(2) Currie, L. M., Hamister, V. C., McPherson, H. G., “Progress in Nuclear Energy,” Series IV, “Technology and Engineering,” R. Hurst, and S. McLain, Eds., pp. 65-107, McGraw-Hill, New York, 1956. (3) Haas, P. A., Ferris, L. M., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 5.234-8 (1966). ( 4 ) Howard, H. C., “Chemistry of Coal Utilization,” H. H. Lowry, Ed., pp. 346-76, Wiley, New York, 1945. (5) Savich, S. R., Howard, H. C., Znd. Eng. Chem. 44, 1409-11 (1952).
(6) Teletzke, G. H., “Wet Air Oxidation,” 56th Annual Meeting, A.I.Ch.E., Houston, Tex., December 1963. RECEIVED for review February 7, 1966 ACCEPTEDJanuary 5, 1967 Division of Industrial and Engineering Chemistry, 150th Meeting, ACS, Atlantic City, N. I., September 1965. Work sponsored by the U. S. Atomic En( vy Commission under contract with the Union Carbide Corp.
EVALUATION OF TEMPERATURE PULSE CHARACTERISTICS AND PULSE TESTING FOR THERMAL DYNAMIC ANALYSIS CLARENCE I. LEWIS, JR.1, DUANE F. BRULEY, AND DANIEL H. H U N T Clemson University, Clemson, S. C. Experimental temperature pulses were generated and studied in terms of normalized frequency content to evaluate optimum pulse characteristics for the determination of reliable frequency response data. All results indicate that total pulse duration is the best criterion for predicting pulse quality with respect to valid data reduction capability. A wetted-wall column was forced using temperature pulses. The pulse-testing data were compared with previously determined direct frequency forced results and mathematical model solutions for laminar and turbulent air flow conditions. The results illustrate the validity of pulse testing but also demonstrate some difficulties and limitations of temperature pulsing for thermal dynamic data recovery.
uLsE-testing techniques were used for the experimental
p thermal dynamic analysis of a wetted-wall column operati n g a s an adiabatic humidifier. The investigation is an extension of previous studies (7) carried out on the same physical system and concerned with the formulation and solution of the system’s mathematical models, in the frequency domain, for both turbulent and laminar air flow conditions. The models were verified by experimental testing, using direct frequency forcing techniques and plotting theoretical and experimental frequency response curves in the form of Bode plots. Although the mathematical models and experimental data agreed well, the experimental frequency range was fixed by physical limitations of the uniquely designed mechanical generator used for producing a harmonic temperature variation a t the inlet to the column. With the apparatus, as described (7), it was difficult to produce a reasonable sinusoidal forcing function at frequencies above 2.5 or below 0.03 radian per second. This frequency range was not sufficient to describe the column dynamics completely, as the maximum obtainable frequency is just slightly greater than the break frequency of the system. To extend the previous investigation the present study was initiated to accomplish essentially three purposes: (1) compare pulse-testing results with those from direct frequency forcing and mathematical formulation for an actual physical system ; (2) extend the frequency range to characterize the thermal dynamics of the wetted-wall column better; and (3) study the effect of arbitrary pulse characteristics in predicting optimum pulse shapes for obtaining the system dynamic response. Present address, Enjay Chemical Co., Baton Rouge, La.
The air-water process investigated is representative of a distributed parameter system involving simultaneous heat and mass transfer. A wetted-wall column, used as the process, provided a definite transfer area and allowed the development of essentially reproducible air and water velocity profiles. Adiabatic humidification conditions, together with the large heat capacity and transfer rates in the liquid phase, relative to those in the gas phase, ensured a constant liquid and interfacial temperature in the column. Hence the analysis of the system was simplified to essentially a one-phase flow problem involving only the air-vapor phase. Frequency Response from Pulse Testing
Pulse testing is a transient response method giving data which can be mathematically reduced to yield frequency response information in terms of magnitude ratio and phase shift. This technique involves disturbing a system, initially operating at steady state, in a pulselike manner and simultaneously recording the pulse traces of the input and output variables. The pulse traces are then reduced to determine the system frequency response. Several authors (2, 4 ) present the mathematics of the pulse-testing method. Based on the mathematics, a digital computer program employing Filon’s (3) method for numerical integration was written for calculating magnitude ratio, phase shift, and normalized frequency content from experimental pulse data. To debug and check the program validity, simple first- and second-order systems were simulated and pulsed on the analog computer. The pulse traces were recorded and then reduced using the digital computer program. The theoretical system VOL. 6
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Figure 1.
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Schematic of pulsing mechanism
1
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0.01
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I
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Figure 3. Experimental wetted section magnitude ratio vs. forcing frequency for turbulent flow center line
POTENTIOMETER NTTEMPERATURE BATH
Figure 2.
