Time-Lag Models for Continuous Stirred Tank and Plug Flow

Article pubs.acs.org/EF

Time-Lag Models for Continuous Stirred Tank and Plug Flow Digesters for Biogas Production Aritra Das,†,‡ Chanchal Mondal,†,‡ and Siddharth G. Chatterjee*,§ †

Department of Chemical Engineering, Jadavpur University, Kolkata, 700032 West Bengal, India Energy Engineering Laboratory, Jadavpur University, Kolkata, 700032 West Bengal, India § Department of Paper and Bioprocess Engineering, SUNY College of Environmental Science and Forestry, Syracuse, New York 13210, United States ‡

S Supporting Information *

ABSTRACT: This work presents time-lag models for biogas production from anaerobic digestion of organic wastes in continuous stirred tank and plug flow digesters. The models have the ability of predicting the sigmoid nature of the volumetric rate of production of biogas with process time that is observed experimentally. The models are calibrated against experimental data on biogas production by anaerobic digestion of mixed vegetable wastes in a compartmental digester. The quality of fit of the models to these data is satisfactory (average root-mean-square error = 9.2%). However, the PFR time-lag model is a better representative of the compartmental digester than the CSTR lag model. The time-lag models have a simple mathematical form whose parameters can be estimated easily and they should be useful for correlating biogas yield with process time, and hence in formulating control schemes for an existing anaerobic digester.

1. INTRODUCTION Anaerobic digestion (AD) is a method of producing biogas, which primarily consists of methane and carbon dioxide, from various types of organic wastes.1 Production of biogas can also be enhanced by using metal catalysts in the digestion process2 and by pretreatment of feedstock.3 For design, scale-up, and control of the AD process, theoretical models of the process are essential. There is a considerable literature on theoretical modeling of AD, some examples of which are the works of Husain,4 Massé and Droste,5 Lokshina et al.,6 and Momoh and Saroj.7 Many models exist which attempt to relate microbial growth, substrate depletion, and biogas generation. Such models are complex and require complicated parameter estimation procedures. Perhaps, the most comprehensive model of AD is the Anaerobic Digestion Model No. 1 (ADM1) which was developed as a result of international collaboration among experts. This structured model requires many input parameters since it attempts to simulate the production of methane by several reaction pathways. Thus, parameter identification and the manipulation of the multitude of stoichiometric and kinetic equations can be difficult according to Yu et al.8 We refer the reader to their paper, and also to the paper of Husain,4 for a comprehensive literature survey on mathematical modeling of AD. Continuous stirred tank reactor (CSTR) and plug flow reactor (PFR) models for AD have also been extensively employed for the design and scale-up of digesters. Using the ADM1 model, Bensmann et al.9 performed a detailed analysis of reactor configurations for improving the efficiency of biogas plants. These included combinations of CSTRs and a combination of a CSTR and a PFR in series. They found that these combinations gave better methane yield compared to a single CSTR but at the expense of the maximum achievable specific methane flow rate, which could be increased by © 2016 American Chemical Society

increasing the solid retention time. According to Yusuf and Ify,10 models of biogas generation from biomass are essentially absent since models that use the concept of the maximum specific growth rate of bacteria and which are valid for short retention times may not be applicable to energy biomass. Moreover, in the case of lingo-cellulosic biomass, there will be complicating factors like mass-transfer limitations, fragmentation, and breakage of biomass particles with progress of AD, etc., which will make the theoretical analysis of AD of such biomass exceedingly complex. In this regard, we note that Momoh and Saroj7 have very recently presented surface-based and water-based-diffusion kinetic models, which attempt to incorporate some of these effects, for correlating experimental data on biogas production from cow manure in batch AD. However, these models contain 4−6 parameters, which have to be obtained by nonlinear regression. The objective of the present work is to present an alternative and simple framework for predicting the rate of biogas production from AD of organic wastes in a single CSTR or a PFR. This is achieved by developing mathematical models, called time-lag models, which focus directly on biogas production rather than on the individual and very complex steps (microbial growth, hydrolysis, acidogenesis, acetogenesis, and methanogenesis) that lead to such production. It has been empirically observed that the volumetric rate of production of biogas or methane as a function of process time in such reactors or digesters is a sigmoid or S-shaped curve, i.e., the rate of change of the volumetric production rate of biogas is low initially, passes through a maximum, and declines subsequently. This feature can be seen in the data taken by Mondal11 on the Received: May 31, 2016 Revised: October 21, 2016 Published: October 24, 2016 10404

DOI: 10.1021/acs.energyfuels.6b01320 Energy Fuels 2016, 30, 10404−10416

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Energy & Fuels

sufficient number of microorganisms in the digester. The loading occurred in a semicontinuous fashion (i.e., once a day). At start-up, 20 mL of bacterial culture, which was prepared with cattle manure procured from a nearby cattle shed, were introduced along with the feed slurry into the digester whose temperature was controlled with a thermostatic controller. In the experiments, the BOD of the influent feed was 45 g L−1. After every 24 h, the BOD of the effluent was measured and the volume of gas produced was determined by the downward displacement of water in an aspirator bottle. This bottle contained a 2% glycerol solution to prevent absorption of carbon dioxide into the water. Before gas collection and BOD measurement, the digester contents were agitated by means of the stirrers. Agitation was also provided intermittently during the experiments, and not continuously in order to lower pumping energy consumption. The volumetric carbon dioxide, carbon monoxide and oxygen contents of the biogas were determined by an Orsat gas analyzer using potassium hydroxide, alkaline pyrogallol and cuprous chloride solutions as absorbents.13 The volume of methane in the biogas was evaluated by difference. A confirmatory test for methane, carbon dioxide, and ammonia was also performed by a gas chromatograph which had a porapak Q & QS column and a thermal conductivity detector (temperature: 60 °C). The flow rate of hydrogen, used as the carrier gas, was maintained at 30 mL min−1 and the temperature of the oven and injection port was 65 °C. Moisture content, volatile matter, ash content, and fixed carbon of the feedstock were determined by the standard methods of the Fuel Research Board and British Standard Institution. The calorific value was determined according to ASTM D2075-77 whereas COD and BOD were measured using the methods proposed by De14 and Clesceri et al.,15 respectively. The percentages of lignin, cellulose, and hemicellulose in the feedstock were determined according to standard AOAC methods.

production of biogas by AD of vegetable wastes in a semicontinuous flow compartmental digester (to be discussed later) and also in the data presented by Jayalakshmi et al.12 on solid-phase AD of kitchen waste in an inclined plug flow digester. The experimental cumulative biogas or methane yield as a function of retention time in batch AD also exhibits such a characteristic.1,6,7 This S-shaped character of the biogas production rate curve can be captured by the time-lag models as will be shown later. The time-lag models presented herein have only three parameters, which can be easily estimated. The models are calibrated against some data of biogas production from AD of mixed vegetable wastes.

