Scaling relations for flare interactions, flame lengths, and crosslighting requirements in large flare fields

Flare fields typically consist of many closely-spaced flares arranged in parallel rows. Each flare in the field produces an axisymmetric buoyant jet flame that for a limited vertical distance develops essentially independent of the other flames in the field. However, at a distance above the flare tips determined by the flare-to-flare spacing s, adjacent flames in the same row merge to form a planar buoyant jet flame. At a further distance determined by the row-to-row spacing S, the planar flames formed from adjacent rows then merge into a single flame. The merging in each row and between rows allows for crosslighting of flares in the field. These interactions, however, significantly complicate prediction of key properties of the resulting flames, including entrainment and mixing rates of the merged flames and the resulting overall flame height L. These aspects of flare fields differ fundamentally and substantially from those of a single flare. Of central importance to flare field design are the combined effects of the flare firing rate and the spacing parameters s and S on the crosslighting ability and the overall flame height. This paper develops fundamental scaling relations for the individual and merged flames in a flare field to allow assessment of flare field performance over a wide range of design and operating parameters, and allow optimal designs that minimize the flare field area and overall flame height.

Presented at the 2007 AFRC-JFRC International Symposium, 22-24 October 2007, Marriott Waikoloa, Hawaii S c a lin g R e la tio n s fo r F la r e I n te r a c tio n s , F la m e L e n g th s , a n d C r o s s lig h tin g R e q u i r e m e n t s in L a r g e F l a r e F ie ld s Werner J.A. Dahm Laboratory for Turbulence & Combustion The University of Michigan Ann Arbor, MI 48109-2140 U SA wdahm@umi ch.edu Coordinated Technologies, LLC Suite 400, 2600 Roseland Ann Arbor, MI 48103-2135 coordinatedtechnologies.com Abstract Flare fields typically consist o f many closely-spaced flares arranged in parallel rows. Each flare in the field produces an axisymmetric buoyant jet flame that for a limited vertical distance develops essentially independent o f the other flames in the field. However, at a distance above the flare tips determined by the flare-to-flare spacing 5 , adjacent flames in the same row merge to form a planar buoyant jet flame. At a further distance determined by the row-to-row spacing S, the planar flames formed from adjacent rows then merge into a single flame. The merging in each row and between rows allows for crosslighting o f flares in the field. These interactions, however, significantly complicate prediction of key properties of the resulting flames, including entrainment and mixing rates o f the merged flames and the resulting overall flame height L. These aspects o f flare fields differ fundamentally and substantially from those o f a single flare. O f central importance to flare field design are the combined effects o f the flare firing rate and the spacing parameters 5 and S on the crosslighting ability and the overall flame height. This paper develops fundamental scaling relations for the individual and merged flames in a flare field to allow assessment o f flare field performance over a wide range o f design and operating parameters, and allow optimal designs that minimize the flare field area and overall flame height. 1. I n t r o d u c t i o n A paper from the 2004 AFRC-JFRC International Symposium (Diez-Garias & Dahm 2004) gave a new, fundamentally-based, and substantially improved integral method for determining the flame length and combustion properties o f the buoyant jet flames produced by isolated "point" flares. The 2004 paper and its extension (Diez-Garias & Dahm 2007) provide a simple yet remarkably accurate means for combustion engineers to predict properties such as flame lengths, entrainment rates, and other key performance parameters o f such individual point flares. However, some gas flaring applications involve large flare fields o f the type shown in Fig. 1 . Such fields are composed o f a large number N o f closely-spaced individual point flare sources arranged in a single row, with a large number M o f such flare rows arranged in parallel to form the entire flare field. Each o f the N x M individual flare sources in the field produces a single buoyant jet flame, and for relatively small distances x above the flare tips these individual flames are unaffected by the other flares in the field. Since the local flow width 5 of these flames 1 increases with x, adjacent flares in each row will interact past a merging distance xm at which the flow width S(x) becomes comparable to the flare-to-flare spacing 5 . At distances x > xm, the adjacent flames in each row will thus merge and thereby form a planar buoyant jet flame. Such planar buoyant jet flames have very different scaling properties than do individual buoyant jet flames produced by isolated "point" flare sources. At distances x > xM for which the local flow width S(x) o f the resulting planar buoyant jet flames becomes comparable to the row-to-row spacing S, the flames from adjacent parallel rows will thus further merge to form a single flame. The flare-to-flare merging within each row and the row-to-row merging within the field are essential to achieve effective crosslighting o f all flares in the field. However, these interactions significantly complicate the prediction o f key properties o f the resulting flames. The entrainment and mixing properties o f the merged flames and the resulting flame lengths are fundamentally different than for a buoyant jet flame produced by a single flare. The combined effects o f the flare firing rate and the spacing parameters 5 and S on the crosslighting ability and the overall flame length o f the flare field are of central importance to flare field design. This paper develops fundamental scaling relations for the individual and merged flames in a flare field to allow assessment o f the flare field performance over a wide range of design and operating parameters. Key practical questions for optimizing the design o f flare fields include: (1) the proper spacing between individual flares to allow for reliable crosslighting along flare rows, (2) the flame length resulting for any given flare spacing, fuel type, and fuel flow rate, and (3) the required minimum spacing between rows that allows sufficient aeration of the flare field to provide the shortest possible flame lengths. This