Znd. Eng. Chem. Res. 1994,33, 1809-1816
1809
GENERAL RESEARCH Combustion in Double Spiral Burners Stephen A. Lloyd Department of Chemical Engineering, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, United Kingdom
Double spiral devices have been employed as combustors for exceedingly lean mixtures (for example, for methane, mixtures of fuel content corresponding to less than 1/5of the lower limit of flammability have been successfully burned), and the leanest mixture combustible is determined by losses. For any given burner an optimum flow exists as a t low flows, losses are critical, and a t high flows there is inadequate area for heat transfer (also a t high flows, large pressure losses occur). Mathematical modeling of the spiral device demonstrates that radiation is significant even a t modest elevations in temperature and that spirals of many turns help control these losses. With metal burners, conduction losses can be critical. Several parameters are presented to identify design limits for spirals including wall emissivity, conductivity, and thickness.
Introduction In the recently published works of Strenger et al. (1990) and Targett et al. (1992)using double spiral devices as incinerators (where catalytic destruction is required to take place on the walls at temperatures up to 900 K) radiation was essentiallyneglected. Previously published work (Lloyd and Weinberg, 1974, 1975) which demonstrated the benefits of double spiral burners as combustors for very lean mixtures showed that, at the limit, heat losses were controlling with radiation heat transfer critical to performance. Radiation can be significant even at modest elevations in temperature with these burners, and ita impact on design is presented here. With metal burners, conduction losses can also be important. Several key parameters are introduced which identify when the onset of complex heat transfer occurs.
Background In order to improve combustion efficiency and control pollutant generation,a series of lean burn devices has been developed that, by recyclingheat from products tokeactant without simulataneous dilution, give independent control of reaction temperatures. They effectivelydivorcereaction rate from initial composition and by burning premixed eliminate unburned products of combustion and allow control of NO,. The heat transfer mechanisms involved with these burners (Jones et al., 1978)include convection-controlled ones (counterflow heat exchangers),radiation-controlled ones (reradiating disks and concentric cylinders), conduction-controlled ones, regenerators, fluidized beds (Lloyd and Weinberg, 1976),and spouted beds (Khoshnoodi and Weinberg, 1978);even the injection of active species from areplasma dischage has been studied (Lloyd,1973;Warris and Weinberg, 1984). The effect of heat losses is demonstrated to be critical, and the use of a double spiral heat exchanger as a burner (Figure 1)offers many advantages including: (i)the possibilityof using continuously variable flow channel dimensionswhich permitted linear velocities/ residence times of the reactants to be adjusted, (ii) lowpressure loss with good heat transfer, (iii) many radiation shields between the central combustor and outside walls,
1 a i r r fuel in
Figure 1. Double spiral heat exchanger configured as a burner.
and (iv) compactness with high ratio of heat transfer area to outside surface area. These burners enable very lean mixtures to be burned, extend the concept of what is a fuel, and have led to the production of an experimental pilot plant device for incineration of solvent waste (Murcar and Boden, 1984). Initial work applied to lean methane and propane air mixtures identifies that the temperature in the reaction zone is approximately constant: the picture for hydrogen combustion is more complex. The lean burn of hydrocarbons was also used to simulate the combustion of very low calorific value fuels when excess oxygen in the air was partially replaced with nitrogen. Solids and heavy liquid fuels (which tend to pyrolyze and form gums) would not be admitted directly to the spiral but could be burned in a fluidized bed located at the center (Lloyd and Weinberg, 1976). Combustion of lean hydrogen-air mixtures was also studied. Potential applications are for clean air environ-
0888-5885/94/2633-1809$04.50/00 1994 American Chemical Society
1810 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
lsyol :*:L4 1200-
1000-
@
N:8,Z=36mm
B
N=lO,Z:36mm
A
N=13,Z=36
Q
N=9,Z=65mm
&
N-8,Z:SSmm
Flow V e l o c i t y ( m / s )
800
2.0
3.0
4.0
7-
6.0
6.0
Figure 2. Variation of measured combustion temperature with cold linear velocity for methane using a number of burners.
menta (as well as “low BTU” gas combustion) as it will both sterilize/incinerate (high-temperature combustion) and offer controlled humidification (from products of combustion via fuel content) in the same process. By predrying the reactants, precise levels of humidification can be obtained.
