INDUSTRIAL AND ENGINEERING CHEMISTRY
964
facilities available, and in granting permission to publish these results; acknowledgment is also made to the Chemical Department, under n.hose auspices the experimental work Tvas carried out, and to the Engineering Department, where the final correlation IKIS made. VOMEUCLATURE
constants in Enskog equation of state specific heat a t constant pressure, P.c.u.,'(lb.I ( ' ( ' , ' I inside diameter of tube, f t . friction factor in Fanning equation for fluid fric.tion in pipes mass velocity, 1b.lchr.j (sq. ft.] gas-side film coefficient of heat transfer, (P.C.U. (hr.'l (sq. ft.) ( " C.1 thermal conductivity, P.c.u. (hr.) (sq. it.: ( " ( ' . it. I molecular weight, Ib.!niole molal heat, capacity at constant prt'~suw, I'.t,.u. (Ib. mole) (" C.) C,p!k. Prandtl number D G l p , Reynolds numtler pressure, atm. heat flow rate bsseti on ga:: teiupcraturv rise. P.c.u. hr. heat flov rate based on condensate, P.c.u. 111.. gas constant in Enskog's equation of stntc inlet temperature of gas,' ( ' . outlet temperature of gas, 'C'. average gas temperature, ' C'. film temperature, C. = .temperature of steam, C. = inside wall temperature C. = absolute temperature, ' h. = over-all coefficient of heat transfer b d e d on inside diameter of tube, P.c u. '(hr.) (sq. ft.) (" C I
t.
t 7'
I.
in t, - f? deviation factoi, gieater than unity, in I?n.kog's equation of state = viscosity, lb./(hr.) (ft.) = density, lb./cu. ft = function of S p , ~
K
,U
p
Q
=
Vol. 39, No. 8
Subscripts f = value of physical property at film temperature t, p = value of phmical property a t pressure p and prevailing temperature LITERATURE CITED
Bsrtlctt. E. P.. Cupples, H. L., and Treniearne, T . H., .l, I n . Chem. Soc.. 50, 1275-88 (1928). Boelter, L. 11.K., Martinelli, R. C.. and Jonassen, Finn, Trans. dm.SOC.V e c h . Engrs., 63, 447-55 (1941). Boyd. J. H., Phys. Rea., 35, 1254-95 (1930). Brunot, d. W..Trans. A m . SOC.Mech. Engrs.. 62, 613-16 (1940).
Colburn. A . P.. Purdue Univ. Eng. Expt. Sta., Res. Ser. Bull. 84 (19421. Colburn, A. P., and Coghlan C. A,, Trans. A m . Soc. IfPch. Engrs., 63, 561-6 (1941). Comings. E. W . , and Egly. It. 5.. ISD. ESG. CHEY..32, 714-18 ~~
~I
(1940).
Deming, K. E.. and Shupe, L. E., Phys. Rea.. 37, 6 3 - 5 4 (1931); 40, 848-59 (1932). Drew, T. B., Koo, E. C.. and Mcildams. W. €I., Trans. .Am. I n s t . Chem. Engrs., 28, 56-72 (1932). Enskog, D., Kungliga Saenska Vetenskapsakademiens Handlingar. 63, No. 1 (1922). Gibson, R . 0.. "Viscosity of Gases at High Pressure," Am?terdam, 1933. Ihhs. T. L., and Ilirst, A . h.,Proc. R o y . Soc. (London), A123, 134-42 (1929). (12.i) International Critical Tables, Vol. V, p. 214 (1929). (13) Kleint, F.. Ber. deut. p h y s i k . G e s . Verhandl., 3, 146-58 (1905). (11)Xfc-idams, TT. 11.. "Heat Transmission," 2nd ed., p. 174 (1942). I b i d . , p. 273. XIichels, A , . and Gibson, R. O., Proc. R o y . SOC. (London). A134, 288-307 (1931). Nikuradse. J.. Forschungshejt, S o . 356 (1932). Susselt, TY,,.Vitt. Forschungsarb., 89 (19101; 2. Ver. deut. Ing.. 53, 1750-5, 1808-12 (1909). Trautz, >I., and Baumann, P. B., Ann. Physil;, [5 1 2, 733-6 (1929). ( P O ) Cj.ehara. 0 . .I..and Watson, K. If..S a t l . Petroleum .\-uios, 36, S o . 40, R-714, 716-20, 7 2 2 (1944). P R E s E s , r E D as part of the Sgmposiurn o n High Pressure Technology before the Division of Industrial and Engineering Chemistry a t the 110th lRIeeting of the AXERICAV CHEXIC.AI. SOCIETY,Chicago, 111.
