- Email: [email protected]
MSWSampling - Software Tool for Solid Wastes Sampling
Copyright e IFAC Automation in Mining, Mineral and Metal Processing, Tokyo, Japan, 2001
MSWSAMPLING - SOFIWARE TOOL FOR SOLID WASTES SAMPLING
Fernando Durio, M. Teresa Carvalho, ADa Soares and Eisa Agante
CVRM - Centro de Geo-sistemas Instituto Superior Tecnico Av. Rovisco Pais, 1000-049 Lisboa, PORTUGAL e-mail: [email protected]
Abstract: The paper describes a software tool, called MSWSampling, to support the evaluation of the solid wastes sampling quality. The software is a random sample generator program that simulates the sampling process of solid wastes, calculating the size of the sample that assures a desired representativeness level with respect to the content of a critical component of the lot to be sampled or the variance of the fundamental error associated to a sample of given size. The solid wastes consists of very different active classes of solid components, which are characterized by their size distribution, specific gravity, weight proportions and shape factors. The results of a case study, the municipal solid wastes that feed a composting plant nearby Lisbon, are presented. Copyright 0 2001 IFAC Keywords: Sampling, Software Tool, Data Processing, Solid Wastes
I.INTRODUCTION
The sampling error is mainly due to the material heterogeneity in terms of constitution, giving rise to the fundamental error (FE), and in terms of time/space distribution, generating the grouping and segregation errocs. I The former one is an intrinsic characteristic of the lot that cannot be canceled. The second one is a function of the fundamental error, of the time/space distribution of the critical component and of the observation scale, i. e., of the sample volume.
The control of processes and circuits is based on variables that are directly measured or estimated. Frequently, those values are taken from samples. Sampling is the operation that consists in the selection, delimitation and extraction, respecting a certain number of rules, of several increments from an heterogeneous lot in order to obtain a batch of matter (sample), which is analyzed to measure (estimate) the critical content of one or several active components.
As the segregation and grouping error is at most equal to the fundamental error, the problem of studying the properties of sampling error is centered around the study of the properties of the fundamental error.
The sampling error, given by the difference between the unknown content of the lot and the measured content of the sample, can not be known, but the statistical properties (moments of the distribution) of that erroc can be estimated and they were fully studied by Pierre Gy for one and zero-dimensional lots. The sampling erroc is mainly the sum of the fundamental (FE) and grouping and segregation errocs (GE), as long as the sampling operation is correct, that is, selection of point increments is probabiIistic with selection probability constant and the delimitation and extraction operations are carried out properly as described in the books of Pierre Gy (1982, 1988) and Francis Pittard (1993).
For all practical purposes, the sampling process must fulfill two properties: correctness (a priori) and rqxesentativeness (a pa;kriori). The sampling process oc the sample, assumed correct, is representative if the
The other errors are mainly due to operational causes. Ifthe human action is correct and the devices are well designed and operated, those errors may be ignored or minimized. 397
variance, equal to first order to the mean square, of the fimdamental error is less or equal to half of the given representativeness level.
takes into 8000mlt the size, mineralogical composition, liberation degree and shape of the particles. This constant is given by
C = f g 1 m d3
There are today some methodologies for solid wastes (SW) characterization (e. g. Waste Analysis Procedure, European Recovery and Recycling Association). As fur as the authors know, those methodologies have not a sound scientific basis in what concerns sampling representativeness. The minimum size of samples to be obtained is not theoretically justified and the quantified sample representativeness is not considered. This can be a very important drawback of those methodologies.
(2)
where f is a shape factor (in ores,j=O.2 for gold ores andj=().5 for fiat ores); g is a constant related with the size distribution (g=O.25 for most ores and g=O.5 for calibrated ores); I is a constant related with the bDetatioo degree of the critical component (/=0 for non bDerated ores and 1= 1 for completely liberated ores, in most cases, 1=0.2); m is a constant related with the constitution of the sampled lot (expressing the mineralogical composition); d is the largest particle size.
