Comparison of the Prediction Accuracy of Total Viable Bacteria Counts in a Batch Balloon Digester Charged with Cow Manure: Multiple Linear Regression and Non-Linear Regression Models

Abstract

Biogas technology is rapidly gaining market penetration, and the type of digesters employed in the harnessing of the biogas from biodegradable waste is crucial in enhancing the total viable bacteria counts. This study focused on the exploration of input parameter (number of days, daily slurry temperature, and pH) and target (total viable bacteria counts) datasets from anaerobic balloon digester charged with cow manure using data acquisition system and standard methods. The predictors were ranked according to their weights of importance to the desired targets using the reliefF test. The complete dataset was randomly partitioned into testing and validated samples at a ratio of 60% and 40%, respectively. The developed non-linear regression model applied on the testing samples was capable of predicting the yield of the total viable bacteria counts with better accuracy as the determination coefficient, mean absolute error, and p-value were 0.959, 0.180, and 0.602, respectively, as opposed to the prediction with the multiple linear regression model that yielded 0.920, 0.206, and 0.514, respectively. The 2D multi-contour surface plots derived from the developed models were used to simulate the variation in the desired targets to each predictor while the others were held constant.

1. Introduction

Waste-to-energy technologies are systematically being developing due to the overwhelming demand for ecological and sustainable energy sources. Biogas produced in an anaerobic digester is a renewable energy source with the capability of contributes to the efficient management of animal waste and provide alternative methods of eliminating pollution by greenhouse gas emissions and reducing diseases caused by water and soil contamination. Research has confirmed that the production of biogas by anaerobic digestion is a promising process, but is financially demanding compared with natural gas. The intensification and optimization of anaerobic digestion processes can contribute to cost reduction. The type of hoods used when using biogas from biodegradable waste is decisive for increasing the total number of viable bacteria, and thus, the overall efficiency of the processes. This article describes and evaluates several proposed and verified mathematical and statistical models for predicting the suitable conditions of anaerobic digestion processes as a waste management method.

Livestock practices are associated with enormous amounts of animal manure and require efficient management to avoid adverse health impact and environmental degradation. The implementation of standard best practices in the efficient disposal and effective management of animal waste as potential remedies to eradicate pollutions from the emission of greenhouse gases and reduction in diseases based on water and soil contamination [1]. Considerable research has confirmed that the utilization of anaerobic digesters can serve as an efficient tool for the proper management of animal manure. Anaerobic digestion occurring in biodigesters results in the generation of biogas and digestate [2]. The biogas generated in anaerobic digesters is a renewable source of energy which can be used for heating, cooking, transportation, and electricity purposes, and contributes to the efficient management of animal waste, resulting in tremendous decreases in microbial loads [3,4,5].

The momentum of utilizing waste for energy is on the rise as the demand of eco-friendly and sustainable sources of energy continues to grow [6]. Research has confirmed that the production of biogas from anaerobic digestion is a promising process, but yet to be cost-efficient with reference to natural gas. Detailed research has been conducted that project the potential cost savings of biogas production from different biodegradable matter and the implementation of anaerobic digestion as proper waste management methods, with the goal of providing investors with informed decisions [6].

Mao et al. [7] conducted a review of the research accomplishments on anaerobic digestion development pertaining to biogas production. Their review elaborates on the factors influencing the efficiency, reactors, and anaerobic digestion processes based on the substrate characteristics. Mao et al. [7] concluded that the factors impacting the efficiency are of paramount importance to anaerobic digestion because they are very significant in the biogas yield and in the determination of the prevailing metabolic conditions for the growth of both acid and methane forming bacteria.

Ciobla et al. [8] conducted research that emphasized the use of agricultural biomass residues through anaerobic fermentation to produce biogas of high quality. Their study exploited wheat bran and a mix of defective ground grains substrates for biogas production; strong conclusions were established from the results that the general characteristics of the biomass used and the process parameters will directly influence the biogas production for each batch with respect to the substrate used.

Grando et al. [9] performed a study to demonstrate the existence of a possible correlation between the academic research and the technology development evolvement from 1990 to 2015, based on the production of biogas from anaerobic digestion processes through a technology perspective. The study also revealed that there was an eminent potential growth of biogas technology in Europe and not solely for energy generation but as fuel and digestates from different kind of waste.

Al-Wahaibi et al. [10] conducted a study on the potential of biogas production from various types of food waste. The authors performed the biogas tests at two set time (24 h and 21 days) intervals, and both results showed very good correlation with using derived polynomial models to predict the yield of biogas and methane during fermentation with the determination coefficient of 0.99 between the theoretical and practical values.

Pramanik et al. [11] conducted a review on the characteristics of food waste, biological process, and biochemical reaction occurring in anaerobic digestion process, the different operational parameters, and classification of the anaerobic process, co-digestion, and pre-treatment of anaerobic digestion process for biogas production. The finding depicted that both co-digestion and pre-treatment processes may enhance the food waste hydrolysis rate and the biogas yield.

The microbial communities in the anaerobic digestion digester are contributing significantly to the production of biogas. The need to establish an in-depth understanding of the microbial compositions, diversity or similarity, metabolic routes, and functional gene arrangements are crucial to optimize microbial productivity and assist in improving anaerobic digestion process [12]. Zhang et al. [12] exploited big-data-based precision fermentation platforms using artificial neural networks and proposed the integration of bioinformatics data of microbial communities with the performance of anaerobic digesters to simplify the utilization of huge metagenomics data.

Ruthiraan et al. [13] conducted an experimental study on the factors and parameters affecting the production of biogas from organic solid waste and concluded that several factors such as temperature, pH, organic loading rate, and hydraulic retention time need to be taking into considerations.

Haryanto [14] conducted a lab-based experimental research of a self-designed anaerobic digester with the aim of determining the impact of the hydraulic retention time on the biogas production using cow manure as feedstock. The author also showed that biogas production was at a stable rate after day 44 and used the modified Gompertz kinetic model to predict the daily biogas production as a function of digestion time.

Wang et al. [15] investigated the effect of temperature on the biogas yield and the microbial community patterns in a dual phase anaerobic digester with cow manure and corn straw as feedstock. The findings confirmed that chemical oxygen demand and volatile fatty acid in the acidogenic phase and biogas yield in the methanogenic phase maintained a higher temperature range (25–35 °C). Furthermore, the authors determined that at temperature above 25 °C, the methane content in the biogas yield could be greater than 50%, whereas a low temperature negatively impacted the performance both in the acidogenic and methanogenic phases.

Nandi et al. [16] conducted a laboratory-scale experiment in Bangladesh to assess the effect of temperature on the anaerobic digestion of cow dung. A reactor with a capacity of 15 L was continuously stirred and the retention period was 30 days. The temperature was set at 5 °C interval from 20 to 45 °C while different parameters (total solids, volatile solids, pH, volatile fatty acids, ammonium–nitrogen, total nitrogen, biogas production rate, and percentage of methane) were determined. A statistical analysis of the experimental processed data using Minitab showed that the optimum process performance based on the methane production and volatile solid degradation were achieved in the reactor at 35.82 °C.

