Ind. Eng. Chem. Res. 1990,29, 1977-1984
1977
Operational Characteristics of a Double-Spiral Heat Exchanger for the Catalytic Incineration of Contaminated Air Mark R. Strenger,t Stuart W. Churchill,*and William B. RetallickI Department of Chemical Engineering, University of Pennsylvania, 220 South 33rd Street, Philadelphia, Pennsylvania 19104
A temperature of about 900 K is sufficient for rapid, catalytic oxidation of contaminants in air such as carbon monoxide, hydrocarbons, organic compounds, tobacco smoke, and microorganisms. It is demonstrated in this work, both theoretically and experimentally, that such a temperature can readily be attained with a thermal input of less than the equivalent of 80 K by the use of a double-spiral heat exchanger. The double spiral is better than other geometries in terms of minimizing heat losses to the surroundings and is more compact. The theoretical solution reveals that for a double-spiral heat exchange an optimal value of UA/wc,the number of thermal transfer units, exists for which the temperature rise for a given heat input is a maximum. This maximum was overlooked in prior experimental and theoretical work. A general correlation is given for the conditions and dimensions which produce this optimal performance.
+
Introduction Minton (1970) asserted that plate-type double-spiral heat exchangers may be advantageous for a number of reasons including enhancement of the heat-transfer coefficient by the centrifugal force, compactness resulting from a shorter undisturbed length for flow, greater resistance to fouling, and relative ease of cleaning. Although not considered by Minton, double-spiral heat exchangers of significant width with many turns would appear to have a particular advantage in high-temperature or cryogenic applications in that the external surface is reduced to the outer turn and the end plates, thereby minimizing the leakage of energy to or from the surroundings. The thermal function of a double-spiral heat exchanger for the catalytic incineration of carbon monoxide, hydrocarbons, organic compounds, aerosols, and microorganisms is to raise the temperature of the air sufficiently so that these contaminants will oxidize rapidly and completely on the catalytically coated surface of the exchanger; 900 K is presumed to be a sufficient temperature for this purpose. Such processing or reprocessing of air appears to have promise for airliner cabins, spacecraft, hospital rooms, industrial clean-rooms,etc. As indicated schematically in Figure 1, cold air enters the exchanger at the periphery and is heated convectively as it flows spirally inward. At the core it is heated incrementally by an external source such as an electrical coil. It then passes spirally outward through a second channel and gives up all of the heat picked up by convection. The exiting stream at the periphery retains only the incremental heat added at the core less any heat losses to the surroundings. (The thermal effects of incineration itself are ordinarily negligible because of the small concentration of the contaminants.) The double spiral shown in Figure 1 is made up of a series of half-cylinders of discretely varying size for ease in fabrication and differs thereby only slightly from a true Archimedean spiral with a constantly changing radius of curvature. The number of turns of the channel through which air passes in proceeding from the periphery to the core is herein designated as n. An equal number of turns are traversed by the exiting stream. The number of double turns, n, in Figure 1 is thus li/z.It may be noted that the
* To whom correspondence should be addressed. ‘Current addreas: 518-1-01,3M Center, 3M Company, St. Paul, MN 55125. Current address: William B. Retallick Associates, 1432 Johnny’s Way, West Chester, PA 19382. 0888-5885/90/2629-1977$02.50/0
channel walls in Figure 1 each constitute n 1/2 or 2 turns, while heat is transferred through only n - 1 / 2 or 1 turn of each wall insofar as heat transfer from the periphery to the surroundings and through the walls of the core is negligible. The area for heat transfer is then 4?rH(n - 1/2)ravg,in general, and 47rHraw for the device in Figure 1. The very close spacing of the turns, which is desirable in terms of a high heat transfer coefficient and compactness, results in a Reynolds number in the laminar regime for any reasonable velocity. Also, a low pressure drop is desirable for a catalytic incinerator for air since the energy supplied to the blower, as well as that added to the core, must be assessed to the process. Hence, laminar flow is preferable, if not essential, as contrasted with conventional applications of double-spiral exchangers for liquids in which the pressure drop is often not an important factor. The thermal figure of merit for a double-spiral heat exchanger, when used for incineration, is
where T , - Ti is the temperature rise of the entering stream of air due to heat transfer and T3- T2is the temperature rise of the air stream in passing through the central core due to an external source of energy (see Figure 1).
