PROCESS CONTROL
Predicting Dynamics of Concentric Pipe eat Exchangers J. M. MOZLEY, Engineering Research Laboratory,
E. I . du Pont de Nemours & Co., Inc., Wilmingfon, Del.
F
OR several decades automatic control has been used in chemical plants to reduce operating costs, to increase product yield and quality, and to maintain a high level of safety. Traditionally, the process equipment for these plants has been designed on a steady-state basis, without much consideration of the dynamic behavior which is so vitally important in determining performance under automatic control. The process design was then handed to instrument personnel who proceeded to specify the instrument “hardware” required. It was hopefully assumed that the built-in flexibility of the instruments, together with a high degree of self-regulation designed into the process equipment would permit “adjustments,” after the plant was constructed, to achieve stable and satisfactory control performance. However, recent trends in process equipment design toward high throughput rates and low holdup of material in process have reduced the self-regulation usually inherent in most chemical process equipment. In addition, the permissible ranges of variation of process operating conditions have been markedly reduced in recent years in order to achieve uniform production of high quality products. These trends have resulted in the evolution of many chemical processes requiring very precise and fast-acting instrumentation systems, which in many cases cannot be built up from off-the-shelf components. Moreover, the economic stakes are getting too large to gamble on being able to correct the instrument and process design after the plant construction is completed. I n the face of these development>s,plant equipment designers as well as instrument personnel, must be supplied with efficient and quantitative methods for ensuring proper control performance in the design stages. This requires that methods for quantitatively predicting the dynamic behavior of chemical process equipment be made available in a form useful for design of automatic control systems. This report presents two methods of dynamic prediction as applied to a specific example from the heat transfer field-a concentric double-pipe heat exchanger. These methods are based on simple mathematical models and passive electrical network analogs. The results predicted by both schemes for a particular heat exchanger mere confirmed by experimental dynamic measurement. Mathematical Formulation of Heat Exchanger Problem The dynamic performance of a concentric pipe heat exchanger may be characterized by the solution of four simultaneous nonlinear partial differential equations. These equations which follow the work of Takahashi (3) are given below with the assumptions: 1. That the fluid temperatures and velocities are uniform across the cross section normal to the direction of flow 2. That the thermal conductivity in the longitudinal direction of the inner wall is constant and in the transverse direction is infinite, a condition valid for thin metal walls 3. That the thermal conductivity in the outer wall in the June 1956
longitudinal direction is zero and in the transverse direction is finite, a condition valid for thick insulated walls Heat balance, inner fluid:
Heat balance, inner wall:
Heat balance, outer fluid:
Heat balance, outer wall: (4)
Boundary conditions:
a_ T, _ dY
- T,o)for y
h,r - (Tz k*
=
0
Obtaining and tabulating solutions of these equations for dynamic analysis and design of heat exchangers is an extremely long and arduous task. Nevertheless, solutions for variations in the inlet temperature of the fluid streams are now available in the literature. None is available as yet for variations in the fluid flow rates-probably because of the difficulty in handling the nonlinear variation of heat transfer coefficient with fluid velocity. Linearized solutions for sinusoidal temperature forcing functions have been obtained by Takahashi (3) and solutions for step forcing functions by Rizika ( 8 ) . These solutions are in somewhat awkward form for plant design purposes, and mere substitution of equipment design constants into the particular solutions involves a great amount of time and expense. Consequently, we have attempted to develop simpler and easier-to-handle mathematical models of heat exchangers that may be applied more rapidly in engineering design and that will still yield calculated results which agree with the rigorous solutions within the limits of usual engineering accuracy Subsequent t o the development of the mathematical approach described here, Takahashi and Paynter (4)developed an ingenious method, based on statistical principles, for approximating the exact solutions we described. Their method gives good agreement with experimental data but is of a higher order of mathematical complexity and not as amenable as ours is to rapid design calculations.