1.1. Mathematical Models Used to Predict Biogas Production in Anaerobic Digestion Process

Mathematical models are mathematical equations, computational programming, or algorithms that can be employed to describe the dynamic behaviour of a physical system or process [17]. All mathematical models comprise input(s) block, model block, and output(s) blocks, whereby the input, model, and output blocks represent the predictors, the mathematic equation or computation algorithm, and the desire response for the model which is capable of describing an actual plant or process [18]. Although kinetic models of differential systems of equations are proven to best predict the biogas production of anaerobic digestion process, these models are complex and challenging to implement. Most mathematical models used in the prediction of performance in anaerobic digestion exploit the mechanistic model derived from anaerobic digestion model no. 1 (ADM1), which is a complex and dynamic model involving four stages (hydraulitic, acidogenic, acetogenic, and methanogenic stages) of the anaerobic digestion process [19]. Despite the complexity of the ADM1 model, it focuses on the optimization of the reaction conditions and kinetics but neglects the impact of the biodegradation of different organic forms of waste. Poggio et al. [20] proposed a simplified model derived from standard ADM no.1 to predict the methane production from organic waste with organic matter and volatile fatty acid as the predictors and the measured and model methane yield gave a determination coefficient of 0.91.

Biswas et al. [21] developed a deterministic mathematical model exploiting differential systems of equations and was used to satisfactorily predict the biogas generation characteristics of anaerobic digesters with vegetable residues as the biodegradable waste. Martinez et al. [22] developed a mathematical model comprising seven systems of differential equations to model the biogas production of an anaerobic digestion of a laboratory-scale plant for slaughterhouse effluents. The mean absolute error was negligible when compared with the smallest volume of biogas produced throughout the experiment. Xia et al. [23] conducted a review of the different ADM no.1 model, artificial neural network, empirical models, machine learning, and computational fluid dynamics models to predict the performance of anaerobic digestion of different forms of biodegradable waste. The authors concluded that there is a need to standardize the mathematical models because it will go a long way in advancing the economic and technical benefits of the implementation of mathematical models to predict behaviours in anaerobic processes in biodigesters.

Multiple linear and non-linear regression models are simple models with the capability of predicting biogas yields with better accuracy while less time is consumed in the implementation process. Mukumba et al. [24] predicted daily methane yields through the utilization of multiple linear regression model with input parameters including the pH, total chemical oxygen demand, digester temperature, total alkalinity, and ratio of ammonium to nitrogen for a fixed dome anaerobic digester that was charged with a 75% cow dung and 25% donkey dung. The determination coefficient obtained with the mathematical model was 0.986 between the measured and modelled methane production, whereas the efficiency expressed as the ratio of the determination coefficients of the experimental measured and the modelled methane yield to that of the optimized and the modelled methane yield was 99.99%.

Obelike et al. [25] developed and validated a robust mathematical equation in the form of a multiple linear regression model to predict the volume of biogas with the input parameters as products of ambient condition and physicochemical properties of the anaerobic digestion of cow manure in an underground batch-type biodigester. The authors showed that the predicted volume of methane production mimic that of the experimental volume of methane production and the determination coefficient was 0.995.

Budianto and Sudjarwo [26] developed a simplified linear regression model to analyse the biogas production rate in a continuous digester. The results revealed that the biogas production rate could increase by recycling a portion of the effluent.

An empirical model based on regression analysis was employed to predict the methane production in an anaerobic digestion of swine wastewater [27]. The mathematical model was developed to establish the correlation between the organic loading rate and the methane production; the determination coefficient between the experimental and predicted methane yield ranged from 0.97 to 0.99.

Addario et al. [28] developed a non-linear regression fitting model employing artificial fuzzy logic model to predict both the biogas yield and the methane fraction with very high accuracies with pH, chemical oxygen demand, volatile fatty acid, alkalinity, and zinc concentration as the input variables based on simulated data from landfill. The authors determined the correlation coefficient between the produced and predicted biogas and methane yields as 0.923 and 0.965, respectively.

Multilinear regression models were exploited to predict the biogas production from the dry anaerobic digestion of organic fraction municipal solid waste with predictors such as the total volatile solid, organic loading rate, hydraulic retention time, C/N ratio, lignin content, and total volatile fatty acids [29]. The prediction based on multiple linear regression showed better accuracy and the determination coefficient between the measured volume of biogas produced to the predicted values was 0.91 when compared with the simple linear regression model whose determination coefficients were less than 0.40 [30].

Non-linear regression models and artificial neural networks were used to accurately predict the biogas production rate using an anaerobic hybrid reactor with input parameters as the chemical oxygen demand, hydraulic retention time, and organic loading rate [31]. The correlation coefficients between the measured and modelled biogas production rate were very high, exceeding 0.965. Dahunsi [32] utilized both single and multiple linear regressions models in order to correlate the chemical composition of the biomass with their methane potentials and obtained average correlations of 0.63 between the measured and the modelled methane potential for the mechanical pre-treatment of lignocelluloses. Research exploring multiple linear and nonlinear regression models as well as optimization methods are critical in demonstrating the influence of physicochemical and weather factors on the biogas production of anaerobic digestion in biodigesters [33]. Most studies conducted to predict the biogas yield in an anaerobic digestion process in biodigesters employing multiple linear and non-linear regression models are based on animal and agricultural waste in fixed-dome batch digesters and in hybrid anaerobic reactors. The study demonstrated a degree of innovation because it integrated the 2D multi-contour surface plots with the developed multiple linear regression model and the non-linear regression model based on the Hougen–Watson model to demonstrate a visual prediction of bacteria counts in a balloon digester charged with cow manure. The Hougen–Watson model is a robust and universal non-linear regression equation suitable to predict dynamic processes, especially the rate of reactions.

In this study, we focused on the biogas yield potential of dairy manure collected from the Fort Hare Farm, Eastern Cape province of South Africa, by comparing the accuracies of the developed multiple linear regression and the non-linear regression models in the prediction of the total viable bacteria counts in an anaerobic balloon digester with the number of days, daily slurry temperature, and pH as the predictors.

1.2. Objectives of the Study

The study sought to accomplish the following objectives:

i.

To develop both multiple linear regression and non-linear regression models to predict the total viable bacteria counts with the number of days, daily slurry temperature and pH as predictors for a balloon digester charged with cow manure.

ii.

To use the 2D multi-contour surface plots derived from the developed mathematical models to illustrate the variation in each of the predictors with the total viable bacteria counts.

iii.

To exploit the 2D multi-contour surface plots for both the multiple linear regression and non-linear regression models to simulate design experiment’s data for each predictor with the desired targets while the others are held constant.

iv.

To compare the prediction accuracies of the multiple linear regression and the non-linear regression models.

2. Methodology of the Study

The study methodology consists of the experimental setup and methods to determine the key parameters needed in the development of the mathematical models.

2.1. Materials and Experimental Setup

Table 1. Materials and sensors used in the study.

