'<1 fed to a typical co"I proce ss ing fluidizer: bed (S kinner, J970); finE' pa rt ic les in the distribution ha ve a co nsiderable dI"ct on the ca lcu la tio n, Similarly, it is not clear at present exac tly ho w dp should be specified in applying the oorrela- tion for jet penetration depth to a bed with a wide range of particle sizes. ACKNOWLEDGMENT This work was carried out at the Research Laboratories of Westinghouse Electric Corporation and was partially sponsored by the Office of Coal Research. The author wishes to thank Dr. D. L. Keaims for the encouragement and a vice he has o ff er ed d ur ing m an y u se fu l d is cus si on s. NOTATION d o je t no zzl e di ameter d p me an particle diam et er D b initia l bubble diamet er g acceleration due to gravity G volumetric gas flow rate L jet penetration depth measured frO'11 nozzle L o jet penetra tion depth measu re d from apex of co ne Uo je t no zzle velocity Vo = initia l bubble vo lume y c length of conical section of jet (Yc/ L = 0.55) y o = distance between apex of cone a d jet nozzle Gr eek Le tt er s o - je t ha lf -an gle P f density of fluid pP ._ den sity of so li ds LITERATURE CITED }L\"J\'. V.. \., V. J . \Iarkht'vka, T . K h .. ". .k hk . .A khna za ro v, a nd D. 1. Oi oc hko , "rllVL·.<,ligatifJll of the Structure of a ;';on-ulli- l(lrll1 Fl uclizr«] Bc d, " Int. Clicm, Eng., 9,263 (1969). Be-hie, L. A., an d 1'. Kehue, "eric! Region in a Fluidized Bed Da v ul -on, 1. F. and D. Harrison, F lu idtzed P artic le" Cam- bridge 1.fni\·. Press, England ( 1963). Crace, J . R., "Viscosity of Fluidized Beds," Can. ]. Chern. Eng., 48,30 (1970). Harriso , D., and L S. Leuna, "Bubble Formation at an Onfice in a Fluidized Bed," 1'rmls. lust. Chem. Engrs., 39, 409 ( L9(1). Kunii, D., and O. Levenspiel, Fluidi zat ion . Eng ineering, Wiley, N ew Y or k ( 1969 ). Markhevka, V.!., V. _~. Basov, T. Kh. Melik-Akhnazar.vv, and D. l;, Orochko, "The Flow _of a_Cas l e t into a Fluidized Bed, Thcor. Found. Chern. Eng., o, 80 1971). l'.'1 t'rr:; I J . \'1. D., "Penetration ot a Horizontal Cas Jet into a F lui di ze d Be d, " Trans. L nst. Chern. Engrs., 49, 189 (1971). Rowe. P. :SO., "Experimental Properties of Bubbles," in J. . F. Davidson and D. Harrison ( Eds.), Fluidization, p. 139, Aca- demic Press, Xew York (1971). Shakhova, :\. :\., "Outflow of Turbulent Jets into a Fluidized Bed," 1n::/1. Fi::.. z i ; 14 ,61 ( 196 8) . Skinner, D. G .. The Fluidized Combustion of COlli, National Coal Board, Lontlon ( 1970). . Yang, W. C'I and D. L. Keairns, "Recirculating Fluidized-Bed Reactor Data l.'tIlizing a Two-Dimensional Cold Model," AIChE Symp. Ser. No. 141, 70, 27 (l970). Zenz, F. A., "Bubb e Formation and Grid Design," in J . \ i P ir ie , ( ed .) , Fluidization, P: 136, lnst. Chern. Eng., London ( 1968). Mo nuscript received Decembe,· 11, 197 4: ; recisior: received January 28 and accepted January 29 , 197' 5. A Generalized Thermodynornic Correlation Based on Three-Parameter Corresponding States a ru NG II( I.rEE The volumetric and ther modynami c Iun ations correlated by Pitzer and co-work ers analyticall y represe nted with improved acc uracy by a modifie d BWR equation of state. The representation provides a smooth transition lower tempera