uctivitv of d
JEROME FKEHLXNG', ROGER E. ECKERT2, AND J. W. WESTWATER University of Illinois, Urbana, Ill.
ECAUSE plastics are often used as heat insulators, a knowledge of their thermal conductivities is desirable. These values are also of importance from the standpoint of establishing the molding properties and curing uniformity of plastics. Many plastics are anisotropic from the standpoint of strengt8hand electrical properties. The object of this investigation was to devise a means for determining whether a particular solid has thermal conductivities which depend on the choice of direction, and if so, to measure the actual values of k along the three major axes. THEORETICAL CON SIDERATION9
Two general methods for measuring the thermal conductivity of a solid are available. The steady-state method employs a steady flow of heat through the solid. A measurement of the heat flow rate and the temperature gradient leads to a calculation of k. The guarded hot-plate apparatus of the American Society for Testing Materials ( I ) , as well as the apparatus of Vernotte ( W ) , Koch (16), Nusselt (H), and others represents an application of this method. A major difficulty arises in trying to prevent heat exchange between the specimen and the surroundings a t edges of the specimen. Insulation can be employed, but common insulators conduct heat appreciably when compared to the plastic under test. For example, 85'% magnesia has a k value of about 0.841 B.t.u. per hour square foot per O F. per foot compared to 0.18 for one of the materials reported herein. The use of so-called guarded plates represents another attempt to get around this difficulty. A second serious difficulty of the steady-state method is that of elimination of air film which may occur between the solid specimen, its source of heat, and its source of heat removal. An air film can offer an additional large resistance to heat tramfer and can lead to erroneous values for thermal conductivity, The A.S.T.M. method has this weakness. The second general method for thermal conductivity measurements is the unsteady-state method. A solid is changed suddenly from one ambient temperature to another. Temperature us. time measurements are taken a t a known position in the solid. From these observations, k can be computed. This nethod can avoid difficulties of insulation and of air-film resistances if properly carried out. An undesirable feature of the method is that t h c density and specific heat of the solid must be known. However, the determination of the latter values is straightforward. In fact, it is possible to design the unsteady-state procedure ( 5 , 8 ) so that C, is obtained simultaneously with k. The unsteady-state method has been widely used for metals, dating back to the work of Forbes (7) in 1864. In recent years many solids have been measured with the Cenco Fitch apparatus (9). This employs the unsteady-state method, but, unfortunately, fails to eliminate completely air films and heat losses. The recent method of Beatty, Armstrong, and Schoenborn ( 3 )uses an unsteady-state method employing a sandwich construction much like that used for many steady-state systems. However, the possible presence of air films and edge losses is pointed out. Present addrws, Standard Oil Development Go., Linden, N. J. * Present addretie, E I . d u Pont de Nemours & Go., Ino., TVilmington, Del. 1
The best scheme for avoiding an air film s e m to be to subst+ tute a condensed steam film for the air film. The thermal resixrance of a condensate film might be expected to be less than the resistance of an air film by a large factor. Common steam has a heat transfer coefficient of about 2000 B.t.u. per hour square foot per 'F. An air fillm would have to be no thicker than 0,0001 incuh if it were to conduct this well. The best scheme for avoiding the possibility of edge heat losses seems to be to avoid a sandwich coustruction and to immerse completely the specimen in the heat transfer medium. These two schemes were used by Gottwald (8) to obtain average values of 12, and were employed also for the present investigation to obtain directional values of IC. One of the fundamental equations for heat transfer is luma n KS Fourier's law
(A)
dt
i 3
=
This equation is applicable wheii heat flows in one direction only. However, if a rectangular solid is receiving heat from all sides, the appropriate equation becomes L .0
L.0
0.9
0.9
0.8 0.Y
0.7
0.6
0.6
0.5
0.5
VI
(c
0.4
0.4
$ * Y
4.
