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Sampling and sub sampling of primary and secondary building materials: a statistical treatise
Environmental Aspects of Constnrction with Waste Materials JJJ.M. Goumans, H A . van der SIoot and l71.G.Aalbers (Editors) 01994 Elsevier Science B.V. AN rights resewed.
835
Sampling and sub sampling of primary and secondary building materials: a statistical treatise A.M.H van der Veen, D.A.G. Nater Materialenbank Nederland MBN, P.O.Box 15 1 , 6470 ED Eygelshoven, the Netherlands Abstract The statistics of sampling and sub sampling of primary and secondary building materials are reviewed. A Monte Carlo model is described, which is capable of describing the sub sampling process of granular materials at a particle level. The model is demonstrated for the socalled cross-riffling process and for distribution heterogeneity. It is concluded from the computational results as well from experiments, that the model is capable of describing the statistics of a sub sampling process. However, the model should be extended in order to study the influence of the distribution of the critical parameter(s) over the sub samples. 1. Introduction
In order to do experiments, reliable starting materials are required. The processes of sample taking and sample preparation from primary and secondary building materials are far from simple. Many steps are involved, and each of them steps may be of influence on the properties of the material. The property of interest, usually called the critical property, should be maintained during the process as good as feasible. When preparing a sample, the only objective is to reduce the amount and the particle size of material in such a way, that a portion suitable for the experiment remains. Under strict controlled conditions, it is possible to do both maintain the property of interest and to make the material suitable for the experiment. In a previous paper [I], an outline over standardised sample preparation and use of reference materials has been given. In this paper, emphasis will be put on the statistical aspects of sample taking and sample preparation. Special attention is being paid to the heterogeneity concept, which is of great importance when sampling construction materials. 2. Sub sampling
When sampling a material, it should be considered that any practical sampling process is to be treated as sampling from a finite population. This observation is very important, since it has great practical implications. After a sample is taken from the population, the properties of the population may have changed. If sampling takes place under strictly controlled conditions, it is possible to minimise this effect, but it may never be excluded that the sampling process itself changes the properties of the population. The effects become stronger as the ratio between the sample mass and the population mass increases. Thus, for sub sampling, the observation of a finite population becomes even more important. If a sub
836 sampling fails, than both the sub sample and the remainder of the original sample become useless. In a sub sampling process, the orders of magnitude of the sub samples and the population (the sample before subdividing) are comparable. Usually ratios between 1:2 and 1:20 are observed. For the preparation of reference materials, it is even more important that the properties of the sub samples are as equal as possible. Although the material being subdivided may be very heterogeneous with respect to its constitution, the objective of a subdividing process is always to distribute the properties of the material equally over the sub samples. If this attempt succeeds, then it is said that the sub samples are homogeneous The distribution heterogeneity (DH), which only exists if there is also constitution heterogeneity (CH) [ 2 ] should be minimised in the sub sampling process. If a bulk material is to be sampled, it is usually desirable to have a sample that amounts a few 100 kg, depending on the sampling conditions. If, for instance, a depot is sampled that contains about 100 tonnes of fly-ash, it is no use to take only a few kilograms of material. A sample of 100 kg for instance allows to design a sampling process that is suitable to the purpose. In practice, a severe problem is how to prepare samples of good quality from such an amount. Any treatment by hand will not meet the quality requirements stated in the introduction. For this type of operations, specially designed large-scale equipment can be used [I]. However, the accuracy of this type of equipment is lower than that of a laboratory riffler. Experiments have shown, that weight differences in the 10 sub samples from the large scale riffler of about 10% may be observed. The differences in particle size distribution, as well as the properties of the sub samples are far less fortunately. The sample size and the particle size distribution may indicate whether a sub sampling step may be successfbl or not. These indicators are not sufficient however when preparing materials for a special purpose. Then the critical parameter should be investigated. In order to guarantee that the samples are homogeneous after sub dividing the 10 sub samples (of 10 kg on average in the example), a special procedure has been developed. Since the process originates from using spinning rifflers, it has been called cross-rij’jing (X-riffling), and it is carried out as follows. The 10 sub samples (labelled #01..#10 in the scheme on the next page) are riffled each into 10 sub samples. After riffling each of the 10 sub samples, a matrix of 100 samples is constructed. The samples are arranged in such a way, that 1. in each column the samples from one sub sample appears 2. in each row from each of the 10 tubes just one sample appears (indicated by the second pair of digits in the scheme) Finally, 10 new sub samples are created by recombination of each of the rows. The recombined samples are labelled #A..#J. The X-riffling process is very efficient with respect to reducing the random error. The Monte Carlo model, to be introduced shortly, has been used to investigate the statistics of this process. The reason for recombining samples from different tubes from the riffler is quite obvious: if samples were taken from one tube (the samples #01.05, #02.05, ,,,,#10.05 for instance), propagation of a systematic error in the riffler is possible.