ture s. It is in a fonn particularly convenient for computer use. The 3- parameter cor respondi ng states principl e as pro- posed by Pitzer and co-workers has been widely used to corr elate the volumetr ic and thermodynar r. ic pr oper ties needed for process design. The original correlations by Pitzer et al., based on that principle, \~'e limited to r educ ed temper atur es above 0.8. Sever al extensions to lower temperatures have appeared in the last five years, Most of these co rr elations are in ta bular or graphical form, difficult to implement on the computer. Also, signiHcant discrepancies appear at the interface (neal' Tr = 0.8) Page 510 May, 1975 AIChE Journal (Vol. 21 No 3) a n d MIC:HAEL G. KESLER Mobil Resee rch and Developmen Co.rporation Post Clffir.e Box 1026 Prineetcn, New Jersey 08540
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fed to a typical co"I processing fluidizer: bed (Skinner,
J970); finE'particles in the distribution have a considerable
dI"ct on the calculation, Similarly, it is not clear at present
exactly how dp should be specified in applying the oorrela-tion for jet penetration depth to a bed with a wide range
of particle sizes.
ACKNOWLEDGMENT
This work was carried out at the Research Laboratories of
Westinghouse Electric Corporation and was partially sponsoredby the Office of Coal Research. The author wishes to thankDr. D. L. Keaims for the encouragement and advice he hasoffered during many useful discussions.
NOTATION
d o jet nozzle diameter
dp mean particle diameter
Db initial bubble diameter
g acceleration due to gravity
G volumetric gas flow rate
L jet penetration depth measured frO'11 nozzle
Lo jet penetration depth measured from apex of cone
Uo jet nozzle velocityVo = initial bubble volume
y c length of conical section of jet (Yc/ L = 0.55)
y o = distance between apex of cone and jet nozzle
Greek Letters
o - jet half-angle
P f density of fluid
pP ._ density of solids
LITER ATU RE CITED
}L\"J\'. V.. \., V. J . \Iarkht'vka, T. Kh.. "..khk ..Akhnazarov, andD. 1. Oi ochko, "rllVL·.<,ligatifJll of the Structure of a ;';on-ulli-
Be-hie, L. A., an d 1'. Kehue, "eric! Region in a Fluidized BedHeactnr,"AIChE I, 19, 1070 ( 1973).
Dav ul-on, 1. F. and D. Harrison, F lu id tzed P artic le" Cam-bridge 1.fni\·. Press, England ( 1963).
Crace, J . R., "Viscosity of Fluidized Beds," Can. ]. Chern. Eng.,48,30 (1970).
Harrison, D., and L S. Leuna, "Bubble Formation at an Onficein a Fluidized Bed," 1'rmls. lust. Chem. Engrs., 39, 409( L9(1).
Kunii, D., and O. Levenspiel, Fluidization. Engineering, Wiley,New York (1969).
Markhevka, V.!., V. _~. Basov, T. Kh. Melik-Akhnazar.vv, and
D. l;, Orochko, "The Flow _of a_Cas l e t into a FluidizedBed, Thcor. Found. Chern. Eng., o, 80 1971).
l ' . ' 1 t'rr:; I J . \'1. D., "Penetration ot a Horizontal Cas Jet into aFluidized Bed," Trans. L nst. Chern . Engrs., 49, 189 (1971).
Rowe. P. : S O . , "Experimental Properties of Bubbles," in J . . F.Davidson and D. Harrison (Eds.), Fluidization, p. 139, Aca-
demic Press, Xew York (1971).Shakhova, :\. :\., "Outflow of Turbulent Jets into a Fluidized
Bed," 1n::/1. Fi::.. zi; 14,61 (1968).Skinner, D. G .. The Fluidized Combustion of COlli, NationalCoal Board, Lontlon ( 1970). .
