I
I121
It may be shown t h a t it is more economical t o use one single unit for multistage flash distillation than several small units to give a n equivalent output of freshwater. The cost of two 30,000-gpd units may be calculated b y following the same procedure outlined above and b y using the same quantity of waste heat. The total relative cost factor resulting from the calculation is 132 which is greater than that for the single 60,000-gpd unit. The capital cost of a multistage flash evaporator computed from the foregoing charts in the size range 5000-60,OOO-gpd output is comparable t o that of a packaged vapor compression unit of the same freshwater output. A comparison of operation costs shows that a multistage flash plant with a waste heat source is least expensive t o operate although a similar distillation plant with a boiler heat source is considerably more expensive to operate than a vapor compression unit. Copcluoions
1
1
2
5
10 20 50 100 Heat Transfer Surface (hundred 9 f t l
ZW
500
Figure 14. Relative evaporator capital cost as function of heat transfer area
value and a vertical line from the 60,000-gpd value. These intersect a t a thermal economy of 4.8 lb/1000 Btu. Step 3. On Figure 13, a line corresponding to a thermal economy of 4.8 lb/1000 B t u is quite flat beyond t h e 20stage point, and 20 stages will be selected. Projecting t o the left gives 70 ft2/1000 gpd or a total of 4200 f t 2for the 60,000 gpd required from t h e unit. Step 4. From Figure 14 the relative cost of equipment is found b y projecting vertically from the heat transfer area of 4200 f t 2 to the line and horizontally to the cost of the unit, Thus, the relative cost factor is 110 for a single 20-stage multistage flash distillation unit.
The study and experimental work indicate the feasibility of operating a multistage flash desalination unit from a variable heat source without manual control. The prototype equipment has been operated on a seawater feed and produced 2500-6000 gpd freshwater utilizing 250,000-600,000 Btu/hr waste heat available from diesel generators having power outputs ranging from 60-150 kw. The simple automatic control system to adjust interstage brine and distillate transfer permits the upit to operate successfully on a wide variety of heat sources. The prototype unit constructed of aluminum appeared to resist corrosion to a greater extent than steel units although further field tests will be required to evaluate ultimately this material. Literafure Cited
Williams, J. R., Nehlsen, W. R., Military Eng., 56, 5-9 (1964). RECEIVED for review June 16, 1969 ACCEPTEDApril 26, 1971
Heat Exchanger Modeling by Conservative Scalar Pulse Testing Charles J. Messa, Gary W. Poehleinl, and Alan S. Foust Department of Chemical Engineering, Lehigh University, Bethlehem, Pa. 18015
Characterization of the dynamic behavior of heat exchangers can be important in startup operations and for the design of modern control systems. Most previous studies of heat exchanger dynamics have been based on single tubes and/or idealized flow patterns-Le., plug flow or total mixing. Unfortunately, most industrial exchangers are more complex. Theoretically, the dynamic behavior of complex geometry industrial exchangers can be characterized b y standard pulse or step test methods using changes in the inlet stream temperature. Such evaluations of a system are not always practical because an undesirable disturbance of the system may be re1
To whom correspondence should be addressed.
466 Ind. Eng. Chem. Process Des. Develop,, Vol. 10, No. 4, 1971
quired. Also, significant damping of the temperature disturbance owing t o heat transfer is likely to lead to low precision measurements. The purpose of this study was to demonstrate that the dynamic response of complex geometry (in our case, helical wound) heat exchangers could be defined b y a mathematical model formulated from basic heat transfer theories, a knowledge of exchanger construction, and experimental fluid mixing data obtained from salt-tracer pulse tests. Coaservative scalar dynamic tests, such as salt solution pulses, do not require substantial disturbance of the heat exchange system and are not affected by thermal capacity elements; in general, the response pulses can be measured more precisely.
A dynamic model has been developed for the shell-side stream of a geometrically complex (helical coil) heat exchanger. Plug flow and axial dispersion sections were used to characterize the fluid-flow pattern. Adjustable parameters in this fluid-flow model were determined by fitting the frequency response data from salt-pulse tracer tests. This flow model was then coupled with heat transfer and thermal capacity terms to model the shell-side frequency response under conditions of heat transfer. The only additional adjustable parameters needed for the heat transfer model were the steady-state coefficients and the amount of heat capacity in the exchanger construction elements. Frequency response characteristics were measured with hotwater pulses under conditions of heat transfer. Satisfactory agreement between the measured responses and model predictions was achieved over a wide range of shell-side flow rates.