837
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3. Segregation
Segregation is the greatest potential risk in the sub sampling process. Segregation is favoured by two factors: 1. a particle size distribution 2 differences in density Segregation due to the particle size distribution is only possible as long as the particles move. The process favoured by density differences is a slow process, but it can proceed even if the material is in rest. There are two ways to circumvent the problem of segregation while sub sampling, in fact mixing or sub sampling with a technique that is not sensitive to differences in constitution or particle size. Mixing will solve only a part of the problems, since any method of sub dividing will allow the particles to move, and thus favouring segregation of the first type. It can be proved that using a vibrating feeder will certainly cause this type of segregation to occur. The Monte Carlo model is also used to investigate the influence of gradually changing properties when subdividing. 4. Monte Carlo model
In order to enable a thorough study on various aspects of the sampling and sub sampling of granular materials, a Monte Carlo model is developed [ 3 ] . The model was intended to be a tool for validation of the cross-riffling procedure, as implemented at MBN for several years. The model was tested first on coals [4], and after that the applicability was tested for other materials. The results of the first computational experiments are that promising, that a fbrther development of the Monte Carlo model is planned. The model developed so far is based on the riffler mechanism. A riffler consists of a head, which turns around an axis. This head allows the material being fed by a continuous feeder to pass one tube at a time. If the rotational velocity of the riffler head and the feeding rate are constant, then it may be expected that all sub samples have about the same properties
838
Inpul
1
N. a v e r a g e . slaridard devlitllon L,Se Po,sso,,,B,no Cpne,.n,o,.s
,,,,",
.~~ -
DrLrrrnine # c1ustcrs
~ i i u s s r a n ene erst or
{_-# _par11cies i z z J &J& ;: D e te 1.111 I" e # "red"
8"' """s'
Omeralor
I
Oulpul
~ i ~ , 1, ,..Algorithm ~ of the Monte
Model
carlo
If the particles stick together, it still must be expected that the average volume of the through-put is constant. The distribution of the average number of particles differs however. Depending on the strength of the clusters, these clusters can be treated as if they were single particles, just of a larger radius. The cluster size will not be distributed normally. Especially at low average cluster sizes, it must be expected that the distribution function of this size will not be symmetrical. The Poisson distribution seems to be suitable for modelling this feature.
Heterogeneity is probably the most complex property to describe. A true quantitative approach would require the knowledge how the various components are distributed over the particles. The theory of Gy [2] provides us with a general treatment on heterogeneity, but this theory is not readily applicable in calculations. In the Monte Carlo model, a method has been sought for, which allows the simulate heterogeneity, without demanding for a true quantitative description. So far, only distribution heterogeneity has been taken into account. The implementation of the distribution heterogeneity (DH) is quite straight forward. DH can simply be regarded as having particles of two types (e.g. red and white particles). In statistics, normally experiments with two possible outcomes are described by means of the binomial distribution. In the MC model the binomial distribution is used in the final step to fix the number of particles of one kind (the number of particles of the other type is fixed then automatically). Figure 1 summarises the algorithm. 5. X-riflling
The first task for the Monte Carlo model was to evaluate the statistics of the X-riffling procedure. From round robin tests, it was known that the process yielded sufficiently homogeneous sub samples. However, one of the questions still remaining was how many times the
839 scheme should be applied before a sufficiently homogeneous batch of samples was prepared. The results of the simulations are shown in figure 2. The figure clearly shows that the standard deviation of the samples is effectively reduced to a level specific for the equipment. The variations after the first cycle are due to the fact, that a stochastic process is being investigated. The various values of n in the figure (where n is the average number of particles falling through a tube per cycle) can be translated to different angular velocities
X-riffling performance at various feeder speeds
10,000 1,000
The larger the value Of n, the greater is the angular velocity. A simulation where the initial standard deviation was lower than that of the riffler yielded exactly the same result as in figure 2. The standard deviation of the samples after the first cycle was equal within the precision of the simulations 6. Blending and heterogeneity
Both blending and distribution heterogeneity can be described accurately by using the binomial distribution. The greater the similarities between the particles in the blend, the better the description will be. I f a material is sampled which can be regarded as a blend (like metal particles in soil, or a mixture of sand and clay in a soil sample), the Monte Carlo simulation f d l y applies. DH increases when the probability to obtain a particle from one part of the sample decreases. Figure 2 shows the results of the MC calculation, where the total number of particles is 100000, the average is 100, the riffler standard
840