INDUSTRIAL AND ENGINEERING CHEMISTRY
I035
ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT Simple Mathematical Models
~4 Heat Exchanger
Simplification was accomplished for the case of input temperature variations by lumping the distributed properties of the heat exchanger into one point in the longitudinal (2) direction, by assuming constant heat transfer coefficient and heat capacity, and by neglecting the thermal storage of the walls. I n the simplest method of lumping, which we will call Model I, it is assumed that the fluid in each side of the exchanger is well mixed, which means that the exit fluid temperature is equal t o the bulk fluid teniperature. The two differential equations describing this case are linear and of the ordinary type:
These equations may be solved readily for 212 under the conditions of sinusoidal variations in T1, with T 2 ,= 0. This expression is one of the transfer functions of the heat exchanger, relating the dynamic variations of the exit cold fluid temperature to variations in the inlet hot fluid temperature.
Some improvement may be obtained in the simplified mathematical prediction of this transfer function by assuming that the bulk temperature of the fluids in each side of the exchanger is equal to the arithmetic mean of the inlet and exit fluid temperatures. The two linear ordinary differential equations describing this case, which we shall call Model 11,then become
&E
These equations may be solved for the same transfer function before, ?'2,JTli ( j w ) n-ith il;, = 0.
Data calculated from this mathemat>icalNodel 11 Ehow better general agreement with the experimental data for the countercurrent flow case, as will be shown later. Horn-ever, discrepancies are still apparent a t the higher frequencies. It may sometimes be necessary t,o calculate dynamic variations in the exit fluid temperature in response to variations in the inlet temperature of the same fluid. Transfer function expressions for T z / T , (ju) ~ for Model I and T*,,/T1~( j w ) for Ilodel I1 have been derived: 1036
--___ Model I
hlodel I1 T2 (ju) = T ,i 0
-
Where the heat exchanger is more truly a lumped parameter system-Le., more compact, such as ~t jacketed vessel or a tank with submerged coils or tubes-these simplified mathematical models should give more accurate results over a wider frcquency range. It is important to note that with the simplifying assumptions used to derive hlodels I and 11, the maximum phase shift possible as the frequency increases is -180'. The phase shift of the heat exchanger in the region of -180" is important because the Etability of a complete control system, of \x-hich the heat exchanger is one part, depends on the sum of the phase shifts of its components. ;\lost instrument components introduce much smaller phase shifts into the control loop than the heat exchanger itself. Thus, the stability of the complete control loop depends significantly on the phnse shift of the heat exchanger. 9 t the resonant frequency of the control loop, n-hich i8 the frequency a t which the heat excha.nger phase shift is -180", the magnitude ratio of the heat eschanger is very small (for example 0.01) but the instruments must make the complete loop gain almosL 1.0 a t the resonant frequency and therefore must contribute a gain of approximately 100. I n reality, the heat exchanger because of its distributed nat'ure is a nonminimum-phase system-that is, thP phase shift does not approach an asymptotic value but a;>proaches negative infinity as the frequency increases. Thus, ai,-. plication of these simplified models to stability calculations of 8, control eystem containing the heat exchanger should be made with careful attention to these considerations. In addition, as a result of the assumptions made in setting them up, Models I and I1 give the mme transfer function for both cocurrent and countercurrent operation of the heat exchanger. The fact that the dynamic hehavior of the heat exchanger is significantly different in t h e t,wo modes of operation is well known and is demonstrated later by the experimental and analog results. It has been possible to show this difference in dynamic behavior betmen cocurrent and countercurrent operation of the heat exchanger by simulating the heat exchanger by means of a passive electrical network analog. Prediction of the dynamic variations in exit temperature with T2 flow variations-for example - (jw)--is also of practicnl Wt
importance in heat exclinnger design. It is somewhat more diEficult, however, t o develop these equally useful relationships because of the nonlinear form of the defining differential equations. However, approximate transfer functions relating output temperature to input floras may be derived in a similar manner by additional linearization of the equations. The final results of this derivation for Model I are:
INDUSTRIAL AND ENGINEERING CKEMISTRY
VOl. 48, No. 6
PROCESS CONTROL Where sensible heat is transported in a flowing fluid, the ratio of the temperature of the fluid t o the sensible heat flow rate, T / Q , may also be considered as a hypothetical resistance of magnitude l/wc. This resistance has no physical significance but is required as a result of the basic definition of resistance as a potential quantity divided by a flow rate quantity. By following this procedure, we can construct the analog for Model I as shown in Figure 2. The direct mathematical equivalence of the physical system and the analog may be seen from comparing the equations describing both systems. The following equations for the analog are exactly the same, except for symbolism, as those given previously for the physical system (Equations 7 and 8).