Impact of Radiation In order to assess the impact of radiation on the design of spiral burners, use is made of a mathematical model which predicts the performance of double spirals burners, treating them as heat exchangers with a combustion chamber at the center. A brief summary of the model is presented in Appendix I. Other modelers of the performance of double spirals as heat exchangers have neglected radiation and tended to work in the turbulent regime. In this application hot gases flow at relatively low velocities with Reynolds numbers in the range 900 to 3500. Thus, radiative transfer between walls is important, laminar flow conditions can exist, and at the burner stability limit, heat loss is critical. For a spiral combustor, burning lean limit mixtures, the gas is essentially transparent so that only the walls take part in the radiative transfer. Analysis of the interchange between two curving walls shows that for a large radius of curvature the net heat flux per unit area approximates to that of two parallel plates
where both walls are presumed gray and of the same emissivity, c. In practice, this is reasonable with error in S of only 5 % for e = 0.8 and for an inner combustor radius of 2 x wall spacing. Usually, higher values for these parameters exist, thereby reducing the error in S. Radiation between turns of the spiral is a potential loss component as the radiation passes from turn to turn until it reaches the outaide wall. Incoming reactants, however, pick up some of this heat by convection. For flows experienced with these devices as combustors (usually operating in the laminar regime), a convective heat transfer coefficient (h,) between the walls and the gas can be computed of the form
where
X = D/(1.857 - 1.836(y/z) + 0 . 9 3 6 @ / ~ ) ~ ) (3) and
+
D = 2yz/(y z ) (4) These expressions were developed from published data for the laminar flow in straight channels (e.g., Shah and London, 1971). The original data were correlated, assuming the fluid properties were those for air, using a “least-squares fit” technique over the temperature range of interest. They are believed to differ from the heat transfer coefficientswhich actually occur in the spiral due to the effect of centrifugal forces (which tends to raise h,) and the high aspect ratio of the channel (which can substantially lower hJ. The mathematical model described in Appendix I produces a family of temperature profiles through the burner for the two gas streams and the walls. These profiles are determined by the burner parameters, in particular the heat released by combustion, Q,. If one then exploits the experimental observation that the temperature in the combustion chamber is essentially constant over a wide flow range, it is possible to produce a theoretical lean stability limit locus for the burner. Figures 2 and 3 give the experimentally observed combustion temperatures for methane (Tf= 1400 K) and propane (!‘f = 1275 K) for a variety of burners. On the basis of these values, Figure 4 gives the theoretical stability limit loci for one burner and the corresponding limit points experimentally observed. Good agreement exists. Of interest is curve c. Here, the convective heat transfer coefficient, h,, was modified so that for in > 1.0 g/s, the convective heat transfer coefficient was programmed to vary according to the relationship h‘, = bh,inai3
(5)
to simulate the transition from laminar to turbulent flow. The value in = 1 was selected as the pivot (Re 1850 based upon inlet conditions) as it permitted a good fit N
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1811
N-8, y x 2-4 x 6 6 m m 1000
* 800
-
600
Flow ( m l / s )
RorlOmm
1400
Tf= 1400 K
y;lmm,Z=SSmm To=T1-300K, N-8
(a) h c ~ l . 7 7 ~ l O - 3 , ~ . O . 8
1200
.N /' =8
0,8 \ E
(b) h ~ - l . 7 7 x I 0 - 3 , b O . S ( c ) h c = 2 / 3 ~ 1 . 7 7 ~ 1 0 - 3 (6x41) h e = h c ( i ) ' l ' (mil)
(d) h e = 2 / 3 ~ 1 . 7 7 ~ 1 0 -( 36'1)
1000
N=10
he=hc(m)"' ( b 1 )
80(
d
'8
S t a b l e Combuition
w
Extinction
4oa
-------\e-I
1.0 20
1.6
Vol W Methane 2'o
150
1.5
mass flow (m,grn.s")
1.0
Figure 5. Variation of temperature rise due to combustion (measure of fuel content) for a final flame temperature of 1400 K with mass flow for spiral burner of fiied flow channel dimensions (5 mm X 55 mm) for N = 8 and 10 turns and two emissivities c = 0.8 and 0.9.
