In dan2cil/tecTools: A collection of small function for Food Technologists

The tempering() function is useful to simulate the crystallization of cocoa butter in different condiction:

• Different percentage of cocoa butter in chocolate
• Different amount of cocoa butter present at the beginning of the simulation
• Different temperature at the beginning of the simulation

Function

Below the sintax of the function tempering():

tempering(BC = 0.42, t0 = 35, f0 = 1, a = 100, n = 3, k = 5)

where:

BC  :   cocoa butter contents (0-1)
t0  :   initial temperature of simulation (raffreddamento)                                  35°C
f0  :   initially present amount of crystals [J/g or % solid fat]                            1%
a   :   value for f as t approaches infinity [J/g or % solid fat]                          100%
n   :   order of reverse reaction, linked to asymmetry of curve [-]                          3
k   :   rate constant [h-1]                                                                  5

Formula used

In the tempering() function we implement the Foubert formula (differenzial => algebraic):

$\frac{dh}{dt} = K * (h^n -h)$    =>    $f = [1-[1+((1-\frac{f_0}{a})-1) e^{-(1-n)K*t}]^{\frac{1}{1-n}}]$

where:

h  :    remaining crystallizable fat at time t [-]  
f  :    amount of crystallization at time t [J/g or % solid fat]  
a  :    value for f as t approaches infinity [J/g or % solid fat]  
f0 :    initially present amount of crystals [J/g or % solid fat]  
K  :    rate constant [h-1 ]  
n  :    order of reverse reaction, linked to asymmetry of curve [-]

Examples

In the following graphic we could compare the crystallizazion courve of three different chocolate, with different cocoa butter contents:

• with 30% of cocoa butter
• with 35% of cocoa butter
• with 40% of cocoa butter

library(ggplot2)
library(tecTools)
x <- seq(0,1,0.025)
y <- tempering(BC=0.42)$temperatura
y2 <- tempering(BC=0.35)$temperatura
y3 <- tempering(BC=0.30)$temperatura
p <- ggplot(data.frame(x, y, y2, y3)) +
  geom_line(aes(x=x,y=y, color='BC 42%')) +
  geom_line(aes(x=x,y=y2, color='BC 35%')) +
  geom_line(aes(x=x,y=y3, color='BC 30%')) +
  scale_color_manual('Cioccolati', values=c('BC 42%'='red', 'BC 35%'='blue', 'BC 30%'='green'))
p

Reference

The function tempering() si basa sul lavoro presentato nei due seguenti articoli ^[Disponibili presso l'autore della libreria], in modo particolare nel primo:

"Modelling of the crystallization kinetics of cocoa butter"
Imogen Foubert; Peter Vanrolleghem; Bert Vanhoutte; Thijs Keersebilck; A. Huyghebaert and Koen Dewettinck

"Modelling isothermal cocoa butter criystallization: influence of temperature and chemical composition"
Imogen FOUBERT

dan2cil/tecTools documentation built on Dec. 19, 2021, 8:04 p.m.