Hydrodynamical modelling of a biogas burner

Energy Convers.Mgmt Vol. 32, No. 4, pp. 395--401,1991 Printed in Great Britain.All rights reserved

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HYDRODYNAMICAL MODELLING OF A BIOGAS BURNER A. C H A N D R A , I G. N. T I W A R I I and Y. P. Y A D A V z t Centre of Energy Studies, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-ll0 016 and 2Department of Physics, C.M. Science College, Darbhanga, Bihar, India (Received 16 April 1990; received for publication 21 January 1991)

Abstraet--Hydrodynamical modelling of a biogas burner is presented. The results of the modelling are useful for optimization of the design parameters of a biogas burner. Biogas burner

Modelling

Design parameters

NOMENCLATURE a I = Cross-sectional area of burner nozzle at entrance of biogas (m2) a2 = Cross-sectional area of nozzle orifice (m2) aD = Cross-sectional area of mixing tube (m2) ap = Cross-sectional area of port (m2) A0 = Burner head area (m2) d2 = Orifice diameter (m) D2 = Diameter of mixing tube (m) hl = Water column height in manometer at nozzle entrance (m) h2 = Water column height in manometer at nozzle orifice (m) m = Port number (dimensionless) n = Stoichiometric ratio (dimensionless) Rh = Ratio of pressure drop to pressure (dimensionless) Rp = Ratio of density of air to gas (dimensionless) Ra = Ratio of cross-sectional area of orifice to port (dimensionless) Rm= Ratio of density of gas to air-gas mixture t = Thickness of nozzle material orifice (m) v = Velocity of entrace of biogas into nozzle (m/s) vs = Velocity of ejection of biogas from orifice (m/s) ma = Mass flow rate of air (kg/s) rhg = Mass flow rate of biogas (kg/s) Greek letters Pa = Density of air (kg/m 3) pg = Density of biogas (kg/m3) Pm= Density of air-gas mixture (kg/m3)

INTRODUCTION C o m m e r c i a l l y a v a i l a b l e b i o g a s b u r n e r s d o n o t give as m u c h h e a t o u t p u t as their r a t e d value [I, 2]. M o r e o v e r , it is also r e p o r t e d t h a t the p e r f o r m a n c e o f these b u r n e r s varies widely u n d e r similar c o n d i t i o n s o f testing [3, 4]. This, however, implies t h a t the design p a r a m e t e r s o f these b u r n e r s are n o t consistent with the scientific a n d technical criteria a p p r o p r i a t e for such low pressure burners. In this view, the design d e v e l o p m e n t o f a biogas b u r n e r a p p e a r s o f w o r t h w h i l e interest. I n this p a p e r , simple m a t h e m a t i c a l m o d e l l i n g o f a b i o g a s b u r n e r has been c a r r i e d o u t following well established laws o f h y d r o d y n a m i c s . T h e results o f the m o d e l l i n g p r o v i d e explicit a n a l y t i c a l expressions for the orifice-aperture o f the b u r n e r nozzle, a p e r t u r e o f the mixing tube, b u r n e r h e a d area, p o r t n u m b e r , m a s s flow rate o f biogas, p r i m a r y air a s p i r a t e d into the mixing tube, a n d flow velocity o f a i r - g a s m i x t u r e as a function o f s o m e dimensionless p a r a m e t e r s , c o m p r i s e d o f pressure i n p u t o f biogas, s t o i c h i o m e t r i c ratio, e t c . . A d d e d a d v a n t a g e s o f these a n a l y t i c a l results are their abilities to evolve a n o p t i m a l b u r n e r design for a given set o f b i o g a s pressure inputs. 395

396

CHANDRA et

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HYDRODYNAMICALMODELLING OF A BIOGASBURNER

Biogo$ su~y

Mixingtube

NozzLe

Fig. ;.

~_

Burner

Schematic of a biogas burner system assembly.

DESIGN

DESCRIPTION

The assembly of the biogs burner system is depicted in Fig. 1. Figure 2 shows the schematic view of a biogas burner. Cross-sections of the various components of the burner, e.g. burner-nozzle, burner body, burner head, burner port, etc. are depicted in Fig. 3. MODELLING Referring to Fig. 2, if hi is the height of water column in the manometer at section A i, the linear velocity of the biogas entering the burner nozzle can be expressed as v =

(l)

Similarly, the linear velocity of the biogas ejected from the nozzle orifice (exit of the burner nozzle at section A:) is vs =

(2)

where h: is the height of water column in the manometer at section A2. As the Math number of the biogas is sufficiently less than one, the compressibility effect in the biogas is rarely of

i? I I P

A~

Nozzle cluct

Nixing tube

Section at A 2

Fig. 2. Schematicof a biogas burner.

CHANDRA et

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HYDRODYNAMICALMODELLING OF A BIOGAS BURNER

397

~ O r i f i c e

t

Biogos A typical biogas burner nozzle

design.