Thermal Conductivity of Gases at High Pressures E. W. COLMINGS AND RI. F. NATHAN Cniversity of Illinois, Vrbnna, I l l .
T
.
HE increasing use of high pressure gas-phase processes has made a knoiTledge of the effect of pressure on the physical properties of gases necessary. 1Iany pressure-volume-temperature measurements with the common gases, as well as equations of state and compressibility factor correlations for these, are available. Within the last few years a considerable amount of high pressure gas-viscosity data has appeared. Comings, Jiayland, and Egly (5)summarized the viscosity data available and correlated the viscosity ratio, as a function of reduced temperature and reduced pressure. Fern experimental measurements of the thermal conductivity of gases a t high pressures have been reported. Comings et al. ( 5 ) pointed out, that as a first) approsimation the correlation of viscosity ratio could be used to predict the thermal conductivity ratio, k / k l , a t high pressure by assuming that the latter ratio is cqual to the viscosity ratio. This folloivs from the transport theory based on the kinetic theory of gases. It r a s realized
that these two ratios are not esactly equal for actual gases at high pressures. The difference between them will eventually need t o be established by accurately determining the thermal conductivity of several gases under high pressures. The thermal conductivity measurements by Sellschopp (15) on carbon dioxide t o 90 atmospheres were used t o compare the two ratios, and the agreement r a 3 not very close ( 5 ) . It has been shown (8) that convection existed in Sellschopp's experiments and his data are therefore not correct. Vargaftik (19) obtained a limited amount of thermal conductivity data on nitrogen up to 90 atmo3pheres. The relation bet--een thermal conductivity and viscosity predicted by the kinetic theory is: k = C Crp. The deviation from this relation becomes progressively greater as the pressure is increased. I n the absence of adequate experimental data another method for predicting the relative magnitudes of the viscosity and thermal conductivity ratios was sought. Enskog (4)derived equations m-hich use the same rntistants to
August 1947
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE I. P, ArrL
\s a t 0‘ C‘
40 60
1.117 l.li7 1 244 1.294 1.507 1,632 1 666 -1.692 1.642 1.639 1,54i
80 100
200 300 400 500 600 800
1000
V? a t SOo C,
40 60 80
100 200
300 400 500 600 800 1000 I? a t 1000 (‘
40 60 80
100 200
300 400 500 800
800
loon CiHn a t 116 4’ C
91
r (”) P aT
6b
( 0 70
61 78 5 i 32 53 80 49 05 40 71
v.%LL-ES O B T A I N E D FROM
0 275 0.559
A 9,965 7.274 5.468 4.645 3.013
0.848
2,625
1.091 1,345 1,502 1,946 2.191
2.547
bpr
G
0 110 0,157 0 222
2,568
2.609 2 . 796
2.925
B 10.364 7.673 5 867
5.044 3.411 3 021 2.943 2,962 3.003 3.188 3.316
1 098 1,146 1,202 1 236 I 422 1 551 1.583 1.600 1.5 i 2 1.583 1.499
0 091 0 114 0.199 0 2x9 0.499 0 773 0 986 1,199 1 358 1.773 1,997
11.848 9.663 5.982 5,169 3,185 2.682 2.547 2.570 2.714 2.821
12.247 10.063 6.381 5.569 3,583 3 079 2.961 2.942 2.965 3.107 3.214
1.07’2 1.117 1,154 1.184 1.362 1.450 1,485 1.499 1.496 1.509 1.444
0 0 0 0 0 0 0
081 124 167 206 462 618 863 1,037 1,201 1.564 1.779
13.162 8.978 6,901 5.809 3.317 2,889 2.616 2,554 2,547 2.630 2.717