n
t
0.3 -2
0.3
e
* (0
0
'2.6
0.3
0.9
,
k ~ e Dimensionless
Figure 1. Evaluation of Midplane Temperature Left ordinate for single blabs
Right ordinate for t w o slabs of thicknesses, 8
906
-
and ns
1.2
April 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
Using the method of Newmann (29-%’1),the final integrated solution becomes
Y = YzYuY, where y = - -ta ts
-t
- to
The above procedure can be accelerated by the use of the Y’/Y lines graphed in Figure 1. These lines show the ratio of the Y value (called Y’)for a slab of thickness, ns, to the Y value for a slab of thickness, s, both taken a t the same value of the abscissa. These lines are very useful for obtaining first approximations to the values of k. For the example above, it W&B shown that the ratio of Y , of block A to Y, of block B occurred. But considering the z dimensions of the two blocks, it is seen that the required ra-
total unaccompliahed temperature change maximum possible temperature change
Thermocouple L e d 6 To Potentiometer
unaccomplished temperature change calculated from X heat flow only t a - tz = yz a maximum possible temperature change and similarly for Yuand Ys Fortunately, the solution for Y., Y,,and Y, are readily available, both in the form of infinite, convergent series, which for Yz i s
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61081
--t Eiross Tubing Air Jet
Trblns
Lucrto Cover
and in the form of graphs of the calculated values. One graphed line in Figure 1 shows the relationship represented by the above equation. The graph can be used to represent k, or k, by changing the z subscripts to y or z. Equations 3 to 6 rest on certain assumptions, the most important of which are: Surface resistances to heat flow are negligible; when the solid is moved from one ambient temperature to another, the surface temperature changes instantly; and no heat crosses any of the three major planes of symmetry-these planes (x-y, 2-q and y z ) receive heat by flow along the planes only, and the axes of symmetry are the principal axes of conductivities. To illustrate the meaning of Equation 3, suppose that a homogeneous cube were heated a t two parallel faces only, with all the other faces insulated, and that its center temperature rose by 50% of its total possible change in a certain time. Then, if heat had been applied a t all six faces for the same time, the cube would have an “unaccomplished” change of temperature of (0.50)q or 12.5%. I n other words, its temperature would have risen by 87.5% of the total possible change. Of course, if k is different in the separate directions, the answer becomes different also. The “average” thermal conductivity for a solid block can be readily found by the unsteady-state method by utilization of the above equations and by assuming that k, = k, = IC,. Note that the particular average value found will depend on the specimen dimensions. Individual thermal conductivities for the different directions in a solid can be found experimentally by using three different specimens, of different sizes, and then by applying a trial-and-error technique to the solution of Equations 3 and 6. One set of k,, k,, and k. values only can satisfy the equations. Tempers, ture us. time data are needed for one position (the center of the block is convenient) for each specimen. The trial-and-error labor can be reduced tremendously by a particular choice of specimen dimensions. It is wise to have two specimens with the same z dimension, two with the same y dimension, and all three with the same z dimension. For example, block A could be 1 X 2 X 3 inches (in z, y, z directions) while B is 1 X 2 X 2 inches, and C i s 1 X 1 X 2 inches. Then for a fixed time, e, Y, of block A equals Yz of block B. Also, Y, of block A equals Y, of block B. From this it follows that the ratio of Y, of block A to Y, of block B is equal to the ratio of Y Ato YB. One then msumes valk, ues of and calculates Ye for each block until the above equalPC, i t y is established. Thus k. is found. In a similar manner, k, and kv are computed.
6”D
Figure 2.
Apparatus for Heating the Speoiniens
-
tio is Y’/Y at n 3/2. This ratio is graphed in Figure 1. Thus one can use the observed Y’/Y value and the observed n value to obtain the abscissa and a first approximation to k.. Table I presents the values needed for the construct,ion of the Y ‘ / Y curves. TABLE I. THEORETICAL RELATIONSHIP” FOR Two SLABSOF THICKNESSES s AND mz Values of Y ’ / Y n = 1.6
1.000
0,004 0.040 0.080 0.120 0.160 0,200 0.320
0.360 0.400 0.620 0.680
a
0.955 0.843 0.714 0.630 0.545 0.373
0.330
1 200 Graphed in Figure 1.