Schematic of system instrumentation
response was then calculated and compared with the pulsetesting results. Excellent agreement was obtained.
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As previously stated, mathematical models were developed and solved in the initial studies (7). The models, because of simplifications resulting from the adiabatic humidification operating conditions of the wetted-wall column, were necessary for the air phase only a t essentially constant water temperature throughout the column. Laminar Air Flow Model. T h e differential equation describing unsteady-state thermal behavior of the air phase is based on the following assumptions: 1. Only x direction velocity components exist.
2. Use of average fluid properties is valid. 3. Temperature is a function of r, x , and 0 only (system is axially symmetric). 4. There is no axial conduction. The equation can be written as:
OPULSE TESTING .DIRECT FREQUENCY FORCING -MATHEMATICAL MODEL
t
I
0.I
0.01
1.0
FREQUENCY,W, RAOIANS/SEC
L
10.0
Figure 4. Experimental wetted section phase angle vs. forcing frequency for turbulent flow center line
1. Uniform velocity and temperature profiles.
2. 3. 4. 5.
Constant mass velocity. Film theory applies. All resistance to heat and mass transfer in the gas film. No axial conduction.
The equation can be written as:
-bt -- _ G-' b_t _2h as
p
bx
psR
t
with boundary conditions as follows: t ( x = 0) = tl t ( x = X) = to
>
for x > 0, R > r 0. Boundary conditions are:
t (r,O,e) = to t (R,x,e) = 0 t (r,x,O) = 0 where to was determined experimentally as a function of radial position. Turbulent Air Flow Model. T h e differential equation describing the unsteady-state thermal behavior of the air phase is based on the assumptions of: 282
l&EC PROCESS DESIGN AND DEVELOPMENT
Equations 1 and 2 were Laplace-transformed and solved in the complex domain for magnitude ratio and phase shift as a function of forcing frequency. Experimental Investigation
Equipment. Bruley and Prados (7) give an equivalent description of the wetted-wall column and the air-water flow system for the column and auxiliary equipment used in the present work. Figures 1 and 2 illustrate the pulsing mechanism and the system instrumentation, respectively. Copper-constantan thermocouples were located at the water and air inlets and outlets. Water temperature was measured
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Figure 7. Experimental wetted section magnitude ratio vs. forcing frequency for laminar flow center line
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Figure 6. Experimental wetted section phase angle vs. forcing frequency for turbulent flow midpoint
Figure 8. Experimental wetted section phase angle vs. forcing frequency for laminar flow center line
with an indicating potentiometer, while air temperatures were traced by use of a highly sensitive dual-channel recording potentiometer. The air temperature thermocouples had time constants of l / q g second and were mounted on adjustable micrometers for accurate positioning within the column. The ice point was used as reference for the water stream thermocouples, while an adjustable constant temperature bath was used as reference for the air stream thermocouples. T h e adjustable temperature bath supplemented the inadequate zero suppression of the recording instrument by bucking the steady-state e.m.f. of the thermocouples. Procedure. After sufficient time had been allowed for the system to attain steady state and for instrumentation warm-up, the inlet air stream was excited with a temperature pulse by impressing a voltage across the coil of the primary pulsing element and observing the inlet temperature trace on the continuous recorder. O n observing peak inlet temperature, the linear valve control lever on the secondary pulsing element was positioned to divert the system air through the unheated tube. This permitted rapid cooling of the inlet air and thus shortened the duration of the exciting pulse. O n approaching the initial steady-state temperature the inlet air stream was again diverted through the heated tube of the secondary pulsing element. Obtaining a suitable pulse was always a trial and error procedure ; however, technique improved with practice. Pulse height and time duration could be varied by varying time duration or magnitude of the impressed voltage on the primary element coil or by varying the length of time the secondary element layer was thrown. Control of pulse characteristics was difficult, since many variables were involved.