T h e coiistruction of the heat' exchanger used for this study, along with appropriate nomenclature, is shown in Figure 1. Complete specifications are given in Table I. Six heat exchangers wit'h different values of the geometric construction variables were tested by llessa (1967, 1968) under steadystate and dynamic coiiditions. Only the results of the dynamic study for a typical exchanger are reported here. -4dditional results and details of the mat'hematical derivations are given by llIessa (1968). Heat Exchanger Dynamics
The operation of a typical heat exchanger is simple to visualize; namely, a process flow stream is contacted with a metal surface across which heat is transferred. The dynamic response is the exit t'emperature as a function of various process disturbances-Le., flow rate or inlet temperature. The dynamic behavior can be represented mathematically in terms of one or more differential equations which incorporate the distribution of residence times (flow pattern) of fluid elemelits moving through the heat exchanger and the dominant heat fluxes and thermal capacity elements within t'he exchanger. Thermal capacity element's and heat fluxes generally can be identified from kiioivledge of heat exchanger design, steadystate heat transfer characteristics, and basic heat transfer theory. On the ot'her hand, characterization of the fluid element residence time distribution is considerably more difficult. Solution of the equations of niot'ion or an assumption of
1
1
v//////////A
- ')
LMANDREL SEALED TO FLOW
WOUND T U B I N G
RADIAL SEPARATION
LDISTRIBUTING COLLECTING~ MANIFOLD
Figure 1.
MANIFOLD
a simplified flow pattern, such as plug flow or total mixing, can be used to describe the residence time distribution of fluid elements in many simple geometries. Unfortunately, for most industrial heat exchangers, the shell-side geometry is complex, making solution of the equations of motion improbable or assumption of a simplified flow pat'tern unrealistic. Consequent'ly, flow patterns through complex geometries must be ascertained experimentally. conservative scalar pulse testing was used to measure the shell-side flow pattern of helical coil heat exchangers. A model in the form of differential equations was curve-fitted to the experiment'al flow pattern data to represent analytically the shell-side flow pattern. The model equations were coupled with the heat transfer terms and solved for the dynamic response under coiidit,ionsof heat' transfer. Measurement of Shell-Side Flow Pattern
Residence time dist,ribution curves were measured using conservative scalar pulse testing techniques which involved injecting a pulse of tracer into the inlet shell stream and measuring the tracer in the exit stream. The tracer was a concentrated salt solution (KaC1-HzO). Inlet and exit electrical conductivities of the shell-side stream mere continuously recorded as a function of t'ime. The experimental flow loop and associated measurement apparatus will not be described here. X detailed description is given by Nessa (1968). Typical input disturbance and output response pulses are shown in Figure 2 . The time between the start of the disturbance and the start of the response is denoted as process dead time. Frequency response characteristics were calculated from each input/output pulse set using Fourier Integral Transforms as described by Hougen (1964) and hIessa et al., (1969). Xeasurements were made over a wide range of flow rates. Steady-state conditions are given in Table 11. The calculated frequency response characteristics are
Table 1. Experimental Heat Exchanger Specifications Winding angle Radial pitch Axial pitch No. of tubes No. of spacer wires Av tube length Tube 0.d. Tube i d . Mandrel diameter No. of tube layers Total heat transfer a r e a H e a t transfer a r e a in bundle Shell free-flow a r e a Pitch =
18.5' 1.29 1 .03 159 113 7 . 8 ft 0 . 2 5 0 in. 0 . 1 5 2 in. 3 . 5 in.
7
80.8 ft2 6 5 . 2 ft2 0 . 1 0 6 ft*
tube diameter f distance between tubes tube diameter
Experimental exchanger design Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1 9 7 1
467
-
2
2 Ia a
I-
z w
0
z
0 0
n
DISTURBANCE
HEAT 0.1
3 a VJ
A
.04
TIME
Figure 2.
