Estimation of Saturated Liquid Heat Capacities K. P. Tyagi Mechanical Engineering Department, Birla Institute of Technology and Science, Pihni (Rajasthan), Mia
For the estimation of saturated liquid heat capacities, the expressions given by Reid and Sobel have been used Exwith a modification in the procedure followed for the estimation of the quantities dHsL/dT and (dQ/d pressions proposed by Lee and Edmister, and Stevens and Thodos, when used in Reid and Sobel expressions, yield values in excellent agreement with experiments.
nsL.
There are several existing methods for estimating the liquid heat capacities. The more important ones are those reported by Bondi (1966); Chueh and Swanson (1973); Reid and Sobel (1965); Reid and Sherwood (1966); Shaw (1966); Johnson and Huang (1955); Watson (1943); Yen and Alexander (1965); Yuan and Stiel (1970). All the correlations available are generally applicable in the room temperature range. The energy-mode method of Sakiadis and Coates (1956) is difficult to use. It is not applicable for hydrocarbons with fewer than five carbon atoms and it cannot ordinarily be used for temperatures above the normal boiling point. For reduced temperatures above T, of 0.7 but below 0.95, the method of Watson as modified by Reid and Sobel appears satisfactory, but significant deviations occur between the experimental values and calculated values. Further, these methods are not consistent with each other. Reid and Sobel (1965) defined the liquid heat phase heat , s ~and , (dQ/dT),L. The capacities in three terms, C p ~ C first term represents the change in enthalpy with temperature at constant pressure; the second term shows the change in enthalpy of the saturated liquid with temperature, along the saturation curve, dH,L/dT; and the third term is the heat necessary to effect a temperature change if the liquid is held in a saturated liquid state. All three terms are in close agreement with each other except near the critical point. The expression for (dQ/dT),L is as follows
(2).
= C'SL =
Cp"
As the 4 functions, Uvand p s ~ were , obtained by graphical computation, which resulted in large deviation from the experimental values. Further, the graphs used by these authors are not suitable for computer calculations. Chueh and Swanson (1973) have modified the method of Shaw (1969) and Johnson and Huang (1955). The authors have estimated the heat capacities for about 140 compounds at 20°C. The agreement between the experimental values and the estimated values was not satisfactory. The following equation was used for saturated liquid heat capacity dX - d(H" - H s g ) --
CSL= -- C," dT a s L
(3) dT dT The derivative of (H" - Hag)was obtained by the following equation d(H" - Hsg) VdUldT - UdV/dT = T, dT V2 where
+
V = 1 F(-ln PwI) In Pvpr= -In P, -d= V
dT
+ 2.3034 - 2.303B/(T - C)
-F.G.(-ln
dU
[ 2.303B ]
= D.E.(PV,,IE dT ( T - C)' = XI((T,- T)/(T, - Ti))" n=a
484
Ind. Eng. Chem., Process Des. Dev., Vol. 14. No. 4, 1975
(54 (5b)
Pvpr)G-l
U = D(P,,)E
+ 40[(41/40) -
The parameter hz, is the difference between compressibility factors of saturated vapor and liquid. The partial derivatives of (H" - Hag)in the functions of 41 and 43 were obtained by Reid and Sobel (1965) by graphical differentiation of the tables of Lydersen et al. (1955), wherein (H" - H)/T, are tabulated as a functions of TI, PI, and 2,. The functions 41 and 43 were plotted against reduced temperature (T, from 0.7 to 0.95) for values of compressibility factors 2, varying from 0.23 to 0.29 at an interval of 0.02. The functions 4 2 and 4 4 were also plotted against T, for various values of Tb,. The parameter hz, was plotted a function of reduced vapor pressure.
