Introduction

The need for biofuels is steadily increasing because of the problems associated with fossil fuel consumption, energy security and formulation of policies favouring the development of renewable energy technology. One means of achieving this is through the bioconversion of organic based materials in a system known as a biorefinery. Biorefineries have the potentials of improving the sustainability of biofuels through the production of bioenergy (biogas) and further recovery of valuable by-products (e.g. biofertilizers) [1]. Several organic matters commonly known as substrates or feedstock e.g. animal dungs (cattle, goat, sheep, pig, poultry, horse, etc.), plant/agricultural by-products (waste paper, rice husks, straws, corn, sugarcane, water hyacinth, etc.), wastewater and solid waste have been shown to be viable for biogas production [2–8]. In this study, cattle dungs collected from an abattoir were used as feedstock in batch bioreactors.

Abattoirs also known as slaughter houses are places where animals (e.g. cattle, horse, sheep, goat, pig, poultry, etc.) are slaughtered and processed for sales to neighbouring communities on a regular basis. Preliminary survey and elementary reconnaissance on several abattoirs within the Port Harcourt metropolis of Rivers State, Nigeria revealed that the capacities (sizes) of abattoirs vary considerably depending on the location and the population they serve. Some abattoirs hold as little as 5 cattle per day while others could hold as much as 180 cattle per day. Furthermore, it was also gathered that cattle dungs were washed and discharged directly into adjoining water bodies. The outcome of this indecent act is a multidimensional environmental problem due to pollution and contamination of the immediate and extended environment. Cattle dung contains pathogens that can be infectious to humans when ingested from drinking polluted water [6]. It is also known to be rich in nutrients such as nitrate and phosphate that may lead to eutrophication of slow flowing surface waters. Additionally, a large quantity of wood is being burnt while processing the slaughtered animals, giving rise to deforestation and air pollution.

In recent times, several researches supported the application of anaerobic digestion of organic matter as an appropriate technology for potential renewable energy (e.g. biogas) and nutrient rich fertilizer recovery, sustainable waste management and pathogen destruction [9–12]. Anaerobic digestion produces lesser pollutants and greenhouse gases than some other waste treatment techniques such as incineration [13], composting [14], and landfilling [15]. It is mainly used for stabilization (treatment) of organic wastes and production of energy in the form of biogas [10, 11, 16]. In an oxygen free environment through the process of hydrolysis, acidogenesis, acetogenesis and methanogenesis, anaerobic microbes such as fermentative bacteria, acetogenic bacteria and methanogenic bacteria, digest biodegradable organic matter into high energy biogas with methane (50–70 %) as potential energy content, carbon dioxide (30–40 %) and other gases such as H2, N2, H2S and O2 in small amount [3, 11, 13, 15, 18]. Anaerobic digestion process for biogas production could be carried out in batch, plug flow and complete mix reactors, with each having its benefits over the other. Batch reactors are gaining widespread application probably because they are easy and cheap to construct, operate and maintain [3, 19–23]. Therefore, its choice in design model development cannot be overemphasised.

The aim of this study was to develop optimized mathematical models, which when applied in conjunction with the modified Gompertz model, would be used as an alternative method for the design and development of batch reactors for biogas production from cattle/livestock dungs in abattoirs.

Mathematical Methods/Model Development

Total and Volatile Solids Estimation for a Typical Abattoir

In order to effectively design biogas reactors for digesting cattle/livestock dungs in abattoirs, the amount of volatile solid (VS) which is a function of the number of cattle received by an abattoir, the quantity of dungs excreted by these cattle per day and the moisture contents of the dung, must first be determined. This could be obtained by estimating the number of cattle an abattoir receives on a daily basis and the amount of dung produced per cattle per day. Usually, an adult cow as often transported to a city or metropolitan abattoirs produces a certain amount of dung daily [20, 24]. Thus, the mass of total solid (TS) in it could be estimated according to Eq. (1).

$$ M_{TS} = \left( {1 - w} \right)M_{Dung} ,\quad w < \, 1 $$
(1)

where w (w/w), represents the moisture content (expressed as a fraction of water) in the sample, M Dung represents the mass of dung produced per cow per day and M TS represents the amount of dry matter (total solids, TS) in the dung.

For an abattoir receiving n number of cattle per day

$$ M_{TS} = \left( {1 - w} \right)nM_{Dung} $$
(2)

Hence, the total amount of dry matter/cow dung obtainable from an animal barn becomes;

$$ M_{TS} = \left( {1 - w} \right)nNM_{Dung} $$
(3)

where N represents the number of days of cow dung accumulation before digestion.

