658
Ind. Eng. Chem. Res. 1992,31, 658-669
Whan, G. A.; Rothfus, R. R. Characteristics of Transition Flow between Parallel Plates. AIChE J. 1959,5,204-209. White, R. R.; Churchill, S. W. Experimental Foundations of Chemical Engineering. AZChE J. 1959,5,354-360. Zel’dovich, Y. B. The Oxidation of Nitrogen in Combustion and Explosions (translated title). Acta Physicochem., USSR 1946,21, 517-628.
Zuzovsky, M.; Brenner, H. Effective Conductivities of Composite Materials Composed of Cubic Arrangementsof Spherical Particlea Embedded in an Isotropic Matrix. J. Appl. Math. Phys. 1977,28, 979-992.
Received for reuiew January 23, 1991 Accepted May 29, 1991
Solutions in Closed Form for a Double-Spiral Heat Exchanger Matthew J. Targett, William B. Retallick,+and Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, 31I A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393
The MACSYMA code for symbolic manipulation was used to obtain solutions in closed form for double-spiral heat exchangers of a few turns, both with and without heat losses to the surroundings. The solutions, which are for an equal rate of flow of the same fluid in both directions, reveal the existence of an optimal number of transfer units (an optimal rate of flow) for which the temperature rise for the heated stream is a maximum for a given thermal input or temperature difference. Such anomalous behavior, which is of obvious importance in the design and operation of double-spiral exchangers, has generally been overlooked or misinterpreted in prior experimental work and numerical solutions. For realistic heat losses to the surroundings, double-spiral exchangers of multiple turns are shown to be superior to true countercurrent exchangers in terms of producing a temperature rise. Minton (1970) asserted that double-spiralheat exchangers might be advantageous for a number of reasons, including enhancement of the heat-transfer coefficient, compactness,greater resistance to fouling, and relative ease of cleaning. Although not mentioned by him,doublespiral heat exchangers of many turns would appear to have a particular advantage in high-temperature and cryogenic applications in that the external area is reduced to the outer curved surface and the end plates, thereby reducing the leakage of energy to the surroundings. Strenger et al. (1990) have examined the use of a double-spiral heat exchanger for the catalytic incineration of low concentrations of contaminates, such as carbon monoxide, hydrocarbons, organic compounds, aerosols, and microorganisms, which may be present in the air of spacecraft, airliner cabins, automobiles, hospital rooms, and industrial clean rooms. In this application the heat losses from a double-pipe heat exchanger would be prohibitive despite the use of the best possible thermal insulation. Numerical solutions of a fairly exact model for doublespiral heat exchangers by Strenger et al. for different numbers of thermal transfer units, N = UA/wc,indicated the presence of a maximum in the thermal figure of merit with respect to the application of the heat exchanger as a catalytic incinerator for air, in which case the fluid flows first inward and then, after being heated incrementally, outward. This criterion is W C ( T- ~Ti) =-T2 - Ti E= (1) Q T3 - T2 where, as shown in Figure 1, T2- Ti is the increase in the temperature of the inwardly flowing stream due to heat transfer and T3- T2is the temperature increase in the fluid resulting from the imposition at the core of an external source of energy Q such as an electrical current through a coil.
* To whom correspondence should be addressed.
t Current address: William B. Retallick Associates, 1432 Johnny’s Way, West Chester, PA 19382.
In conventional heat exchangers in which the “entering” temperatures TIand T3are specified rather than Ti and the difference T3 - T2,the thermal effectiveness
is more commonly chosen as the measure of performance. , in a double-spiral catalytic incinFor ( w c ) , = - ( w c ) ~as erator, eq 2 reduces to (3)
Another related dimensionless quantity has also been used as a criterion of performance for noncountercurrent heat exchangers, namely, the correction factor F, which equals the ratio of the total amount of heat transfer to that for a true countercurrent heat exchanger. Since the latter equals U A ( A T ) , , (4)
For the operation of Figure 1and negligible losses to the surroundings,(AT), = T3- T2= T4- Ti,and eq 4 reduces to (5)
In all of the above expressions the heat capacities are implied to be mean values and in the event of eqs 3 and 5 to be equal for the two streams. For conditions such that eqs 3 and 5 are applicable, the quantities E, E, and Fobviously following from one another. As shown subsequently, E is a much more sensitive measure of the performance than e or F. Prior experimental work for double-spiral heat exchangers has almost wholly been for turbulent flow, and generally has been miscorrelated in terms of a mean overall
0 1992 American Chemical Society oaaa-5aa5/92/2~31-0~5a~03.00/0
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 659 lnsulation
Exif S t r e a m ,
x lnsulation
+
I
Cold
Figure 1. Schematic of a double-spiral heat exchanger with opposing inlet and outlet (in this sketch np = 1).
