Modelling Cow-Dung and Grass-Clippings Isothermal Continuous Stirred Co-digester for Biogas Production using Modified Gompertz Rate Equation

The continuous search in renewable energy sources for industrial and domestic utilization is very imperative and motivated this study. The study of Cow- dung and grass clippings Co-digestion are important due to the fact that successful outcome of the studies will provide a basis for waste minimization and enhance renewable energy production for global consumption. This work therefore is focused by exploiting the modified Gompertz kinetic equation in developing design models for the simulation of continuous stirred  co-digester(CSC) at isothermal condition. Co-digester functional dimensions such as volume, length, space time, space velocity and heat generation per unit volume were developed for Continuous Stirred Co-digester type..The developed models were simulated using Matlab codes programming technique using design basis of 50,000 metric tons of biogas per annum at 37oC isothermal condition. The developed performance models were solved numerically using MATLAB version 7.1 within the operational limit of conversion degree, XA = 0.1 to 0.9.The results obtained showed that increase in fractional dimensions of Co-digester volume VR, length LR and Space time SV­ increases with increase in fractional conversion at constant radius.  Results of space velocity (SV­) and heat generation/per unit volume (q) showed inverse characteristic behavior as increase in fractional conversion decreases space velocity and heat generation per unit volumes.. Further careful examination of the results, demonstrated that at optimal yield of 0.9 degree of conversion, Co-digester volume value of 10.0m3 at constant radius was feasible. The results as obtained in this work proved dependable relationship with fractional conversion. 
 


I. INTRODUCTION
Researchers globally have resolved in tackling the high potential of global warming problems resulting from the utilization of non-renewable energy sources for industrial and domestic use.These have led to increased awareness search [1], [7], [9], [11] for alternative renewable energy sources, such as biogas and bio-fuels production; that expected to elevate global economy and minimize global warming problems.In addition, the significance of this study will also encompass the following; [7].It will reduce the environmental nuisance of cow-dung and grass clippings waste contamination problems [2].The CH4 gas produced will be useful for both industrial and domestic application [11].The digester models to be developed in this work will be generalized and can be applied with a known kinetic model to simulate the functional dimensions of any digester type.
To this end, it is necessary to research on the possibilities of developing various Co-digester types in the treatment and production of bio-gas using all domestic and industrial wastes routes.Major expected achievable deliverables in this waste recycling process are waste minimization, renewable product formation, boost global economy and of course enhances employment capacity.Hence, this research is a prelude to the adoption of modified Gompertz rate equation sighted in [8] works of Cow-Dung and Grass Clipping combination as raw materials for renewable energy sources.
In order to establish the design models with respect to the Isothermal continuous stirred co-digester unit, it is relevant to appraise quantitatively and qualitatively the scientific aspect of cow-dung and grass-clippings as related to properties, structure, and reaction mechanisms.Interestingly, cow-dung and grass-clippings are readily available materials that can be used to produce bio-gas through the biodegradability in a co-digester process.

C. Stoichiometry of Grass Clippings and Cow-Dung Reaction
The possible stoichiometric equation describing grass clipping cow-dung reaction is invoked using induction principles of reaction mechanism and proposed as;

II. KINETIC RATE ANALYSIS
Numerous kinetic rate analysis was postulated regarding bio-gas production from various municipal and industrial waste types.[16]- [18] worked on optimizing biogas production from anaerobic co-digestion of chicken manure and organic fraction of municipal solid waste.In this work, the use of biochemical methane potential with 4:1 ratio with optimal biodegradation rates gave optimum biogas (methane) yield with 15 days' retention time.Since it was an experimental research work, the materials were sourced Their findings gave reasonable amount of moisture content at 37 0 C and pH 7 which after Buswell formula was used to determine the percentage of methane gave 40 -70% (CH4) and 30 -60% (CO2) and other traces of other elements.Thus, their work gave optimum biogas and methane yield at ratio of 1:1 but optimum biogdegradability is at 4:1 ratio.
Abdulsalam and Yusuf [1] investigated the kinetic of biogas produced from cow and elephant dungs using residual substrate concentration approach.In the batch biodigester, anaerobically the best biogas production of 3.92 x 10 -4 g/cm 3 and average yield of 0.0845 over a period of 33 days follow a kinetic shifting order of (0 -1).Also, the codigestion of elephant and cow-dung of equal proportion can increase the yield of biogas.

