Thermic Model to Predict Biogas Production in Unheated Fixed-Dome

Article pubs.acs.org/est

Thermic Model to Predict Biogas Production in Unheated FixedDome Digesters Buried in the Ground Georgina Terradas-Ill,† Cuong H. Pham,‡ Jin M. Triolo,*,† Jaime Martí-Herrero,§ and Sven G. Sommer† †

Institute of Chemical Engineering, Biotechnology and Environmental Technology, Faculty of Engineering, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark ‡ Ministry of Agriculture and Rural Development, National Institute of Animal Science, Thuyphuong, Tuliem, Hanoi, Vietnam § Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Building Energy and Environment Group, Edifici GAIA (TR14), C/Rambla Sant Nebridi 22, 08222 Terrassa, Barcelona, Spain S Supporting Information *

ABSTRACT: In many developing countries, simple biogas digesters are used to produce energy for domestic purposes from anaerobic digestion of animal manure. We developed a simple, one-dimensional (1-D), thermal model with easily available input data for unheated, unstirred, uninsulated, fixeddome digesters buried in the soil to study heat transfer between biogas digester and its surroundings. The predicted temperatures in the dome, biogas, and slurry inside the digester and the resulting biogas production are presented and validated. The model was well able to estimate digester temperature (linear slope nearly 1, R2 = 0.96). Model validation for methane production gave root-mean-square error (RMSE) of 54.4 L CH4 digester−1 day−1 and relative-root-mean-square errors (rRMSEP(%)) of 35.4%. The validation result was considerably improved if only using winter data (RMSE = 26.1 L CH4 digester−1 day−1; rRMSEP(%) = 17.7%). The model performed satisfactorily in light of the uncertainties attached to it. Since unheated digesters suffer critically low methane production during the winter, the model could be particularly useful for assessing methane production and for improving the ability of unheated digesters to provide sufficient energy during cold periods.

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INTRODUCTION Production of biogas in anaerobic digesters is one of the most efficient technologies for supplying clean and renewable energy from biomass with high water content. In developing countries, cheap and simple digesters are particularly important as a national energy infrastructure may be weak or nonexisting.1 In addition, biogas production from animal manure is useful for recycling nitrogen (N) and phosphorus (P), while also reducing greenhouse gas emissions from manure management.1−3 Because of their advantages for poor households, to date millions of simple biogas digesters have been constructed on small and medium-sized animal farms in developing countries. In Asia, more than 30 million biogas digesters are now in use, the most popular designs being the fixed-dome digester and the floating-drum digester,4 while the low-cost tubular digester are most popular in South America,5,6 with some models being adapted to cold climate through addition of insulation and greenhouses.7 Most of the biogas digesters are buried underground to ensure a constant temperature and are not heated, stirred, or insulated (simple biogas digesters). These simple digesters are fully functional in tropical or subtropical climate zones, but temperatures below 20 °C during winter can lead to low biogas production.8 This is because of slow growth of the microorganisms responsible for the digestion of organic matter at low temperatures.9,10 As a consequence, biogas © 2014 American Chemical Society

production cannot cover the energy demand during winter periods.10 There is an urgent need to develop a new design of a simple and reliable biogas digester that produces the required energy throughout the year. The focus must be on increasing digester temperature, which can be achieved by better insulation, solar heating by means of a greenhouse or heat exchanger, or using some of the gas produced to heat the digester.10−12 When designing technology to solve the problem, environmental conditions, availability of biomass, and the amount of heating required need to be taken into account. This information can be obtained by quantifying and modeling heat exchange between the biogas digester and its surroundings. The average temperature in a fixed-dome digester can be described by a simple algorithm, assuming that the soil temperature is similar to groundwater temperature.11 For lagoon digesters, a heat transfer model has been developed to predict the energy required to operate the digester at a specific temperature.13 Heat losses from lagoon digesters can be reduced by designing the construction with the help of a 3-D model that simulates heat transfer.14 Heat transfer models can Received: Revised: Accepted: Published: 3253

July 19, 2013 February 11, 2014 February 11, 2014 February 11, 2014 dx.doi.org/10.1021/es403215w | Environ. Sci. Technol. 2014, 48, 3253−3262

Environmental Science & Technology

Article

also support the design of plug-flow plastic tubular digesters with passive solar heating provided by a greenhouse by identifying the most important pathways of heat transfer.12 The models of Perrigault and co-workers,12 Gebremedhin and coworkers,13 and Wu and Bibeau14 all assume that soil temperature varies sinusoidally at all depths. In practice, the temperature in the soil varies as determined by soil characteristics, soil cover, and climate.15,16 Therefore, there is a need to extend the model of Perrigault and coworkers12 with a submodel to compute the apparent temperature at the soil surface from data on air temperature since, in developing countries, data on soil surface temperature are not always available. The aim of the present study was, therefore, to develop a new heat transfer model that can estimate digester temperature and biogas production in a small, nonheated, fixed-dome digester that is completely buried, except for the cover. The model is intended for use in regions where the end-user of the biogas digester and the biogas adviser do not have access to detailed meteorological data.

