Ind. Eng. Chem. Res. 1987,26, 2445-2449
2445
Viscosity of Crude Oil Blends Jasem M. AI-Besharah,*Omar A. Salman, and Saed A. Akashah Kuwait Institute for Scientific Research, Petroleum, Petrochemicals Materials Division, 13109 Safat, Kuwait
T h e viscosities of light, medium, and heavy crude oils and the viscosities of their blends were experimentally measured over a temperture range of 10-60 "C. These viscosity data were used to develop a new method for predicting the viscosity of crude oil blends based on known component viscosity. In addition, ASTM D341 and Refutas index methods were evaluated for predicting viscosities of crude oil blends. It was found that the new method gave the best representation of experimental results. The deviation in most cases was less than 6%. There is no doubt that knowledge of physical, chemical, and thermochemical properties is an essential step preceding the design of any efficient chemical process. An important physical property is viscosity, for which many empirical correlations have been successfully developed for pure components. Generalized viscosity correlations are also available for mixtures of gases, nonpolar liquids, and petroleum products (Dean and Stiel, 1965; Ried and Sherwood, 1966; Stiel and Thodos, 1961; Lydersen et al., 1955). None of these correlations, however, were tested on petroleum crude blends. It was therefore decided to evaluate two common methods used for petroleum products and to develop an alternative method for predicting viscosities of crude oil blends. Dean and Stiel (1965) developed a method to calculate the viscosity of nonpolar gas mixtures at moderate and high pressures. Their method is based on known molecular weight and critical constants of each component. Calculated viscosities were in excellent agreement with reported ones. For pure liquids at moderate temperatures, the following correlations were developed by Orrick and Erbar (1972): In (7/p)M = A B/T (1)
+
A = (6.95 + 0.21n) + GC(A)
(2)
B = 275 + 99n + GC(B) (3) where 7 is the viscosity, p is the liquid density, M is the molecular weight, T is the absolute temperature, n is the number of carbon atoms, and GC is the group contribution. For evaluating liquid viscosities at high temperatures, Letsou and Stiel (1973) suggested a corresponding state approach that uses the Pitzer acentric factor (wo): 7e = (74O
+ wo(174'
(4)
where (7~)'
0.015174 - 0.012135Tr + 0.00752',2
( 7 ~=) 0.042552 ~ - 0.076741: e
+ 0.0340T:
= T,'/6/(1M1/2P:/3)
(5) (6) (7)
T,, M , and P, are the critical temperature, molecular weight, and critical pressure, respectively. Lobe (1973) proposed the following correlation for the viscosity of liquid mixtures: v, = ~lvle+@z + &v2e@lal (8)
where a, = -1.7 In vl/vl
a2 = 0.27 In v 2 / v 1 + (1.3 In v z / v 1 ) ' / 2
41 = XlVl/(XlV,
+ X2V2)
(9)
(10)
0888-5885/87/2626-2445$01.50/0
Table I. Weight Fraction of Light, Medium, and Heavy Crudes for Each Blend blend light medium heavy 1
2 3 4 5 6 7 8
0.30 0.30 0.40 0.55 0.20 0.10 0.70 0.10
0.30 0.50 0.50 0.40 0.30 0.70 0.10 0.20
0.40 0.20 0.10 0.05 0.50 0.20 0.20 0.70
Table 11. Experimental Viscosities of Light, Medium, and Heavy Crude Oils at Various Temperatures kinematic viscosity, cSt temp, "C light medium heavy 10 20 30 40 60
14.22 9.51 7.14 5.96 4.01
163.40 81.46 51.81 34.63 17.76
2193.10 887.90 426.40 223.50 78.71
v, x , and V are the kinematic viscosity, mole fraction, and liquid molar volume, respectively. Another method that has been successful in predicting liquid mixture viscosities is the group solution model (Wedlake and Ratcliff, 1973). The model is based on the fact that real mixtures deviate from ideality, giving rise to an excess quantity. The excess quantity is attributed to group contribution and structural contribution. The group contribution is a result of the interactions between groups that constitute the molecules, and the structural contribution results from the manner in which the groups are joined together to form the molecules. In this paper, we present a model for predicting viscosities of crude oil blends. The model is compared with two methods, ASTM D341 (ASTM, 1983a) and the Refutas index (British Petroleum, 1947), which are used to estimate viscosities of hydrocarbon mixtures.