Yang, W. C'I and D. L. Keairns, "Recirculating Fluidized-BedReactor Data l.'tIlizing a Two-Dimensional Cold Model,"AIChE Symp. Ser. No . 141,70,27 (l970).
Zenz, F. A., "Bubble Formation and Grid Design," in J . \iPirie, (ed.), Fluidization, P: 136, lnst. Chern. Eng., London( 1968).
Mo nuscript received Decembe,· 11, 1974:; recisior: received January
28 and accepted January 29, 197'5.
A Generalized Thermodynornic Correlation
Based on Three-Parameter Corresponding
States
aru NG II( I.rEE
The volumetric and thermodynamic Iun ations correlated by Pitzer and
co-workers analytically represented with improved accuracy by a modified
BWR equation of state. The representation provides a smooth transition
between the original tables of Pitzer et al. and more recent extensions to
lower temperatures. It is in a fonn particularly convenient for computer use.
The 3-parameter corresponding states principle as pro-
posed by Pitzer and co-workers has been widely used to
correlate the volumetric and thermodynarr.ic properties
needed for process design. The original correlations by
Pitzer et al., based on that principle, \~'e limited to
reduced temperatures above 0.8. Several extensions to
lower temperatures have appeared in the last five years,
Most of these correlations are in tabular or graphical form,
difficult to implement on the computer. Also, signiHcant
discrepancies appear at the interface (neal' Tr = 0.8)
'1 mpressibihty f Lro~' and t'le denved'. 1: ) •rmc p operties of other substance." m addition
to [limp or octane.
Bnefiv, the work consisted of the following:
1. Modification of Benedict et al. equation of state(1940) as represented by Equation (3).2. Fitting the constants in Equation (;3) \.Ising experi-
.mental data on enthalpy, P-V-T, and the second virialcoefficient.
S. Derivation of a new vapor pressure equation and itsuse to derive an equation for estimating acentric factors.4. Use of a new set of mixing rules to define critical
temperatures and pressures and acentric factors of mix-tures.
These steps are described below in further detail.
DESCR IPTION OF W ORK
The compressibility factors of both the simple fluid Z(O)
and the reference fluid z(n have been represented by thefollowing reduced form of a modified BWR equation ofstate;
(PrVr) BCD
Z=-- =1+-+--+--r. v, Vr2 Vr5
where
B = b, - b21Tr - ha lTr2 - b41Tr3 (4)
C = CI - c21Tr + calTra (5 )
D = d, + d21Tr (6)
In determining the constants in these equations, the fol-lowing constraints, Equations (7) and (8), were usedalong with the data shown in Tables 3 and 4.
i" = t': (at saturated condition) (7)
( a P r ) ( a2P r )
aV r r, = aV r2 Tr = = 0(at critical point) (8)
TABLE 2. COMPARISON OF REDUCED VAPOR PRESSURE
-LogPr(O) -Log PrellCarruth- Carruth ..
Tr This work Pitzer et al. Kobayashi Hsi-Lu This work Pitzer et al . Kobayashi Hsi-Lu
59.7%-40.3% 13 0.58-1.04 0.46-3.25 2600 Lenoir et al. (1970)
n-pentane-cyclohexane( 1971)1.2%-38.8% 19 0.83-1.06 0.38-2.66 2600 Lenoir et al.
n-pentane-n-hexadecane58.7%-41.3% 26 0.55-0.98 0.08-4.22 7400 Lenoir and Hipkin (1970)
Benzene-n-hexadecane
58.1 %-41.9% 18 0.74-0.87 0.31-4.4 4600 Hayworth et al. (1971)
• Data used .in correlational work.
5 - 5°+In( ;0 ) = In(Z) b, +b31Tr2 +2b4/T r3
R Vr
CI - 2c31Tr3 d1(12)
2Vr2----+2E
5Vr5
where
(15)
(a P r ) T T { 2 B 3C 13D- =-- 1+-+--+--sv, r, Vr2 v, Vr2 Vrs
+ ~ v 4 2 [3/3 + { 5 - 2 ( , 8T; r
+ _ ! _ ) l2_ ] exp (- -~)}Vr2 j Vr2 . V
r2
To calculate any of the quantities given in Equations
(9) through (16), given a T and P for the fluid ofinterest, the following procedure, illustrated by the en-thalpy departure function, should be followed.