To analyze all the heat transfers to the digester, three energy balances are relevant: one for the cover, one for the biogas, and one for the slurry. These heat balances can be written as follows: 0 = Q Solar,C + Q rad, ∞− C + Q cnv, ∞− C + Q cover,G − C + Q rad,C − S

mGC P,G

mSC P,S

(1)

dTGas = Q cover,G − C + Q cnv,G − S + Q walls,G − gr dt

dTSlurry dt

(2)

= Q rad,C − S + Q cnv,G − S + Q walls,S − gr

+ Q floor,S − gr + Q load

(3)

where all heat transfers are expressed in watts. In eq 1, QSolar,C is the heat gain by the cover of the digester from the solar radiation during daytime, Qrad,∞−C and Qcnv,∞−C are the heat exchanges between the cover and the ambient air by radiation and convection, respectively. The heat transfer between the cover and the biogas by conduction and convection is represented by Qcover,G−C and the heat that is transferred from the slurry to the cover by radiation is Qrad,C−S. In eq 2, the accumulated energy of the gas is computed by the product of the mass of the gas (mG), the heat capacity of the gas (CP,G) and the temperature variation with time (Tgas, °C). Qcnv,G−S is the convective heat transfers from the gas to the slurry. Qwalls,G−gr is the heat loss from the gas through the walls of the digester to the soil. In the third equation, the energy accumulated by the slurry is calculated. Qwalls,S−gr and Qfloor,S‑gr are the heat losses from the slurry through, respectively, the walls and the floor of the digester to the soil and Qload is the heat loss from the daily load of slurry from the storage tank. Solar Radiation. Gebremedhin and co-workers13 computed solar radiation heat flux (W m−2) as the sum of beam radiation (Sb), diffuse radiation (Sd), and albedo radiation (Sr). Here we assumed that there is no albedo radiation, that is, Sr = 0, because the cover of the digester is at ground level. Thus, the heat flux of solar radiation (q″s ) at any time t is calculated as

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METHOD The basic principle of this model is to use simple and accessible data on the heat gain of the soil and the digester from solar radiation and on the ambient air temperature to compute the heat exchange through soil, biogas cover, walls, and floor. The thermal effect of the slurry feeding and the variation in gas volume over time are also considered. In this way, the heat transfer, process temperature, and biogas production in a simple, small biogas digester buried in soil, could be simulated to provide decision support for digester construction at a specific site. The model to predict biogas production was developed using the mathematical software tool Matlab. The inputs to the model were: hourly air temperature and wind velocity, geometric characteristics of the digester, daily information on digester feeding, and weekly methane fermentation conditions. The model was validated with data from a recent study of biogas production in a fixed-dome digester in Hanoi, Vietnam.17 Theoretical Calculations. The model is based on an energy balance, which consists of a thermal, one-dimensional model. The assumptions used in developing the model were as follows: (1) The digester completely buried, except for the top cover, which is exposed to the environment. (2) The cover and walls of the digester are flat plates, an assumption made to facilitate computation of convection. (3) Microbial heat generation is negligible.18 (4) The fiber glass construction material of the digester is considered to have no thermal inertia. (5) There is no gradient in slurry temperature inside the digester, as shown in studies of temperature gradients in Danish slurry stores covered with insulating leca pebbles.19 (6) The thermal and hydraulic characteristics of the slurry fed to the digester are similar to the characteristics of the treated slurry, with the exception of the temperature.12 (7) Evaporative and convective losses from the surface of the slurry inside the digester are negligible, because they have an insignificant influence on digestate temperature compared with heat transfer from soil and air.11 (8) Infrared heat radiation is not absorbed by gases.12 (9) The temperature of the outer wall of the digester at a given depth is equal to the soil temperature at that depth. (10) Soil temperature is constant beyond a certain depth.16

qs″(t ) = S b + Sd

(4)

The beam radiation Sb to a horizontal surface is determined by the product of direct solar irradiance on a surface perpendicular to the beam (Sp) and the sine of the sun elevation angle (θ) S b = Sp sin θ

(5)

Sp can be expressed as SP = amSP0

(6)

where a is the coefficient of transmissivity (Table 1). The optical air-mass number (m) is calculated as Table 1. Values for the Coefficient of Transmissivity Used in Solar Radiation Computations

a

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atmosphere characteristics

value of transmissivity coefficient (a)

very clear clear daysa hazy and smoggy

0.90 0.84 0.6

Typical value. dx.doi.org/10.1021/es403215w | Environ. Sci. Technol. 2014, 48, 3253−3262