In the recycling activity, for instance, it is common that both buyers and sellers of recyclable materials check the conformity of the same lot with the predefined specifications regarding certain components. If there is not a common, theoretically based, methodology for sampling and an equal value of sample representativeness, there is place for disagreement.
Assuming that the lot weight is much larger than the sample weight (ML » Ms), the size of the sample, given the variance of the fimdamental error, r(FE) , for the content of critical component, is
C
M =-s s2(FE)
The well known relationship between the variance of the fundamental error and the mass of the sample, developed by Pierre Gy for particulate material of mineral origin, is not applicable to the sampling of solid wastes because of the different shapes factors, size and specific gravities of the component classes.
(3)
2
For a specified representativeness level, 0 0 , the
condition2s2(FE)~0~ implies that the minimum sample size must be at least 2Ms.
This papa- describes a software tool - MSWSampling inspired on the theory of P. Gy, that permits to compute the variance of the fimdamental error for a given sample size, in the particular case of solid wastes. It is a computational statistics program that simulates the sampling process of a lot of SW, previously characterized, calculating the minimum size of the sample that assures a desired representativeness level with respect to the aitical component or evaluates the error variance associated with the size of the sample collected.
2. FUNDAMENTAL ERROR
The basic equation (2), developed by Gy taking into account some specific regularities observed in the results of the si.m-density analysis of particulate matter of mineral origin (the so called ores), can not be used to calculate the constant C of a critical solid waste component. In municipal solid wastes (MSW), the characterization methodologies define about 30 completely different components such as organic residues, packages of different polymers, cans, newspapers, etc. The constant C must, therefore, be calculated for the diffef'eDt canponents considering the components shape, specific gravity and size distribution.
P. Gy, taking into account the theoretical statistical models, developed the sampling basic equation, defined as follows:
3. THEORETICAL FOUNDATIONS OF THE METHODOLOGY
s2(FE) = C (_1 - _1 ) Ms ML
(1)
The methodology developed is based on sampling the multinomial distribution, the probability model associated to sampling with replacement from a finite multi-classes population (Johnsoo et al, 1997). As a matter of fact, the correct probability model is the multivariate hypergeometric distribution associated to sampling without replacement. However, when the ~oo si1Je, NL , is not known exactly but it is very large compared to the sample size, n, it seems reasonable that sampling without replacement is not
In equation (1), ~(FE) is the variance of the fundamental error, corresponding to the critical component, Ms and ML are the masses of the sample and of the lot, respectively, and C (units of mass) is a constant, recently named heterogeneity invariant of the lot, related with the characteristics of the ore, that
398
too much different than sampling with replacement, and hence the multivariate hypergeometric distribution should be well-approximated by the multinomial.
and p' the selection probability.
Consider a multi-classes population of Wlits, in which each unit is one of k classes. Denoting by p; (i = 1,2, ... , k) the probability of occurrence of class i units, the joint probability of the number (random variables) of class i units in one sample, N; (i=I,2, ... ,k), in a series of n independent trials, is given by (Johnson, et al,I997):
The software developed (Duriio, 2000) is a program based on the Monte Carlo simulation technique (Fishman, 1996) that permits to compute the representativeness of samples, by generating a large number of random samples of size n from a multinomial distribution with previously estimated parameters using experimental data acquired in a previous simple characterization of the sw.
4. MSWSAMPLING SOFfWARE TOOL
The parameters of the program are: I) the probabilities of occurrence of the k classes of components, 2) the shape factors for the different components, 3) the size distriOOtioo of each class and 4) the specific gravities of the k component classes.
(4)
n;~O,
k
k
;cl
;_1
L n;=n, p;~O, LP;=I
The inputs of the program are: I) the components designation, 2) the number of samples to be generated, 3) the number of units per sample and 4) the name of the output file of the simulation results.