(4)
+ PT + yT2 + 6T3+ tT4
(54 (5e) (50 (5g)
A, B, and C are Antoine constants. The values of D, E, F, and G in the above equations, as given by these authors as a functions of Z, are plotted in Figure 1. It is not possible to interpolate or extrapolate the values of the functions D, E, F, and G, accurately corresponding to any value of 2, below 0.23 or for any other value of Z,, that lies between the given limits. The constants a,8, y, 6, and t may be obtained when latent heats are available over a wide temperature range. It becomes difficult to predict the values of constant n when sufficient experimental data are not available. It is observed that the saturated liquid heat capacities can be estimated more accurately if the term (H" - Hsg)in the expressions of Reid and Sobel and in the expression of Chueh and Swanson can be expressed as a function of reduced pressure and reduced temperature. In this paper simple analytical procedures are presented, using the equations for enthalpy departure from their ideal
state, suggested by Lee and Edmister (1971) and Stevens and Thodos (1963) for predicting the functions dH,LldT and (dQ/dT),L. Method 1. The function dH,L/dT, as defined in eq 3 may be written as
X
D
. E A F G
DATA FROM CHUECH 6 SWANSON (1973)
+
dHs~ - ddT - H O ) CPo (6) dT Lee and Edmister (1971) described the following generalized equation for isothermal enthalpy difference for pure liquids CSL = -- -(H,L
H,L - Ho = A2 - A3T, R TC (A6 - A7T,
- 2A4Tr3- 6A5Tr7+
- 2AsTr3)P, - 3AgTr4Pr2+
(AloTr2 + A11 +
- 3A13Tr4Pr2) (7)
where A,+ are the generalized constants defined as: A1 = 6.32873; A2 = -8.45167; A3 = -6.90287; A4 = 1.87895; A5 = -0.33448; A6 = -0.018706; A7 = -0.286517; As = 0.18940; A9 = -0.002584; A10 = 8.7015; A11 = -11.201; A12 = -0.05044; A13 = 0.002255. By differentiating eq 7 with respect to temperature, the function d/dT(HsL - HO) is derived as d -(H,L dT
-
H O )
= R(-A3
Pr(-A7
- 6A4Tr2 - 42&Tr6 +
Figure 1. Constants D, E, F, and G vs. Z,: X, D;0, E; A, F; m, G (data from Chuech and Swanson, 1973).
+ 45.652, - 48.75ZC2 k = 7.401 - 39.652, + 43.75ZC2
m = -7.424
(2A10Tr - 12A13Tr3Pr2))) (8)
m
41 = - - (mT, + k ) ( l / P ) - l
(16)
43 = 0.0
(17)
P
The ideal heat capacity CPo is calculated by the method suggested by Rihani and Doraiswamy (1965) in the form
C," = a + b T + c T 2 + d T 3
(9)
where a, b, c, and d are constants. By substituting eq 8 and 9 in eq 6, C,L may be calculated. Method 2. Stevens and Thodos (1963) described the relation for enthalpy departure for saturated liquid as
where a , 6 , and y are related to the critical compressibility factor 2, as follows
P = -335.52
+ 2559.52, - 5O12.5Zc2 + 2686.52, - E1237.52,~
y = 252,
-
csL
=
lop (a
r
-
the method of Lu
+ Y ( l - T,) + W(l - Tr)l/3
(19)
Pc
The parameters Y and Ware defined as
W = 1.75238
+ 0.74293~
(20b)
+ a + bT + cT2 + dT3
Ole)
(13)
Method 3. In this method the relationship given by Stevens and Thodos (1963) for enthalpy departure of saturated vapor from the ideal gas state, is used for calculating the functions 41 and 43. Their relation is 1 ,
p , ~ ,= p. = 1
p , ~ ,by
(204
- 2.25
where the parameters p , m, and k are defined as
(iii) obtain reduced liquid density et al. (1973).