Therefore, the mass of volatile solid (VS) obtainable in a typical abattoir holding/barn becomes:

$$ M_{VS} = f_{VS} M_{TS} $$
(4a)
$$ M_{VS} = \left( {1 - w} \right)nNf_{VS} M_{Dung} $$
(4b)

where f VS represents the fraction of VS in a mass of cow dung.

Sizing the Biodigester

Let ρ TS and x be the dry density and % (w/w) of total solid (TS) in the digester respectively;

Then,

$$ V_{TS} = \frac{{M_{TS} }}{{\rho_{TS} }} $$
(5)

where (V TS ) represents the volume of total solid in the digester; and (noting that 1 ml of water weighs 1 g),

$$ \frac{x}{100} = \frac{{M_{TS} }}{{M_{TS} + V_{w} }} $$
(6)

For which,

$$ V_{w} = \frac{{M_{TS} \left( {100 - x} \right)}}{x} $$
(7)

where V w is the volume of water in the digester.

Note that V w consists of two components: (1) the volume of water/moisture (V m ) in the dung, and (2) the volume of water required (Vw R ) to dilute the substrate to the required concentration.

Therefore,

$$ V_{m} = wV_{TS} = \frac{{wM_{TS} }}{{\rho_{TS}}} $$
(8a)

and

$$ V_{wR} = V_{w} - V_{m} $$
(8b)

Substituting Eqs. (7) and (8a) into Eq. (8b) and simplifying further, one obtains,

$$ V_{wR} = M_{TS} \left( {\frac{100 - x}{x} - \frac{w}{{\rho_{TS} }}} \right) $$
(9)

Therefore, the working volume of digester V D becomes,

$$ V_{D} = V_{TS} + V_{w} $$
(10)

Substituting Eqs. (5) and (7) into Eq. (10) and collecting like terms

$$ V_{D} = M_{TS} \left( {\frac{100 - x}{x} + \frac{1}{{\rho_{TS} }}} \right) $$
(11)

By dividing Eq. (9) by Eq. (11),

$$ \frac{{V_{wR} }}{{V_{D} }} = \frac{{\frac{100 - x}{x} - \frac{w}{{\rho_{TS} }}}}{{\frac{100 - x}{x} + \frac{1}{{\rho_{TS} }}}} = R_{v} $$
(12)

where R v is a dimensionless ratio, which also represents the fraction of dilution water in the digester.

In the anaerobic digestion of organic feedstock, one of the most important parameters to look out for is the volatile solid concentration.

Mathematically, the concentration of volatile solid [VS] in the digester can be represented by,

$$ \left[ {VS} \right] = \frac{{M_{VS} }}{{V_{D} }} $$
(13)

Substituting Eqs. (4a) and (11) into Eq. (13) and further simplifying yields,

$$ \left[ {VS} \right] = \frac{{f_{VS} }}{{\left( {\frac{100 - x}{x} + \frac{1}{{\rho_{TS} }}} \right)}} $$
(14)

Sizing the Gas Chamber: An Application of the Modified Gompertz Model

Biogas yield (y t ) is defined as the ratio of the volume of gas produced to the mass of volatile solid loaded.

Mathematically,

$$ y_{t} = \frac{Volume\,of\,gas\,produced}{mass \,of\,volatile \,solid \,loaded } = \frac{{V_{G} }}{{M_{VS} }} $$
(15)

Therefore,

$$ V_{G} = y_{t} M_{VS} $$
(16)

where V G represent the volume of gas produced and M VS represents the mass of volatile solid loaded.

The modified Gompertz kinetic model (Eq. 17) has been widely used in prediction biogas production from the anaerobic digestion of dung slurry [21, 23, 25–27].

$$ y_{t} = y_{m} EXP\left\{ { - EXP\left[ {\frac{{R_{m} e}}{{y_{m} }}\left( {\lambda - t} \right) + 1} \right]} \right\} $$
(17)

where y t represents the experimental/predicted biogas yield (L/kg VS) obtainable after time t, y m represents the maximum biogas yield potential (L/kg VS), R m represents the maximum biogas production rate (L/kg VS/day), λ represents the lag phase or minimum time taken to produce biogas (days), and e = 2.718281828. The unknown variables y m , R m and λ could be determined by non-linear regression analysis.

Substituting Eq. (17) into Eq. (16),

$$ V_{G\left( t \right)} = y_{m} M_{VS} EXP\left\{ { - EXP\left[ {\frac{{R_{m} e}}{{y_{m} }}\left( {\lambda - t} \right) + 1} \right]} \right\} $$
(18)

It is pertinent to note that Eq. (18) represents the cumulative volume of gas produced after time t (say at the end of a production day). The daily volume of gas produced is obtained by subtracting the volume of gas produced at t−1 from that produced after time t, i.e.

$$ V_{G} = V_{G\left( t \right)} - V_{{G\left( {t - 1} \right)}} \left( {\text{L}} \right). $$
(19)

where V G represents the daily gas production.