In most other applications, the two “entering” temperatures, Ti and T3,are specified rather than T I and the difference T3- T2. Also, the two streams may be different fluids at different mass rates of flow, resulting in the parameter C = - ( w c ) ~ / ( w c ) , .The appropriate measure of performance is then usually taken to be the thermal e f fectiveness: 6’
- ( ~ c .. ) ; ( T-qTi) .. ( ~ ~ ) m i n (-TT3I )
For - ( w c ) ~= (wc),,as in Figure 1, eq 2 reduces to (3)
One other measure of performance has also been used, namely, the correction factor for the logarithmic-mean temperature difference: (4)
F is a measure of the performance as compared to a true 0 1990 American Chemical Society
1978 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 Insulation
Figure 1. Schematic of a double-spiral heat exchanger with opposing inlet and outlet (in this sketch n = 1.5).
countercurrent exchanger. For the operation of Figure 1 and no heat losses, = T3- T2= T4- T,,and eq 4 reduces to
(An1,
T2 - Tl where N = UA/wc is called the number of thermal transfer units. The heat capacities in all of the above expressions are implied to be mean values. The objective of the work reported herein has been to characterize the thermal behavior of a double-spiral heat exchanger in this particular application. Analytical and numerical solutions were developed for limiting and representative conditions, respectively, and experiments were carried out to test the validity of these solutions.
Prior Work Previous work on double-spiralheat exchangers is rather limited and specialized. The experimental studies have generally been for nearly square cross sections rather than for the large aspect ratio considered herein (that is like a slice of a jelly roll instead of the entire roll). They have also been limited almost exclusively to the turbulent regime. The several theoretical studies have included large aspect ratios and have nominally been for the laminar regime but, as will be shown, have generally not encompassed the conditions of importance in catalytic incineration. Experimental Investigations. The double-spiralheat exchanger was originally introduced by Rosenblad in Sweden in 1930, according to Coons et al. (1947), who presented widely scattered experimental data for the friction factor and Nusselt number for laminar, transitional, and turbulent flow of a number of fluids in a number of such devices. For some unexplained reason, their measured pressure drops fell significantly below the predicted values for a straight pipe. Their experimental Nusselt numbers significantly exceeded the theoretical values for a straight pipe in the laminar regime but agreed reasonably with predicted values for the turbulent regime. A transition to turbulent flow was observed to occur at Re = 1400-1800. Baird et al. (1957) passed water through both sides of a Rosenblad exchanger for Re from 9000 to 70000, and
determined heat-transfer coefficients 40% higher than those predicted for a straight channel. Tangri and Jayaraman (1962) made similar measurements with water in a double spiral with unequally sized channels for Re > 3000 and obtained values exceeding the predicted values for a straight channel by approximately 60%. Buonopane and Troupe (1970) correlated their own data for Nu for water in both sides of a double-spiral heat exchanger at Re from 4000 to 91 000 with an expression which differs negligibly from a widely accepted one for straight channels. More recently, Morimoto and Hotta (1986) measured heattransfer coefficients for Re from 2.5 X lo3to lo4 in several double-spiral heat exchangers with different radii of curvature and different angular locations of the inlet and outlet at the periphery. Their measured values, which exceeded the predicted values for a straight channel by 20-50%, increased with decreasing radius of curvature and varied ,somewhat with the inlet/outlet configuration. Thus all of the above experiments, except for a few by Coons et al., were confined to the turbulent regime. Also, all of the reported heat-transfer coefficients were based on the postulate of a logarithmic-mean difference in temperature, and all of the Nusselt numbers and Reynolds numbers were based on the hydraulic diameter. The increased heat-transfer coefficients were undoubtedly due all or in part to the secondary flow induced by the centrifugal force. Theoretical Investigations. Bounopane and Troupe (1970) solved a differential model iteratively by a Runge-Kutta technique for an exchanger with six double turns for equal and nonequal countercurrent flows. They present profiles of the mixed-mean temperature of the fluid streams for n = 6, as well as a plot of E versus N for several values of C and an unspecified value of n (=1.5?). Their computed temperature profiles agree well with their measured values. Jones et al. (1978) derived a numerical solution for the combustion of a mixture of low-heating-valuegas and air at the core of a double-spiral heat exchanger with the same configuration of flow as in Figure 1, except for an inlet and outlet at the same location on the periphery. They took into account heat losses and wall-to-wall radiation and used several different expressions for the variation of the heat-transfer coefficient with the mass rate of flow in order to simulate both the laminar and the turbulent regimes. To obtain numerical results, they represented the temperature profiles of the walls and the streams of gas by fourth-degree polynomials and evaluated the corresponding coefficients iteratively. They present a plot of