-
-
.02-
Typical disturbance and response
.01
FLOW
'
'
PATTERN MODEL I
I
I
t
I
1
I
01
Salt water pulses
plotted in Figures 3 and 4, in terms of the dimensionless frequency, wrB. The parameter, T E ( = V E / F ) , is inversely proportional to the shell-side volumetric flow rate. The coiistant of proportionality, V E ,was determined from a curve fit of the shell-side flow pattern inodel to the frequency response data and represents a n effective flow volume. Use of the dimensionless frequency allows dat,a for all flow rates to be easily plotted on a single graph. The data fall in a narrow band illustrating that increased turbulent mixing caused by increased flow rate is offset by a lower residence time. Flow Pattern Model
A schematic representatioii of a helical coil heat exchanger (HCHE) is shown in Figure 5. The fluid elements move through the shell side serially and encounter multiple regions of different geometric configurations. The recognizable flow regions are an entrance section (ab) connecting the shell flange to the shell, an end section region (bc) where the flow is subjected to a rapid expansion and where t'he flow volume is relatively free of tubes, an end section region (cd) where the flow contacts the tube pigtails which transport tube-side fluid from the tube manifold to the tube bundle, a bundle section (de) where the tubes are tightly wound and where the bulk of the heat transfer takes place, and three exit sections (ef, fg, and gh) which are identical to the first three sections (cd, bc, and a b ) . The hypothesis that fluid elements move through each of the sections of the heat exchanger serially as opposed to parallel flow paths is support'ed by the experimental observation that a pulse-like disturbance results in a single-peaked response. Volumes for each of t'hese sections can be calculated from heat exchanger coiistruction data. Calculated volumes and the
total measured volume, VT, of the heat exchanger are shown in Table 111. The volumes of sect'ions bc and cd are difficult to calculate separately. Consequently, the sum of these two volumes is shown in Table I11 as V a d . Vo is t'he volume outside the heat' exchanger and between the measuring probes. The analyt'ical models chosen to represent the flow through each of the sections were the plug flow model and the axial dispersion model. The plug flow model represents flow through a region in which each group of fluid elements entering the region a t the same time leave the region together a t some later time, T ( = V j F ) , where V = volume of the region and F = volumetric flow through the region. The differential equatioi? and transfer function for a plug flow region are:
B.C.(Z
=
Tt;'
0,
c = C,)
= e-TS
(2)
The dependent variables represent deviations from the steady-state values. Consequently, the initial conditions assumed for all differential equatioiis have the respective systems a t steady state prior to any dist,urbance. Process disturbances are not damped in a plug flow region. The process response is equivalent to the process disturbance
Table II. Conditions for Salt Tracer Runs Flow rate, CFM
Run
1s 2s 3s
3.38 3.38 9.98
Flow rate, CFM
Run
4s 5s 6s
HEAT
I
Table 111. Calculated Volume Parameters Ft3 V T = 1.138 T'& = 0.212
V,*
=
v,,= 0 022
Vbd
=
V,, = 0 . 4 4 0
V,
=
RUN RUN
IS
Q 0
RUN RUN
3s 4s
+
13.29 19.10 32.67 -2001
-240 .01
EXCHANGER I
0
2s
b
- FLOW
PATTERN
I
I
I
.02
.04
.OS
1
1
0.1
"
0.2
Figure 4.
Frequency response data
Salt tracer flow pattern study
468
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
I I .04 .06
TE
0 032
I
MODEL
I
. LO
Table IV. Transfer Functions of Shell-Side Flow Stream
t Figure
5.
Heat exchanger sections
with the exception t,hat response is displaced in time by T units. From a study of the measured frequency response characterist,iey (Figures 3 and 4),a plug flow model is not a n adequate representation of the flow through the entire exchanger since sigiiificaiit damping of the process disturbance occurs. The axial dispersion model includes the effect of turbulent mixing superimposed on the bulk flow. Significant mixing (depending 011 the value of the dimensionless mixing coefficient, D*) can occur in an axial dispersion region. Wheii D* = 0.0, the model reduces to the plug flow model. The differential equation and transfer functioii for an axial dispersion region
CY
=
(1
+4
71: =
TESD*j"Z
Vz;'F
Table V. Fitted Volume Parameters
Ft3 Vn Vz D*
_e-alD*
where