A. Devices and Materials Employed in Both the Physical and Chemical Analysis
Item	Device and Materials	Quantity
1	Centrifugal tubes	10
2	Nessler’s reagent	-
3	Hexios, Thermo-Spectronics Spectrometer	1
4	Distilled water	-
5	Dish	1
6	Electric oven	1
7	PHH-SD1 pH meter	1
8	TMC6-HD copper pipe temperature sensors	4
9	Portable biogas analyser, IRCD4	1
10	Tryptic soy broth medium	-
11	Physical saline	-
12	Triplicates of microbiological media (nutrient agar and anaerobic agar)	-
13	Fridge	1
14	Weight balance	1
15	Incubator	1
16	muffle furnace	1
B. Devices, Sensors, and Materials Used in the Experimental Setup
Item	Device and Materials	Quantity
1	Balloon digester	1
2	Feedstock of cow manure (slurry)	2500 L
2	An open surface concrete structure	1
3	A black wooden board (insulation cover)	1
4	Portable biogas analyser, IRCD4	1
5	TMC6-HD copper pipe temperature sensors	4
6	Hobo-UX120 four external channel data logger	1
7	PHH-SD1 pH meter	1
8	Slurry measuring cylinder (250 L)	1
9	ZAN-TECHS gas flow meters with data logger	1
10	Control flow valve	1
11	Biogas circulation pump	1
12	Connecting biogas tubing or pipe	1
13	Biogas collection chamber	1
14	Weatherproof data loggers’ enclosure	1
15	A zinc roof open structure (protection of digester, ambient temperature sensor and data loggers)	1

shows the materials, sensors, data loggers, and technological device used in the study.

The balloon digester was accommodated within a concrete structure of 8 m³ by volume and partition into three sectors, including an influent tank of dimensions 0.95 m by 0.89 m by 0.83 m, a bioreactor tank of length 3.25 m and breadth 2 m and an effluent tank of 1.2 m³ by volume. Figure 1 shows a schematic diagram of the balloon digester embedded in the concrete structure and the installed temperature sensors, gas flow transducer, and pH electrode installed at precise location of the balloon digester. The different data loggers were housed in a weatherproof enclosure to shield from unfavourable weather conditions. Five consecutive days (26–30 June 2015) were used to charge the balloon digester with a uniform mixture of dairy cow manure and water at a ratio of 1:1. The adopted mixing ratio of the cow manure to water in a bid to prepare the slurry was ascertained relative to the moisture content of the waste [34]. The physicochemical parameters (pH, moisture, percentage of total solids, percentage of volatile solids, ammonium level and percentage of ash content) of the undigested waste were determined before the preparation of the slurry. The bottom and intermediate temperature sensors (labelled 1 and 2) measure the average temperature of the slurry, whereas the temperature sensors within the biogas region in the balloon digester (labelled 3) and in the vicinity of the concrete structure (labelled 4) measured the temperature of the biogas and ambient temperature. All the temperature sensors were connected to an external four-channel data logger (labelled 5). The pH electrode (labelled 25) was inserted in the slurry and connected to the pH meter (labelled 24), which recorded the measured pH of the slurry. The biogas flow meter (labelled 8) was installed along the connecting tubing transporting the biogas produced from the balloon digester (labelled 18) to the biogas collection chamber (labelled 22). The cumulative volume of the biogas was recorded and stored in the inbuilt data logger of the biogas flow meter (labelled 9). The transducer of the biogas analyser (labelled 6) sensed and analysed the composition of the biogas produced from the anaerobic digestion in the balloon digester and the percentage of biogas (methane and carbon dioxide) where stored in the biogas analyser (labelled 7). The open and close control valve (labelled 19) regulated the flow rate of biogas from the balloon digester to the biogas collection chamber. The gas circulation pump (labelled 21) provided the appropriate pressure for biogas to be transported from the balloon digester to the biogas collection chamber. The balloon digester was monitored for six months (July to December 2015) while operating in a batch mode. Samples were withdrawn on a daily basis for analysis of the microbial loads, while the pH and temperature of the slurry were continually monitored in 1 min intervals and the average daily values were determined for the entire hydraulic retention time.

2.2. Methods

2.2.1. Raw Anaerobic Digestion Material (Cow Manure)

Equal volumes of five samples of fresh cow manure (labelled as sample’s A–E) with a cumulative volume of 2500 L over consecutive weekdays (26–30 June 2015) were used to charge the balloon digester and the cow waste was procured from the Fort Hare Diary Farm, Alice.

2.2.2. Physicochemical Analysis of the Slurry Samples

i.

Calculation of the ammonium (NH4) level of sample

The technique by Ziganshin et al. [35] was employed in the determination of ammonium level. Each of the five samples were centrifuged at the rate of 2000× g in 20 min and the supernatant was decanted and coloured with Nessler’s reagent in accordance with the procedure of Manyi-Loh et al. [36]. The wavelength of the absorbance of the coloured solutions was determined to be 425 mm with the aid of Hexios thermo-spectronics spectrometer. The calculation of the ammonium level was performed by applying the standard (ammonia solution) with a concentration of 0.909 g/mL. In addition, distilled water was utilized as a blank to neutralize the absorbance of water in the samples and standard.

ii.

Determination of percentage of moisture content of samples

The percentage of total moisture content was determined by the APHA method [37] and applying the same procedure described by Fridh et al. [38]. The method involved weighing the samples in a dish and drying in an oven at a temperature of 105 °C overnight. Both the weight of the dish and sample before drying and that of the dish and sample after drying were measured with a mass balance. The percentage of moisture content was evaluated using the APHA standard equation shown in Equation (1).

$$\% \mathrm{moisture} \mathrm{content}=\frac{\left({\mathrm{m}}_2-{\mathrm{m}}_1\right)-\left({\mathrm{m}}_3-{\mathrm{m}}_1\right)}{\left({\mathrm{m}}_2-{\mathrm{m}}_1\right)}\times 100\mathbf{}$$

(1)

where ${\mathrm{m}}_1$ is the mass in grams of the empty dish, ${\mathrm{m}}_2$ is the mass in grams of the sample and empty dish before drying, and ${\mathrm{m}}_3$ is the mass in grams of the sample and empty dish after drying.

iii.

Determination of percentage of dry matter (total solids)

The approach of Bradley [39] was used to determine the percentage of total solids. The method involved determining the weight of a sample (W_S) in a dish and again measuring the weight of the dry matter (W_DM) after drying at 105 °C in an oven over a 24 h period. The percentage of total solids was calculated using the standard equation from the APHA method, and is given in Equation (2).

$$\% \mathrm{total} \mathrm{solids}=\frac{{\mathrm{W}}_{\mathrm{DM}}}{{\mathrm{W}}_{\mathrm{S}}}\times 100$$

(2)

iv.

Determination of percentage of volatile solid content and ash content

After the determination of the total solids, the sample dried overnight was combusted using a muffle furnace at 550 °C for a 1h period. The residual weight of ash and the dish was recorded, and the percentage of volatile solid was derived using the standard method called APHA; the calculation formula is given in Equation (3).

$$\% \mathrm{volatile} \mathrm{solid}=\frac{\left({\mathrm{W}}_{\mathrm{DM}}-{\mathrm{W}}_{\mathrm{ash}}\right)}{{\mathrm{W}}_{\mathrm{DM}}}\times 100$$

(3)

In addition, the percentage of ash content can be obtained from Equation (4), as postulated by Van Wychen et al. [40], following the APHA method [37].

$$\% \mathrm{ash} \mathrm{content}=\left(\frac{\left({\mathrm{m}}_4-{\mathrm{m}}_1\right)}{\left({\mathrm{m}}_2-{\mathrm{m}}_1\right)}\times 100\right)\times \frac{100}{100-\%\mathrm{moisture} }$$

(4)

where m₁ is the mass in grams of the empty dish, m₂ is the mass in grams of the sample and empty dish before drying, and m₄ is the mass in grams of the ash and empty dish.