heat-transfer coefficient based on a log-mean difference in temperature rather than in terms of E, e, or F. The resulting values reported for the mean overall heat-transfer coefficient have, not surprisingly, demonstrated considerable scatter and poor correlation with respect to the better known results for straight channels. A number of theoretical solutions, generally by numerical methods, have been derived for discrete numbers of turns, particular peripheral locations of the inlets and outlets, and a range of values of N . Such results have generally been expressed in terms of e or F. Since this prior experimental and theoretical work has recently been reviewed by Strenger et al. (1990), only those features which relate directly to the solutions developed herein will be mentioned. The numerical solutions of Strenger et al. (1990) predict a maximum in E (and therefore in e) with increasing N and a possible but undetermined finite value of E (and therefore of e) for asymptotically large N . These two important characteristics were not identified in any of the prior experimental and theoretical work, although a plot of the computed values of Buonopane and Troupe (1970) suggests a possible maximum in e beyond the range of their calculations and one of the corresponding curves of Jones et al. (1978) appears to go through a maximum. Bes (1987) inferred from numerical solutions for exchangers with both inlets and outlets at the same angular location that e (and therefore E) approaches a maximum value asymptotically as N increases. Chowdhury et al. (1985) [also see Martin et al. (1986) and Martin (1988)l correlated their extensive computed values for a number of geometries (numbers of turns for flow and locations of the inlets and outlets) very successfully with the equivalent of E = nf tanh \ N / n f ) (6) which corresponds to the exact solution for a countercurrent cascade of identical cocurrent heat exchangers. Equation 6, which was based on and proposed only for relatively small values of N / n f ,predicts an asymptotic value of E = N for small N j n f and, beyond its proposed range of applicability, an asymptotic value of E = nf for N l n f -. It does not predict a maximum in E. The objective of the work presented herein has been to test critically the predictions by Strenger et al. of the existence of a maximum and of a possible finite asymptotic value of E for a double-spiral heat exchanger, when operated as shown in Figure 1, by developing solutions in closed form for limiting and idealized conditions.
-
=f
.
Y t;k)
x=l
x=o
Figure 2. Notation for model of a one-turn spiral exchanger with opposing inlet and outlet (spiral 1).
Mathematical Models The following primary idealizations were made in constructing the mathematical models whose solutions are presented herein: (1) fully developed flow of both streams throughout the exchanger; (2) fully developed forced convection throughout the exchanger; (3) uniform and equal physical properties for both streams; (4) negligible radiative heat transfer; ( 5 ) negligible end effects, except for heat losses, due to the finite axial length of the passages; (6) negligible thickness, and hence negligible thermal resistance, for the curved surfaces; (7) negligible thermal conduction in the direction of flow in either the fluids or the walls; (8) negligible heat losses from the outer curved surface to the surroundings; (9) negligible heat exchange between the inner curved surface and the fluid in the core; (10) negligible variation of the local overall heat-transfer coefficient throughout the exchanger (here "overall" implies the coefficient for exchange between the fluids in adjacent passages); (11)negligible variation of the area for heat transfer per unit length of passage through the heat exchanger, which implies that the turns have a very large radius of curvature with respect to the spacing between the walls; (12) representation of the heat losses from the end walls in terms of an overall coefficient between the fluid in each passage and the surroundings (conduction through the spiral walls to the end plates is implied to be encompassed by this coefficient, and radial conduction in the end plates is implied to be negligible). One-Turn Exchanger with Opposed Inlet and Outlet A double-spiral heat exchanger with one complete turn (nf = 1) for flow in each direction and with the external inlet and outlet on opposite sides is illustrated in Figure 2. Analysis of this diagram indicates that three complete turns of metal (n, = 3) are required for its construction and that the total surface area A through which heat is transferred consists of one turn (n, = 1). The designation of the size of the devices considered herein is arbitrarily based on the latter quantity, that is, the number of turns of metal through which heat is transferred from one stream to the other. Model. Separate energy balances for each fluid must be written for each half-turn owing to the different conditions thereof. These balances can be expressed in dimensionless form as
660 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
(7) 4*0
1
(9)
with the boundary conditions t’l(0) = 0 t’l{72J = ti(721
T’1172) = till) + 1
0.0
’ 0.
I
N
T’i(0) = T’2(1)
I
I
a.
=m wc
10.
1
s.
I 20.
Figure 3. Thermal figure of merit for spiral 1 (A)and for a true
Here
countercurrent exchanger (M) with no heat losses.
47radU0- 2dU0 N = -2 7 r a ~=- -U A a=-=wc wc ’ 27ralU 1U Solution for No Heat Losses to the Surroundings. For the limiting case of perfect insulation on the ends of the double-spiral heat exchanger, a in the above model is simply set to zero. The solution of this reduced model, which can readily be obtained by elementary methods, is N(2N 4)x t; = (15)
+
w+4 (2W)+ N ( - W + 4 ) ~ t’z = w+4 (2N + 4 ) + N(2N + 4)x
(16)
T‘1 =
Ti =
W+4 (W - 2N + 4 ) + N(-2N
w+4
+4
(17) ) ~
(18)
If follows that
E
t’,(lJ =
4N w+4
Setting the derivative of E with respect to N to zero indicates a maximum value of E = 1 at N = 2. For asymptotically small N the dependence approaches that for a true countercurrent exchanger, namely E=N (20) while for asymptotically large N E -,4 / N (21) which goes to zero in the limit. The complete dependence of E on N is plotted in Figure 3 along with the above asymptotes. The maximum value of E for the doublespiral exchanger is only half the value for a true countercurrent exchanger at the same value of N = 2. (Martin and -workers (Chowdhwy et aL, 1985, Martin et al., 1986; Martin, 1988)) derived the equivalent of eq 19 but did not note the consequence mentioned above.) The explanation for this behavior is provided by the temperature profiles which are described immediately below. The temperature profiles are shown for N = 1, 2, and 20 in the top row of Figure 4. In Figure 4 for N = 1 and