13.562 9.378 7.301 6.208 3.716 3.287 3.013 2.950 2.942 3.024 3.110
1,055 1.079 1.103 1.131 1,224 1.311 1.360 1.374 1,378 1.398
0 065
16,227
11,170 9,736
1.378
0.097 0.113 0.169 0 334 0.527 0 703 0 841 0 976 1.277 1 514
2.630 2 568 2.556 2 613
16.627 11.569 10.136 7.259 4.453 3.496 3.155 3.026 2,964 2.950 3.007
1,905 1,396 1.037 0 789 0.593 0 411 n . 245
1,901 1 429 1.106 0.922 0.807 0.669 0.490
2.773 2.588 2,546 2,587 2.654 2.804 3 216
3.166 2.982 2.942 2 983 3.051 3,201 3.614
1.972 1,454 1,082 0.802 0.581 0 397 0 235
2,003 1.530 1.198 0.953 0.769 0.614 0.433
2.824 2.619 2.547 2.575 2.686 2.896 3.439
3,216 3,013 2.942 2.971 3.083 3.294 3 838
2.565
6.860
4 054 3.099 2.758
relate pressure n i t h density and tempeiature, p p i v i t h density, and klX-1 n i t h density. The equation for p / p 1 has been shown ‘4,11, I S ) to agree ne11 nith the viicosity measurmients on car-
-4 method
for predicting the thermal conducti7ity of This requires only a hnowledge of the critical pressure, the critical temperature, and the thermal conductility a t atmospheric pressure and the teniperature in question for the gas. EniLop’s equation of state and his two equations relating riscosity ratio with densit4 and thermal conductisity ratio n i t h density are used as a basis. The constants in all these equations are the same. These constants are eialuated from aiailable P-F‘-T data and high pressure gas-\iscosity data, and are then used to predict the thermal conductility ratio a t high pressures. The calculated ratios are arranged as a function of reduced pressure and reduced temperature. The effect of pressure on the Prandtl number ratio has also been determined. A graph is presented showing this ratio as a function of reduced pressure and reduced temperature. a gas a t high pressures is described.
96s
P-T’-T DATA
B 8 680 7,370 0 2265 6.146 0 2974 5.308 0 3770 4.641 0 4799 4.116 0.6129 3.648 0 8161 3.077 1465 11.946 0.1342 CHI a t looo C 9,253 0.1787 8.021 0,2233 6,624 6 225 0.2926 5.648 5 249 0.3706 4.890 4 491 0.4711 4 278 3 879 0,6020 3,743 3 344 0,8025 3,100 1 703 1.445 3.411 3 020 3.856 C2H4 a t 8.25‘C. 3.468 3.077 4.034 3.722 3.333 4.796 4.025 3.638 5.702 4.338 3 954 6.663 4.671 4.288 7.716 3,442 3 044 0,5203 GHa a t 30° C . 3.152 2.755 0,706 0,840 3,028 2.631 0.838 1.156 3,242 2.850 2.053 3 298 3,419 3.029 2.374 3,873 3 491 3.101 2.495 207,63 4.080 3.717 3,329 2,863 277.68 4.788 3.996 3,609 3.291 383.31 5,645 4.295 3 910 3.733 525,82 6,575 4,606 47.647 4 807 CzHd a t 50@C . 0.2569 3,582 3 184 68.179 0.5041 3.182 2 785 84.180 0.8288 3.011 96,909 2.614 1,1737 2.942 112.754 2 546 1,6060 2.953 126.614 2 558 1.9325 3,093 2 700 168.997 2.7018 3.270 2 879 218.50 3,355 3.433. 3 042 270.72 3,899 3,478 3 087 289.05 4,054 3.710 3 322 337.37 4.774 3,972 3 585 495.80 5.598 4.258 3 872 656.44 6.499 5.982 0.216 49.779 0 1941 5.588 C2Ht a t 95O C 5.099 0,271 58.935 4.700 0 2481 3,775 90,452 0 4866 0.447 3.377 3.238 121,201 0 8107 0.646 2.840 3.011 0,866 149,726 2,614 1 1737 2.941 1.121 185.22 2,546 1 6166 1.315 2.957 2 562 213 808 1 9490 290,261 1.731 3,089 2,696 2 6918 3,253 2,073 368 60 2,861 3 320 3.396 444.96 2,333 3.005 3 826 2,421 3,447 470.70 3.056 3 993 3.639 2.738 585.03 3,250 4 633 a D a t a i n range i n which isometric passes through a n inflection pointtherefore, not very accurate.