E versus N for n = 5 and 8 and two axial lengths. They note that thermal radiation decreases the performance and suggest the use of walls of low emissivity. They include a curve representing the solution for a parallel-plate exchanger. It falls significantly below the curves for double-spiral exchangers as N increases, owing to the greater heat losses to the surroundings. Cieslinski and Bes (1983) used Hermite polynomials to develop a solution for a double spiral with entry and exit on opposite sides of the periphery, as in Figure 1. They present an additive correction to t for true countercurrent exchange for N up to 6, a series of values of C, and a series of values of n up to 16. Zaleski and Lachowski (1984) used the method of characteristics to solve the set of differential equations representing the 20 double turns of an exchanger designed to cool sulfuric acid with water. Using the Laplace transform, they developed a solution for the thermal behavior during start up, taking into account the heat ca-
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1979 pacity of the walls as well as that of the fluids. However, they only present transient temperature profiles of the fluid streams for several different conditions. Choudhury et al. (1985) obtained numerical solutions using a Runge-Kutta method for a series of values of C through double-spiral exchangers with three different angular locations of the exit and inlet at the periphery and core. They also investigated the geometrical effect of the radius of curvature by varying the ratio d/rc Their results for C = -1 (equal, countercurrent flow) are tabulated and presented graphically in terms of F versus N / n . They observed that such a plot was relatively insensitive to N and n separately, particularly for small N and large n, and showed that this relationship can be approximated by n N F = - tanh N n They note that eq 6 is the exact solution for a countercurrent cascade of n identical concurrent heat exchangers. They conclude that location of the inlets and exits at both the core and the periphery a t the same angular position results in the largest value of F but find the difference to be negligible for n > 10. The ratio d / r , was found to be a negligible parameter for values less than 0.1. (The paper by Martin et al. (1986) constitutes a slightly abbreviated version of that of Choudhury et al.) Morimoto and Hotta (1986) derived an analytical solution for the limiting case of one fluid at constant temperature (C = 0 or m ) and present an algorithm for numerical calculations for finite C. They present a very involved empirical correlation for F. Theoretical solutions do not appear to have been derived for the friction factor or the heat-transfer coefficient in laminar flow through a spiral passage of rectangular cross section. A variety of solutions have, however, been derived for channels of rectangular cross section with a constant radius of curvature. The solutions for the velocity field, which are all limited to breadth-to-spacing ratios ( H f d ) of 5 or less, indicate the occurrence of two secondary circulations (see, for example, Cheng and Akiyama (1970)). As the Dean number, Dn = Re(D/r,)1/2, increases, additional smaller vortices are superimposed on the larger ones (see, for example, Winters (1987)). fRe and Nu, which are independent of Re in fully developed laminar flow in a straight channel, increase with the Dean number. However, the magnitude of this increase is predicted by Cheng and Akiyama to decrease as the aspect ratio H / d increases and to be less than 5% for fRe and 7% for Nu at H l d = 5 and Dn = 100.
Mathematical Model The apparatus used in the experiments was constructed of a series of concentric half-cylinders, as in Figure 1. This discrete geometrical form, which has been used in most prior analyses, experiments, and practical devices, was therefore adopted for the modeling. In the experimental work, the external inlet and outlet were located at opposite sides, as in Figure 1, for convenience in the modeling, even though Choudhury et al. (1985) and others have found this arrangement to produce a slightly lower F or 6 than for both inlets and outlets at the same angular position. General Model. The following primary idealizations are implied in the model: (1)fully developed flow, (2) fully developed convection, (3) uniform physical properties (tHereby excluding natural convection), (4) negligible radiative transfer, (5) negligible drag on the end walls, (6) negligible conduction in the plane of the end walls, (7) negligible thickness of the spiral surfaces, and hence negligible resistance to transfer across the surface and
negligible conduction in the plane of the surface, (8) negligible conduction in the fluid in the direction flow, and (9) negligible heat transfer across the end walls and across the inner and outer surfaces of the exchanger. Postulates 1and 2 imply neglect of developing convection in the two entrances (from the outside and from the core), as well as inwardly due to strengthening of the secondary motion. The theoretical results of Cheng and Akiyama (1970) suggest that the latter effect is negligible in the present application. It may be noted that (except for the greater rate of strengthening of the secondary motion, the greater entrance effects, and the effect of the variation in the Reynolds number with temperature) the above postulates are all applicable to turbulent flow as well as laminar flow. Postulates 1 and 2 would be expected to result in a smaller temperature rise (smaller