a = 0, a large, uniform temperature difference, even larger than the dimensionless temperature increase of unity in the core, exists over the first half-turn from the outside while a uniform dimensionless temperature difference of less than unity occurs over the second half-turn. The average of these two dimensionless temperature differences is less than unity, which results in the slightly lower figure of merit (E = relative to that for a true countercurrent exchanger (E = 1/2). In Figure 4 for N = 20 and a = 0, a very small, uniform temperature difference is observed to occur over the first half-turn from the outside and a sliihtly smaller, negative difference over the second half-turn. Most of the heat transferred to the entering stream in the first half-turn is returned to the exiting stream in the second half-turn because of this crossover of the temperatures. Thus, increasing N by increasing the surface area per turn, increasing the heat-transfer coefficient, or decreasing the rate of flow degrades the performance in this regime. In Figure 4 for the optimal number of transfer units, N = 2 and a = 0, the dimensionless temperature difference is seen to be unity in the first half-turn from the outside, just as in a true countercurrent exchanger, but zero in the second half-turn, thereby accounting for the value of E = 1as compared to E = 2. The optimal number of transfer units in the double-spiral exchanger of one turn corresponds to the incipient condition for crossover of the temperatures of the two streams in the second half-turn. Solution for Finite Heat Loss to the Surroundings. The above solution for a one-turn double-spiral heat exchanger reveals that such a device is inferior in thermal performance for the same number of transfer units to a true countercurrent heat exchanger in the idealized case of negligible heat losses. As illustrated below, this inferiority in thermal performance for negligible heat losses also occure for double-spiral heat exchangers with multiple turns. However, double-spiral heat exchangers of sufficient multiple turns prove to be superior if a realistic allowance is made for the effect of heat losses, aa now considered. Although a solution of the above model for a finite value of a can be obtained by the method of elimination, the process is very tedious. Accordingly, the MACSYMA (Symbolics, Inc., Cambridge, MA) code, which carries out mathematical operations symbolically, was u t i l i for the manipulations. The resulting solution, which is somewhat detailed, is given in the Appendix under the label spiral 1. Illustrative results obtained by numerical evaluation
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 661 spiral
N-1.0
1 rt-0.00
1
splral 1 a-0.00
'*O
splral 1 a-0.00
N-2.0
7-T
c en
-
0.0
E
0.0
X = number 01 turns through spiral
X = number of turns through spiral
0.0
0.0
-
0.2
splral
0.0
0.4
1 a-0.01
0;.
1;O
I
I
I
I
I
0.2
0.4
0.0
0.8
1.0
- spiral
N-1.0
';I-"'" -/ y-y-/ c e-
c e-1.0
-
4 -
1.0
X = number of turns through spiral
,.o
-
y-fO
e e-1.0
3.0
._spiral
1 a-0.10
N-1
- spiral
-
1.0
-
I- I-" y-fO-/ '.O
-
// -
2.0
-
1.0
t
0.0
I
I
I
5.0
_I
0.0
1 a-0.10
N-1
t
0.8
1.0
-
spiral 1 a-0.01
N-20.0
1.0
x=
,.o
number of turns through spiral
- spiral
1 a-0.10
N-20.0
?-To - -
I- h"
1.0
1 a9l.00 N-0.S
-
0.0
c-"
.5
-
0.4
qf-- L-
--
0.2
X = number O f turns through spiral
N-1.9
4
- rplrol
c e" fl
-
,.o
:-lY
-
spiral
,eo
1 a=l.00 N10.4
7-T-
c e-
-
1 a-0.01
0.0 0.0
X = number of turns through spiral
.O
-/ /
'*o
c
1.0
N-20.0
,.o
- spiral
2.0
-
1
a i l .OO N - 1 5 . 0
- j /
-
of this expression are plotted in Figure 5. The effect of heat losses, as represented by a,is seen to result in the same qualitative dependence of E on N but to reduce the maximum value and to shift ita occurrence to smaller values of N. "he corresponding temperature profiles are illustrated in the lower three rows of Figure 4 for a = 0,0.01,0.1,and
"
1
1
1.0
0.0
:
1.0 and for N less than, equal to, and greater than the optimal value. The effect of the heat losses is seen in Figures 4 and 5 to be greatest for large N.
Solutions for Multiple-Turn Exchangers Closed-form solutions for the four multiple-turn exchanger sketched in Figure 6 (spirals 2-5), with and
662 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 *.O
3.0
1
I
*'O
0.0
s. N
0.
=m wc
IO.
1
s.
ZO.
-
'
I
0.
5.
I I 0.
N
Figure 5. Thermal figure of merit for spiral 1 with heat losses [a = 0.00(A),0.01 (01, 0.10 (m), and 1.00 (V)].
Splral
1 t
7
Splral 2
outlet
Spin/ 3
splrrl 4
(N + 2fi)eTm/6]]/[ Spiral
6 (true countercurrent)
without heat losses, were also obtained using MACSYMA. The results are summarized below. For spiral 2, which consists of one and a half turns for flow in each direction and two complete turns for heat transfer, the figure of merit for no heat losses is given by
E=
4[eNJ2- 11
+
5eNI2- 8eNI4 5
As illustrated in Figure 7, E for spiral 2 goes through a
s.
20.
maximum of 4 / 3 at N = 4 In 2 = 2.77 but, in contrast to spiral 1, approaches a finite asymptotic value of 4 / 5 for large N. The corresponding temperature profiles for N = 1,2.77, and 20 are plotted in Figure 8. The upper curves starting at x = 0 and x = l J 2 represent the "inner" and "outer" exiting streams, respectively, as seen by the entering stream at each position. The outer exiting stream is of course simply the inner exiting stream displaced by one turn. In Figure 8 for N less than the optimal number of transfer units, N = 1 and a = 0, the entering stream exchanges heat on its inner wall over the first half-turn from the outside, on both of its walls in the second halfturn, and on its outer wall in the third half-turn. In Figure 8 for the optimal number of transfer units, N = 2.77 and CY = 0, a crossover of temperatures occurs over part of the second and third half-turns. Over the first portion of the second half-turn and the latter portion of the third halfturn, the entering stream is heated on both its inner and outer walls, but in between it loses heat to the outer exiting stream. In Figure 8 for a large value of transfer units, N = 20 and CY = 0, the temperature profiles are quite irregular, and the entering stream loses heat to the exiting stream through its outer wall over most of the second and third half-turns. For spiral 3, which consists of two turns for flow in each direction, and three total turns for heat transfer, the figure of merit for negligible heat losses is given by
[
Spiral 5
I
I ?