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Table 2. Parameters for the Calculation of Heat Transfer between Soil, Digester, and Atmosphere parameter density thermal conductivity specific heat thermal diffusivity dynamic viscosity kinematic viscosity absorptivity emissivity

ρ λ Cp,s αs μ υ

units (SI)

cover

digester

slurry

air

biogas

kg·m−3 W·m−1·K−1 J·kg−1·K−1 ×10−6 m2·s−1 ×10−5 Pa·s ×10−6 m2·s−1

220 0.035 795 0.2

220 0.035 795 0.2

1.205 0.026 1010 211.2 1.82 15.11

1.156 0.026 1682.2 498.1 1.16 11.93

0.75 0.75

0.75 0.75

1000 0.605 4.179 0.149 97.72 0.98 0.67 0.67

ε

⎛P⎞ m = ⎜ ⎟ /sin θ ⎝ P0 ⎠

hcnv, i − j = (7)

⎧ 0.664Re1/2Pr1/3 Re < 5 × 105 ⎫ ⎬ Nu = ⎨ ⎪ ⎪ ⎩ 0.037Re 4/5Pr1/3 Re > 5 × 105 ⎭ ⎪

360(284 + dn) 365

Re =

(9)

∫t

tss

qS″(t )dt

sr

(16)

(17)

and (2) the lower surface of hot plate or upper surface of cold plate Nu = {0.27RaL1/4

(11)

105 ≤ RaL ≤ 1010}

(18)

The Rayleigh number is defined with the following equation, where the characteristic length (L) for a horizontal plate is the ratio between the area and the perimeter of the plate

Convective Heat Transfer in the Digester. The biogas contained in the digester can gain or lose heat by convection within the digester with the slurry and the walls. The slurry exchanges heat by convection with the gas, the walls, and the floor of the digester. The convective heat transfer (Qcnv,i−j, W) between two elements is described in the equation below. The value for absorptivity is given in Table 2. Q cnv, i − j = hcnv, i − jAi (Ti − Tj)

ρ u windL u windL = air υair μair

1/4 ⎧ 104 ≤ RaL ≤ 107 ⎫ ⎪ 0.54RaL ⎪ ⎬ Nu = ⎨ ⎪ 1/2 7 11 ⎪ ⎩ 0.15RaL 10 ≤ RaL ≤ 10 ⎭

(10)

Because the solar radiation flux (W m−2) is a function of the time of day, integration of qs″(t) from sunrise (tsr) to sunset (tss) gives the total solar radiation flux during the day ″ = qS,day

(15)

For internal flows, the computation of the Nusselt number must take into account two conditions based on the Rayleigh number (Ra):22 (1) upper surface of hot plate or lower surface of cold plate

Campbell and Norman21 estimated the diffuse radiation component as Sd = 0.3SP0(1 − am)sin θ

⎪

where the Prandtl number is assumed to be 0.70 and the Reynolds number is calculated as

(8)

Duffie and Beckman20 described solar declination (δ) by the following expression, where dn is the day of the year, from 1 to 365 δ = 23.45 sin

(14)

For a horizontal plate with an external flow, as in the case of the digester cover and the environment, the Nusselt number is a function of the Reynolds (Re) and Prandtl (Pr) numbers as follows:13

where P is atmospheric pressure (Pa) at the location of the digester and P0 is that at sea level. Spo is the extraterrestrial flux density normal to the solar beam and was assumed here to be 1360 W m−2.13 The sun elevation angle (θ) depends on latitude ( f i) and solar declination (δ), expressed in degrees and standard local time (t) sin θ = sin fi sin δ + cos fi cos δ cos(15(t0 − t ))

Nu·λf L

RaL =

gβf (Ti − Tj)L3 αf υf

(19)

For vertical plates, the Nusselt number depends on Rayleigh and Prandtl numbers as follows, where the characteristic length (L) is the height of the surface:

(13)

⎧ ⎪ 0.68 + ⎪ ⎪ ⎪ ⎪ Nu = ⎨ ⎡ ⎪⎢ ⎪⎢ ⎪ ⎢0.825 + ⎪⎢ ⎪⎣ ⎩

To simplify the computation, the heat transfer coefficients by convection (hcnv,i−j, W·m−2·K−1) were determined by considering the gas and slurry volumes as rectangular cavities. Therefore the contact surfaces (Ai) between the elements were considered to be horizontal plates and the walls (the contact surfaces between the digestate in the digester and the soil) as vertical plates.22 The convective heat transfer coefficients are calculated by the Nusselt number (Nu) of each contact surface between element i and j, the thermal conductivity of the fluid (λf, W·m−1·K−1) and the characteristic length (L, m) of the element

0.67RaL1/4 ⎛ 1+ ⎝

⎜

9/16 ⎞4/9

( 0.492 Pr )

⎟

⎠

⎤2 ⎥ 0.387RaL1/6 ⎥ 9/27 ⎥ 9/16 ⎛ ⎞ 0.492 ⎥ ⎜1 + ⎟ Pr ⎝ ⎠ ⎦

( )

⎫ RaL ≤ 109 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ 9⎪ RaL ≥ 10 ⎪ ⎪ ⎪ ⎭ (20)

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Heat Exchange through the Digester. The heat exchange through the cover, walls, and floor of the digester is influenced by the thermal resistance of the materials used to construct the digester. For our model, we divided the heat transfer into two parts to distinguish between the energy balance of the gas and the slurry.

