Ind. Eng. Chem. Fundam. 1981, 20,216-220
210
x = longitudinal coordinate (flow direction)
y = transverse coordinate yo = initial transverse particle position y* = amplitudes of particle excursions from streamline Greek Symbols a = see eq 24 { = phase
f = particle flux density
y = see eq 27 9 = efficiency
vc = gas viscosity corrected for Cunningham factor = (8.854 X F/m) X = wavelength of magnetic structure eo
F = permeability po = 4?r X lo-’ H/m w = angular frequency of
electric field
Alexander, J. C.; Melcher, J. R. I&. € q ~Chem. . Fun&“ 1977, 76, 311. Davey, K. R. Ph.D. Theds, Department of Electrical Engheedng and Com puter Science, M.I.T., Camkidgs, MA 02139, 1979. Davies, J. T. “Twbulence Phenomena-An Intmdudon to the E m Transfer of Momentum, Mass and Heat Transfer, Partlcularty at Interfaces”; Academic Press: New York, 1972 p 91. Deutsch. W. Ann. Phys. 1922, 68,335. Hoburg, J. F.; Penney, G. W. “Parallel and Twbuknt Fbw precrpitetw Efflclencles for Distributions of Partide MoMlities”; Proceedings of IEEE-Industrlal Appllcatkns Society Annual hhtlng, Cleveland, 1979. Inculet, I. 1.; Castle, G. S. P. ASMAE J. 1971, 73,47. Lewk, R. E. “Powder Cloud Devebpment Mechanisms"; Tokyo Symposlwn ’77on Photo- and Electro-Imaging,Soclety of Photographlc Scientists and Englneers, 1977. hu,H. Steub Reinhalt. Lun, 1909, 20, IO. Oberteuffer, J. IE€ Trans. Magn. 1974,MAG. IO, 223. Roblnson, M. Atmos. Envkon. 1987, 7, 193. Roblnson, M. J . AtPo&. COntrdAssoc. 1968, 18, 235. White, H. J. ”IndusMal Electrostatic Precipitation”; Addlson-Wesley: R e a m , MA, 1963 p 164. Zahedi, K.; Melcher, J. R. J. AkPoih~t.ConholAssuc. 1976, 26, 345.
Literature Cited
Received for review May 19, 1980 Accepted March 23,1981
Alexander, J. C. Ph.D. Thesis, Department of Electrical Englneerlng and Computer Sclence, M.I.T.,Cambridge, MA 02139,1976.
Precise Measurements of the Thermal Conductivity of Toluene and n-Heptane by the Absolute Transient Hot-wire Method Yujl Nagasaka” and Aklra Nagashlma Department of Mechankal €ngineering, Keio University, 3-14- 1 Hlyoshi, Yokohama, Japan
For the purpose of obtaining precise liquid thermal conductivity data as standard reference values, a new highprecision instrument, which is based upon the absolute transient hot-wire method, has been developed. A newly built digital system is used in this instrument for recording the transient temperature rise, which eliminates the conventional difficulty of this method. The thermal conductMtbs of toluene and n-heptane, reference substances suggested for the liquid thermai conductivity standard, have been measured in the temperature range 0-90 O C at atmospheric pressure. The accuracy of the present measurements is estimated as f0.5%.
Introduction As for the measurements of the fluid thermal conductivity in the fields of science and technology, there are two typical needs for its accuracy. Firstly, the accuracy of a few percent is sufficient for the calculation and designing in the industrial heat transfer problems. Secondly, we need the accuracy of, at least, better than 1%for calibrating the relative instrument or examining the reliability and the limit of theoretical prediction. From the first standpoint, a considerable amount of data for liquids important for industrial use has been accumulated in recent years. However, the data with the accuracy of less than 1%are rather scarce until quite recently. It is very difficult and exhaustive to measure the thermal conductivity of a large number of liquids absolutely and accurately in wide temperature and pressure ranges. However, if we try to determine the thermal conductivity in reference to the established value of another material, the apparatus and its operation will become simpler and easier. The accuracy of such relative measurements depends therefore in the first place on the reliability of the thermal conductivity value of the reference substance. Besides, although many types of empirical and semiempirical equations for predicting the thermal conductivity have been proposed in the past, not all of these equations have been examined against high accuracy data. Before
the theoretical treatise could be developed, reliable experimental data on the liquid thermal conductivity must be available. In the case of liquids, the thermal conductivity is a property which has been considered to be one of the most difficult to measure accurately. The difficulty associated with the measurement of that property is the onset of natural convection in the sample liquid. This problem is serious in steady-state methods because the presence of convection cannot be directly detected during the measurement. The transient hot-wire method, on the other hand, has the advantage that the effect of natural convection can be practically eliminated in its operation. Historically, this type of method was fmt wed by Stdhane and Pyk (1931). In the early stage of investigations, most efforts have been directed toward improving the theory. Because of the difficulty of recording the temperature rise of the wire, the accuracy of this method has not been better than that of steady-state methods. The transient method has been considered as an easy technique to measure the thermal conductivity due to the ease of construction and short measurement time. However, the accuracy of this method has been improved recently as a result of the development of electronic instruments and it has been gaining in reputation as a precision method. Recent works on measurements of the thermal conductivity of fluids by
0196-4313/81/ 1020-0216$01.25/0 0 1981 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 217