2. MATERIALS AND METHODS 2.1. Materials. For the benefit of the reader, the materials, analytical instruments and characteristics of the feedstock used by Mondal11 are reported in Tables S1, S2 and S3, respectively, under Supporting Information. 2.2. Sample Collection and Preparation. Vegetable wastes were procured from local markets, which were identified as the source of easiest availability. The wastes were graded, sorted, chopped into small pieces, and finally subjected to sun drying for 8 h daily for 2 d consecutively in order to remove extraneous moisture. After sufficient drying, the dried matter was passed through a Wiley mill for particle size reduction. The undersized particles were kept in airtight containers until further use. These containers were maintained at 5 °C to prevent biological degradation, with the composition of the feedstock (Table S3) being checked periodically in order to ensure that it did not change with storage time. 2.3. Experimental Procedures. Experiments were carried out in a three-stage compartmental digester, a schematic of which is shown in Figure 1. The capacity of the digester, which was equipped with a

3. TIME-LAG MODELS FOR BIOGAS PRODUCTION IN CONTINUOUS STIRRED TANK AND PLUG FLOW DIGESTERS As mentioned previously, the typical experimental volumetric production rate of biogas is a sigmoid shaped curve when it is plotted as a function of process time in the digester, thereby indicating that the rate of change of the biogas production rate reaches a maximum from an initial low value, with a subsequent decline. The time-lag models for biogas production in continuous and plug flow digesters developed in this section attempt to capture this feature through a delay parameter τ, which has the units of time. In formulating the models, the following criterion is aimed at. The models should be simple and contain a minimum number of independent parameters, which can be easily estimated from experimental data. As will be seen later, the time-lag models developed in this work fulfill this criterion. We note that the concept of time delay has been used in modeling phenomena, such as population dynamics, forest fires, wavefronts in reaction diffusion systems,16,17 and circadian rhythms.18 A well-known example of a time-delay model is the hyperbolic reaction diffusion equation This equation can be derived by combining a time-delayed form of Fick’s law of diffusion with the law of conservation of mass. In chemical reactor analysis, the concepts of the continuous stirred tank reactor and plug flow reactor are often used in order to simplify calculations. In the former, the reactor contents are assumed to be perfectly mixed, which implies that concentrations within the reactor are the same as those in the outlet stream from the reactor. In the latter, the reaction mixture is assumed to travel through the reactor in the form of “plugs” (i.e., in a rod-like fashion) so that there is no mixing or dispersion along the length of the reactor. Thus, these two ideal

Figure 1. Schematic of the three-stage compartmental digester used by Mondal.11 motor driven stirrer introduced into each compartment, was 10 L. The opening between compartments was constructed in such a way so as to allow the flow of slurry inside the digester to occur in a zigzag pattern. The entire digester was placed in a thermostatic bath for maintaining the slurry temperature at the desired level. The experimental runs were performed at different digestion temperatures. Three gas outlets, one corresponding to each compartment, were connected to a single unit from which gas was collected for measurement of its volume (at approximately 30 °C and atmospheric pressure) by the waterdisplacement method, and also for determining its composition by gas chromatography. The pH of the effluent was checked frequently in order to ensure that it was within the optimum range of 6.7−7.1. Whenever a drop or rise in pH was observed, a proportionate amount of lime or carbonic acid was added to the digester contents in order to maintain the pH within the proper range. A prepared sample of slurry of biomass particles of a fixed particle size was fed into the digester, which was initially filled with water, at a fixed volumetric loading rate. The loading rate was gradually increased until it reached a value of 1 L/d in order to ensure that there was a 10405


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where k is a pseudo first-order rate constant that is an indication of the overall rate of the complex biochemical reactions between the microorganisms and substrate, including mass-transfer limitations, that result in the production of biogas and θHR (= V/Q) is the hydraulic retention time (HRT) of the liquid in the digester. According to eq 1, the chemical reaction and the inflow/outflow terms do not affect the accumulation term instantaneously (i.e., at time θ) but only after τ units of time have elapsed (i.e., at time θ + τ); thus a time delay has been introduced between cause and effect, which is more realistic. More elaboration on the lag time τ will be given later. Expanding the accumulation term in eq 1 in a Taylor series and retaining only the first two terms of this series transforms eq 1 into the following second-order differential equation:

reactors are expected to circumscribe the behavior of an actual reactor. For the same residence time of the reaction mixture inside the reactor, the PFR achieves a higher conversion of the reactant than the CSTR for reaction orders greater than zero.19 Figure 2a,b shows schematics of the CSTR and PFR, respectively.

τ

S d 2S dS + + k modS = in 2 dθ θHR dθ

(2)

where kmod is a modified rate constant given by k mod = k +

1 θHR

(3)

Eq 2 is analogous to the equation that describes the vibrations of a spring-mass system. The first three terms on the left-hand side of the equation represent the “inertia” (with τ being the system “mass”), “damping”, and “spring force,” respectively, while the right-hand-side is the “external force.” We note that for τ = 0 (i.e., no lag), eq 2 reduces to the standard model of a stirred tank reactor with a first-order chemical reaction. The characteristic equation corresponding to eq 2 is τm2 + m + k mod = 0

whose roots are given by

Figure 2. Schematics of a CSTR (a) and PFR (b) for producing biogas by AD of an organic waste.

m1,2 =

−1 ±

1 − 4k modτ 2τ

(5)

It is observed from eq 5 that the roots can be real and unequal, real and equal, or complex depending upon the magnitude of the quantity 1 − 4kmodτ. If 1 − 4kmodτ < 0, the roots will be complex that will lead to oscillatory behavior of S with θ, which is not realistic; hence this case is discarded. If 1 − 4kmodτ > 0, the roots will be real and unequal; however, the general solution for S(θ) will then contain two independent parameters k and τ. If 1 − 4kmodτ = 0, then both roots will be real and equal, which gives

3.1. Continuous Stirred Tank Reactor. Consider a CSTR with a working volume V into and out of which flows a liquid at a constant volumetric flow rate Q. The reactor or digester is assumed to contain a sufficient number of microorganisms, which produce biogas by consuming the substrate. This assumption can be fulfilled by any of the following experimental conditions: (1) there is a separate inoculum stream continuously introduced into the digester, (2) there is an adequate number of microorganisms in the digester feed, or (3) the liquid flow rate through the digester is sufficiently small so that the growth of cells balances cell death and cell washout. The feed liquid has a substrate concentration of Sin while the outflow liquid has a substrate concentration of S at modified process time θ, which is the difference between the actual time t that has passed since the beginning of the process and the initial incubation period tincb of the microorganisms (typically less than 3 days) during which there is negligible production of biogas, i.e., θ = t − tincb. The contents of the digester are assumed to be well mixed, hence the concentration of substrate inside it is also S. Assuming that the substrate is consumed by a first-order reaction, the mass balance for substrate in the digester is written as ⎛ dS ⎞ S −S ⎜ ⎟ = in − kS ⎝ dθ ⎠ θ + τ θHR

(4)

τ=

1 4k mod

(6)

and the general solution for S(θ) will contain only one parameter (i.e., either k or τ). By invoking the criterion of simplicity (i.e., minimum number of model parameters), we will consider only this case, which, in the field of process control, is referred to as the critically damped situation.20 Also, as will be shown later, this will allow us to correlate empirical data on the volumetric rate of production of biogas as a function of process time satisfactorily, with the parameters being obtained in a straightforward manner. It follows from eq 6 that the lag time τ is inversely proportional to the modified rate constant kmod. It then follows from eqs 3 and 6 that the higher the value of the rate constant k and smaller the value of the HRT, the lower is the lag time, and vice versa.