Figure 7. Thermal figure of merit for spirals 1 (v),2 (o), 3 (m), 4 (A),5 (e), and 6 ( 0 )with no heat losses.
E = 24[ (N + 2fi)efwl2
Figure 6. Designation of spiral configurations.
=E wc
efW13
+ (y ) e f W i 3 + (W + 4 f i
+ 72)efW/2 +
+ (W + 4 f i N + 72)erwIe
1
(23)
As illustrated in Figure 7, E for spiral 3 goes through a maximum of 1.82 at N = 3.9 and approaches zero as N increases. The corresponding temperature profiles for N less than, equal to, and greater than the optimal value are plotted in Figure 9. The behavior is qualitatively similar to that for spiral 2, except that heat is lost outwardly from the entering stream over the entire inner turn and a half. This difference is associated with the relative locations of the inlet and exit for the entering stream, which are the same for spirals 1and 3 but opposite for spiral 2.
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 663 *.O
1.O
0.0
1
2 a=O.OO
spiral
N-1 .O
I.0
1
rpiral 2 a-0.00
rpiral 2 a-0.00
N-2.77
EE
N-20.0
1 .O
0.0
0.s
*.a
1.0
0.0
2.0
X = number of turns through spiral
0.0 0.0
os os
*.a *.a
1.0 1.0
y 0.s
0.0
2.0 2.0
0.0
X = number o f turns fhrough spiral
1.a
1.0
P.O
X = number 01 turns through spiral
Figure 8. Temperature profiles of entering stream (+) and exiting stream (+) for spiral 2 with three values of N = UA/wc and no heat losses. spiral 3 a-0.00
*.O
0.0
2.a X = number 0 1 turns through spiral
0.0
0.9
1.s
1.0
2.0
4.0
E7y c-
1.o
0.0
1
N1ZO.O
I--2.0
1.o
0.0
0.3
1.0
1.a
2.0
2.s
0.0 0.0
0.s
1.0
1.a
2.0
a.s
X = number of turns through spiral
X = number ot turns through spiral
Figure 9. Temperature profiles of entering stream (--) and exiting stream (+) for spiral 3 with three values of N = UA/wc and no heat losses.
Solutions are not possible with MACSYMA for more than three turns of heated surface with the outer inlet and exit on opposite sides. Solutions are possible, however, for up to five turns of heated surface if the outer inlet and exit are on the same side of the spiral, because energy balances can then be written for full turns rather than half turns. The behavior for one turn with the inlet and exit at the same angular position, labeled spiral 6 in Figure 6, is, for the idealized conditions postulated herein, the same ea that for a true countercurrent exchanger with equal rates of flow. For negligible heat losses to the surroundings
-. T,
Figure 10. Schematic of a true countercurrent parallel-plate exchanger.
I 0.
I
3.
N
E=
1
B coth {j3N + a
(29)
where 0 = [a(2 + a)]1/2.This expression for E increases ~ p) for any a. monotonically with N from N to I / ( + Equations 27-29 are also applicable for a parallel-plate exchanger, such as that illustrated in Figure 10, if a
=
u,/u
(30)
=m wc
10.
I
ia.
I
10.
Figure 11. Thermal figure of merit for a true countercurrent exchanger with heat losses [a= 0.00 ( O ) , 0.01 (A),0.10 (W), and 1.00
(Wl.
A plot of E 88 a function of N is shown in Figure 11 for a = 0,0.01, 0.1, and 1. For spiral 4 with two turns in each direction for flow and three turns of surface for heat transfer, just as for spiral 3, but with all of the inleta and outlets at the same angular
664 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
temperatures of the entering stream and of the outer exiting stream are equal or almost equal on the mean. As a consequence, the area for heat transfer to the entering stream is effectively reduced to that of the inner wall, that is, it is cut in half. The number of transfer units N as defined herein is based on the active area for heat transfer through both walls. This definition was chosen in order that the figure of merit E converges to that for a true countercurrent exchanger for N 0. In a true countercurrent exchanger, as illustrated in Figure 10 and for spiral 6 in Figure 6, heat is transferred to the entering stream through only one of the two walls. If a double-spiral exchanger is compared with a true countercurrent exchanger on the basis of only the inner wall of the entering passage, N is reduced by a factor of 2 and all of the curves of E versus N , except for the true countercurrent exchanger (spiral 6) are shifted to the left by that factor. For small N the double-spiral exchanger then has a higher value of E, and for the optimal value of N , an equal or only slightly lower value. The existence of a maximum in E for a finite value of N means that a further increase in the overall heat-transfer coefficient, a further increase in the average circumference of the turns, or a further decrease in the mass rate of flow will actually degrade the performance. This aspect of the behavior is particularly significant because it is counterintuitive. Also, the existence of an optimal value of N implies a criticality in the design and operation of a double-spiral exchanger. The occurrence of a finite asymptotic value of E for increasing N is shown above to depend on the relative location of the inner and outer inlets and exits. The necessary condition for a finite asymptotic value of E appears to be the location of the inner and outer inlets at the same angular position. This result is of less practical importance than the confirmation of the existence of an optimal value of N but is nonetheless of intrinsic interest. The asymptotic values of E for large N , which follow directly from the solutions in closed form, are difficult if not virtually impossible to determine by finite-difference calculations, as already noted by Strenger et al. (1990). The temperature distributions shown in Figures 8,9,13, and 14 for spirals 2, 3,4, and 5 , respectively, are surprisingly irregular. Such behavior is primarily a consequence of the crossover in temperature, if one occurs, but also a result of heat transfer on only one side of two half-turns or one complete turn of both the entering and exiting streams, depending on the location of the inlets and outlets. As the number of turns increases, the irregularities arising from the outer and inner channels and half-channels which experience heat transfer on only one side becomes less significant, as suggested by the temperature distributions for spiral 5, and can be expected to be confined to the outer axid inner turn or two. The temperature differences for the intermediate turns can be expected to be relatively uniform, although not of course equal on the inner and outer sides of a passage. Effect of Idealizations in the Model. The solutions derived herein are applicable for any N = UA/wcinsofar as the product U A does not vary through the exchanger. Changes of E with N per the equations and curves herein can be implemented by varying A, w,and/or U and do not necessarily reflect a single change in operating conditions or design. For example, for a given design, a point on a curve of E versus N at large N may correspond to laminar flow whereas a point on the same curve at small N may correspond to turbulent flow owing to a larger value of w. Similarly, the same point on a curve may correspond to
-
0.0
9.
