333
Id.Eng. Chem. Fundam. 1981, 20, 333-336
Laser Flash Method for Measuring Thermal Conductivity of Liquids. Application to Molten Salts Yutaka Tada,' Makoto Harada," Masataka Tanlgakl, and Wataru Eguchl Institute of Atomic Energy, Kyoto universny, by, Kyoto 6 I 7, Japan
A new variation of the laser flash method for measuring the thermal conductivity of molten salts was developed. The sample liquid was sandwiched between a thin metal disk and a sample holder without using any liquid container. This ensured onedimensional heat flow downward through the sample liquid. In this method, the thermal CondUctMty of liquids could easily be obtained without using any reference materials and without the measurement of the total input heat energy of the laser beam and of the liquid layer thickness. The minor heat loss to the surroundings could easily be obtained experimentally. The data obtained for sodium and potasslum nitrates between 320 and 390 O C agree well with those published in the literature. The mean deviation, 3.8 % , was only slightly larger than that in the cases of water and toluene near room temperature, 2.6%. This suggests that the present method is suitable for measurement at higher temperatures.
Introduction The use of high-temperature liquid materials has increased recently and the measurement of their thermal properties becomes increasingly important. Though the thermal conductivities of these materials are vital for understanding the behavior of these materials and also for plant design, there exists no reliable method established so far for measuring them, especially for molten salts. The significant heat loss a t high temperatures makes the use of steady-state methods difficult. That is, although the difficulty of heat loss estimation requires reference materials, there is no reference material available for high temperatures. The hot wire transient method for ionic molten salts may cause considerable error resulting from convection due to ita geometry and also by the high electrical conductance of the salts. Utilization of another useful method with stepwise heating by electric current is questionable for molten salts because of the high electrical conductance. The laser flash method developed by Parker et al. (19611, which has been applied to liquid mercury by Schriempf (1972), is promising because it does not have the above difficulties. This method, however, does present difficulty when applied to molten salts because the thermal conductivities of these liquids are much lower than those of liquid metals and the heat flow through the container wall cannot be neglected. We have developed a new method applicable to liquids of low thermal conductivity and have shown that it could be used for liquids near room temperature with satisfactory accuracy (Tada et al., 1978). In this method, the sample liquid is sandwiched between a thin metal disk and a sample holder without using any liquid container. This ensures that heat absorbed on the metal disk flows downward one-dimensionally through the sample liquid and minimizes thermal disturbances due to convection. The thermal conductivity of the sample liquid is obtained from the temperature response of the metal disk due to the heat discharged into the liquid withoot using the value of the liquid layer thickness. The aim of the present work is to demonstrate that this method is applicable to molten salts at elevated temperature.
Principle of Measurement Consider a system composed of three layers as shown in Figure 1. The metal disk receiving the laser beam is designated as the first layer, the thickness of which is 1. The sample liquid under the metal disk and the inert gas above the disk are designated as the second and the third layers, respectively. When t < 0, the system is in thermal equilibrium state and at t = 0 the impulse laser beam is flashed onto the top surface of the first layer. The expression describing the temperature response of the system was obtained and is represented in the report of Tada et al. (1978). Here, four ideal conditions were postulated (1) one-dimensional heat flow; (2) semiinfinite cylindrical shape of the second and the third layers; (3) negligible contact resistance between the first and the second layers; (4) negligible change of the physical properties due to the small temperature variation. The first layer is a thin metal disk with high thermal conductivity. Therefore, the temperature of the fmt layer becomes uniform in a short time and the response thereafter is expressed (1)
hi = (pih,Cpi)1/2/(p1CplZ) i = 2,3
(3)
To = Q/(PlC,lO
(4)
(2)
Cpi,and hi are the density, specific heat capacity, and thermal conductivity of the ith layer, respectively. Q is the input heat energy per unit area. The exact solution and the condition for the applicability of eq 1 are given in the report by Tada et al. (1978). With eq 1,the thermal conductivity of the sample liquid is obtained from the measured temperature response of the metal disk without using the thermal conductivity of the metal disk and the thickness of the sample liquid. Note that there is no constant characteristic of the apparatus used. This makes the absolute measurement of the thermal conductivity possible without using any reference materials. Experimental Apparatus and Procedure The experimental layout is the same as that reported previously (Tada et al., 1978). The diagram of the experimental apparatus is shown in Figure 2, and the detailed
p,,
Department of Environmental Chemistry and Technology, Faculty of Engineering, Tottori University, Tottori 680, Japan. 0196-4313/81/ 1020-0333$01.25/0
T l ( t )/ To= exp(h2t) erfc(W2) h = hz + h3
0
1981 American Chemical Society
334
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 L a s e r beam irci l a ) e r ( l n c r t
-------- ----__ &
1st
layer['lctnl
p n ~ j
disk)
2nd 1nyeriSa:npie I icjuld)
%
Sample holder.