T-O(ju)= w1
I
3 '2
- (jw) =
WP
+ U'ICI + U A ] c z ( T z-~ Tz) + U A ] [ J l z ~ z ( j w+) t&ci + U A ] - ( U A ) *
[J!fici(ja)
+
I ~ ~ I c I ( ~CJICI w)
where the bar indicates average value over range of operating conditions.
I
Hat water Out
c
b~~~~,~
C d d Water Out
T.C.
I
1
Rei c
I-in. Schsd. 40 Steel Ptpe
Re,
Hot Woter In
-
Re2
Cold Water In
Cez
d Vz
-dt
TIC.
(Vi
- V,)=
Vii (13)
+ Vz +
l k i n . Sshed. 40 Steel Pipe
Figure 1 .
Physical diagram of heat exchanger, Model i
The dynamic parts of the transfer functions are identical to those for temperature inputs; however, the steady-state parts are modified. These transfer functions have not been checked against experimental frequency response data.
Passive Electrical Network Simulation of Heat Exchanger I n prediction of the dynamics of a piece of physical equipment, i t is often easier to proceed by constructing an electrical analog or model of the real system and then studying the behavior of the analog, The procedure involves selecting electrical components and wiring them properly into an electrical network so that the behavior of the network, when expressed mathematically, is identical with or sufficiently approximates the behavior of the original system.
Figure 2.
Electrical analog of heat exchanger, Model I
For example, if we consider t h a t the heat exchanger may be suitably characterized by our Model I, the electrical network analog for this case may be quickly derived from fundamental engineering considerations. The physical equivalent of the heat exchanger described by Model I is given in Figure 1. By making tcmperature analogous to voltage, and heat flow analogous to current, thermal capacitance becomes analogous t o electrical capacitlance, and thermal resistance becomes analogous t o electrical resistance. Thus, wherever heat is transferred across a thermal resistance in the physical system, we place an electrical resistor of the proper value in the analog, and wherever heat is stored in the physical system, we place an electrical capacitor in the analog. June 1956
By equating the coefficients of the equations describing t h e physical and analog systems, we may find the equivalences existing between the properties of the physical system and the values of the analog components. Thermal Resistance,
Electrical Resistance,
Ri 1 -
Re Rei
WlCl
1
Rez
wzc2
Re,
1 -
Electrical Capacitance, C, C*l C,Z
Thermal Capacitance, C* Mici
UA
-
Mzcz
The actual numerical relationships between the thermal and electrical values depend on the units in which these values are expressed. This brings up the problem of "scaling" the analog. Briefly, the scaling procedure is carried out so that convenient and readily available voltage and current sources may be used t o supply the analog and so that standard types of electrical components may be used to construct the analog. Scaling is often used t o change the time scale of the analog relative t o the real physical system, since it is often convenient t o make the analog events occur faster or slower than in the physical system. Three scale factors must be specified for the analogs used to study the dynamics of heat exchangers. These are: 1. The resistance scale factor, nl, which relates electrical resistance to thermal resistance in the equation, R, = nlRt 2 . The time scale factor, nz, which relates analog time t o real time in the equation, ts = nztr 3. The potential scale factor, m, which relates electrical voltage t o temperature in the equation, V = nsT
The scale factors for capacitance, current, and frequency are also fixed by specification of these three factors as may be seen from the relationships which follow.
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
c,
=
1
n3
2 Ct
i=-Q
ni
.fa
= ~
f
t
7Ll
Once the analog has been constructed in accordance with the equivalences, it is an easy matter to study the dynamic behavior of the analog over wide ranges in important design variables b y merely adjusting variable resistors and capacitors.