paper provides simple yet accurate fundamentally-based methods for addressing these questions related to the design and performance o f large flare fields. The results reveal how the separation between flare rows affects the aeration o f the overall flare field, and provide the optimal flare spacing 5 within rows and separation S between rows to obtain the minimum possible flame length and the minimum total area o f the flare field. The fundamental scaling relations developed in this paper can give combustion engineers a simple yet accurate means to address the most important issues associated with the design of large flare fields. The presentation is organized as follows. The basic flow regions involved in flare field flames are identified in §2. The scaling laws for axisymmetric turbulent jet flames that form between the flare exit plane (x = 0) and the distance xm at which merging occurs between adjacent flames in a flare row are developed in §3. Scaling laws associated with the merging process are developed in §4. The scaling laws for planar buoyant jet flames are developed in §5, including the planar jet limit (§5.1), the planar plume limit (§5.2), the planar buoyant jet flow (§5.3), and the resulting planar buoyant jet flame scaling (§5.4). Scaling laws associated with merging between adjacent flare rows are developed in §6. Finally, the implications o f the present results for optimum design o f large flare fields is discussed in §7. Conclusions concerning the use o f these scaling laws for optimizing the design and operating performance of practical flare fields are given in §8. Readers who are less interested in the technical rigor associated with the derivation o f these scaling laws should focus on §2, §7 and §8. 2 . F lo w R e g i o n s I n v o lv e d in F l a r e F ie ld F l a m e s The approach used herein is based on decomposing the flare field, as well as the flames produced by individual flares within the field, into a set o f distinct regions for which fundamental scaling 2 laws either exist or can be readily obtained. The resulting flow regions are defined by planes shown schematically in Fig. 2, at which fundamental changes in the underlying flow field occurs. The flare tips define the "exit plane", at which the vertical coordinate x = 0 . The ground plane is located at a distance H below the exit plane. The flare field is analyzed in terms o f the flow produced by a set o f "far-field equivalent point sources" located at the "virtual origin" a distance xe below the exit plane, as described in §3. Each o f these sources initially produces an axisymmetric jet flame, with the resulting flow, mixing and combustion properties given in §3. Although in principle the effect of buoyancy acting on these axisymmetric jet flames over these relatively could be accounted for as described in Diez-Garias & Dahm (2004, 2007), in most practical circumstances these effects will remain negligible within the region between the exit plane (x = 0) and the first merging plane (x = xm) in Fig. 2. At x = xm, the adjacent flames in each row interact and merge as described in §4 to form a single planar jet flame. Above this merging plane, the effects o f buoyancy may no longer be negligible, and thus the flow, mixing and combustion properties in this region must explicitly account for buoyancy. The underlying flow for x > xm is thus analyzed in §5 as a planar buoyant jet flame, including the corresponding virtual origin. This section extends the axisymmetric buoyant jet flame analysis o f Diez-Garias & Dahm (2004, 2007) to address the planar buoyant jet flames produced by each row in a flare field. At a second merging plane, located at x = xM, the planar buoyant jet flames produced by adjacent rows in the flare field interact and merge as described in § 6 to form a single flame. 3 . S c a lin g in t h e A x i s y m m e t r i c J e t F l a m e R e g i o n (x < x m) The analysis follows a similar approach as Diez-Garias & Dahm (2004, 2007), and uses similar notation as defined therein. These papers should be referred to as background information for the present analysis. As shown therein, the flow produced by an actual source located at x = 0 that introduces an exit mass flux mE and exit momentum flux JE into a surrounding fluid with density is equivalent to the flow produced by a theoretical "point" source located at a distance xE upstream o f the exit plane that introduces the same momentum flux J 0 = JE into the same surrounding fluid with density , but has zero mass flux, namely m0 = 0. The resulting axisymmetric turbulent jet flow at any distance x from the exit plane (x = 0 ) is self-similar and characterized entirely by the local flow width 8 (x) and centerline velocity uc(x). The scaling laws for 8 and uc can then be deduced directly on dimensional grounds as 5 = ( C5 ) j • (x + XE) uc = ( cu)j • ( v p# r ( * + O ) r , (i* ) where the virtual origin (x = -x E) is determined by yfn / 2 I, ! 0.262 2 ( ) j " 0.36 3 mE (2 ) (c, ). - 7.2 (3 a,b,c) are fundamental scaling constants obtained from analysis o f experimental data o f Papanicolaou & List (1988). Here d* is the "far field equivalent source diameter"; from its definition in (2), the equivalent diameter for a circular nozzle with diameter d 0 issuing fluid with uniform exit density pE and uniform exit velocity UE gives the classical result d* = (p£ / ) d0 . As also shown in Diez-Garias & Dahm (2004, 2007), the mass flux o f the resulting axisymmetric turbulent jet flow is then m ( x ) = A ( c u ) j ( C ) 2 ( p - J e ) 1/2 ( x + x e ) , (4) where from (2) it can be readily verified that m(0) = mE . The mass flux in (4) allows the mass entrainment rate E = dm / dx into the jet to be obtained as E = I ( c )j (C )2 ( p . J e ) ' /2 , (5) which is independent o f the downstream location x . The scaling laws in (1) - (5) were rigorously derived for nonreacting axisymmetric turbulent jets. However, as shown by Tacina & Dahm (2000), a general equivalence principle allows the inertial effects o f the density reduction due to heat release by combustion to be rigorously accounted for by replacing the actual surrounding density in these scaling laws with the effective value p " determined by the adiabatic flame temperature Ts and the mole fraction X 5 of jet fluid in a stoichiometric mixture as \ p f = p^ Teff J " "i where T0 is the temperature o f the jet fluid. The scaling laws for the exothermically reacting axisymmetric jet flames between the flare exit plane x = 0 and the first merging plane x = xm are thus given by (1)-(5) with p^ replaced by p 5 from ( 6 ). The linear growth in (1a) for such axisymmetric jet flames can be seen in Fig. 3. Note that the entrainment rate E in (5) determines the aeration requirements for a single isolated flare operating in this jet limit. 