The extensive numerical modelling of transport phenomena was performed for the melt growth of mercury cadmium telluride. To increase the fidelity of modelling the kinematic viscosity of liquid Hg(1-x)Cd(x)Te was determined at various compositions in the range of x between 0 and 0.2 and at temperatures around and below the respective melting point. The phase diagram of Hg(1-x)Cd(x)Te shows that for this range the melting point varies from 670 C for pure HgTe to 790 C at x=0.2. The vapor pressure above the melt varies correspondingly from 15 to about 40 atm. Hence, the measurement of viscosities in this system requires a technique that allows for combinations of high temperatures and pressures. In addition, a closed isothermal system is required. The high pressure melt container must also be inert to molten Hg(1-x)Cd(x)Te to avoid possible errors from contamination of the liquid. A novel technique that largely circumvents the above experimental problems is described. Its theory is also presented

Alexander, J. Iwan D.

Jin, Wei-Qing

Rosenberger, Franz

English

NASA Technical Reports Server

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8.1
"Viscosity Measurements at High Temperature and Pressure: A Novel Technique"
(NASA-CR-}a7239) VISCQSITY MFASU_EMENTS AT
HIGH TEMPERATURE AN_ HIGH PR£SSURE: A NnVEL
TEC!-INrQUE (Alabdma Univ.) L2 p CSCL 20C
N91-}3967
Uncl as
G3/70 0309068
https://ntrs.nasa.gov/search.jsp?R=19910004654 2020-03-19T20:35:10+00:00Z
VISCOSITY MEASUREMENTS AT HIGH TEMPERATURE AND
HIGH PRESSURE - A NOVEL TECHNIQUE
Wei-qing Jin, J. Iwan D. Alexander and Franz Rosenberger
Center for Microgravity and Materials Research
University of Alabama in Huntsville, Huntsville, AL 35899
1. Introduction
Most modelling of transport phenomena in materials processing from fluid phases suffers
from the lack of reliable data for the transport properties under typical process conditions. Our
group is involved in extensive numerical modelling efforts for the melt growth of mercury
cadmium telluride. To-date the viscosity of this liquid material has only been estimated [1]. To
increase the fidelity of our modelling, we set out to determine the kinematic viscosity of liquid
Hgl.xCdxTe at various compositions in the range 0<x<0.2 and at temperatures around and below
the respective melting point. The phase diagram of Hgl_xCdxTe [2] shows that for this range the
melting point varies from 670°C for pure HgTe to 790°C at x = 0.2. The vapor pressure above the
melt varies correspondingly from 15 atm to about 40 atm. Hence, the measurement of viscosities in
this system requires a technique that allows for combinations of high temperatures and pressures.
In addition, a closed isothermal system is required. Otherwise, due to the large disparity of the
fugacities of the two binary constituents [2], rapid and uncontrolled compositional shifts in the
liquid would occur during a measurement. The high pressure melt containfr must also be inert to
molten Hgl.xCdxTe to avoid possible errors from contamination of the liquid.
Similar problems were faced earlier by Kakimoto and Hibiya who determined the viscosity
of molten GaAs employing the oscillating cup method [3]. The accuracy of this method, however,
tends to be limited by the temperature and load dependence of the torsional spring constant of the
suspension fiber used. In addition, it is difficult to design a suspension system for the oscillating
cup that ensures purely rotational motion.
2In this paper, we describe a novel technique that largely circumvents the above
experimental problems. The principle of this technique and its theory are presented in Section 2.
The actual instrumentation is described in Section 3, together with viscosity measurements of room
temperature liquids, which illustrate the precision and accuracy of the method. The planned
extension of the technique to high temperatures and pressures is touched upon in Section 4.
2. Principle and theory of the technique
As sketched in Fig. 1, the viscometer cell consists of two cylindrical containers (radii rl
and r2) that are connected at the bottom by a capillary tube of radius r and length L. The liquid of
interest is initially filled into the cylinders to unequal heights. The resulting pressure head forces
the liquid to flow through the capillary until equal heights are attained. The mass transfer associated
with this equilibration is recorded by differential weighing. This is done with an electrobalance
which, advantageously, keeps the cell horizontally positioned at all times. The kinematic viscosity
of the liquid can be obtained from the time-dependent weight shift between the two cylinders. Note
that the cell used in this technique resembles a classic viscosity measuring arrangement devised by
Ostwald[4].
Ah o
I
' 2 r1
lu==
I Ah
..... iill
. e,.L_,,.. -,
/'.,/
_,,/',¢.
|
: : E|