Figure 1. Schematic diagram of geometry.
Figure 3. Measuring setup and cell in detail (see caption of Figure 2 for details).
O
b
Figure 4. Procedure of forming cylindrical sample liquid layer.
Figure 2. Experimental apparatus. Construction of measuring device and liquid reservoir: 1, liquid reservoir; 2, cylindrical quartz mantle; 3, electric furnace; 4, programmable controller; 5, 6, 7, chromel-alumel thermocouples;8, support;9, handle; 10, cell cover; 11, suspender;12, upper lid of mantle, 13, 13', optical glass window; 14, nickel wiper; 15, glass wool; 16, nitrogen gas cylinder; 17, brick insulator; 18, sample holder; 19, nickel rods; 20, metal disk 21, nickel wires; 22, nickel rods; 23, 23', sample liquid.
description of the essential parta in Figure 3. The liquid reservoir (1)is placed on the support (8)with the insulator (17) between them. The measuring device is fixed to the upper lid (12) of the mantle by the suspender (11).These are sealed in the cylindrical quartz mantle (2) and are located at the center of the electric furnace (3), which is regulated by the controller (4). The mantle has a window (13) for the passage of the laser beam and gas lines for evacuating or filling nitrogen gas into the system. The liquid reservoir can be moved vertically against the fixed measuring device by the motion of the handle (9) attached to the support. The sample holder (18)which is made of nickel is fixed to the suspender (11)by three nickel rods (19). The metal disk (20) is suspended by three nickel wires (21) of 0.05 mm diameter about 1mm above the top of the sample holder. The temperature of the top surface of the disk is measured by a chromel-alumel thermocouple (5) of 0.05 mm diameter spot-welded on the top surface of the disk. The temperature of the liquid in the reservoir and of the furnace wall are measured by the other chro-
mel-alumel thermocouples (6) and (7), respectively. Two kinds of metal disks were employed as the first layer; a nickel disk 0.0250 cm thick and 0.655 cm in diameter (purity min. 99.9 wt %) and a silver disk 0.0250 cm thick and 0.600 cm in diameter (purity min. 99.999 wt %). In using the nickel disk, the temperature near its Curie point at 358 "C was avoided. The molten salts used were sodium nitrate and potassium nitrate. Both were of reagent grade (purity min. 99.5%). The salt powder wa8 melted under nitrogen atmosphere in the cell placed at the lower position shown in Figure 4a. After evacuation of the mantle for the release of gas bubbles present in the fused salt, the cell was raised until the liquid level just touched the metal disk as shown in Figure 4b. Then the nickel wiper (14) was turned to break the liquid surface so that the liquid poured in between the metal disk and the sample holder. If any bubbles were observed under the disk, they were removed by turning the wiper again. During this operation, the furnace was opened for direct observation. The cell was lowered again to the position shown in Figure 4c, leaving the cylindrical liquid layer of about 1mm in thickness sandwiched by the metal disk and the sample holder. The furnace was then closed and the whole system was allowed to be at the desired temperature before the triggering of the laser beam onto the metal disk. The temperature response was then detected by the same procedure reported previously (Tada et al., 1978). Results and Discussion Figure 5 shows the typical temperature responses of the top surface of the nickel disk. The upper (a) and the lower (b) curves correspond to the responses without the sample liquid and with sample liquid (Ni-NaNO:, system), respectively. The response without sample liquid was taken for the estimation of the heat loss from the disk to the surroundings. The surface temperature of the disk rises up to the maximum value almost instantaneously, due to the impulse laser flash, and decreases sharply due to the rapid heat flow in the metal disk. Thereafter, it decreases
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981
335
1 1,O -
0
1
0.5 t I S 1
+&Ov5
Figure 5. Representative temperature responses of the top surface of the metal disk at 338 "C. The metal disk is nickel (0.655 cm in diameter and 0.0250 cm in thickness) and the sample liquid is sodium nitrate. Curve a: without liquid; curve b with liquid.