A411models used in this paper iiicorporate axial dispersion sectioiis and assume plug flow a t the entrance to the axial dispersioii sectioiis. Uouiidary condition 1 states that fluid element.; approach the boundary of an axial dispersion region by bulk flow and leave the boundary into the axial dispersioii regioii by both bulk Aow and longitudinal turbulent mixiiig. Boundary condition 2 was chosen for coiivenience of solution. Further tliscussioii of the axial dispersioii model and boundary conditiolis can be found in Danckwerts (1953), Bishoff and Leveiispiel (1962), niid Vincent et al. (1961). The application of these models to the various shell sections is constrained by two points: The total volume characteristic of the combined models should be less than or equal t'o the measured total exchanger volume (the difference is regions of stagnant flow) , and the resultiiig transfer function must represent both the dynamic amplitude ratio and the phase angle which have been measured experimentally. The volume outside the tightly wound bundle represents 82% of the total exchanger flow volume. I n addition, the mean cross sectional area for flow in the end sectioiis is twice t h a t of the bundle section. Consequently, t,he residence time outside the bundle should be much greater than in the bundle
= =
=
0.185 0.269 0.4
V B 8, Veir
= = =
0.212 0.032 1.098
and predominately affect the measured residence time distribut'ion. Axial dispersion equations were chosen to represent end sections cd and fg. The volume of each of these sect,ions is V , and the mixing coefficient D*. Plug flow equations were used to represent sections ab, bc, fg, and gh. The total volume of all these sections is 2 Vn.The bundle \vas represented by a plug flow section of volume V,. Ideally, the axial disperFioii model mould be a more realistic picture of the bundle. However, because of the large influence of the end sections on residence time, the mixing coefficient' for the bundle could not be extracted from the data. Table IV cont'ains the transfer function for each of the sections and for the entire shell-side flow stream. Only end section volumes V, contribute to the damping of conservative scalar process disturbances. The model contains the following parameters: V o , V E ,V B ,and D*. The bundle volume, VB,was assumed equal t'o the bundle volume calculated from heat exchanger construction data. The values of V, and D * were obtained from a curve fit of the dynamic amplit,ude ratio (DAR) of the system transfer function to the experimental DAR data. The volume, Vo,\vas obtained from a curve fit of the phase angle of the system transfer function to the experimental phase angle data. Values of these parameters are shown in Table V. The effective exchanger volume (= 2 V o 2 V E V B - V,) is slight,ly less than the measured volume V T .The difference is attributed to regions of stagnant flow a t the entrance and exit of the end sections of the heat exchanger. The model shown in Figures 3 and 4 by solid lines fits the experimental data well which is expected in view of the iiumber of adjustable parameters. The fit was obtained by using templates. Consequently, statistical parameters were not calculated t o measure the goodness of fit. The additional accuracy obtained by using mathematical curve fitting techniques was not justified since the absolute magnitude of the parameters was not of prime interest. The true test of
+
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
+
469
I
Table VI.
I
l
I
l
I
I
I 1
Conditions for Temperature Tracer Runs
Run
Flow rate, CFM
Run
1T 2T
14 62 9 80
3T 4T
Flow rote, CFM
18 87 18 87
x
HEAT EXCHANGER I -120
the model is how well it predicts the dynamic response under conditions of heat transfer.
RUN RUN RUN RUN
0 Q
w
3
A
IT 2T 3T 4T
Thermal Pulse Testing
Thermal pulse disturbances can be used as ail alternative to salt water pulses in measuring the shell-side flow pattern. Several disadvantages to the use of thermal pulses include the following problems: The steady-state operat'ion of the heat exchanger has to be upset (as olr,,osed to injecting a mass tracer, such as salt water, which does iiot significantly affect either the mass flow or the steady-Atate heat transfer) ; the observed response curves are influenced b y thermal capacity elements-Le., heat exchanger metal-and the response curves are measured with less precision because of extended low strength response. Residence time distribution curves were measured with thermal pulses and compared with those obtained from the salt water pulses. The experiments were performed wit'h the tube side filled with stagnant air. I n general, thermal disturbance and response pulses were of the same type as those obtained from the salt water tests. However, the tail of the response pulses was generally of longer duration and lower strength. The flow rates employed are listed in Table VI, and the frequency response characteristics are shown in Figures 6 and 7. The measured dead times are plotted in Figure 8. The dead times for thermal pulses are equivalent to those for salt water pulses (Figure 8) since that dead time is strongly dependent on the bulk flow. On the other hand, the att'enuation of high frequency componeiits is dependent 011 the system thermal capacity, which is much greater for a temperature pulse because of heat exchanger metal. The flow model developed from the salt water tests is shown in Figures 6 and 7 as solid line 1. Comparison of this model with experimental thermal data shows the influence of thermal capacity elements. The equations in Table VI1 apply if: The dimensionless mixing coefficient for mass transport D* is equivalent to that for thermal transport D * r ; the heat exchanger metal call be