0.291 0.199 0.122 0.0247
n - 2 0.999 0.846 0.593
0.424 0.310 0.229 0.0927 0.0693 0.0618 0.0218 0.00142 0.00652
APPARATUS AND PROCEDURE
The method consisted of observing the temperature us. time relationship for plastic blocks which were suddenly immersed in an atmosphere of steam. The specimens were originally at a uniform temperature which corresponded to room temperature. The steam bath consisted of a hollow brass cylinder, 5 inches in diameter and 7 inches high. Steam entered and left through l/4inch standard pipe nipples. Condensate flowed out with the leaving steam. A loose-fitting cover was utilized for introduction of the test specimen. A slow venting of steam occurred a t the cover, also. The steam was a t atmospheric pressure. Figure 2 illustrates the apparatus.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44. No. 4
5.(
TABLE 11. MATERIALS TESTED
a
4.r
A
B C
3.c
D
3.2
Material Paper-base phenolic laminate, Insurok XXX-T-640 Cloth-base phenolic laminate, Insurok C-T-601 Hydrolysed lignocellulose, Benelux 70 Cloth-base phenolic laminate, Lamiooid C-6030
Measured Specific Gravity 1.344 t o 1.351 1.336 t o 1.344 1.405 to 1.408
Noniinal Measured Thickness Cp Used, Inch
1.310
0.363 0.334 0.326 0.361
1 / ~ ,2/4*1
1 1 1/2
by measurement of its weight and dimensions. The specific heat was found by use of a standard calorimeter.
r;
i
DISCUSSION O F RESULTS
ld 2.E
2 .c I
1.4 5
I
T
I I 15
ne, Min.
Figure 3. Observed Heating Curves for Insurolr XXX-T-640
The thermal conductivities along the three major axes of the four plastic materials are given in Table 111. As might be expected for laminated solids, each material is decidedly anisotropic from a heat-flow standpoint. I n general, the thermal conductivities parallel to the layers are much higher than the conductivities perpendicular to the layers. For the paper-base phenolic material, the two values were different by a ratio of nearly 6 to 1. Even for the most nearly homogeneous of the test materials, those of lignin-cellulose composition, the values differed by about 40'%. Thus, if a laminated material is used as a thermal insulator, it should be oriented so that its laminations are perpendicular to the direction of expected heat losses. Fortunately, this practice can be followed conveniently. The thermal conductivity in the lengthwise direction was about equal to the k value in the crosswise directions for each of the sheet materials except for the Lamicoid. This is reasonable, for most laminates are built up with the reinforcing material being oriented equally in the two major directions. The Lamicoid material is presumed to be composed of cloth filler layers which mere oriented in one direction. Such orientation is sometimes used in order to obtain finished sheets with extra high strength in one certain direction. For this particular test material, the orientation produced a IC value for the 2/ direction which wm nearly triple the value in the z direction. It is recognized that for laminated materials, the k values parallel to the layers is some sort of average of the conductivities of the two materials, filler and binder, which make up the system. It is not clear what kind of average this value is. The difficulty arises because the individual layers, especially of the filler, are usually anisotropic themselves. These directional k values were determined as outlined previously. By way of illustration, the temperature-time data for the paper-base laminate are given in Figure 3. The measured electromotive force values for the steam for specimens Al, A4, and A5 were 5.278, 5.269, and 5.270 mv., respectively; the steam temperature range, therefore, was from 211.63' to 211.93' F. for these three runs, Specimen A1 had dimensions of 0.973 X 3.055 x 2.953 inches, A4 measured 0.480 X 2.973 X 1.973 inches, and
The test specimens were blocks of several types of plastic described in Table 11. The specimen thicknesses ranged from l/* to 1 inch; the width and length measurements were between 1.9 and 3.2 inches. The resulting volumes ranged from 2 to 9 cubic inches. Each specimen was prepared for temperature measurement before insertion into the test chamber. Temperatures were determined by use of an iron-constantan thermocouple 10cated a t the geometric center of the block. The bole diameter, 0.067 inch, was adequate for 30 gage wire. A Leeds and Northrup precision-type potentiometer was used to observe the thermocouple voltage. A reference junction a t 32' F. was connected in the circuit in the usual manner. It was desirable to prevent contact between the wires and the steam. Such contact would lead to heat transfer along the wires to the junction and could result in erroneous thermocouple readings. It was decided to substitute air for steam a t the wire itself. This was accomplished by fitting a small glass tube around the wires and through the lid of the steam bath. Air from the laboratory line was kept in this tube at a slight positive pressure. I n fact, an actual slow circulation of air was maintained by the use of a small metal tube extending part way down into the glass tube. The glass tube was sealed against the plastic specimen by means of a tiny rubber gasket. A slight pressure was used to maintain contact. This seal also served to prevent air from entering the thermocouple well. The air-immersed thermocouple leads were felt to be superior to steam-immersed ones, because the possible gain of heat from steam heating was thought to be much greater than the possible loss of heat from air cooling. After the specimen was ready for temperature measurements, it was put in the steam chest and placed upon a knife-edge supoort. The steam chest was tipped so that :he ;;per face of the specimen made an angle of 30" with the horizontal. This served to TABLE 111. THERMAL CONDUCTIVITY RESULTS ensure adeauate drainage of condensate from the Thermal Conductivities solid, The steam wasturned on and time and OF temperature observations were started as soon as Cal./sq. em.