and were recorded photographically as shown by the sample given in Figure 4. The values of amplitude ratio and phase shift versus frequencS were scaled from the photographs and were plotted in standard frequency-response form. These analog frequency-response results for both cocurrent and countercurrent flow conditions are given in Figures 5 and 6. ~~~c~~~~~~~ of Heat Exchanger and
Cocurrent
Flow
Dynamic Response The heat exchanger used to check our prediction methods experimentally was a concentric double-pipe type, constructed of 1-inch and 1' 12inch Schediile 40 steel pipe approximately 15 feet it long. Details of its construction are given in Cold Fluid I Figure 7 Bare thermocouples were located as indicated to permit continuous measurement of the inlet and outlet temperatures of both streams. Special care was used to ensure a fast temperatuie measuring and recording system. The thermoHot Fluid In ___c couples used were 30-gage iron-constantan duplex v 11 types having a time constant of less than 0.1 see under the prevailing flow conditions. Tempe1 ature recordings were made on strip-chart record@ Represents .--I ing potentiometers having full scale travel times of 1 see. Thus, the measuring and recording equipFigure 3. Five-lumped section electrical analog of concentric doublement, being- several orders of magnitude faster in pipe heat exchanger response than the heat exchanger, had a negligible effect on the dynamic measurements. Water w a s used as the heat transfer mediumhot water in the central pipe and cold water in the annulus. SineOf course, the analog of Model I would probably never be wave variations in inlet hot water temperature, in the frequency built and tested, since the mathematical solutions to the dynamic range 0 to 11.5 cyclcs/min., werc generated by the apparatus equations are so simple and readily available. However, this shown in Figure 8. simple analog of the heat exchanger mag be regarded as a building The temperature wave form produced by the generator was not block from which we may construct a more useful analog, whose perfectly sinusoidal and did not need t o be, since any irregularities dynamic behavior more closcly approaches the true behavior of were quickly filtered out by the heat exchanger. the heat exchanger as described by Equations 1 to 4. T h a t is, the distributed-parameter heat exchanger may be approximated fairly accurately by a number of lumped systems such as Model I connected together. An electrical analog derived in this manner, having five lumped sections in the longitudinal direction, is given in Figure 3. This circuit is not an exact analogy, since the inclusion of active elements would be required. The results, however, are such t h a t the additional complexity of the exact analogy is not warranted. I n this analog, the thermal storage and t h e longitudinal flov of heat in the metal pipe walls are also taken into account. The thermal values equivalent to the electrical resistances and capacitances shown in Figure 3 are calculated from the relationships 1
Lunn 1
1
LUWDZ I
Lump3
I
Lump4
I
Lump5
I
-
Q
I
cs =
M2~2 ~
Ra = 5w2Cz 1
=
m 1 ,
5
This analog was scaled to give convenient values of resistance and capacitance and to permit using a standard audio-oscillator (20 t o 200,000 cycles/sec. range) to introduce the variablefrequency sinusoidal input voltage. The input and output sinen-ave voltages-analogous t o input and output temperatures in t h e actual system-were displayed on a double-beam oscilloscope
1038
Figure 4.
Photograph o f analog frequency-response results
The frequency response of the heat exchanger outlet cold water temperature to inlet hot water temperature disturbances with all flom-s constant was determined experimentally for both cocurrent and countercurrent operation. The conditions used in these tests were as follows: 1. Peak-to-peak temperature amplitude of inlet hot r a t e r : This varied as a function of frequency. Average value was approximately 5' C.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL 1.0
1.0
0.1
0.1
.01
.01
.001
,001
0
0
.90
-90
-180
-180
.270 .01
*1
1.0
10.
100.
.01
0.1
Frequency, c./min.
Figure 5. Frequency response of heat exchangercomparison of analog and experimental results
1.0
100.
10.
Frequency, s./min.
Figure 6. Frequency response of heat exchangercomparison of analog and experimental results (countercurrent flow)
2. Static temperature level of inlet hot water: Two values were used, 50" and 70" C., with no apparent effect on results. 3. Flow rate of inlet hot water: Approximately 1900 lb./hr. 4. Flow rate of inlet cold water: Approximately 1900 Ib./hr. 5. Temperature of inlet cold water: 10" C.
exchanger by electrical network analog, as described previously, permits much greater accuracy, particularly in phase, a t the higher frequencies.
These results in frequency response form were compared with the analog results in Figures 5 and 6.