Experimental Points
BO(
ds
2.6
Figure 4. Stability characteristics for one spiral burner using methane and propane fuels. Experimental points are marked @ = stable combustion and = extinction. Loci are predictions from the model described in Appendix 1.
between the model and experimental results for methane. The value for b = 213 was selected as this gave coincident results for h,as predicted for laminar conditions according to eq 2 and the results for turbulent flow predicted by the Dittus-Boelter relationship. Radiation heat transfer, however, is temperature, emissivity, and geometry dependent; hence, the effect of radiation as a loss component decreases with increasing
flow rate as it is then a smaller fraction of the gas energy content. This explains partly the characteristic bulge (minima) in the experimental lean limit curve since, at low flows, wall and end losses dominate and, at high flows, and inadequate heat transfer area exists. Increasing the number of turns will increase the heat transfer area and also reduce radiation losses, but sideways losses, unless controlled, will also start to take effect. Hence, the impact of increasing turns on the location of the "bulge" is not immediately obvious: it is also influenced by burner depth. Figure 5 gives the locus for two spiral burners ( N = 8 and N = 10) and two emissivities (e = 0.8 and e = 0.9 (Haigh and Chojnowski, 1975)) of the variation in temperature rise due to combustion (QR)versus mass flow. A final combustion temperature of 1400 K was employed. The model illustrates clearly that for a given set of parameters an optimum flow (minimum QR)exists and the impact of radiation (via e) on it. Figure 6, for the same set of conditions, gives the
1812 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 E
can be made between the two figures. The impact of radiation is reduced, yet the model still gives a variation in core temperature of roughly 200 K. Key Parameters for Estimating the Impact of Losses One key parameter (R1) is the ratio of heat loss by radiation to heat recovered by convection at any point.
-
Figure 6. Variation of exit temperature for spiral burner of fixed flow channel dimensions (5 mm X 55 mm) for N = 8 and 10 turns for two emissivities t = 0.8 and 0.9.
1100
IQQO
Inserting suitable values shows that for R1 = 1, Ti 750 K. This is a comparativelylow value and is dependent on the values selected for other parameters such as Tg,Tj. Values of Ti significantlylower than 750 K can be obtained by varying Tg, Tj as there is no necessity for Tg to lie midway between Ti, Tj, and the expression is dominated by !P terms. (See Appendix I1 for further details.) At any given element, for R1= 1,the heat picked up by convection and the heat transferred by radiation are nominally the same. Before radiation can be neglected, we require R1 to be small (say, less than 0.05),and this requires Ti to lie in the range 350-450 K. In order to neglect radiation at temperatures up to 900 K, h, would need to increase by roughly 2 orders of magnitude above those which are typically encountered with spirals, and though this could be achieved in high Re turbulent flow, it would be associated with high pressure losses. For a multiturn spiral, two further parameters, R2 and R3, can be developed.
R2 = heat transferred radially by radiation heat exchanged (recirculated)
(7)
N= 8 600
4
Figure 7. Temperature profile through burner for t = 0.8 and variation of combustion temperature with emissivity at low flows (h= 0.6).
predicted exit temperature with flow. The large influence of emissivity at low flows (broadening of loci) compared to high flows illustrates the importance of radiation at low flows. Similarly, the impact of number of turns is more significant at high flows (through substantial extra surface area) than at low flows where radiation is reduced proportionately to the reciprocal of the number of turns (l/N). Again, an optimum flow for maximum heat transfer exists, and this does not occur at rit = 1. Figure 7 gives the predicted temperature profile though a burner at "low flown and shows the characteristic temperature pinch which exists for low flows toward the midpoint of the radius vector. It presents, for identical conditions, the combustion temperature which could exist (neglecting dissociation) as the walls emissivity is varied. At this flow the potential variations is massive (c = 0 corresponds to effectively neglecting radiation). Figure 8 presents the results at "high flow": identical values for h, were employed so that a direct comparison
S(T;' - T t ) CyIL)for a multiturn spiral hc(T2- To)
(8)
A N / ( ~ N A identifies ,) the benefits of the compact spiral exchanger. The parameter R3 is given by R3 = heat transferred radially by radiation heat released centrally by combustion
- ( S / W ( T ; '- T:MN mfQ
(9) (10)
which helps identify the impact of radiation on minimum heat release. Inserting typical values for 1%methane-air mixture gives R2 0.06 and R3 0.36. Radiation has been controlled in terms of heat recirculated but is still a significant loss term when expressed as a fraction of the fuel content at the lean limit. The above parameters focus on radial losses, since the spiral can be made deep or even formed into a tonus.