Experimental Procedure The kinematic viscosities of light crude oil (36 "API), medium crude oil (24.7 OAPI), and heavy crude oil (14.8 "API) were measured at 10, 20, 30, 40, and 60 "C. The ASTM D445/IP71 (1983b) standard method was followed for all viscosity measurements. The time required for a volume of fluid to travel under gravity a certain distance through a calibrated glass capillary viscometer is measured. The viscometer used in all measurements is of type BS/IP U-tube. A water bath controlled to fO.O1 "C was used. Bath temperature was monitored with a calibrated thermometer. Ternary blends were prepared by mixing light, medium, and heavy crude oils at different weight fractions (Table I). Kinematic viscosities of blends were also measured 0 1987 American Chemical Society
2446 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 Table 111. Experimental Viscosities of Blends at Various Temperatures kinematic viscosity, cSt temp, O C blend 1 blend 2 blend 3 blend 4 blend 5 88.12 68.25 36.25 207.16 10 122.1 19.81 20 70.75 48.47 34.32 110.3 13.57 28.35 22.03 62.71 30 37.25 9.76 39.43 40 25.56 18.93 14.08 60 13.88 10.34 8.39 6.53 19.70 Table IV. Calculated Values of a L2, aZ3,and a Is as a Function of Temperature temp, "C 10 20 30 40 60
a12
a23
a13
-0.569 1.188 -1.493 -0.567 -0.730
-3.869 -3.566 -3.303 -3.269 -2.315
-0.769 -1.791 -0.553 -1.227 -0.990
at 10,20,30,40, and 60 "C using ASTM D445/IP7 (Tables I1 and 111).
Results and Discussion The ideal viscosity of a mixture can be defined by (Ratcliff and Khan, 1971)
where v is the absolute viscosity of the mixture, vi is the absolute viscosity of component i, and xi is the weight fraction of component i. For real mixtures, an excess function must be included to account for deviation from ideality. Thus, 11 can be modified to give
where (In Y ) is~ the excess function. The excess function for a ternary mixture can be expressed by a three-parameter equation as (In v)E = uI2x1x2+ a23x2x3
+ a13x1x3
(13)
blend 6 171.78 83.38 50.51 31.98 16.27
blend 7 27.09 16.01 11.59 8.43 5.61
blend 8 499.36 234.40 127.59 73.77 32.40
where uij is the interaction parameter between components i and j . To determine the value of these parameters, the viscosities of the pure components and three blends must be measured experimentally. Equations 12 and 13 can then be used to calculate corresponding interaction parameters. Table IV presents calculated values of u12,~ 2 3 ,and a13 at five temperatures: 10,20,30,40, and 60 "C. The values are based on measured viscosities of blends 6-8. To check the validity of this model, viscosities of blends 1-5 were calculated by eq 12 and 13. Calculated and measured viscosities are in good agreement for blend 1,in which the deviation is less than 8.1% at all temperatures (Table V). For other blends, however, the deviation, particularly at low temperatures, is rather high. For example, the percentage error in predicted viscosity for blend 3 at 20 O C is 26.1. Furthermore, it may be seen that the lowest deviations were obtained at higher temperatures (40 and 60 "C). A fourth interaction parameter (~123)was added to eq 13 to account for interactions between components 1, 2, and 3: (In v ) =~ u12x1x2+ a23x2x3
+ a13x1x3 + a123qX2x3
(14)
The four parameters can be calculated by solving a set of four simultaneous equations and using experimental data on blends 1,6,7, and 8 (Table VI). On the basis of these results, the viscosities of blends 2,3,4, and 5 were calculated by eq 12 and 14. It can be seen from Table V that predicted and measured viscosities are in excellent agreement. The deviation in most cases is less than 6 % .