E ) and postulates 4 and 6-9 in a greater temperature rise (larger E ) than in a real double spiral, and some compensation may occur. The net effects of postulates 3 and 5 are not obvious. The feature that distinguishes the performance of a double-spiral exchanger from a conventional one is the occurrence of heat transfer between the entering (or exiting) stream and the fluid moving countercurrently in both the adjacent inner and adjacent outer turn. The energy balance for a differential angular sector of the entering fluid stream can thus be written as wc dTp = uq+idAq+i(Tp+i - Tp) + uq-1dAq-i(Tp-1- Tp) (7) where Tpis the mixed-mean temperature of the entering fluid at any location in the pth half-turn; TP+,and T p l are the mixed-mean temperatures of the exiting fluid in the adjacent inward and outward turns, respectively; U +1 and Uq-l are the overall heat-transfer coefficients !or transfer between T +1 and Tp and between T and Tp, respectively; and dtfq+, = Hr d8 and dA,;, = Eiq-1d8are the corresponding differenti$keas of the inner and outer walls of the pth half-turn of the entering channel. The radii of these walls, rq+l and rq-l,decrease discretely by d a t the end of each half-turn. Equation 7 is applicable for the outermost half-turn of the channel if Uq-l is set equal to zero and similarly for the innermost half-turn if U,,, is set to zero. Equation 7, including the reduced forms for the innermost and outermost half-turns, is applicable to the exiting stream if w is replaced by -w. In order to obtain numerical solutions, eq 7 was approximated by the following implicit finite difference expression:
Tpf- T p = HA8 -[rq+lUq+l(Tp+l’ - T,’) wc
+ rq-lUq-l(Tp-l’- T,’)]
(8)
where the absence of a prime indicates that value a t an angle 8 and a prime the value a t 8 A8. This results in 4nn/A8 simultaneous algebraic equations to be solved. Matrix inversion was used for this purpose. If U is postulated to be constant, eq 8 can be restated in dimensionless form as t p f- t, = N[R,+l(t,+l’- t p f )+ Rq-l(tp-l’- t,’)]A8 (9)
+
where t = (T - T l ) / ( T 3- T 2 )and R = r / l , 1 being the combined total length of the portion of the two surfaces through which heat is exchanged. From eq 9 it follows that E is a function only of N for a given geometrical design, and the location which is in turn defined by n, rJd, H/d, of the inlets and outlets. That is, insofar as this model is
1980 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
8
6 li
ii
0
2
N =-UA
i
wc
Figure 2. Figure of merit for a double-spiral heat exchanger (n = 8.75, d / r c = 0.2, opposing inlet and outlet): (e)experimental values based on net input of heat; (0) experimental values based on total parallel-plate input of heat; (- - -) true countercurrent exchanger; exchanger with a = 0.233. (-e)
valid, solutions are applicable for turbulent as well as laminar flow. Of course, as noted above, postulate 2 is severely strained in the turbulent regime owing to the significant development of the secondary motion with decreasing radius. Simplified Model. As rc increases, the effect of curvature, for a given d and n, decreases. In the limit of negligible curvature, eq 7 can be written in dimensionless form as where 2 = N(2z/1) and 2.211 is the fractional distance along one of the heated surfaces. Equation 10 must be applied for both fluids for each half-turn, leading to a set of 4n simultaneous, linear algebraic equations as compared to 4 n ~ l A Oequations for the model represented by eq 9. However, the primary merit of this very idealized model is that it can be solved in closed form for small n, thereby giving some insight into the functional dependence of E on N .
Numerical Solutions The principal calculations were, in conformity to the experimental work, for n = 8.75 and d l r , = 4/20 = 0.2. Five angular increments, Ae, per half-turn were used in the calculations, resulting in 175 simultaneous algebraic equations. Calculations were also carried out with 10 angular increments per half-turn, but the results did not differ significantly. The computed values of E are plotted versus N in Figure 2. The principal feature of this result is the maximum in E of -7.5 at N = 20. The explanation of such behavior is revealed in the plots of dimensionless temperature versus i, the number of turns from the inlet traversed by the fluid, as shown in Figures 3, 4, and 5 for N = 5, 20, and 79, respectively. In these plots, the curve for the outer exiting stream is simply that for the inner exiting stream shifted one turn to the right. For very high rates of flow corresponding to very small N , the change in the temperature of the fluid streams is small, and the full temperature difference at the core exists throughout the exchanger. In the limit of w m (or N 0 ) ,the behavior approaches that of a true countercurrent exchanger and E-N (11)
-
-
-
4
6
8
1
0
number of tums from outside
Figure 3. Computed temperature profiles of gas in a double-spiral heat exchanger for N = UA/wc = 5 (same geometrical confwration as in Figure 2): (-) entering stream; outer exiting stream; (- - -) inner exiting stream. (e-)
lo
1
II
,
T -Tl t=T3 -Tz
0
2 i
-
4
6
8
10
number of turns from outside
Figure 4. Computed temperature profiles of gas in a double-spiral heat exchanger for N = UA/wc = 20 (same geometrical configuration and line code as in Figure 3).
'i
I
I
f
5
0
2 i
-
4
6
6
10
number of tums fmm outside
Figure 5. Computed temperature profiles of gas in a double-spiral heat exchanger for N = UA/wc = 79 (same geometrical configuration and line code as in Figure 3).