4 . M e r g i n g o f A d j a c e n t F l a m e s in a F l a r e R o w (x ~ x m) The jet flames produced by each o f the N x M individual flare sources in the field will follow the scaling laws in §3 until the local flow width 8 (x) o f adjacent flames in each flare row becomes comparable to the flare-to-flare spacing 5 , as indicated in Fig. 4. Denoting the merging location as xm, this occurs at ) = Cm • S , (7) ^ 2.5 . (8 ) where From (1a) the flare merging location Xm is thus given by Xm S _ _ Cm ( C S ) j (9 ) XE s ' From the values for the scaling constants noted above, this gives 4 36 ! 6 .9 ------ - . s s/d * x (10) As evident from Fig. 3, adjacent flames at x = xm may not appear to be visibly merged as judged by their luminous "boundary", however the entire jet flow including the luminous flame and the invisible flow o f nonluminous hot gases around this will be effectively merged at this point. Above this point, the flow, mixing and combustion properties produced by each flare source will no longer correspond to those o f an isolated axisymmetric jet flame. The result in (9) also allows determination o f the maximum separation Smax for adjacent flares in a flare row to achieve crosslighting. As shown in Tacina & Dahm (2000), the flame length L for an isolated axisymmetric turbulent jet flame is given by T e(11) where 9 is the stoichiometric oxidizer-to-fuel mass ratio; e.g., for a methane flame burning in air, 9 = 17.2. If the adjacent flames in a flare row have not merged by x = L, then they will not crosslight. Thus the maximum separation smax at which crosslighting within a flare row will occur corresponds to xm = L , and thus is given by f meff\1/2 smax d* + 10 ^ ^ (1 + # ) T \ Ii (C ) , (C ) (12) J From the values for the scaling constants noted above, this gives smax d* / 1 rc \ 1/2 T ef 3.6 + 4.3(1 + " ) (13) It must be recognized, however, that this is an absolute maximum separation value at which crosslighting could occur. For separations approaching this value, adjacent flares will crosslight near their flame tips, where buoyancy effects that were ignored in (1)-(13) will be significant. Such buoyancy effects on 8 (x) and the resulting effect on the maximum separation s can be obtained from the results in Diez-Garias & Dahm (2004, 2007). 5 . S c a lin g in t h e P l a n a r B u o y a n t J e t F l a m e R e g i o n ( x m < x < x M ) For x > xm, the flow field produced by individual adjacent axisymmetric jets in a flare row will have effectively merged into a single planar flow, as indicated in Fig. 5. Using tildes to denote parameters associated with the resulting planar flow, the momentum flux per unit flare row length is then JE = (JE/ s ), and the mass flux per unit flare row length is mE = ( m( xm) / s ), where m(xm) is the jet mass flux at the merging plane. From (4) and (9), mE is given by n E ! h Cn ( Cu ) j ( )j ( # $ J E )12 . (14) Assuming that the air that has been entrained into the jet up to the merging plane xm will have burned a corresponding stoichiometric amount o f the fuel being carried by the jet, then the total heat that has been released by combustion occurring in the jet up to the merging plane is 5 Qe = - [m( X m ) " m E (15) ] where q is the heating value o f the fuel, namely the amount o f heat released per unit mass o f fuel burned, and all other parameters have been defined above. The heat flux QE in turn corresponds to a buoyancy flux g Be = (16) Qe KCpT! The resulting total buoyancy flux across the merging plane per unit flare row length, namely Be ! Be / s , is thus f \ gq BE = 1 [ mE - (mE / 5)] , (16) As indicated in Fig. 5, the merging plane can thus be treated as a new "exit" plane that discharges mass flux mE , momentum flux JE and buoyancy flux BE per unit length. The result is a planar buoyant jet flow, for which scaling laws can be developed in a manner analogous to the axisymmetric buoyant jet flow treated in Diez-Garias & Dahm (2004, 2007). These scaling laws are obtained by first considering the planar jet limit that applies when BE ! 0 , as well as the planar plume limit that applies when JE ! 0 , as indicated in Fig. 6 . The planar buoyant jet then corresponds to the case when both JE and BE are nonzero. 5.1 Scaling in the Planar Jet Limit We first consider the case where BE = 0 , which corresponds to the planar turbulent jet limit as indicated in Fig. 6a. The flow produced by a planar source at x = 0 that introduces an exit mass flux mE and exit momentum flux JEinto a surrounding fluid with density is equivalent to the flow produced by a theoretical "line" source located at a distance xEupstream o f the exit plane that introduces the same momentum flux J0 = JE into the surrounding fluid with density , but has zero mass flux, namely m 0 = 0 . The resulting planar turbulent jet flow at any distance x from the exit plane (x = 0 ) is also self-similar and characterized entirely by the local flow width 8 (x) and centerline velocity uc(x). The scaling laws for 8 and uc in planar turbulent jets can then be deduced directly on dimensional grounds as 5 (17a) . ^E) = ( ^5 ) : • (*x + +" uc = ( Cu )j • ( -^E / P~ ) 12 where the virtual origin is / _ \-l /2 x+ + X. e ) (X (17*) 2 E (18) 1 (^ , (c5) j and where ! ! %f (") d" = (& / a f ) 1/2 ( 0.512 (C6 ). " 0.42 6 (cu). « 2.4 (19a,*,c) are fundamental scaling constants obtained from analysis o f experimental data o f Gutmark & Wygnanski (1976). Here b * is the "far field equivalent slot width". From its definition in (18), the equivalent width for a slot o f width b0 issuing fluid with uniform exit density pE and uniform exit velocity UE gives b* = (p 0 / p „) b0. The scaling laws in (17)-(19) apply to nonreacting planar turbulent jets. The general equivalence principle o f Tacina & Dahm (2000) allows inertial effects o f heat release to be accounted for by replacing the density p„ with the effective value p " in ( 6 ). The mass flux o f the planar turbulent jet flames is thus m ( x ) ! A (cu)j (cs )j (pe JE) 1/2 (x + xE) 1/2 , (20) where from (18) it can be readily verified that m (0) = mE. The mass flux in (20) allows the mass entrainment rate E = dm / dx into the planar jet flame to be obtained as Unlike in axisymmetric turbulent jet flames, where the entrainment rate E(x) in (5) was independent o f the downstream location x, in planar turbulent jet flame the entrainment rate in (21) decreases with increasing downstream location x. Thus