0.
7 0.
ts.
20.
N =!&
WC
Figure 12. Thermal figure of merit for spiral 4 with heat losses [a = 0.00 (o), 0.01 (A),0.10(B),and 1.00 (Wl.
location, the figure of merit for negligible heat losses is given by
E=
+ 6 tanh (&V/6l + N tanh { f i N / 6 )
2fiN 3fi
(31)
As indicated in Figure 7, for spiral 4 with no heat losses E increases monotonically (in contrast with all of the other multiple-turn spirals) to an asymptotic value of 2 4 2 = 2.83. The solution for E for finite heat losses is deferred to the Appendix because of its detail, but representative values are plotted in Figure 12. Surprisingly, a maximum is observed for all of these finite values of a. The corresponding temperature distributions for small, intermediate, and large values of N are plotted in Figure 13 (for a = 0 only two temperature distributions are plotted due to the monotonic increase of E with N. Spiral 5 has the same configuration as spiral 4 but with three turns in each direction for flow and five total turns of surface for heat transfer. For this configuration, even the solution for E for negligible heating is deferred to the Appendix because if its complexity. As indicated in Figure 7, a maximum value of E = 3.16 occurs at N = 7.7 and the asymptotic value for large N is (5d3+ 6 ) / ( 2 ( 1 +d3)) = 2.683 for no heat losses. The corresponding temperature distributions for N less than, equal to, and greater than the optimal value are plotted in Figure 14. Discussion Significance of Results. The closed-form solutions presented above confirm the existence of an optimal value of N at which a maximum in E is attained. The only exceptions encountered in this investigation were for spiral 4 in the limiting case of negligible heat losses and in the trivial case of spiral 6, which degenerates to a true countercurrent heat exchanger when the effects of curvature are neglected. The solutions for the temperature profiles reveal that the finite, optimal value of N is a consequence of a crossover of the temperature of the entering stream and that of the exiting stream in the adjacent outer passage. For N less than the optimal value, the entering stream in heated from both sides. For N greater than the optimal value, the entering stream is heated on its inner wall but transfers some of this energy back to the exiting stream through its outer wall. A t the optimal value of N the
Ind. Eng. Chem. Res., Vol. 31,No. 3,1992 666
1
spiral
4
a 90.00
N-1 .O
{spiral
4 a90.00
l.oe *.O
4.0
N-20.0
p2.0
-
1.O
0.0
0.0
1 .o
0.0
2.0
1
splral
*.O
4
a -0.01
2.0
X = number o f turns through spiral
~ .o - i
aplrol 4 a-0.01
N-6.2
aplrai
4 a=O.oi
N120.0
I
I I 1
0.0
1.o
2.0
0:o 1 .o 2.0 X = number of turns through spiral
X = number of turns through spiral
4.0
n
1 .o
0:o
X = number of turns through spiral
- splral
4 tr9O.lO
aplral
4 a-1 .OO
N - 1 .O 4.0
spiral 4 a-0.1 0 N - 3 . 0
i
aplral 4 a-1.00
N-1 .O
2.0
-
1.o
0.0
2.0
X = number of turns through spiral
4.0
N-1.2
- aplral
i
spiral
*.O
4 a-0.10
4
a= 1.OO
N-20.0
N-20.0
f2.0
1 .o
0.0
0.0 0.0
1 .o
2.0
0.0 0.0
1.o
2.0
0.0
1 .o
2.0
x = number of turns through Spiral Figure 13. Temperature profiles of entering stream (+) and exiting stream (+) for spiral 4 with values of N = UA/wc and heat losses aa indicated. X = number of turns through spiral
X = number of turns through spiral
laminar flow for one geometry and to turbulent flow for another. In practice, some variation in UA through the exchanger can be expected owing to development of the velocity profile and the dimensionless temperature profile, to changes in the ratio of the heat flux densities on the opposing sides of a channel, and to intensification of the degree of secondary motion and a decrease in the area per
unit distance in the inward direction. These deviations from ideality can be expected to be most significant near the ends of the passages and for heat exchangers with a high degree of curvature (a small core). Radiative transfer, which was neglected completely in the modeling, can be reasoned to decrease the performance, i.e., to lower E, since some energy is then transferred outward from the core to the periphery without incrsasing
666 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 5.0 4.0
]spiral
s a-0.00
~
9 .O 1
.O
0.0
4.0
-
1.0
-
0.0 0.0
1 .o
2.0
3.0
X = number of turns through spiral
5 a-0.00
5.0 -spiral
N-7.7
I I 1 .o 2.0 3.0 X = number o f turns through spiral
0.0
9
SO
-spiral
4.0
-
0.0
b a-0.00 N 9 2 0 . 0
* .O
2.0
SO
X = number of turns through spiral
Figure 14. Temperature profiles of entering stream (+) and exiting stream (+) for spiral 5 with three values of N = UA/wcwith no heat losses.