0.4
t
t
L
0
O,4
0,2
p y
2.2=
+-
2.0-
-
I
o
I
0.2
t
~
"
0.4 p / 2
'
'
0.6 s1/2
]
o.a
'
' 1.0
G.8
]
Figure 7. Replot of the data in Figure 5 (curve b).
i
1
0,6
2 [ s1/2
I
Figure 6. TI@)against t1/2(without sample liquid). Slope and intercept show -4h8,T0/(a1/2) and To,respectively.
gradually with the lapse of time due to the heat loss from the metal disk to the surroundings. The possible causes of this heat loss are by conduction through suspending and thermocouple wires and convective, conductive, and radiative heat transport through the gaseous phases. Among these, the heat conduction through wires was shown to be negligible in the previous report since doubling the number of the suspending wires did not give any signifcant difference in the result. Radiative heat transport through the gaseous phase is small enough to be neglected because the maximum temperature rise of the metal disk is only about 2 K at 300 to 400 "C. The remaining conductive and convective heat transport through the gaseous phase were shown to be treated as an effective conduction (Tada et al., 1978)
T l ( t ) / T o= 1 - ( 2 / r 1 9 (2h3,eff,)t1/2 + O[(2h3,en,t1/2)2] (5)
Here, the heat loss from the periphery of the metal disk was neglected. Higher order terms of 2h3,eff,t1/2 were neis much smaller than unity glected in eq 5, because h3M.t1/2 when t < 0.5 s. Note that both the second and the third phases are gas phases in this case in contrast to the actual measurement with sample liquid, where only the upper third phase is gas. The plotted points of Tl(t)against t1/2lie on a straight line as shown in Figure 6. This indicates that the poswas tulates for eq 5 are reasonable. The value of h3,eff. obtained in this case to be 0.032 s-1/2 from the slope of the straight line in the figure. This value is considerably larger than the h value estimated for the case of pure conduction, 0.0048 s-lY2. The difference of the two values may arise from the convective heat transport due to gas phase turbulence. The value of h,, which shows the heat transfer to the third layer, however, is much smaller than the value of h2,1.011 d2 as obtained below, which shows the heat conduction to the second molten salt layer. Thus, it is again reasonable to estimate the heat loss as an effective heat conduction. The value of h was found to change slightly whenever the e x p e r i m e n x p p a r a t u s was reset because of minor changes in the disk orientation and the angle. It was,
therefore, determined for each run prior to adding the sample liquid. The linearity shown in Figure 6 was always obtained. The temperature of the metal disk in the case with sample liquid (curve b in Figure 5) initially behaves in a similar fashion as in the case without sample liquid (curve a), and thereafter decreases much more rapidly due to the heat conduction to the sample liquid. The data in the range of 50 to 500 ms were fit to eq 1 by the least-squares method and the resultant h and Tovalues were found to be 1.043 s-lI2and 2.52 K, respectively. Subtraction of h,, value from h gives h2 to be 1.011 s-lI2 and the thermal conductivity of molten salt, X2, is calculated to be 5.28 X 1 V W cm-l K-l. Values of p2, Cp2,and C were taken from the literature [p2: Jam et al. (1972);Cpl(&i): Kelley (1949); C (Ag): Stull and Sinke (1956); C 2(NaN0)3): Janz et (1979); Cp2(KN03): Clark (1973!] and p l l was determined by measuring the mass of the metal disk. Figure 7 shows the plot of the response curve obtained in Figure 5 with sample liquid. The ordinate is scaled such that the relation between T l ( t ) / T oand t1/2gives a straight line according to eq 1. The solid line is the one calculated with the above values of h and Toand agrees well with the observed data. The convection in the liquid phase was neglected in the above treatment and few words should be necessary on this account. Convection in the liquid phase can be neglected if the product Ramd is less than 5 X lo4 cm, where Ra is the Rayleigh number and d is the liquid thickness in centimeters (Michels and Sengers, 1962). The thickness is about 0.1 cm and the temperature difference between the top and the bottom of the sample liquid is about 2 K. Thus, the value of R a d is about 3.5 cm, which satisfies the above condition. Thus, convection in the sample liquid need not be considered. The thickness of