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| --- | |
| title: "Determination of oxygen mass transfer coefficient (KLa) in batch reactor airation" | |
| author: "Rafael Ventura" | |
| date: "9th July of 2017" | |
| output: | |
| html_document: default | |
| pdf_document: default | |
| word_document: default | |
| --- | |
| ```{r setup, include=FALSE} | |
| knitr::opts_chunk$set(echo = TRUE) | |
| ``` | |
| ## Objectives | |
| Exercise on the statement of a model, programming in AQUASIM, and devising a strategy for parameter fitting. | |
| Assignment is part of evaluation of Modelling Sanitation Systems Online course. | |
| ````{r dataset, fig.width = 6, fig.height = 3, fig.fullwidth = TRUE, fig.cap = "Endogenous respiration- phase I", warning=FALSE, cache=TRUE, echo=FALSE} | |
| library(ggplot2) | |
| library(readxl) | |
| wd <- getwd() | |
| wd <- gsub("/","\\\\", wd) | |
| filepath <- paste(wd,"\\","Data.xls", sep="") | |
| dataset <- read_excel(filepath, sheet = "Data_all_and_calcul (2)", skip = 5) | |
| dataset$time_hh <- as.numeric(dataset$time_hhmmss - as.POSIXct("1899-12-31 14:51:40 UTC"))/60 | |
| ggplot(dataset, aes(time_hh,doc_mgl,fill=phase))+geom_line(aes(colour=phase, group=phase)) | |
| ``` | |
| ## Procedure | |
| - First, oxygen uptake rate is obtained from data by fitting a linear model to data in phase I | |
| ```{r OUR2, fig.width = 8, fig.height = 4, fig.fullwidth = TRUE, fig.cap = "QO2 estimation- Phase 1 data", warning=FALSE, cache=TRUE,echo=FALSE} | |
| library(readxl) | |
| library(lattice) | |
| wd <- getwd() | |
| wd <- gsub("/","\\\\", wd) | |
| filepath <- paste(wd,"\\","Data.xls", sep="") | |
| dataset <- read_excel(filepath, sheet = "Data_all_and_calcul (2)", skip = 5) | |
| dataset$time_hh <- as.numeric(dataset$time_hhmmss - as.POSIXct("1899-12-31 14:51:40 UTC"))/60 | |
| p1_dataset <- subset(dataset,dataset$phase==1) | |
| OUR_lm <- lm(doc_mgl ~ time_hh, data = p1_dataset) | |
| # print(OUR_lm) | |
| tp <- sprintf("%s=%.1f + %.3f %s, r2=%.4f", all.vars(formula(OUR_lm))[1], | |
| coef(OUR_lm)[1], coef(OUR_lm)[2], all.vars(formula(OUR_lm))[2], summary(OUR_lm)$r.squared) | |
| mypanel<-function(x,y,...){ | |
| panel.xyplot(x, y, ...) | |
| panel.abline(OUR_lm) | |
| panel.text(0.3,4,labels=tp) | |
| } | |
| xyplot(doc_mgl ~ time_hh, data = p1_dataset, panel = mypanel) | |
| print(paste("QO2 = ", as.character(round(OUR_lm$coefficients[2],3))," [mg/L-h]", "")) | |
| ``` | |
| - Second, initial condition for SO (SO_ini) for AQUASIM simulation is read from data at phase II, by retrieving the minimum value in phase II. | |
| ```{r SO_ini, cache= TRUE, warning=FALSE, echo=FALSE} | |
| library(readxl) | |
| wd <- getwd() | |
| wd <- gsub("/","\\\\", wd) | |
| filepath <- paste(wd,"\\","Data.xls", sep="") | |
| dataset <- read_excel(filepath, sheet = "Data_all_and_calcul (2)", skip = 5) | |
| p2_dataset <- subset(dataset,dataset$phase==2) | |
| SO_ini <- min(p2_dataset$doc_mgl) | |
| print(paste("SO_ini = ", as.character(SO_ini)," [mg/L]", sep = "")) | |
| ``` | |
| - Third, dissolved Oxygen concentration at equilibrium is read from data in phase II as the maximum value: | |
| ```{r SO_eq, warning=FALSE, echo= FALSE} | |
| library(readxl) | |
| wd <- getwd() | |
| wd <- gsub("/","\\\\", wd) | |
| filepath <- paste(wd,"\\","Data.xls", sep="") | |
| dataset <- read_excel(filepath, sheet = "Data_all_and_calcul (2)", skip = 5) | |
| p2_dataset <- subset(dataset,dataset$phase==2) | |
| SO_eq <- max(p2_dataset$doc_mgl) | |
| print(paste("SO_eq = ", as.character(SO_eq)," [mg/L]", sep = "")) | |
| ``` | |
| - Fourth, we obtained SO_sat from data in phase III, when SO reached its maximum | |