Soil Temperature Profile. The heat exchange through the digester walls below the soil surface and through the floor depends on the temperature of the soil, which varies with time and depth, and is influenced by soil characteristics and climate conditions. Hillel23 defined the following equation to compute the daily average soil temperature at different depths. The model combines the effect of the annual temperature variation and the effect of diurnal changes to improve the simple mathematical representation of thermal fluctuation, which assumes that the daily average temperature is the same for all depths

Q m,i−j =

hcnv, i − j

⎛ −z ⎞ ⎛ 2π π⎞ z = Tsurf,av + A 0 exp⎜ ⎟sin⎜ (t − t0) − − ⎟ ⎝ d ⎠ ⎝ 365 d 2⎠ (21)

where t is the time considered (day), Tgr,t(z) is the daily average soil temperature (°C) at any depth (z, m), and Tsurf,av is the average temperature (°C) of the soil surface; A0 is the amplitude of the surface temperature fluctuation (°C), calculated as half the difference between the maximum mean and the minimum mean for the ambient temperature of one year, and the constant d is a characteristic depth (m), called damping depth, and is expressed as a function of the thermal diffusivity of the soil (αs, m2 s−1). The radial frequency ω = 2π/ 365, and the value of thermal diffusivity is given in Table 3. d=

parameter

⎪

λ isol, j

(24)

(26)

Pt − 1 Tt · Pt Tt − 1

πdGas 2 (3rdig − dGas) 3

(27)

(28)

Radiative Heat Transfer. A difference in temperature between two surfaces causes heat transfer by long-wave radiation. For the heat transfer between the cover surface and the environment, the atmosphere is considered a blackbody. According to Swinbank,25 the atmosphere temperature (T∞, K) is expressed as T∞ = 0.0552Tair 3/2

(29)

The radiative heat transfer is then expressed as the product of cover emissivity (εcover), cover surface (Acover, m2), the Stefan− Boltzmann constant (σ), and the difference between the cover and atmosphere temperatures raised to the fourth power Q rad, ∞− C = εcover ·Acover ·σ ·(T∞4 − Tcover 4)

(30)

However, as there is a difference in temperature between the slurry and the cover, there is radiative heat transfer from the slurry to the cover not influenced by the gas, because it is assumed to be transparent. Assuming that the two surfaces define the simplest example of an enclosure as proposed by Incropera and co-workers,22 eq 31 has been used to compute this heat transfer. For the digester studied, the second term of the denominator, which takes into account the effect of the insulation, is zero

⎧Tair(s1 + (1 − s1)exp( − s2(LAI − LAI ref ))) Tair ≥ 0 ⎫ ⎬ Tsurf = ⎨ Tair < 0 ⎭ ⎩ ssnow Tair (23) ⎪

Aj , i (Ti − Tj)

A wall,S = 4πrdig 2 − 2πrdigdgas

Vgas =

Since the collection of temperature data at the soil surface is complex and measuring instruments are rarely available in the countries where simple digesters are mostly installed,12 following the method of Wu and Nofziger,24 suggested that the annual mean ambient temperature should be increased by 2 °C to estimate the annual mean of the soil surface temperature. This method is used by Perrigault et al.12 For the present work, a submodel to compute the temperature at the soil surface has also been considered to compare both paths. Kätterer and Andrén16 presented the following equation for calculating the apparent temperature at the soil surface (Tsurf, °C) when the ambient air temperature (Tair, °C) is known ⎪

L isol, j

(25)

Vgas, t = Vgas, t − 1·

8 × 10−7 0.95 0.40 0.20 3.0 0.0

⎪

λj

+

A wall,G = 2πrdigdgas

value αs s1 s2 ssnow LAIref LAI

Lj

where dgas is obtained by equaling the following equations for gas volume

(22)

Table 3. Parameters Used for the Calculation of Soil Surface and Soil Temperature thermal diffusivity (m2·s−1) constant s1 constant s2 constant ssnow leaf area index of reference leaf area index