the sample liquid was about 1 mm. The temperature rise at the bottom of the liquid layer calculated at t = 0.5 s is less than 0.005T0K. This ensures the semiinfinite treatment of the liquid layer used in the present work is valid. The thermal conductivities of sodium nitrate and potassium nitrate obtained in the present work are shown in Figure 8, together with data published in the literature (Bloom et al., 1965; Gustafsson, 1968; McDonald and Davies, 1970; Turnbull, 1961; White and Davies, 1967; Kat0 et al., 1977). The agreement of the present results with the published data is satisfactory. The mean deviation of the data obtained in this work is 3.890, which is slightly larger than 2.6% for water and toluene near mom temperatures (Tada et al., 1978). This suggests that the
d
lnd. Eng. Chem. Fundam. lQ81. 20, 336-340
338
Acknowledgment The authors acknowledge with thanks the financial support of the Ministry of Education, Japan, a Grant in Aid for Fundamental Scientific Research. Nomenclature C, = specific heat capacity, J g-’ K-’ h = thermal conductivity parameter, s-’I2 1 = thickness of the first layer, cm Q = input heat energy per unit area, J cm-2 Ra = Rayleigh number, dimensionless I’, = temperature rise of first layer, K ?;, = Q / b i C p i O , K n = position, cm Greek Letters X = thermal conductivity, W cm-I K-’ 0 = temperature of measurement, O C p = density, g Subscripts eff. = effective i = ith layer
Literature Cited
present laser flash method is a powerful method for measuring thermal conductivities of liquids with lower thermal conductivities, such as molten salts, at much higher temperatures. Conclusion The thermal conductivities of molten salts at elevated temperatures were measured easily and with the same degree of accuracy as those near room temperature by use of a laser flash method. The method does not require any reference materials or the measurement of the total input energy of the laser beam and the liquid layer thickness.
Bloom, ti.; Doroszkowskl, A,; Tricklebank, S. B. Aust. J. Chem. 1865, 18, 1171. Clark, R. P. J. Chem. fng. Date1879, 18, 67. Gustafsson, S.E.; Halllng, N. 0.;Kjellander, R. A. E, 2. Netwlorsch. 1868, 23A, 682. Janz, G. L.; Allen, C. B.; Bansal, N. P.; Murphy, R. M.; Tomkln8, R. P. T. “Physical Properties Data CompilaUon Relevalent to Energy Storage. 11. Maken Satts”; 1979, NSRDS-NBS 61. Part 11. Janz, G. L.; Krebs, U.; Siegenthaler, H. F.; Tomkins, R. P. T. phys. Chem. Ref. Data, 1872, 1, 678. Kato, Y.: Kobayashi, K.; Arakl, N.; Furukawa, K. J. phys. E., Scl. Instrum. 1877, 10, 921. Kelley, K. K. ”Contributions to the Data on Theoretical Metallurgy”;Weshlng ton., I949 . - .-. McDonald, J.; Davles, H. T. J. phys. Chem. 1870, 74, 725. Mlchels. A.; Sengers, J. V. physlce 1862, 28, 1238. Parker, W. J.: Jenklns, R. J.; Butler, C. P.; Abbott, G. L. J. Appl. phys. 1861, 32,1679. Schrlempf, J. T. Rev. Sei. Instrum. 1872, 43, 781. Stun, D. R.; Slnke, G. C. A&. Chem. Ser. 1856, 18. Tada, Y.: Mrada, M.; Tanigakl, M.; Eguchl, W. Rev. Scl. Instrum. 1878, 49, 1305. Turnbun, A. G. Aust. J. Sci. 1861, 12, 30. White, L. R.; Davles, H. T. J. Chem. php. 1867, 47, 5433.
Received for review December 15, 1980 Accepted May 28,1981
Simple probabilistic Approach to Enzyme Deactivations AJH Sadana Chemical Engineering Divkbn, Natbnal Chemical Laboratory, Poona 4 1 1008, India
A theoretical deactivation model is developed utilizing a simple probabilistic approach involving requlred bond breakage prior to enzyme deactivation. The time course of a deactivation reaction may be described by the sum of two or more exponential terms even if no stable intermediates are present. A “flow-chart” type approach Is presented to help model enzyme deactivation data exhibtting, in general, firstorder kinetics for each step but other than flrst-order overall kinetics.
Introduction An analysis of enzyme deactivation kinetics is an important consideration, especially when designing continuous reactors. It would be helpful to know how much denaturation is required to abolish activity (Reiner, 1969). Often a first-order model 0 196-43 13/8 1/ 1020-0336$0 1.25/0
_-dE = kdE dt
(1)
adequately represents enzyme deactivation kinetics, where E is the enzyme activity. Herein, a “single-hit” type of curve or the disruption of a single bond or structure is 0 198 1 American Chemical Society