Table V. Experimental and Calculated Viscosities of Blends 1-5 Using the Three-Parameter and Four-Parameter Models three-parameter four-parameter expression expression blend t e m n "C exvtl calcd dev % calcd dev % 1 10 122.1 120.84 -1.03 70.75 20 65.05 -8.05 37.25 30 36.56 -1.85 25.56 40 23.85 -6.68 13.88 60 12.99 -6.44 2 10 88.12 -10.79 78.61 97.17 10.27 20 51.81 48.47 6.89 51.46 6.17 2.57 30 24.22 28.35 -14.57 29.08 40 18.25 18.93 -3.59 3.54 19.60 6.09 60 10.26 10.34 -0.77 10.97 10 68.25 56.89 -16.65 67.60 -0.95 43.28 32.51 26.09 20 34.32 -5.27 17.81 -19.16 20.90 -5.13 30 22.03 14.89 40 14.08 13.62 -3.27 5.80 8.39 8.41 60 8.06 -3.93 0.23 10 36.25 38.83 7.13 44.56 22.92 20 19.81 32.43 63.71 20.49 3.44 30 13.57 -5.32 14.79 9.02 12.85 40 -2.22 9.76 11.54 9.54 18.32 60 5.96 -8.70 6.68 6.53 2.38 5 10 207.16 183.95 -11.20 209.16 0.965 110.3 92.02 -16.57 113.67 3.06 20 52.71 -4.91 62.71 -15.95 59.63 30 32.39 -17.85 38.63 -2.03 40 39.43 -1.88 17.01 -13.66 19.33 60 19.7
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2447 Table VI. Calculated Values of a 12, Function of Temperature temp, O C alo a71 10 0.178 -1.259 -1.483 20 -1.279 -0.916 30 -0.714 -1.198 40 -1.595 60 -1.391 -1.078
a239
a 13, and B 125 as a
all -4.442 -4.499 -3.274 -3.515 -2.511
-4-parameter ASTM -*-
ales
----REFUTAS x Experimental
1.978 10.586 -0.298 5.204 4.475
As mentioned earlier, the modified ASTM D341 method and Refutas index method are used to predict viscosities of petroleum blends only. The validity of these methods for predicting viscosities of crude oil blends is discussed below. The ASTM D341 method is based on conventional viscosity-temperature charts. The kinematic viscosity of a binary mixture can be determined at any temperature provided the kinematic viscosities at two temperatures are known for each component. Standard charts are available covering wide viscosity and temperature ranges. The procedure to obtain blend viscosities requires that viscosity-temperature lines of components be plotted and then blended by linear proportioning at a constant viscosity (measuring the distance along a horizontal line). Since measuring the distance on the chart is time consuming and inaccurate, Twu and Bulls (1981) used a modified Walther's equation (Walther, 1937) to predict blend viscosity: In In (v + 0.7) = m In T + b (15) where v is the kinematic viscosity in centistokes, T i s the absolute temperature, and m and b are constants for a particular oil. The two parameters m and b can be calculated from two viscosity measurements. To calculate the viscosity of a multicomponent mixture, the mixture is split into a series of binary blends. This method was used to predict the viscosities of crude oil blends 1-5 (Table VII). It can easily be seen that the deviation is rather high in most cases. For example, at 60 OC, the error is greater than 23% for all blends except blend 5. I t can therefore be concluded that the ASTM
01
0
10
20 Temperature
30
40
50
A
BO
('C
Figure 1. Experimental and predicted viscosities as a function of temperature for blend 2.