The physicochemical characterization of the waste is presented in

Table 2. Sample characterization based on utilizing feedstock of cow manure.

Parameters	Sample A	Sample B	Sample C	Sample D	Sample E	Mean	Standard Deviation
Percentage of moisture content	89.12	91.23	91.19	89.96	90.78	90.38	0.886
Percentage of total solids	10.88	8.77	8.81	10.04	9.22	9.55	0.975
Percentage of volatile solids	65.76	68.25	72.65	71.55	73.04	70.25	3.138
Percentage of ash content	34.24	31.75	27.35	28.45	26.96	29.75	3.138
Ammonium (NH4) level (mg/mL)	2.19	2.01	2.25	2.28	2.20	2.18	0.105
pH	6.83	6.82	6.59	6.76	6.46	6.69	0.162

. The study was conducted in a six-month period (July to December 2015) at the Physics Department, University of Fort Hare, Alice Campus, Eastern Cape, South Africa.

v.

Determination of pH, ambient temperature, slurry temperature and biogas yield

The average daily ambient temperature was measured using the hobo copper pipe temperature sensor (labelled 4), and the slurry temperature was measured using the hobo copper pipe temperature sensors (labelled 1 and 2). All the temperature sensors were connected to a hobo four-external channel data logger (labelled 5). The pH of slurry was measured using the pH electrode (labelled 25) inserted at the bottom of the balloon digester and the recorded pH was stored in the PHH-SD1 pH meter (labelled 24) which was connected to the pH electrode by specialized cables. The percentages of the composition of the biogas produced were sensed by the transducer (labelled 6) and analysed (methane (65.8%) and carbon dioxide (31.2%)) by a portable biogas analyser, IRCD4 (labelled 7). The cumulative volume of biogas produced over the six-month hydraulic retention time (July to December 2015) was measured by the biogas flow meter (labelled 8) and stored by the data logger ZAN-TECHS gas flow data logger (labelled 9). Furthermore, triplicate determination was conducted on each of the parameters.

vi.

Microbial analysis of samples

The method employed by Sahlström [41] was used to determine the total viable bacteria counts for the undigested and withdrawn samples during digestion of the slurry. Each sample was aseptically collected and poured into a trptic soy broth medium in sterile centrifuge tubes for onward transportation to the laboratory based on best practices. The samples were analysed immediately once arrived the laboratory. The determination of the total viable bacteria counts was conducted following subsequent procedure: 1 g of each sample was consecutively diluted tenfold in 9 mL of sterile saline. The range of dilution of 10⁻¹–10⁻⁵ was spread in triplicates in different microbial media (nutrient agar to obtain total aerobic bacteria counts and anaerobic agar to obtain total anaerobic bacteria counts, respectively). Above all, the inoculated plates were incubated at 37 °C for 24 h for the aerobic and anaerobic bacteria counts. At the end of the incubation, the number of emergent colonies on each plate was counted and recorded, with each value representing the average of the triplicate plating in accordance with the method used by Bodhidatta et al. [42].

2.2.3. Statistical Analysis

The analytical processing and statistical test were conducted with MATLAB software version 2021a by applying the procedural standard methods [43]. The one-way analysis of variance (ANOVA) test was utilized to check on any significant difference between the testing data of the targets and the model outputs as well as the validation data of the targets and the predicted values using the derived mathematical models (multiple linear and non-linear regression models) based on their p-values [44,45]. The reliefF algorithm was employed to rank the contribution of the predictors (number of days, daily slurry temperature, and pH) according to the weight of importance to the desired output (total viable bacteria counts) [46].

2.2.4. Development of Non-Linear Regression Model

A non-linear fitting model (Hougen’s model) with the dynamics of a non-linear regression model was developed using number of days (x1), daily slurry temperature (x2), and pH (x3) as the predictors, while the log of total viable bacteria counts (y) was the desired response. The reliefF algorithm was employed to rank the contribution of the predictors (number of days, daily slurry temperature, and pH) according to the weight of importance to the output (log of total viable bacteria counts) [47]. The determination coefficient, mean absolute error, and p-value were used to test the accuracy of the model with reference to the predictions (model outputs) and the targets (experimental determined log of total viable bacteria counts) [48]. 2D multi-contour surface plots were developed to simulate the production of the total viable bacteria counts with each of the predictor using the derived non-linear regression model [49].

The developed non-linear regression model used to predict the log of total bacteria counts with the number of days, daily slurry temperature, and pH as predictors is given in Equation (5).

$$\mathrm{y}=\frac{{\mathsf{\beta }}_1\mathrm{x}2-\mathrm{x}3/{\mathsf{\beta }}_5}{1+{\mathsf{\beta }}_2\mathrm{x}1+{\mathsf{\beta }}_3\mathrm{x}2+{\mathsf{\beta }}_4\mathrm{x}3}$$

(5)

where y is the Log of total bacterial counts, x1 is the number of days, x2 is the daily slurry temperature, x3 is the pH, and ${\mathsf{\beta }}_1$, ${\mathsf{\beta }}_2$, ${\mathsf{\beta }}_3$, ${\mathsf{\beta }}_4$ and ${\mathsf{\beta }}_5$ are scaling factors.

The mathematical model was developed in MATLAB and by considering the modelled equation as the optimization function, whereby y represented the desired response (log of total viable bacteria counts) and x1, x2, and x3 represented the predictors (number of days, daily slurry temperature, and pH, respectively) while ${\mathsf{\beta }}_1$, ${\mathsf{\beta }}_2$, ${\mathsf{\beta }}_3$, ${\mathsf{\beta }}_4$, and ${\mathsf{\beta }}_5$ were the given scaling parameters attributed to specific input variables. The values of the scaling parameters were determined by performing non-linear function optimization with the modelled equation as the optimization function using the trained (testing) dataset for both the input and output parameters obtained from the experimental data. The optimization algorithm was implemented by choosing initial values for the input and output parameters. A computation iteration was executed by running the optimization function with chosen initial values for x1, x2, x3, and y to form the initial condition. The iteration stopped when the model outputs (predicted log of total bacteria counts (yp) highly accurately mimicked the actual targets (determined log of total viable bacteria counts (yo) and the correct scaling values of each of the scaling parameters are computed.

Table 3. Input and output parameters of the developed non-linear regression model.

Input Parameters	Input Symbols	Scaling Attribute	Scaling Values	Output
Constant			1	Log of total bacteria counts (y) Determination coefficient (r2) = 0.959, Mean absolute error (MAE) = 0.180, p-value = 0.602
Number of days	x1	${\mathsf{\beta }}_2$	0.0047
Daily slurry temperature	x2	${\mathsf{\beta }}_1$ ${\mathsf{\beta }}_3$	0.4947 0.0521
pH	x3	${\mathsf{\beta }}_4$ ${\mathsf{\beta }}_5$	−0.1445 2.2002

shows the inputs and output parameters and the scaling values derived from the developed non-linear regression model. The input variable x1 was associated with a single scaling attribute (${\mathsf{\beta }}_2$) and x2 was assigned with two scaling parameters (${\mathsf{\beta }}_1$ and ${\mathsf{\beta }}_3$), while x3 was attributed two scaling parameters (${\mathsf{\beta }}_4$ and ${\mathsf{\beta }}_5$). The determination coefficient, mean absolute error, and p-value between the log of total viable bacterial counts (yo) and the predicted log of total viable bacteria counts (yp) based on the testing dataset were 0.959, 0.180 and 0.602, respectively. The high value of the determination coefficient, which is close to 1, is an indication that the developed non-linear regression model demonstrates a very good agreement between the predicted outputs (yp) and the actual targets (yo). The large p-value (0.602), which is greater than 0.05 (threshold value), was obtained between the predicted outputs (yp) and the actual targets (yo) and confirmed that there was no significant difference between the two groups within a 95% confidence level. The mean absolute error between the predicted outputs and the actual targets was smaller than the minimum value of the actual target. Therefore, these very good values for the determination coefficient and mean absolute error provide adequate reason for the utilization of the developed non-linear regression model.