the temperature of the entering fluid proportionately. Accounting for radiation would grossly complicate the analysis, not only due to the fourth-power dependence on temperature, but also because energy balances would then need to be written for the walls as well as for the fluid streams. Energy balances for the walls were avoided in the above modeling by the implicit invocation of Ohm’s law for heat transfer from one stream to another, i.e., by the use of an overall local coefficient for convective heat transfer. In their rigorous form the energy balances for the walls would be integrodifferential equations. The integrals might, however, be avoided by neglecting the variation in the temperature of an opposing wall over the field of view of any point on the surface, including reflections. In any event, neglect of radiative transfer was obviously necessary to obtain solutions in closed form. Although the neglect of radiation results in an overestimate of E, particularly for operation at high temperatures, the dependence of E on N, a,and the location of the inlets and outlets would not be expected to change qualitatively. The analyses herein imply axial uniformity in the temperature of the fluid and walls and in the velocity of the fluid. Some axial gradient in temperature must exist owing to the heat losses from the end walls and the lower velocity in their vicinity. Some loss in heat transfer between the fluid streams must result, but such minor effects can probably be compensated for by a slight increase in the effective overall heat-transfer coefficient from the fluid through the end walls to the surroundings. A more severe effect is radial conduction in the end walls owing to the temperature gradient from the core to the outer curved surface. The process degrades the performance as does thermal radiation. The above-mentioned heat-transfer coefficient for heat losses may be increased for compensation. Heat losses from the outer curved surface to the surroundings were neglected in the modeling because their accounting precludes the attainment of solutions in closed form for double-spirals of multiple turns. Conventional wisdom suggests that heat losses from an exchanger can be reduced to perhaps 5% of the heat transferred by the use of thermal insulation on the external surface. A much lower fractional of less than say 1% would be expected for a double-spiral exchanger since the outer surface is at a relatively low temperature. However, in most of the applications cited for double-spiral exchangers, the relevant measure of the heat losses is as a fraction of the heat input at the core rather than aa a fraction of the heat transferred. The fractional heat loss on the conventional basis is thereby multiplied by E. For an exchanger of many turns, for which E may be as high as 10 or 20,even a 1% loss on the conventional basis would be very significant. Ac-
cordingly, heat losses from the outer curved surface, although not significant in the exchangers of a limited number of turns and hence a small value of E as analyzed herein, should also be taken into account, presumably by numerical analysis, for exchangers of many turns and large
E. Comparisons with Prior Solutions. The numerical solutions of Chowdhury et al. (1985)incorporate all of the idealizations made herein except that curvature was taken into account in the area per turn by choosing the ratio of the width of a passage to the inner diameter d / r to be 0.06. However, they assert that these results differed by less than 1%from those in which curvature was neglected. Their calculations were for all combinations of nr = 3,4, 6,8,12,and16andN=1/2,1,2,3,4,5,6,7,8,9,and10, with inlets and exits both as in spiral 1 and spiral 4. Their tabulated values agree closely with the predictions of the expressions herein insofar as the conditione coincide. However, the predictions of eq 6 differ considerably from the solutions herein even over its purported range of validity, particularly when the comparison is made in terms of E rather than F. Equation 6 does not predict a maximum in E even when one occurs within ita purported range of validity. Finally, the predicted dependence of F on N/n, alone is refuted by the solutions herein, even within its purported range of validity. The numerical solutions of Ci6slidski and Bes (1983) appear to incorporate the same idealizations as made in the closed-form solutions herein. Their results, which are for an opposing outer inlet and exit, are presented graphically in plots equivalent to (N/(1 + N)) - (E/(1+ E ) ) versus N from 0 to 6 with nr = 2, 4,6,8, 12, and 16 as a parameter. These values appear to be in only fair agreement with eq 23,which represents the only condition of overlap. Ci6slifiski and Bes assert that the behavior of the double-spiral differs negligibly from that for a true countercurrent exchange since their maximum computed value of ( N / ( N + 1))- (E/(E+ l)),which occurs for nf = 2 and N = 6, is only 0.16. This comparison is somewhat misleading in that the corresponding values of E for the double-spiral and the true countercurrent exchanger are 2.26 and 6,respectively. The anomalous behavior (the absence of a maximum) for spiral 4 with no heat losses is apparently an artifact of the small portion of the heat-transfer surface for which a crossover in temperature occurs. No prior solutions were identified that take heat losses to the surroundings into account. Conclusions For double-spiral heat exchangers of several turns operated with heating at the core and reversed flow, an op-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 667 timal value of the number of transfer units, N , exists for which the figure of merit, E, is a maximum. For very large N the figure of merit may approach zero or a finite value, depending on the configuration of the inlets and exits. For double-spiral exchangers of only a few turns, the temperature profiles are very irregular. As the number of turns increases, this irregularity is confined to the outer and inner turns. In the idealized case of negligible heat losses