-
-
Conduction Heat Losses Conduction heat losses impact the design of metal burners. For a thin spiral (or a series of concentric
Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1813
1500
1000
500
0.2
0.4
oa
R'Rnl,,
as or
,
1.0
E
Figure 8. Temperature profile through burner for t = 0.8 and variation of combustion temperature with emissivity at high flows (rh = 1.6).
cylinders) the heat loss along the width into the sides is important. For a long spiral or torrus, the heat loss along the Archimedean length is of interest. Parameters considered here are as follows:
Ideally, all of the above parameters should be small. Their examination identifies key features which should be addressed; e.g., t should be thin and L should be large (unless sideways losses start to dominate). However, values for N ,L, and A, are dictated by other considerations such as the heat to be recirculated (A,) and control of radiation (A?. The above expressions are most useful in determining the minimum wall thickness, t. Inserting suitable values in the above expressions showsthat to keep H small (less than 0.051, large values of t can still be tolerated (up to 50 mm) before heat loss along the spiral is critical, yet for sideways losses for thin spirals (or concentric cylinders) an upper value of t of only 0.02 mm can exist.
Use of Ceramic Materials heat lost by conduction Hcl = heat convectively transferred (recirculated) He2
(11)
heat lost by conduction = preheat of gas (recirculated heat) heat lost by conduction = heat released centrally by combustion
(13)
For sideways analysis (thin spiral or concentric cylinders) one obtained
(14)
For the deep spiral (or torrus) we get
The above parameters illustrate the benefits of manufacturing spirals from ceramics as their emissivities and conductivities are generally low. Examination of the R, range of parameters shows that they are all linearly dependent on S which is related to t in eq 1. Thus, the lower the value of E, the greater the improvement in S,hence reduction in R-other parameters being constant. Returning to our previous example where for R1 = 1and E = 0.8,Ti 750 K, if c was halved, Le., c = 0.4, and all other parameters remained unchanged then Ti 1400K. Similarly, for the example of R1 = 0.05 and t = 0.8,Tj 450 K. If now t = 0.4 then Ti 815 K. Thus, the peak temperature could increase by approximately 809% before radiation became significant as a loss component. However, the impact of emissivity is flow dependent, as is graphically illustrated by Figures 7 and 8. Thus, though reducing emissivity will invariably reduce radiation losses, its impact is more significant at low flows than high flows. In a similar fashion, the impact of conductivity can be seen from examination of the H parameters which are all linearly dependent on the thermal conductivity of the walls, k. Reducing k will directly influence the permissible minimum wall thickness, t . However, if the value (klt) becomes toolow, then an undesirable temperature gradient will exist across the wall and impede the spiral's performance as a heat exchanger.
-
-
- -
Overall Performance of Spiral The overall behavior of the spiral burner as a heat exchanger (and hence its ability to combust poor fuels) can be examined in terms of the effectiveness, E, heat capacity ratio, p, and the number of transfer units, NTU.
1814 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
~
0
~
2
4
NTU
8
8
10
12
Figure 9. Variation of effectiveness, E, with NTU for a spiral.
Conventionally,
-
J$=
NTU =
heat transferred (recirculated) max theoretically possible to transfer
SudA hA mCp 2mCp N-
Figure 9 shows the variation of E with NTU for a 10-turn spiral burner for Tf = 1400 K. These predicted curves are again presented for the extreme values of emissivity oft = 0.0 and t = 1.0 together with the comparison of a counterflow (convective) heat exchanger (lossless) for the two extremes of p = 0.0 and p = 1.0. The importance of radiation can be seen. Conclusions When using the spiral burner as a combustion device, radiation is significant as the core temperature is elevated above the adiabatic combustion predicted from the energy content of the fuel alone. This radiation is a "loss" component transferring heat between walls of the spiral, passing through the reacting gases which are essentially transparent: this is especially significant a t low flows. However, if radiation heat transfer is combined with convective heat transfer mechanism, accurate modeling of the spiral can be achieved. Even comparatively simple evaluations of this convective component give acceptable results. In order for the spiral to be used a t the leanest of mixtures, low emissivity materials are required. This is demonstrated by the model for though it is possible to design to make conduction losses negligible,radiation heat transfer will always exist due to the elevated temperatures required for complete combustion. For mechanical reasons, most experimental work was performed using metal models whose oxidized surfaces had always exhibited high emissivity,though some cursorywork using ceramic devices did appear promising. Like others, we have found the use of spiral exchangers as combustion devices to be excellent, but in predicting performance, radiation losses dominate and must be minimized and conduction losses must be controlled.
