ENGINEERING AND PROCESS DEVELOPMENT
Design Charts for Symmetric Regenerators A. M. PEISER'
AND
J. LEHNERZ N. Y.
Hydrocarbon Research, Inc., New York,
T""
methods of transferring heat between countercurrently flowing gases are by means of the heat exchanger and the heat regenerator. I n a regenerator, as opposed to a heat exchanger, heat is not transferred continuously between the gases. Instead, the gases flow alternately through t,he same passage and heat is alternately stored in and released by the packing. Thus, regenerator operation consists of two phases:
1. The gas-cooling phase, in rvhich the hot gas flows and heat is transferred from the hot gas to the packing 2 . The gas-heating phase, in which the cold gas flom and heat is transferred from the packing to the cold gas After the unit has been in operation for a while, a periodic state is reached in which the amount of heat transferred during a heat.ing or cooling phase is the same. The time interval consisting of two successive phases is called a period. The exchanger may evidently be regarded as the limit,ing case of the regenerator as the period approaches zero. Regenerators are used in high and IOTV temperature applications, which are quite different process-wise. A typical high temperature application is an air preheater in an open hearth furnace. Air is heated from 100" to perhaps 1500" F. against flue gas, which is brought down, for example, from 2100" to 600" F. X o att,empt is .made t,o secure close approach between the gas and air temperatures. The regenerator is used primarily as an economizer. Therefore, crude calculation methods would suffice to set the regenerator design, since only a low thermal efficiency is to be achieved. The low temperature application is quit'e different. Regenerators have been used as a component, of the Linde-Frank1 cycle for making oxygen. Air a t 100" F. is cooled to about -300" F. against its products of distillation, nitrogen and oxygen. The temperature approach is necessarily small (perhaps 10' F . ) in comparison with the gas temperature rise, since large teniperature differences are reflect,ed in an increased refrigeration load, which must be made up by the expander, and appear ultimately as a higher pressure at the discharge of the air compressor. The balance of pumping versus heat exchanger costs favors a close approach and a large heat. exchange surface. The design of the regenerator must be calculated closely, since it contributes P O much to the econoniics of the process. The two applications are tied together by the fact that the same laws of heat transfer and the same physical constraints are valid in both caseep. I n other words, t,hough the two physical situat'ions are quite dissimilar, the mathematical problems (differential equations plus boundary conditions) are identical. After the mathematical problem has been solved and the solution reduced Present address, Consultant, 35 Wing Lane, W-antagh, N. Y. Present address, Department of Mathematics, Unlversity of Ponnsylvania, Philadelphia, Pa. 1
2
2166
to dimensionless charts, the desigiiei n-ill u w the same (.hart for both applications but ~ 1 1 1refer to different regions of it for each case. The mathematical analysiq of the heat transfer perfoimance of an actual regenerator is very coniplicated and would be virtually impossible t o cariy out by usual computing methods. This consideration led to the construction of an idealized model, called the symmetric regenerator In a symmetiic regenerator, the heating and cooling phases are balanced n-ith respect to flow rate5 and physical pioperties of the gases, with the result that the mean gas-to-gas temperature diff erence (averaged over a period) is the same at any point in the regenerator. and the swing in hot gas temperature a t the cold end is equal to the swing in cold gas temperature at the hot end. This can be accomplished, for example, bv supposing that the same gas flows in both phases and that the flow rates and phase durations are the same. I n addition, the following simplifying assumptions are made: The tempeiatures of packing and gai are constant over a section normal to gas flow; both gas and packing have negligible thermal conductivity in the direction of gas flow; the density of the gas is small compared t o packing density; specific heats of packing and gas and gas film heat transfer coefficient are intiependent of temperature condensation or evaporation of diluent from the gas stream docs not take place. no heat is transferred by radiation reversal of gas streamc: is arcomplished instantaneously. The equations of regeneration for the -yrnmetric case have been solved ( I ) , and formulas have been obtained which relate thc four design quantities (see Komenclature): S, P , A T , and 6T. The curves in Figure 1 have been computed on the basiq of these formulas. Some of the formulas are presented later. Examples of the use of Figure 1 are also given JTThere useful relative recultq are obtained for a sample symmetric regenerator. Performance of Symmetric Regenerafor Is Based on Four Design Factors
