“catalyst water” during the extended regeneration period, a similar sample of used catalyst was regenerated in the same way and then treated with a moisture-laden stream of air for 14 hours at 325°C. prior to subsequent testing with ethyl acetate. The conversion was even lower (21%), suggesting that surface dehydration is not the source of the decreased activity on regeneration. One further experiment was made to obtain additional evidence on this question. A fresh sample of SK-500 was predried in situ a t 450°C., instead of the usual 275“C., and then tested a t 268°C. The activity was even higher (58% conversion) than that observed in the standard procedure. This confirms that any dehydration occurring a t 450°C. is not harmful, and suggests that the lower conversion after regeneration is associated with the initial coke deposition or with the burning off of the coke. The reason for this is not known.
Literature Cited
Baier, R. W., Weller, S. W., IND. ENG.CHEM.PROCESS DESIGNDEVELOP6, 380 (1967). Beecher, R., Voorhies, A., Eberly, P., Ind. Eng. Chem. Product Res. Develop. 7, 203 (1968). DePuy, C. H., King, R. W., Chem. Revs. 60, 431 (1960). Hanneman, W. W., Porter, R. S., J . Org. Chem. 29, 2996 (1964). Hansford, R. C., Ward, J. W., J . Catalysis 13, 316 (1969). Hurd, C. D., Blunck, F. H., J . A m . Chem. SOC.60, 2419 (1938). Mailhe, A., Chem. Ztg. 37, 778 (1913). Miale, J. N., Chen, N. Y., Weisz, P. B., J . Catalysis 6, 278 (1966). O’Neal, H. E., Benson, S. W., J . Phys. Chem. 71, 2903 (1967). Sashihara, T. F., Syverson, A., IND.ENG.CHEM.PROCESS 5 , 392 (1966). DESIGNDEVELOP. Shah, K. S., Weller, S. W., unpublished results, 1969. Ward, J. W., J . Catalysis 9, 225 (1967). Ward, J. W., J . Catalysis 10, 34 (1968). Wheeler, A., in Emmett, P. H., “Catalysis,” Vol. 2, pp. 105-65, Reinhold, New York, 1955.
Acknowledgment
The authors are indebted to James Wei for a very helpful discussion of the problem of pore diffusivity in zeolites.
RECEIVED for review May 6, 1969 ACCEPTED October 27, 1969
HEAT TRANSFER TO NON-NEWTONIAN PSEUDOPLASTIC FLUIDS IN AGITATED VESSELS 0 . C .
S A N D A L L
A N D
K .
G .
P A T E L ’
Department of Chemical and Nuclear Engineering, Unicersity of California, Santa Barbara, Calif. 93106 Heat transfer to non-Newtonian pseudoplastic fluids in jacketed agitated vessels was studied experimentally for two types of impellers, a six-flat-blade Sieder disk turbine and an anchor agitator. The data were correlated with a SiederTate type equation, based on the use of an effective viscosity in the expressions for the generalized Reynolds and Prandtl numbers. The effective viscosity was determined by application of the relationship between impeller speed and the average shear rate existing in the vessel as proposed by previous investigators in correlating power consumption data for the agitation of pseudoplastic fluids. For a variation in f l o w behavior index of 0.35 to 1.0 approximately 100 data points were obtained for each impeller. For the turbine impeller the data extended over Reynolds numbers from 80 to 93,000 and Prandtl numbers of 2.1 to 644. Data for the anchor agitator were obtained for a variation in Reynolds number of 300 to 90,000 and a Prandtl number range of 2.1 to 644. The resulting Nprl” Nvi,c-0.12 for the turbine agitator correlating equations are: NNU= 0.482 NRe2’3 and NNu= 0.31 5 NRe2’3 N,r1’3Nvi,,-0.12 for the anchor agitator.