while merging o f the axisymmetric jet flames is advantageous because it allows for crosslighting, it causes a reduction with x in the subsequent entrainment o f air into the planar flow above the merging plane, and thus will create an increase in the overall flame length L o f the flare field. The shortest overall flame length will thus occur when merging is delayed as far above the flare tips as possible. If crosslighting is to be accomplished, then the minimum flame length occurs when the flare-to-flare spacing 5 and the row-to-row separation S are both equal to 5 max in (13). An even shorter flame length is possible if the crosslighting requirement is dropped, so that 5 and S can each be made slightly larger than 5max, and thus each flare source will produce an isolated axisymmetric buoyant jet flame o f the type in Diez-Garias & Dahm (2004, 2007). However while such an arrangement minimizes the flame length, it will not minimize area A = ( N - 1) 5 • ( M - 1) S (22) required for the flare field. The optimization o f flare field design with respect to minimizing flame length and the required field area based on the scaling laws developed herein in discussed in greater detail in §7. 5.2 Scaling in the Planar Plum e Limit We next consider the case in Fig. 6 b, where BE ! 0 but JE = 0 , which corresponds to the planar turbulent plume limit. The flow produced by a planar source at x = 0 that introduces an exit mass flux mE and exit buoyancy flux BEinto a surrounding fluid with density p„ is equivalent to the flow produced by a theoretical "line" source located at a distance xEupstream o f the exit plane that introduces the same buoyancy flux B0 = BE into the surrounding fluid with density p „ , but has zero mass flux, namely m 0 = 0 . The resulting planar turbulent plume flow at any distance x from the exit plane (x = 0 ) is also self-similar and characterized entirely by the local flow width 8 (x) and centerline velocity uc(x). The scaling laws for 8 and uc in planar turbulent plumes can then also be deduced directly on dimensional grounds as 7 ! Uc (23a) ) p • (* + x £ ) = ( Cu )p • (V (23b) P„ j"'' , where the virtual origin is ~ Xe = (m E 1 P # ) 1 (24) A ( ! ) p ( c 5 ) p ( B0IP # ) 1/3 and where +$ ! ! %f (") d" = (& / a f ) 1/2 ( 0.512 (C6)p " 0.42 (cu)p ! XXX (25a,b,c) are scaling constants obtained from analysis o f experimental data o f Kotsovinos & List (1976). From (23b), in planar turbulent plumes the centerline velocity does not decrease with increasing distance from the buoyancy source. The increase in total momentum flux in the flow due to the buoyancy body force is exactly offset by the increase in flow width 8 , with the consequence that the momentum flux density (i.e., the velocity) is independent o f downstream distance x. The mass flux of the planar turbulent plume is then m (x ) / Ii (cu) p p (A )/ P- ) 1 3 (x + Xe ) , (26) where from (24) it can be verified that m (0) = rnE. The mass flux in (26) allows the mass entrainment rate E = dm / dx into the plume to be obtained as E ( x )l p . # i t (c, )r (c$ )p (JS„/p. ) ‘' 5 , (27) which is independent o f the downstream location x. While the momentum flux o f the line source at the virtual origin is zero, namely J0 = 0 , at the exit plane x = 0 the cumulative effect o f the buoyancy body force gives the momentum flux there as (c ) (J / P .) >>‘e / P . ) ( V P . ) ,,S . (28) *1 5.3 Scaling of Nonreacting Planar Buoyant Jets We now use the planar jet and plume scalings in §§ 5.1 and 5.2 to obtain the scalings for nonreacting planar buoyant jets in Fig. 6 c. Since the flow is nonreacting, there is no heat releaed by combustion and thus the buoyancy flux B( x ) remains constant at the source value, namely B(x ) = Be = B0 . Because buoyant jets have an additional source parameter (either B0 or J0) relative to the jet or plume limits, the scalings for 5 and uc involve an additional length scale l* and velocity scale u*. These are the only such quantities that can be formed from J0, B0 and p„, and therefore must be l* ! ( } (V P" l/3 P#) and 8 u* ! ( A )/ P- ) 1 3 (29a,b) Note that the velocity scale u* in (29b) does not depend on J0 . On dimensional grounds, the resulting scalings for 8 and uc in planar buoyant jets must thus be - = f (X) l* and u* = f u(X) (30a,b) where X + XE (31) The functions f 8 and f u must recover the jet-limit scalings as B0 ! 0 , for which l* and thus ! " 0 , and recover the plume-limit scalings as J0 ! 0 , for which l* ! 0 and thus ! " # . As a result, in the jet limit %" 0 the scalings in (17a,b) require f (X) # f e )j X and fu ! (cu)j X" 1' 2 (32a,b) while in the plume limit %" # the scalings in (23a,b) require f (X) # (c5 )p " and fu ! p" 0 . (33a,b) Comparing (17a) and (19b) with (23a) and (25b), it is apparent that in planar buoyant jets the flow width is identical in both limits. Moreover, as can be seen in Fig. 7, data from measurements in planar buoyant jets by Kotsovinos & List (1977) clearly show that (cs ) is the same for all %between the jet and plume limits. Consequently, for all % f (") = <VX (34) and thus for all x 5 = c5 - (x + ) (35) with c 5 " 0.42 . (36) The constancy o f cs for all x throughout planar buoyant jets allows the a d hoc "Morton entrainment hypothesis", which is used as the basis for essentially all other integral approaches for buoyant flows, to be replaced by this constant cs . This allows a similar integral method to be formulated for planar turbulent buoyant jets as for the axisymmetric turbulent buoyant jets in Diez-Garias & Dahm (2004, 2007). From the 8 (x) scaling in (35) and (36), the integral equation for the momentum flux J(x) allows a corresponding integral equation for the centerline velocity uc(x) in terms o f the buoyancy flux B(x). The procedure is analogous to that in Diez-Garias & Dahm (2004, 2007). Thus $ ! u 2dy = $ gAp dy (37) dx from which $ h (!) d ! d ) B(*) i - J{ * ) = ------ ---------------------dx U r c J f ( ! ) h (!) d ! 9 (38) which can be written as 1 dJ J dx i~ ~x V3 1 B(x) (39) 4/ J (x ) Mc From the definition o f the momentum flux J(x) J (x ) = I 2 u2 c (x ) c# • (x + xE) (40) and thus the left-hand side o f (39) can be written as 1 d J _ 2 duc + 1 J dx uc d x (41) x Here +$ ! ^ %f 2(") d " = (& / 2 a f j1' 2 (42a) +$ ! 3 " J%h(") J "M= V (&/ ha h)1/2 (42*) +$ (42c) Using (39), (40) and (41) then gives 2 du uc dx x ------ 1---- = ! B(x) / (43) uc2 x where (44) For a nonreacting planar buoyant jet, B (x ) = B0 and then (43) can be written as 2 f dX X " (Xf2 (45) ) _1 which is equivalent to f ) 3 = I ex'1'- (46) fu = [ o + C x-#3-' ] 1:3 . (47) d- ( dX 2 and thus integrates to The integration constant C is found by taking the jet-limit %" 0 , which gives (48) Rewriting (48) in uc and x gives u c ! C l/3 ( V P~ ) (* + *£ ) and comparing with the jet-limit scaling in (17*) shows that 10 (49) c = (c )j . (50) As a check, note that taking the plume-limit %" # in (47) then gives fu ! " (51) " ‘ "3 (B , l p . )" . 