In order to obtain an idea of how a regenerator operates, consider a two-stream symmetric exchanger working between temperatures TI' and T," and having a total length, L. The temperature profiles of the two gas streams and the packing material will then appear as in Figure 2a. Now, shut off the cold gas and begin to operate the exchanger as a regcnerator. Heat enters the packing, causing its temperature to rise. The hot gas also rises in temperature, and it leaves the regenerator a t a higher and higher temperature as time goes on. The temperatures near the hot end reach the incoming gas temperature first, producing a characteristic flat region as shown in Figure 2h. On reversal and introduction of the cold gas the temperature lines move in the opposite direction, Figure 2c.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Voi. 45, No. 10
ENGINEERING AND PROCESS DEVELOPMENT
S=DlMENSlONLESS SIZE
AT=DIMENSIONLESS MEAN
P=DIMENSIONLESS PHASE
6 T=DIMENSIONLESS
GAS TEMPERATURE DIFFERENCE
SWING IN EXIT GAS TEMPERATURE
SEE NOMENCLATURE
d
Figure 1.
October 1953
Design Chart for Symmetric Regenerator
INDUSTRIAL AND ENGINEERING CHEMISTRY
2167
ENGINEERING AND PROCESS DEVELOPMENT
(d) REGENERATOR: GAS COOLING PHASE. NUMBERS ON CURVES REPRESENT ELAPSED TIME AS PERCENTAGE OF PHASE DURATION.
REGENERATOR : GAS COOLING PHASE. INFINITE HEAT TRANSFER COEFFICIENT.
REGENERATOR: GAS COOLING
-' 0
L
L
' V
Td' PACKING TEMP. REGENERATOR:
IFigure 2. 7
, , . , ) , , , .
Figure 3.
, , . , ( , , , ,
,,/,,,,., ,,,,I,,,,
, / , , I , , , ,
,,,,,,,,,
, , , , I , , / ,
,
1
~
,
,
,
,
,
,
Ratio of Swing to Mean Temperature Difference as Functions of Phase Duration
Figure 2d shows a typical temperature history of the hot gas during the gas cooling phase obtained from the mathematical solutiun. The temperature profiles move from right to left a t a steady rate and with practically no chitnge of shape. At a given time the profile is a nearly linear function of distance along the regenerator until it reaches the highest possible temperature, l'z'';it then continues a t Tz"to the hot end. The idealized model suggested by Figure 2d is shown in Figure 2e, where the temperature profiles are broken lines. These profiles would be realized in the
2168
L-Lo
Lo
Sample Regenerator Temperature Profiles
limiting case of infinite heat transfer coefficient; the gas and packing temperatures would then be equal. One of the principal problems encountered in regenerator design is the determination of the surface required to maintain a specified mean gas-to-gas temperature difference. We now obtain a first approximation to the surface requirement on the basis of the crude regenerator model described above. Consider first a symmetric exchanger of length Lo, total surface a&, with w mole:: per hour of identical gases flowing, and maintaininga gas temperature difference (see Nomenclature) of ( T z " -
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING AND PROCESS DEVELOPMENT 6T for curves of constant P and AT. If any two of the four quantities appearing in Figure 1 are specified, the remaining two quantities can be determined from the curves. Note that the 0 corresponds to asymptotic value of S for a given AT as 6T the exchanger, described by Equation 1. Solutions of the equations of regeneration have been obtained in three separate regions of the S, P plane-namely, ( a ) small P ; ( b ) large P with S much less than P; and ( c ) large P with S much greater than P. The authors have been unable to obtain solu-
T I"). The temperature profiles will appear as in Figure 2a. From elementary exchanger theory, the length and gas temperature difference are known to be related by Tz" - TI" = Tz'
- TI'
= 2 (T2"
-
- TI')
huLo 2 + -
(1)
WY
c
*
The temperature difference between gas and packing is just half of this. If we want a regenerator, operating with phase duration e, to have the same gas temGOLO Y - G A & - - F perature difference (averaged over a period), GAS HEATING it will be necessary t o increase the length to, say, L. The packing temperature profile which is suggested by models 2a and 2e is shown in 2f; a t the beginning of the cooling phase, the packing temperature is ABC and a t the end of the cooling phase, it is ADC. The amount of heat entering the packing during the phase is cm times the area of the parallelogram ABCD or cm(L - LO)times (Tz" - 2'2'). The quantity of heat removed from the hot gas These two quantities is w y e times ( Tz"must be equal, so that
q x '
HOT GAS
HOT END
Fz').
