Thermal Diffusivity of Low Conductivity Materials PAUL K. CHUNG AND MELBOURNE L. JACKSON' U. S. Naval Ordnance Test Station, Inyokern, China Lake, Calif.
T
Equation 3 is a straight line when log Y is plotted against 0. From the slope of the line, b, it is possible t o calculate the thermal diffusivity, a , by the equation
HERMAL diffusivity has the same basic significance in
unsteady-state heat conduction that thermal conductivity has in steady-state conduction. Thermal diffusivitg (designated by a ) is made up of three physical properties: thermal conductivity, heat capacity, and density, so that normally three separate determinations are needed t o establish the thermal diffusivity. Values so obtained are subject to the experimental errors of each method. The usual method for the determination of conductivity, the guarded-hot-plate method, requires the attainment of steady-state conditions and the determination of the rate of heat flow. It is time-consuming and requires a specimen of appreciable size and thickness. Direct determination of thermal diffusivity is desirable. Such a method was desired to provide thermal diffusivity data for evaluation of the heating and cooling characteristics of rocket motors. An unsteady-state method was developed which gives the thermal diffusivity directly, is rapid and versatile, and uses only simple and inexpensive apparatus. MATHEMATICAL DEVELOPMENT AND METHOD
The method is based on unsteady-state conduction where the temperature is a function of the heating or cooling time. For simplicity in design of the apparatus, a specimen was chosen in the form of a solid cylinder. The basic differential equation for unsteady-state heat conduction in a cylinder is:
It can be shown (3)that this equation has a solution
--=t E
y=
m
Y=
1
to to
ti
J1
xv
[J,Z
(XV)
(")
+ J:
(xv)]
[exp.
- (x?
6t h limit -- = - - ( t o - t ) k ?+rm 6' ti
at e =
e,,.
This series converges rapidly, as indicated by Figure 3, and after a comparatively short time of about 3 to 5 minutes all terms after the first become negligible, as shown by numerical calculation. Equation 2 t'hen reduces to the general form
Y
=
A IO-bS
X?
where xi is the smallest positive root of the transcendental equation
The value of x is seen to be depondent on the dimensionless group m = ( k / h T ~ ) commonly , known as the resistance ratio. When nz = 0, X I has a limiting value of 2.405; this condition can be attained by a proper choice of experimental conditions. The derivation of the equations imposes certain conditions which must be fulfilled experimentally. Equation 1 provides only for radial heat flow and requires, theoretically, that the cylinder be infinitely long. Using the method of Olson and Schultz ( 6 ) )a cylinder with L I D = 8 can be shown t o fulfill this requirement, and a cylinder with L/D = 4 would be less accurate by only 1 part in 10,000. Equation 2 requires t h a t the initial temperature of the solid be uniform before the specimen is suddenly exposed to a different ambient temperature. Equation 5 assumes the resistance ratio to be constant, which is most easily accoinplished by making m = 0, or practically so. The magnitude of the error introduced by assuming ?tz = 0 can be estimated. The value of the surface film coefficient, h, is obtained by use of Equation 25 of ( 4 ) if water is the cooling medium. The value of ~nis determined and the corresponding value of X I is obtained from Jacob's Table 13-9 (3). Using this procedure for the apparatussand materials described, the error introduced in cy by assuming the limiting value of XI was estimated to be less than 1% a t a water velocity of 1 (foot)/ (2) (second), less than 0.5% a t 2 (feet)/(second), and negligible a t 5 (leet)/(second). It was considered best to select flow conditions where the limiting value of XI could be used with accuracy and for which fluctuations in flow rate would have negligible effect. I n determining the thermal diffusivity, the temperature a t some point in the spccimen is observed as a function of time. A plot of log Y us. B is made, the slope is evaluated for the straight portion, and the diffusivity is calculated from Equation 4. It is not necessary to know the location of the thermocouple, because cooling lines for various positions in the specimen are parallel. Equation 2 shows that the slopc of the cooling curve is independent of the position ratio, n, which is also shown graphically by MeAdains' charts (4). Furthermore, the method does not require a knowledge of the rate of heat flow or the attainment of steady-state conditions of temperature, except that the fluid trmperaturc about the spccimen must be constant during the determination.