(1) 10406


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Energy & Fuels The initial conditions for eq 2 are assumed to be the following: S = S0 at θ = 0

δ* =

⎛ S 1⎞ 1 = ⎜S0* − ⎟(1 + 2αθ*)e−2αθ * + ⎝ Sin α⎠ α

(9)

⎛ S 1⎞ 1 = ⎜S0* − ⎟e−αθ * + ⎝ Sin α⎠ α

(19)

⎡⎛ δ 1⎞ 1⎤ = (α − 1)⎢⎜S0* − ⎟e−αθ * + ⎥ ⎝ ⎠ ⎣ β α α⎦

(20)

and δ* = (10)

For 0 ≤ S0* ≪ 1/α, eqs 19 and 20 reduce to

where S0 Sin

(11)

α = 1 + kθHR

(12)

S0* =

θ θHR

R su = kVS

(14)

Assuming that the rate of biogas (or methane) production Rbg is directly proportional to Rsu, we have R bg = βR su = βkVS

r* =

V

= βkS

r* =

QSin

=

r* = (16)

r* =

⎛1 ⎞ dδ * = α(α − 1)⎜ − S0*⎟e−αθ * ⎝α ⎠ dθ *

(22)

⎛ α − 1 ⎞ −2αθ * dδ * ⎟θ *e = 4α 2⎜ ⎝ α ⎠ dθ *

(21A)

dδ * = (α − 1)e−αθ * dθ *

(22A)

It can be inferred from eqs 21 or 21A of the lag model that the r* versus θ* curve is bell-shaped, which will be demonstrated later. It starts from a value of zero at θ* = 0, reaches a maximum at a value of θ* = θmax * , and then declines to zero as θ* → ∞. The value of θ*max can be obtained by setting dr*/dθ* = 0 using eq 21 and is given by

(17)

Eq 17 can be expressed in dimensionless form by using eqs 10 and 16 as follows: δ* =

(21)

and

γVθHR Sin

⎛1 ⎞ dδ * = 4α 2(α − 1)⎜ − S0*⎟θ*e−2αθ * ⎝α ⎠ dθ *

For 0 ≤ S*0 ≪ 1/α, these equations simplify to

Also let δ be the volume of biogas (or methane) produced per unit mass of substrate input into the digester, i.e., γVV

(20A)

and

(15)

where β is the volume of biogas (or methane) produced per unit mass of substrate consumed. Let γV be the volumetric rate of biogas (or methane) production per unit volume of liquid in the digester, i.e., γV =

⎛ α − 1⎞ −αθ * ⎜ ⎟(1 − e ) ⎝ α ⎠

It is interesting to compare eqs 10A and 18A of the lag model with eqs 19A and 20A of the no-lag model The dimensionless rate r* of substrate consumption per unit mass of substrate input into the digester for the lag and no-lag models can be obtained from eqs 18 and 20, respectively, as

(13)

The rate of substrate consumption or utilization Rsu is given by

R bg

(19A)

and δ* =

θ* =

1 (1 − e−αθ *) α

S* =

and

δ=

(18A)

S* =

Eq 9 can be cast into the following dimensionless form: S* =

⎛ α − 1⎞ −2αθ * ⎜ ⎟[1 − (1 + 2αθ *)e ] ⎝ α ⎠

It can be easily shown that for the no lag (i.e., τ = 0) first-order CSTR model, equations analogous to eqs 10 and 18 are given by

(8)

with S0 being the initial substrate concentration in the digester. The solution of eq 2 subject to eqs 6 to 8 is ⎛S ⎞ S 1 1 =⎜ 0 − ⎟(1 + 2k modθ)e−2k modθ + Sin 1 + kθHR ⎠ 1 + kθHR ⎝ Sin

(10A)

and

(7)

and dS = 0 at θ = 0 dθ

1 [1 − (1 + 2αθ*)e−2αθ *] α

S* =

⎡⎛ δ 1⎞ 1⎤ = (α − 1)⎢⎜S0* − ⎟(1 + 2αθ*)e−2αθ * + ⎥ ⎝ ⎠ ⎣ β α α⎦

* = θmax

(18)

The quantity δ* represents the mass of substrate consumed in producing biogas per unit mass of substrate input into the digester. As steady state is approached), δ(θ → ∞) → βkθHR/ (1 + kθHR). Thus, β and δ can be looked upon as the intrinsic and operational yields of biogas, respectively. Further, as the quantity kθHR → ∞, the steady-state operational yield and the intrinsic yield approach one another. For 0 ≤ S0* ≪ 1/α, eqs 10 and 18 simplify to

θmax 1 = θHR 2α

(23)

where θmax is the value of the modified retention time at the inflection (or critical) point of the lag-model curve (see Figure 3to be discussed later). Upon substituting eq 23 into eq 21 we get: * = rmax 10407

⎛1 ⎞ 2 α(α − 1)⎜ − S0*⎟ ⎝α ⎠ e

(24) DOI: 10.1021/acs.energyfuels.6b01320 Energy Fuels 2016, 30, 10404−10416

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Energy & Fuels S = Sin at x = 0 for θ > 0

Note that no outlet boundary condition is necessary at x = L because the axial dispersion term has been neglected in eq 26. The solution of eq 27, subject to eqs 28 and 29, can be derived by taking Laplace transforms of these equations with respect to the space variable xan outline of the derivation is provided in the Appendix. However, unlike the case of the CSTR, the lag time τ can only be defined implicitly (i.e., not explicitly) for a nonzero value of S0 (see eq A4 in the Appendix). An explicit relation for τ can be provided only for S0 = 0; this relation is

Figure 3. Substrate consumption in a CSTR as a function of modified process time as predicted by the no-lag and lag models in dimensionless coordinates (α = 2).