: L = 2r2 '
Figure 1. Schematic diagram of viscometer
3The theoryis basedon thepremisethattheexchangeof massbetweenthetwo reservoirs
(seefig. 1) takesplaceby atime-dependentPoiseuilleflow throughthenarrowcapillary.Theflow
is drivenby a pressuregradientG(t) which is takento beuniform in thecapillary.Thepressure
gradientis givenby
G(t) = p g Ah(t)
L ' (1)
where p is the constant density of the fluid, g is the gravitational acceleration and Ah(t) is the
liquid head.
For a unidirectional flow driven by a uniform time-dependent pressure gradient, the
creeping flow assumption is valid provided that its Reynolds number is much smaller than unity at
all times, i.e.
Re=_ <<1 , (2)
V
where v = rl/p the kinematic viscosity, with rl the dynamic viscosity. The reference velocity Vrnax
arises in response to the initial pressure head APmax associated with the initial liquid head AIM at
to, the time of the fast differential (height) measurement. In a PoiseuiUe flow through the capillary
Vmax = _ APmax r2x p g Aho
8 _ L = 811 L (3)
Consequently, the limiting condition (2) becomes
Re g Aho r2= << 1 (4)
8 v 2
Provided that (4) is satisfied, the volume flow rate at time t can then be written as a modified form
of Poiseuille's relation [5]
Q(t) = _ AP(t) (5)
_qL
where AP(t) is the time-dependent pressure difference between the two ends of the capillary.
4Themassof liquid in containerI is
Ml(t)--Ml(t0) + AMI(t) , (6)
with a similar expression for the mass in container 2. The mass difference between the two
containers is then
AM(t)= MI(0 - M2(t) = &M0 + 2AMI(t) , (7)
where the 2AM1 term follows from the fact that the total mass is constant (AMI(0 = - AM2(t) ),
and the difference in mass between the two containers at t=t0 is
zkMo = Ml(to) - M2(to) (8)
The liquid head zMa(t) between containers 1 and 2 can be expressed as
Ah(t) = hi(t) - h2(t) = Aho + (_ + r_) Ahl(t) , (9)
where we have used the identity _Ahl(t) = - _2Ah2(t), which follows from the conservation of
mass. The changes in liquid level Ahl and Ah2 are as defined in Fig.1.
The rate of change of the mass difference between the two containers, using (5), is
dAM(t)
= 2.___..1,t,dAM¢_= 2 p Q(t) ="a'a-:-AP(t)-'a
dt dt 4vL
(10)
The pressure difference AP(t) is
AP(t) = p g Ah(t) = g(_ + r_) AMCt)
where we have used equations (6) and (7) together with the identifies
(11)
2p_
AMo = Aho , (12)
5and
AMI(t) = O x R 2 Ahl(t) (13)
The equation for rate of change of the mass difference AM(t) is obtained upon substitution of (1 1)
into (10) and takes the form
dAM(t) _ g r4(r21 + 4) AM(t) , (14)
dt 8vt,r]l 
with AM(t0) = AM0. This has the solution
where the characteristic time 1: is
AM(t) = AM0exp(- t ) (15)
x = 8vL_ (16)
r4g(_ +fi)
Fig.2 illustrates this exponential decay of the mass difference between the initially unequally f'tlled
containers.
1.2
1.0 t
0.8
O3
<
0.6
03
O3
UJ
'J 0.4Z
_o
03
z 0.2
LLI
D 0.0
0 1 2 3 4 5
DIMENSIONLESS TIME (t / "c)
Figure 2. Normalized mass difference AM(t) / AMo versus normalized time t/'_
6The kinematic viscosity can be obtained from equations (15) and (16) and is given
by
v = - r4g (_+r2) t / In (AM/AMo) (17)
For the special case that rl = r2 =R, equation (17) simplifies to
r4g
v = t / In (AM/AMo) = - 13t/In (AM/AMo) , (18)
4LR 2
where the viscometer cell constant is 13= r4_g
4 L R 2 (19)
3. Apparatus and low temperature measurements
In order to determine the precision and accuracy of this method, we have performed
viscosity measurements at 24 °C with water, at 25 °C with ethanol and carbon tetrachloride and at
22 °C with glacial acetic acid. The experimental arrangement used consisted of a viscometry cell
suspended from an electrobalance (Cahn model 1000), see Fig. 3, the output of which was fed to a
chart recorder. We used a glass cell with R = 1.42 cm, L = 9.5 cm, and r = 0.0431 cm. Besides
the capillary connecting the bottom part of the cylinders, the cell was reinforced by a rod
connecting their upper parts. Holes in the upper shoulders of the cylinders facilitated cleaning and
filling.
Figure 3. Photograph of viscosity measure-
ment cell suspended from electrobalance.
ORIGINAL PAGE IS
OF POOR QUALITY
7The measurement procedure was as follows. After filling the horizontally positioned cell to a
certain level, one of the hang-down wires from the balance was raised and the liquid was allowed
to equilibrate to the new levels in the tilted ceil. On rapid re-levelling of the cell, there is then a
liquid head which drives mass transfer between the containers. The associated differential weight
change (AM/AM0), which is directly computed by the data processing system of the
electrobalance, was then recorded versus time.
Figure 4 shows a chart recording obtained with CC14. From this curve, values of
-In(AM/AM0) and -t/In (AM/AM0) are calculated for increasing times after the first reading is