Equation 11is represented by the straight line with a slope of unity in Figure 2. For a large but finite rate of flow,
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1981 Table I. Supplemental Computations l6
7.5 7.5 8.75 8.75 12.0 18.0
0.20 0.10 0.25 0.20 0.25 0.25
17.2 15.3 20.9 19.7 26.4 40.2
6.63 6.85 7.48 7.59 10.02 14.57
as illustrated in Figure 3 for N = 5, the temperature of the outer exiting stream falls between the temperature of the entering stream and the next inner exiting stream, and the combined heat transfer is less than in the limiting case of N 0. Hence E falls somewhat below the asymptotic line representing eq 11. For a fourfold decrease in flow, as represented in Figure 4 for N = 20, the outer adjacent exiting stream has essentially the same temperature as the entering stream over much of the exchanger, and the total heat transfer is effectively cut in half. For a further fourfold decrease in flow, as represented in Figure 5 for N = 79, the outer, adjacent exiting stream is actually at a lower temperature than the entering stream, and some of the heat transferred to the entering stream through the inner wall of the channel is transferred back to the exiting stream through the outer wall of the channel, thus resulting in a decrease in E. An alternative interpretation of Figures 3-5 and hence of Figure 2 is that the greater the value of N the greater the amount of heat given up outwardly to the entering stream and hence the greater the downward shift of the curve for the next outer exiting turn relative to the inner exiting turn. Were it not for end effects due to transfer through only one wall of the inner and outer half-turns, the maximum value of E would be expected to be half that given by eq 11. This maximum, which is of critical importance for the design and operation of a catalytic incinerator, has apparently not been recognized by any of the prior investigators, although a plot of the numerical solutions of Bounopane and Troupe (1970) suggests a maximum in t a t a value of N beyond their calculations, and one of the curves of Jones et al. (1978) appears to demonstrate a maximum. The calculations of Cielinski and Bes (1983) and of Choudhury et al. (1985) do not extend to sufficiently large values of N to produce a maximum. Equation 6, the correlating equation of Choudhury et al., approaches a maximum value in E only as N and therefore is quite misleading in this respect even though it provides a very good representation for small N a n d large n. A maximum in E implies a maximum in t and a point of inflection in F a t the same value of N . The temperature distributions of Figures 3, 4, and 5 explain the rapid rate of convergence of the numerical calculations in that only a single increment per half-turn would be necessary for the essentially linear temperature profiles and constant temperature differences in all but the innermost and outermost turns. Additional calculations were carried out to determine the effects of spacing and the number of turns. The results for the maximum in E, which is the most important condition for the application considered herein, and for the corresponding optimal value of N are summarized in Table I. The effect of d l r , on the maximum value of E is seen to be quite small. On the other hand, the maximum value of E is seen in Figure 6 to increase linearly with n. (Equation 6 predicts a linear dependence of E on n for large n but not, of course, the correct values.) The optimal value of N is also plotted in Figure 6. This plot constitutes a general solution for E,, and Noptfor all n and all
-
-
0)
1
6
8
10
12
14
16
18
20
n = number of double turns
Figure 6. Maximum figure of merit and optimal number of thermal transfer units for double-spiral heat exchangers with opposing inlet and outlet and d l r , = 0.25.
practical values of d / r , and H l d .
Analytical Solution The solution of the set of four equations represented by eq 10 for only one turn in each direction (n = 1) gives E = - 4N 4 + w Equation 12 reveals a maximum value of E = 1 a t N = 2 followed by a decrease to zero as N increases. In this limiting geometry, heat transfer occurs only through one side of a channel and only for one half-turn. On the other hand, for n = 312 the solution gives
which indicates a maximum value of E = 413 at N = 4 In 2, followed by a decrease to an asymptotic value of E = 415 as N increases. Unfortunately, the algebra required to obtain such a solution is closed form rapidly becomes intractable as n increases. The merit of eqs 12 and 13, and hence of the simplified model represented by eq 10, is in demonstrating that the maximum in Figure 2 is neither a consequence of finite curvature nor of the numerical method of solution. In addition, the existence of a finite asymptotic value of E for large N is implied for all N > 1 for exchangers in which the entrances a t the core and the outside are on opposite sides, also implied is an asymptotic value of zero for entrances on the same side. This limiting behavior was impossible to establish numerically.
Solution for a Parallel-Plate Exchanger The relevant comparison of the double-spiral heat exchanger as an incinerator is with an externally insulated parallel-plate exchanger with countercurrent flow and an externally heated return loop. The figure of merit for such an exchanger can readily be shown to be e@N- e-@N E = @(eon+ e-@N)+ a(epN- e-@N) (14) where @ = [a(2
+ CY)]^/^;
a = h,/U
with h, designating the coefficient for heat transfer from the exchanger to the surroundings through the insulation.
1982 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
-
As N m, E increases monotonically and approaches an asymptotic value of (/3 CY)^/^. A curve representing eq 14 is included in Figure 2 for a representative value of cy = 0.233, corresponding to air as the fluid and the use of 25 mm of good insulation. The resulting asymptotic value of E is 1.048. The relative advantage of a double spiral when operated a t the optimal value of N is quite evident. The actual advantage is somewhat less than indicated because heat losses were not taken into account in the model for the double spiral; however, these losses are much less significant because of the greatly reduced external area and the lower average external temperature.