Advantages and Limitations of Present Methods
Comparison of Prediction Methods The frequency-response data relating exit cold water temperature t o inlet hot water temperature were calculated by the Model I expression and are compared to the actual frequency response data for countercurrent flow operation in Figure 9. The agreement in both magnitude ratio and phase is only fair. When compared to the actual frequency-response data for cocurrent flow operation, Model I gives about the same agreement in magnitude ratio and somewhat better agreement in phase. The same frequency response irrformation was calculated by the Model I1 expression and is compared to the experimental frequency response data for countercurrent operation in Figure 10. The agreement in magnitude and phase is very good to the frequency where the actual phase shift approaches -180". When compared to the actual frequency response data for cocurrent operation, Model I1 gives somewhat better agreement in magnitude and poorer agreement in phase. I n general, however, both Models I and I1 give fairly reasonable engineering predictions of the static and dynamic behavior of the exchanger for temperature inputs. As previously noted, the maximum phase shift possible in Models I and I1 as the frequency increases is - 180'. This results in increasingly large error in the phase shift predicted by these simple models a t the higher frequencies. Simulation of the heat June 1956
The expressions derived from the simple mathematical models of the heat exchanger have the advantage of being in analytical form. The denominator for each model has two real roots in terms of the complex operator, j w , and may be expressed as the product of two factors. Therefore, the frequency-response plots may be developed rapidly, without laborious point-by-point plotting, through the use of standard nondimensional graphs available in almost any book on servomechanism theory. I n addition, in this factored form, information on the heat exchanger dynamics may readily be combined with instrument information in predicting the performance of a complete control loop containing the heat exchanger. The principal disadvantage of the
W I AT T l i l
-
WI AT TI
. W 2 A t T2
~~
Figure 7.
Concentric double-pipe heat exchanger
INDUSTRIAL AND ENGINEERING CHEMISTRY
1039
ENGINEERING, DESIGN. AND PROCESS DEVELOPMENT
Figure 8.
Temperature sine-wave generator
simplified mathematical models lies in the phase error a t and beyond -180’. The analog prediction of heat exchanger dynamics gives much better agreement, particularly in phase, a t the higher frequencies where the simple models break down. The electrical analog may be constructed from relatively inexpensive, adjustable resistors and capacitors. The resistors used in this work were of the radio volume-control type, and the capacitors were decade units made u p from a number of individual “paper” and “electrolytic” capacitors. Construction of the ana& x-ith adjustable
components permits rapid determination of the effect of changes in design, such as size, geometry, and inaterials of construction, on the dynamic performance of the heat exchanger. The analog approach, however, requires actual construction of an electrical network and physical measurement of its frequency response, which takes somewhat more time than the simple mathematical approach. The choice between the analog and the mathematical approaches for predicting heat exchanger dynamic behavior is usually dictated by accuracy requirements. The simple mathe-
1.0
.-.-
.1
0
d
-0 aJ
E: m
2
.01
0
0
dr V
3 D
-90
-90
?i -180 .01
01
1.0
10.
100.
Frequency, e./rnin.
Figure 9. Frequency response of heat exchangercomparison of calculated results, Model I, and experimental results (countercurrent flow)
1040
.1P@
.ni
.1
1.0
10.
100.
Frequency, c./min.
Figure 10. Frequency response of heat exchangercomparison of calculated results, Model II, und experimental results (countercurrent ¶ow)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL matical models give only a rough semiquantitative estimate of the dynamic behavior of a long distributed parameter heat exchanger but should give excellent results for relatively welllumped systems such as stirred, jacketed vessels. The analog method gives much better prediction of the dynamic behavior of distributed heat exchangers but is surely not justified for study of a compact, well-lumped heat exchange system. This work has demonstrated clearly that the transfer functions and corresponding frequency-response characteristics of heat exchangers depend not only on exchanger design variables but also on the properties of the materials flowing through the exchanger. Thus, i t is not possible to ascribe a definite dynamic behavior to a particular exchanger design, as is common with motors, amplifiers, and other components in the servomechanism field. This means t h a t it is not reasonable to assume that manufacturers of heat exchangers and other similar types of processing equipment will be able to furnish dynamic specifications t o the user. Rather, the user must determine the dynamics of this type equipment for each application on the basis of its process use.
Additional Work Needed
It would be of considerable interest to test the applicability of the simple prediction methods described here to other heat exchangers of more intricate geometry, such as multipass, shelland-tube, or cross-flow types. Perhaps no more sophisticated methods than those described would be required t o predict, within engineering requirements, the dynamic response of these types of exchanger t o temperature inputs. More experimental dynamic data on these more complex types of heat exchanger would be welcomed. I n addition, experimental work is needed to determine the validity and range of applicability of the relationships describing the dynamic behavior of heat exchangers when subjected to flow changes. It is expected that the transfer functions for flow inputs will not agree as well as those for temperature inputs because of the more extreme linearization of the equations. Nomenclature =
heat transfer area, sq. ft.