(a)

- Burner

Cross-section of the burner body

Cross-section of the burner head

(b)

(c)

Fig. 3. Schematicof various components of a biogas burner. importance, and therefore, it is fair to approximate the biogas as an incompressible fluid. The mass continuity equation for an incompressible fluid demands

a,v = a2v+.

(3)

Equations (1)-(3) give

h: = (al yh,. \as~

(4)

Now,

2__ 2

Ah= h l ( ~ 2 at)

(5a)

where Ah = h i - h2 = pressure differential along section AIA2 of the burner nozzle. Here, two cases arise: Case I: for a, > a2, Ah corresponds to a pressure drop during the flow of the biogas from section A~ to A2. In this case, equation (5a) can be expressed as (a~ - a 2) Ah = - h i a~ (5b) Here, the negative sign signifies that Ah is a depression in the height of water column in the manometer. Case II: when a~ < a:, Ah would correspond to a pressure gain thereby signifying an elevation of water column in the manometer. Taking a~ = as + Aa and neglecting the square of the differential, equation (5a) reduces to

Ah = - 2h~Aa. a2

(6a)

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CHANDRA et al.: HYDRODYNAMICAL MODELLING OF A BIOGAS BURNER

It follows from equation (6a) that, for a given pressure, the pressure drop is directly proportional to the opening cross-section area of the mixing tube (Fig. 2), which aspirates primary air into the mixing tube, and is inversely proportional to the orifice area of the burner nozzle. Moreover, equation (6a) is useful for selection of the opening aperture of the mixing tube for a given range of biogas pressure input and nozzle orifice. In terms of the diameter of the nozzle orifice, equation (6a) can also be expressed as Ad Ah = - 2hi -~-2"

(6b)

Mass flow rate of primary air aspirated into the mixing tube The mass flow rate of primary air aspirated into the mixing tube can be written as ~"a= 4 [D2 -- (d2 + 2t)2]vaPa

(7)

where D2 = diameter of the mixing tube t = thickness of the material of the nozzle orifice. However, it is the pressure drop (Ah) around the nozzle orifice that is responsible for the aspiration of primary air into the mixing tube. Hence, the balance of energy gives the flow velocity of the primary air as Va =

Putting the value of v~ from equation (8) into equation (7), one can obtain the mass flow rate of air as a function of pressure drop, i.e. m~ = 4 [D2 - (d2 + 2t):]p,x/~Ah.

(9)

Considering only the effective diameter of the nozzle orifice, i.e. neglecting the thickness (t) with respect to the diameter (d~), equation (9) reduces to F~/a = 4 ( 0 2 -- M~)PaN/2gAh"

(10)

Mass flow ratio of air to biogas (stoichiometric ratio) A proper mixture of air and biogas is essentially required for complete combustion of the biogas. The mass flow rate of biogas ejected from the nozzle orifice will be given by tits = --4- X~zgnz Pg"

(11)

Now, the stoichiometric ratio

m8 i.e.

n = RhRp

D: - dz

where Rh -- A ~

P, Pg

(12)

CHANDRA et al.: HYDRODYNAMICALMODELLINGOF A BIOGASBURNER

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Further, equation (12) gives D 2 J 1 + ~n -~2=

(13)

where R = RhRp.

Equation (13) stands for the ratio of the diameters of mixing tube to nozzle orifice as a function of dimensionless parameters. This expression is useful for fixation of the diameters of mixing tube as well as nozzle orifice for a range of values of stoichiometric ratio, pressure input of biogas and pressure drop around the nozzle orifice. In terms of cross-sectional area of the mixing tube and the orifice, equation (13) can be written as a2 Flow velocity of air-gas mixture

The momentum and the mass continuity equations for the burner can be expressed, respectively, as ( aDvaO.)V. + ( a2vgp~)vg = (Aov~om)vm

(15)

aDVap. + a2v~o~ = AovmPm.

(16)

On algebraic manipulation, equations (15) and (16) give the expression for the flow velocity of the air-gas mixture as Vm=Vg

[l + (R + n)Rh) l+R+n

(17)

Burner head area

The burner head area (A0) is determined from the mass continuity equation (16) as A0

a2Rml(1 + R + n)

+ (R + n)R h

(18)

where Rm = P_~s P~ Port number

If m is the port number and ap is the cross-sectional area of each port, m

RaRm.l(1 + n + R) 2 + (n + R)Rh

(19)

where

Rtt ad ~

--

a t,

RESULTS AND DISCUSSION

Equations (1) and (2) are the expressions for the flow velocity of the biogas entering the nozzle and leaving the orifice, respectively. Depending upon the nozzle geometry, a discharge coefficient can be introduced in the expression for the flow velocity of biogas ejected from the orifice [5]. For a known pressure of biogas, the flow velocity of biogas ejected from the orifice can be determined with the help of equation (2). The primary air which is required for the burning of biogas is aspirated into the mixing tube through the opening around the orifice, i.e. at the inlet of the mixing ECM