+

where i is gas or slurry, j is the digester cover or ground, m is cover, wall or floor, hcnv,i−j is the convective heat transfer coefficient, L is the thickness (m) of the construction material, and λ is its thermal conductivity (W m−1 K−1, Table 2). Equation 24 considers the thermal resistance from insulation. However, for the present study, this term is null because, as mentioned previously, the digester is not insulated. To compute heat transfer through the wall, the variation in the area (m2) of the two contact surfaces (Aj,i) during the process of biogas production owing to gas volume fluctuations is taken into account. Assuming a spherical shape for the fixeddome digester and treating biogas as an ideal gas, the contact surface of the walls for the gas and for the slurry volumes can be written as a function of the height of the gas volume (dgas) as

Tgr, t(z)

2·αs·3600·24 ω

1 1

which includes the effect of snow cover (ssnow) and surface cover during the growing season (LAI), where s1, s2, and ssnow are constants and LAI is the leaf area index (Table 3). 3256



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Table 4. Variables Used in the Calculation of Biogas Production13a month parameter inflow rate HRT VS (S0) dry matter (DM) kinetic constant (K) a

units 3

−1

m day days kg VS m−3 kg VS m−3

Aug

Sept

Oct

Nov

Dec

Jan

Feb

0.136 37.0 11.3 22 0.636

0.136 38.0 9.1 17 0.633

0.144 30.0 7.4 12 0.630

0.140 30.1 4.0 6 0.625

0.133 41.9 4.0 7 0.626

0.091 67.6 7.8 10 0.632

0.076 41.0 10.8 11 0.636

The biochemical methane production potential (B0) is 356 LCH4 kg−1(VS).27

k = 0.6 + 0.0206e0.051·S0

Q rad,C − S = {1/(Lj /λj) + (L isol, j /λ isol, j)

Operation of a Full-Scale Anaerobic Digester in Hanoi, Vietnam. A full-scale digester was installed in July 2012 at an altitude of 100 m.a.s.l at the Pig Research Centre, National Institute of Animal Science, Thuyphuong, Tuliem, Hanoi, Vietnam. The digester is made from fiberglass and is uninsulated. The total volume is 7 m3 and working capacity 5 m3. Pig slurry was used as feedstock. HRT was 30.0−67.6 days, the feeding rate being 0.076−0.136 m3 day−1 in the period July 2012 until February 2013. Both DM and VS were fairly low, as can be seen in Table 4 because pig slurry contains more urine than cow manure and because more cleaning water is used. In the present study, the pig slurry was diluted with cleaning water at a ratio of 1:2. The parameters that were used to validate the model were recorded by Pham and co-workers17 between July 2012 and February 2013 and they were as follows: At an interval of 30 min, temperature was measured at 1 m above soil surface, in the digester at a depth of 1, 1.4, and 1.8 m, and into the soil at 1.4 m. Methane production rate was measured every two to three days, and mass flow, composition, and temperature of the slurry was measured when the slurry was added to the digester. Data on wind velocity were obtained from the National Center for Hydro-Meteorological Forecasting, Vietnam, which is located 5 km from the experimental site. For the first two months of adding inoculum and biomass to the digester, fermentation was unstable and therefore measurements from this period were not used for model validation. To provide precise measurements of the biogas production rate, the volume of gas produced and thus released during measurements has to be significant, and therefore measurements are carried out at intervals of between two and seven days from October 2012 until February 2013. Consequently, we decided to use cumulative weekly production of biogas for model validation.

+ {1/[(σ (Tcover + Tslurry )(Tcover 2 + Tslurry 2)) /((((1 − εs))Acover )/(εsA surf,C − S) + (1/FC − S) + ((1 − εc)/εc))])]}Acover (Tcover − Tslurry ) (31)

where FC−S is the view factor from the cover to the slurry, L and λ are the thickness (m) and the thermal conductivity (W m−1 K−1) of the material, respectively, σ is the Stefan−Boltzmann constant (W m−2 K−4), ε is the emissivity of the surface considered (Table 2), and Acover and Asurf,C−S (m2) are the area of the cover surface and the slurry surface, respectively. Heat Transfer from Slurry Inflow. Gebremedhin and coworkers13 described the influence of slurry feeding on the temperature of the slurry inside the digester by heat transfer as m Q load = load Cp,S(Tslurry − Ts_in) (32) dt where mload is the slurry feeding mass flow (kg), Cp,s is the slurry specific heat (J·kg−1·K−1), Tslurry is temperature (K), and Ts_in is the temperature of the slurry fed (K). The loading of the reactor (Qload) may vary. In the present study slurry is fed to the digester once every day (dt). Biogas Production. Anaerobic digestion does not operate under a constant temperature condition in unheated and simple digesters, rather the temperature conditions can vary from psychrophilic to mesophilic due to poor insulation. Pham and co-workers17 developed algorithms for microbial growth and kinetic factors for a range of temperature between 10 and 30 °C. The maximum specific growth rate (μm) of microorganisms in pig manure as a function of the temperature of the slurry inside the digester was determined with the following equation: μm = 0.0039e 0.1188Tslurry