The system was pulsed for both turbulent and laminar air flow conditions, and for center line and midpoint thermocouple positions in both cases. The steady-state operating conditions were essentially the same as those employed in the initial studies (7). For each case, several pulses of different amplitude were reduced and the frequency response data compared to ensure that the pulses were within the linear operating range of the column. Presentation of Results
Frequency Response Curves. Pulse data for laminar air flow and turbulent air flow a t both midpoint and center line positions in the column were reduced and plotted as Bode plots with the initial theoretical and direct frequency forced results (7). The plots are for the wetted section only (Figures 3 through IO). The results for the two turbulent flow runs and the laminar flow center line run (Figures 3 through 8) compared favorably, in that the pulse data closely followed those of the previous work. Deviations a t the higher frequencies were due to low normalized frequency content of the exciting pulse a t those frequencies. The data started to deviate where S(w), fell below 0.3 to 0.2, thus confirming the findings of previous investigators (2, 4). Slight vertical displacement of the curves was assumed to be due to inability to reproduce previous operating conditions exactly. Since the system is nonlinear and assumed to be operating in a region of linearity, identical VOL. 6
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Figure 9. Experimental wetted section magnitude ratio vs. forcing frequency for laminar flow midpoint
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90 II O I30 150 PULSE AMPLITUDE, MICROVOLTS
170
Figure 1 1. Normalized frequency content vs. pulse amplitude of experimental pulses with nearly equal total time duration
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reproduction of operating conditions would be required to obtain identical results. In the laminar flow midpoint case (Figures 9 and 10) vertical displacement of both magnitude ratio and phase angle curves was greater than in any of the previous three cases. The difference in operating point was probably a minor factor; however, more important in this case were the shape of the temperature profile across the column and the position of the outlet thermocouple. Experimentally, it was noticed that the profile across the column was not quite symmetrical. This condition was probably caused by a slightly warped inlet section tube. Since the profile was unsymmetrical, the inlet and outlet thermocouples would not have been measuring the same temperatures as for a symmetrical profile. Also, with the parabolic profile in laminar flow, exact thermocouple positioning a t the midpoint would be more critical than for any other of the experimental cases. Extension of Data. T h e frequency range of the dynamic response curves could not be extended. I n fact, reliable results were not obtained a t frequencies as high as those in the previous work. T h e problem was attributed to the lack of normalized frequency content in the experimental pulse. Frequency content became insufficient at frequencies of 0.57 to 0.74 radian per second. However, results were plotted out to 3.0 radians per second to show the scatter of calculated points beyond the useful limits of S(w),. Points that do fall along the actual curve after S ( U ) ~< 0.3 are unpredictable. 284
I & E C P R O C E S S DESIGN A N D DEVELOPMENT
Pulse Characteristics. Pulse characteristics were investigated to optimize certain features of the temperature pulse in an attempt to get reliable data a t higher frequencies for more fully testing the mathematical model. The general configuration (shape) of the pulse was restricted by the nature of the pulsing mechanism, but certain characteristics could be varied. Pulse amplitude, pulse strength, rise and decay time, and total pulse duration were studied as they affected normalized frequency content. For pulses of nearly equal total pulse duration (Figure l l ) , increasing the amplitude of the pulse did not increase the normalized frequency content. Thus the reliable linear system dynamics can be found using a pulse amplitude that is just high enough for the noise-distinguishable pulse to be read accurately. A higher pulse amplitude will not increase the reliable frequency range of the dynamic results and will eventually excite system nonlinearities. Pulse strength or area under the pulse curve provided no indication of increasing the reliable frequency range from the standpoint of normalized frequency content. Several pulses (Figure 12) for which the pulse strength was more than doubled while total pulse duration was held constant make plain that normalized frequency content was not affected by increasing the pulse strength. Investigation of the effect of rise and decay time (5) on frequency content and normalized frequency content of experimental pulses resulted only in a directional correlation between normalized frequency content and decay time (Figure 13). Frequency content does not appear to be a valid criterion for judging the usefulness of randomly generated experimental pulses. Normalized frequency content as a function of total pulse time duration (Figure 14) shows that as time duration decreases, the normalized frequency content increases. I t is evident that the apparent correlation between normalized frequency and decay time results because the decay time approaches the total pulse time duration. When using pulses of relatively long time duration compared to the rise or decay time, the pulse duration seems to be the overriding effect which limits the normalized frequency content, and hence, the highest frequency at which reliable dynamic data can be extracted.
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Figure 14. Normalized frequency content vs. experimental pulse duration IMPULSE FUNCTION
Experimental Pulse Shapes as Compared to Mathematically Expressible Forms. Figure 15 presents the normalized frequency content of a typical experimental pulse as a function of pulse duration times forcing frequency. The figure compares the typical experimental pulse to several mathematically expressible pulses as presented by Clements and Schnelle ( 2 ) . The experimental pulse was seen to be better than a rectangular pulse, but not so good as the weighted, displaced cosine or cubed triangular pulse as far as normalized frequency content considerations were concerned. At low values of uTp (wTp < 7) the experimental pulse compared closely to the half-sine, ramp, and triangular pulse as presented by Clements and Schnelle (2). The latter pulses were not shown in Figure 15, to avoid cluttering. Figure 15 suggests that pulses of the experimental type, with respect to general shape, are apparently satisfactory for some real systems analysis.