(18)
Y = 0.73098 + 0.28908~
P = 10 -(a
PTr)(l'Y)-l
AZ, = [ l - (1/(P,*Tb,3)]1/2
(llb)
- PT,)(l/Y)-' (12) dT 7 By substituting eq 12 and 9 in eq 6, the parameter C,L yields as H O )
In eq 1, other functions are obtained as: (i) obtain 40,4 2 , and 44 by the help of eq 2.a, 2.c, and 2.e, respectively; (ii) obtain AZv by the following relation
(lla)
By differentiating eq 10 with respect to temperature, the term dldT(H,L - H") may be written as d -(H,L
(154
From eq 14 and 15, the functions $1 and y3 are derived as
- 6AsTr2 - 12A9Tr3Pr2)+
a = -317.49
(15b)
The reduced critical density p, is replaced by scaling as 1= -
v,,=
v0.6
(21) 0.3862 - 0.0866~ The quantity vO.6 is the molar volume of the liquid evaluated a t the reduced temperature of 0.6. By substituting the 4 functions, AZv, p , ~ , , and CPo in eq 1the parameter C,L may be calculated. Table I lists the various compounds for which the liquid heat capacities are obtained by the present three methods. These calculated values are compared with experimental values. The liquid heat capacities for ethylene glycols calculated by present methods are compared with experimental values as shown in Figure 2. All the compounds listed in Table I were tested by Johnson and Huang method a t 2OoC and an average error of 5.4% was observed. The method of Sakiadis and Coates was tested only for those compounds in which carbon atoms are more than 5 , and for temperatures below normal boiling point. In this method an average error of 0.7% was observed. Pc
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
485
Table I. Deviations between the Calculated and Experimental Saturated Liauid Heat CaDacities Absolute % deviation No. of
(11
(2)
(3)
Data source
Freon 11
23
1.79
2.10
2.11
Freon 12
22
1.64
2.54
2.30
Freon 13 Freon 14 Freon 2 1
29" 17" 30
2.08 2.82 1.59
2.47 3.17 1.41
3.12 3.10 1.79
Freon 22
30
2.12
2.44
2.56
Freon 113
24
2.24
2.98
3.25
Methyl bromide
20
2.63
3.42
3.79
Ethyl bromide
20
2.11
2.71
2.68
For maldehyde
27
1.97
2.19
2.17
Acetyl chloride
25"
1.08
1.26
1.25
Methyl ethyl ketone
15"
1.33
1.4
1.5
Methyl ether
55
1.04
1.03
1.1
Ethyl ether
55
0.79
0.87
0.97
Propyl ether Butyl ether Methyl methacrylate Isopropyl acetate Ethyl formate Dim ethy 1 acetamide
40' 50' 40" 40' 40" 49
1.o 0.97 1.34 1.63 1.26 0.63
1.9 1.16 1.45 1.52 1.2 0.87
1.89 1.21 1.62 1.6 1.27 0.94
Dimethylformamide
49
0.84
0.99
1.o
Aniline
45
1.46
2.18
2.20
Pyridine
50"
1.8
1.9
2 .o
Benzene
55
1.4
1.5
1.51
Ethyl benzene
54
1.05
1.4
1.6
Propyl benzene
50
0.85
0.90
0.86
Toluene
38
1.44
1.47
1.51
Methyl chloride
44
1.5
1.5
1.6
Acetone 1-Butanol Ethanol 1-Propanol
20 25 25
2.02 0.98 1.02 2.12
2.15 1.10
2.13 1.10
2.4
2.7
Gallant (1968a), Benning and McHarness (1940) Gallant (1968a), Air Conditioning Data Book (1955) Gallant (1968a) Gallant (1968a) Gallant (1968b), Benning and McHarness (1940) Gallant (1968b) Benning and McHarness (1940) Gallant (1968b) Benning and McHarness (1940) Gallant (1968~) Timmermans (1950) Gallant (1968c) Timmer mans (1950) Gallant (1968d3, Kobe and Pennington (1950) Gallant (1968e), T immer mans (1950) Gallant (1968f), Timmermans (1950) Gallant (1968g), Kennedy et al. (1941) Gallant (1968g), Timmermans (1950) Gallant (19686) Gallant (1968g) Gallant (1968h) Gallant (1968i) Gallant (19681') Gallant (1969a), Du Pont Bulletin (a) Gallant (1969a), Du Pont Bulletin (b) Gallant (1969a), Timmermans (1950) Gallant (1969a), Timmermans (1950) Gallant (1968b), Timmermans (1950) Gallant (1968b), Guthrie (1944) Gallant (1968b), Messerly (1965) Gallant (1968c), Timmermans (1950) Awbery and Griffith (1940) Kelley (1929) Parks (1925) Kelley (1929) Parks and Huffman (1926)
Substance
406
Method
data point
7
1.o
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
1.o
Reduced temperature range 0.4 3-0 -81 0.40-0.80 0.48-0.98 0.7-0.98 5 0.49-0.84 0.5 1-0.91 0.46-0.80 0.41-0.80 0.4-0.8 5 0.49-0.80 0.4 5-0.90 0.4 2-0.71 0.44-0.9 5 0.44-0.90 0.44-0.85 0.44-0.8 5 0.4 7-0.7 0 0.46-0.80 0.51-0.80 0.4 2-0.85 0.46-0.78 0.464.78 0.4 5-0.84 0.44-0.82 0.44-0.90 0.42-0.90 0.46-0.80 0.4 7-0.8 1 0.4 5-0.65 0.4 5-0.65 0.4 6-0.74 0.51-0.60
Table I (Continued) Absolute % deviation No. of
Method
data point
Substance
(1)
(2)
(3)
Data source
Reduced temperature range
Eaucken and Hauck 0.69-0.9 5 (1928) n-Heptane 35 1.28 1.30 1.31 Douglas et al. (1954) 0.42-0.9 5 Ball (1954) Sulfur dioxide 14 1.3 1.32 1.34 Brass and Lamb (1957) 0.45-0.87 Diethyl oxalate 20 1.94 2.08 2.13 Timmermans (1950) 0.50-0.87 Diethyl adipate 20 1.88 2.18 2.19 Timmermans (1950) 0.46-0.79 Ethyl laurate 15 1.54 1.59 1.59 Timmer mans (1950) 0.49-0.84 Triethylene glycol 16’ 2.01 2.34 2.6 Curme (1953),Bulletin, 0.47-0.80 dimethyl ether Carbide Chemical Tetraethylene glycol 16’ 1.43 1.6 1.62 Curme (1953),Bulletin, 0.45-0.80 Carbide Chemical Triethylene glycol 2o b 1.27 1.51 1.53 Curme (1953),Bulletin, 0.44-0.90 Dow Chemical Co. Tetraethylene glycol 20b 1.19 1.25 1.24 Curme (1953),Bulletin, 0.44-0.90 dimethyl ether Carbide Chemicals Ethylene glycol 17’ 2.4 2.86 3.12 Curme (1953),Bulletin, 0.44-0.95 Dow Chemical Co. Diethylene 12b 2.51 2.74 2.95 Curme (19531,Bulletin, 0.44-0.80 Dow Chemical Co. 1,4-Butanediol 12c 1.9 1.92 2 .o Not available 0.48-0.74 2,3-Butanediol 12c 1.76 1.83 1.83 Not available 0.48-0.74 a For these compounds, the liquid heat capacity data are available at room temperature. For other temperatures these have been extrapolated by the standard methods. The measurements of the critical properties of the polyols are very difficult because these begin decomposing even below their boiling points. Consequently the critical property data taken for the estimation of the heat capacity have been estimated by the methods of Lydersen (1955). The heat capacity was determined a t 20°C by Johnson and Huang method and extrapolated for other temperatures. 10
CO2
0,68
1
+
X
0 . 6 1 A
CALCULATED BY METHOD CALCUATED BY METHOD CALCUATED BY METHOD EXPERIMENTAL VALUE
313
353 TEMPERATURE
1.46
While testing the method of Chueh and Swanson, great difficulty was observed for calculating the parameter F in their equation for compounds having Z, between 0.25 and 0.27. Great uncertainty arises in this case. The compounds having 2, less than 0.23 were not tested by this method. An average error of 11.2% was observed for other compounds.
I
2 3
I
273
1.5
1.42
O K
-
!
!