-see.- "C.a B.t.u./sq. ft.-hr.--' ft. om. the first contact occurred. Code At the conclusion of each run, the thermoMaterial Number b% ku kr kz kv kI couple was pulled from its position and was placed A Insurok XXX-T-640 0.00058Ci 0,00363 0,00353 0.143 0.865 0.856 in the raw steam. This served to establish the B Insurok C-T-601 0,000789 0.00161 0.00161 0.191 0.300 0.390 steam temperature and to verify the thermocouple C Benelux 70 0.000862 0.001145 0.00110 0.208 0.278 0.264 D Lamicoid (3-6030 0.000748 0.00330 0.000945 0 . 1 8 1 0 . 8 0 0 0.229 calibration. The thermocouple had been pre0 x refers t o direction perpendicular to laminations: 21 and P refer t o directions parallel viously calibrated a t three fixed temperatures,-one to laminations. being the boiling point of water. The density of each specimen was determined
April 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
A5 measured 0.973 X 2.922 X 1.947 inches. Values of the temperatures at a particular t i m e 4 0 0 seconds for this case-were utilized to arrive at the three values of k. It will be noted that the "equal" dimensions of the three blocks were not quite exactly equal. This is immaterial; these dimensions were assumed equal in order to make possible a speedy estimate of the k values. Then the true dimensions were used to arrive at the values of k given in Table 111. This procedure is based on an assumption that k is independent of specimen temperature. The k values obtained are for an average temperature and not for an instantaneous temperature. Since k need not be independent of temperature always, a check for the calculated k was made for each case as follows: The values of k from the 400-second observations were used to calculate a predicted midpoint temperature for each specimen a t some different time, say 200 seconds. For all results herein the check appeared excellent, and the calculated temperatures agreed with the observed ones to within 0.45" F. It is concluded that k is independent of temperature for these materials in the range of 75" to 212" F. In addition, Beatty et al. (3)have presented a few results indicating that k does not change significantly with temperature. Additional experimental work is needed on this subject. Another possible objection to the method is that it seems to be based on an assumption that k is independent of specimen thickness. Actually, one could choose three specimens from one sheet of original material and could thus avoid this objection if so desired. This was the practice for three of the four materials tested, aa is indicated in Table 11. However, for the Grade XXX material, samples of mixed thicknesses were used. No effect of thickness on the value of k could be discerned, This effect would have appeared when the check calculation described above was carried out, In fact, if the check had failed to give calculated midpoint temperatures that agreed with the observed temperatures, then the effect of temperature and of thickness on k would have been confounded and one would not know the magnitude of either effect. As a further check on the nondependence of k on specimen thickness, a 3/4-inch thick specimen of the Grade XXX material was prepared from a 1-inch specimen by machining off l/4 inch from one face of the thicker material. This specimen +as substituted for an ordinary a/,-inch specimen during a check determination. The temperature reading was well within 1' F. of the prior value, and it is concluded that k is independent of specimen thickness for this material. Apparently there is no measurable effect from the use of special finish coatings on the materials. Such effect would have shown up as being an effectof specimen thickness. The values of k perpendicular to the laminations all fall within the ranges given in the Modern Plastics Encyclopedia Charts (18). Presumably, the published values were obtained in such a way as to minimize the effects of heat flow in the y and z directions. Similarly, the cross-lamination value of k for the paperbase laminate, 0.143 B.t.u. per hour square foot per ' F. agrees with the value obtained by Beatty et al. (9),0.147, using unidirectional heat flow. The condensate film is believed not to cause perious errors in the determination of k. Steam condensate films at this university give heat transfer coefficients that average about 3000 B.t.u. per hour square foot per O F. But even if a very low coefficient of 1000 were assumed, the worst possible error this could cause in the reported values of k would be no greater than an error caused by a mistake of 1% in measuring the specimen thickness. Actually, the real error caused by the film resistance was undoubtedly much less than this. Heat transfer along the thermocouple wire is believed not to cause measurable error either. Eckert (6) has shown that the midpoint temperature us. time curves for the same specimen tested with various wires, air-insulated as described above, are hardly distinguishable as long as the wire sizes do not exceed 30-gage.