Nomenclature A, = cross-sectional area of metal walls AN = external (perimeter) area of exchanger A, = area of wall of heat exchanger spiral a,,(=) = coefficients of polynomial describing temperature profile b = multiplier to convective heat transfer coefficient C p = specific heat of reactants (which for lean mixtures N specific heat of products) D = equivalent diameter f = fractional fuel content gT = external heat loss coefficient H d = conduction coefficient h = convective heat transfer coefficient (c = overall; 1 = reactants; 2 = products) k = thermal conductivity L = length of spiral m = mass flow (g/s) N = number of turns of spiral Q = calorific value of the fuel Qc = heat released by combustion QR = adiabatic temperature rise due to combustion Q L = total heat lost from the spiral burner R = radiation coefficient R , = radius of central combustion chamber r = radius (1, 2 refer to first and second spiral) S = radiation coefficient ( d ( 2 - e)) t = wall thickness T = temperature (i, j = walls; a-d = walls for model; 1 , 2 = cold and hot gas streams; g = gas; f = flame; e = exit; o = ambient) W , = radiation flux between walls i, j y,2 = rectangular flow channel dimensions (width X height) c = emissivity 0 = angular displacement p = ratio of specific heats u = Stefan-Boltzmann constant (56.7 X 10-9 W m-2 K4) Appendix I Outline of Technique To Predict Temperature Profile through Double Spiral Burners. Mathematical analysis of the spiral as a burner assumes that the spiral behaves as a heat exchanger with counterflow heat exchange between products and reactants and complete combustion occurring at the core. Heat losses occur from the outer periphery and the sides. The burner geometry is assumed to follow that of an Archimedean spiral. Independent heat balances are performed on (i) the walls of the central combustion chamber, (ii) each interior wall, (iii) outer wall where losses occur through the edge, and (iv) each gas stream. In addition, further heat balances are required for the burning gas in the central combustion chamber and overall heat balances for the burner. Here, we are interested in the double spiral as an exchanger where radiation transfer occurs between turns. The walls are arranged as shown in Figure 10. Performing heat balances on the wall at radius r2,i and neglecting the temperature gradient across the walls of the spiral gives
Similarly, for the wall at rl,i, the heat balance gives
Ind. Eng. Chem. Res., Vol. 33, No. 7 , 1994 1815 Hence, the above equations reduce to
I
I
T &,*-I
I
I I
ICombustion Chamber Figure 10. (Pi
at the combustion chamber (i = 0) and the outer wall (i = N). Similarly consider any element of gas a t a radius, r, between the angular locations 8 and 8 + d8. Performing a heat balance on a cold gas element at radius r, whose temperature rises between T I and T I + dT1, yields
+
hC,Tli rJ d8 h1(Tci- Tlc)= hCp(Tli+ dT1,J+ r,,iZ d8 hl,i(Tl,i- Td,J+ (rl,: - r2,?)dB gT(T1,i- To) where gT allows for heat losses from the sides of the spiral. Rearranging the above (dB is negative) and assuming that the temperature gradient across the wall is negligible Tb,i we obtain SO that Tc,i
- Y)Zh,i(Tdi-l-T2,J + y(2ri- Y)gT(To- T2,i)l (A81
Heat balance on the central combustion chamber yields
T2,0- Tl,o(6 = l r ) = temp rise due to combustn = 21rRc3h1,~(T2,~ - 'b,O) --21rR&~ --Qc (7'2,o - To) (A91 %,2
"CP2
"CP,Z
where the first term on the right-hand side = Qc/(mC,) = adiabatic temperature rise from fuel, the second term is the heat transferred from the combustion chamber into the spiral and the third term is the heat lost from the top and bottom of the spiral. An overall heat balance yields
where QL is the total heat lost from the spiral obtained byintegrating over the whole external surface area, i.e.: Similarly, a heat balance can be performed on the hot gas stream
2rR:gT(Tf- To) (All) The above equation can be simultaneously solved along with the boundary conditions to yield a solution of form T , = T(r). However, the equations are strongly nonlinear, and hc,C,, etc. are also functions of temperature; hence, for each temperature (two gas streams and two walls) we assume a solution of the form Equations Al-A4 will hold for all values of i, except a t the central combustion chamber (i = 0) and outer turn (i = N) where special conditions will apply. Since the spacing of the turns is constant ri+l H ri + y
similarly
for an Archimedean spiral.
with, from symmetry aT(,,/ar = 0 (r = 0). Equation A12 can be expanded to yield four coefficients for each of the four expressions for T . With eq A5-Al1 plus boundary conditions all valid at any radius vector, sufficient expressions can be developed to uniquely solve for the 16 coefficients. Since any value of radius vector (angular value of 8) can be arbitrarily selected, the technique employed was to pick 10 values for the vector giving an overdeterminancy of the coefficients and then selecting optimum values for the coefficients using a least-squares fit technique.