-240
I
I
I
I
0.1
D4 .06
02
.01
QJ
Figure
I
I
2
.4
I
.6
7. Frequency response data
Temperature tracer flow pattern study
lumped as a distributed mass in each section of the exchanger; the metal is distributed only in end volumes V E and bundle V B ;the conduct~ion(radial) within the metal and a t a right angle to t'he flow is extremely rapid; the conduction (axial) wit'hin the metal and parallel to the flow is not significant; and the heat transfer coefficient is the same in all sections. The additional parameters of these equations include metal volume V , and heat' transfer area A s . These parameters were calculat'ed from heat exchanger construction data. Solid lines 2 , 3 , and 4 in Figures 6 and 7 represent solution of these equations with only parameters obtained from the salt water flow niodel and calculated from construction data. The difference between Models 2 , 3, and 4 is that Model 4 includes the metal volume in the shell of the exchanger in addition to the tube metal rolume. Model 3 includes only the metal in the tubes and Model 2 only the metal in the tubes in the bundle section. The importance of considering all metal in the exchanger is clearly emphasized when Models 2 , 3, and 4 are compared with t'he experimental data. Further study of these models and the experimental data s h o w that Model 4 deviat'es considerably from the data at high frequencies. The most reliable fit must occur a t those frequencies attenuated the least because these frequencies 9
I
I
I
I
10
20
30
40
HEAT EXCHANGER I 0
RUN
IT
0
RUN
2T
0
RUN 3T RUN 4T
A
.01
.06
THERMAL CAPACITY FLOW MODELS
,I
.2
.4
.6
I.
2.
4.
6.
10.
-
00
UJ TE
Figure 6.
pattern study
470 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
50
SHELL FLOW (cfm)
Frequency response data
Temperature tracer flow
1.0
TE
Figure 8.
Dead time measurements
Table VII. Transfer Functions of Shell-Side Flow Streams
Table IX. Steady-State Operating Conditions Heat transfer study
Temperature pulses
wShel, = TFD*TFE*TFB*TFE*TF~ TFn
= e-'DS
ha Shell flow rate, CFM
Run no.
1H
17 17 12 20 9
2H
3H 4H 5H
TFE =
~
(rl - r Z ) e r l + r 2 [rl (1 - D*rz)erl]- [r2 ( I - D*rl)er*l
Table VIII. Heat Exchanger Transfer Function
Under heat transfer conditions
64 64 94 68 49
ht
ft2, O F Btu/hr
Tube flow rate, CFM
ft2, O F Btu/hr
1451 1477 1119 1683 857
4 87 4 71
1556 1519 1501 1486 1442
4 71 4 60 4 50
have the greatest influence upon the time domain response. Model 4 does not represent the dat'a over the entire range of frequencies because of the assumptions inherent in the model. The assumption of no significant axial conduction in relationship to radial conduction was checked by the solution of equations similar to those in Table VI1 but which cont,ained axial conduction terms. Solution of these equatioiis showed no significant effect of axial conduction. Ot'her assumptions of t'he model were not checked theoretically. However, the effect' of finite radial conduction in the metal of different metals (tubes and shell) and of different heat transfer coefficients in the various sections of the exchanger can be significant in determining effective thermal capacitance. In addition, the effective thermal capacity decreases with higher frequency disturbances and appareiit'ly with the measured data. Comparison of Xodels 1 and 4 shows t h a t thermal capacity effects can be readily included in a flow model. Furthermore, flow patterns obtained from thermal pulse testing must include the influence of all t'hermnl capacity elements. Heat Exchanger Model
End sections (plug flow)
rl, r2
=
1 i: (1
E n d sections (dispersion with thermal capacity)
+4D * ~ E ) ~ " , 2 D*
Center bundle (plug flow with heat transfer and thermal capacity)
The mathematical model predicting the dynamic response of the heat exchanger under conditions of heat transfer was developed from the measured flow patt'ern (salt water pulses), the appropriate thermal capacity elements (same as l\lodel4), and basic considerations of heat transfer to the tube-side fluid. Turbulent', countercurrent plug flow was assumed for the tube-side fluid since the tube-side Reynolds Kumbers were large. The transfer function of the shell-side fluid under conditions of heat, transfer is shown in Table VIII. Frequency response characteristics were calculated from this model. Subsequent'ly, the dynamic response of the heat' exchanger was measured experiment'ally. Table IX gives the steady-st'ate
0 la a w
0
3
c_ E!
t 0
9z *0 .06
.I
I
I
.P
.4
I l l .6
I.