deviation is 10, and the number of sub samples is also 10. At the y-axis, the standard deviation divided by the 'binomial' probability is plotted. This correction is made to correct for the number of particles involved for which the probability applies. Two important observations have been made when testing the modelling of DH. First of all, the riffler mechanism allows to treat materials with gradually changing properties. This observation is very important, since it implicates that segregation will not be of influence on the riffling process. Secondly, when the total number of particles is increased, the relative standard deviation decreases. Depending on the requirements of the samples, the minimum sample size can be determined. It is a clear demonstration of a well-known fact, that an insufficient amount of material will not result in sensible analytical data. 7. Discussion and conclusions
The Monte Carlo model under development provides a flexible basis for sampling and sub sampling modelling. Depending on the assumptions made when sub sampling, the algorithm can be modified to be applicable for other equipment. One of the main issues is how to translate the results from the Monte Carlo model to the real world. One of the features to be implemented is to study how the value of the critical parameter is affected by (1) the distribution of this parameter in the material, (2) the distribution heterogeneity, and (3) blending. The approach of the model becomes somewhat different when implementing these features, since the 'analysis' of the sub samples has to take place also by the Monte Carlo principle. On the other hand however, the model comes closer to the real world, and thus the quality of the predictions will become better. Parallel to the development of the model, the knowledge of how certain sources of errors are translated into the final results on analysis will help to improve the quality of environmental measurements. The model is at this stage already a valuable tool in design, development and validation of sampling and sub sampling strategies. Additional refinements of the model in this direction will certainly be beneficial to our knowledge on how to sample in environmental studies. 8. Literature
1 . F.J.M. Lamers, G.J de Groot, "Standard sample preparation and reference samples as a
tool for determination of the environmental quality of building materials", Waste Materials in Construction, Proceedings of the WASCON Conference 1991, Elsevier, Amsterdam 1991, pp 375-378. 2. P.M. Gy, "Sampling of particulate materials", Elsevier, Amsterdam 1982. 3 . A.M.H. van der Veen, "Computer simulations of subsampling with use of spinning rifflers", final report, SBN/MBN Eygelshoven, the Netherlands 4. A.M.H. van der Veen, D.A.G. Nater, "Sample preparation from bulk materials: an overview", Proc. Third Rolduc Smyposium on Coal Science, Elsevier, Amsterdam pp 1-7
835
Sampling and sub sampling of primary and secondary building materials: a statistical treatise A.M.H van der Veen, D.A.G. Nater Materialenbank Nederland MBN, P.O.Box 15 1 , 6470 ED Eygelshoven, the Netherlands Abstract The statistics of sampling and sub sampling of primary and secondary building materials are reviewed. A Monte Carlo model is described, which is capable of describing the sub sampling process of granular materials at a particle level. The model is demonstrated for the socalled cross-riffling process and for distribution heterogeneity. It is concluded from the computational results as well from experiments, that the model is capable of describing the statistics of a sub sampling process. However, the model should be extended in order to study the influence of the distribution of the critical parameter(s) over the sub samples. 1. Introduction
In order to do experiments, reliable starting materials are required. The processes of sample taking and sample preparation from primary and secondary building materials are far from simple. Many steps are involved, and each of them steps may be of influence on the properties of the material. The property of interest, usually called the critical property, should be maintained during the process as good as feasible. When preparing a sample, the only objective is to reduce the amount and the particle size of material in such a way, that a portion suitable for the experiment remains. Under strict controlled conditions, it is possible to do both maintain the property of interest and to make the material suitable for the experiment. In a previous paper [I], an outline over standardised sample preparation and use of reference materials has been given. In this paper, emphasis will be put on the statistical aspects of sample taking and sample preparation. Special attention is being paid to the heterogeneity concept, which is of great importance when sampling construction materials. 2. Sub sampling
When sampling a material, it should be considered that any practical sampling process is to be treated as sampling from a finite population. This observation is very important, since it has great practical implications. After a sample is taken from the population, the properties of the population may have changed. If sampling takes place under strictly controlled conditions, it is possible to minimise this effect, but it may never be excluded that the sampling process itself changes the properties of the population. The effects become stronger as the ratio between the sample mass and the population mass increases. Thus, for sub sampling, the observation of a finite population becomes even more important. If a sub