$)] [J, (x. k)]
for the boundary condition,
and initial condit,ion, t =
2.303 rZ b a=----
(3)
where
and EXPERIMENTAL EQUIPJIENT AND PROCEDURE
Present address, Department Idaho, Mosoow, Idaho. 1
cilemioai ~
~
,
~vniversity i ~ of ~
~
The experimental apparatus was designed to accommodate a slender ~ i cylinder ~ ~ ,and to meet the requirements given above. Figure 1 shows the arrangement of the apparatus.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
2564
The test specimen, A , is suspended in the heat-transfer cell, B. Heating is accomplished by steam from the pressure regulator, C, entering the cell a t port D and leaving through port E . Cooling water is supplied through the quick-opening valve, F , control valve G, and rotameter H . Thermocouples are indicated by the symbol TC. TC
TC
___
STEAM
TC-
and show satisfactory reproducibility The greatest variation from the mean is only 1%. For comparison, specimens from the same sheet of material were submitted to the Kational Bureau of Standards for the determination of thermal conductivity ( I ) , specific heat (drop method and a Bunsen ice calorimeter), and density, with the results shown in Table 111. The thermal diffusivity calrulated from the data in Table I11 is 0.00417, which is about 37, below the average value obtained by the unsteady-state method. The conductivity was estimated by the National Bureau of Standards t o be IOM (because of warping of the flat specimens) by not more than 3%. The specific heat R as considered to be correct within &2y0. Assuming the specific heat figure to be correct and the conductivity figure to be 3% low, the thermal diffusivity would be within 1% of the average figure obtained by the unsteady-state method and within instrumental errors. Tests were also made with extruded rods of Lucite and pol)Etyrene. The results of successive runs for the same material THERMOCOUPLE ENCASED IN :TUBING SEALED AT JUNCTION
WATER MA1 N
Figure 1.
Vol. 46, No. 12
Schematic Arrangement of E q u i p m e n t
The detailed construction of the heat-transfer cell is shown in Figure 2. The test specimen, attached to the cover plate by means of three screws, is machined from sheet or bar stock int'o a rod approximately 1 inch in diameter and 8 inches in 1engt)h. A small hole is drilled longitudinally through the center of the specimcn to a depth of 4 inches for placement of a thermocouple. A neoprene gasket is inserted between the specimen and the cover plate to act as a seal. The thermocouples viere compared to a standard thermocouple calibrated by the Bureau of Standards and were found to agree within the accuracy of the recording instrument to better than 10.5" F. A multipoint Brown Elcctronilr high-speed recording potentiometer having a temperature range of -50" to 250' E'. was used. I n a typical run, steam was admitted to the top port to bring the specimen to a uniform t'emperat'ure. When the thermocouple in the specimen reached the temperature of the steam, heating was stopped and cooling water was admitted immediately in the reverse direction. A complete test cycle, heating and cooling, lasted approxiinately 45 minutes, and 15 to 20 minutes x a s usually long enough to establish the slope of the log Y-8 curve. I n testing rocket propellant', heating by steam was unsstisfactory because of cracking of the test sample. h satisfactory method was to bring the specimen (with cover plate attached) t o a uniform temperature in a nearby oven and transfer the specimen quickly to the cell. The propellant specimen had to be supported during heat,ing to prevent, distortion.
fl WATER OUTLET $'BRASS PIPE COVER PLATE
STD I(BRASS
A
SPECIMEN,
v
SPACER PLATE
THERMOCOUPLE ENCASED IN f COPPER TUBING SEALED AT JUNCTI ON
WATER INLET
h
$7=4
L
Figure 2.
Details of Heat-Transfer Cell
EXPERIMENTAL RESULTS
A typical cooling curve for the Plexiglas specimen is shown in Figure 3 and experiment,al data are given in Table I. The thermal diffusivities calculated from this data are given in Table I1
TABLE 11. THERMAL DIPFUSIVITY OF P L A ~ TMArEnIA1,s IC Material Plexiglas
Slope, hIin.-' 0.112 0.110 0.109
FOR COOLIKG TABLE I. ABBREVIATED TEMPERATURE-TIME DATA OF PLEXIGLAS SPECIWEN
Time,
Min.
Tiial 1, tr = 220 to t 220
0
1 2 3
5 7 Y 11 13 15 17
55:2 51.1 50.3 49.5 49.4 49.4 49. 49.r 49.9 49.8
G
220
206.1 187.3
Tiial 3, I I = 220 to t 220 61
56.7
139.0 103.4
54.3 51.0 51.0
82.0
51.0
69.0 61.2 56.8 54.3
51.0 51.2 51.2 51.3
220.2 206.0 187.5
140.6 106,O 85.0
Lucite
o.ino5
Polystyrene
0.102 0.102
o
Double-base propellant
7 1 .I
83.8 58.8 56.0
0.09m 0.1005
a
103
0.0815 0.0823 0 0813
Thermal Diffusivity, Sq Foot/Hour Unsteady-state method Literature 0.00438 0,00431 0.00429 A v . 0.00483
0.00417a
0.00417 0.00414 0.00417 A v . 0.00416
0.00374 -0.00361 b
0,00424 0.00424 0 00427 A v . 0.00425 0.00338 0.00342 0.00338 Av. 0.00330
Calculated from National Bureau of Standards data.
b (5).
0.00245-0.00380b
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1954
Figure 3.