This is the multinomial distribution with parameters (n, PI>Jl2, ..... ,pJ. The probabilities of occurrence (numeric concentrations) can be estimated from the weights, volumes, shape factors, specific gravities and size distributions of the k classes present in one preliminary sample and using the maximum likelihood method.
The outputs of the program are 1) the average weight of the sample and corresponding absolute and relative standard deviation, and 2) the weight percentage of the components and corresponding absolute and relative standard deviation. It is displayed, as well, the histogram of the weight percentage of the selected active component and the corresponding theoretical normal density function (see fig. I).
The procedure to generate random samples of size n from the multinomial distribution is done by using a chain of binomial random variables, as follows (Johnson et ai, 1997, Fishman, G. S., 1996):
When the user modifies the sample size, the program recalculates the weight of the sample and the fundamental error's absolute and relative standard deviation.
N] as a binomial (n,p]) variable, N2 as a binomial (n-Nl>pj(l-p]» variable, then
N3asa binomial (n-N]-N2,pJ(l-P]-P2» variable, and soon.
5. CASE STUDY The software was used in the sampling of the MSW that feed a composting plant situated nearby Lisbon, treating MSW from an area corresponding to about one million people. The MSW were first characterized in 28 classes of components. For this characterization, the MSW that feed the plant were sampled with a 45 minutes sampling period. The samples were weighed and characterized. Each one of the components was separated by size (using screens of20, 40, 80 and 120 mm of aperture). For each size fraction, it was detamined the weight, the volume and the number of objects, taking into account the average shape and volume of the elements.
After generating the number of class i Wlits in the sample, NI> N1 , ••• ..Nt, the number of class i Wlits in size fraction j is generated by sampling the multinomial distriOOtion with parametes (N;, PiI' P/1> ... ,p;ItC) with PiI' Pj], ... ,p;1tC the probabilities of occurrence of ne size classes of class i units. Then, the numbers of class (iJ) units are converted to masses using the shape factors and specific gravities. Finally, the content of a given active component is obtained by dividing the mass of class i Wlits by the mass of all class units.
The sample size, n, can be user-defined or it can be generated from sampling a binomial random variable with parameters (Nb p') with NL the population size
399
Histograma de teores das amostras 35r----....,.----.----~--__,_--___,
Jornais
30 ------- ~ ------- :-----
Re-.istasIlMos Cartao
I
Caitao canelado Telrapack Outros papeis
I
I
I
I
,
,
I
I
---- ~ - -- ---, I
I
I
--- --- -------------
PEA[)
------- ,I~ -------~ ,, -
PVC
PET
, , , , , -------~-------~ , , , _______ l... ___ _ , ,
Fame pIastico Tubos/copos.Ivasos _I Outros pI6stic:os _0_" _embI!l. 0_ ..=.J ~
5
- - -----~---
!.-----,, ,
o4
Peao an,ab~ lkp)
reo( pc.,cle,~ p;)
4.5
5 5.5 Tear ponderal (%)
516.9sss 1
6
6.5
3.3551
J
0.3010
Fig. I. View of the user interface The MSWSamp/ing software tool was used and several simulatioos were made. Fig. 2 shows the relationship between the sample weight and variance of the fimdamental error, considering the organic residues of the MSW as the critical component. The coefficients of the simple linear regression model, In(Ms)=-c(xln[s2(FE)]+In(C), were adjusted to the simulated results (diamond markers in the graphic), giving the final relationship:
In Ms = -0.934 In s2(FE)
+
Considering both the fundamental and grouping and segregation errors, but neglecting all other sources of errors, a sample of 10kg should be obtained. However, if the tolerated error is 10%, a sample of 40kg should be prepared.
6. CONCLUSIONS A software tool developed to help the evaluation of the quality of the sampling of solid wastes was presented. The program computes the size of the sample that assures a desired representativeness level with respect to a critical component of the lot to be sampled or it calculates the error variance associated with a predefined sample size.