I
2.
4.
6.
wr E
Figure
9.
Frequency response data Heat transfer study
Ind. Eng. Chem. Process Des. Develop., Vol. 10,
No. 4, 1971 471
Nomenclature
0
A A, B.C.
-40
C
- 80
1
RUN RUN
0 Q
I
\
HEAT EXCHANGER I A RUN IH RUN 2H
1
W
4H
HEAT TRANSFER
MODEL
.I
.2
c,
.04 .06
.02
.01
.4
.6
1.0
w =E
Figure 10.
Frequency response data Heat transfer study
r "
I DISTURBANCE
W
= = = =
D*
=
D p'
= =
h
=
.1
=
k L
= = =
S -240
= heat transfer area, f t 2
TI; t V
= =
=
X
=
z
=
cross-sectional flow area in bundle, ft2 boundary condition concentration of salt tracer, lb,/ft3 heat capacity, Btu/lb, O F dimensionles mixing coefficient = DA ,/LF ciprocal of Pe Xo. turbulent mixing coefficient flow rate, C F M heat transfer coefficient, Utu/hr, f t 2 , O F
=
re-
47
thermal conductivity, Btu/hr, ft, O length of vessel, ft Laplace variable transfer function time, miri volume, it3 distance, ft dimensionless distance
F
G R E E KLETTERS = frequency, rad/sec = density, Ib/ft3 7 = time parameter, sec 0 = temperature deviation from steady state, O w p
F
a =I
t-
4
a
I
-I Figure 1 1 .
DT2
-I
TIME
Typical disturbance and response Temperature pulses
operating coiiditions under which the data were obtained. Figures 9 and 10 illustrate a comparison of the measured frequency response with that calculated from the heat exchanger model. Node1 predictions (solid lines) in Figures 9 and 10 were determined, a priori, from the flow model and the theoretical heat transfer considerations previously outlined. No adjustable parameters were used to fit the model to the heat. transfer frequency response data. Comparison of the model to the data shows t h a t t'he phase angle prediction is more accurate than the amplitude rat'io prediction and that the low flow rat,e runs deviate most widely. The phase angle is predicted more accurately because it is largely determined by the process dead time. Typical response curves, under dynamic heat transfer conditions, contained two peaks as shown in Figure 11. These results were baffling a t first, but subsequently the small first peak was shown to result from the flow pulse caused by the injection of the temperature tracer. Injection of a tracer pulse in an incompressible system which has a temperature gradient will cause the whole profile to slide forward. Heat transfer effects will eventually return the system to the original profile. This happened rapidly for our exchangers, and we were able to separate temperat'ure induced dynamic response from that induced by the flow disturbance. The slight overlap could be responsible for some of the data scatter, especially a t the lower flow rates. This study shows that a simple experiment, which does not upset the steady-state operation of a heat exchanger, can be a powerful tool in developing a useful dynamic model. 472
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
SUBSCRIPTS B = tube bundle D = plug flow region E = end section s = shell side t = tube side w = tube wall literature Cited
Bishoff, K. B., Levenspiel, O., Chem. Eng. Sei., 17, 245-57 (1962). Danckwerts, P. W., ibid., 2 , 1 (1953). Hougen, J. O., Chem. Eng. Progr. dlonogr. Ser., 4 , 60 (1964). hIessa, C. J., LIS thesis, Lehigh University, Bethlehem, Pa., 10~7
LIe&a,' C. J., PhD thesis, Lehigh University, Bethlehem, Pa.,
1968.
RIessa, C. J., Luyben, W. L., Poehlein, G. W., 2nd. Eng. Chem. Fundam., 8 (4), 743-8 (1969). Vincent, G. C., Hougen, J. O., Dreifke, G. E., Chem. Eng. Progr., 57 ( 7 ) ,4.5 (1961).
I~I:CI:IVI*:D for review September 11, 1969 ACCF~PTED May 18, 1971
~-
Correction
T H E R M A L CRACKING OF ISOBUTANE. KINETICS AND PRODUCT DISTRIBUTIONS I n this article by Alfons G. Uuekens and Gilbert F. Froment [IND.ESG. CHEM.PROCI:SS DES.DEVELOP., 10, 309 (1971)], the last equation on page 314 should read =
loge(ksec/ktert)
log& - ( E / R T )
The paragraph directly beneath this equat'ion should read, "Our data lead to lo& , . in fair agreement with log,A. . . ,
. J J