(16)
P erge 514 M ery, 1975 AIChE Jeurnel (Vol. 21,
1. As described earlier, determine Vy and Z(O) for the
simple fluid at the T; and P; appropriate for the fluid ofinterest. Employ Equation (11)and, with the simple fluidconstants in Table 1, calculate (H - H O ) / R T c . Call this[(H - HO)/RTc](OJ. In this calculation, Z in Equation(11) isZ(O)_
2. Repeat step 1, using the same T; and P r, but em-ploying the reference fluid constants from Table 1. A newVT and Z(T) will be obtained. With these, Equation (11)
allows the calculation of [ ( H - H O ) /R T c ] ( T ) . In thiscalculation, Z in Equation (J I) is Z(T).
3. Determine the enthalpy departure function for thefluid of interest from
[ ( H - H O ) /R T c ] = [ ( H - H O ) /R T c ] ( O ) + ( W l « / T »
{ [ H · _ H O ) I R l 'c ] ( r J - [ ( H - H O ) /R T c ]C O l }
~~ ...... , Carruth and Kobayashi (H.l73),an Wilhoit and Zwolinski (1971) to r heavy hydrocarbons.
the equation is given below:
(17)
This equation satisfies:
1. The definition of acentric factor, UJ = -log (P/)- 1at T T = 0.7.2. The Riedel condition dal dT T = 0 at T T = 1.0.
3. The critical point requirement, P r" = 1.0 at 1 'T =1.0.
Equation (17) gives the following relation between
Riedel's parameter and the acentric factor;
L J ! _<; .. ;; r .. ;. o ;~ ~ . .. .. . . : > , : a ' M I : 18)
(20)
(21)
(22)
(23)
EVALUATION AND DISCUSSION
Figures I through 4 compare the results obtained f::omthe proposed correlations with those of other methodsin calculating compressibility factors and enthalpy de-partures. As can be seen, the proposed analytical method
is in good agreement with other tabular correlations forsimple fluids. The comparison of this correlation withexperimental data for real fluids is also favorable asshown in Table 3 and 4. Similar c.greement was also
AIChE Journal (Vol. 21, No.3)
observed for entropy values and fugacity coefficients. On
the other hand, the agreement in t' . deviation functionsas defined by Pitzer and CO-WOI' is onlv fair. Thedisagreement, however, is probably due to inaccuraciesin the original deviation functions rather than in thoseof new correlations. As shown in Table 3, this method was
successfully applied to 1 'T and P T as high as 8.7 and 31,respectively.Since Pitzer's method, as adopted in this work, is less
accurate around the two-phase region, the compressibility
factors of saturated vapor and liquid were also evaluated.These are given in Table 3. As expected, the accuracy inthis case is slightly poorer than that of sub cooled orsuperheated fluids. In this connection it should be pointedout that the values at saturation conditions were obtainedby extrapolating isotherms into the saturation envelopesof the simple and reference fluids. This is in line withthe Curl and Pitzer's observation (1958) that fluids with
higher acentric factors have lower reduced vapor pres-
sures at the same reduced temperatures.The second virial coefficients obtained from Equation
( 4) and the equivalent form of Equation (2) are com-pared in Figures 5 and 6 with those of other correlations(Pitzer and Curl, 19.57; Tsonopoulos, 1974). Again the
( 19)
values obtained from Equation (4) compare favorably
with literature data.Isobaric heat capacity equation, Equation (14) ,. was
also evaluated using the methane and propane data of
-··-oT~(h- -t4~~(tf6~------t--~i-----;4i---t-, - - c . t - - - ; ' C " C ' ! - ! : ; : 9 JO '
R(DUCED PfO:ESSURE
[
H * -_ H ] ( 0 )Fig. 3. Comparison of
RT c
~ , ' , ," -~'""-."