BATCH heat
transfer to fluids in agitated vessels is a common industrial process. The research reported in this paper is concerned with heat transfer in agitated vessels to non-Newtonian pseudoplastic fluids whose rheological behavior conforms to the power-law model. T
= Ky”
’ Present address, Cosmodyne Corp., Torrance, Calif. Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970
There have been several studies of heat transfer to Newtonian fluids in agitated vessels. The pioneering investigation of Chilton, Drew, and Jebens (1944) was for a paddle agitator. These workers were able to correlate their data with an equation of the form
(1)
N,” = CNR,”NPrbNvlKc
(2) 139
The exponents on the Reynolds number and the Prandtl number in Equation 2 reported by Chilton et al. were: a = ?4 and b = %. Most of the subsequent investigations report the same values for a and b. I n particular, for the majority of studies involving the two impellers used in this work, the turbine (Brooks and Su, 1959; Chapman et al., 1964; Cummings and West, 1950; Strek, 1962; Uhl, 1954) and the anchor (Uhl, 1954; Uhl and Voznick, 1960), the exponent on the Reynolds numbers has been reported as 2/3 and the exponent on the Prandtl number as %. The exponent on the viscosity ratio number, which takes into account the variation in the physical properties of the system due to nonisothermal conditions has been the subject of some uncertainty in the literature, and similarly there is no consensus on the exact value of the coefficient, C, in Equation 2. This coefficient, however, is expected to vary with the geometry of the systems studied. Recently some attention has been given to heat transfer to nowNewtonian fluids in agitated vessels. Carreau, Charest, and Corneille (1966) studied heat transfer to aqueous solutions of CMC and Carbopol using a four-bladed, 45" pitched turbine as the agitator. The rheological behavior allows these solutions to be classified as pseudoplastic liquids and their shear stress-strain rate relationship can be fairly well represented by the power-law model. The approach taken by these investigators in correlating their data was to transform the conventional Reynolds, Prandtl, and viscosity ratio numbers used for Newtonian fluids in corresponding dimensionless groups for non-Newtonian fluids. They found that their data were best correlated by defining a differential viscosity a t infinite shear rate
Y-
-
(3)
Infinite shearing rate was taken to be 500 set.-' This differential viscosity was used in the Prandtl number, and the viscosity ratio number and a generalized Reynolds number similar to that proposed by Metzner (1956) for pseudoplastic fluids flowing in conduits were used in their final correlation. Pavlushenko and Gluz (1966) used dimensional analysis to obtain the appropriate dimensionless groups for correlating heat transfer during the mixing of non-Newtonian power-law fluids. Their analysis resulted in an equation of the same form as Equation 2 in which an equivalent viscosity is used. /.Le
= K(4nN)" ~
(4)
The factor 4n in Equation 4 is based on an analogy between the motion of fluid between concentric cylinders and in a mixing vessel. Hagedorn and Salamone (1967) used dimensional analysis of the mass, momentum, and energy balance equations to correlate their data on heat transfer to pseudoplastic power-law fluids in agitated vessels. The resulting relationship is
The values for the constants in Equation 5 are different for each of the four types of impellers investigated. 140
COOLING WATER L
STEAM
4
T.C. 7 CONDENSATE
Figure 1 . Experimental apparatus
Experimental
Two types of impellers were investigated in this study, a six-flat-blade disk turbine and an anchor, chosen because the turbine was taken to be representative of the various kinds of nonproximity agitators used for mixing liquids of low consistency and the anchor impeller was considered representative of the proximity class of agitators used for mixing liquids of higher consistency. A schematic sketch of the experimental apparatus is shown in Figure 1. The experiment was designed so that heating and cooling of the vessel fluid could be studied by introducing either steam or cooling water into the vessel jacket. The agitator drive was provided by a g - h p . variable speed motor. The jacketed vessel was constructed by soldering together two stainless steel beakers having inside diameters of 7.22 and 7.91 inches. T o ensure a uniform distribution of fluid inside the jacket, it was equipped with eight vertical baffles and the steam or cooling water was introduced through a %-inch copper tubing ring a t the top of the jacket and was uniformly distributed throughout the jacket through sixty 0.03-inchdiameter holes drilled in the ring. The jacket fluid entered the ring through an O-ring seal. For the experiments using the turbine impeller the vessel was baffled with four equally spaced baffles. Baffles could not be used for the experiments with anchor agitator, since the clearance between the agitator and the wall was too small. The important geometrical parameters of the vessel and the two impellers used are given in Table I. For both the heating and cooling experiments air was removed from the vessel jacket before each run. To maintain a constant concentration of Carbopol in the vessel solutions, water was periodically added to the batch to make up for evaporation losses during the runs. Cooling water rates were measured for each cooling run and several steam condensation rates were measured so that energy balances could be made as a check on the consistency of the data. Temperature Measurement. All temperatures were measured using 30-gage iron-constantan thermocouples aqd recorded using a 12-point Leeds & Northrup millivolt recorder. Four thermocouples (2, 3, 4, and 5 ) were used to measure the bulk fluid temperature, positioned so that their arithmetic average would give the cup-mixing temperature. The vessel wall temperature was measured by two thermocouples (1 and 12) positioned 180" apart and located % and ?4 of the distance from the bottom of Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970
Wall temperatures were measured; however, as a check on the reasonableness of the measurements, the data were taken in such a way that modified Wilson plots could be constructed for the Newtonian fluids used as a check on the measured wall temperatures. The temperature difference between the wall and the bulk fluid as calculated from the Wilson plot intercepts agreed with the measured temperature difference to within an average deviation of 3.6"C. (13%). This was taken to be a satisfactory agreement with the wall temperature measurements. The reproducibility of the data wa5 confirmed by repeating eight runs with an average deviation of 9.57~. This compares favorably with an estimated experimental error of 12.6% (Patel, 1969).