CHI a t
C.
43 829 52 826 65 504 70 367 93 445 111.052 135 293 203 379 44,523 57.251 69,676 87.616 106 362 129.061 156.593 195.961 312 491 108.65 119.35 173.39 259.67 381.70 552.36 57.952 67.483a 73,56fja 151.6760 192,647
0 1803
0.136 0.165 0.209 0 . 255 0.312 0.381 0,480 0,776 0.094 0.126 0.149 0.189 0.234 0.288 0.356 0.455 0,751 2.360 2.456 2,870 3.334 3.796 4.274 0.547
8 281 6,971 5 747 4 909 4 242 3 717 3 250 2.680 11 546 Si 854 621
bon dioxide and nitrogen. Leipunsky ( 9 ) investigated the applicability of this equation to the viscosity of several gases. These equations cannot be used to predict thermal conductivity or viscosity ratios from P-V-T data alone. It is possible to evaluate the constants in them by using both P-V-T data and viscosity data, and then to predict the thermal conduetivity. This has been done, and the calculated thermal conductivity ratios for nitrogen agree with Vargaftik’s data n-ithin almut lC6over the range of his data. The method has been used to predict k / k l ratios over a considerable range of temperatures and pressures, and these have been arranged in the same form as the viscosity ratio correlation as a function of reduced pressure P,, a t various constant values of reduced temperature, Tr, 1IETHOD OF CALCULATION
The equations developed by Enskog based on the kinetic theory are as follows: Equation of state for a static gas relating P , p , and T ,
P
+
UP*
=
+
~ ( 1 bp4
Equation relating viscosity ratio and density,
(1)
INDUSTRIAL AND ENGINEERING CHEMISTRY
966
Vol. 39, No. 8
4.0
3.0
b 2.0
1.00
100
200
300
400
500
700
600
800
900
1000
P, ATMOSPHERES
Figure 1.
Effect of Pre3.sure on Constant b for Seteral G:i*e:
+
p' = sl' g. Tlie residual r \vas defined as the difference t x ~ ~ tween actual pressure P and p' or
P Residual
T
sl'
=
+g
T
r
was then plotted agaiiiqt tc,niperature and
i5 )
($) ~ v m
ruhtained graphica!ly from the smoothed curves.
(g)u
]vas calculated from the q u a t i o n
Tliib method of u&ig residuals i u highly accurate sincc s is usually largc comp:trcd t o
($>),,.abl le I gives calculated values of
7
-
IO
Equation relating thermal ratio and density,
klh
=
bp
- + 1.2 + 0.7574bpx)
(b,lX
9.0 8.0
conductivity
7.0
-
6.0
-
5.0
-
4.0
-
(3)
If it is assumed that r is a function of density only and t h a t a and b are constants, the term bpx may be evaluated along a n isometric from Equation 1. Thus,
-
-3.0 -
5 The values of
r&)
determined by Demirig
and Shupe (6) for nitrogen based on Bartlett's P-V-T data ( I , Z) n-ere used for this gas. The values for propane, ethylenc. and methane Tvere calculated graphically from the P-V-T data of Beattie et a!. ( 3 ) , hlichels and Geldermans (IO),and Miclieli and Sederbragt ( l a ) , respectively, using a method of residuals (df ). -\ straight-line relation was used to connect the ends of a n isometric. The equation of this line \\as
-
$22.0
U
I
C
1.0 2l
0.9 10
2.0
RUXJCED Figure 2.