to the surroundings, the performance of a double-spiral heat exchanger is inferior to that of a true countercurrent exchanger for the same number of transfer units. If the number of transfer units is based on the total surface area through which heat is transferred, the maximum value of E for a double-spiral exchanger is approximately half that for a true countercurrent exchanger. If realistic losses of heat to the surroundings are taken into account the figure of merit for a true countercurrent exchanger approaches a fairly constrained asymptotic value as N increases, whereas the figure of merit of a doublespiral exchanger at the optimal value of N increases indefinitely as the number of turns is increased. This superior performance results from the exposure of only the outer curved surface and the end walls of the double spiral to the surroundings as compared to the entire surface for the true countercurrent exchanger. Although the comparisons herein are with a parallel-plate, countercurrent exchanger, the same relative behavior would be expected for a conventional double-pipe exchanger. Prior investigators, other than Strenger et al. (1990), apparently overlooked the existence of an optimal rate of flow (an optimal number of transfer units) because of confinement of their studies, both experimental and theoretical, to values of N less than that required for such behavior. This limitation of prior work to small values of N was because of the perceived thermodynamic advantage of attaining a value of F which does not differ greatly from unity. However, as indicated by eq 20, a small value of N limits the maximum rise in temperature which can be attained. The rise in temperature is more important than the thermodynamic efficiency in the applications cited herein, such as catalytic incineration. The derivation of solutions in closed form for idealized conditions is demonstrated in this instance to reveal qualitative and even quantitative aspecta of behavior which were not apparent from experimental work and numerical solutions for discrete conditions. The MACSYMA code proved useful even though the conditions for which solutions could be obtained were quite restrained. The choice of variables for presentation of the results proved critical in this instance. The insensitivity of the correction factor F and the effectiveness t as compared to the figure of merit E was responsible in part for the failure of prior analysts to detect the optimal value of N even though it was encompassed in their computations. Furthermore, the choice of F as a dependent variable is responsible for the somewhat misleading correlation of computer values by eq 6, and the choice of e is responsible for the erroneous conclusion that the behavior of doublespiral exchangers does not differ significantly from true countercurrent ones.
Acknowledgment This work was supported by the National Science Foundation under SBIR Grant IS1 8619609. We appreciate the perceptive and constructive comments of all three reviewers.
Nomenclature A = total area for heat transfer between fluid streams, m2 a = mean radius of spiral, m a,, = inner radius of spiral, m
c = specific heat capacity, J/(kg.K) d = spacing between spiral walls, m E = (T2- T,) / (2'3 - T2)= thermal figure of merit F = wc(T2- Tl)/UA(AT)h = correction factor for deviation
from log-mean temperature difference 1 = axial length of double spiral, m N = U A / w c = number of thermal transfer unita nf = number of turns through which entering (and exiting)
fluid passes n, = number of turns of metal in double spiral n4 = number of turns through which heat is transferred from one fluid stream to the other q = heat flux imposed at core, W w = mass rate of flow, kg/s T = local mixed-mean temperature of exiting stream, K T,= ambient temperature, K T2 = temperature of stream entering the core, K T3 = temperature of stream leaving the core, K T4= temperature of stream leaving the double spiral, K t = local, mixed-mean temperature of entering stream, K T'= (T - T1)/(T3- T2)= dimensionless, local, mixed-mean temperature of exiting stream t = ( t - T,) / (2'3 - T2)= dimensionless, local, mixed-mean temperature of entering stream (AT), = logarithmicmean temperature difference between the entering and exiting streams X = number of turns through spiral z = fraction distance through a turn of the double spiral traveled by the entering stream U = local, overall coefficient of heat transfer between adjacent fluid streams, W/(m2.K) U,= local, overall coefficient of heat transfer between a fluid stream and the surroundings, W/(m2.K) Greek Symbols a = 2d Us/ 1U for ends of double spiral, =
Us/U for parallelplate exchanger = [a(2 + cY)]'/2 t = (T2- T1)/(T3 - TI) = thermal effectiveness of a doublespiral heat exchanger
Subscripts c = colder (entering) stream min = minimum value
Appendix Spiral 1 with heat losses from ends:
668 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
Spiral 5:
where here only
r = +(CY + 2) Spiral 4 with heat losses from ends:
[(
15t-18)N-(1845-18)
(*
1
e(zJF5W
+
+
N + 3 6 6 ) e[(2+\r9ysW + (126A4
(*N+36)e(1'5)N+
[("i
('5$+18)N-(18-6+18)
+84)
(+N
[
za4
+
+~
( - 2 ~ 3
- 4a2 - 2a) + &(-e)+
l6 ) N
+ 5 6 -61 +
11
+
(e) N
+ ( 5 4+ 6)l
Spiral 6 double-pipe with heat losses:
em& - 1
E=
+ file"& + (-a + fi)
(a
where here only
r
= a2+ 4a + 2
+ 2a 8 = 2a3 + 6a2 + 6a + 2 A = 2a2 + 4a + 3 A = a2
T = 2a3 + 8a2+ loa + 4
a = 1 0 ~ 3+ 1 5 ~ 2+ 7a + 1
+ 6a + 2 ! l = 4a2 + 8a + 4
@ = 6a2
where here only A = a2
+ 2a
Literature Cited Bes, T. Eine Methode der thermischen Berechnung von Gegen- und Gleichstrom-Spiralwarmeaustauschern. WbrmeStoffiibertragung 1987,2I,301-309. Buonopane, R. A,; Troupe, R. A. Analytical and Experimental Studies in a Spiral Heat Exchanger. R o c . Fourth Znt. Heat Transfer Conf. (Paris-Versailles)1970,I , 1-11 (paper HE 2.5). Chowdhury, K.; Linkmeyer, H.; Bassiouny, K.; Martin, K. Analytical Studies on the Temperature Distribution in Spiral Plate Heat Exchangers: StraightforwardDesign Formulae for Efficiency and Mean Temperature Difference. Chem. Eng. Process 1985, 19, 183-190.