393
433
Figure 2. Liquid heat capacity of ethylene glycols: 0 , calculated by method 1; +, calculated by method 2; *, calculated by method 3; A,experimental value.
The methods of Reid and Sobel and Chueh and Swanson were tested for the compounds of Table I. In the method of Reid and Sobel an average error of more than 10%was observed for the compounds having 2, less than 0.23. The overall average error of 8.7% was observed for all the compounds listed in Table I for reduced temperature range of 0.7 to 0.95.
Conclusion On the basis of the evaluation it has been observed that the heat capacities of saturated liquids estimated by the present methods are in agreement with reasonable accuracy with the experimental values, for a wide range of temperature. The values of C,L and C‘,L, computed by methods 2 and 3 are nearly equal except the critical point, a deviation of up to 8.3% occurs for some of the compounds tested. Method 1 is slightly superior among these methods. Following are the salient points of the present methods. (1) These methods are applicable to a reduced temperature range of 0.45 to 0.98. (2) These methods can be used for a variety of organic liquids. (3) These methods are continuous and avoid any error in graphical computation. (4) These methods are easy for computer calculations and do not require any table or graph. ( 5 ) The saturated liquid heat capacities predicted by these methods are in close agreement with the experimental values.
The author is thankful to V. Shanker of the Chemical Engineering Department, B.I.T.S., Pilani (Raj.) for his valuable comments. Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
487
Nomenclature C = heat capacity, cal/g-mol O K H = enthalpy, cal/g-mol O K P = pressure, atm Q = heat,cal/g-mol R = gas constant, 1.987 cal/g-mol OK T = temperature, OK V = volume,cc/g-mol 2 = compressibility factor Greek Letters a,P,y,S,c = constants A = enthalpy of vaporization, cal/g-mol p = density, g-mol/cc w = acentricfactor
Superscripts O = idealgasstate Subscripts b = quantity measured a t normal boiling point c = quantity measured a t critical point g = gasorvaporstate L = liquidstate p = quantity measured a t constant pressure r = reduced parameter s = saturated state sc = scaling property sg = saturatedvapor sL = saturated liquid vp = vaporpressure
Literature Cited Awbery, J. H., Griffith, E., Proc. fhys. SOC.London, 52, 770 (1940). "Air Conditioning, Refrigerating Data Book, Design Volume", The Am. SOC. Refrig. Engrs., New York N.Y., 1955. Ball, A. F., J. Res. Nat. Bur. Stand., 53, 139 (1954). Benning, A. F., McHarness, R. C., Ind. Eng. Chem., 32, 814 (1940). Bondi, A,. Ind. Eng. Chem., Fundam., 5,442 (1966). Brass, R., Lamb, J., Proc. Roy. SOC.,Ser A,. 243, 94 (1957).
Chueh, C. F.,Swanson. A. C., Can. J. Chem. Eng., 51, 596 (1973). Curme, G. P., "Glycols", Reinhold, New Yo&, N.Y., 1953. "Dimethyl Acetamide", Product Bulletin of Du Pont Co. (a). "Dimethyl Formamide", Product Bulletin of Du Pont Co. (b). Douglas, T. B., et al., J. Res. Nat. Bur. Stand., 53, 139 (1954). Eaucken, A., Hauck, F. 2.. J. fhys. Chem., 134, 161 (1928). Gallant, R. B., Hydrocarbon Process.. 47(1), 135 (1968a). Gallant, R. B., Hydrocarbon Process., 47(2), 113 (1968b). Gallant, R. B., Hydrocarbon Process.. 47(4), 128 (1968~). Gallant, R. B., Hydrocarbon Process.. 47(5), 151 (1968d). Gallant, R. B., Hydrocarbon Process.. 47(7), 141 (1968e). Gallant, R. B., Hydrocarbon Process.. 