3.0
4.0 5.0 6.0 7.0 809.0 K)
PRESSURE, Pr
Comparison of Experimental Viscosity Ratio w - i t h Calculated Thermal Conductivity Ratio
August
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1947
OBTAIXED FROM I TABLE 111. I-ALL-E~
r
967
~ASD P-TT-T ~ ~DATA ~ FOR ~XITROGCS, ~ ~PROPASE, ~ ASD - NETHASE
and ~ - J ’ - Tdata liatcd in Table 11 ~.,-erc~ p tcj~ d thermal conductivity ratio for several gases a t constant reduced The temperatures. The curves are compared with those of the nieascalculate the values of b which are givcn in Tables I11 and I\-, ured viscosity ratio. Figure 3 gives the thermal conductivity b = p / p i ‘p.4 (7 ) Tables 111 anti IV also list the calculated thermal conductivity ,ratio, k ‘ k , , and ( k ’ k l ) / ( p / p i ) = B , / d . This ratio varies in the range considered from 1.0 to 1.16. The thermal conductivity ratio, k,’?:],Jyas then calculated from Equation 3 or
8.0
k!ki = bpB using the values of b obtained from viscosity and P-T’-T data and R from P-V-T data, both a t the same temperature and pressure ratio is evaluated.
.
CALCULATED RESULTS
For a given temperature the calculated values of b remain relatively constant n-ith a variation in pressure escept near, the critical pressure. I n the latter rcgion the calculations are quit(, sensitive to small errors, a 1% error in.
(g)vproducing an
error in the calculated viscosity ratio when using a constant value of b. As the pressure is increased, the viscosity ratio beconies less sensitive to errors in
(?$)?and b tends to remain constant
over a considerable pressure range. Hon-ever, as the pressure is increased, especially a t low reduced temperatures, a slight variation appears in b which follows a definite trend. Figurc 1 shows the typical variation of b with pressure for several gases at constant temperatures as calculated from viscosity data. KO attempt was made to calculate the thermal conductivity ratio beyond a retluced pressure of 10, except in the case of nitrogen where viscosity data were available to nearly 1000 atmospheres. This limit was set because of the variation in b at the higher pressures .in the region of low reduced temperatures. Table T’ compares Vargaftik’s experimental values (29) of the thermal conductivity of nitrogen expressed as k ’kl lvith these ratios calculated by Enskog’s equation. The calculated values agree with the experimental within 1%. Figure 2 shows the effect of reduced pressure on the calculated
3,0
Figure 3.
Calculated Thermal Conductivity Ratio t’s. Reduced Temperature
968
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE 11'. Temp.,
VaLLJES CALCUL4TED FROM \rISCOSITY AND
C.
P,Atm.
8.25
108.65 119.35 173,39 259.67 381.70 552.36
30
50
5.782 6.012 6,954 7.946 8.985 10.18
57,952 67,483 7 3 566 151 676 192.647 207 63 277 68 383 31 5 2 5 82
'1.385b 1.684b 2,049b
47 68 84 96 112 126 168 218 270 289 337 495 656
1.183b 1.391b 1,6746 2.031b 2.4436 2.756b
647 179 180 909 754 614 997 50 72 05 37 80
7.iQC
1.1301 1.355/ 1.604~' 2,0141 2.3531 2.6251 3.333/ 3.9551 4.528f 4.706f 5.2501 6.171 7.041
0 281 0 366 0 469 0 562 0.663 0.736 0.909 1.062 1.193 1.231 1.406 1.611 1.838
4 ??b 3.805 3.57b ;3 6?b 3.69b 3.75b
c___c_d
05
49.779 5x,9:35 90.45.' 121,201 149.726
a
3 966 3.63b 3.626
1.111 1 150 1,348 1 628 1 952 2 29 2 54
3 13 3 64 4 OB 4 20 4 75
0.304 0.314 0.388 0 478 0 ,562 0 663 0.737 0 908 1.056 1 178 1.219 1.376
U.395 0 530 0.651 1.196 1.341 1.392 1.574 1.796 2,038
4.05C 3.61C 3.33: 3.99 4.16C 4.17C 4.24c 4.22C d