2.2.5. Development of Multiple Linear Regression Model

A multiple linear regression model is a data driven model and exploits the least squares regression method to determine both the forcing and scaling constants associated with the input parameters of the developed model. The developed model used the number of days (x1), daily slurry temperature (x2), and pH (x3) as the predictors, while the log of total viable bacteria counts (y) was the desired response. The determination coefficient, mean absolute error, and p-value were used to test the accuracy of the model based on the prediction values (model outputs) and the targets. 2D multi-contour surface plots were developed to simulate the total viable bacteria counts with each of the predictors using the derived multiple linear regression model [40].

The mathematical model equation for the multiple linear regression model contains all the selected predictors expressed independently and in a linear form, as shown in Equation (6).

$$\mathrm{y}={\mathsf{\gamma }}_1+{\mathsf{\gamma }}_2\mathrm{x}1+{\mathsf{\gamma }}_3\mathrm{x}2+{\mathsf{\gamma }}_4\mathrm{x}3$$

(6)

where x1 is the number of days, x2 is the daily slurry temperature, x3 is the pH, y is the log of total viable bacteria counts, ${\mathsf{\gamma }}_1$ is the forcing constant, and ${\mathsf{\gamma }}_2$, ${\mathsf{\gamma }}_3$, and ${\mathsf{\gamma }}_4$ are scaling parameters.

The mathematical model was developed in MATLAB, the multiple linear regression model exploited the least square method, and y represented the desired response (log of total bacteria counts), while x1, x2, and x3, represented the predictors (number of days, daily slurry temperature, and pH, respectively) with ${\mathsf{\gamma }}_1$, ${\mathsf{\gamma }}_2$, ${\mathsf{\gamma }}_3$, and ${\mathsf{\gamma }}_4$ as given scaling constants associated with specific input variables. The values of the scaling constants were determined by performing a regression analysis with the trained dataset for the inputs (lump parameter and chosen predictors) and output parameter (log of total viable bacteria counts). The regress algorithm was performed between the input and output parameters. The values of the scaling constants were determined upon the execution of the regression function in MATLAB.

Table 4. Inputs and output parameters of the developed multiple linear regression model.

Inputs	Input Symbols	Scaling Attribute	Scaling Values	Output
Constant	1	${\mathsf{\gamma }}_1$	1.2067	Log of total bacteria counts (y) Determination coefficient (r2) = 0.920, Mean absolute error (MAE) = 0.206, p-value = 0.514
Number of days	x1	${\mathsf{\gamma }}_2$	−0.0172
Daily slurry temperature	x2	${\mathsf{\gamma }}_3$	0.1205
pH	x3	${\mathsf{\gamma }}_4$	0.3871

shows the inputs, outputs, and scaling values determined from the developed multiple linear regression model. The input variables x1, x2, and x3 were associated with the scaling attributes (${\mathsf{\gamma }}_2,{ \mathsf{\gamma }}_3,{ \mathrm{and} \mathsf{\gamma }}_4$) and the lump parameter with the scaling constant (${\mathsf{\gamma }}_1)$. The determination coefficient, mean absolute error, and p-value between the measured log of total viable bacteria counts (yo) and the predicted log of total viable bacteria counts (yp) for the testing dataset were 0.920, 0.206, and 0.514, respectively. The value of the determination coefficient was close to 1 and insinuated that the developed multiple linear regression model could provide a good prediction with negligible difference between the predicted outputs (yp) and the actual targets (yo). The large p-value was significantly greater than 0.05, revealing that no significant difference existed between the predicted outputs (yp) and the actual targets (yo) within a 95% confidence level. The mean absolute error between the predicted model outputs and the actual targets was smaller with reference to the minimum value of the actual targets and confirmed the suitability of the developed multiple linear regression model. The developed model was acceptable for the prediction of the log of total viable bacterial counts, especially if the validation of the model yielded acceptable values for the determination coefficient, mean absolute error, and p-value.

2.2.6. Measurement Accuracies and Uncertainties

The calculated uncertainties from the determined parameters with referenced to the result of the error measurements obtained from the set of independent variables, is given by Equation (7) [50].

$${\mathrm{w}}_{\mathrm{r}}={\left[{\left[{\mathrm{w}}_1\frac{\partial \mathrm{R}}{\partial {\mathrm{X}}_1}\right]}^2+{\left[{\mathrm{w}}_2\frac{\partial \mathrm{R}}{\partial {\mathrm{X}}_2}\right]}^2+\dots \dots \dots \dots +{\left[{\mathrm{w}}_{\mathrm{n}}\frac{\partial \mathrm{R}}{\partial {\mathrm{X}}_{\mathrm{n}}}\right]}^2\right]}^{\frac{1}{2}}$$

(7)

where $\mathrm{R}$ is the given function, ${\mathrm{w}}_{\mathrm{r}}$ is the total uncertainty, ${\mathrm{X}}_1,{ \mathrm{X}}_2, \dots ,{\mathrm{X}}_{\mathrm{n}}$ are independent variables, and ${\mathrm{w}}_1,{ \mathrm{w}}_2, \dots ,{\mathrm{w}}_{\mathrm{n}}$ are uncertainties in the independent variables.

The type A uncertainties were measured for the statistical means and standard deviations obtained from the recorded measurements [51] and are shown in

Table 5. Uncertainties of the measured and derived quantities.

Quantity	Type A Uncertainty	Type B Uncertainty	Combined Uncertainty
Ambient temperature (°C)	±0.200	±0.120	±0.320
Biogas flow rates measurements (L/min)	±0.010	±0.006	±0.016
pH measurements	±0.130	±0.003	±0.133
Slurry temperature (°C)	±0.200	±0.120	±0.320
Percentage total solids content	±0.850	±0.105	±0.955
Percentage total volatile content	±0.850	±1.405	±2.255
Percentage ash content	±0.850	±1.203	±2.053
Ammonium level (mg/mL)	±0.060	±0.020	±0.080

. In addition, the type B uncertainties were attributed to the accuracy of the sensors or by the utilization of Equation (7) to obtain the derived uncertainty of the quantities (percentage of total solids, percentage of volatile solids, and percentage of ash content). Table 4 shows the combined uncertainties of the prescribed quantities (both Type A and Type B uncertainties).