In (0.0493) (5)
The constant C was estimated for all of the components. Values of C range from 0.04 Kg (Fines fraction) to 18.84 Kg (PVC fraction).
Results of a case study, the sampling of municipal solid wastes that feed a composting plant nearby Lisbon, are presented. The software has a wider applicability than the one presented in the case study. The authors are presently working in its application to circuits sampling, in the calculation of plant efficiencies and the control.
The size of the sample that assures a desired representativeness with respect to a critical component can now be calculated. Supposing that the critical component for the composting plant is the organic residues and that the desired sample representativeness level is 20010, a sample with a minimum weight of 5kg should be collected.
400
:= 7.0 ell :!!. 6.0
i
5.0 ~ 4.0 Q) 3.0 Q. 2.0 E : 1.0 .s 0.0 + - - - - - - , - - - - - , - - - - - - - - - , - - - - - , - - - - - - - , 5.0 6.0 7.0 8.0 9.0 10.0
-In Fundcmental &l'or V.-lance Fig.2.Minimum sample size as a function of the fundamental error's variance (simulated results and regression analysis function) considering the organic fraction the critical component.
ACKNOWLEDGEMENTS The authors wish to acknowledge TRATOLIXO for their support in the experimental work and for the permission to publish the results presented in the paper.
REFERENCES
Durio, F. (2000). MSW User's Guide. Fishman, G. S. (1996). Monte Carlo - Concepts, Algorithms and Applications, Sprioger-Verlag. Gy, P. (1982). Sampling of Particulate Materials. Theory and Practice, Elsevier Scientific. Gy, P. (1988). Heterogeneite, Echantilonnage, Homogeneisation, Masson. Johnsoo, N.L., K~ S., Balakrishnan (1997). Discrete Multivariale Distributions, John Wiley and Sons, Inc. Pitard, F. (1993). Pierre Gys Sampling Theory and Sampling Practice, Second Edition, CRC Press Waste Analysis Procedure, ERRA (European Recovery & Recycling Association), 1993.
401
MSWSAMPLING - SOFIWARE TOOL FOR SOLID WASTES SAMPLING
Fernando Durio, M. Teresa Carvalho, ADa Soares and Eisa Agante
CVRM - Centro de Geo-sistemas Instituto Superior Tecnico Av. Rovisco Pais, 1000-049 Lisboa, PORTUGAL e-mail: [email protected]
Abstract: The paper describes a software tool, called MSWSampling, to support the evaluation of the solid wastes sampling quality. The software is a random sample generator program that simulates the sampling process of solid wastes, calculating the size of the sample that assures a desired representativeness level with respect to the content of a critical component of the lot to be sampled or the variance of the fundamental error associated to a sample of given size. The solid wastes consists of very different active classes of solid components, which are characterized by their size distribution, specific gravity, weight proportions and shape factors. The results of a case study, the municipal solid wastes that feed a composting plant nearby Lisbon, are presented. Copyright 0 2001 IFAC Keywords: Sampling, Software Tool, Data Processing, Solid Wastes
I.INTRODUCTION
The sampling error is mainly due to the material heterogeneity in terms of constitution, giving rise to the fundamental error (FE), and in terms of time/space distribution, generating the grouping and segregation errocs. I The former one is an intrinsic characteristic of the lot that cannot be canceled. The second one is a function of the fundamental error, of the time/space distribution of the critical component and of the observation scale, i. e., of the sample volume.
The control of processes and circuits is based on variables that are directly measured or estimated. Frequently, those values are taken from samples. Sampling is the operation that consists in the selection, delimitation and extraction, respecting a certain number of rules, of several increments from an heterogeneous lot in order to obtain a batch of matter (sample), which is analyzed to measure (estimate) the critical content of one or several active components.
As the segregation and grouping error is at most equal to the fundamental error, the problem of studying the properties of sampling error is centered around the study of the properties of the fundamental error.
