_{i=1}^{i=I} \dot{V}\_{i,\mathbf{R}} \sum\_{n=1}^{n=N} e\_{\mathbf{j},\mathbf{R},\mathbf{j},\mathbf{n}} \tag{12}$$

where:.

*mj*,*R*—total emission of the *<sup>j</sup>*-th product, with *<sup>R</sup>*-th air separation, g or mg; .

*mj*,*R*(*t*)—emission stream of the *j*-th product, with *R*-th air separation, at time *t*, g/min or mg/min;

Δ*t*—time step, min.

2.1.3. Determination of the Relative Uncertainty of the Assessment of Gaseous Emissions of Combustion Products

The relative uncertainty of the assessment of the total emissions of gaseous products (exhaust components) can be derived from the formula:

$$\frac{\delta\_{m\_{j,R}}}{m\_{j,R}} = \sqrt{\left(\frac{\delta\_{\Delta t}}{\Delta t}\right)^2 + \left(\frac{\delta\_{\dot{V}}}{\dot{V}}\right)^2 + \left(\sum\_{i=1}^{i=I} N\_i\right)\left(\frac{\delta\_\varepsilon}{\varepsilon\_m}\right)^2} \tag{13}$$

where:

*δmj*,*<sup>R</sup>* —absolute error of the total emission assessment of the *j*-th exhaust component for the *R*-th variant of air separation, g or mg;

*δ*Δ*t*, *δ* . *<sup>V</sup>*, *δe*—absolute errors in the assessment of the time step, dry exhaust gas stream, exhaust gas component concentration, respectively: min, m3/min, g/min or mg/min.

Taking into account the assumptions that the air stream and the dry flue gas stream are equal, as specified in Section 2.1.2, one should take into account the error of the method. Thus, the value *<sup>δ</sup>* . *V*. *<sup>V</sup>* in the Equation (10) can be determined from the formula:

$$\frac{\delta\_{\dot{V}}}{\dot{V}} = \sqrt{\left(\frac{\delta\_{ex}}{\dot{V}}\right)^{2} + \left(\frac{\delta\_{m}}{\dot{V}}\right)^{2}},\tag{14}$$

where:

*δex*, *δm*—absolute error of the determination method of dry flue gas stream and the measurement of air stream, m3/min.

In order to determine the error of the method, stoichiometric calculations were carried out to determine the ratio α of the air stream to the dry flue gas stream, for exemplary fuels and exemplary values of excess air ratio λ. The obtained values are given in Table 1.

As the table shows, the error of the method for the analyzed exemplary solid fuels for λ = 1.0 is less than 4%, and it decreases with the increase of the excess air ratio. For the value λ = 1.6, it does not exceed 2.5%. For the conducted research, it was 1.6%.

The remaining uncertainty assessments are based on relative measurement errors. For the presented studies, they were less than 6%.

The relative uncertainty of the total emission of gaseous combustion products, determined according to the Equation (13), was less than 8%.


**Table 1.** Calculated values of α (ratio of air stream to dry exhaust gas stream) for the exemplary fuels and for the values λ = 1.0; 1.4; 1.6 (λ—excess air ratio).

\*—fuel used in the presented research.

#### 2.1.4. Determination of the Share of Combustible Parts in the Slag

In order to determine the proportion of combustible parts in the slag, the slag was collected from the furnace chamber, from which, three samples weighing about 10 g were collected after it was homogenized. In the samples, the combustible parts were determined in accordance with PN-Z-15008-03:1993 [40]. The share of combustible parts was accepted as the average value of these determinations. The average value of these determinations was accepted as the share of combustible parts.

As to the share of combustible parts in the slag, the determination uncertainty of this value was 5%. This value was based on the determinations carried out in an additional series of 7 samples taken from the furnace chamber carried out for one of the tests. This value was defined as the quotient of the standard deviation related to the mean value.