τ=

+ S0(1 + 2kθ)e−2kθ H(x − 2uθ) for θ ≤ 0.5θHR

∂S = 0 at θ = 0 for x ≥ 0 S = S0 and ∂θ

(31A)

and S(x , θ ) = Sine−(k / u)x for θ > 0.5θHR

(25)

(31B)

where H(x) and δ(x) are the Heavyside unit step and Dirac delta functions, respectively, and θHR(= V/Q = L/u) is the HRT in the PFR. For the PFR, the rate of substrate consumption Rsu is given by

R su = kVSavg

(32)

where Savg is the average substrate concentration in the digester, which can be obtained from Savg(θ ) =

1 L

∫0

L

S(x , θ )dx

(33)

Similar to the case of the CSTR, we assume that the rate of biogas production Rbg is proportional to Rsu, i.e.,

R bg = βR su = βkVSavg

(34)

where β is a constant of proportionality (volume of biogas produced per unit mass of substrate consumed). The volumetric rate of biogas production per unit volume of liquid in the PFR is given by

(26)

γV =

where k is a pseudo first-order rate constant, x is the axial distance measured from the inlet (x = 0) of the digester, and u the axial velocity of the feed. Once again a time delay has been introduced into the accumulation term. The characteristic feature of the solution of a hyperbolic differential equation, such as eq 26, is a shock wave or front at which the solution is undefined or discontinuous; this will be discussed later. By a similar process as done for the case of the CSTR, eq 26 is converted into the following equation: ∂ 2S ∂S ∂S τ 2 + +u + kS = 0 ∂θ ∂x ∂θ which is to be solved subject to the conditions:

(30)

S(x , θ) = Sine−(k / u)x[1 − H(x − 2uθ)] + 2uθe−2kθ(S0 − Sin)δ(x − 2uθ)

We note from eqs 24 and 25 that r*max is a function of α and S*0 . Thus, for S0 = 0, the greater the HRT and the reaction rate constant, the greater is the maximum volume of biogas produced per unit mass of substrate input into the digester, which is in accord with physical intuition. The ratio of r*max of the lag model to that of the no-lag model is 2/e = 0.7358. Thus, the effect of the lag is to suppress the maximum rate of substrate consumption (i.e., biogas production) by about 26.4%. 3.2. Plug Flow Reactor. We consider a PFR of length L and a working volume of V into and out of which flows a liquid (containing a substrate) at a constant volumetric flow rate of Q. As in the case of the CSTR, the PFR is assumed to contain a sufficient number of microorganisms. The feed liquid has a substrate concentration of Sin while the outflow liquid has a substrate concentration of Sout at modified process time θ. Assuming that the substrate is consumed by a first-order reaction, the mass balance for substrate in the PFR is expressed as ⎛ ∂S ⎞ ∂S ⎜ ⎟ +u + kS = 0 ⎝ ∂θ ⎠θ + τ ∂x

1 4k

The solution for the substrate concentration S(x,θ) is given by

where r*max is the (dimensionless) maximum rate of substrate consumption. Thus, at the critical point of the lag model, r*max = 0.7358 for S0* = 0 and α = 2. For the no-lag model, r* = rmax * at θ* = 0 and it follows from eq 22 that * = α(α − 1)⎛⎜ 1 − S0*⎞⎟ rmax ⎝α ⎠

(29)

R bg V

= βkSavg

(35)

while the expression for δ (volume of biogas produced per unit mass of substrate input into the PFR) is given by eq 17. Using that equation along with eqs 31A, 31B, 33, and 35, it can be shown that the expression for δ*(= δ/β) is given by * − 2θ*) δ* = 1 − e−2θ * + 2θ*e−2θ *(S0* − 1) + S0*(1 + 2θ*)e−2θ *(θHR * for θ* ≤ 0.5θHR *

*

* e−θ HR (S0* − 1) for θ* > 0.5θHR * =1 − e−θ HR + θHR

(36)

(27)

where θ* = kθ

(37)

and

(28)

* = kθHR θHR

and 10408

(38) DOI: 10.1021/acs.energyfuels.6b01320 Energy Fuels 2016, 30, 10404−10416

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A value for k is guessed and the experimental γV is plotted against the term within square brackets on the right-hand side of either eq 16A or eq 16B. The value of k which yields a straight line on this plot is the optimum value of k. The value of β can then be calculated from the slope of this line. However, if steady-state measurements of δ and δ* are available, then β = δss/δss* where δss and δss* are steady-state values of the volume of biogas (or methane) produced per unit mass of substrate input into the digester and fractional substrate consumption, respectively. In this case, k is the only unknown parameter which can be obtained from eq 16A or 16B as discussed earlier. We note that δss* = (Sin − S)/Sin for the CSTR and (Sin − Sout)/ Sin for the PFR.

with S*0 being defined by eq 11. Note that the dimensionless modified process time θ* defined by eq 37 is different from that used in the case of the CSTR (see eq 13). It is also observed from eq 36 that the PFR reaches steady state when θ = 0.5θHR (i.e., in a finite time period). Under this condition, the value of the initial substrate concentration S0 affects the value of δ, unlike in the case of the CSTR where this effect becomes vanishingly small as steady state is approached (i.e., as θ → ∞). Further, as the quantity kθHR → ∞, the steady-state operational yield obtained from eq 36 and the intrinsic yield approach one another. For 0 ≤ S0* ≪ 1, eq 36 reduces to * δ* = 1 − (1 + 2θ*)e−2θ * for θ* ≤ 0.5θHR * * )e−θHR * =1 − (1 + θHR for θ* > 0.5θHR

4. RESULTS AND DISCUSSION 4.1. Continuous Stirred Tank Reactor (CSTR). Eq 18A of the CSTR lag model is plotted in Figure 3 for α = 2. The Sshaped nature of the dimensionless substrate consumption curve and an inflection point on it are clearly visible. For comparison, Figure 3 also shows the standard first-order (i.e., no lag) model, i.e., eq 20A for the same value of α. The lag model predicts a lower value of δ* compared to the no-lag model until approximately θ* = 0.6 after which it predicts a higher value of δ*. For both models, δ* → 0.5[=(α − 1)/α = (2−1)/2 = 1/2] as θ* → ∞., i.e., 50% of the substrate in the feed to the reactor is converted into biogas at steady state. Eqs 21A (lag model) and 22A (no-lag model) are plotted in Figure 4 for α = 2 which reveals the radical difference in the

(36A)

It can be shown that for the no-lag (i.e., τ = 0) model, equations analogous to eqs 31A, 31B, and 36 are given by S(x , θ ) = Sine−(k / u)x for 0 ≤ x < uθ and θ ≤ θHR = S0e−kθ for uθ < x ≤ L and θ ≤ θHR (39A)

S(x , θ ) = Sine−(k / u)x for 0 ≤ x ≤ L and θ > θHR

(39B)

and * − θ*) for θ* ≤ θHR * δ* = 1 − e−θ * + S0*e−θ *(θHR * * =1 − e−θHR for θ* > θHR

(40)

For 0 ≤ S0* ≪ 1, eq 40 reduces to * δ* = 1 − e−θ * for θ* ≤ θHR * * = 1 − e−θHR for θ* > θHR

(40A)

Once again it is interesting to compare eqs 36 and 36A of the lag model with their no-lag counterparts (eqs 40 and 40A). Comparing eq 18 A for the CSTR with eq 36A for the PFR it is seen that there is a remarkable similarity in these mathematical expressions for the dimensionless substrate consumption rate δ* for the lag model. The same feature can be observed when one compares eqs 20A and 40A of the no-lag model. The three parameters of the time-lag models tincb, k, and β can be estimated as follows: The incubation time tincb during which there is negligible production of biogas depends on the nature of the substrate, microbial culture, and experimental conditions, and as previously mentioned, is usually less than 3 days. For all calculations in this work, a value of tincb = 1.5 d was used. For illustrative purposes, we consider the case when S0 = 0. Combining eqs 12, 13, 17, and 18A yields the following expression for the CSTR: γV =