(temperature = 10 − 30 °C)

■

(33)

RESULTS AND DISCUSSION Taking into account the principles of heat transfer and energy conservation, the model predicts the heat absorbed by the digester from solar radiation, the thermal influence of daily feeding and the heat losses through the digester cover, walls and floor. The model was run using the parameters shown in Tables 1−4. The predictions of the model are validated against experimental data. The measured data provided are very useful because they show the evolution of temperature over a long period and not only the diurnal cycles of the temperatures, permitting an overview of the annual thermal regime. Daily Soil Surface Temperature. Figure 1 shows the predicted daily soil surface temperature versus the measured temperature using the model of Kätterer and Andrén16 and a

This equation is used in the model of Chen and Hashimoto26 to predict methane production in this project. Their model includes the effect of solid retention in the unstirred digester and estimates the methane production rate as follows: γ=

⎞ ⎛ B0 ·S0 ⎞⎛ k ⎜ ⎟⎜1 − ⎟⎟ ⎜ ⎝ HRT ⎠⎝ HRT·μm − 1 + k ⎠

(35)

(34)

where the values of overall production yield (B0, L(CH4)· kg(VS)−1), inflow volatile solids (VS) concentration (S0, kg(VS)·m−3), and hydraulic retention time (HRT, days) are reported in Table 4. In this study k is a kinetic constant (dimensionless) which depends on the concentration of organic components in the feedstock (S0) and can be estimated as 3257



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the soil temperature has a huge importance in the thermal performance of this kind of digesters, we have related calculated soil temperature to the experimental by a linear regression (Figure 2, line B). Using this relation, the soil temperature

Figure 1. Predicted versus measured daily soil surface temperature. Prediction carried out using the model of Kätterer and Andrén16 and the model used by Perrigault et al.12

submodel used by Perrigault and co-workers,12 who assumed that the soil surface temperature is the annual mean of the ambient air temperature plus 2 °C, as proposed by Wu and Nofziger.24 The authors found that R2 of both models was high and that slope and intercept were also similar with slope being close to 1 (Figure 1). While the soil temperature profile proposed by Hillel23 requires knowledge of the soil surface temperature, the Kätterer and Andrén model16 is useful (R2 = 0.86 and slope 1.08) but has a need of parameters that are not always available as the LAI. The submodel23 used by Perrigault and co-workers12 was likewise able to reduce the complexity of the computations (R2 = 0.86 and slope 0.97) requiring only data from the annual ambient temperature, without the need of other parameters. In the present study the soil surface temperature has been estimated considering daily, weekly and monthly mean ambient temperature plus 2 °C, getting R2 of 0.86, 0.88, and 0.88, respectively, with slopes of 0.97, 0.92, and 0.88 (not shown in the figure). Soil temperature has a vertical temperature gradient, and it was therefore decided to show temperatures at a halfway digester depth (1.4 m) and at the soil surface. Soil Temperature. The Hillel23 model to estimate the soil temperature needs the surface soil temperature as input data. In order to validate the Hillel model, the soil surface temperature model of Kätterer and Andrén16 and the one used by Perrigault et al.12 have been used as input. Perrigault et al.12 use the annual mean ambient temperature plus 2 °C, and we have also evaluated what would occur if daily, weekly or monthly mean ambient temperature is used instead of the annual one. The results for the soil temperature at a depth of 1.4 m applying the Hillel model, when Kätterer and Andrén soil surface temperature model16 is used, fits the measured soil temperature data with R2 = 0.88 and a slope of 0.35. In the case where daily, weekly, and monthly mean ambient temperature, increased by 2 °C, has been employed, R2 varies from 0.62 to 0.65 with slopes over 1. The increase of 2 °C of the ambient temperature, used to estimate the soil surface temperature reports the best results with a R2 of 0.88 and a slope of 0.64, as recommended by Perrigault and co-workers.12 The results indicate that small changes in the soil surface temperature estimation produce a huge difference in the soil temperature prediction. Finally, as

Figure 2. Validation of predicted temperatures from the model proposed to compute soil temperature: A as regression line for the model of Kätterer and Andrén16 (y = 0.35x + 17.35, R2 = 0.88); B for the model of annual mean air T + 2 °C (y = 0.64x + 8.83, R2 = 0.88); C for best regression line (y = x); D for the model of weekly mean air T + 2 °C (y = 1.33x − 9.51, R2 = 0.62); E for the monthly mean air T + 2 °C (y = 1.32x − 9.42, R2 = 0.65).