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Experimental Tests The experimental exchanger was fabricated from corrugated stainless steel foil with a washcoat of activated alumina impregnated with platinum. The thickness of the metal foil was 63.5 pm. The corrugations, which were used to absorb thermal expansion and avoid buckling, had a height of about 1 mm. The channels in both directions consisted of 8.75 turns. The inner radius was 20 mm, the spacing was 4 mm, and the axial length was 90 mm. Thus, dlr, was 0.2 and H l d was 22.5. One end of the spiral was cemented into grooves in a 6-mm-thick ceramic plate. The other end was embedded in a puddle of ceramic cement in a shallow stainless steel dish, thus forming an end plate when the cement solidified. An electric resistance heater in the form of a finned cylinder was used to heat the air during its passage through the core. The wattage of the heater was determined from measurements with a voltmeter and an ammeter. The temperature profile on the two spiral surfaces was measured with thermocouples, the rate of flow of air with a rotameter, and the overall pressure drop with a manometer. The outer half-turns of the spiral and the end plates were insulated. The experimental measurements, although reasonably precise, suffered from two shortcomings. Firstly, the experimental values of N extended only up to about 10 and hence did not encompass the regime of the predicted maximum in E. This limitation occurred because the possibility of a maximum was not realized at the time the experiments were designed and carried out. Secondly, heat losses to the surroundings, although less than 10% of the energy which was transferred, were about 70% of the electrical input of energy. This latter relative loss, which was a consequence of the effectiveness of the device, precluded the unambiguous evaluation of E as wc(T2 T J / Q . Based on the total heat input, the experimental value of E for N = 9.5 was about 2.5, which is below the predicted value of 6.3, whereas for the net heat input the experimental value of E was about 8.5. Fixed values of E for both methods of evaluation were obtained for total heat inputs ranging from 50 to 140 W since the temperature difference T2- T1rose proportionately. This behavior confirmed the predicted one-to-one correspondence between E and N despite a significant but fixed fractional loss of heat. Also, the experimental device did produce temperatures at the core as high as 630 "C,which confirms its potential as an incinerator. Representative measurements of the temperature profiles on the spiral surfaces are shown in Figure 7 for N = 9.5. (The value on the inner wall for 5.5 inward turns appears to be in error, presumably because of mislocation of the thermocouple.) The greater upward curvature of the measured surface temperature in Figure 7 as compared with the computed values of the fluid temperature in Figure 4 suggests that the heat-transfer coefficient increased toward the core. (The shape of the curves in both of these plots is influenced by the use of the number of
500 400
1
1
0
I
+x
2
-
4
6
8
1
0
i number of turns from outside
Figure 7. Measured wall and gas temperatures (n = 8.75, d / r c = 0.2, opposing inlet and outlet, N = 9.5): (+) inner wall; ( X ) outer wall; (8)gas.
double turns rather than the surface area as the abscissa.) The measured pressure drops were about twice that predicted for laminar flow between parallel plates by using the expression
The discrepancy is presumably due in part to the secondary motion and corrugations but probably in the main to the effect of slight variations in the spacing of the channels. Because of the d3 term in the denominator of eq 15, such variations can have a significant effect. However, the measured pressure drops correspond to a power requirement of less than 1% of the electrical power supplied to the core. Interpretation In many applications, a value of F approaching unity is economically desirable. This can be achieved by operating at a sufficiently low value of N (high rate of flow) such that the deviation from the linear regime of Figure 2 is minimal. Such a high rate of flow in a double spiral generally results in operation in the turbulent regime. This high duty is achieved a t the expense of a large pressure drop and a limited increase in temperature per turn. In catalytic incineration (or in the homogeneous combustion of a low-heating-value gas, as discussed by Jones et al. (1978) and Churchill and Tepper (1985)), a large increase in temperature is more important than a high value of F. The value of N which produces the maximum in E (and in t) is then optimal. Insofar as the geometry is fixed and U is invariant (as in the laminar regime), this optimal value of N must be attained by the choice of the appropriate rate of flow. The results of the experimental investigations of double-spiral heat exchangers in the turbulent regime have generally been expressed in terms of a heat-transfer coefficient based on the logarithmic-mean temperature difference. If the experimental values of N were sufficiently large to cause a deviation from the linear regime represented by eq 11, the so-derived heat-transfer coefficients would incorporate the equivalent of the correction factor F. However, these prior experiments have generally been for very low N . The deviations are then primarily indicative of the effect of secondary motion. In order to use Figure 2 or its equivalent for other values of n and d/rcr or Figure 6, which holds for essentially all
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1983 conditions, the overall heat-transfer coefficient must be estimated. Figures 3-5 indicate uniform but unequal temperature differences and hence uniform but unequal heat flux densities over the inner and outer surfaces of the channels, except for the innermost and outermost turns. The theoretical solution for fully developed convection in laminar flow between parallel plates with uniform but unequal heat flux densities can be expressed as