909
Wires larger than 30-gage are not recommended. His results include wire sizes up to 24-gage, and hole sizes up to 0.063 inch. A limitation of the test method is evident when materials which absorb water during the test period are tested. For such materials, k could change during the course of a run. Water absorption studies were made on each of the materials reported on this paper. Not one showed a measurable increase in weight when immersed in an atmosphere of steam for 1000 seconds. This period of time is 30% longer than the greatest time interval used for calculation of the conductivity values. Of course, for absorbent materials a water-impervious coating of high thermal conductance could be used. Condensed aluminum films offer the most obvious possible solution. Reproducibility of the k data was excellent. Duplicate temperature-time curves agreed within 1" F. The density and heat capacity observations also ohecked satisfactorily. Duplicate density values and duplicate C;, values deviated no more than 0.5% in any case. CONCLUSIONS
The results obtained indicate that directional conductivities of anisotropic materials can be obtained readily by the unsteadystate method. Laminated plastics have much higher thermal conductivities parallel to the layers than perpendicular to the layers. The ratio of the directional values of k may be as high as 6 to 1. The thermal conductivities along the two major axes parallel to the laminations may be nearly equal or may be quite different, depending upon the orientation of the filler material. ACKNOWLEDGMENT
Appreciation is extended to H. B. Clark for helpful advice. Gratitude is extended to the Masonite Corp., the Mica Insulator Co., and the Richardson Co., who supplied the test materials. NOMENCLATURE
C , = specific heat, B.t.u. per pound per ' F. e = base of natural logarithms, 2.718 k = thtrmal conductivity, B.t.u. per hour square foot per F. per foot k,, k,, k. = thermal conductivity in the 2,y, and z directions n ratio of thicknesses (or widths or lengths) of two slabs s = half the thickness of a sl,a;b, feet t = temperature at a point, F. to = uniform temperature of a specimen at the start of heating, O F. ts = surface temperature, " F. t, = temperature at a point receiving heat flow in z direction only, F. 2, y, z = coordinate axes. Axis 2 is perpendicular to the layers if the solid is a laminate t* - t Y = temperature difference ratio = - dimensionless t, - to' YA, Y B = temperature difference ratio, as defined above, for specimen A and for specimen B Y,, Y,, Y , = temperature difference ratio for heat flowing along r axis only, or y or z axis only. Y' = temperature difference ratio for slab of thickness, ns p = density, pounds per cubic foot 6 = time, seconds =i
BIBLIOGRAPHY
(11 American Society for Testing Materials, "Standards," Vol. 111, p. 304,1949. (2) Angstroem, A. J., Ann. Physik u. Chern., 123,628(1864). (3)Beatty, K. O.,Jr., Armstrong, A. A , , Jr., Schoenborn, E. M., IND.ENG.CHEM.,42,1527 (1950). (4)Bosomworth, G.P.,Ibid., 33,568 (1941). (5) Carslaw, H. S., and Jaeger, J. C., "Conduction of Heat in Solids," London, Oxford University Press, 1947. (6) Eckert, R. E., M. S. thesis, University of Illinois, 1948. (7)Forbes, J. D.,Trans. Roy. SOC.Edinburgh, 24,73 (1867).