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Conclusions
Normalized frequency content is a valid criterion for predicting the reliability of pulse testing results. Pulse results are
TRIANOULAR PULSE
w TP
Figure 15. Normalized frequency content of several pulse curves vs. pulse duration times forcing frequency VOL. 6
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generally unreliable below normalized frequency contents of 0.3 to 0.2 when Filon’s (3) method is used for data reduction. At normalized frequency contents above 0.2 to 0.3, pulse testing results are in excellent agreement with those obtained by direct frequency forcing and mathematical model solution. Excessive experimental pulse time, due to pulse tailing, was the major cause of the inability to extend the dynamic frequency range using pulse testing techniques. Pulsing temperature limits the high frequency results obtainable because the thermal capacity of the physical system results in energy storage and therefore excessive pulse tailing or time duration. For a given pulse shape, the normalized frequency content does not appear to be affected by pulse amplitude or pulse strength. For pulses of long time duration, the time duration rather than rise or decay times was found to be the best criterion for predicting the relative normalized frequency content of the pulses. Further Work
Although many investigators have done an excellent job in advancing the methodology of pulse testing to its present state, there is still a need for further refinement and improvement. Although the method appears to be adequate for the dynamic analysis of most physical systems, difficulties might be encountered in special cases such as the one discussed. Further studies on in-plant pulse-generating techniques along with studies concerning pulse types (one-sided, two-sided, multiple, etc.) and their characteristics with respect to normalized frequency content could be helpful in the practical application of pulse-testing techniques. Also there is a need for further study of existing data reduction schemes with the intent of improving present methods or developing new ones. Studies of the above nature are at present in progress a t Clemson University.
Acknowledgment
The authors are grateful to the National Science Foundation for support of this work. Nomenclature
G’ h
=
MR R r s t
= = = = =
T,
=
mass velocity of air, lb./(hr.)(sq. ft.)
= convective conductance, B.t.u./(hr.) (sq. ft.) (”
u
= = S(w), =
i(w) x
=
X
=
F.)
magnitude ratio tube radius, ft. variable radius, ft. humid heat capacity, B.t.u./(lb.)(” F.) dynamic temperature, ” F. time duration of pulse, sec. local velocity, ft./hr. frequency content normalized frequency content axial distance, ft. total wetted section length, ft.
GREEKLETTERS = thermal diffusivity, sq. ft./hr. 6 = time, hr. p = density, lb./cu. ft. 4 = phase shift, deg. w = frequency, rad./sec. or cycles/hr. CY
literature Cited (1) Bruley, D. F., Prados, J. W., A.Z.Ch.E. J. 10, No. 5, 612
(1964). (2) Clements, W.C., Jr., Schnelle, K. B., Jr., IND.ENG. CHEM. DEVELOP. 2,94 (1963). PROCESS DESIGN (3) Filon, L. G.N., Proc. Roy. SOC.Edinburg 49, 38-47 (1928-29). (4) Hougen, J. O., “Experiences and Experiment with Process Dynamics,” Chem. Eng. Progr. Mono. 60, No. 4 (1964). (5) Middleton, R. C., “Pulse Circuit Technology,” pp. 11-13, Bobbs-Merrill, New York, 1964.
RECEIVED for review April 4, 1966 ACCEPTED January 9, 1967 59th National Meeting, American Institute of Chemical Engineers, Columbus, Ohio, May 1966. Study carried out under NSF Contract GP-3027.
REACTION KINETICS OF CYCLOHEXANOLACETIC ACID ESTERIFICATION D. J.
McCRACKEN AND P. F. D I C K S O N
Department of Chemical and Petroleum ReJining Engineering, Colorado!School of Mines, Golden, Colo.
kinetic studies of alcohol-acid esterifications are cited in the literature. Goldschmidt (6) developed a n equation which related the rate constant to the initial reactant acid concentration, ester concentration, catalyst concentration, protonated alcohol concentration, and time. He tested the equation for a number of esterification reactions a t 25’ C. and found it successful in obtaining the rate constant. Smith (74, 75) confirmed Goldschmidt’s equation for normal aliphatic acids in methanol catalyzed by hydrochloric acid over the range 20” to 50” C. I n these and other studies (7, 8, 9, 73, 76, 77, ZO), either sealed glass ampoules or a ground-glass-stoppered flask was used as the reactor. The studies were also similar in that the alcohol was always in excess, and the organic acid concentration was initially constant. A high alcohol-to-acid ratio was found to result in a UMEROUS
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desirable high yield. previous studies:
A number of conclusions developed from
Organic acid-alcohol esterification is catalyzed by a strong mineral acid. The rate constant is a function of catalyst concentration, temperature, and alcohol-acid molar ratio. The equilibrium constant is strongly influenced by the alcohol-acid molar ratio, weakly influenced by temperature, and not influenced by the catalyst concentration. The reaction is second-order with respect to the organic acid concentration. I n previous experiments involving equimolar cyclohexanol and acetic acid, this author found the rate constant to be a function of the initial reactant concentration as well as temperature and catalyst concentration, and the reaction to be third order. T h e present work involved the study of the