0.30; 0.411 0,536 0.647 0,766 0.849 1.041 1,207 1.346 1.387 1.571 1.785 2.020
4 031
3.7l.f 3.42f 3 58f 3.55f
3.52f
3.6,f 3.72/ 3.80/ 3.821 3 73f 0 0
-------
DATA FOR E T H Y L E K E
1.337 1,383 1.576 1,809 2.059 2.334
4.88 4.90 4.93 4.86 4.79
0,350 0.464 0.565 1.052 1.188 1 286 1.410 1 622 1.856
1.4170 1.6730 1.88OC 4.196C 4.944c 5.160C 6.973C 6.84iC
44
1,184 1.227 1 411 1.636 1 .S i 6 2.143
P-V-T
7.77 8.79 9.86 11.09
0.326 0.340 0.434 0.545 ____0.647
__
h
0.766 0.850 1.040 1.200 1.331 1 374 1.541
__-
1 60: 1 91
1.57b 1 93b
2 36b
2 liC 4 i T C 5 58C
5.80C 6 6 i C 758C 8 5 6 ~ 1 24,' 1 5?/ 1 83/
130b 1 86b 1.9lb
2.345 2.836 7.18b
2 3?f z.:2/
3 031 3 821 4 4'3/ 5 11/ 5 301 . ~ . _ _ _ _ _5 _ 86/ __. 6 84f 7 741
c _
3.65 3 67 3 48 3.41 3 47
1.13 1.13 1.12 1.11 1.10 1.09
6.53 6.77
.-
1.1.3 1.14' 1 1 14< 1 13 1 13 1.12
1.10 1 0:)
1 12 1 14 1 15 1 16 1 15 1 15 1.14 1.13 1 1:J 1 12 1 11 1 10
xhere the riumeriLa1 conatant is that predicted t~\ Enskog. Table VI gives I* as obtairicd by using the values of b calculated from viscosity data The valut'c obtained are closer to 1.4 X 10-7 than to 1.872 X lo-' Figure 4 i. a plot o f :
1 IJi 1 06 1 1' 1.14
__
A.32
iatio ab a iuriction of reduced temperature a t constant ieduced pressures. C r o s ~plot5 such as this were uwd to construct the final correlation. The value b represents the actual volume of the molecules as given by Enskog'q equation (9). This equation, expressed in terms of reduced temperature, k:
G-,
.._._A
1.19 1 25 1 51 1.86 2 25 2.64 2.97 3.59 4 14 4.59 4 74
Vol. 39, No. 8
1 ,1 5 1 16 I 15 1 15 1 14 1 13 1 13 1 12
~_.
Beyond the range of viscosity d a t a , b v a s taken a3 4.75. b Values calculated with aid of CzH4 viscosity d a t a of Comings et czi. ( 6 ) . 6 Values calculated with aid of CBHa viscosity d a t a of Smith and Brown ( 1 6 ) . d Beyond the range of viscosity d a t a b was taken as 4.20. Data in t h e range in which the iso&tric goes through a n inflection point; therefore, Ratio aFunction of Reduced Pressure and Reduced Temperature
reduced temperature. This figure may be used to predict the effect of pressure on the thermal conductivity of any . gas - when its critical tempekature, critical pressure, and thermal conrluctivii y a t atmospheric pressure are knox7.n. Ex2mPLE. The use of the correlations for predicting thermal conductivity is illustrated by the follon5ng csaniple for ethylene at 150' F. and 300 atmoaphcres absolutc:
arid from Comings et nl. (6)
It Eolloms''Iat
?/
= 1
' j "f?
(MC,, -jL
-
= ~'~if',,
T,, MC,,)
(14)
P, = 50.9 atrn.; T, = 509" It k,
X-
=
=
0.0140
0.0140 X 4.81
(:ORRELiIIO\
=
l3.t.u. hr. X ft. X
O.OF45
O
F.
Bt u hr. X f f
F
OF PRAVDTL SU3IBER RATIO
In the dczign of heat transfcr equipment a t high pressure, it is often necessary t o knon. thc tffcict of pressure on the Prandtl numher, CL>p' k . Saunders (14) fourid it possible to correlate heat transfer data on nitrogen from 1 to 1000 atmospheres by assuming C,p/k independent of pres-ure a t relatively high reduced temperatures. This number for nitrogen a t 0", 50", and 100" C. From 1 to ,1000 atmospheres was calculated using the k,'kI ratios obtained above and the C, data of Deming and Shupe. These values, listed in Table VII, show a slight increase with pres-
I
4 2 3 REDUCED PRESSURE,Pr
5
6
7
8
910
Figure 6. Variation i n Prandtl Number Ratio with Reduced Pressure. and Reduced Temperature (Uot a General Correlation)
970
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE VII. EFFECT OF PRESSURE ON THE PRAADTL NUMBER Temp.,
c.