INDUSTRIAL AND ENGINEERING CHEMISTRY
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(8) Gottwald, F., Kunstcffe,46,248 (1939). (9) Harvalik, Z. V., Reo. Sci. Instruments, 18,815 (1947). (10) Ingersoll. L. R., Zobel, 0. J., Ingersoll, A. C., “Heat Conduction,” New York, McGraw-Hill Book Co., Inc., 1948. (11) Jacob, M., “Heat Transfer,” Vol. I, p. 2, New York, John Wiley &Sons, Inc., 1949. (12) Ibid., pp. 266,274,285,288. (13) Ibid., pp. 278,281,286-7, 289-91. (14) King, R. W., Phys. Em.,6 , No.2 , 4 3 7 (1915). (15) Koch, Kunstoffe, 31, 135 (1941). (16) McAdams, W. H., “Heat Transmission,” 2nd ed., pp. 4, 32-5, New York, McGraw-Hill Rock Co., Inc., 1942.
Physica
Vol. 44, No. 4
(17) Ibid., pp. 36-7. (18) iModern Plastics Encyolopedia Charts, “Chart 2, Plastic Ploperties,” New York, Modern Plastics. 1950. (19) Newmann, A. B., IND.ENQ.CHEW,28,545 (1936). (20) Newmann, A. B., Trans.Am. Inst. Chem. Engrs., 24,44 (1930). (21) Ibid., 27, 310 (1931). (22) Nusselt, W., Forsch. CfebeiteIngenieurw., No. 64 (1909). (23) Vernotte, P., Compl. rend., 204,563 (1937). (24) Worthing and Holliday, “Heat,” New York, John Wiley Sons, Inc., 1948. RECEIVED for review July 2, 1951.
ACCEPTEDNovember 2 7 , 1951.
erties of Ammonia
AlWMONIUM NITRATE-AMMONIA-WATER AND UREA-AMMONIA-WATER AND D. P. SCHUTZ Allied Chemical &Dye Corp., Hopewell, Va. The Solvay Process Division, Nitrogen Section,
E. A. WORTHINGTON’, R. C. DATIN,
N SPITE of the very considerable use of ammonium nitrateammonia-water and urea-ammonia-water solutions in the fertilizer industry, relatively little information regarding their physical properties has been published. Vapor pressure, density, and solubility data for the systems ammonia-water, ammonium nitrate-water, ammonium nitrate-ammonia, and urea-water are compiled in the International Critical Tables (8-4) and in Seidell’s “Solubilities” (10). Additional data for the pertinent binary systems were obtained by Adams and Gibson (1). Kracek (6), Scholl and Davis (Q), Mittasch, Kuss, and Schlueter (7), Shultz and Elmore (11), and Sanders and Young (8) have determined densities and vapor pressures of ammonium nitrate-ammonia-water solutions, and Janecke (6) has reported vapor pressure and solubility data for the system urwammonia-water. The present paper reports data for the solubilities, vapor pressures, and densities of ammonium nitrate-ammonia-water and ureaammonia-water solutions in the ranges 0 to 75% ammonium nitrate, 0 to 65% urea, and 0 to 65% ammonia. The atudy includes and covers a much wider area than the range of current commercial interest.
the boiling point of ammonia to -80” C. was dried over anhydrous magnesium perchlorate. After weighing, the ammonia was again solidified, the tube was opened, and the required amounts of ammonium nitrate or urea were weighed into the tube. The desired quantity of water was then added to the tube by volume measurement from a pipet graduated in unitti of 0.05 ml. The tube after sealing was reweighed and geiierally found to check within 0.01 gram in 10 grams-the sum of the weights of the several materials added; otherwise, the tube waB discarded. The triangular diagrams, Figures 1 and 2, were constructed from plots of dissolution temperature 2)s. percentage of ammonium
SOLUBILITIES
Solubilities were determined by measuring the temperature a t which the last trace of solid dissolved in the appropriate mixtures as they were gradually warmed in sealed glass tubes immersed and rotated in a controlled temperature bath. The temperature of the latter bath containing water or methanol, controlled t o 3 ~ 0 . 0 5C., ~ was raised very slowly in 0.1” C increments. Heavy-walled glms tubes were filled with thc mixtures by distilling about 2 ml. of anhydrous ammonia into the tube a t atmospheric pressure, the aotual uantity of ammonia being determined by we& after sealing the tube while immersed in a dry ice-acetone bath. Air introduced owing to the temperature change from 1 Present address, Kaiser Aluminum and Chemical Corp., Permsnente, Csiif.
Figure 1. Solubility in the System Ammonium NitratwAmmoniaWater