1/3 which can be rewritten in uc as Uc ! (52) Comparing with the plume-limit scaling in (23 b) then shows that (C. )P . (53) Thus 1/3 Uc ( Cu ) 3 ( V P" )32 (* + ) " 3/2 + ( Cu ) p ( B 0 f P " ) (54) The result in (54) is the exact solution for the centerline velocity decay in planar turbulent buoyant jets without heat release due to combustion. Figure 8 compares the result in (54) obtained from the present scaling analysis with the experimental data in nonreacting planar buoyant jets o f Kotsovinos & List (1977). It can be seen that the agreement is excellent, consistent with the fact that (54) was obtained from a completely fundamental and rigorous analysis with no a d hoc assumptions. From uc in (54) and 8 in (35) and (36), the scalings for the mass flux m(x ) , and the momentum flux J (x ) can be obtained via m(x ) ! %p udy = p$uc& %/ ( ' ) d ' = I p$uc8 J (x ) ^ %Pu2dy & p$u2 c 8 %f 2( ( ) d ( = I2 p$u2 c' . (55) (56) Figures 9 and 10 compare the resulting m(x) and / ( x ) with the corresponding experimental values in nonreacting planar buoyant jets reported by Kotsovinos & List (1977). Consistent with the agreement seen for 8 from (35) and (36) in Fig. 7 and the agreement seen for uc from (54) in Fig. 8 , the scaling for the mass and momentum fluxes are also seen to be excellent. As a result, the local mass entrainment rate E (x ) into the flow, namely E (x ) = din(x) / d x = Ixc 5 p$uc , (57) which is related to the aeration requirements o f the flare field above the first merging plane, will also be comparably accurate. Moreover, the centerline conserved scalar scaling ! c(x) can be directly obtained from the mass flux m(x) in (55) by an analogous procedure as used in Diez-Garias & Dahm (2004, 2007). For the planar buoyant jet, the mass flux o f scalar m^ across any horizontal plane is 11 +% m^" u d y '# % u c( ! c &f ( ) ) h ()) d ) = I4 p ^ S ^ , (58) which must remain constant for any conserved scalar. As a result, L_ = mE ! e I4 m (x) where ! E = m^ / mE is the value o f the scalar at source. conserved scalar scaling is thus (59) From (55), the resulting centerline (60) ! E 14 P # Uc5 In the following section, the integral equation for u in (43) that led to the result in (54) for nonreacting planar buoyant jets, for which B(x ) = B0 , will be solved for exothermically reacting planar buoyant jet flames, for which B (x ) will no longer be constant due to the heat released by combustion. 5.3 Scaling of Exotherm ically Reacting Planar Buoyant Jets We now extend the scaling for nonreacting planar buoyant jets from the previous section to develop the scaling laws for exothermically reacting planar buoyant jets. There are two key differences between the reacting and nonreacting flows that must be properly accounted for. The first is the inertial effect o f the reduced densities that result from the elevated temperatures in the reacting flow. As noted at the end o f §3, and in detail by Tacina & Dahm (2000), Dahm (2005) and Diez-Garias & Dahm (2004, 2007), this inertial effect can be rigorously accounted for the "general equivalence principle", the result o f which is that the density p„ in the scaling laws for the nonreacting flow are replaced by the effective density p " in ( 6 ) to obtain the corresponding scaling law for the reacting flow. Thus the scaling law for 8 (x) in (35) and (36) and the integral equation for uc(x) in (43) in nonreacting flows can be directly extended to reacting flows merely by replacing pOTwith p"f. The second is the additional buoyancy produced by the exothermic reactions occurring in the flow, as a consequence o f which the buoyancy flux B (x ) in the integral equation for uc(x) in (43) is no longer constant at the source value B0 . However, as shown in Diez-Garias & Dahm (2004, 2007), the relation between the buoyancy flux B(x ) and the heat flux Q(x) in (16), and the relation between the rate o f increase dm / dx in the mass flux and the rate o f increase dQ / dx in heat flux, allows B(x ) to be determined as shown below. We begin with the integral equation for Uc(x) in (43), namely duc uc B (x ) / j -----------1--------= ! ----------- 2" dx u„ x x the derivation o f which applies in the reacting flow when pOT is replaced with p"f, as shown. Next we write the relation in (16) between the buoyancy flux and the heat flux as 12 dB dx f \ dQ g dx . c T v p ~J (62) and the relation between the rate o f increase in heat flux and the rate o f increase in mass flux as dQ k, din ---- ~ ! h c-----dx dx (63) where ! hc is the mass-specific heat o f combustion, and the proportionality enters because the local molecular mixing rate (and thus the local combustion rate) is about one-third the local entrainment rate din / d x , as noted in Diez-Garias & Dahm (2007). Thus d B _ ^ dm dx (64) dx where n = (65) In (64), we now use m(x) from (55) and 8 (x) in (35) and (36), both o f which apply in the reacting flow when pOTis replaced with p f , to obtain d_ (B / p f ) = ^ c# n d dx (6 6 ) ( x • uc ) , dx which can be directly integrated to get ( B (x ) / p f ) = i i c 8 n [ ( x • uc( x ) ) - ( xE • ucE) ] + ( b e / p f ) , (67) where ucE = uc(xE). N ow multiply (61) by u2 c and rearrange to obtain d / 1/2 \ 3 3 _ - (x uc ) = -! x 1'2 ( B ( x ) / p f ) dx^ c> 2 , (6 8 ) ~ 3 /~ a /i c n xEUcE - 2 ! ( b e / p f ) (69) and upon substituting from (67) we get 3 d (x m uc ) 3 = - a ! c 5 # x 3I\ - x m dxK c 2 1 5 c 2 Using uc = u* ■ f u(#) and x = l* # we can write this in the dimensionless centerline velocity and vertical coordinate as d (%m f u) 3 = " X 3l2f u - X 1 2 " X eL (X e ) - 3 (c 2 )p (70) where A - 3 I,c#(c )' n . ( J / &> 4/ 3 21 P (Be / p f ) 13 (71) and where we have substituted ! = (cu) 3 from (53). Following an analogous procedure as in Diez-Garias & Dahm (2007), we put this in a more compact form by defining eL (XE) - 3 ( Cuff (72a, b) which allows (70) to be written as a simple integral equation, namely d w = "%W a a, 1/3 # b7 !., 1/2 . - d! (73) The result in (73) can be easily integrated in a spreadsheet, and allows the uc(x) in exothermically reacting planar buoyant jet flames to be obtained in a manner analogous as for axisymmetric buoyant jet flames in Diez-Garias & Dahm (2004, 2007). Key differences are in the definition of A in (71) and b in (72b), and in the length scale l* and velocity scale u* in (29a,b), as well as in the constants cs and (cu) . The scalings for the mass flux m(x) in (55), the momentum flux J (x ) in (56), the entrainment rate E (x ) in (57) and the conserved scalar scaling ! c(x) in (60) then follow directly from 8 (x) in (35) and (36) and from uc(x) obtained via (73). The properties o f the planar buoyant jets that form from merging o f the individual "point" flares in a flare row are thus fully determined above. The flame length L then follows directly from these, via the same procedure as shown in Diez-Garias & Dahm (2004, 2007), namely ! c(L ) = " / (1 + # ) where 9 is the stoichiometric mass ratio o f ambient fluid to jet fluid. 