Directive 2005/55/EC of the European Parliament and of the Council (repealed)Show full title

6. CALCULATION OF THE SMOKE VALUES

6.1. Bessel algorithm

The Bessel algorithm shall be used to compute the 1 s average values from the instantaneous smoke readings, converted in accordance with Section 6.3.1. The algorithm emulates a low pass second order filter, and its use requires iterative calculations to determine the coefficients. These coefficients are a function of the response time of the opacimeter system and the sampling rate. Therefore, Section 6.1.1 must be repeated whenever the system response time and/or sampling rate changes.

6.1.1. Calculation of filter response time and Bessel constants

The required Bessel response time (t_F) is a function of the physical and electrical response times of the opacimeter system, as specified in Annex III, Appendix 4, Section 5.2.4, and shall be calculated by the following equation:

where:

t_p

=

physical response time, s

t_e

=

electrical response time, s

The calculations for estimating the filter cut-off frequency (f_c) are based on a step input 0 to 1 in ≤ 0,01 s (see Annex VII). The response time is defined as the time between when the Bessel output reaches 10 % (t₁₀) and when it reaches 90 % (t₉₀) of this step function. This must be obtained by iterating on f_c until t₉₀-t₁₀≈t_F. The first iteration for f_c is given by the following formula:

The Bessel constants E and K shall be calculated by the following equations:

where:

D

=

0,618034

Δt

=

Ω

=

6.1.2. Calculation of the Bessel algorithm

Using the values of E and K, the 1 s Bessel averaged response to a step input S_i shall be calculated as follows:

where:

S_i-2

=

S_i-1 = 0

S_i

=

1

Y_i-2

=

Y_i-1 = 0

The times t₁₀ and t₉₀ shall be interpolated. The difference in time between t₉₀ and t₁₀ defines the response time t_F for that value of f_c. If this response time is not close enough to the required response time, iteration shall be continued until the actual response time is within 1 % of the required response as follows:

6.2. Data evaluation

The smoke measurement values shall be sampled with a minimum rate of 20 Hz.

6.3. Determination of smoke

6.3.1. Data conversion

Since the basic measurement unit of all opacimeters is transmittance, the smoke values shall be converted from transmittance (τ) to the light absorption coefficient (k) as follows:

and

where:

k

=

light absorption coefficient, m^-1

L_A

=

effective optical path length, as submitted by instrument manufacturer, m

N

=

opacity, %

τ

=

transmittance, %

The conversion shall be applied, before any further data processing is made.

6.3.2. Calculation of Bessel averaged smoke

The proper cut-off frequency f_c is the one that produces the required filter response time t_F. Once this frequency has been determined through the iterative process of Section 6.1.1, the proper Bessel algorithm constants E and K shall be calculated. The Bessel algorithm shall then be applied to the instantaneous smoke trace (k-value), as described in Section 6.1.2:

The Bessel algorithm is recursive in nature. Thus, it needs some initial input values of S_i-1 and S_i-2 and initial output values Y_i-1 and Y_i-2 to get the algorithm started. These may be assumed to be 0.

For each load step of the three speeds A, B and C, the maximum 1s value Y_max shall be selected from the individual Y_i values of each smoke trace.

6.3.3. Final result

The mean smoke values (SV) from each cycle (test speed) shall be calculated as follows:

For test speed A:

SV_A = (Y_max1,A + Y_max2,A + Y_max3,A) / 3

For test speed B:

SV_B = (Y_max1,B + Y_max2,B + Y_max3,B) / 3

For test speed C:

SV_C = (Y_max1,C + Y_max2,C + Y_max3,C) / 3

where:

Y_max1, Y_max2, Y_max3

=

highest 1 s Bessel averaged smoke value at each of the three load steps

The final value shall be calculated as follows:

SV = (0,43 x SV_A) + (0,56 x SV_B) + (0,01 x SV_C)