The sampling error, given by the difference between the unknown content of the lot and the measured content of the sample, can not be known, but the statistical properties (moments of the distribution) of that erroc can be estimated and they were fully studied by Pierre Gy for one and zero-dimensional lots. The sampling erroc is mainly the sum of the fundamental (FE) and grouping and segregation errocs (GE), as long as the sampling operation is correct, that is, selection of point increments is probabiIistic with selection probability constant and the delimitation and extraction operations are carried out properly as described in the books of Pierre Gy (1982, 1988) and Francis Pittard (1993).
For all practical purposes, the sampling process must fulfill two properties: correctness (a priori) and rqxesentativeness (a pa;kriori). The sampling process oc the sample, assumed correct, is representative if the
The other errors are mainly due to operational causes. Ifthe human action is correct and the devices are well designed and operated, those errors may be ignored or minimized. 397
variance, equal to first order to the mean square, of the fimdamental error is less or equal to half of the given representativeness level.
takes into 8000mlt the size, mineralogical composition, liberation degree and shape of the particles. This constant is given by
C = f g 1 m d3
There are today some methodologies for solid wastes (SW) characterization (e. g. Waste Analysis Procedure, European Recovery and Recycling Association). As fur as the authors know, those methodologies have not a sound scientific basis in what concerns sampling representativeness. The minimum size of samples to be obtained is not theoretically justified and the quantified sample representativeness is not considered. This can be a very important drawback of those methodologies.
(2)
where f is a shape factor (in ores,j=O.2 for gold ores andj=().5 for fiat ores); g is a constant related with the size distribution (g=O.25 for most ores and g=O.5 for calibrated ores); I is a constant related with the bDetatioo degree of the critical component (/=0 for non bDerated ores and 1= 1 for completely liberated ores, in most cases, 1=0.2); m is a constant related with the constitution of the sampled lot (expressing the mineralogical composition); d is the largest particle size.
In the recycling activity, for instance, it is common that both buyers and sellers of recyclable materials check the conformity of the same lot with the predefined specifications regarding certain components. If there is not a common, theoretically based, methodology for sampling and an equal value of sample representativeness, there is place for disagreement.
Assuming that the lot weight is much larger than the sample weight (ML » Ms), the size of the sample, given the variance of the fimdamental error, r(FE) , for the content of critical component, is
C
M =-s s2(FE)
The well known relationship between the variance of the fundamental error and the mass of the sample, developed by Pierre Gy for particulate material of mineral origin, is not applicable to the sampling of solid wastes because of the different shapes factors, size and specific gravities of the component classes.
(3)
2
For a specified representativeness level, 0 0 , the
condition2s2(FE)~0~ implies that the minimum sample size must be at least 2Ms.
This papa- describes a software tool - MSWSampling inspired on the theory of P. Gy, that permits to compute the variance of the fimdamental error for a given sample size, in the particular case of solid wastes. It is a computational statistics program that simulates the sampling process of a lot of SW, previously characterized, calculating the minimum size of the sample that assures a desired representativeness level with respect to the aitical component or evaluates the error variance associated with the size of the sample collected.
2. FUNDAMENTAL ERROR
The basic equation (2), developed by Gy taking into account some specific regularities observed in the results of the si.m-density analysis of particulate matter of mineral origin (the so called ores), can not be used to calculate the constant C of a critical solid waste component. In municipal solid wastes (MSW), the characterization methodologies define about 30 completely different components such as organic residues, packages of different polymers, cans, newspapers, etc. The constant C must, therefore, be calculated for the diffef'eDt canponents considering the components shape, specific gravity and size distribution.