Figure 4. Rate of substrate consumption in a CSTR as a function of modified process time as predicted by the no-lag and lag models in dimensionless coordinates (α = 2).

rate of substrate consumption r* (and hence in the rate of biogas production) between the lag and no-lag models. As indicated earlier, the lag-model r* versus θ* curve is bellshaped; it starts from a value of zero at θ* = 0, reaches a maximum at a value of θ* = θmax * = 0.25, and then declines to zero as θ* → ∞. In sharp contrast, the no-lag substrate consumption rate shows a continuous decrease with θ*, which ranges from a maximum value of 1 at θ* = 0 to a value of zero as θ* → ∞. 4.2. Plug Flow Reactor (PFR). Eq 31A of the lag model is qualitatively sketched in Figure 5 assuming Sin > S0. We see that there is a substrate concentration shock front which travels through the PFR at a speed equal to twice the liquid velocity. (This situation is analogous to the case of a sound wave that is riding on a moving mass of fluid, which is traveling at a speed smaller than that of the wave.) When this front reaches the end of the tube at x = L, a steady-state concentration profile is

βkSin [1 − {1 + 2(1/θHR + k)θ}e−2(1/ θHR + k)θ ] 1 + kθHR (16A)

Similarly, by using eqs 17, 36A, 37, and 38 and gives the following expression for the PFR: γV =

βSin [1 − (1 + 2kθ)e−2kθ ] for θ ≤ 0.5θHR θHR

(16B) 10409


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Figure 7. Substrate consumption in a PFR as a function of modified process time as predicted by the no-lag and lag models in * = 3). dimensionless coordinates (θHR

Figure 5. Substrate concentration profile in a PFR as predicted by the time-lag model [eq 31A, Sin > S0].

established in the PFR (see eq 31B). Thus, unlike the CSTR which theoretically takes an infinite amount of time to reach steady-state operation, it takes only a finite time period for the PFR to attain this state. A qualitative sketch of eq 39A (no-lag model) is shown in Figure 6 assuming Sin > S0. However, in this case, the

wastewater, which was operated at a range of HRTs and a temperature of 35 ± 1 °C (308 ± 1 K) . They used the Monod and Contois models, each of which has 4 parameters, to fit their experimental data on substrate consumption and found that the Contois model provided better predictions for the effluent substrate concentration. However, they did not analyze their data on biogas production. Their data, which were obtained under steady-state conditions and are shown in Table 1, were Table 1. Experimental Data of Hu et al.21 on Substrate Consumption and Biogas Production by AD of Molasses Wastewater in a CSTR at 35 ± 1 °C (308 ± 1 K) under Steady-State Conditionsa

Figure 6. Substrate concentration profile in a PFR as predicted by the no-lag model [eq 39A, Sin > S0].

concentration shock front travels through the digester at a speed equal to that of the liquid. Also, comparing Figures 5 and 6 reveals that the concentration levels ahead of the shock front are different for the lag and no-lag models. The increased speed of the shock front in the case of the lag model (compared to the no-lag case) is due to the retardation of substrate consumption in the lag model. Figure 7 shows the dimensionless substrate consumption δ* as a function of modified process time θ* for the PFR no-lag and lag models using a value of θHR * = 3. The lag-model curve shows an inflection point, which is not present on the no-lag curve. According to the lag model (eq 36A), the PFR will reach steady-state at a value of θHR * = 1.5 when δ* = 0.8. However, the no-lag model (eq 40A) predicts a greater steady-state value of δ* = 0.95 achieved at θ*HR = 3, which is twice the time required compared to that for the lag model. As indicated earlier, the quicker attainment of steady state in the lag model is due to delayed substrate consumption, which is manifested in an increased speed of the shock front. 4.3. Study of AD of Sulfate-Rich Wastewaters by Hu et al.21 in a CSTR under Steady-State Conditions. Hu et al.21 investigated the kinetics of AD in a CSTR fed with molasses

run no.

1

2

3

4

5

HRT (d) effluent substrate conc. S (g SCOD L−1) COD removal (%) total gas yield γV (L biogas L−1 liquid d−1) methane yield (L CH4 g−1 COD removed) volumetric liquid flow rate Q (L d−1)b δss (L biogas g−1 SCOD input)b total gas yield (L biogas g−1 SCOD removed)b methane (%)b δss (L CH4 g−1 SCOD input)b δss* (g SCOD removed g−1 SCOD input)b β = δss/δss* (L biogas g−1 SCOD removed)b

7.96 0.297

6.12 0.449

5.08 0.560

4.04 0.703

2.40 1.422

96 0.446

94 0.598

92 0.724

90 0.908

81 1.217

0.370

0.358

0.355

0.361

0.303

0.628

0.817

0.984

1.238

2.083

0.486

0.501

0.504

0.503

0.400

0.507

0.534

0.546

0.556

0.497

73 0.355

67 0.336

65.1 0.328

64.9 0.326

61 0.244

0.959

0.938

0.923

0.904

0.805

0.507

0.534

0.546

0.556

0.497

a V = 5 L and Sin = 7.3 g SCOD L−1. experimental data.

b

Calculated from the

used by us in order to see if the simple first-order reaction mechanism assumed in this work could fit the data on substrate consumption and biogas productivity, and if so, to obtain some estimates of the two parameters k (first-order rate constant) and β (volume of biogas produced per unit mass of substrate consumed). The values of these parameters were used by us to simulate the unsteady-state operation of a CSTR and a PFR using the time-lag models, which is presented in Section 4.4. The data in Table 1 show that although γV increases by a factor 10410


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lower the HRT, the higher is the value of γV, i.e., they have an inverse relation. From Figures 8 and 9 it is evident that the firstorder reaction mechanism represents the experimental substrate consumption and biogas production data of Hu et al.21 satisfactorily. Since the root-mean-square (RMS) error in the predicted value of γV calculated from eq 42 was much smaller than that in the predicted value of S obtained from eq 41, 6% as opposed to 16%, the values of β = 0.531 Lg1− and k = 2.168 d−1 were used in unsteady-state calculations to be presented next. 4.4. Simulation of the Unsteady-State Operation of a CSTR and a PFR using the Time-Lag Models. Figures 10

of 2.7 as the HRT is reduced by a factor of 3.3, the variation of δss (L biogas g−1 SCOD input) with HRT is much smaller. After an initial increase of δss by about 4% as HRT is reduced to 5.08 d from 7.96 d, it falls by 18% as HRT is further decreased to 2.4 d. The percentage of methane in the biogas produced continuously decreases as HRT is reduced due to insufficient time available for the methanogenesis reactions. The yield of methane (L CH4 g−1 SCOD input) decreases by 31% as the HRT is reduced from 7.96 to 2.4 d. For a CSTR operating at steady state, both the lag and no-lag models predict the same values for S (effluent substrate concentration) and γV (volumetric rate of biogas production per unit volume of liquid in the digester)see eqs 9, 12, 16, and 19. From these equations it follows that Sin − S = kθHR S

(41)

and

θ 1 1 = + HR γV βkSin βSin

(42)

Eq 41 and the substrate consumption data shown in Table 1 were used to estimate k with the corresponding plot shown in Figure 8, which yields a value of k = 2.623 d−1. The average Figure 10. Theoretical prediction of the volume of biogas produced per unit mass of substrate input by the CSTR and PFR time-lag models using the data of Hu et al.21 shown in Table 1 (run no. 1). V = 5 L, Sin = 7.3 g SCOD L−1, S0 = 0 g SCOD L−1 (assumed), θHR = 7.96 d, β = 0.531 L g−1, and k = 2.168 d−1.