3. Results and Discussion

3.1. Profiles of Ambient Temperature and Input and Output Parameters during Anaerobic Digestion

The variations in the daily average ambient temperature, slurry temperature, and pH are presented on the left y-axis, whereas that of log of total viable bacteria counts is on the right y-axis in Figure 2. The dynamics of the weather condition, the input and output parameters, demonstrated rapid fluctuation throughout the anaerobic digestion process. Even though the slurry temperature of digester may be influenced by ambient temperature, the profiles of the average daily ambient temperature and slurry temperature showed no distinct correlation. This can be attributed to the interactive effect from other input factors such as pH, total volatile fatty acids, biological oxygen demand, insulation of balloon digester, etc. Observation based on the displayed profiles revealed that the anaerobic digestion process occurred in two regimes. The first regime is associated with the psychrophilic process (<20 °C) over the first 60 days (two months), and the second regime illustrates the mesophilic process (20 °C < slurry temperature < 30 °C) for the remaining 119 days (4 months). These results agree with the findings by Cioabla et al. [8], although they conducted their research using agricultural vegetable residues. Further observation confirmed that, during the psychrophilic regime, the ambient temperature, slurry temperature, and pH varied from 18.1 to 21.2 °C, 17 to 19.23 °C, and 5.85 to 6.60, respectively, whereas the log of total viable bacteria counts ranged from 5.00 to 6.02 cfu/g. Furthermore, in the mesophilic regime, the ambient temperature, slurry temperature, and pH fluctuated within the ranges of 21.04–22.58 °C, 20.9–25.5 °C, and 6.1–7.67, respectively, whereas the log of total viable bacteria counts ranged from 3.38 to 5.40 cfu/g.

3.2. Characteristics of the Cow Manure and Physicochemical Properties during Digestion

The results presented in Table 2, based on the percentage of total solids, total volatile solids, and ash content, justified that the cow manure possessed a substantial biodegradable constituent with the capability of undergoing digestion by the microorganisms in the anaerobic digester and generating biogas [52]. Figure 2 revealed that at the initial state (before the commencement of charging), the total aerobic and anaerobic counts were very high. These results affirmed that cow manure is a favourable feedstock (substrate) for biogas production because the rumen microbes are of vital importance in anaerobic digestion, which is responsible for the degradation of the organic composition of the cow manure [53]. The high level of microbial loads, shown in Figure 2, justified that public health control measures need to be reinforced because manure can act as a potential source of water and soil pollution. The failure to treat manure before it is released into the environment can pose high risk and act as a potential source of both soil and water pollution. The digestate can be utilized in the agricultural field in the enhancement of crop yields [54]. The absolute present of high microbial loads in cow manure is enough reason to be concerned. The high microbial loads serve as an eminent threat to living animals and humans because infection is likely probable due to the feasible mode of transmission [55].

More so, the enhancement of the performance of anaerobic bacteria is governed by the temperature and pH of the slurry contained in the digester. The waste from livestock, especially dairy cow manure, has been proven to have high buffering alkaline potential when digested by microorganisms [56]. It is important to mention that throughout this study, the pH of the digester remained uncontrolled. Figure 2 shows that during the first two months of the anaerobic digestion, there was a decrease in the pH medium (pH drop from 6.60 to 5.85), which may be attributed to the high concentrations of volatile fatty acids, bicarbonate alkalinity, and carbon dioxide, which are the resulting end products of the first and second phases of the anaerobic digestion process (hydrolysis and acidogenesis). The findings agreed with those from most studies conducted on the performance of anaerobic digestion in balloon digester; as the process progresses, the volatile fatty acids are metabolized and the pH eventually increases to the appropriate buffering level (neutral pH) favourable to producing biogas [57]. Research has revealed that both acidogenic and methanogenic bacteria have optimum pH conditions for metabolism [58]. The methanogens are extremely sensitive and thrive efficiently within the pH range from 6.6 to 7.6 [59]. In addition, the metabolic rate of microorganisms is influenced by the temperature and can alter the effectiveness of anaerobic microbes in the process of biogas production. Figure 2 also shows that the daily variation in the slurry temperature during the anaerobic digestion is not significantly affected by the ambient temperature. The profile reveals the existence of two regimes: the psychrophilic process (temperature < 20 °C) occurring during the first two months, and mesophilic process (20 °C < slurry temperature < 30 °C) occurring during the last four months of the anaerobic digestion [60].

3.3. ReliefF Test Used in Ranking of the Predictors to the Total Viable Bacteria Counts

A reliefF test is a statistical technique used to rank predictors based on the weight of importance to the desired response (targets) by employing the parametric regression method. The reliefF algorithm is used in MATLAB by executing the command [Rank Weight] = reliefF(x,y,k), which returns the order of ranking of the predictors and their respective weights of importance to the desired response. The quantities x and y represent the predictors and outputs, respectively, while k is the number of potential predictors (x) influencing the output (y) and, by default, is set to 10. The weighting ranged from −1 to 1 and negative weights ascribed to predictors insinuated that the input parameters were secondary factors, while positive weights attributed to input parameters revealed that the input parameters were primary factors. More positive weight values implied that the parameters significantly contributed to the desired output. Figure 3 shows a pie chart of the predictors (number of days, daily slurry temperature, and pH) and the weights of importance to the desired output (log of total bacteria counts). The predictors were primary factors and the number of days contributing the most with a weight of 0.106 (50% contribution). The least influential factor was daily slurry temperature, with a weight of 0.024 (17% contribution); the weighting for the pH was 0.038 (33% contribution).

3.4. 2D Multi-Contour Surface Plots to Simulate the Variation in the Response with Each Predictor

2D multi-contour surface plots are developed from the derived mathematical models to simulate the variation in the output with specific input parameters while the other predictors are held constant [61]. Each of the 2D plots represents predictors on the x-axis and the desired response on the y-axis. A solid green graph (line or curve) is used to demonstrate the variation in the desired response with changes in the specific input parameter with the other input parameters held at constant values, while the two broken red graphs (lines or curves) between the solid green graph represent the 95% confidence bounds.

2D Multi-Contour Surface Plots with the Developed Models

Figure 4a,b show the 2D multi-contour surface plots of the predictors (number of days (x1), daily slurry temperature (x2), and pH (x3)) and the desired response (log of total viable bacteria counts (Y1)), for the non-linear and multiple linear regression models, respectively. The green solid graphs show the rate of change of the predicted output (Y1) with a specific predictor while the other predictors are invariant. The red broken graphs between the solid green graphs represent the 95% confidence bounds. The simulated 2D multiple plots of total viable bacteria counts (Y1) with the number of days (x1) while the other two predictors remain constant (x2 = 20 °C, and x3 = 6.5), Figure 4a, show that the rate of change in predicted log of total viable bacteria counts with the number of days is negative; Figure 4b also shows a linear pattern for x1 and predicted total bacteria counts with the two other input parameters at x2 = 20 °C and x3 = 6.5. The green solid curve can be approximated with a negative slope of −0.0146 cfu/g/day for the variation in the targets with the number of days in Figure 4a, whereas in Figure 4b, the gradient was −0.0172 cfu/g/day and was equal to the scaling value attributed to the input parameter (x1) in the developed multiple linear regression model. The rate of change in the predicted log of total viable bacteria counts (Y1) with slurry temperature (x2) with unchanged predictors (x1 = 71 and x3 = 6.5), Figure 4a, can be approximated to a slope of 0.1205 cfu/g/°C, whereas in Figure 4b, the gradient is 0.1649 cfu/g/°C, with the other input parameters unchanged (x1 = 71 and x3 = 6.5). Again, the gradient was same as the scaling value associated with the input parameter x2 from the developed multiple linear regression model. In addition, the rate of change in the predicted log of total viable bacteria counts (Y1) and pH (x3) in Figure 4a with the predictors (x1 = 71 and x2 = 20 °C) staying unchanged, had a positive slope of 0.1771 cfu/g, whereas in Figure 4b, the gradient of the bacteria count (Y1) and the pH (x3) was 0.3871 cfu/g and corresponded with the scaling value for the pH (x3) from the multiple linear regression model.