P. Gy, taking into account the theoretical statistical models, developed the sampling basic equation, defined as follows:
3. THEORETICAL FOUNDATIONS OF THE METHODOLOGY
s2(FE) = C (_1 - _1 ) Ms ML
(1)
The methodology developed is based on sampling the multinomial distribution, the probability model associated to sampling with replacement from a finite multi-classes population (Johnsoo et al, 1997). As a matter of fact, the correct probability model is the multivariate hypergeometric distribution associated to sampling without replacement. However, when the ~oo si1Je, NL , is not known exactly but it is very large compared to the sample size, n, it seems reasonable that sampling without replacement is not
In equation (1), ~(FE) is the variance of the fundamental error, corresponding to the critical component, Ms and ML are the masses of the sample and of the lot, respectively, and C (units of mass) is a constant, recently named heterogeneity invariant of the lot, related with the characteristics of the ore, that
398
too much different than sampling with replacement, and hence the multivariate hypergeometric distribution should be well-approximated by the multinomial.
and p' the selection probability.
Consider a multi-classes population of Wlits, in which each unit is one of k classes. Denoting by p; (i = 1,2, ... , k) the probability of occurrence of class i units, the joint probability of the number (random variables) of class i units in one sample, N; (i=I,2, ... ,k), in a series of n independent trials, is given by (Johnson, et al,I997):
The software developed (Duriio, 2000) is a program based on the Monte Carlo simulation technique (Fishman, 1996) that permits to compute the representativeness of samples, by generating a large number of random samples of size n from a multinomial distribution with previously estimated parameters using experimental data acquired in a previous simple characterization of the sw.
4. MSWSAMPLING SOFfWARE TOOL
The parameters of the program are: I) the probabilities of occurrence of the k classes of components, 2) the shape factors for the different components, 3) the size distriOOtioo of each class and 4) the specific gravities of the k component classes.
(4)
n;~O,
k
k
;cl
;_1
L n;=n, p;~O, LP;=I
The inputs of the program are: I) the components designation, 2) the number of samples to be generated, 3) the number of units per sample and 4) the name of the output file of the simulation results.
This is the multinomial distribution with parameters (n, PI>Jl2, ..... ,pJ. The probabilities of occurrence (numeric concentrations) can be estimated from the weights, volumes, shape factors, specific gravities and size distributions of the k classes present in one preliminary sample and using the maximum likelihood method.
The outputs of the program are 1) the average weight of the sample and corresponding absolute and relative standard deviation, and 2) the weight percentage of the components and corresponding absolute and relative standard deviation. It is displayed, as well, the histogram of the weight percentage of the selected active component and the corresponding theoretical normal density function (see fig. I).
The procedure to generate random samples of size n from the multinomial distribution is done by using a chain of binomial random variables, as follows (Johnson et ai, 1997, Fishman, G. S., 1996):
When the user modifies the sample size, the program recalculates the weight of the sample and the fundamental error's absolute and relative standard deviation.
N] as a binomial (n,p]) variable, N2 as a binomial (n-Nl>pj(l-p]» variable, then
N3asa binomial (n-N]-N2,pJ(l-P]-P2» variable, and soon.
5. CASE STUDY The software was used in the sampling of the MSW that feed a composting plant situated nearby Lisbon, treating MSW from an area corresponding to about one million people. The MSW were first characterized in 28 classes of components. For this characterization, the MSW that feed the plant were sampled with a 45 minutes sampling period. The samples were weighed and characterized. Each one of the components was separated by size (using screens of20, 40, 80 and 120 mm of aperture). For each size fraction, it was detamined the weight, the volume and the number of objects, taking into account the average shape and volume of the elements.
After generating the number of class i Wlits in the sample, NI> N1 , ••• ..Nt, the number of class i Wlits in size fraction j is generated by sampling the multinomial distriOOtion with parametes (N;, PiI' P/1> ... ,p;ItC) with PiI' Pj], ... ,p;1tC the probabilities of occurrence of ne size classes of class i units. Then, the numbers of class (iJ) units are converted to masses using the shape factors and specific gravities. Finally, the content of a given active component is obtained by dividing the mass of class i Wlits by the mass of all class units.