Table I . Geometrical Characteristics of Experimental Equipment
Heat transfer vessel Tank diameter, inches Wall thickness, inch Jacket space, inch Heat transfer area, sq. ft.
7.218 0.045 0.35 1.42
Turbine impeller Diameter, D., inches Kumber of blades Blade length, inch Blade width, inch Disk diameter, inches Distance from vessel bottom, inches Number of baffles Baffle width, inch
2.5 6 0.625 0.500 1.875 2.5 4 0.75
Anchor impeller Diameter, D,, inches Blade width, inch Blade height, inches Wall clearance, inch
7.093 0.750 6.5 0.0625
Correlation of Results
the vessel. The wall thermocouples were soldered into grooves cut in the wall, and to prevent corrosion they were coated with a thin layer of Thermon cement grade T-85 having the same thermal conductivity as stainless steel. The inlet and outlet jacket fluid temperatures were measured with thermocouples 6 and 7. All thermocouples were calibrated against a N.B.S. standardized thermometer. Physical Properties. All of the non-Newtonian fluids used were aqueous Carbopol-934 solutions. Their rheological behavior was measured with a Brookfield Synchro-Lectric viscometer using cylindrical bobs. The density was measured by weighing a known volume. Heat capacities (Ferment, 1962; Friend, 1959) and the other pertinent physical property, the thermal conductivity (Friend, 1959; Vaughn, 1956), were obtained from the literature.
Results
For the two impellers used in this investigation 215 data points were obtained; 97 runs were made using the anchor agitator and 118 runs were made with the turbine agitator. From the data obtained, heat transfer coefficients were calculated using the equation h = q / A ( T , - Tb) (6)
Since the correlation of power consumption data for the agitation of non-Newtonian fluids has met with a fair degree of success, it was felt that the approach taken to this problem might lead to an appropriate correlation for heat transfer. In the correlation of power data in agitated vessels for non-Newtonian power-law fluids, Metzner and Otto (1957) proposed that an average shear rate exists in the agitated vessel and is proportional to the rotational speed of the impeller y. = A,N (8) Metzner and Otto (1957), Calderbank and Moo-Young (1959), and Metzner et al. (1961) used this definition of the average shear rate to define an effective viscosity for power-law fluids pe =
K ( A , N ) "-
(9)
With this relationship for the effective viscosity these authors successfully reduced their power data for several pseudoplastic power-law fluids to a single curve. The constant, A , , was fitted so that the power number for the non-Newtonian fluid agitation is the same function of the Reynolds number as for Newtonian fluids in the laminar region. I n a subsequent paper Calderbank and Moo-Young (1961) further extended this Metzner-Otto approach to allow the average shear rate t o be a function of the flow behavior index, E . The basis for their development is that for the laminar flow of power-law fluids in pipes Metzner (1956) has proposed a generalized Reynolds number as
The rate of heat transfer was calculated from
(7) The heat transfer coefficient for each run was determined a t a bulk temperature of 60°C., except for heating runs using water as the vessel fluid. A bulk temperature of 85°C. was used for these runs; the higher temperature was necessitated by the high rate of heat transfer to water during the startup of a heating run. Equation 7 neglects the rate of temperature rise due to viscous dissipation of energy in the vessel fluid and the rate of heat transfer to the impeller and impeller shaft. I t can be shown, however, that under the conditions of these experiments these simplifications lead to a negligible error (Patel, 1969). Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970