where hl is the coefficient for transfer between wall-1 and the fluid stream; j, and j 2 are the heat flux densities from wall-1 and wall-2, respectively, to the fluid, In the limit of N 0, j, = j, and
-
hid = 70 = 4.118 k 17
In the limit of N
-
00,
such that j2
---+
-jl
hld - 4 2 k At the optimal value of N , such that j, 4 0
hld
35 13
- 4 - =
k
2.692
The overall coefficient is, of course, approximately equal to h1/2. The geometric effect of curvature, per the approximation of Langmuir (1912), is given in this application by
where the + and - refer to the outer and inner sides of the wall, respectively. Even for the largest value of d / r c = 0.25 and the smallest value of (hd/k), = 2 of this investigation, the correction factor for U is only about 0.1%. The effect of d f rc as expressed through the surface area was shown in Table I to be somewhat greater but still negligible. The quantitative effect of secondary motion in a double spiral has not yet been resolved, but, on the basis of the results of Cheng and Akiyama (1970) for a curved rectangular channel, it is estimated to be less than 5% for Dean numbers less than 100. If a double-spiral heat exchanger such as the one of this investigation were to be operated a t the optimal value of N , Figure 6 and the heat-transfer coefficient given by eq 19 could be used to determine the required rate of flow. If the equipment were operated at some other N , the values of j, and j2determined explicitly or implicitly in the numerical calculations leading to E must be used with eq 16 to determine U and hence w. Figures 2 and 6 are nominally applicable to turbulent flow with the appropriate mean value of U , which must take into account not only the effect of the secondary motion but also the effect of the change in the secondary motion with radius. Unfortunately, the dependence of the heat-transfer coefficient on Dn is even less well known than in the laminar regime. Conclusion A double-spiral exchanger is found on the basis of a theoretical model to be an effective device for heating a fluid to a high temperature with a minimum external input of energy. The principal unexpected finding is that a
maximum in temperature may be attained at a particular rate of flow. This represents the optimal operating condition for the use of a double-spiral heat exchanger for catalytic incineration. The existence of this maximum has been overlooked in all previous investigations, both theoretical and experimental, apparently because of a focus on low values of the number of thermal-transfer units and thereby a high thermal efficiency. The number of transfer units N for which the maximum figure of merit E was predicted to occur could not be attained in the experimental apparatus. Also, the value of E could not be determined unambiguously because of the significant heat losses relative to the electrical input at the core. Nevertheless, the level of temperature required for catalytic incineration of ordinary contaminants in air was readily obtained with this experimental device. For example, temperatures of air as high as 900 K were attained at the core with a total heat input equivalent to 236 K and a net heat input equivalent to 71 K. The double spiral is more effective in this application than conventional exchangers because of the reduced external area and the low temperature of the outer external surface, thereby minimizing heat losses to the surroundings. Measured pressure drops, which should be minimal in this application, were about twice the predicted value, presumably because of variations in the spacing between the walls of the channels. Acknowledgment This work was supported by the National Science Foundation under SBIR Grant ISI8650638. The analytical solution (eqs 12 and 13) was derived by Matthew Targett. We also wish to express our appreciation for the constructive criticisms of the anonymous reviewers. Nomenclature A = H1,area through which heat is exchanged, m2 c = mean specific heat capacity of fluid, J/(kgK) c = -(wc)i/(wc), d = spacing between adjacent walls of spiral, m D = hydraulic diameter: 4 times cross section divided by perimeter, m E = (T2- T , ) / ( T ,- T 2 ) ,figure of merit f = (2/n)(-dp/d0)(Dd2wp/w2)= Fanning friction factor F = correction factor for logarithmic-mean temperature difference h = heat-transfer coefficient, W/(m2.K) H = breadth of passages (axial length of exchanger), m i = index of the number of turns from the inlet traversed by the fluid j = heat flux density, W/m2 k = thermal conductivity of fluid, W/(m-K) 1 = length of surface through which heat is transferred, m lf = total combined length of channels through which the entering fluid passes, m n = number of spiral turns through which the entering fluid passes N = (UA)/(wc),number of thermal transfer units N u = h D / k , Nusselt number p = pressure, Pa Q = thermal input at core, W r = radius, m ravg= average radius of the channel walls, m R = r/l Re = ( D w ) / ( d H F ) ,Reynolds number t = ( T - T,)/ ( T3- T 2 ) ,dimensionless temperature T = mixed-mean temperature of air, K T I = temperature of air entering exchanger, K
1984 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
T , = temperature of air entering the core, K T3 = temperature of air leaving the core, K T4 = temperature of air leaving the exchanger, K U = overall heat-transfer coefficient from one fluid to the other, W/(m2-K) w = mass rate of flow, kg/s t = distance along the heated surface, m Z = =N(2z/l) Greek Symbols a = h,/U
b = [a(2+ t = ( T 2- T 1 ) / ( T 3 T2),thermal effectiveness 6 = angular coordinate, rad v = kinematic viscosity of fluid, m2/s
Subscripts 1 = on wall-1
2 = on wall-2 c = at inner surface of spiral (at core) cy1 = for a cylindrical surface e = exiting flat = for a flat surface i = entering lm = logarithmic-mean value max = maximum value min = minimum value o = at outer surface of spiral (at outside of exchanger) opt = optimal value p = index of a turn for entering fluid p + 1 = index of adjacent inner turn for exiting fluid p - 1 = index of adjacent outer turn for exiting fluid q + 1 = index of outer wall for turn p q - 1 = index of inner wall for turn p Registry No. Al, 7429-90-5;Pt, 7440-06-4.