calculated by the Hillel model and using the recommendation from Perrigault and co-workers for soil surface temperature, has been corrected by T′gr = (Tgr − 8.8258)/0.6373 (Figure 2, line B). The subsequent results were run with the corrected soil temperature. Digester Temperatures. The average slurry temperature related to depths were 24.8 (±3.7) °C at 1 m, 24.5 (±3.6) °C at 1.4 m, and 24.4 (±3.5) °C at 1.8 m, respectively, and the mean standard deviation of the temperature at the three depths was 0.5 °C, which is in a good accordance with assumption 5, mentioning that there is no temperature gradient in the digester (Figure 3). Therefore, to achieve a unique, experimental slurry temperature, an average slurry temperature was computed to compare with the temperature predicted from the model. Figure 4 shows the time evolution for the predicted and the measured digester temperatures over seven months on the left, and the validation results on the right, showing a good agreement (R2 = 0.96, slope 0.96). The soil cover has a daily temperature amplitude of 30 °C because of the effect of solar radiation and ambient temperature (Figure 5), while the slurry temperature keeps an almost constant temperature, showing a huge thermal inertia. The steady thermal behavior of slurry is very important because it permits the computation of the volume of methane produced inside the digester. The model shows that the daily manure feeding of the digester appears not to affect the overall slurry temperature. Soil temperature computed was also found to correlate well with digester temperature, indicating that a soil temperature can be used to predict the temperature inside a 3258



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measured at half hourly intervals, thus production of biogas was calculated using these data and then transformed to give the daily production for the given week. A plot of predicted methane production against measured production showed that prediction was fairly accurate (Figure 6), that is, the R2 value for prediction of biogas production was 0.65. The reason for the lower precision in October could be that the digester, before the monitoring period, was not loaded with the frequency desirable, and this is reflected in the scattering values that reach a steady state after the first month once the load profile is kept constant. The precision of the model was high during the winter period, although there were outliers and tendency for underestimation, which may be an effect of changes in composition of excreta between summer and winter due to changes in diet. Furthermore, the measured values of methane production fluctuated more than those predicted by the model. This could be due to instability in the digester caused by fluctuations in the anaerobic conditions in themselves caused by changes in, for example, VS concentration, feeding rate, etc. Validation of the model for prediction of methane production was performed as described by Triolo and coworkers,28,29 using the root-mean-square error of prediction (RMSE) to assess the quality of the model. The relative rootmean-square error (rRMSEP (%)) was also used to assess relative model error, according to Madsen and co-workers30 and Triolo and co-workers.29 RMSE was 54.4 L CH4 digester−1 day−1 and rRMSEP(%) was high at 35.4%. The biogas production model17 had a model error at RMSE 23.3 NL CH4 digester−1 day−1 and the relative model error RRMSE was 18.6%. The model error of this model is affected by the error of both the biogas production model and the thermic model. Hence, the uncertainty of this model is partly due to the precision of the biogas production model and to the precision of the thermic model. In comparison the relative model error (rRMSEP) was 13.7% in a study where Raju and co-workers31 concluded that a near-infrared reflectance spectroscopic method (NIRs) gave good model results when predicting biogas production potentials and Triolo and co-workers32 reported that with a rRMSEP(%) at 14.0% their NIRs model was moderately successful in predicting biogas production potential. For weekly and monthly methane production, the relevant model error was lower, at 27.9% and 20.9%, respectively. The validation result was considerably improved when only the data for the winter season (December to February) were included (RMSE = 26.1 L CH4 digester−1 day−1; rRMSEP(%) = 17.7%).

Figure 3. Temperatures at three different depths inside the digester and air temperature within the period from the 12th to the 16th of August.

digester that is unheated, unstirred, without insulation and buried in the ground. The close relationship of soil temperature with the measured temperature inside the digester is evidenced by a regression coefficient of 0.90 and linear slope of 0.87. Hence, for this type of digester (buried, uninsulated, and with a poor solar passive heating design), a simple way to predict the slurry temperature is to assess or predict the soil temperature. The prediction of thermal behavior of low-cost digesters relies on using as little input data as possible, and this affects the results. However, a simple, one-dimensional thermal balance model, where each digester component is characterized by a unique temperature, can be used to give an acceptable prediction of the digester temperatures. For insulated or solar-heated digesters, it appears that a simple 1-D model can be used to study the optimization of the system based on material, thickness of insulation, solar radiation caption, or heating provided by the daily feedstock. Validation of the Methane Production Model. The validation covers a period with air temperatures between 25 and 35 °C in October and 15 and 25 °C in February, which is reflecting the high and low air temperatures measured over the year in the region. Furthermore, biogas production was determined at three days to weekly intervals, because the higher pressure and larger release of biogas gave more precise measurements of the production. The temperature was

Figure 4. Left: Predicted and measured daily mean digester temperature from July 2012 until February 2013. Right: Predicted versus measured digester temperature. The measured digester temperature is a weighted average of the measurements in the digester.17 3259



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Figure 5. (a) Measured ambient air temperature, (b) predicted cover temperature using ambient air temperature, and (c) predicted temperature of slurry and soil using ambient air temperature.

production more precisely from December after the microbes are well adapted.” Nevertheless, the results of the validation are positive, taking the uncertainty throughout the modelling process into account. Since unheated digesters without active heating or solar passive design suffer critically low methane production during the winter season, the model presented here could be a particularly useful tool for assessing methane production and for improving the ability of unheated digesters to provide sufficient energy during cold periods.