this method include those by Pittman (1968), Haarman (1969), Mani (1971), de Groot et al. (1974), and Castro et al. (1977). The purpose of the present study is to develop a highprecision transient hot-wire instrument to measure the thermal conductivity of liquids and to obtain the precise data which can be the reference standard for calibrating the relative instrument. In the present measurements the thermal conductivities of toluene and n-heptane, which have been suggested for the reference materials of the liquid thermal conductivity, have been measured in the temperature range 0-90 OC at atmospheric pressure. The accuracy of the measurements is estimated as i0.5%. Principle of Measurement Essentially, the apparatus consists of a vertical thin platinum wire immersed in the sample liquid. Initially at equilibrium conditions, a constant heat production is suddenly generated electrically in the wire. This causes a rise of temperature both in the wire and in the sample liquid. Then the wire acts both as an electrical heating element and a resistance thermometer, and the thermal conductivity of the sample liquid is determined by the constant heat production per unit length q and the time evolution of the temperature of the wire. In the simplest mathematical model of the transient hot-wire method, the temperature at time t referred to the initial value is given by (Carslaw and Jaeger, 1959) 4
AT(a,t) = - E i ( u 2 / 4 ~ t ) 4ax
where X is the thermal conductivity of the sample liquid, u is the radius of the wire, K is the thermal diffusivity, Ei
is the exponential integral, and t is the time after the start of heating. The assumptions made in the above derivation are: (1)infinitely thin and long wire, (2) liquid sample of infinite extent, (3) constant heat production of the wire, (4) uniform properties of the sample, and (5) no convection and radiation. When the value of u 2 / 4 ~ist sufficiently small, eq 1can be expanded as 9
AT(a,t) = - h ( 4 ~ t / a ~ C ) 4ah
where C = expy = 1.781..., and y is Euler’s constant. By differentiating this equation with respect to In t , the thermal conductivity of the sample liquid can be expressed as A=
9/4a
(3)
dAT/d In t
Thus eq 3 provides the basis of the transient hot-wire method of the thermal conductivity measurement. Although the ideal conditions described above cannot be satisfied in the experimental apparatus, it is designed so that the errors due to departure from this idealized model are negligibly small. Since a considerable amount of literature already exists on this analysis (Pittman, 1968; Healy et al., 1976), we introduce here only the final form of the analysis and magnitudes of the major errors. Finite Properties of the Wire. The error due to the assumption of a heat source of zero diamater can be estimated using the equation (Healy et al., 1976) AT=-
In
4Kt U2 w - 1 a2 + - + --In 2Kt 2Kt W U2c
4Kt -a2C
8
where A, and K, are the thermal conductivity and the thermal diffusivity of the wire, respectively, and w = pC,/(pC,),. In the present apparatus the magnitude of this error was 0.03%. End Effects. The electrical connection of the wire necessarily implies the presence of heat loss to the connection leads and two-dimensional heat flow near the end of the wire. Although the approximate analysis for these end effects exists as suggested in some studies (Haarman, 1969; Healy et al., 1976),it is not possible to estimate the error analytically because the experimental boundary conditions cannot be expressed mathematically. Thus it is adopted practically to compensate for this effect experimentally. There are two such methods: the shortand-long wire method and the potential lead method. In the present apparatus the latter method has been used to eliminate this effect. Finite Outer Boundary. The sample liquid is filled in the cylindrical vessel of radius b in the experimental apparatus. A correction to the recorded temperature rise is important at long time and can be examined by using the following equation (Fischer, 1939)
Here g, are the consecutive roots of Jo(gn)= 0. Jois the zeroth-order Bessel function of the fmt kind and Yois the zeroth-order Bessel function of the second kind. In the present measurement b / u = 670 and Kt/U2 N 7000 (at the final time of the measurement), and thus the correction due to this effect was not needed. Temperature Dependence of the Liquid Property. Considering the fact that the properties of liquid change with temperature and the maximum value of (l/X)(ah/ aT), of liquid is about 0.3%, a further correction to the value of experimental temperature is needed. The analysis for this point represents that when the initial temperature of the liquid is Tb,the temperature to which the measured thermal conductivity refers, T,, is defined as (Pittman, 1968) 1 Tr = Tb + i ( A T ( t J + AT(t2)} (6) where tl and t 2indicate the times at the start and the end of the run, respectively. Time Dependence of the Heat Production. In the basic assumption of the derivation of eq 1, heat production per unit length of the wire is constant. In practice, the resistance of the wire and the electric current through the wire vary during the experimental run and thus this assumption cannot be satisfied. However, since the temperature rise of the wire was sufficiently small in the present measurement, deviations from constant q were less than 0.04%. Experimental Apparatus One of the most essential points of the measurement based upon the transient hot-wire method is how to record the small temperature rise of the wire as a function of time accurately. In other words, the limiting factor in the precision of the measurement has been the difficulty of recording the temperature rise of the wire of 1-2 K in a few seconds. In the present instrument this problem has been solved by using high-precision digital equipment. A schematic diagram of the electrical circuit used to measure the thermal conductivity is dhown in Figure 1. Here, R , represents the resistance of the thin platinum wire which is enclosed in a liquid-filled cell. Before the start of a run the double bridge is balanced with a small
218
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 D V.M.