47(8), 127 (1968f). Gallant, R. B., Hydrocarbon Process., 47(9), 269 (19680). 223 (1968h). Gallant, R. B., Hydrocarbon Process., 47(1 l), Gallant, R. B., Hydrocarbon Process., 47(12), 89 (19681). Gallant, R. B., Hydrocarbon Process., 48(9), 199 (1969a). 263 (1969b). Gallant, R. B., Hydrocarbon Process.. 4 8 f l l), Gallant, R. B., Hydrocarbon Process., 48(12), 113 (1969~). "Glycols: Property and Uses", The Dow Chemical Co. Product Bulletin. "Glycols", Union Carbide Chemicals Co. Product Bulletin. Guthrie. G. B., Jr., et al., J. Am. Chem. SOC., 86, 2120 (1944). Johnson, A. I., Huang. C. J.. Can. J. Techno/., 33, 421 (1955). Kennedy, R. M., et al., J. Am. Chem. SOC., 83, 2267 (1941). Kelley, K. K., J. Am. Chem. SOC.,51, 180, 779, 1145(1929). Kobe, K. A,, Pennington, R. E., Pet. Refiner, 29(9), 135 (1950). Lee, B., Edmister, W. C., Ind. Eng. Chem., Fundam., 10, 229 (1971). Lydersen, A. L., College of Engineering, University of Wisconsin, Engineering Experimental Station, Report No. 3, April 1955. Lydersen, A. L.. et al.. Engineering Experimental Station, University of Wisconsin, Report No. 4, Oct 1955. Lu, 9. C. Y., et al., J. Chem. Eng. Data, 18, 241 (1973). Messerly, J. F., et al., J. Phys. Chem., 89, 4304 (1965). Parks, G. S..J. Am. Chem. SOC.,47, 338 (1925). Parks, G. S., Huffman, H. M., J. Am. Chem. SOC.,48, 2788 (1926). Reid, R. C., Sobel J. E., hd. Eng. Chem., Fundam., 4, 328 (1965). Reid, R . C., Sherwood, T. K.. "The Properties of Gases and Liquids", 2nd ed, pp 283-297, McGraw-Hill, New York. N.Y.. 1966. Rihani, D. N., Doraiswamy, L. K , hd. Eng. Chem., Fundam., 4, 17 (1965). Sakiadis, B. C., Coates, J., A./.Ch.E. J., 2, 88 (1956). Shaw, R. J., J. Chem. Eng. Data, 14, 461 (1969). Stevens, W. F.,Thodos, G., A.LCh.E. J., 9, 293 (1963). Timmermans, J., "Physico-Chemical Constants of Pure Organic Compounds", Elsevier. New York, N.Y., 1950. Watson, K. M., hd. Eng. Chem., 35, 398 (1943). Yen, L. C., Alexander, R. E., A./.Ch.E. J., 11, 334(1965). Yuan, T. F., Stiel, L. I., Ind. Eng. Chem., Fundam., 9, 393 (1970).
Received for review November 15,1974 Accepted May 29,1975
Gas Absorption by Newtonian and Nan-Newtonian Fluids in Sparged Agitated Vessels Hideharu Yagi and Fumitake Yoshida' Chemical EngineeringDepartment, Kyoto University, Kyoto, Japan
On the basis of experimental data for oxygen desorption from two Newtonian and two nowNewtonian fluids of various concentrations, a dimensionless correlation for kLa in sparged agitated vessels is proposed. In addition to ordinary liquid properties and operating parameters, the correlation includes the Deborah number, which is defined as the product of the characteristic material time and the agitator speed and represents the viscoelastic behavior of non-Newtonian fluids.
Introduction Sparged agitated vessels are often used for various gasliquid reactions and for aerobic fermentation. A number of correlations have been proposed for the rate of gas absorption in sparged agitated vessels. Most of those correlations are applicable only to Newtonian fluids, despite the fact 488
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
that polymer solutions and fermentation broths often show non-Newtonian behavior. Perez and Sandal1 (1974) correlated kLa data for carbon dioxide absorption into water and a 0.25 wt % Carbopol solution, a non-Newtonian fluid, using the apparent liquid viscosity which was defined by Metzner and Otto (1957) and was related to the impeller