| ```{r SO_sat, warning=FALSE, echo=FALSE} | |
| library(readxl) | |
| wd <- getwd() | |
| wd <- gsub("/","\\\\", wd) | |
| filepath <- paste(wd,"\\","Data.xls", sep="") | |
| dataset <- read_excel(filepath, sheet = "Data_all_and_calcul (2)", skip = 5) | |
| p3_dataset <- subset(dataset,dataset$phase==3) | |
| SO_sat <- max(p3_dataset$doc_mgl) | |
| print(paste("SO_sat = ", as.character(SO_sat)," [mg/L]", sep = "")) | |
| ``` | |
| As substrate is depleted, Biomass undergoes endogenous respiration process in phase 1. As stated in problem definition, OUR is constant. | |
| As $\frac{dSO}{dt} = KLa*(SO_{sat} - SO) - Q2O*SO$, at equilibrium $\frac{dSO}{dt} = 0$, hence $KLa = \frac {Q2O}{SO_{sat} - SO_{eq}}$ | |
| ```{r Kla, warning=FALSE, echo=FALSE } | |
| print(paste("KLa = ",as.character(round(20.376/(5.6198-4.14821),3))," [1/h]")) | |
| ``` | |
| **Simulation with AQUASIM** | |
| State variable are dissolved oxygen concentration in liquid phase(SO) and oxygen concentration in gas phase, both in [mg/L]. Process variable is time in [h]. | |
| Process are defined as two reactions: | |
| $K_{La}*(SO_sat - SO)$ for oxygen mass transfer during airation, | |
| and $-QO2*SO$ for endogenous respiration (decay). | |
| Biomass Heterotrophic growth is not considered as substrate is depleted. For airation, KLa and SO_sat are set to be 13,846 [1/h] and 5,6198 [mg/L], from calculations mentioned above. Reactor volumen is set to be 1,64 [L]. Petersen matrix is shown below: | |
| ```{r petersen, echo=FALSE, warning= FALSE} | |
| # c("Component","XH", "XI", "SO", "SG") | |
| library("knitr") | |
| petersen <- data.frame(process=rep(NA,2), SO=rep(NA,2), Expr=rep(NA,2)) | |
| airation <- c("Airation", 1,"KLa*(SO_sat - SO)") | |
| decay <- c("Decay",-1,"QO2" ) | |
| petersen[1,] <- airation | |
| petersen[2,] <- decay | |
| knitr::kable( | |
| petersen, caption = 'Petersen matrix.') | |
| ``` | |
| For endogenous respiration (decay) process's reaction rate, QO2 was estimated above. | |
| A compartment is defined as a CSRT with airation and decay process activated. An inflow of air to compartment with no water flow but gas with O2 concentration (SG) equal to 203 [mg/L]. SG is calculated using the Ideal gas law: $SG = PO/RT$ at with appropriate units. | |
| Neither diffusive nor adventive link are considered. | |
| Simulation time interval simulation was set to be 0.001 [h] ranging and 400 steps. | |
| ##Sensitivity Analysis## | |
| Sensitivity analysis was done for SO_sat, QO2, SO_ini, and KLa. SO showed to increase as SO_sat increases and decreased as KLa increased. A changein QO2 and SO_ini don't change SO as the other parameters do. Plot at the left and side shows the sensitivity analysis before and after parameter estimation, respectively. | |
|  | |
| ##Model parameters adjustment## | |
| The strategy is fitting SO_sat and KLa, simultaneouly. Results are shown in Table 2. | |
|  | |
| ```{r AQUASIM resuts, echo=FALSE, warning= FALSE} | |
| parameters <- data.frame(parameter=rep(NA,2), Initial_Estimation=rep(NA,2) , Fitted_Value=rep(NA,4), ChiSqr=rep(NA,4)) | |
| SO_ini <- c("SO_ini [mg/L]", "0,189951", "0,189951", NA) | |
| QO2 <- c("Q2O [mg/L-h]", "-20,376447", "-20,376447", NA) | |
| SO_sat <- c("SO_sat [mg/L]", "5,6198", "4,1992", "5,0473533") | |
| KLa <- c("KLa [1/h]", "13,846", "15,000", "5,0473533") | |
| parameters[1,] <- SO_ini | |
| parameters[2,] <- QO2 | |
| parameters[3,] <- SO_sat | |
| parameters[4,] <- KLa | |
| knitr::kable( | |
| parameters, caption = 'Parameter fitting with AQUASIM.') | |
| ``` | |
| Simulation file can be found at the following link: Dropbox (https://www.dropbox.com/s/jt96iv8p5yep26w/oxygen-mass-transfer-sensitivity-analysis-2.lis?dl=0) |