Tr
0
2.17
50
2.56
100
2.96
OF
NITROGEN
p, Atm.
CP (6). Cal./Mole
1 20 40 60 80 100 200 300 400 500 600 800 1000 1 40 80 100 200 300 400 500 600 800 1000 1 20 40 60 80 100 200 300 400 500 600 800 1000
6.91 7.20 7.51 7.83 8.15 8.44 9.25 9.46 9.39 9.28 9.21 9.11 9.04 6.92 7.29 7.63 7.78 8.38 8.69 8.80 8.82 8.81 8.75 8.70 6.94 7.06 7.18 7.29 7.40 7.50 7.95 8.27 8.47 8.07 8.61
8.64 8.71
k 0.706 0.727 0.737 0.758 0,776 0,793 0.834 0,839 0.830 0.821 0.817 0.816 0.814 0.700 0.716 0.729 0.736 0.766 0.769 0.773 0,772 0.771 0,769 0,768 0.693 0.696 0.700 0.703 0.711 0.708 0.723 0.732 0,739 0.744 0.745 0.747 0.756
Vol. 39, No. 8
takes account of the dependence of the latter ratio on MC,, which Figure 6 does not. SOX1ENC LATURE
1.00 1.02 1.04 1.07 1.10 1.12 1.18 1.19 1.18 1.16 1.16 1.16 1.15 1.00 1.02 1.04 1.05 1.08 1.10 1.10 1.10 1.10 1.10 1.10 1.00 1.00 1.01 1.01 1.03 1.02 1.04 1.06 1.07 1.07 1.08 1.08 1.09
'4
=
1
- f 0.8
bx
+ 0.7614bpz
a = constant
b B
actual volume of inolecules 1 1.2 0.7574bpz bpx = coiiytant = specific heat a t constant pressure = Ypccific heat at constant volume 9 = constant k = thernial conductivity a t given pressure and temperature k l = thcinial conductivity a t 1 atmosphere and temperature of k '$1 = inol(uh.r weight P = p w s u r e , atmospherer: P, = reduced pressure = I ' / P c , where P, is critical pressure R = gas constant, (cc.)(atm.)/" K./inole r = pressure residual, atmospheres s = constant 7 ' = absolute temperature, ' K. T , = rrduced temperature = T'/TC, where Yc is critical temperature v = specific volume, cc./gram x = inolecular collision probability factor density, grams,/cc. P = viscosity at given pressure and temperature !-l= iL1 = vi;.conity a t I atmosphere and temperature of p =
=
+
+
LITERATURE CITED Therefore the Prandtl number ratio is a function of the reduced pressure, the reduced temperature, and the molal heat capacity of the separate gases a t atmospheric pressure. Although i t has just been shown t h a t the Prandtl number ratio Is not a function of reduced temperature and reduced pressure alone, an approximate correlation was prepared on this basis. It indicates the magnitude of the variations in the Prandtl number with pressure. Values of MC, were calculated from the correlation by Watson and Smith for several gases, and the ratio C,/C,, was determined. Prandtl number ratios were then evaluated from the predicted viscosity and thermal conductivity ratios, and are shown in Figure 6 as a function of reduced pressure a t several constant values of reduced temperature. Figure 6 is of limited utility as a generalized correlation since it does not indicate the variation with MC,,, and also because the Watson and Smith correlation predict values of M C , - MC,, which may be aa much as 50% in error a t some points. At temperatures just above the critical, the Prandtl number increases considerably with increase in pressure b u t a t somewhat higher temperatures, the effect of pressure on C,r/k is slight. T h e Prandtl number ratio may be predicted for a given gas with greater certainty than by using Figure 6. This method consists of predicting the three ratios p / p 1 , k / k l , and C,/C,, (calculated from the correlations of M C , - MC,, and the experimental value of MC,,) from the separate correlations, and combining these to give the desired Prandtl number ratio. This method
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PRESESTED as p a r t of the Syi~iposiumon High Pressure Technology before t h e Division of Industrial a n d Engineering Chemistry at the 110th Meeting of t h e .$MF:RIC.AY C r i t ~ r r c . ~S~O.C I E T YChicago. , Ill.