Table 6. Design experimental results from 2D simulated plots using the developed models with daily slurry temperature and pH held constant (x2 = 20 °C and x3 = 6.5).

Chosen Input x1 (Number of Days)	Predicted y with 2D Simulation Response Surface Model (Log of Bacteria counts)	Predicted y 2D Simulation with Multiple Linear Regression Model (Log of Bacteria Counts)	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts
10	6.036	5.961	6.035	5.960
30	5.580	5.617	5.579	5.616
50	5.188	5.273	5.187	5.272
70	4.847	4.929	4.847	4.928
90	4.548	4.585	4.548	4.584
110	4.284	4.241	4.284	4.240
130	4.049	3.897	4.049	3.896
150	3.839	3.553	3.838	3.552
170	3.649	3.209	3.649	3.208

shows the design-simulated results of the sample dataset obtained from the generated 2D multi-contour surface plots of number of days (x1) and the modelled log of total viable bacteria counts (Y1) based on the non-linear regression model and the multiple linear regression with x2 = 20 °C and x3 = 6.5 remaining unchanged. It was observed that the outputs for the 2D simulation with the non-linear regression model and multiple linear regression model were in perfect agreement with their corresponding modelled outputs determined using the derived non-linear and multiple linear regression models. Furthermore, both predicted model outputs for the non-linear and multiple linear regression models were very closed to each other. These strong correlations between the results from the 2D simulations and the mathematical models confirmed that the simulation with the 2D multi-contour surface plots can be used in accurately predicting the total bacteria counts.

Table 7. Design experimental results from 2D simulated plots using the developed models with number of days and pH held constant (x1 = 70 and x3 = 6.5).

Chosen Input x2 (Daily Slurry Temperature)	Predicted y with Response Surface Model (Log of Bacteria Counts)	Predicted y with Multiple Linear Regression Model (Log of Bacteria Counts)	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts
17	4.277	4.567	4.277	4.567
18	4.482	4.688	4.482	4.687
19	4.672	4.808	4.671	4.808
20	4.842	4.928	4.847	4.929
21	5.010	5.049	5.010	5.049
22	5.162	5.170	5.163	5.169
23	5.305	5.290	5.304	5.290
24	5.438	5.411	5.437	5.410
25	5.563	5.531	5.562	5.531

shows the design-simulated results of the sample dataset obtained from the generated 2D multi-contour surface plots of daily slurry temperature (x2) and the modelled log of bacteria counts based on the developed non-linear and multiple linear regression models whereby x1 = 71 and x3 = 6.5 are held constant. It was observed that the 2D simulation prediction and modelled outputs based on the non-linear regression model had an excellent agreement and same was the case with the multiple linear regression model. In addition, both results from either the non-linear regression or multiple linear regression from the derived simulation using the 2D multi-contour surface plots shows no significant difference. Therefore, the two simulations using the 2D multi-contour surface plots accurately predicted the total bacteria counts; however, the non-linear regression yielded better predictions based on the determination coefficient and mean absolute values.

Table 8. Design experimental results from 2D simulated plots using the developed models with number of days and daily slurry temperature constant (x1 = 70 and x2 = 20 °C).

Chosen Input x3 (pH)	Predicted y with Response Surface Model (Log of Bacteria Counts)	Predicted y with Multiple Linear Regression Model (Log of Bacteria Counts)	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts	Modelled with Non-Linear Regression Model Total Log of Bacteria Counts
5.50	4.691	4.542	4.691	4.541
5.75	4.727	4.639	4.727	4.638
6.00	4.765	4.735	4.765	4.735
6.25	4.805	4.832	4.805	4.832
6.50	4.847	4.929	4.847	4.928
6.75	4.891	5.026	4.891	5.025
7.00	4.938	5.122	4.937	5.122
7.25	4.987	5.219	4.986	5.219
7.50	5.038	5.316	5.038	5.315

shows the design-simulated results of the sample dataset obtained from the generated 2D multi-contour surface plots of pH (x3) and the predicted log of total viable bacteria counts (Y1) with the non-linear regression and multiple linear regression models, whereby x1 = 71 and x2 = 20 °C remained unchanged. It was observed that the predicted and modelled outputs based on the non-linear regression model are of no significant difference and same was the case when comparing the predicted and modelled outputs with referenced to the multiple linear regression model. The predicted outputs with the non-linear regression model show negligible differences from the predicted outputs using the multiple linear regression model, but yielded better results due to the very good values for the determination coefficient and mean absolute error.

3.5. Testing of the Accuracy of the Developed Models

The dataset of the input and out parameters during the six months (179 days) of hydraulic retention time of the anaerobic digestion of cow manure in a balloon digester were randomly split into testing and validation datasets, at a ratio of 60% and 40%, respectively. The dataset (107 test samples) of the measured input parameters (number of days, daily slurry temperature, and pH), were used in the derivation of the model equations to determine the predicted outputs (log of total viable bacteria counts) called the model outputs, whereas the 107 values of the experimental log of total viable bacteria counts were used as the targets. The targets and the model outputs are compared to verify the accuracy of the developed models based on the determination coefficients, mean absolute errors, and the p-values.

3.5.1. Testing of the Accuracy from Non-Linear and Multiple Linear Regression Models

Figure 5a,b show the data of the determined log of total viable bacteria counts (targets) for the test dataset and the modelled graph from the model outputs of the non-linear and multiple linear regression models, respectively. It was observed that the model outputs accurately predicted the experimental targets with reference to the two models. The determination coefficient, mean absolute error, and p-value between the targets and model outputs for the non-linear regression model were 0.959, 0.180, and 0.602, respectively, whereas the outputs for the multiple linear regression model they were 0.920, 0.206, and 0.514, respectively. The determination coefficient and the mean absolute error for the non-linear regression model were better than the corresponding values with the multiple linear regression model. Hence, the prediction with the non-linear regression model was more accurate compared with the prediction with the multiple linear regression model. The p-values for both models were significantly greater than the threshold value of 0.05, which confirmed that no significant difference existed between the targets and the model outputs for the two models. Therefore, these developed models gave good predictions and can be accepted in the prediction of the log of total viable bacterial counts.

3.5.2. ANOVA Statistics to Confirm Model Accuracy with Testing Dataset

Table 9. ANOVA results for the test dataset of targets and model outputs.