836 sampling fails, than both the sub sample and the remainder of the original sample become useless. In a sub sampling process, the orders of magnitude of the sub samples and the population (the sample before subdividing) are comparable. Usually ratios between 1:2 and 1:20 are observed. For the preparation of reference materials, it is even more important that the properties of the sub samples are as equal as possible. Although the material being subdivided may be very heterogeneous with respect to its constitution, the objective of a subdividing process is always to distribute the properties of the material equally over the sub samples. If this attempt succeeds, then it is said that the sub samples are homogeneous The distribution heterogeneity (DH), which only exists if there is also constitution heterogeneity (CH) [ 2 ] should be minimised in the sub sampling process. If a bulk material is to be sampled, it is usually desirable to have a sample that amounts a few 100 kg, depending on the sampling conditions. If, for instance, a depot is sampled that contains about 100 tonnes of fly-ash, it is no use to take only a few kilograms of material. A sample of 100 kg for instance allows to design a sampling process that is suitable to the purpose. In practice, a severe problem is how to prepare samples of good quality from such an amount. Any treatment by hand will not meet the quality requirements stated in the introduction. For this type of operations, specially designed large-scale equipment can be used [I]. However, the accuracy of this type of equipment is lower than that of a laboratory riffler. Experiments have shown, that weight differences in the 10 sub samples from the large scale riffler of about 10% may be observed. The differences in particle size distribution, as well as the properties of the sub samples are far less fortunately. The sample size and the particle size distribution may indicate whether a sub sampling step may be successfbl or not. These indicators are not sufficient however when preparing materials for a special purpose. Then the critical parameter should be investigated. In order to guarantee that the samples are homogeneous after sub dividing the 10 sub samples (of 10 kg on average in the example), a special procedure has been developed. Since the process originates from using spinning rifflers, it has been called cross-rij’jing (X-riffling), and it is carried out as follows. The 10 sub samples (labelled #01..#10 in the scheme on the next page) are riffled each into 10 sub samples. After riffling each of the 10 sub samples, a matrix of 100 samples is constructed. The samples are arranged in such a way, that 1. in each column the samples from one sub sample appears 2. in each row from each of the 10 tubes just one sample appears (indicated by the second pair of digits in the scheme) Finally, 10 new sub samples are created by recombination of each of the rows. The recombined samples are labelled #A..#J. The X-riffling process is very efficient with respect to reducing the random error. The Monte Carlo model, to be introduced shortly, has been used to investigate the statistics of this process. The reason for recombining samples from different tubes from the riffler is quite obvious: if samples were taken from one tube (the samples #01.05, #02.05, ,,,,#10.05 for instance), propagation of a systematic error in the riffler is possible.
837
u
u
u
u
u
u
u
u
u
u
3. Segregation
Segregation is the greatest potential risk in the sub sampling process. Segregation is favoured by two factors: 1. a particle size distribution 2 differences in density Segregation due to the particle size distribution is only possible as long as the particles move. The process favoured by density differences is a slow process, but it can proceed even if the material is in rest. There are two ways to circumvent the problem of segregation while sub sampling, in fact mixing or sub sampling with a technique that is not sensitive to differences in constitution or particle size. Mixing will solve only a part of the problems, since any method of sub dividing will allow the particles to move, and thus favouring segregation of the first type. It can be proved that using a vibrating feeder will certainly cause this type of segregation to occur. The Monte Carlo model is also used to investigate the influence of gradually changing properties when subdividing. 4. Monte Carlo model
In order to enable a thorough study on various aspects of the sampling and sub sampling of granular materials, a Monte Carlo model is developed [ 3 ] . The model was intended to be a tool for validation of the cross-riffling procedure, as implemented at MBN for several years. The model was tested first on coals [4], and after that the applicability was tested for other materials. The results of the first computational experiments are that promising, that a fbrther development of the Monte Carlo model is planned. The model developed so far is based on the riffler mechanism. A riffler consists of a head, which turns around an axis. This head allows the material being fed by a continuous feeder to pass one tube at a time. If the rotational velocity of the riffler head and the feeding rate are constant, then it may be expected that all sub samples have about the same properties
838
Inpul
1
N. a v e r a g e . slaridard devlitllon L,Se Po,sso,,,B,no Cpne,.n,o,.s
,,,,",
.~~ -
DrLrrrnine # c1ustcrs
~ i i u s s r a n ene erst or
{_-# _par11cies i z z J &J& ;: D e te 1.111 I" e # "red"
8"' """s'
Omeralor
I
Oulpul
~ i ~ , 1, ,..Algorithm ~ of the Monte
Model
carlo
If the particles stick together, it still must be expected that the average volume of the through-put is constant. The distribution of the average number of particles differs however. Depending on the strength of the clusters, these clusters can be treated as if they were single particles, just of a larger radius. The cluster size will not be distributed normally. Especially at low average cluster sizes, it must be expected that the distribution function of this size will not be symmetrical. The Poisson distribution seems to be suitable for modelling this feature.