Figure 8. Experimental data (Hu et al.21) of steady-state substrate consumption in a CSTR fed with molasses wastewater as a function of HRT used to estimate k from eq 41. The RMS error in the prediction of S is 16%.

value of β, calculated using the experimental data in Table 1, is 0.528 L g−1. Eq 42 and the substrate consumption data (Table 1) were also used to estimate both β and k simultaneously. Figure 9 shows the resulting plot from which β = 0.531 L g−1 and k = 2.168 d−1. Thus, the values of k estimated by these two distinct methods differ by about 17% while the values of β are very close to one another. Figure 9 also demonstrates that the

Figure 11. Theoretical prediction of the volume of biogas produced per unit mass of substrate input by the CSTR and PFR time-lag models using the data of Hu et al.21 shown in Table 1 (run no. 5). V = 5 L, Sin = 7.3 g SCOD L−1, S0 = 0 g SCOD L−1 (assumed), θHR = 2.40 d, β = 0.531 L g−1, and k = 2.168 d−1.

and 11 show theoretical predictions of the volume of biogas produced per unit mass of substrate input in the unsteady-state operation of a CSTR and a PFR (both having the same working volume) by the time-lag models (eqs 18A and 36A) using the data of Hu et al.21 (which, as indicated earlier, were taken under steady-state conditions) shown in Table 1 for runs 1 and 5, respectively. For both digesters, after an initial lag period, biogas production increases continuously until it reaches a steady-state value. For HRT = 7.96 d (Figure 10), the CSTR and PFR curves virtually coincide until a value of θ = 0.5 d after which the PFR outperforms the CSTR by approximately 6% as

Figure 9. Experimental data (Hu et al.21) of steady-state volumetric biogas production rate in a CSTR fed with molasses wastewater as a function of HRT used to estimate β and k from eq 42. The RMS error in the prediction of γV is 6%. 10411


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Energy & Fuels steady-state is approached. The steady-state values of δ are 0.502 and 0.531 L biogas g−1 SCOD input, which represent 95 and 100% substrate conversion, for the CSTR and PFR, respectively. For a lower HRT of 2.40 d (Figure 11), initially the performance of CSTR is marginally better than the PFR until θ = 0.35 d; thereafter, biogas production by the PFR outstrips that by the CSTR. The steady-state values of δ are 0.445 and 0.513 L biogas g−1 SCOD input (which represent 84 and 97% substrate conversion) for the CSTR and PFR, respectively, which is a difference of about 15%. The PFR reaches steady-state in 1.2 d while it takes approximately 2 d for the CSTR to do the same. It is also observed that as the HRT is reduced by a factor of 3.3, the steady-state values of δ decrease by 11.4 and 3.4% for the CSTR and PFR, respectively. It can therefore be concluded that, especially for short HRTs, the PFR is to be preferred over the CSTR for enhanced biogas productivity. The reduced biogas productivity of the CSTR can be attributed to the effect of liquid back mixing. 4.5. Comparison of the Time-Lag Models with the Experimental Data of Mondal11 on AD of Mixed Vegetable Wastes in a Compartmental Digester. We now turn to the unsteady-state experimental data of Mondal11 on the yield of methane produced by AD of vegetable wastes in a compartmental digester, which was described earlier. As will be discussed later, quasi steady-state operation of this digester was achieved in a finite time period. Although this type of digester is a hybrid of a CSTR and a PFR, eq 18A for the CSTR and eq 36A for the PFR were both applied to these data in order to test the ability of these two limiting models to correlate the data. (In this context, we note that Skiadas et al.22 modeled a four-compartment periodic anaerobic compartmental reactor as a combination of four CSTRs in series.) The daily methane yield was calculated from the following equation: CH4 yield = QSinδ

Figure 13. Comparison of the PFR time-lag model (eqs 36A and 43) with the experimental data of Mondal11 on the yield of methane produced by AD of vegetable wastes in a compartmental digester. Experimental conditions and model parameter values are reported in Table 3.

Table 2. CSTR Time-Lag Model Parameters for the Experimental Data of Mondal11 on Biogas Production by AD of Vegetable Wastes in a Compartmental Digestera temperature (K)

301

323

328

tincb (d) k (d−1) β (L g−1) RMS error of fit (%)

1.5 0.07 0.441 9.2

1.5 0.19 0.571 11.6

1.5 0.19 0.269 7.5

V = 10 L, Sin = 45 g BOD L−1, Q (assumed equal to the volumetric loading rate) = 1 L d−1, slurry concentration = 6% solids and size of feed particles = 1.09 × 10−3 m. a

Table 3. PFR Time-Lag Model Parameters for the Experimental Data of Mondal11 on Biogas Production by AD of Vegetable Wastes in a Compartmental Digestera

(43)

temperature (K)

301

323

328

tincb (d) k (d−1) β (L g−1) RMS error of fit (%)

1.5 0.17 0.182 9.2

1.5 0.29 0.374 11.6

1.5 0.28 0.180 6.3

where δ was obtained from either eq 18A or eq 36A. The fits of the CSTR and PFR time-lag models to the methane yield data of Mondal11 are shown in Figures 12 and 13, respectively. Experimental conditions and values of model parameters, along with RMS errors of fit, are reported in Tables 2 and 3. It is evident from these figures that both models can capture the S-shaped nature of the experimentally measured

V = 10 L, Sin = 45 g BOD L−1, Q (assumed equal to the volumetric loading rate) = 1 L d−1, slurry concentration = 6% solids and size of feed particles = 1.09 × 10−3 m.