For fluids in agitated vessels the corresponding form of this generalized Reynolds number as proposed by Calderbank and Moo-Young is
where B is a constant replacing the factor 8 in Equation 10 and is determined experimentally. With this definition of the Reynolds number the expressions for the average shear rate and the effective viscosity become:
4n Ya =
BN
.I
1-n
(12)
141
1000 1
I
I 1 1 l l 1 ~
I
I
I
I
I
I I I l l 1
t
I
I
I
I l l (
Y-1 -1
Figure 2. Heat transfer correlation for anchor agitator 0 Water n = 1 .O
b
Glycerol n = 1.0
0 0.5% Carbopol n
= 0.887
's 0.6% Carbopol n = 0.742 A 0.75% Carbopol n = 0.675
a 1.0% Carbopol n
=
0.348
0
IO
Calderbank and Moo-Young have shown that Equations 8 and 12 for the average shear rate are approximately equivalent over a wide range of n. For pseudoplastic fluids they experimentally determined the value of B by matching the power number curves for Newtonian and for non-Newtonian fluids. For the two agitators used in this work their reported values are:
B = 11 f 10% for turbines and
B = 9.5
+ s9 -s 21
=t10%
for anchor agitators.
With this definition of the effective viscosity a generalized Prandtl number can be defined as
On the assumption that n is constant with temperature (a good assumption for the solutions used in this work), the viscosity ratio number becomes
I
I
I I I I I I
I
I
I
I
1
I l l l l
I
I
I
I l l /
agreement with the usual values found for these exponents, 2/3 and % (Table 11). However, the exponent on the viscosity ratio number is considerably less than the generally reported value of -0.14. There is, however, some disagreement in the literature on the exact value of this exponent. Uhl (1954) finds -0.24 for turbine agitators and -0.18 for anchor agitators. Chapman et al. (1964) also report -0.24 for turbine agitators. These values are all higher than the commonly reported value of -0.14 as initially obtained by Sieder and Tate (1936) for heat transfer in pipe flow. On the other hand, Petersen and Christiansen (1966) report the value of c to be -0.10 for heat transfer to pseudoplastic fluids in transitional and turbulent flow in pipes. Malina and Sparrow (1964) obtained a value of -0.05 for heat transfer to Newtonian fluids in turbulent pipe flow. In this work the range in K , / K was 1.4 to 0.45 if the data for glycerol are excluded. With the exclusion of the glycerol data, the K , J K range is rather narrow and therefore not much confidence can be placed in the value of c found in this work. When a and b are assumed to be 2/3 and %, respectively, the exponent on the viscosity ratio number becomes -0.12 Table II. Constants for Equation 16
With these definitions of the appropriate dimensionless variables the heat transfer data were fitted to the form of the equation used for correlating heat transfer data for Newtonian fluids in agitated vessels. NKU
= CNReaNPrbNvlscC
(16)
Linear regression analysis was used to fit the constants in this equation. Two cases were considered for each agitator; in the first all constants in Equation 16 were fitted and in the other a and b were taken to be 2/3 and %, respectively. The values chosen for a and b have been fairly well established for heat transfer to Newtonian fluids. Table I1 summarizes the results of the data fitting and Table I11 summarizes the range of the experimental variables. When all constants are fitted, the exponents on the Reynolds number and the Prandtl number are in fair 142
A'o. of
Impeller
C
a
b
c
Ac. Dec.
Anchor Turbine
0.408. 0.833
0.651 0.594
0.30 0.34
-0.08 -0.02
11.8 16.2
97 118
Anchor Turbine
0.315 0.482
%
!?
'5
3'3
-0.12 -0.12
11.3 18.3
118
Data Pts.
9i
Table 111. Range of Variables
Turbine Speed Flow behavior index, n Flow consistency index, K %U
%, NPI "'visc
(non-Newtonian), lb.: sec. ft.