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CHANDRA et al.: HYDRODYNAMICALMODELLING OF A BIOGAS BURNER

tube. The pressure drop at the nozzle orifice is solely responsible for the aspiration of primary air into the mixing tube. Therefore, it is interesting to express the cross-sectional area of the opening aperture, aspirating primary air into the mixing tube, in terms of pressure drop (Ah), biogas pressure input (hi) and the nozzle orifice area. It has been expressed explicitly by equations (6a) and (6b). For a given set of values of these parameters, the opening-aperture around the nozzle-orifice can easily be determined using equations (7), (9) and (10) which stand for the mass flow rate of primary air aspirated into the mixing tube. The flow velocity of primary air can be determined by equation (8). The mass flow rate of biogas ejected from the nozzle orifice has been expressed by equation (11). The stoichiometric ratio for air to biogas, expressed by relation (12), is obtained using expressions (10) and (11). This relation is useful in choosing the amount of primary air required for complete combustion of the biogas. Further, the ratio of the diameters of the mixing tube and nozzle orifice, and their corresponding cross-sectional areas are expressed by equations (13) and (14), respectively, in terms of some dimensional parameters, such as stoichiometric ratio (n), ratio of pressure drop to pressure at orifice (Rh) and the ratio of the density of air to biogas (Rp). Equation (17) depicts the flow velocity of the air-gas mixture as a function of the flow velocity of biogas and the above defined dimensionless parameters, R(=RhRp) and n. It follows from this expression, i.e. equation (17), that, for a given value of these dimensionless parameters, the flow velocity of the air-gas mixture is directly proportional to the flow velocity of biogas, which implies that the greater the flow velocity of biogas, the greater would be the flow velocity of the air-gas mixture. Equation (17) is helpful for estimating the flame size and rate of flame propagation as well, which is essentially required in selection of the pan-position over the burner head. Thus, equation (17) is also useful for the design of the supporting structure, i.e. pan-stand height over the burner head, so that optimum utilization of the burner heat output could be made at the use end. Selection of the burner head area is also an important design aspect which depends upon the nozzle orifice area in addition to the dimensionless parameters R and n. The burner head area is expressed here by equation (18). It follows from equation 08) that, for given values of the dimensionless parameters R and n, the burner head area is directly proportional to the nozzle orifice area, which is an expected result. For uniform distribution of the flame over the bottom of the cooking pan, a number of ports (i.e. small holes) instead of one big hole, is desirable. In this view, a number of ports are made over the burner head. The port numbers can be determined by equation (19). It is obvious from this equation that, apart from the dimensionless numbers R, n and Rm (ratio of the density of gas to the air-gas mixture), the port number also depends upon one more dimensionless number R a which is the ratio of the cross-sectional area of the orifice to the port.

CONCLUSIONS The model is simple, as well as general, for a gas burner and provides explicit expressions for the design parameters of a burner. For a given set of input parameters, e.g. biogas pressure and pressure drop across the nozzle orifice, a range of design parameters of a biogas burner, namely

orifice, interspace between orifice and the mixing tube for aspiration of primary air, burner head area, part number, etc., can easily be selected with the help of the analytical expressions presented herewith. As the model comprises a term, n, standing for air to biogas ratio, it could also be used for simulation of the burner with respect to the stoichiometric ratio. The stoichiometric ratio will depend upon the composition of the fuel gas. In the case of biogas, its methane content will vary with various parameters, e.g. feed stock, temperature, etc. of the biogas plant; hence, a universal burner design could not be the most efficient under all conditions. The dimensions of the supporting structure of the burner, e.g. pan size and its position over the burner head, will depend upon the flame length which could easily be estimated using this model [equation (17)]. Thus, the model is useful for the design of a biogas burner and its supporting system for optimum utilization. Acknowledgement--This work was supported by The Department of Non-conventional Energy Sources, Ministry of Energy, Government of India.

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REFERENCES 1. M. C. Jain, Ind. Farming (April 1980). 2. A. Chandra, A. Kumar and Y. P. Yadav, Comparative performance of some biogas burners. 5th Annual Convention of BioEnergy Society of India, Badodra (1988). 3. A. Chandra, Y. P. Yadav, A. Kumar and V. K. Srivastava, Performance testing of some biogas burners. International Workshop on Renewable Energyfor Rural Development, National Physical Laboratory, New Delhi (1-3 February 1989). 4. A. Chandra, G. N. Tiwari, V. K. Srivastava and Y. P. Yadav, Energy Convers. Mgmt 32, 353 (1991). 5. J. Griswold, Fuels, Combustion and Furnaces, Chemical Engineering Series. McGraw-Hill, New York (1946).