COLD END
Tirnox.
Now introduce dimensionless variables.
Let
,
TII
DIFFERENCE
7
I
huL hue S = -andP = cm
WY
(3)
.-b
GAS mATlNG.
GAS COOLINQ
-
___.(
TIME
Figure 4. Notation for Regenerator Temperatures S is a L'dimensionlesssize" and P is a "dimensionless phase duration" for the regenerator. It can be shown from the equations of regenertions in the region S a P and have resorted to numerical interation that, when S and P are specified, the performance of the regenerator is completely determined. We scale the tempolation to complete Figure 1. perature range (T2" - TI') down to a unit interval and are thus For the mean temperature difference, AT led t o the notion of a dimensionless mean gas temperature dif2 1 - e-8 ference, AT, and a dimensionless swing in exit gas temperature, AT = P 2 +terms of order P4 (5a) S 2 + 6(S 2)z 6T. In dimensionless units, Equations 1 and 2 become, respectively, S AT=l-P Aff = 2 A and L - LQ= PL/S AT = R -P 2 SLOIL
+
0
-
+
*
+
Thus,
L -- l - zP
Lo _
.
valid in regions (a), ( b ) , and ( e ) , respectively. The formula for the swing, 6T, in region ( a ) is quite complicated. Here the results are given for regions ( b ) and ( c ) only:
and we obtain, finally,
AT =
2 2+S-P
(4)
The above equation, being derived from a crude model of the packing temperature profile, cannot be expected to be accurate. We shall see, however, that Equation 4 is approximately correct for very large values of S and P (S > P). This is to be expected in view of Equation 3 and the fact that the profiles (Figures 2e and 2f) correspond t o infinite heat transfer coefficient. Any Two Unknown Factors Can Be Established from Any Two Known Factors
The relationships among the four dimensionless quantities-S,
P, AT, and 6T are shown in Figure 1, in which S is plotted against October 1953
where A is defined in Equation 5c. Several facts concerning these relationships are worthy of mention : 1. The value of AT obtained in Equation 4 from the crude regenerator model is asymptotically correct as S, P + m , S >>P.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING AND PROCESS DEVELOPMENT This can be seen in Equation 5c, since A 1 under these conditions. This is not unexpected, since S = m , P = m correspond to infinite heat transfer coefficient. However, writing Equation 4 in the form --f
2 AT = s + 2
2p + m
+ terms of order Pz
we see from Equation 5a that the crude model is not adequate in the case P -.,0, the "near-exchanger" case. 2. I n region ( c ) , the ratio of swing to mean temperature difference is essentially independent of S . For, combining Equations 5c and 6c
4 d G (1
same. We therefore enter the chart a t S = 150 and locate the curve P = 80 = 2 X 40. This gives the curve AT = 0.0216 and the value 6T = 0.150, d i i c h give the percentages above. This example and some others are summarized in the following tablc: Change in Variable changed Phase duration Surface/running ft. Regenerator length
Change, yo 100 20
+
++ 20
AX,
%
+35 - 16 - 19
6T,yo +go - 9 -20
Suppose it is required to maintain the AT at 6.4" F., but decrease 8T to 25" F., a 20% decrease. Enter the chart a t 6T = 0.063 and locate the curve AT = 0.016, then read S = 138, P = 25. Thus S must he decreased by 8% and P decreased by 38%. This can be done. for example, by decreasing regenerator length by 8% and decreasing phase duration by 38%. Instead of changing the phase time, one could increase the mas6 per unit length of the packing. As another example, if it were desired to maintain the 61' a t its value of 31.5 F.,but decrease AT to 5.2' F. ( a decrease of 20%), it could be done by increasing S by 34% and increasing P by 60%.
