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#' @title Jossens Isotherm Linear Analysis
#' @name jossens.LM
#' @description The Jossens isotherm model predicts a simple equation based on
#' the energy distribution of adsorbate-adsorbent interactions at adsorption
#' sites. This model assumes that the adsorbent has heterogeneous surface with
#' respect to the interactions it has with the adsorbate.
#' @param Ce the numerical value for the equilibrium capacity
#' @param Qe the numerical value for the adsorbed capacity
#' @import nls2
#' @import Metrics
#' @import stats
#' @import ggplot2
#' @return the linear regression, parameters for the Jossens isotherm, and model error
#' analysis
#' @examples Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
#' @examples Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
#' @examples jossens.LM(Ce, Qe)
#' @author Paul Angelo C. Manlapaz
#' @author Chester C. Deocaris
#' @references Jossens, L., Prausnitz, J. M., Fritz, W., Schlunder, E. U.,
#' and Myers, A. L. (1978) <doi:10.1016/0009-2509(78)85015-5> Thermodynamics of
#' multi-solute adsorption from dilute aqueous solutions.
#' Chemical Engineering Science, 33(8), 1097-1106.
#' @export
#'
# Building the Jossens isotherm linear form
jossens.LM <- function(Ce,Qe) {
x1 <- Qe
y1 <- Ce
data <- data.frame(x1, y1)
# Obtaining the model exponent
### Jossens isotherm nonlinear equation
fit1 <- (y1 ~ (x1/H)*(exp(J*(x1^Nj))))
### Setting of starting values
start1 <- data.frame(H = c(1, 500), J = c(1, 100), Nj = c(0, 1))
### Fitting of the Jossens isotherm via nls2
fit2 <- nls2::nls2(fit1, start = start1,data=data,
control = nls.control(maxiter = 100 , warnOnly = TRUE),
algorithm = "port")
param <- summary(fit2)
expModel <- param$coefficients[3]
# Establishing Jossens isotherm linear form
x <- Ce^expModel
y <- log(Ce/Qe)
rhs <- function (x, H, J) {
-log(H) + J*x
}
# Jossens isotherm linear fitting
fit3 <- lm(y~x)
print("Jossens Isotherm Linear Analysis")
print(summary(fit3))
### y = a + bx
c <- summary(fit3)
a <- c$coefficients[1]
b <- c$coefficients[2]
### Parameter values calculation
J <- b
print("J")
print(J)
H <- -exp(a)
print("H")
print(H)
Nj <- expModel
print("Nj")
print(Nj)
# -------------------------------------
print("Aikake Information Criterion")
print(AIC(fit3))
print("Bayesian Information Criterion")
print(BIC(fit3))
# Error analysis of the Jossens isotherm linear model
errors <- function(y) {
rmse <- Metrics::rmse(y, predict(fit3))
mae <- Metrics::mae(y, predict(fit3))
mse <- Metrics::mse(y, predict(fit3))
rae <- Metrics::rae(y, predict(fit3))
N <- nrow(na.omit(data))
SE <- sqrt((sum(y-predict(fit2))^2)/(N-2))
colnames(y) <- rownames(y) <- colnames(y)
list("Relative Mean Squared Error" = rmse,
"Mean Absolute Error" = mae,
"Mean Squared Error" = mse,
"Relative Absolute Error" = rae,
"Standard Error for the Regression S" = SE)
}
a<- errors(y)
print(a)
# Graphical representation of the Jossens isotherm linear model
#### Plot details
ggplot2::theme_set(ggplot2::theme_bw(10))
ggplot2::ggplot(data, ggplot2::aes(x = x, y = y)) + ggplot2::geom_point(color ="#3498DB" ) +
ggplot2::geom_smooth(formula = y ~ x, method = "lm", se = F, color = "#D35400" ) +
ggplot2::labs(x = expression(paste("Qe"^"Nj")),
y ="ln(Ce/Qe)",
title = "Jossens Isotherm Linear Model",
caption = "PUPAIM") +
ggplot2::theme(plot.title=ggplot2::element_text(hjust = 0.5))
}
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