Figure 12. Comparison of the CSTR time-lag model (eqs 18A and 43) with the experimental data of Mondal11 on the yield of methane produced by AD of vegetable wastes in a compartmental digester. Experimental conditions and model parameter values are reported in Table 2.

methane yield as a function of process time in a satisfactory manner. Figure 12 shows the fit of the CSTR time-lag model (eqs 18A and 43) to the experimental data of Mondal11 at three different operating temperatures. As the temperature rises from 301 to 323 K, there is a drastic increase in methane yield. This is reflected in greater values of k and β (Table 2), with the increase in k (171%) being much more significant than the increase in β (29.5%). With a further increase of temperature to 328 K, there is a large reduction in the methane yield with the k value being unaffected but with a decrease in β by 53% compared to the case when the temperature was 323 K. The value of β at 328 K is also lower by 39% compared to that at 301 K. This behavior may be attributed to the death and a severe degradation in the ability of microorganisms to produce biogas at high thermophilic temperatures. It was mentioned earlier that in the experiments of Mondal11 steady state was practically attained in a finite time period. This

a

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steady state, which can be seen by comparing Figures 10 and 13. We note that the horizontally drawn lines in Figure 13 represent the steady-state values of the methane yield. As can be observed from eq 36A, the PFR theoretically achieves steady-state at θ = 0.5θHR. The data for 301 K indicate that steady-state was actually reached in 12 d whereas the PFR timelag model predicts a value of tincb + 0.5θHR = 1.5 d + 0.5 × 10 d = 6.5 d, which is about 46% lower. (The PFR no-lag model predicts a value of tincb + θHR = 1.5 d + 10 d = 11.5d, which is much closer to the experimental value of 12 d.) However, for the data at 323 and 328 K, the experimentally observed value for the time taken to reach steady state was 8 d, which is much closer to theoretical prediction of 6.5 d by the PFR time-lag model. Thus, the PFR lag model, which is an idealized concept, is not able to completely capture the performance of the compartmental digester used by Mondal.11 This is a common feature of a scientific model which is an interpretative and partial description of a phenomenon.23 It can be speculated that the axial-dispersion model would give more consistent results. However, this would then involve the axial dispersion coefficient as an extra parameter, which would depend in a complicated fashion on the internal geometry of the baffled digester used by Mondal11 since that would dictate the nature of liquid flow occurring inside it. We also examined some data available in the literature on biogas production in PFR-type digesters.12,24,25 Unfortunately, we were unable to use these data in our modeling work due to limitations in the way these data were taken (e.g., changing organic loading rate and HRT during an experimental run). Figures 14 and 15 show theoretical predictions of δ as a function of process time by the CSTR and PFR time-lag

state is represented by the last data point on the experimental trajectory (see Figure 12) for each temperature. Thus, at the temperatures of 301, 323, and 328 K, steady state was reached in 12, 8, and 8 d, respectively. Also, the experimental steadystate yields of methane at these three temperatures were 6.88, 14.85, and 6.97 L d−1, respectively. However, the theoretical curves in Figure 12 indicate that the methane yield continues to increase somewhat beyond the experimental steady-state value. Since the attainment of steady state in a definite time period is a feature of plug flow behavior, we also fitted the PFR time-lag model (eqs 36A and 43) to the experimental data of Mondal11 assuming that the model extends all the way up to the last data point at each temperature. This is shown in Figure 13 and values of the model parameters are reported in Table 3. The numerical magnitudes of the constants k and β are different for the PFR lag model when compared to the CSTR lag model as can be seen by comparing Tables 2 and 3. With an increase in temperature from 301 to 323 K, k and β increase by 71% and 105%, respectively, with the increase in β being much more significant compared to the case of the CSTR lag model. With a further rise in temperature to 328 K, the value of k is virtually unaffected but β decreases by 52%, which trends are similar to the CSTR case. However, the value of β at 328 K is very close to that at 301 K, which is in sharp contrast to the CSTR lag model. In order to find out which model is a better representative of the compartmental digester, we estimated β directly from the steady-state substrate consumption and biogas production data of Mondal,11 as was done earlier with the data of Hu et al.21 The results are shown in Table 4 where it is seen that the values Table 4. Values of β Estimated Directly from Steady-State Experimental Data (Mondal11) on Substrate Consumption and Methane Yield Obtained by AD of Vegetable Wastes in a Compartmental Digestera temperature (K) effluent concentration S (at steady state) (g BOD L−1) δss (L CH4 g−1 BOD input) δ*ss (g BOD removed g−1 BOD input) β = δss/δss* (L CH4 g−1 BOD removed)

301

323

328

1.3

1.452

1.429

0.153 0.971 0.158

0.330 0.968 0.341

0.155 0.968 0.160

V = 10 L, Sin = 45 g BOD L−1, Q (assumed equal to the volumetric loading rate) = 1 L d−1, slurry concentration = 6% solids and size of feed particles = 1.09 × 10−3 m. a

of β at the three temperatures are much closer to the corresponding values for the PFR model (Table 3) than to those for the CSTR model (Table 2), which were estimated, as discussed earlier, by fitting the time-lag models to unsteadystate methane yield data. It can thus be concluded that the PFR lag model is a better representative of the compartmental reactor used by Mondal11 than the CSTR lag model. The average value of k for the data shown in Figure 13 is 0.25 d−1 (for the PFR model), which is only 11.4% of the corresponding value for the molasses wastewater used by Hu et al.21 It can be hypothesized that the soluble nature of the substrate and good mixing were responsible for the high value of k in the experiments of Hu et al.21 whereas in the experiments of Mondal11 there were considerable external and internal mass transfer resistances due to the nature of the reactor and feed, which consisted of a slurry of biomass particles. Also, a high value of k will lead to a faster approach to

Figure 14. Theoretical prediction of methane yield per unit mass of BOD input by the CSTR time-lag model (eq 18A). Values of the model parameters are reported in Table 2.

models, respectively, using the values of tincb, k, and β in Tables 2 and 3, which were obtained by analyzing the experimental data of Mondal11 on methane yield. We note that δ is a more universal measure of biogas productivity than γV since its range of variation is expected to be much lower. The steady-state values of δ in the experiments of Mondal11 lie in the range of 0.153 to 0.33 L CH4 g−1 BOD input, which can be compared to the range of 0.244 to 0.355 L CH4 g−1 SCOD input (Table 1) in the work of Hu et al.21 4.6. Limitations of the Time-Lag Models. According to Owamah and Izinyon,26 who presented two simple models for biogas production, the majority of biogas kinetic models available in the literature were developed for low energy 10413


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experimental data of Mondal11 on biogas production by AD of mixed vegetable wastes in a compartmental digester was satisfactory (average RMS error = 9.2%). However, the PFR lag model is a better representative of the compartmental reactor than the CSTR lag model. The time-lag models contain only 3 parameters which can be estimated easily, and they offer a simple way of correlating biogas yield with process time, and hence in formulating control schemes for an existing anaerobic digester. For future work, we suggest the following: (1) validation of the time-lag models with a wider range of feed stocks like market refuse, organic kitchen wastes, farm wastes, yard wastes, sewage, manure, etc.,1 and (2) testing the ability of the models in predicting influences of operating conditions like feed concentration and flow rate on biogas productivity under dynamic conditions.

Figure 15. Theoretical prediction of methane yield per unit mass of BOD input by the PFR time-lag model (eq 36A). It has been assumed that steady state is reached in 12, 8, and 8 d at 301, 323, and 328 K (in accord with the experiments of Mondal11), respectively. Values of the model parameters are reported in Table 3.