Ind. Eng. Chem. Res. 1992,31,669-681 Cikliiiski, P. J.; Bes, T. Analytical Heat Transfer Studies in a Spiral Plate Exchanger. R o c . XVZth Int. Congr. Refrig. (Paris) 1983, IZ, 449-454 (paper B.1-198). Jones, A. R.; Lloyd, S. A.; Weinberg, S. A. Combustion in Heat Exchangers. Proc. R. SOC.London, Ser. A 1978, 360, 97-115. Martin, H. Wdrrneiibertrager; Georg Thieme Verlag: Stuttgart, Federal Republic of Germany, 1988. Martin, H.; Chowdhury, K.; Linkmeyer, H.; Bassiouny, M. K. Straightforward Design Formulae for Efficiency and Mean Temperature Difference in Spiral Plate Heat Exchangers. Proceedings of the Eighth International Heat Transfer Conference (Sun Francisco);Hemisphere Publishing: Washington, DC, 1986; Vol.
669
VI, pp 2193-2797. Minton, P. Designing Spiral-Plate Exchangers. Chem. Eng. Progr. 1970, 77, 103-112. Strenger, M. R.; Churchill, S. W.; Retallick, W. B. Operational Characteristicsof a Double-Spiral Heat Exchanger for the Catalytic Incineration of Contaminated Air. Znd. Eng. Chem. Res. 1990,29, 1977-1984. Received for review January 22, 1991 Revised manuscript received August 5, 1991 Accepted August 13, 1991
Decay of Turbulence in a Tube following a Combustion-Generated Step in Temperature Lance R.Collins+and Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
An essentially abrupt change in temperature owing to thermally stabilized combustion produces a correspondingly abrupt decrease in Reynolds number from the 3000-6000 range down to 650-1550 for an ethane-air flame in a 9.5-mm channel. Since the Reynolds number downstream from the flame front is below the transitional value for a tube (approximately 2100), the turbulence decays with distance from the flame front. The decay of turbulence and the consequent development of a parabolic velocity profile were investigated theoretically using a modified k-t model of Jones and Launder. The predictions of the velocity a t the centerline and axial gradient of pressure are in qualitative agreement with experimental results throughout the system, although the predicted approach to the asymptotic value for laminar flow was faster than that which was observed. This discrepancy identifies a nonphysical characteristic of the modified k-c model which was utilized in the current study.
Introduction Theoretical and experimental studies of the laminarization of turbulent flows go back to the early investigations of the decay of turbulence behind a screen in a wind tunnel (for a review of that behavior see Batchelor (1953) or Hinze (1975)). Because the kinetic energy of turbulence is continually being degraded to thermal energy as a result of viscous dissipation, there must be a mechanism to convert continually the energy of the mean flow. Since the mean flow behind screens in wind tunnels is generally uniform, and does not include a mechanism for producing additional turbulent kinetic energy, the preexisting energy decays. This decay is usually modeled in terms of a power law in space (or equivalently in time if the frame of reference is moving at the mean velocity). Wall-bounded flows have also been observed to laminarize under certain circumstances. Perhaps the most important example is boundary layer flow undergoing an acceleration due to a favorable pressure gradient (e.g., Pate1 and Head (1968), Launder and Jones (1969),Jones and Launder (1972a,b),Narasimha and Sreenivasan (1973,1979),and more recently Spalart (1986)). Examples of accelerated boundary layers include external flows over curved surfaces and internal flows in converging ducts. The dimensionless grouping that is usually assumed to control laminarization is the parameter of acceleration K. If K is greater than 3 X lo”, the boundary layer will generally revert to a laminar one, implying a decrease in thickness, thereby resulting in very *Author to whom inquiries should be addressed. Current address: Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802.
different rates of heat and mass transfer to the surface. The laminarization of internal flows has also been studied. For example, laminarization of fully developed turbulent flow of a gas in a tube heated at the wall was investigated by McEligot et al. (1970). In this case, the flow laminarizes because the viscosity of the gas increases with temperature and thereby cam the Reynolds number to decrease. They showed that the parameter which determines whether the flow will laminarize is closely related to the parameter of acceleration, K, thereby relating the laminarization in their system to that of the accelerated boundary layer. Laminarization in the thermally stabilized burner (TSB) is closely related to that in the heated pipe of McEligot et al. (1970). However, before discussing the process of interest further, it is worthwhile to provide a general description of thermally stabilized combustion. The TSB consists of one or more ceramic tubes 1cm or greater in diameter, through which premixed ethane and air are fed at one end and hot burned gases exit at the other. Separating the cold reactants from the hot products is a very thin (on the order of millimeters) zone of reaction whose axial location for a fixed fuel/air ratio and rate of flow is determined by an overall energy balance. All combustors recirculate some of the energy released by the reactions back to the cold reactant gases, heating them to the point of ignition. Typically this thermal feedback is accomplished by physical backmixing between burned products and unburned reactants as, for example, with a bluff-body stabilized flame. The TSB, by contrast, accomplishes this thermal feedback by two mechanisms that involve the wall of the channel. Energy from the hot products of combustion downstream from the flame front convects to the wall, which in turn radiates (-60%) and
QS88-5885/92/2631-QS69$03.QQ/Q0 1992 American Chemical Society