+SF)
and it is clear from the definition of A that, for sufficiently large S , the quantity (S - P ) (1 - A ) is essentially independent of S . This is illustrated in Figure 3, in which the ratio 6T/AT is plotted against P for curves of constant S. For sufficiently large S, a single curve suffices to relate 6T/AT and P. As a result, this figure may be used conveniently in conjunction with Figure 1 to determine regenerator performance. 3. For a fixed S, the ratio 6T/AT attains its maximum value for P approximately equal to S, as shown in Figure 3. I n regenerator design, the region S = P might well be avoided, in view of the relatively large swing that is attained for a given mean temperature difference. On the line S = P , me have, almost exactly, 62' = 0.62. 4. For a fixed P , as S increases from zero, the swing increases to a maximum value of nearly unity, and then decreases steadily. From Equation 6b, one can determine that the maximum swing occurs a t about S = 0.18 P. No significance can be attached to this point, since, as shown in Figure I, the maxima are extremely broad. Equation 6b shows that 6T approaches zero as S -., 0. The case S = 0 can be characterized by either zero surface area or infinite gas flow rate-that is, it represents a regenerator in which the temperatures do not change during a phase. Accordingly, there is little practical interest in the case of small S . Examples illustrate Function of the Charts
The following illustrations indicate how the chart of Figure 1 may be used t o obtain relative results for a symmetric regenerator. ..~
Let us consider a typical low temperature application in the range S = 150, P = 40, with a temperature spread of 400" F. From Figure 1 AT = 0.016 and 6T = 0.079. T h e e values in O F. are AT = 6.4' F. and ST = 31.6' F. If the phase duration were doubled, the AT mould increase by 35%, and the swing would almost double. This is from the chart as follows: Doubling the phase means doubling 8. Thus P is doubled, but S remains the
Nomenclature (See Figure 4)
S
= dimensionless size = h u L / w y = dimensionless phase duration = huF)/cm. AX = dimensionless mean gas temperature difference
P
=
Tz" - Ti" - Tz' - Ti' Tz" - Ti' Tz" - Ti'
6T = dimensionless swing in exit gas temperature =
TIumax ___
Tz"
- TI'' min - Ti'
Ts' max "2''
- Tz' - TI'
min
Any consistent set of units may be employed for the physical quantities following: h = gas-to-packing heat transfer coefficient u = heat transfer surface per unit length of regenerator L = regenerator length w = mass flow rate of gas y = specific heat of gas e = phase duration c = specific heat of packing m = mass of packing per unit length of regenerator T = gas temperature: subscripts 1, 2 refer to cold, hot gas, respectively; primes (') and double primes (") refer to cold and hot ends of regenerator, respectively. T = mean gas temperature, averaged over appropriate phase Literature Cited (1) Peiser, -4. M., and Lehner, J., Quart. A p p l i e d Mathematics, t o be published; Hausen, H., 2. angew. Math. u. Mechanik., 9,173-99 (1929); Nusselt, W., 2. Ver. deut. Ing., pp. 1052-4 (July 1928). RECEIVED for review Januaiy ZS, 1953.
ACCEPTEDJuly 15, 1953.
END OF ENGINEERING AND PROCESS DEVELOPMENT SECTION
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