The sample size, n, can be user-defined or it can be generated from sampling a binomial random variable with parameters (Nb p') with NL the population size
399
Histograma de teores das amostras 35r----....,.----.----~--__,_--___,
Jornais
30 ------- ~ ------- :-----
Re-.istasIlMos Cartao
I
Caitao canelado Telrapack Outros papeis
I
I
I
I
,
,
I
I
---- ~ - -- ---, I
I
I
--- --- -------------
PEA[)
------- ,I~ -------~ ,, -
PVC
PET
, , , , , -------~-------~ , , , _______ l... ___ _ , ,
Fame pIastico Tubos/copos.Ivasos _I Outros pI6stic:os _0_" _embI!l. 0_ ..=.J ~
5
- - -----~---
!.-----,, ,
o4
Peao an,ab~ lkp)
reo( pc.,cle,~ p;)
4.5
5 5.5 Tear ponderal (%)
516.9sss 1
6
6.5
3.3551
J
0.3010
Fig. I. View of the user interface The MSWSamp/ing software tool was used and several simulatioos were made. Fig. 2 shows the relationship between the sample weight and variance of the fimdamental error, considering the organic residues of the MSW as the critical component. The coefficients of the simple linear regression model, In(Ms)=-c(xln[s2(FE)]+In(C), were adjusted to the simulated results (diamond markers in the graphic), giving the final relationship:
In Ms = -0.934 In s2(FE)
+
Considering both the fundamental and grouping and segregation errors, but neglecting all other sources of errors, a sample of 10kg should be obtained. However, if the tolerated error is 10%, a sample of 40kg should be prepared.
6. CONCLUSIONS A software tool developed to help the evaluation of the quality of the sampling of solid wastes was presented. The program computes the size of the sample that assures a desired representativeness level with respect to a critical component of the lot to be sampled or it calculates the error variance associated with a predefined sample size.
In (0.0493) (5)
The constant C was estimated for all of the components. Values of C range from 0.04 Kg (Fines fraction) to 18.84 Kg (PVC fraction).
Results of a case study, the sampling of municipal solid wastes that feed a composting plant nearby Lisbon, are presented. The software has a wider applicability than the one presented in the case study. The authors are presently working in its application to circuits sampling, in the calculation of plant efficiencies and the control.
The size of the sample that assures a desired representativeness with respect to a critical component can now be calculated. Supposing that the critical component for the composting plant is the organic residues and that the desired sample representativeness level is 20010, a sample with a minimum weight of 5kg should be collected.
400
:= 7.0 ell :!!. 6.0
i
5.0 ~ 4.0 Q) 3.0 Q. 2.0 E : 1.0 .s 0.0 + - - - - - - , - - - - - , - - - - - - - - - , - - - - - , - - - - - - - , 5.0 6.0 7.0 8.0 9.0 10.0
-In Fundcmental &l'or V.-lance Fig.2.Minimum sample size as a function of the fundamental error's variance (simulated results and regression analysis function) considering the organic fraction the critical component.
ACKNOWLEDGEMENTS The authors wish to acknowledge TRATOLIXO for their support in the experimental work and for the permission to publish the results presented in the paper.
REFERENCES
Durio, F. (2000). MSW User's Guide. Fishman, G. S. (1996). Monte Carlo - Concepts, Algorithms and Applications, Sprioger-Verlag. Gy, P. (1982). Sampling of Particulate Materials. Theory and Practice, Elsevier Scientific. Gy, P. (1988). Heterogeneite, Echantilonnage, Homogeneisation, Masson. Johnsoo, N.L., K~ S., Balakrishnan (1997). Discrete Multivariale Distributions, John Wiley and Sons, Inc. Pitard, F. (1993). Pierre Gys Sampling Theory and Sampling Practice, Second Edition, CRC Press Waste Analysis Procedure, ERRA (European Recovery & Recycling Association), 1993.
401