A. Modified Gompertz Kinetic Expression
Interestingly, Etuwe, C.N., et al [10] worked on the development of mathematical models and application of the modified Gompertz model for Designing Batch Biogas Reactors.They got yield optimal (yt) of 60 -80g.VS/L at 42 days' period.Comparatively, the modified Gompertz formula gave minimum time    of 9.7 -12 days and maximum yield (ym) of 68 and 86ml/g at operating temperature of 37 0 C and pH of 7.
Further work was reported on the kinetic of biogas rate from cow dung and grass clippings resulting to biogas production on a laboratory scale level [15], [8].Their works resulted in the development of a concise modified rate expression as follows: Fig. 3. Hypothetical Methanogenic process Gompertz [8], [23], [25], [26] considered a Pseudofirst order reaction process for the decomposition of cow-dung to CO2 and CH4 for the methonogenic bacteria process as shown below: The rate expression for the methanogenesis process gives; Since it is a Pseudofirst order reaction, n = 1 hence, (4) becomes; In terms of fractional conversion, XA Where; k= Rate constant which is a function of time and it represent pre-exponential factor (A) in the modified Gompertz equation as;

 
A r  =Depleting rate of reactant and it is represented in the modified Gompertz equation as   Equation ( 7) represent the rate law used for bio-gas production which will be exploited for the design of the continuous stirred co-digester.L mol XA = Fractional conversion of converted reactant to product Since the works reviewed used batch wise laboratory scale method to establish the Gompertz rate equation, it is worthwhile to adopt the modified rate equation to establish large scale design for various co-digester types.Therefore, this work is focused on the application of Gompertz modified rate equation to design an Isothermal Continuous Stirred Co-digester (CSC) for bio-gas production using cowdung and grass-clipping as raw materials.

A. Materials
The materials exploited in this work encompassed mainly desk top research component such as; Laptop, Mathlab software, work spreadsheet and other associated computational tools for the simulation of the developed models below.

B. Development of Design Models
Applying simple material balance equation, [22], is utilized for the development of the modeling equation thus; Input = output + Rate of disappearance + Accumulation.In this work consideration is based on the fact that flow is in steady state at isothermal condition with inform stirring and negligible pressure drop within the within the co-digester system, then;

C. Co-Digester Volume Model
Consider the schematic representation of a hypothetical continuous stirred co-digester with feeds and product as shown below.
Combining ( 8) to (9) yields At steady state, the accumulation term is equal to zero, i.e.

D. Co-digester Length (LR) Model
Since the reactor is cylindrical, volume of a cylindrical reactor is given as Where: VR = Volume of reactor (m 3 ) R = Radius of reactor (m) LR = Length of reactor (m)  = Constant Comparing (13) into ( 14) yields,

E. Co-digester Space Time (CSTR) Model
This is defined as the ratio of reactor volume to volumetric flow rate.thus

F. Co-digester Space Velocity (SV)
This is defined as the reciprocal of space time and mathematically modeled as.

G. Co-digester Heat Generated Per Unit Volume
Exploiting the works [2], heat generation per unit volume model is derived as; Where q = quantity of heat generated per unit volume of the co-digester.

A. Stoichiometric mole computations
The essential operating design parameters for the simulation process were computed from the summarized stoichiometric equation.Thus; (27) Corrosion Allowance e = 1.25mm [24] 8.
Design Temperature T = 150 0 C [24] Flow Chart of Programme for CSC Table II shows the results of the Mathlab simulations to obtain functional parameters of CSTR with respect to XA (fractional conversion).

A. Effect of Fractional Conversion
Fig. 6.Rate of reaction against fractional conversion Fig. 6 shows the rate of reaction with fractional conversion.The rate of reaction is inversely proportional to the fractional conversion.This means that at higher fractional conversion, the rate of reaction is very small and high volume of methane gas.Comparatively, the volume of CSTR is high at XA=0.9.Generally, the volume of the reactors increases exponentially as fractional conversion increases.Fig. 10 shows the variation of space time of reactors with fractional conversion.From the graph, the space time increases as fractional conversion increases.For instance, at XA=0.9, τPFR=5.46secs.