Anchor
63-1060 17-95 0.35-1.0 0.35-1.0 0.01-8.0 0.01-8.0 92-1100 83-1020 80-93,100 320-89,600 2.1-644 2.1-644 0.2-11.1 0.2-11.1 0.0018-0.052 0.0028-0.027
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970
1000
Figure 3. Heat transfer correlation for turbine impeller Water n = 1.0
N U 0 I
.L"
:
>
Z -bo a
L
z
IO0
Glycerol n = 1.0
-0 0.059'0 Carbopol n = 0.930 J j 0.5% Carbopol n = 0.887 A 0.75% Carbopol n = 0.675 0 1.0% Carbopol n = 0.348
\
3
z z
IO I o3
IO2
I o4
for both agitators and the average deviation of the experimental data from the correlation remains approximately the same as for the case when all constants are fitted. Since the exponents on the Reynolds number and the Prandtl number have been fairly well established for heat transfer to Newtonian fluids in agitated vessels, it is desirable to retain this form of the equation for nonNewtonian fluid heat transfer. Thus the recommended design equations are:
N,,,
= 0.315 NRe2 3Npr1 'Nvisc -0 12
Io5
Conclusions
A correlation has been developed which successfully predicts heat transfer coefficients for pseudoplastic fluids in agitated vessels. The equations developed use data from and are based on previously proposed methods for eorrelating the rate of viscous dissipation of energy in non-Newtonian fluid agitation. The dependence of the Nusselt number on the viscosity ratio number was not conclusively determined in this work and further experiments designed to investigate this effect would be useful.
for t h e anchor a g i t a t o r Acknowledgment
and
NNu= 0.482 NRe2I3 NPr1'3 N v18c . -'"' for the turbine agitator The experimental data for each agitator are plotted according to these equations in Figures 2 and 3. Comparison w i t h Other Work
To compare the correlation developed in this work with previously published correlations for heat transfer to nonKewtonian fluids in agitated vessels, the average deviation of the Nusselt numbers as predicted by each correlation from the experimental Nusselt numbers found in this work was calculated. Since the pre-exponential constant, C, is a function of the geometry of the system, the procedure employed was to determine a value of C for each correlation by fitting by regression analysis the experimental Nusselt numbers of this research to the Reynolds, Prandtl, and viscosity ratio number terms of each equation. The results of these calculations are shown in Table IV. Table IV. Comparison with Other Heat Transfer Correlations
Turbine
c Carreau et al. (1966) Pavlushenko and Gluz (1966) Hagedorn and Salamone (1967) This work
0.469 0.475 0.903 0.482
Au. Deu., %E
37.0 18.7 26.8 18.3
Anchor Au. Deu , ( 2 %
...
...
0.398 0.263 0.315
14.1 19.6 11.3
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970
The statistical analysis of the data was facilitated by the provision of computer time by the Computer Center of the University of California, Santa Barbara. Nomenclature
A = area for heat transfer, sq. ft. A , = proportionality constant in average shear rate expression a = constant in Equation 16 a' = constant in Equation 5 B = constant in Equation 11 b = constant in Equation 16 b' = constant in Equation 5 C = constant in Equation 16 c = constant in Equation 16 c' = constant in Equation 5 c, = heat capacity, B.t.u./lb. F. D, = agitator diameter, ft. Dt = tank diameter, ft. d = constant in Equation 5 d , = pipe diameter, ft. e = constant in Equation 5 f = constant in Equation 5 g = constant in Equation 5 h = heat transfer coefficient, B.t.u./hr. sq. ft. OF. K = consistency index, lb./ft. sec.' K , = consistency index a t wall temperature, lb./ft. sec.' " k = thermal conductivity, B.t.u./" F. ft. hr. m = mass of vessel fluid, lb. N = angular speed, rev./sec. N," = Nusselt number = hD, k N P r = Prandtl number defined by Equation 14 ~
143
Prandtl number used in Equation 5 = C,KN”-‘/k Reynolds number defined by Equation 10 Reynolds number used in Equation 5 = pD;NZ ‘ J K viscosity ratio number defined by Equation 15 flow behavior index heat flux, B.t.u./hr. sq. ft. Dt Do cup-mixing temperature, O F. wall temperature, O F. time, hr. velocity, ft. / sec. width of agitator, ft. shear rate, sec. ’ average shear rate, sec. density, lb. / cu. ft. shear stress, lb./ft. sec.2 differential viscosity a t infinite shear, lb./sec. ft. effective viscositv. lb. ’sec. ft. Literature Cited
Brooks, G.. Su, G., Chem. Eng. Progr. 55, ( l o ) , 54 (1959). Calderbank, P. H., Moo-Young, M. B., Trans. Znst. Chem. Eng. 37, 26 (1959). Calderbank, P. H., Moo-Young, M. B., Trans. Znst. Chem. Erg. 39, 338 (1961). Carreau, P., Charest, G., Corneille, J. L., Can. J . Chem. Eng. 44, 3 (1966). Chapman, F . S., Dallenbach, H., Holland, F. A,, Trans. Znst. Chem. Eng. 42, T398 (1964). Chilton, T. H., Drew, T. B., Jebens, R. H., Znd. Eng. Chem. 36, 510 (1944).