Statistics Based on Targets and Model Outputs Using the Non-Linear Regression Model
Source	Sum of Square (SS)	Degree of Freedom (df)	Mean Square (MS)	F	Prob > F
Columns	0.0674	1	0.0674	0.27	0.6017
Error	53.2762	216	0.2467
Total	53.3436	217
Statistics Based on Targets and Model Outputs with the Multiple Linear Regression Model
Source	Sum of Square (SS)	Degree of Freedom (df)	Mean Square (MS)	F	Prob > F
Columns	0.1115	1	0.1115	0.43	0.5140
Error	56.3962	216	0.2611
Total	56.5077	217

shows the ANOVA results for the targets (experimental log of total viable bacteria counts) and the model outputs with the dataset for the testing samples using the non-linear and multiple linear regression models. The columns and error under the source are presented between the groups (columns = 2) and within groups (error = 216). The degree of freedom within columns is equal to the number between groups minus 1 (columns −1), and was equal to 1. The degree of freedom of the error is equal to the degree of freedom within the groups and was 216. The total degree of freedom is the sum of the degree of freedom between columns and errors and was 217. The groups of targets and model outputs with the non-linear regression model revealed that the mean square of the columns, which is the ratio of the sum of square and the degree of freedoms for columns, was 0.0674 (0.0674/1), whereas for the multiple linear regression, it was 0.1115 (0.1115/1). The mean square of the error for the non-linear regression model, which is the ratio of sum of the square and the degree of freedom for error, was 0.2467 (53.2762/216), whereas for the multiple linear regression it was 0.2611 (56.3962/216). The ratio of the mean square of the columns and error for the non-linear regression model was 0.27 (0.0674/0.2467) and was equivalent to the F-statistic, whereas for the multiple regression model it was 0.43 (0.1115/0.2611). The p-value is equal to the probability of the F-statistic; for the non-linear regression model it was 0.602, whereas for the multiple linear regression model, the p-value was 0.5140. The p-values (0.602 and 0.5140) were significantly greater than the threshold value (0.05); therefore, no mean significant difference existed between the targets and predicted model outputs for both the non-linear and multiple linear regression models. In addition, the means and medians were 4.7146 cfu/g and 4.655 cfu/g for the target group and 4.679 cfu/g and 4.636 cfu/g for the model output group for the non-linear regression model, respectively, whereas for the multiple linear regression, the mean and medians for the target group were 4.7146 cfu/g and 4.655 cfu/g and for the model output group they were 4.665 cfu/g and 4.671 cfu/g, respectively.

3.6. Validation of the Developed Mathematical Models

The 72 values (validation dataset) of the measured input parameters (number of days, daily slurry temperature, and pH) were used in the validation of the developed models to determine the validated predicting outputs (log of total bacteria counts), called the model outputs, whereas the 72 values of the experimental log of total viable bacterial counts were called the validating targets. The targets and model outputs were compared to validate the developed models with respect to the determination coefficients, the mean absolute error, and the p-values.

Table 10. Sample test dataset used in the validation of the mathematical models.

Number of Days (x1)	Slurry Temperature (x2)	PH (x3)	Determined Log of Bacteria Counts (y)	Predicted Log of Total Viable Bacteria Counts with Response Surface Model	Predicted Log of Total Viable Bacteria with Multiple Linear Regression Model
2	18.358	5.847	5.682	5.731	5.648
11	17.236	5.496	5.102	5.217	5.222
16	17.212	5.790	5.077	5.182	5.246
20	18.439	5.716	5.188	5.309	5.297
31	18.059	5.954	5.170	5.078	5.154
36	18.235	5.922	5.222	5.009	5.077
52	18.019	5.949	4.704	4.692	4.787
73	23.758	7.367	5.554	5.542	5.666
80	21.691	6.580	5.005	4.976	4.992
95	23.954	6.505	5.291	5.069	4.977
123	24.270	7.48	4.990	4.628	4.472
145	22.786	7.519	4.383	4.408	4.369
153	25.875	7.550	4.811	4.741	4.616
161	25.040	7.550	4.180	4.290	4.148
168	23.178	7.534	4.123	4.214	4.027

shows some sample validation data of the determined log of total viable bacteria counts (targets) and the model outputs based on the non-linear and multiple linear regression models. The model outputs and the experimental determined targets exhibit a strong correlation. The determination coefficient, mean absolute error, and p-value between the validating targets and the model outputs for the non-linear regression model were 0.974, 0.145, and 0.543, respectively, whereas for the multiple regression model they were 0.946, 0.177, and 0.523, respectively. The determination coefficients, mean absolute errors, and p-values are in the acceptable ranges for better predictions. Therefore, from the accuracy achieved with the validation dataset, the developed non-linear and multiple linear regression models can predict the log of total viable bacterial counts in the anaerobic digestion of cow manure in a balloon digester. Based on the exceptional values for the criteria parameters for model accuracy, it can be affirmed that the non-linear regression model gave better predictions when compared with predictions using the multiple linear regression model.

ANOVA Statistics to Confirm the Accuracy of the Developed Models with the Validation Dataset

The degrees of freedom between and within groups were 1 and 142, respectively. The total degree of freedom was 143. The groups of targets and model outputs with the non-linear regression model had a mean square of columns and error of 0.966 and 0.2603, respectively, whereas with the multiple linear regression model they were 0.1080 and 0.2706, respectively. The F-statistic and the p-value were 0.370 and 0.5434 for the non-linear regression model and 0.40 and 0.5286 for the multiple linear regression model. The p-values (0.5434 and 0.5286) were larger than the threshold value (0.05) and confirmed that no mean significant difference existed between the targets and predicted model outputs.

The distributions between the measured and modelled outputs from the non-regression model were normally distributed with the mean and median, as 4.7166 cfu/g and 4.6363 cfu/g for the targets, and 4.666 cfu/g and 4.5706 cfu/g for the model outputs, respectively. In addition, with the multiple linear regression model, the mean and median values for the validated targets were 4.7166 cfu/g and 4.6363 cfu/g, and 4.663 cfu/g and 4.648 cfu/g for the modelled outputs.

Hence, based on the validation results, the developed non-linear regression and multiple linear regression models were suitable for the prediction of the log of total bacterial counts in the anaerobic digestion of cow manure in a balloon reactor.

4. Conclusions

The profiles of daily slurry temperature during the retention period of anaerobic digestion in a balloon digester charged with cow manure exhibited two temperature regimes, which were in the psychrophilic and mesophilic temperature ranges. The physicochemical properties of the cow manure with reference to the percentage of total solids, volatile solids, ash contents, and ammonium levels prior to the anaerobic digestion in the balloon digester justified that adequate biodegradable matter existed in the manure. The study affirmed that the number of days, daily slurry temperature, and pH used as predictors of the total viable bacteria counts were all primary factors, with the number of days contributing the most by the weight’s ranking. Furthermore, it could be confirmed that over the hydraulic retention time of anaerobic digestion, the cumulative volume of biogas production was 6.75 m³, with percentages of methane and carbon dioxide of 65.8% and 31.2%, respectively, in proportion to the summative total viable bacteria counts. The developed multiple linear and non-linear regression models were capable of predicting the total viable bacteria counts using the chosen set of input parameters with acceptable accuracies in the balloon digester made of heat-sealed plastic and charged with cow waste. The determination coefficients, mean absolute errors, and p-values for the targets and models outputs for both the testing and validation dataset based on the non-linear regression and the multiple linear regression models were in acceptable ranges. The prediction using the developed non-linear regression model was better than the corresponding prediction from the multiple linear regression model based on the values of determination coefficients, mean absolute errors, and p-values. The derived simulated 2D multi-contour surface plots from the developed models were explicitly utilized to demonstrate the variation in each predictor while the others were held constant. The results from the 2D multi-contour surface plots and in conjunction with the model outputs revealed that increases in both the slurry temperature and pH lead to increases in the total viable bacteria counts.