taken. Note that tO can be taken at any point after the balance has stabilized after the re-leveUing of
the cell. The results shown in Table 1 illustrate that the mass transfer indeed follows an exponential
function with high precision, as can be seen from the small variations in the values of the last
column.
Table 1: Experimental values of-In(AM/AM0) and -t/In(AM/AM0) for CC14 at 25 oc
t (sec) AM/AMo -In (AM/AM0) -t/In (AM/AM0)
66 0.60 0.51 129
90 0.50 0.69 130
101 0.45 0.80 126
118 0.40 0.91 130
135 0.35 1.05 129
154 0.30 1.20 128
The reproducibility of the measurements from run to run falls within the fluctuations of the
-t fin (AM/AM 0) values obtained within a run. Table 2 displays the weight transfer results obtained
from several runs for the four liquids investigated.
In order to evaluate these data for the kinematic viscosity, the cell constant 13needs to be
determined, see eq. (18). We have first evaluated eq.(19) for the geometrical dimensions used.
8Figure 4. Chartrecording of weight transfer versus time obtained with CC14 at 25°C.
ORIGiI'_ID,L PAGE IS
OF POOR QUALITY
This resulted in 13geom = 4.44 x 10 "5 cm2/s 2. Note, however, that due to the fourth power depen-
dence of [3 on the capillary radius an uncertainty of 1% in r translates into an error of 4% in [3 and,
thus, in v. Considering all dimensions (r, L, R) entering the evaluation ofeq. (19), we estimate the
total uncertainty to be _+5%. Hence, we have also determined the cell constant from eq. (18) and
our data for water (Table 2), for which the dynamic viscosity is known with high accuracy [6].
Thus we obtained a _calib = 4.52x10 "5 cm2/s 2.
Table 2: Experimental values of -t / In (AM/AM0) [sec] for four liquids
Water /-/20 E&an_ Carbon _trac_oride Glad, acetic acid
C2H60 CC_ C_I'{402
202 313 129 324
201 310 130 320
199 313 126 319
202 316 130 322
204 310 129 318
207 313 128 322
204 306 132 324
201 310 130 323
207 308 131 319
204 310 132 322
207 311 132 318
204 309 129 319
In Table 3 we have listed the kinematic viscosities deduced from the weight transfer data of
Table 2, using both cell constants. Percent deviations from the v -values (last column) computed
from the published results for rl and p (for which no accuracy statement could be found) are
given in parentheses. The deviations for the upper three liquids shown in Table 3 are 2% or less.
Thus, from the results for these three liquids we conclude that our technique is very accurate.
However, our result for glacial acetic acid deviates by +28% from the value deduced from the
10
literaturedata.This maybedueto alackof purityof theacidweused.Glacialaceticacid israther
hydroscopic.Addition of water to the pure acid, which can occur by absorptionin a moist
atmosphere,changestheviscositysignificantly[7].
Table 3: Comparisonof experimentalresultsfor kinematicviscosities(in cm2/s)obtainedin this
studyandcorrespondingvaluescomputedfrom literaturedatafor r1andp.
v from
Liquid [3geom 13calib rl[6] p[7] v (calc.)
10-3Nsm-2 gcm-3 cm2s-1
Water 0.00906 0.8985
H20 (24°C) (- 1.8%) (+0.1%)
0.9973 0.009221
Ethanol 0.0138 0.00140 1.081 0.7852 0.01376
C2H60 (25°C) (+0.3%) (+1.7%) (+0.5%)
Carbon tetrachloride 0.00577 0.00587 0.9116 1.5867 0.005746
CC14 (25°C) (+0.4%) (+2.1%) (+0.5 %)
Glacial Acetic Acid 0.0143 0.0145 1.1004
C2H402 (22oc) (+27%) (+28%) (+1.0%)
1.0477 0.01050
4. Extension of technique to high pressures and temperatures
Encouraged by the favorable results obtained at atmospheric pressure and ambient
temperature, we plan to employ this technique for studies on liquid mercury-cadmium-telluride.
Note that the arrangement described above is well suited for operation at high temperatures and
pressures since a cell, constructed of heavy wall silica tubing, can be hermetically sealed and hung
on non-oxidizing wires into a high temperature furnace. Thus, the sensitive electrobalance can be
well isolated thermally. In order to prevent a differential pressure build-up between the cylinders
in the cell during mass transfer, the upper rod (Fig. 3) will be replaced by another capillary. A split
tube furnace has been ordered, that will facilitate the mounting of the cell on the hang-down wires.
11
References
[11
[2]
[3]
[4]
[5]
[61
[7]
D. H. Kim and R. A. Brown, J. Crystal Growth 96, 609 (1989)
J. C. Brice and P. Capper, J. Crystal Growth 75, 395 (1986)
K.Kakimoto and T. Hibiya, Appl. Phys.Lett., 50, 1249 (1987)
Wo. Ostwald, Z. Phy. Chem., 111, 62 (1924)
R. B. Bird, W. E. Stewart and E. N. Lightfoot (ed.), Transport Phenomena (John Wiley
& Sons Press, New York 1960) p. 46
D.S. Viswanath and G. Natarajan (ed.), Data Book on the Viscosity of Liquids
(Hemisphere Publishing Corporation, New York 1989)
R. C. Weast (ed.), Handbook of Chemistry and Physics (C R C Press, Florida 1980)


Viscosity measurements at high temperature and high pressure: A novel technique

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