Rosin - Rammler Distribution Functions

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Description

Rosin - Rammler model of particle-size distribution and cumulative undersize and oversize distributions used to obtain approximation of of powders or granular materials originated by grinding.

Usage

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drr(x, ex, xs)
orr(x, ex, xs)
urr(x, ex, xs)

Arguments

x

particle size, equivalent particle diameter

ex

Rosin - Rammler exponent, measure of the uniformity of grinding

xs

finesse of grinding, that width of mesh associated with a remainder equal to exp(-1) ~ 0.3679

Details

Following functions are used, based on Rosin - Rammler mathematical model of particle-size distribution, for approximation of size distribution.

drr is Rosin - Rammler probability density function
urr is Rosin - Rammler cumulative distribution function (CDF) representing undersize mass fraction
orr is Rosin - Rammler complementary CDF representing oversize mass fraction ie. relative remainder on the sieve with the mesh size x

Rosin - Rammler model (1933) is the Weibull distribution which was proposed by Weibull in 1939, and Weibull distribution functions are part of R.
So the user can use stats::dweibull(x,shape=ex,scale=xs) the same way as drr, and use Weibull distribution functions provided by stats package for deeper analysis.
Similarly, stats::pweibull(x,shape=ex,scale=xs) can be used the same way as urr or
stats::pweibull(x,shape=ex,scale=xs,lower.tail=F) the same way as orr.

Value

Both urr and orr returns value of distribution function.
Function drr returns density.

References

Rinne, H. (2008) The Weibull Distribution: A Handbook, chapter 1.1.2. Taylor & Francis.

See Also

Weibull, plot.std, summary.std

Examples

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## The function drr is currently defined as
#  function (x, ex, xs) 
#  {
#      (ex/xs) * (x/xs)^(ex - 1) * exp(-(x/xs)^ex)
#  }

## The function urr is currently defined as
#  function (x, ex, xs) 
#  {
#      1 - exp(-(x/xs)^ex)
#  }

## The function orr is currently defined as
#  function (x, ex, xs) 
#  {
#      exp(-(x/xs)^ex)
#  }


x <- c(1,5,10,50,100)
ex <- 1.386
xs <- 178
stats::dweibull(x,shape=ex,scale=xs)
drr(x,ex,xs)
stats::pweibull(x,shape=ex,scale=xs)
urr(x,ex,xs)
stats::pweibull(x,shape=ex,scale=xs,lower.tail=FALSE)
orr(x,ex,xs)