F. Effect of Fractional Conversion on Space Velocity of Reactors
Fig. 11.Space velocity of reactors against Fractional conversion Fig. 11 depicts the variation of space velocity of continuous stirred co-digester with fractional conversion.At same condition, the space velocity of reactors varies inversely to fractional conversion.At XA= 0.9, SVCSTR = 0.055 -1

G. Effect of Fractional Conversion on Heat Generated of Reactors
Fig. 12.Heat generated per unit volume against Fractional conversion Fig. 12 shows the variation of heat generated per unit volume of reactors versus fractional conversion.Increase in fractional conversion results to decrease in heat generated per unit volume.qCSTRis far less than qPFR due to high volume produced from CSTR than PFR.At XA=0.9, qCSTR=0.426kw/m 3 and qPFR=1.67kw/m 3 .Very small heat is generated per unit for CSTR.

A. Conclusion
Hypothetical design equations for the modeling of Isothermal Continuous Stirred Co-digester unit is developed using cow-dung and grass clippings as source of raw materials.The developed models were capable of computing digester volume, length, space time space velocity, heat generation per unit volume and other associated engineering dimensions as a function of fractional conversion and parametric product design basics.
The results obtained were quit comparable and in-line with design expectations as volume, length and space time increases with increase in fractional conversion.While space velocity and heat generation per unit volume increase with decrease in fractional conversion.

B. Recommendations
This project was limited to the development of hypothetical design models for Continuous Stirred flow cow-dung and grass clippings Co-digester unit for biogas production at Isothermal condition using Gompertz kinetic model.Therefore, authors I recommend that further work should be carried out on the following;  Cow-dung and grass clippings co-digester units should be developed for the simulation of the functional design dimensions using Gompertz kinetic model at non-isothermal condition of continuous and plug flow co-digester units at steady state. Isothermal condition of plug flow co-digester for cow-dung and grass clippings  Cow-dung and grass clippings co-digester heat exchanger units should be developed for the simulation of the functional design dimensions using Gompertz kinetic model at both Isothermal and Nonisothermal condition. Evaluate the performance criteria for all co-digester types in other to ascertain the optimal product routes.

Fig. 2 .
Fig. 2. Typical Grass Clipping processes The following physical characteristics are identified with grass clipping; Colour: Green Density = 91 -227.445kg/m 3 (Garden Grass) 150 -450 for Garden Trees Structure of Grass Clipping locally and then subjected to laboratory investigation for results and discussions.The methanogenesis of the process gave researched on the design models for Anaerobic Batch Digesters producing Biogas from municipal solid waste.The use of Microsoft Visual Basic version 6.0 computer programme with fractional conversion range get empirical optimum yield of 20% in the batch digester.A lot of literatures reviews were carried out on waste management in Port Harcourt.Their materials were 5 batch-wise anaerobic digesters of 5 litres for the experimental set-up.The digesters were the loaded with 2kg of organic municipal solid waste which was diluted to 26.7% total solids concentration and end up with the usual Monod rate equation.In addition, other researchers also limited their works to the laboratory analysis of possible methane production resulting to the use of Monod rate equation[12]-[18] rate, day-1  [S] = Concentration of limiting substrate, mg/L Ks = Half-saturation constant (i.e.concentration of S when 2 max where; A = Biogas production potential Y = k(t) = cumulative of specific biogas production  = Maximum biogas production rate (d -1 ) e = Mathematical constant (2.718282) t = Cumulative time for biogas production (days)  = Lag-phase period CAO = Initial concentration of the glucose;

Fig. 7 .
Fig. 7. Volume of reactors (CSTR) versus Fractional conversion Fig. 7 depicts the volume of reactors continuous stirred tank reactor varying with fractional conversion.Comparatively, the volume of CSTR is high at XA=0.9.Generally, the volume of the reactors increases exponentially as fractional conversion increases.

9 ComputeFig. 8 .
Fig. 8. Diameter of reactors against Fractional conversion Fig. 8 shows the diameter of the reactor varying with fractional conversion.The diameter of the reactors (CSTR) generally increases as fractional conversion.

Fig. 9 .
Fig. 9. Length of reactors against Fractional conversion Fig. 9 depicts the length of the reactors varying with fractional conversion.The length increases as fractional conversion reaches maximum value of LCSTR=3.71m

Fig. 10 .
Fig. 10.Space time of reactors against Fractional conversion

TABLE I :
OPERATING PARAMETERS DETERMINATION

TABLE II :
FUNCTIONAL PARAMETERS RESULTS OF CSTR WITH