6 . M e r g i n g o f A d j a c e n t F l a r e R o w s (x ~ x M ) The flares in each flare row will follow the scaling laws for planar buoyant jet flames in §5 until the local flow width 8 (x) becomes comparable to the row-to-row spacing S. The merging o f flare rows follows similar considerations as does the merging discussed in §4 between individual flares within a row. The same merging criterion applies in determining the height o f the second merging plane x = xM in Fig. 2, namely XM) = Cm • 5 . Since from (35) and (36) follows directly that 8 (74) = cs - x everywhere between the first and second merging planes, it ^ =- Xm (75) S and thus from (9) XM _ Cm _ (76) s '(c )," * . With increasing vertical distance, as row merging is approached and finally occurs essentially no more air can be entrained into the flare field except along its outer periphery. The remaining 2/3 o f all the air that has been entrained up to that point in the jet - but has not yet mixed at the molecular level with the fuel and thus has not yet been consumed by combustion (see discussion immediately preceding Eq. 64 above) - must be sufficient to complete the combustion. Thus, if the mean flame tip determined by the conserved scalar scaling in the planar buoyant jet flames in 14 exceeds the vertical distance at which row merging occurs, then the remaining flame past this second merging plane can be expected to become long and probably very unstable. In practical flare field design, this condition must be avoided. §6 7 . I m p l ic a tio n s f o r F l a r e F ie ld D e s ig n The results in the previous sections provide considerable insights into optimum design o f large flare fields, and present methods that allow quantitative prediction o f the overall flame length via relatively simple yet fundamentally-based and highly accurate spreadsheet calculations. These methods can, for the first time, allow the required calculations needed to optimize the layout o f a flare field for any particular installation. Previously, optimal layout o f a flare field could be based on (1) empirical rules o f thumb derived from observations o f flame lengths o f individual flares or from experience with prior fields, or (2) a relatively large set o f CFD simulations o f various flare field layouts to determine an optimal design. Regarding the former, empirical rules o f thumb for the flame length variation o f individual flares in terms o f their design and operating parameters are o f limited relevance to flare fields, since the flare-to-flare merging that characterizes such fields not only invalidates these rules but also causes fundamentally different scaling o f the flame length with design and operating parameters o f each flare in the field (see §5). Regarding the latter, CFD simulations o f an " 5 x S unit cell" in a flare field can be performed over a range o f firing rates, however the accuracy o f such simulations is no better than that o f the fundamental scaling methods developed herein (e.g., see the accuracy evident in Figs. 12-15 o f Diez-Garias & Dahm 2007). Moreover, owing to the relatively long run times that can be required for simulations with sufficient accuracy to be useful for flare field optimization, the large number o f parametric variations required to identify an optimum design essentially prohibits such an approach. By comparison, the scaling laws and the integral equation for planar buoyant jet flame length developed herein allow rapid parametric studies with an accuracy at least as good as that o f CFD simulations. There are five main considerations involved in optimum layout o f a flare field: ( 1 ) the maximum firing rate at which the field is to be operated, which sets the exit mass flux mE o f each o f the identical flares in the field and thereby influences the flame length L, (2) the total area A = (N - 1) s ■ ( M - 1) S occupied by the field, where N and M respectively are the number o f flares in each row and the number o f rows in the field, s is the flare-to-flare spacing in each row, and S is the row-to-row spacing in the field, (3) the perimeter P = 2 [(N - 1 ) 5 + (M - 1)5] o f the field, which is the total length o f the fence needed to surround the field and thus for a given fence height is directly related to the cost of the fence, (4) the fence height L needed to protect the flames from crosswinds and obscure them from view, which is typically equal to the flame length L that results from the scaling laws in §5, and (5) the total number ( N ■ M ) o f flares in the field, since for a given maximum firing rate this determines the per-flare exit mass flux mE and thereby affects to locations o f the first merging 15 plane and the relative buoyancy in the planar buoyant jet flames between the first and second merging planes. The field area and the fence length and height are major factors in determining the total cost o f installing the field. In general, the fence cost increases dramatically with fence height, and thus even a relatively small increase in fence height produces a substantial increase in fence cost. Since fence height is proportional to - and in many cases equal to - the flame height, this places a premium on minimizing the length L o f the flames. As noted in §5, the flame length is determined by the vertical distance x needed to entrain the stoichiometric amount o f air required to complete the combustion. Moreover as is evident from §5 the entrainment rate E (x ) o f the planar buoyant jet flames - which form once the flames within each row have merged as noted in §4 - is substantially lower than that o f the axisymmetric buoyant jet flames, and thus the shortest flame length is obtained when the flare-to-flare spacing s and the row-to-row spacing S are large enough to avoid any merging. This, however, will not minimize the flare field area, nor will it minimize the fence length P. Moreover, it will also