The maximum value of Eq. (19), VG(max) represents the maximum volume of gas produced daily and it is the key parameter in designing the gas chamber.

Materials and Methods

Substrate Collection and Preparation

The substrate collection was in accordance with the methods applied by [3, 11, 23]. About 5 kg of fresh cow dung was collected from an abattoir is Choba community in Port Harcourt, Rivers State, Nigeria. The dung was air-dried for 21 days to preserve its microbial population. The stomach chambers of ruminant animals such as cattle, goat, sheep, horse, etc. are known to contain microbes that are required for digestion of their dung under anaerobic conditions [6]. As such, no addition of inoculum was required. The dried dung was then crushed with mortar and pestle to ensure homogeneity and subsequently analysed for moisture, volatile solid, carbon and nitrogen contents according to [28] using an English made muffle furnace-Carbolite model LMF 4. The respective moisture, volatile solid, carbon and nitrogen contents were found to be 14.2, 70.40, 7.14 and 0.28 % per mass of TS. Thus, the carbon to nitrogen ration was calculated to be about 25.50.

Experimental Procedures

Biogas production from cow dung was studied in five (5) anaerobic digesters labelled OP1–OP5 at ambient temperatures and in accordance with the methods described by [3, 17, 23]. An electronic weighing balance Mettler model PN163, manufactured in Switzerland with specification range between 0.10 mg and 500 g was used in weighing the dried dung. Thereafter, the dung was mixed with 250 ml of water inside a 500 ml Buckner flask and corked to exclude air, thereby creating the anaerobic condition required for biomethenation of biomass. The total solid concentration ranged from 8 to 10 % at 0.5 increments as suggested by [29] for low solid loading. The set up was monitored for forty-two (42) days at an ambient temperature with a mean value of 30 ± 3 °C. Within Port Harcourt metropolis the mean annual temperature ranges from 22.54 to 31.03 °C [30]. The procedure was stopped when gas production reduced significantly and in some cases when it stopped completely. The reactors were agitated once daily and biogas production was measured by brine displacement method [11, 12]. Figure 1 shows the cumulative biogas yield from the five (5) digesters.

Fig. 1
figure 1

Comparative plot of cumulative biogas yield from digesters OP1–OP5

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Results and Discussion

Effects of Total Solid on Biogas Yield

In this study, it was observed that biogas production (yield) tend to follow the general sigmoid function (S-curve) as shown in Fig. 1. A similar pattern was observed by [6, 10, 11, 17, 21, 22, 26]. This arises from the specific growth rate pattern of methanogenic bacteria in batch reactors. Gas production was delayed at the beginning due to the slow growth (lag phase) of methanogenic bacteria. After about 2–4 days, gas production began rapidly due to the exponential growth of microorganisms in the reactor. However, gas production started declining after about 28–32 days. At this stage, microbial growth could be predicted to be in the stationary phase (i.e. death rate is in equilibrium with growth rate). At about 42 days of incubation, biogas production dropped significantly and stopped completely is some reactors. This could be attributed to the decline in the population of methanogens because of the decline or complete exhaustion of the limiting nutrient. In all five reactors, biogas yields were observed to increase with increasing total solid (TS) concentrations. The minimum cumulative biogas yield was recorded in digester OP1 containing 8.0 % TS while the maximum value was measured in digester OP5 containing 10.0 % TS (Fig. 1). This is in agreement with the work of [6].

Model Calibration and Accuracy

The modified Gompertz model was calibrated using experimental data. The parameters: y m , R m and λ we estimated using the non-linear regression approach made possible by the Solver ToolPak of Microsoft Excel 2013. The modified Gompertz model fitted/predicted the experimental data with over 99 % accuracy (R 2 = 99.6–99.9 %). Similar levels of accuracy were observed in the works of [22, 26].