= inside surface area of inner tube, sq. ft. = outside surface area of inner tube, sq. ft.
inside surface area of outer tube, sq. ft. ft. = cross-sectional area of outsid; $pe, sq. ft. = heat capacity of fluid in inner tube per unit length, P.c.u./(ft. j( O C . ) = heat capacity of fluid in annulus per unit length, P.c.u./ (ft.)(" C.) = electrical capacitance, farads = heat capacity of inner tube wall per unit length, P.c.u./ =
= cross-sectional area of inside tiae. sa.
I C 1 \I 0 r'l \ tLU.J( u.i
= thermal capacitance, P.c.u./" =
= = =
= =
C.
heat capacity of fluid in inner tube, P.c.u./(lb.)(" C.) heat capacity of fluid in annulus, P.c.u./(lb.)(" C.) heat capacity of inside pipe, P.c.u./(lb. j( O C.) heat capacity of outside pipe, P.c.u./(lb.)( O C.) frequency, cycles/min. individual heat transfer coefficient inside the inner tube, P.c.u./(min.)(sq. ft.)(" C.)
June 1956
hz
= individual heat transfer coefficient outside the inner
h,
=
i
=
tube, P.c.u./(min.)(sq. ft.)(', C.) individual heat transfer coefficient inside the outer tube, P.c:u./(min.)(sq. ft.)(' C.) electrical current, amp.
=2/'=1 kl = thermal conductivity of inside pipe, P.c.u./(min.)(sq. ft.)(" C./ft.) kz = thermal conductivity of outside pipe, P.c.u./(min. j(sq. ft.)( ' C./ft.) = thermal conductivity of outer wall per unit length, P.c.u./(ft.)(min.)( C.) L = total length of heat transfer surface, ft. M I = holdup of fluid in inner tube, lb. Mp = holdup of fluid in annulus, lb. M,I = total mass of inside pipe, lb. M,z = total mass of outside pipe, lb. n 1 1 . 2 , 3 = scale factors Q = heat flux, P.c.u./min. R , = electrical resistance, ohms R t = thermal resistance, C./(P.c.u./min.) r = over-all thickness of outer wall, ft. T = temperature, " C. T I = temperature of fluid in inner tube, C. Tp = temperature of fluid in annulus, O C. Th = temperature of inner wall, O C. T, = temperature of outer wall, O C. T," = T , a t w = 0 t = time, min. U = over-all heat transfer coefficient, P.c.u. /( min.)( sq. f t . X O C.) u1 = velocity of'fluid in inner tube, ft./min. uz = velocity of fluid in annulus, ft./min. = voltage, volts = flow rate of fluid in inner tube, lb./min. = flow rate of fluid in annulus, Ib./min. = distance in longitudinal direction, ft. = X / L = dimensionless distance in longitudinal direction = distance in transverse direction, ft. = Y / r = dimensionless distance in transverse direction, measured from inside surface of outer wall = thermal diffusivity of outside wall? sq. ft./min. = L/ul = transportation time of fluid in inner tube, min. = t/Bl = dimensionless time = frequency, radians/min. = 2xf
j
O
O
Subscripts 1 = inside pipe or fluid inside inner Fiipe 2 = outside pipe or fluid in annulus c = cross-section of pipe wall e = electrical quantity h = inner wall i = inlet fluid conditions o = outlet fluid conditions s = outer wall t = thermal quantity w = pipe wall
Refere n ces Brown, G. S., Campbell, D. P., "Principles of Servomechanisms," Wiley, New York, 1948. (2) Rizika, J. W., Trans. Am. SOC.Mech. Engrs. 76, 411-20 (1954). (3) Takahashi, Y.,in "Automatic and Manual Control" (A. Tustin, editor), pp. 235-48, Academic Press, New York, 1952. (4) Takahashi, Y., Paynter, H. M., Am. SOC.Mech. Engrs., Paper 55-SA-50. (1)
RECEIVED for review January 21, 1956.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED March 14, 1956.
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