1816 Ind. Eng. Chem. Res., Vol. 33, No. 7,1994
Appendix I1 Some Thoughts on R1. By definition
R, =
S(T: - T;) h,(Ti - Tg)
where S=-
Ut
2 - 6
Now
+
Ti4 - T; = (Ti2 + T:)(Ti Tj)(Ti- Ti) (B2) If one makes the assumption that Tglies midway between Ti and Tj then T g= (Ti + Tj)/2
033)
T i - T , = (Ti-Tj)/2
(B4)
and In addition
h, =
1.68 X 10"
X
Tgo74
Substituting B3 into B5 we get
h, = k'(Ti + Tj)0.74
036)
where
k' =
1
X lo" 20.74 = constant for any given spiral
Substituting B2 and B6 into B1 yields
R, =
S(T? + T:>(Ti + Tj)(Ti- Ti) k'(Ti + Tj)0.74(Ti - Tj)/2
Hence, for R1 to be small then Ti and Tj must be small. Returning again to eq B1 and inserting suitable values into the equations we obtain: y X z = flow channel dimensions = 4 mm
:. D = 7.46 mm and X = 4.31 mm = 0.004 31 m
X
55 mm
If Ti = 750 K and Tj = 650 K then T g= 700 K and h, = 49.7 W/m2 K. Similarly, assuming t = 0.8 s = 3.78 x lo4 and 3.78 X 104(7504 - 6504) = 2.1 R, = 49.7(750 - 700) Similarly if Ti = 400 K, T j = 350 K then h, = 31.5 and 8, = 0.05 Literature Cited Arbib, H. A.; Levy, A. Comb. Sci. Tech. 1982,28,83. Haigh, C. P.; Chojnowski, B. Hemispherical Emissivity of Clean Furnace Tubes. J. Znst. Fuel 1978,48,139. Jones, A. R.; Lloyd, S. A.; Weinberg, F. J. Combustion in Heat Exchangers. Proc. R. SOC.London A 1978,360,97. Khoshnoodi,M.; Weinberg,F. J. Combustion in Spouted Beds.Comb. Flame 1978,33, 11. Lloyd, S.A. Study Report No. 1 to Shell. Private communication, 1973. Lloyd, S. A.; Weinberg, F. J. A Burner fore Mixtures of Very Low Heat Content. Nature (London) 1974,251,47. Lloyd, S. A,; Weinberg,F. J. Limitato Energy Release and Utiliiation from Chemical Fuels. Nature (London) (1975)257,367. Lloyd,S.A.; Weinberg,F. J.ARecirculatingFluidinedBedCombuetor for Extended Flow Ranges. Comb. Flame 1976,27,391. Murcar, W. G.; Boden, J. C. Heat Recovery from Solvent Extraction. In Solvent Problems in Industry; Kakabadee, G.,Ed.; Elsevier: New York, 1984. Shah, R. K.; London, A. L. Laminar Flow Forced Convection Heat Transfer and Flow Friction in Straight and Curved Ducts-S Summary of Analytical Solutions. Tr. No. 75,Stanford University, CA, 1971. Strenger, M. R.; Churchill, S. W.; Retallick, W. B. Operational Characteristics of a Double Spiral Heat Exchangerfor the Catalytic Incineration of Contaminated Air. Znd. Eng. Chem. Res. 1990,29, 1977. Targett, M.J.; Retallick, W. B.; Churchill, S. W. Solutions in a Closed Form for a Double Spiral Exchanger. Znd. Eng. Chem. Res. 1992, 31, 658. Warris, A. M.;Weinberg, F. J. Ignition and Flame Stabilization by Plasma Jete in Fast Gas Streams. 20th International Symposium on Combustion, 1984 p 1825. Received for review July 1, 1993 Accepted August 10,1993. Abstract published in Advance ACS Abstracts, October 1, 1993.