Kinetic Study of Anaerobic Degradation Process

Experimental biogas yield (y t−exp ) from the five digesters ranged from 60.45 to 66.82 ml/gVS averaging at about 64.71 ± 1.61 ml/g VSloaded. The predicted biogas yield (y t−pred ) based on the modified Gompertz model ranged from 63.26 to 70.98 ml/g VS with an average of 67.24 ± 1.71 ml/g VSloaded. Also, the maximum biogas yield potentials (y m ) were found to be between 68.51 and 86.78 ml/g VSgVSloaded and the average value was 78.02 ± 3.58 ml/g VSloaded. As for the maximum biogas yield rate (R m ), the values ranged from 2.50 to 2.97 ml/g VSloaded/day with an average value of 2.73 ± 0.09 ml/g VSloaded/day. Finally, the lag phase (i.e. the time between experimental setup and actually biogas production) was between 9.73 and 12.01 days with an average value of about 10.93 ± 0.43 days. In all five digesters, the coefficient of determination (goodness of fit), R 2 of the modified Gompertz model ranged from 99.6 to 99.9 %, with an average value of 0.9971 ± 0.0005. It was observed that amongst all five digesters the highest experimental biogas yield of 68.82 ml/g VSloaded was recorded in digester OP5 which contained 10.0 % TS, while the highest predicted biogas yield of 70.98 ml/g VSloaded was recorded in digester OP4, which contained 9.5 % TS. The highest value of maximum biogas yield potential of 86.78 ml/g VSloaded was recorded in digester OP5 and the highest maximum biogas production rate of 2.87 ml/g VSloaded was observed in digester OP4. Finally, the maximum lag of 11.57 days was recorded in digester OP5. In all five digesters, the modified Gompertz model best fitted the biogas data of digester OP5. Through the methods of non-linear regression analysis (executed with Microsoft Excel 2013 Trendline Function), the maximum biogas yield potential (y m ) was found to be related to the volatile solid concentration according to Eq. (20) with a goodness of fit (R2) of 0.9168 (Fig. 2).

$$ y_{m} = - 0.0266\left[ {VS} \right]^{2} + 4.8396\left[ {VS} \right] - 129.1 $$
(22)
Fig. 2
figure 2

Relationship between volatile solid concentration and maximum biogas yield potential

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Model Limitations and Application

Equations (1)–(3) and (4b) are strictly for estimating the amount of dungs accumulated from cattle/livestock in an abattoir over time. These are basic for the determination of the mass of total solids (M TS ). It is pertinent to note that before embarking on the design of reactors, experimental procedures should be carried out with dung samples from a typical abattoir/specific livestock species for the purpose of determining y m , R m and λ according to Eq. (18). In this way, misleading results that might arise from generalisation/estimation of design parameters could be eliminated.

For design purpose, a family of curves have been produced from Eqs. (10), (13), and (14) see Figs. 3, 4 and 5 respectively. Theses curves, when combined with Eqs. (18)–(20), would lead to the design of an optimal biogas plant consisting of a bioreactor and a gas holder.

Fig. 3
figure 3

Relationship between TS, %TS and working volume of digester

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Fig. 4
figure 4

Relationship between moisture content (%), %TS and RV. \( R_{v} = \frac{Volume\,of\,water\,required}{Working\,volume\,of\,digester} \)

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Fig. 5
figure 5

Relationship between x (%TS), f VS and volatile solid concentration

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Design of Biogas Plant for Abattoirs

The choice of the design adapted here is that in which the gas hold is separated from the biogas digester. Authors [20] and [24] reported that the average amount of dung produced by a head of cow on a daily basis ranges from 9 to 15 kg, while the average density of the dried cow dung was found to be about 680 kg/m3. The dung in the form of slurry would be fed into the reactor with pumps if the plant is built above the surface of the ground. Otherwise, gravity flow should be employed when the system is built below ground surface.

Design Parameters

Assumptions

Let M Dung  = 10 kg/cow/day, n = 12 cows, N = 42 days, f VS  = 70 % = 0.7 and w = 80 % = 0.8.

Table 1 shows a summary of the biogas tank design. Figure 6 is a plot of daily biogas production from a plant designed with the parameters of Table 1. The maximum daily biogas production was found to be 1925.57 L.

Table 1 Summary of design parameters
Full size table
Fig. 6
figure 6

Daily biogas productions from a single unit biogas plant

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Conclusions

Anaerobic digestion offers a clean and cheap technology for the production of biogas from organic substrates such as cow dungs. The quantity of dungs accumulated in abattoirs due to the amount of cattle they receive is large enough for the development of biogas reactors in situ. The ambient temperatures within Port Harcourt and other cities in Nigeria are appropriate for the anaerobic digestion of cattle dungs as a means of waste management with gas production. This could supplement or completely meet the daily energy requirements for abattoirs. Furthermore, the applicability of the modified Gompertz model in the design of batch biogas reactor has been revealed in the study. Most importantly, the appropriate application of the developed models would lead to an optimal design of a biogas plant digesting cattle/livestock dungs with biogas and organic fertilizers production for farmers and gardeners alike.