Heterogeneity is probably the most complex property to describe. A true quantitative approach would require the knowledge how the various components are distributed over the particles. The theory of Gy [2] provides us with a general treatment on heterogeneity, but this theory is not readily applicable in calculations. In the Monte Carlo model, a method has been sought for, which allows the simulate heterogeneity, without demanding for a true quantitative description. So far, only distribution heterogeneity has been taken into account. The implementation of the distribution heterogeneity (DH) is quite straight forward. DH can simply be regarded as having particles of two types (e.g. red and white particles). In statistics, normally experiments with two possible outcomes are described by means of the binomial distribution. In the MC model the binomial distribution is used in the final step to fix the number of particles of one kind (the number of particles of the other type is fixed then automatically). Figure 1 summarises the algorithm. 5. X-riflling
The first task for the Monte Carlo model was to evaluate the statistics of the X-riffling procedure. From round robin tests, it was known that the process yielded sufficiently homogeneous sub samples. However, one of the questions still remaining was how many times the
839 scheme should be applied before a sufficiently homogeneous batch of samples was prepared. The results of the simulations are shown in figure 2. The figure clearly shows that the standard deviation of the samples is effectively reduced to a level specific for the equipment. The variations after the first cycle are due to the fact, that a stochastic process is being investigated. The various values of n in the figure (where n is the average number of particles falling through a tube per cycle) can be translated to different angular velocities
X-riffling performance at various feeder speeds
10,000 1,000
The larger the value Of n, the greater is the angular velocity. A simulation where the initial standard deviation was lower than that of the riffler yielded exactly the same result as in figure 2. The standard deviation of the samples after the first cycle was equal within the precision of the simulations 6. Blending and heterogeneity
Both blending and distribution heterogeneity can be described accurately by using the binomial distribution. The greater the similarities between the particles in the blend, the better the description will be. I f a material is sampled which can be regarded as a blend (like metal particles in soil, or a mixture of sand and clay in a soil sample), the Monte Carlo simulation f d l y applies. DH increases when the probability to obtain a particle from one part of the sample decreases. Figure 2 shows the results of the MC calculation, where the total number of particles is 100000, the average is 100, the riffler standard
840
deviation is 10, and the number of sub samples is also 10. At the y-axis, the standard deviation divided by the 'binomial' probability is plotted. This correction is made to correct for the number of particles involved for which the probability applies. Two important observations have been made when testing the modelling of DH. First of all, the riffler mechanism allows to treat materials with gradually changing properties. This observation is very important, since it implicates that segregation will not be of influence on the riffling process. Secondly, when the total number of particles is increased, the relative standard deviation decreases. Depending on the requirements of the samples, the minimum sample size can be determined. It is a clear demonstration of a well-known fact, that an insufficient amount of material will not result in sensible analytical data. 7. Discussion and conclusions
The Monte Carlo model under development provides a flexible basis for sampling and sub sampling modelling. Depending on the assumptions made when sub sampling, the algorithm can be modified to be applicable for other equipment. One of the main issues is how to translate the results from the Monte Carlo model to the real world. One of the features to be implemented is to study how the value of the critical parameter is affected by (1) the distribution of this parameter in the material, (2) the distribution heterogeneity, and (3) blending. The approach of the model becomes somewhat different when implementing these features, since the 'analysis' of the sub samples has to take place also by the Monte Carlo principle. On the other hand however, the model comes closer to the real world, and thus the quality of the predictions will become better. Parallel to the development of the model, the knowledge of how certain sources of errors are translated into the final results on analysis will help to improve the quality of environmental measurements. The model is at this stage already a valuable tool in design, development and validation of sampling and sub sampling strategies. Additional refinements of the model in this direction will certainly be beneficial to our knowledge on how to sample in environmental studies. 8. Literature
1 . F.J.M. Lamers, G.J de Groot, "Standard sample preparation and reference samples as a
tool for determination of the environmental quality of building materials", Waste Materials in Construction, Proceedings of the WASCON Conference 1991, Elsevier, Amsterdam 1991, pp 375-378. 2. P.M. Gy, "Sampling of particulate materials", Elsevier, Amsterdam 1982. 3 . A.M.H. van der Veen, "Computer simulations of subsampling with use of spinning rifflers", final report, SBN/MBN Eygelshoven, the Netherlands 4. A.M.H. van der Veen, D.A.G. Nater, "Sample preparation from bulk materials: an overview", Proc. Third Rolduc Smyposium on Coal Science, Elsevier, Amsterdam pp 1-7