i ~ ~ ~ - - - - - : - - - - ; - - - - - - - -:o CLlr i and Pit ze r 11958 )
• Chao OI" IdGr ll tlnkorn (1971 )
- Thi$Work
O J 0.2 03 04O~o\;-t,o~'g+---+----i----+-+-+-+~R ED UC ED P RE SS UR E
[H * -- H - 10)
Fig. 4. Comparison ofRTc •
0,---------------
'"0EI
co-ISO;;
Eo
af
-200
-250
-30080 100
T his W ork
Trcropoulos (19741
P it ze r a nd C url (1957)
Byrne, ot ct. (1968)
We ir , e t 01. (1967)
160
TEMPERATURE O K
Fig. 5. Second virial coefficients of simple fluids.
220
Page 526 May, 1975 AUChE Journal (Vol. 21, No.3)
-1500r-----r·---r-----r-----r~1
/
-1700
~ -1800
E,
sS -1900
£Ii
-2000
This Work
Tsonopoulos (1974)
Pi tzer and Curl (1957)
o McGlashan andPotter (1962)
370 42080 400 41090
TEMPERATURE 01(
Fig. 6. Secend virial coeff icient of n-octene.
It should be pointed out that the critical propertiesused throughout this work, excel,lt for the critical pressureof 1-pentene l P ; = 39.9 atm. from Reid and Sherwood,1966) were taken from the API Data Book, while theacentric factors were obtained from Equation (19).
ACKNOWLEDGMENT
The authors are grateful to Mobil Research and Develop-ment Corporation for the permission to publish this paper.
N')TATION
bI>b2, ba , b4 = constants as given in Table 1
Cl> C2, Ca, C4 = constants as given in Table 1dhd2 = constants as given in Table 1B, C, D = coefficients in Equation (3)Cp = isobaric heat capacityC" ;::: isochoric heat capacityE = defined by Equation (10)
f = fugacity
H enthalpyP = pressure
reference pressure for ideal gas state entropygas constantentropytemperaturevolume
PcVIRTcmolar compositioncompressibility factor
Greek Letters
a Riedel's parameterf3 constant as given in Table 1
i, i. k = = component identificationr = reduced property
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Carruth, G. F., and R. Kobayashi, "Vapor Pressure of NormalParaffins Ethane Through n-Decane From Their Triple Pointsto about 10 mm Hg," J. Chern. Eng. Data, 18, 115 (1973).
Chao, K. C., and R. A. Greenkom, "Enthalpy and Entropy ofNon-polar Liquids at Low Temperatures," Nat. Gas Proc,
Assoc., Research Report RR-3, Tulsa, Okla. (1971).------., O. Olabisi, and B. H. Hensel, "Fugacity andVapor Pressure of Non-polar Liquids at Low Temperatures,"AlChE]., 17,353 (1971).
Chappelow, C. C., P. S. Snyder, and J. Winnick, "Density ofLiquid n-Octane," J. Chern. Eng. Data, 16,440 (1971).
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Ind. Eng. Chem., 50, 265 (1958).Dawson, P. P., Jr., I. H. Silberberg, and J. J. McKetta, "Volu-.metric Behavior, Vapor Pressures, and Critical Properties ofNeopentane," ]. Chern. Eng. Data, 18, '7 (1973).
Day, H. O. and W. A. Felsmg, "The Compressibility of Pen-tene-l," J. Am. Chern. Soc., 73, 4839 (1951).
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Hayworth, K. E., J. M. Lenoir, and H. G. Hipkin, "Enthalpiesof Mixtures of Benzene and Hexadecane," J. Chern. Eng.Data, 16,276 (1971).
Hsi, C., and B. C. Y. Lu, "Generalized flP, ZT and Zp Valuesfor Liquids at Low Reduced Temperatures," AIChE J. , 20,
109 (1974).