■

APPENDIX This appendix presents a brief derivation of the solution of eq 27 subject to eqs 28 and 29. Taking Laplace transforms of these equations with respect to the space variable x gives

substrates like wastewater and are not suitable for complex, high energy substrates like food waste and maize husk. The simplicity of the time-lag models presented in this work is an advantage, but one which is achieved at the cost of not being able to capture the complexities of the AD process. As indicated earlier, the parameter k is a pseudo first-order rate constant which is a measure of the overall rate of the biochemical reactions between the microorganisms and substrate that lead to the production of biogas. It also includes the effects of both external and internal mass-transfer resistances to the transfer of reactants and products between the biomass particles and the surrounding liquid. Thus, the value of k will depend in a complicated fashion on the nature of microorganisms (usually a consortium), type, structure, and size of the biomass particles, nature of the liquid flow in the reactor, etc. Moreover, the biomass particles undergo a reduction in size and disintegrate as they are consumed by the microorganisms. Also, there will be competitive and synergistic effects between the different species of microbes in the consortium. To account for all of these myriad phenomena in a mathematical model will be formidable. Thus, although the lumped-parameter time-lag models may not be as technically sophisticated as the distributed-parameter or mechanistic approach taken in the modeling of heterogeneous chemical reactors, they nevertheless have the qualities of harmony and simplicity and enable one to grasp the totality of the process of AD of an organic waste in a flow reactor. It was found that both lag models (CSTR and PFR), which are constructive in nature, could fit the experimental data on biogas production, albeit with somewhat different values of the parameters k and β. This is to be expected since, as pointed out by Chatterjee,27 a common feature of a constructive model or theory is that there can be more than one such theory based on different speculative constructions or hypotheses that can correlate the (same) experimental data.

τ

d 2S ̅ dS ̅ + + (k + pu)S ̅ = uSin 2 dθ dθ

(A1)

S0 dS ̅ = 0 at θ = 0 and p dθ

(A2)

with S̅ =

Here p is the Laplace transform parameter and S̅ is defined as S ̅ (p , θ ) =

∫0

∞

S(x , θ )e−pxdx

(A3)

The case of repeated or equal roots of the characteristic equation of eq A1 occurs if: τ=

1 4(k + pu)

(A4)

For this case, the solution of eq A1 subject to eq A2 is given by ⎛S uSin ⎞ uSin S ̅(p , θ) = ⎜ 0 − ⎟[1 + 2(k + pu)θ ]e−2(k + pu)θ + k + pu ⎠ k + pu ⎝p

(A5)

The inversion of eq A5 yields eq 31A. As mentioned earlier, an explicit expression for τ can only be provided for the case of S0 = 0 (i.e., zero initial concentration of substrate in the PFR). Integrating eq 27 over the length L of the PFR and using eq 33 gives τ

5. CONCLUSIONS This work presented time-lag models for biogas production from AD of organic wastes in continuous stirred tank and plug flow digesters. The models can capture the sigmoid nature of the volumetric rate of production of biogas with process time that is observed experimentally. The first-order reaction mechanism assumed in the models was adequate in explaining the experimental data on biogas production taken by Hu et al.21 and Mondal.11 The quality of fit of the lag models to the

d 2Savg dθ

2

+

dSavg dθ

+ kSavg =

1 θHR

[Sin − S(L , θ )]

(A6)

We now consider the time interval 0 < θ < 0.5θHR within which the concentration shock front has not yet reached the end of the PFR (x = L), i.e., the PFR has not yet attained steady state. For S0 = 0, the outlet concentration S(L,θ) = 0 during this time interval as can be seen from eq 31A and Figure 5. By setting S0 = 0 in eq 31A and using eqs 33 and A6 it can be shown that eq 30 results. 10414


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■

δ(x) = Dirac delta function δss = steady-state value of δ, L g−1 δ* = mass of substrate consumed in producing biogas or methane per unit mass of substrate input into the digester (= δ/β) δ*ss = fractional substrate consumption at steady state [= (Sin − S)/Sin for the CSTR and (Sin − Sout)/Sin for the PFR] θ = modified retention time (= t − tincb), d θHR = hydraulic retention time of the liquid in the digester (= V/Q), d θmax = value of the modified retention time at the inflection point of the CSTR δ versus θ relation θ* = dimensionless modified retention time (= θ/θHR for the CSTR and kθ for the PFR) θHR * = dimensionless hydraulic retention time of the liquid in the PFR (= kθHR) θ*max = given by eq 23 τ = delay or lag parameter, d

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.energyfuels.6b01320. Materials (Table S1), analytical instruments (Table S2), and characteristics of the feedstock (Table S3) (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*Telephone: +1-315-470-6517; Fax: +1-315-470-6945; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE H(x) = Heavyside unit step function k = pseudo first-order reaction rate constant, d−1 kmod = modified rate constant defined by eq 3, d−1 L = length of the PFR, cm or m m = roots of eq 4, d−1 m1,2 = given by eq 5, d−1 p = Laplace transform parameter Q = volumetric flow rate of liquid through the digester, L d−1 r* = dimensionless rate of substrate consumption per unit mass of substrate input into the CSTR rmax * = dimensionless maximum rate of substrate consumption per unit mass of substrate input into the CSTR Rbg = rate of biogas or methane production, L d−1 Rsu = rate of substrate consumption, g d−1 S(θ) = substrate concentration of the liquid in the CSTR at modified process time θ, g L−1 S(x,θ) = substrate concentration of the liquid at location x in the PFR at modified process time θ, g L−1 S(̅ p, θ) = Laplace transform of S(x,θ) defined by eq A3 Savg(θ) = average substrate concentration of the liquid in the PFR at modified process time θ (defined by eq 33), g L−1 Sin = substrate concentration of the liquid feed to the digester, g L−1 Sout = substrate concentration of the liquid effluent from the PFR, g L−1 S0 = initial substrate concentration of the liquid in the digester, g L−1 S* = dimensionless substrate concentration of the liquid in the CSTR (S/Sin) S0* = dimensionless initial substrate concentration of the liquid in the digester (S0/Sin) t = process time, d tincb = initial incubation period of microorganisms, d u = liquid velocity in the PFR, cm d−1 or m d−1 V = volume of the digester, L x = axial distance in the PFR measured from its inlet, cm or m

Abbreviations

■

AD:anaerobic digestion ADM1:anaerobic digestion model No. 1 BOD:biological oxygen demand COD:chemical oxygen demand CSTR:continuous stirred tank reactor HRT:hydraulic retention time of the liquid in the digester PFR:plug flow reactor RMS:root-mean-square SCOD:soluble chemical oxygen demand

REFERENCES

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Greek Letters

α = defined by eq 12 β = volume of biogas or methane produced per unit mass of substrate consumed, L g−1 γV = volumetric rate of biogas or methane production per unit volume of liquid in the digester, L L−1 d−1 δ = volume of biogas or methane produced per unit mass of substrate input into the digester, L g−1 10415


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Time-Lag Models for Continuous Stirred Tank and Plug Flow

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