D341 method cannot predict the viscosities of crude oil mixtures with high accuracy. Similar results were obtained when the Refutas index method was used on crude oil blends. This method was developed to predict blend viscosities of all petroleum components from gasoline to vacuum residue. The Refutas function or blending index is given by the relationship
I = f ( v ) = 23.097
+ 33.468 log log (v + 0.8)
(16)
where I is the Refutas index and v is the kinematic viscosity. To determine the viscosity of a blend by this method, Refutas index of components is determined based on known viscosities. The Refutas index of the blend is then
Table VII. Experimental and Predicted Viscosities for Blends 1-5 Using ASTM D341 and Refutas Index Methods ASTM D341 Refutas index blend temp, " C exptl calcd dev % calcd dev % 1 10 122.1 142.00 16.29 114.22 -6.45 20 59.41 70.75 -16.03 61.60 -12.93 30 25.90 34.69 37.25 46.90 -6.88 40 -15.45 25.58 21.61 25.56 32.10 -26.13 18.87 60 10.25 13.88 16.50 2 10 88.12 79.59 -9.67 95.30 8.72 42.41 20 48.47 -12.51 47.70 -1.59 30 28.35 25.30 -10.74 34.10 20.28 40 16.09 24.10 27.31 18.93 -15.01 7.89 25.72 60 10.34 -28.64 13.00 10 52.55 61.00 68.25 -23.00 -10.62 34.32 -15.67 -3.26 20 28.94 33.24 6.67 22.03 -19.33 30 17.77 23.50 11.59 22.86 14.08 -17.65 40 17.30 5.93 9.92 18.22 -29.31 60 8.39 22.94 10 36.25 -22.91 38.60 6.48 22.76 17.66 20 19.81 -10.88 14.58 30 13.57 -11.29 19.38 16.20 12.04 28.13 8.55 40 9.76 -12.32 12.50 60 -24.35 7.52 15.24 4.94 6.53 253.25 5 10 207.16 22.25 245.00 18.26 123.64 20 110.3 12.09 93.80 -14.80 30 73.20 16.72 68.31 62.71 8.93 40 2.61 47.60 20.21 40.46 39.43 22.70 15.22 17.67 -10.29 60 19.70
2448 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
- 4-parameter -.- - - - AR ES FT UMT A S
- 4A4SaTr aMm e t e r
-.-
-
70
----
REFUTAS Experimental
X
x Experimental
-
60
- -
; 50 Y)
v
I
2. L
'2 4 0 v Y)
.-
> .U
30-
n
3
20-
10
0
0
I
! I
10
20
40
30
Temperature ("C
b
J
I
b
50
I
1
1
4
10
20
30
40
1
50
60
Temperature ("C
60
)
Figure 2. Experimental and predicted viscosities as a function of temperature for blend 3.
Figure 4. Experimental and predicted viscosities as a function of temperature for blend 5.
45-
40
-
-4-parameter
-.35
-
30
-
-
ASTM _ - x_ _Experimental REFUTAS
9
'i
1 Y)
? 25-
--
r 0 . . l
:: 2 0 >
.-u
5
I5
-
10
-
.-
Y
'.Blend4 5 -
01 0
Figure 5. Deviations of the calculated kinematic viscosity by the four-parameterequation for blends 2-5 a~ a function of temperature.
A
10
20
30
Tamperature
40
50
60
i'C 1
Figure 3. Experimental and predicted viscosities as a function of temperature for blend 4.
calculated, based on the weight fraction of each component:
I@=
2i IiWi
(17)
where IB is the Refutas index of the blend and Wi is the weight fraction of component i. Once I, is determined, the blend viscosity can be read directly from a corresponding table or deduced from eq 16. This method was used to predict the viscosities of blends 1 , 2 , 3 , 4 ,and 7 at 10,20,30,40,and 60 "C. As presented in Table VII, calculated and measured viscosities are not
in good agreement. The deviation for most cases is greater than 15%. Finally, a comparison between the three methods, ASTM D341, Refutas index, and the four-parameter model, is shown in Figures 1-4. The figures present experimental and calculated viscosities for blends 2-5 at a temperature range of 10-60 "C. It is clearly demonstrated that the four-parameter model gives the best representation of experimental data over the whole temperature range. It may also be seen that the greatest deviations in the ASTM D341 and Refutas index methods are at low temperatures. Figure 5 presents the deviations of the calculated kinematic viscosity by the four-parameter equation of blends 2, 3, 4, and 5 as a function of temperature.
Ind. Eng. Chem. Res. 1987,26, 2449-2454
2449
Acknowledgment
Literature Cited
We express our gratitude to Kuwait Institute for Scientific Research for the financial support during the course of this project.