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ASSOCIATED CONTENT

S Supporting Information *

Figure 6. Predicted versus measured methane production rates over the period Oct 2012−Feb 2013.

Details of the biogas production model (ref 17). This information is available free of charge via the Internet at http://pubs.acs.org.

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These validation results indicate that further research and development of a more elaborate model are needed to predict methane production, including hidden factors that may improve the model precision, especially at the change of season when the change in temperature from mesophilic to psychrophilic or psychrotrophic conditions and vice versa can affect the microbial community. Thus lowering the temperature leads to a decrease in the maximum specific growth and affects substrate utilization rate.33,34 At ca. 25°C a rapid decrease of mesophilic microbial activity occurs35 and this temperature presents the edge between psychrophilic and mesophilic conditions. Moreover temperature shock is not included in the model, while it includes the change in biogas production of microorganisms that is adapted to a gradual decrease in temperature.36,37 For this reason the model predicts biogas

AUTHOR INFORMATION

Corresponding Author

*Tel: +45 4117 8867. Fax: +45 6550 7359. E-mail: jmt@kbm. sdu.dk. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was funded by the Danish Ministry of Foreign Affairs through a grant to the SUSANE research project (sustainable, sanitary, and efficient management of animal manure for plant nutrition) and the financial assistance of Hivos. 3260



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Article

NOMENCLATURE

A0 Acover Awall,G Awall,S Asurf,G−S a B0 cp dGas dn dt fi g HRT hcnv,i−j k L LAI m mG mload mS P Pr Qcnv,i−j Qcover, Qwall, Qfloor Qload Qrad,i−j QSolar q″s Ra Re rdig Nu Sb Sd Sr Sp Sp0 S0 ssnow, s1, s2 Tair Tcover Tfloor Tgas Tgr,t(z) Ts_in Tslurry

Tsurf,av T∞ t trs tss t0

amplitude of the surface temperature fluctuation (K) area of the cover surface (m2) area of the wall surface in contact with the gas (m2) area of the wall surface in contact with the slurry (m2) area of the contact surface between the gas and the slurry (m2) coefficient of transmissivity biochemical methane production (LCH4· kgVS−1) at 1 atm pressure and 0 °C specific heat (J·kg−1·K−1) height of gas volume (m) Julian date time interval (s) latitude (degrees) gravitational acceleration (m·s−2) hydraulic retention time (days) convective heat transfer coefficient (W·m−2· K−1) kinetic constant characteristic length (m) leaf area index optical air-mass number mass flow rate of the gas (kg) slurry feeding mass flow (kg) mass flow rate of the slurry (kg) gas pressure (Pa) Prandlt number convective heat transfer between element i and j (W) heat transfer by conduction and convection through the digester (W) heat losses from daily manure feeding (W) radiative heat transfer between element i and j (W) heat gain from solar radiation (W) heat flux of solar radiation (W·m−2) Rayleight number Reynolds number radius of the digester (m) Nusselt number beam radiation (W·m−2) diffuse radiation (W·m−2) Albedo radiation (W·m−2) direct solar irradiance (W·m−2) extraterrestrial flux density normal to the solar beam (W·m−2) inflow VS concentration (kgVS·m−3) constants of soil cover effect air temperature (K) temperature of the cover (K) temperature of the digester’s floor (K) temperature of the gas (K) daily average soil temperature at any depth (K) temperature of the daily manure feeding (K) temperature of the slurry inside the digester (K)

uwind Vgas z α β γ δ ε λ μ μm ρ σ υ θ ω

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average temperature of the soil surface (K) sky temperature (K) current time (s) sunrise time sunset time day of the year with the lowest temperature (day) wind velocity (m·s−1) gas volume (m3) depth (m) absorbance for cover, thermal diffusivity for the cover (m2·s−1) expansion coefficient (K−1) methane production (L CH4 ·digester −1 · day−1) at 1 atm pressure and 0 °C solar declination (degrees) coefficient of emissivity thermal conductivity (W·m−1·K−1) dynamic viscosity (Pa·s) maximum specific growth rate (days−1) density (kg·m−3) Stephan−Boltzmann constant (W·m−2·K−4) kinematic viscosity (m2·s−1) sun elevation angle (degrees) radial frequency (rad)

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Thermic Model to Predict Biogas Production in Unheated Fixed-Dome

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