Jones, J. L., Jr., D. T. Mage, R. C. Faulkner, Jr., and L. D.Katz, "Measurement of the Thermodynamic Properties ofGases at Low Temperature and High Pressure-Methane,"
Chern. Eng. Prog. Symp. Ser. No. 44, 59,52 (1963).Lenoir, J . M., K. E. Hayworth, and H. G. Hipkin, "Enthalpiesof Benzene and Mixtures of Benzene with n-Octane," J.Chern. Eng. Data, 16,280 (1971).
Lenoir, J. M., and H. G. Hipkin, "Enthalpies of Mixtures ofn-Hexadecane and n-Pentane," ibid., 15,368 (1970).
Lenoir, J. M., G. K. Kuravila, and H. G. Hipkin, "Enthalpiesof Cyclohexane and Mixtures of n-Pentane and Cyclohex-ane," ibid., 16,271 (1971).
Lenoir, J. M., C. J. Rebert, and H. G. Hipkin, "Enthalpy ofcis-2-Pentene and a Mixture with n-Pe ntane," ibid., 16, 401
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tane," iu«, 15, 23 (1970).
.. ,
n-Pentane," ibid .• 15,26 (1970).Lu, B. C.-Y, C. Hsi, S. D. Chang, and A. Tsang, "VolumetricProperties of Normal Fluids at Low Temperatures: An ex-tension of Pitzer's Generalized Correlation," AlChE J., 19,
748 (1973).~lcGlashan, M. L., and D. J . B. Potter, "An Apparatus for theMeasurement of the Second Vi rial Coefficients of Vapors;
the Second Virial Coefficients of Some n-Alkanes and ofSome Mixtures of n-Alkanes," Proc. Royal Soc. Ser. A, 267,
478 (1962).Michels, A., J. M. Levelt, and W. DeGroaff, "CompressibilityIsotherms of Argon at Temperatures Between -25°C and
Issac, and at Densities up to 640 Amagat," Physica, 24,659 (1958).
Passut, C. A., and R. P. Danner, "Acentric Factor. A ValuableCorrelating Parameter for the Properties of Hydrocarbons,"Ind. Eng. Chern. Process Design Deoelop., 12, 365 (1973).
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Pitzer, K. S., D. Z. Lippmann, R. F. Curl, Jr., C. ,\1. Huggins,and D. E. Peterson, "The Volumetric and ThermodvnamicProperties of Fluids. II. Compressibility Factor, Vapor Pres-sure and Entropy of Vaporization," ibid., 77, 3433 (1955).
Prydz, R., and R. D. Goodwin, "Experimental Melting andVapor Pres-ures of Methane," ]. Chern. Thermodynamics,4, 127 ( 1972 ) .
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---., "volumetric and Phase Behavior in the Hv-drogen-n-hexane System," AICilE J., a , 262 (1957). '
------., "Phase Equilibria in Hydrocarbon Systems:Volumetric and Phase Behavior of the Propane-n-DecaneSystem, J. Cliem. Eng. Data, 11, 17 (1966).
Reamer, H. H., B. H. Sage, and W. N. Lacey, "Phase Equilibriain Hydrocarbon Systems: Volumetric and Phase Behavior ofEthane-rr-Pentane System, ibid., 5, 55 ( 1960).
Reid, R. C., and T. K . Sherwood, The Properties of Gases andLiquids, ~lcGraw-HiIl, New York (1966).
Riedel, L., "Eine Neue Universelle Dampfdruckformel," Chem.
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dqnanuc Properties of Hydrocarbons and Related COIII-
pounds, Carnegie Press, Pittsburgh, Penn. (1953).
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Tsonopoulos, C., "An Empirical Correlation of Second VirialCoefficients," AICilE J ., 20, 26:3 (1974).
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Mrunsct ipt received November 20, 1974; rev is ion receiced JamUlf!)