ASTM D341, The American Society for Testing and Materials, 1983a. ASTM D445/IP71, The American Society for Testing and Materials, 193b. British Petroleum, Sunbury Report 3282, UK, 1947. Dean, D. E.; Stiel, L. I. AZChE J. 1965, 11(3), 526. Letsou, A.; Stiel, L. I. AIChE J. 1973, 19, 409. Lobe, V. M. M.S. Thesis, University of Rochester, New York, 1973. Lydersen, A. L.; Greenkorn, R. A.; Hougen, 0. A. Engineering Experimental Station Report 4,1955; College of Engineering, University of Wisconsin, Madison, WI. Orrick, C.; Erbar, J. A. Reported in Properties of Gases and Liquids, 3rd ed.; Ried, R. C., Prausnitz, J. M., Sherwood, T. K., Eds.; McGraw Hill New York, 1972. Ratcliff, G. A.; Khan, M. A. Can. J. Chem. Eng. 1971, 49, 125. Reid, R. C.; Sherwood, T. T. The Properties of Gases and Liquids, 2nd ed.; McGraw-Hill: New York, 1966; pp 431-440. Stiel, L. I.; Thodos, G. AZChE J. 1961, 7, 611. Twu, C. H.; Bulls, J. W. Hydrocarbon Process. 1981, 5, 217. Walther, C. Proc. 2nd World Pet. Congr. 1937, 2, 133. Wedlake, G. D.; Ratcliff, G. A. Can. J. Chem. Eng. 1973, 51, 511.
Nomenclature aij = interaction parameters ASTM = American Society for Testing and Materials GC = group contribution parameter I = Refutas index or blending index la= Refutas index of blend M = molecular weight n = number of carbon atoms P, = critical pressure T,= critical temperature x = mole fraction V = liquid molar volume W = weight fraction w o = Pitzer acentric factor Greek Symbols c$~ = liquid molar volume fraction v = kinematic viscosity 7 = absolute viscosity p = density
Received for review July 28, 1986 Revised manuscript received August 17, 1987 Accepted September 4, 1987
Analytical Design Equations for Multisolute Reverse Osmosis Systems Srinivas Palanki and Sharad K. Gupta* Chemical Engineering Department, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Analytical equations are developed for tubular, spiral-wound, and plate-and-frame reverse osmosis (RO) modules to predict the membrane channel length for achieving a given fractional solvent recovery for a dilute feed with two highly rejected solutes. Extension to a large number of noninteracting solutes is straightforward. Analytical equations to predict the permeate solute concentrations are also developed. These design equations are simple to use and can be applied for both laminar and turbulent flow conditions. On the basis of these design equations, some approximate design equations are also derived. The predicted dimensionless length of the module and the average permeate concentrations are compared with the results obtained by numerical integration and found to be in excellent agreement. The above results are also compared with literature values. These design equations can be applied for systems where the solution-diffusion model is applicable. Reverse osmosis (RO) has rapidly emerged as a commercially attractive microsolute separation process. Applications of RO range from desalination, the dominant commercial application, to separation of products in bioreactions. Such widening activity suggests the need for development of simple and accurate design equations for making rapid process design calculations. Numerical design procedure and results for single-solute RO are available in the work of Ohya and Sourirajan (1971). Sirkar et al. (1982)developed a simple analytical design equation for single-solute RO. The flux equations were simplified by using a truncated series for the term exp(Nl/kliC) in the expression for concentration polarization, and then explicit expressions for solute and solvent fluxes were obtained. This approach was extended to multicomponent solute processes by Prasad and Sirkar (1984,1985).Prasad and Sirkar used the linear approximation of exp(Nl/kliC) for N,/kliC S 0.2 and quadratic approximation of exp(Nl/kEC)for Nl/k& 6 1. The design equations for N,/k,C > 1 may also be derived by using better approximations for exp(N1/ k&). However, the 0888-5885/87/2626-2449$01.50/0
resulting equations become very complex to integrate, and it may be easier to use numerical procedures. Recently Gupta (1985)developed an analytical procedure for a single-solute RO system where the use of a truncated series for the exponential term is not required. Thus, the resulting equations are valid for all values of Nl / k1, c. In this study, the design equations for plate-and-frame, spiral-wound, and tubular modules are developed by an analytical technique similar to that used by Gupta (1985). It is shown that the resulting design equations are accurate and simple to use. On the basis of these design equations, further approximate solutions are also obtained and compared with those obtained by Prasad and Sirkar (1985). Development of Design Equations We consider the case of two noninteracting solute species in the feed solution as in Prasad and Sirkar (1984). We assume that the solution diffusion model is applicable for transport across the membrane. In addition, we assume that the total molar density, C, of the solution is constant, 0 1987 American Chemical Society