not minimize the total number o f flares in the field, which also contributes significantly to the total cost o f installing the field. For any given installation, the optimal flare field design depends on the relative balance between minimizing the field area A, minimizing the fence height L, minimizing the fence length P, and minimizing the total number N • M o f flares. Because the weight that must be assigned to the first factor (A ) relative to the factors (L and P ) and the number o f flares differs from one installation to another, it is not possible to develop a single optimal design that can be rescaled to any particular application. Instead, the optimal layout that minimizes the total installation cost must be determined for each application, based on the relative importance o f minimizing the field area A . If the field area A is unconstrained and the only objective is to minimize the installation cost while ensuring crosslighting o f all flares in the field, then s = S and s is determined from (13). In that case, the flares are essentially independent and the number o f flares is chosen so that the flame length L resulting from the per-flare firing rate mE will minimize the total fence cost. The field would thus be square (N = M and s = S) and the optimal number o f flares depends on the relative premium cost o f fence height over fence length. Since the flames are essentially independent and merge only near their flame tips to ensure crosslighting, the flame length L would be obtained from the scaling laws for axisymmetric buoyant jet flames in Diez-Garias & Dahm (2004, 2007). When A is constrained, the optimization requires s ! S . The flames will then interact upsteam o f their flame tips, which will assist with crosslighting but requires the length o f the resulting planar buoyant jet flames to be determined using the corresponding integral equation laid out herein. The number o f flares in the optimum layout is then determined by the flame length L resulting from the per-flare firing rate mE that will minimize the total o f the fence cost and the cost o f all the flares. This optimum is dependent on the maximum firing rate envisioned for the particular field under consideration, together with the marginal costs o f each additional flare in the field, each additional foot o f fence height, each additional foot o f fence length, and each additional square foot o f field area. For a given application, the number o f combinations that would need to be explored to come close to an optimal layout is very large, and for this reason 16 CFD simulations to determine the optimum are essentially prohibitive. However, a spreadsheet can easily solve the integral equation for the flame length for any given configuration, and thus can readily explore essentially the entire parameter space in sufficiently small increments to determine the optimal flare field design. 8 . C o n c l u d i n g R e m a r k s Scaling laws for the local flow width 8 (x) and centerline velocity uc(x) o f the interacting flares in a large flare field have been developed herein to provide a simple yet accurate spreadsheet-level determination o f the overall flame length L for any given total firing rate, flare-to-flare spacing s, and row-to-row spacing S . The results build on the previous integral equation approach developed in Diez-Garias & Dahm (2004, 2007), which determined the flow, mixing and combustion properties o f isolated "point" flares. That approach developed a rigorous integral equation for the centerline velocity Uc(x) based on the experimentally demonstrated invariance o f the growth rate d ! / d x = c5 o f axisymmetric buoyant jets for all downstream locations between the jet-like and plume-like limits. This replaced the ad hoc "Morton entrainment hypothesis" that is used in essentially all other integral approaches for analyzing buoyant flows. Together with the general equivalence principle o f Tacina & Dahm (2000), which rigorously addresses the inertial effects o f heat release to allow the scaling laws for nonreacting turbulent shear flows to be extended to exothermically reacting turbulent shear flows, this led to an integral equation for the centerline velocity Uc(x) and a corresponding integral approach for finding the flame length L for any given application. The integral approach o f Diez-Garias & Dahm (2004, 2007) was here extended to analyze the planar buoyant jet flames that form from merging o f flares within each row in a flare field. Scaling laws for the location o f this first merging plane were developed, and the equivalent source conditions for the planar buoyant jet flow that applies past this merging plane were obtained based on the experimentally-demonstrated invariance o f the planar buoyant jet growth rate d ! / d x = c5 . The resulting integral equation for the centerline velocity uc(x) was developed, and an explicit solution was obtained for the case of a nonreacting planar buoyant jet that showed excellent agreement with the experimental data o f Kotsovinos & List (1977). For the exothermically reacting planar buoyant jet, the rate o f increase in the buoyancy flux d B (x ) / dx was expressed via the rate o f increase in the heat flux d Q (x ) / d x via the entrainment rate dm / d x and the mass-specific heat o f combustion. This led to a simple integral equation for the centerline velocity that can be readily solved by a spreadsheet, and from which the flame length can be determined as shown in Diez-Garias & Dahm (2004, 2007). Since the present approach paralleled that in the earlier paper, the accuracy o f the present results should be essentially the same in that earlier paper, where excellent agreement was found with experimental measurements in buoyant jet flames. Such agreement is not surprising, given that the only approximations involved in this approach are the invariance o f cs , the equivalence principle, and the self-similarity o f buoyant jet flows, all o f which are strongly supported by experimental observations. This reduces the problem o f finding the flame length L for any given flare field design and operating conditions to a simple spreadsheet calculation. The resulting calculation can be performed far more quickly and with comparable or higher accuracy than a corresponding CFD calculation. Such spreadsheet-level calculations thus permit optimization o f the flare field 17 design for any given application, by allowing essentially the entire parameter space to be evaluated in sufficiently small increments to determine the optimal flare field design. For the maximum firing rate envisioned for