L4
D
Power upply
RS
/t-i
p Mercury relay
Ra
Figure 1. Schematic diagram of electrical circuit.
current and the resistance of the wire is measured. In the present study, the time-dependent temperature rise of the wire is measured in the following manner. After initiating the heating current which is provided by a constanbvoltage source to the platinum wire, the temperature rise of the wire is converted to transient voltages using the out-ofbalance of the double bridge circuit. These transient voltages are measured using an integrating digital voltmeter (DVM), which is triggered externally by a timing control, with a repetition rate of 6/s and the resolution of 0.1 pV. The time base for the measurement is generated by the 120-kHz crystal oscillator. The measured voltage data are memorized in a digital memory which has a memory size of 17 bits X 256 words. The 100-kQvariable resistor RB is used for stabilizing the current through the wire. Before the start of a run the current to be used in the run is allowed to flow through RBand the current is switched by a mercury-wetted relay from RB to the wire resistance R,. The final element of this circuit is a 1 4 standard resistor R, and the current is measured by the voltage across it. In this electrical circuit, the working equation, which is equivalent to eq 3, for the calculation of the thermal conductivity is written as
(7) 4rLdTR dR RS + S/ d Ad Inv t where S is an internal resistance of the double bridge, L is the length of the platinum wire between potential leads, dR/dT is the temperature coefficient of the wire resistance, and I is the current through the wire. Figure 2 shows the hot-wire cell assembly. This cell is designed to operate the measurement up to 500 bar (50 MPa) and from 0 to 130 OC. The platinum wire of 15 pm diameter (6),and of about 80 mm length is used as a heat source with two potential leads (12) spot-welded at positions 15 mm from each end of the wire. After the wire was mounted in the cell it was annealed and then tightened by means of the adjustment screw (4). Each of the four platinum leads is connected to the bridge circuit through the terminal (9) sealed with Teflon packing (10). A copper cylinder (7) ensures a nearly uniform vertical temperature distribution of the inner pressure vessel. The indicates temperature difference did not exceed 0.05 K during the experiments. The entire cell assembly was immersed in a liquid thermostatic bath attaining the temperature stability of st0.05 K, and the temperature of the bath was measured with a standard thermometer. The calibration of the resistance-temperature relation of the platinum wire was carried out in the temperature A=---
V
/I/ A u Figure 2. Cell assembly: 1, flange; 2, vessel; 3, conductor; 4,adjustment screw; 5, spring; 6, Pt wire & 15 pm; 7,Cu cylinder; 8, Pt leads & 500 pm; 9, terminal;10, Teflon packing; 11, Cu packing; 12, potential lea% 13, Bakelite elements. 0.2y
I
-021 0
1
I
I
O
Figure 3. Deviations of AT,, Toluene at 81.66 "C.
I
I
I
2
3
Time/s
I
I
4
from least-squarea straight line.