Cummings, G. H., West, A. S., Znd. Eng. Chem. 42, 2303 (1950). Ferment, G., M. S.thesis in chemical engineering, Newark College of Engineering, Newark, X. J., 1962. Friend, P. S.,M. A. Sci. thesis, University of Delaware, 1959. Hagedorn, D. W., Salamone, J. J., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 6, 469 (1967). Malina, J. A., Sparrow, E. M., Chem. Eng. Sci 19, 957 (1964). Metzner, A. B., “Advances in Chemical Engineering,” Vol. 1, Academic Press, New York, 1956. Metzner, A. B., Feehs, R. H., Ramos, H. L., Otto, R . E., Tuthill, J. D., A.1.Ch.E. J . 7,31 (1961). Metzner, A. B., Otto, R. E., A.1.Ch.E. J . 3, 3 (1957). Patel, K. G., M. S. thesis in chemical engineering, University of California, Santa Barbara, 1969. Pavlushenko, I . S., Gluz, M. D., J . Appl. Chem. ( U S S R ) 21, 2146 (1966). Pavlushenko, I. S., Gluz, M. D., J . A p p l . Chem. (Ll!%‘Rj 21, 2323 (1966). Petersen, A. W., Christiansen, E . B., A.1.Ch.E. J . 12, 221 (1966). Siedert E* N . t Tate, G. E., Eng. 289 1429 (1936). Stosowana 6, 329 Uhl, V. W., Chem. Eng. Progr. Symp. Ser. 51, 93 (1954). Uhl, V. W., Voznick, H. P., Chem. Eng. Progr. 56, 78 (1960). Vaughn, R. D., Ph.D. thesis, University of Delaware, 1956. RECEIVED for review May 2, 1969 ACCEPTED August 8, 1969 F.j
CALCULATION OF NET DEVIATIONS FROM CONSISTENCY IN
LOW PRESSURE VAPOR-LIQUID EQUILIBRIUM DATA JOHN
FRIEND’,
W I L L I A M A .
S C H E L L E R ,A N D
J A M E S
H . W E B E R
University of Nebraska, Lincoln, Neb. 68508 In studies of vapor-liquid equilibrium relationships, the quantities P-T-y-x
are
usually measured. Utilizing any three of these and the Gibbs-Duhem expression, the fourth can be calculated. The difference between the calculated and the experimental value then represents the net deviation from consistency in the fourth variable and gives the engineer desiring to produce a safe design a clearer picture of uncertainties than do the reported experimental errors in the measurements. Equations permit evaluation of the net deviation in each of the four measured quantities for systems investigated under low pressure and isothermal conditions. Application is illustrated on four binary systems of differing characteristics.
WHEN osing
experimentally determined vapor-liquid equilibrium data, the design engineer is concerned about the degree of accuracy of the reported values of temperature, pressure, and compositions. The accuracy of these experimental measurements is limited by the capability of the instruments and equipment and the care and skill of the experimenter. Also, errors may be system-
’ Present address, Continental Oil Co., Ponca City, Okla. 144
74601
atic or random in nature. T o check for thermodynamic consistency, the Gibbs-Duhem equation-or a semitheoretical solution of it-is employed. Frequently, although not always, errors reflected as inconsistencies can be determined in this manner. While consistency checks can and should be made and the design engineer has information on the magnitude of the experimental error, these, in themselves, do not indicate to him the largest possible deviation between Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970