the particular flare field under consideration, together with the marginal costs o f each additional flare in the field, each additional foot o f fence height, each additional foot o f fence length, and each additional square foot o f field area, this allows the optimal flare field layout to be determined that minimizes the total installation cost. Acknowledgem ents The work presented herein was performed at Coordinated Technologies and in the Laboratory for Turbulence & Combustion (LTC) at the University o f Michigan. Raymond Liang assisted with the production o f several figures. Technical discussions with Roberto Ruiz and with I-Ping Chung o f John Zink Co. led to the original questions that this paper has addressed. References Dahm, W.J.A. (2005) Effects o f heat release on turbulent shear flows. Part 2. Turbulent mixing layers and the equivalence principle. Journal o f Fluid Mechanics, Vol. 540, pp. 1-19. Diez, F.J. & Dahm, W.J.A. (2004) Improved prediction o f buoyancy effects on flame length and combustion properties o f flares. Proceedings o f the AFRC-JFRC 2004 Joint International Combustion Symposium, 10-13 October 2004, Maui, HI. Diez-Garias, F.J. and Dahm, W.J.A. (2007) Effects o f heat release on turbulent shear flows. Part 3. Buoyancy effects due to heat release in jets and plumes. Journal o f Fluid Mechanics, Vol. 575, pp. 221-255. Gutmark, E. and Wygnanski, I.J. (1976) The planar turbulent jet. Journal o f Fluid Mechanics, Vol. 73, pp. 465-495. Kotsovinos, N.E. and List E.J. (1977) Plane turbulent buoyant jets. Part I. Integral properties. Journal o f Fluid Mechanics, Vol. 81, pp. 25-44. Tacina, K.M. and Dahm, W.J.A. (2000) Effects o f heat release on turbulent shear flows. Part 1. A general equivalence principle for nonbuoyant flows and its application to turbulent jet flames. Journal o f Fluid Mechanics, Vol. 415, pp. 23 - 44. 18 Presented at the 2007 AFRC-JFRC International Symposium, 22-24 October 2007, Marriott Waikoloa, Hawaii Fig. 1. Schematic indicating basic layout of a typical flare field, consisting of individual "point" flares arranged in parallel rows with flare-to-flare spacing 5 and row-to-row spacing S. Flare tips are located a height H above ground plane to provide for adequate aeration. Overall flame length produced by the field is L. Note that typical separations s/L and S/L are smaller than shown here (for clarity) to ensure effective crosslighting between adjacent flares in same row, and between adjacent rows in flare field. Fig. 2. Schematic showing planes separating key regions in flare field analysis. Exit plane defined by flare tips (x = 0) is located at distance H above ground plane. Merging between adjacent point flares in same row occurs at first merging plane (x = xm), and merging between adjacent flare rows occurs at second merging plane (x = xM). 19 Presented at the 2007 AFRC-JFRC International Symposium, 22-24 October 2007, Marriott Waikoloa, Hawaii Fig. 3. Photographs showing (a) instantaneous flame luminosity, (*) long-time exposure of flame luminosity, (c) instantaneous schlieren image o f hot flame gases, and (d) instantaneous shadowgraph image o f hot flame gases. Linear increase in jet flame width 8 = (c8). • (x + xE) is not evident in instantaneous flame boundary in (a), but is readily seen in (*)-d) and determines flare-to-flare crosslighting ability via contact o f hot gases from one flare to next. Fig. 4. Schematic indicating first merging between adjacent jet flames in a flare row at x = xm determined by flare-to-flare spacing s. Mass flux m(xm) and associated buoyancy flux B(xm) at merging location xm determine effective exit conditions ( mE, JE, BE) for planar buoyant jet flame that results for x > xm as indicated in Fig. 5. 20 Fig. 5. Planar buoyant jet flow produced above first merging plane x = xm by mass flux per unit length mE, momentum flux per unit length J E, and buoyancy flux per unit length BE from jet flames at x < xm. uc uc uc Fig. 6. Scaling limits for planar buoyant jet flow, showing (a) planar jet flow produced by mass and momentum fluxes per unit length ( mE, J E), (b) planar plume flow produced by mass and buoyancy fluxes per unit length ( mE, BE), and (c) planar buoyant jet flow produced by mass flux per unit length mE , momentum f u x per unit length J E, and buoyancy ^ux per unit length BE. 21 Presented at the 2007 AFRC-JFRC International Symposium, 22-24 October 2007, Marriott Waikoloa, Hawaii d! dx 1.E+00 (x + XE) / l * Fig. 7. Experimental results for growth rate (cs ) " d ! / d x o f nonreacting planar buoyant jets between jet-like limit and plume-like limit, from Kotsovinos & List (1977), showing invariance o f (c8) with x. Conditions for each set o f symbols are defined in Fig. 3 o f Kotsovinos & List (1977). Dashed line corresponds to (cs ) = 0.42 for 8 as defined herein. (x + xE) / 1* Fig. 8. Comparison o f exact scaling law obtained in (54) for centerline velocity uc(x)/u* in nonreacting planar buoyant jets (solid line) with experimental results from Kotsovinos & List (1977) (symbols). Symbols are defined in Fig. 3 o f Kotsovinos & List. 22 1 .E+02 1 .E+01 m (x ) u *l * 1 .E+00 1 .E-01 1.E-02 -----‘-........... 1----- *-........... 1----- *-........... 1----- *-............ ...... ‘-........... 1.E-03 1.E-02 1 .E-01 1 .E+00 1 .E+01 1 .E+02 (X + XE) / l * Fig. 9. Comparison o f exact scaling law for mass flux m(x) obtained via (55) from 8 (x) in (35) and (36) and uc(x) in (54) (solid line) with corresponding experimental results from Kotsovinos & List (1977) (symbols). Symbols are defined in Fig. 3 o f Kotsovinos & List. 1.E-03 1.E-02 1 .E-01 1 .E+00 1.E+01 1 .E+02 (x + xE) / 1* Fig. 10. Comparison o f exact scaling law for momentum flux J (x ) obtained via (55) from 8 (x) in (35) and (36) and uc(x) in (54) (solid line) with corresponding experimental results from Kotsovinos & List (1977) (symbols). Symbols are defined in Fig. 3 o f Kotsovinos & List. 23 Presented at the 2007 AFRC-JFRC International Symposium, 22-24 October 2007, Marriott Waikoloa, Hawaii ( X + X E ) / 1' Fig. 11. Comparison o f exact scaling law for centerline conserved scalar value £c(x) obtained via (60) from 8 (x) in (35) and (36) and uc(x) in (54) (solid line) with corresponding experimental results from Kotsovinos & List (1977) (symbols). Symbols are defined in Fig. 3 o f Kotsovinos & List. Quantity reported by Kotsovinos & List is "(x + xE) ! c(x )$ , as shown. Flame length L for planar boyant jet flames is determined by ! c(L) = k / ( 1 + # ) , where 9 is the stoichiometric mass ratio o f ambient fluid to jet fluid, as discussed in Diez-Garias & Dahm (2004, 2007). 24