Literature Cited Baird, M. H. I.; McCrae, W.; Rumford, F.; Slesser, C. G. M. Some Considerations on Heat Transfer in Spiral Plate Heat Exchangers. Chem. Eng. Sci. 1957, 7, 112-115. Buonopane, R. A.; Troupe, R. A. Analytical and Experimental Studies in a Spiral Heat Exchanger. Fourth Proc. Znt. Heat
Transfer Conf. (Paris-Versailles) 1970, I (Paper HE 2.5), 1-11. Cheng, K. C.; Akiyama, M. Laminar Forced Convection Heat Transfer in Curved Rectangular Channels. Int. J. Heat Mass Transfer 1970, 13, 471-490. Choudhury, K.; Linkmeyer, H.; Bassiouny, K.; Martin, H. Analytical Studies on the Temperature Distribution in Spiral Plate Heat Exchangers: Straightforward Design Formulae for Efficiency and Mean Temperature Difference. Chem. Eng. Process 1985, 19, 183-190. Churchill, S. W.; Tepper, P. The Characteristics of a Single-Fluid, Double- or Triple-Pass Heat Exchanger and Combustor for LowHeating-Value Gases. Znd. Eng. Chem. Process Des. Deu. 1985, 24, 542-550. Cieslinski, P. J.; Bes, T. Analytical Heat Transfer Studies in a Spiral Plate Exchanger. Proc. XVIth Znt. Congr. Refrig. (Paris) 1983, IZ (Paper B.1-198), 449-454. Coons, K. W.; Hargis, A. M.; Hewes, P. Q.;Weems, F. T. Spiral Heat Exchanger-Heat-Transfer Characteristics. Chem. Eng. Prog. 1947, 43, 405-414. Jones, A. R.; Lloyd, S. A.; Weinberg, F. J. Combustion in Heat Exchangers. Proc. R. SOC.London, Ser. A 1978,360,97-115. Langmuir, I. Convection and Conduction of Heat in Gases. Phys. Reo. 1912, 34, 401. Martin, H.; Choudhury, K.; Linkmeyer, H.; Bassiouny, M. K. Straightforward Design Formulae for Efficiency and Mean Temperature Difference in Spiral Plate Heat Exchangers. Proc. Eighth Znt. Heat Transfer Conf. (San Francisco); Hemisphere Publishing: Washington, DC, 1986; Vol. VI, pp 2793-2797. Minton, P. Designing Spiral-Plate Exchangers. Chem. Eng. Prog. 1970, 77, 103-112.Morimoto. E.: Hotta. K. Studv of the Geometric Structure and Heat Transfer Characteristics of a Spiral Plate Heat Exchanger. Nippon Kikai Gakkai Ronbunshu 1986,52,926-933; English translation Heat Transfer-Jpn. Res. 1988, 17, 53-71. Tangri, N. N.; Jayaraman, R. Heat Transfer Studies on a Spiral Plate Heat Exchanger. Trans. Inst. Chem. Eng. 1962, 40, 161-168. Winters, K. A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section. J . Fluid Mech. 1987,180,343-369. Zaleski, T.; Lachowski, A. Znzymieria Chemiczna i Procesowa 1984, 5,559-577; English translation Unsteady Temperature Profiles in Parallel-Flow Spiral Heat Exchangers. Znt. Chem. Eng. 1987,27, 556-565. Received for review July 3, 1989 Revised manuscript received March 29, 1990 Accepted April 4, 1990
ADDITIONS AND CORRECTIONS Calcium Oxide Sintering in Atmospheres Containing Water and Carbon Dioxide [Volume 28, Number 4, Page 4931. Robert H. Borgwardt Page 497. The second term on the right-hand side of eqs 7 and 8 should be preceded by In and should be enclosed in brackets. The correct equations are In 7 H l O = 0.002622"+ In [(ln P H l o - 1.39)/11.1] (7) In yCo2= 0.0034T + In [(ln Pco2- 1.948)/44.9] (8) The error is typographical only and does not affect the calculated results presented in the article.