range C--90 O C during the course of the experiments. The resulting data were correlated with a maximum deviation of 0.05% by the equation R(T) = 25.1150 + 9.9258 X 10-2(T- 273.15) - 1.29 X 10-'(T - 273.15j2 (8) where R is the resistance of the platinum wire in f2 and T is the absolute temperature. Estimation of Accuracy The magnitudes of the main factors which determine the accuracy of the thermal conductivity measurements were estimated as follows. The length of the platinum wire between the potential leads was measured by a cathetometer and the measurement was accurate to 0.02%. The uncertainty in the temperature coefficientof the resistance dR/dT, which was obtained by differentiating eq 8 with respect to T,was estimated as within 0.08%. The current through the wire was measured by the DVM and the uncertainty in the measurement was 0.05%. The possible error in the slope of the voltage (or temperature)-log time relationship includes the uncertainties introduced by electrical noise and the timing of the voltage measurements. Figure 3 shows the deviations of experimental data from lineality predicted by eq 2 for a typical run. The maximum deviation is less than 0.2% and the accuracy of dAV/d ln t is estimated as 0.17%. Finally, accounting for other small factors, the overall accuracy of the present
Id.Eng. Chem. Fundam., Vol. 20, No. 3, 1981 218 t /s
Table I. Experimental Thermal Conductivity Data for n-Heptane temp, "C
4,W/m
1.67 16.66 31.66 46.56 61.81 77.00 92.05
0.2700 0.2841 0.3066 0.3148 0.3205 0.3243 0.3301
Table 11. Experimental Thermal Conductivity Data for Toluene temp, "C 4,W/m 1.43 23.74 43.90 67.85 81.73
h, W/m
I
K
10
1 '
I
'
" '
0.1286 0.1 236 0.1192 0.1155 0.1114 0.1069 0.1021
I
P 11.5
i
h , W/m K
0.26 29 0.2898 0.2982 0.3277 0.3326
0.1374 0.1308 0.1251 0.1180 0.1140
thermal conductivity measurements is 0.5%. Results and Discussion The apparatus described here has been used to measure the thermal conductivities of toluene and n-heptane, which have been suggested as the standard reference materials of the liquid thermal conductivity (Riedel, 1951; Ziebland, 1961; Castro et al., 1977), in the temperature range 0-90 "C at atmospheric pressure. In the present measurements the toluene and n-heptane used were spectrograde samples which had a stated purity of better than 99%, and the samples were taken directly from the bottles. Figure 4 shows a typical recording of the temperature rise of the wire. As indicated in this figure, the start of natural convection is detected as a curvature in the plot of AT against In t. Consequently, the convection effect on the measurement of the thermal conductivity is eliminated by using the data before the start of natural convection. Table I and Table I1 list the measured data for the thermal conductivities of n-heptane and toluene at atmospheric pressure, respectively. Each datum represented here is the mean value of four to nine runs. The temperature rise of the platinum wire during one run was 1.7-2.2 K and the measuring time was 0.4-4.4 s. A period of about 30 min was allowed between runs. The experimental data for toluene and n-heptane have been represented by the following linear equations n-heptane: X = 0.1287,, - 2.858 X 104(T - 273.15) (9) toluene: X = 0.137& - 2.914 X 10-4(T - 273.15) (10) where Tis the absolute temperature and X is the thermal
- 3 - 2 - 1
0
1 in t
2
3
4
Figure 4. Typical recording of the temperature rise of the wire. Toluene at 19.72 O C , q = 0.3326 W/m.
conductivity in W/m K. The departure of the experimental data from these equations is *0.2% with a standard deviation which is consistent with the estimated accuracy. Figure 5 compares the present experimental data for n-heptane with those in earlier work. The data reported by Jobst (1964), which have an estimated uncertainty of leas than 2%,are consistent with the present results within mutual accuracy. The data of Castro et al. (1977) deviate systematically by about 1%, which still lie within the mutual accuracy of 0.5%, and ita temperature coefficient agrees within 1.4%. The present results for toluene are compared with earlier work in Figure 6. As shown in this figure, the present results are in general rather lower than previous work, which can be explained for the elimination of the convective effect. The data of Castro et al. (19771, obtained with the transient hot-wire method, agree with those of present work within the claimed accuracy. However, the data of Poltz and Jugel (1967) with the parallel-plate method and Mani (1971) with the transient hot-wire method are about 1% lower than the present data systematically. The reason for this difference is that they proposed and applied the correction for radiative heat transfer effects due to infrared absorption of toluene as shown in the investigation of Saito et al. (1976). In the present measurement, this effect could not be detected clearly. Therefore further investigation of the problem of radiative heat transfer is left for future study. In Table I11 the measured thermal conductivity of toluene at 0 OC, the temperature coefficient of the thermal
Table 111. Summary of Previous Thermal Conductivity Measurements of Toluene author (year) Bridgman (19 23) Riedel(l951) Challoner and Powell (1956) Ziebland (1961) Horrocks and McLaughlin (1963) Venart (1965) Poltz and Jugel (1967) Pittman (1968) Tree and Leidenfrost (1969) Mani (1971) Castro et al. (1977) present work
type of apparatusa
temp range,
"C
claimed accuracy, %
2 1, 2, 3 1
30,75 -80, 2 0 , 8 0 0-80 -15-11 2 25-61 15-80 25-55 -92-123 0-100 -91-101 10-50 0-80
1