Typical Cooling Curve Plexiglas specimen
TABLE 111. PROPERTIES OF PLEXI~LAS Property k, (B.t.u.)/(hour-sq.foot-" F./foot) Cp, (B.t.u.j/(lb.-oF , j p, (lb.)/(ou. foot)
Mean Value 0.0992 0.334 73.4
Temp. Mean Range, E". Temp., F. 51- 93 72 32-122 77 51- 93 72
showed good agreement. No single determination for the Lucite or polystyrene sample deviated by more than 0.5% from the mean for either set of determinations. These specimens were available only in rod form and were therefore unsuited for thermal conductivity determinations by the guarded-hot-plate method. Independent verification of test results, such as was made for Plexiglas, was therefore not possible. A comparison with values reported in the literature was inconclusive, because physical and thermal properties of mill-run materials vary considerably. Experimental results and a range of values reported by manufacturers (6) are given in Table 11. The unsteady-state method of determining thermal diffusivity was applied to rocket propellant, for which the apparatus was originally developed. A specimen of double-base propellant was machined from extruded stock into cylinders. The thermocouple was inserted in the center hole, where it was embedded with a drop of acetone. The heating procedure was modified as previously mentioned to avoid distortion of the soft material. Results of successive runs, given in Table 11, again show good agreement between determinations. The maximum deviation from the average is less than 1%. The thermal properties of this type of propellant have not been determined by other methods and no basis of comparison is available.
2565
series (Equation 2) have no effect. Presumably, the slight curvature is caused by a temperature effect. The calculations could be refined by taking the slope of the tangent to the curve a t a given point to evaluate the diffusivity for a more limited average temperature, although this was not done for the data presented. A change in the diameter of the test specimen might exaggerate this curvature, permitting such a procedure to give accurate results. The method has special merit for use with propellant and other materials for which prolonged heating a t elevated temperatures is undesirable. The method does not require the attainment of a steady-state condition and is therefore rapid. The test cell is simple to construct and a nonrecording potentiometer could be used. The rotameter could be eliminated, if it were established that a moderately high water rate could be maintained. Good thermal contact between specimen and plates, which is essential to the guarded-hot-plate method, is not a problem with the unsteady-state method. The cell need not be insulated: because heat exchange with the surroundings is of no importance, provided that the ambient fluid remains a t a constant temperature. This is the case for water a t a suitable rate of flow and for condensing steam. The placing of the thermocouple in t h e center of the specimen is not necessary, but represents a convenient location. Delay in quenching after the specimen has been brought to an equilibrium temperature may produce a time delay, resulting in an unknown ambient temperature. However, if the delay is minimized, the exact point of zero time need not be known, as this mill result only in a shift of the position of the line and will not alter the slope. Actually m need not be zero if it can be maintained constant a t a known value, but in practice it is easier to make m constant by making it equal zero. The equipment required could be simplified by using room temperature as the initial temperature of the specimen, and heating (rather than cooling) the specimen by steam or hot water. The method as used, however, was more rapid for making a number of determinations of the same specimen. Thermal conductivity could be determined indirectly by this method by determining the diffusivity, heat capacity, and density, and solving Equation 4 for k. This procedure would be especially suited for materials which warp badly or for shapes such as extruded rod which cannot be obtained in the form of large flat specimens. Shapes other than cylindrical, such as square, could be used and values of X are available for other cross sections ( 3 ) . Ball (8), working with canned foods, reported that the experimental curves closely approximated the theoretical lines for most canned products. This suggests that the unsteady-state method could be extended to the determination of thermal diffusivity of pasty or granular materials if they were suitably encased. The method, as currently employed, did not give accurate results for high-conductivity materials, such as metals. Cooling curves could not be accurately established because of the very rapid cooling with water. The use of air as the coolant permitted a suitable cooling rate, but the air rose appreciably in temperature on passage through the cell and the condition of constant ambient temperature was not fulfilled. This could probably be corrected to a large extent by placing a water jacket around the cell and maintaining the same temperature for the water as for the air. Experimental data obtained for steel and brass specimens using water or air for cooling (without water cooling of the jacket) showed distinct curvatures and considerable scattering of the points. SUMMARY
DISCUSSION
Equation 2 was derived with the assumption that the thermal diffusivity was constant, whereas actually it is a function of the temperature. I n plotting Figure 3, a slight curvature was observed in the region where the second and third terms of the
-4n unsteady-state heat transfer method is proposed for the direct determination of the thermal diffusivity of a solid, and shown to be accurate, rapid, and simple, and t o offer advantages over conventional methods in which three factors (thermal conductivity, heat capacity, and density) must be determined
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INDUSTRIAL AND ENGINEERING CHEMISTRY
separately. The method is especially useful for materials which cannot be held a t elevated temperatures for extended periods of time, aiid may be adaptable to pasty and granular materials. ‘The method as described is suited only t o materials of low thermal conductivity. Experimental data are given for three types of plastic materials and for a double-base rocket propellant.
n r
ACKNOWLEDGMENT
Y
The assistance of J. H. Kiegand, Rocket Department, U. S. Naval Ordnance Test Station, is gratefully acknowledged.
A
= a constant defined in text; in Figure 3 it is the intercept a t 6 = 0 of the tangent to the st,raight port,ion of the
b
= an exponent defined in the text;
log Y-0 curve, dimensionless
graphically, it is the
slope of the log 1’-0 curve, hr. -l heat capacity at constant pressure (B.t.u.)/(lb.-” F.) = base of natural logarithm e = surface heat transfer coefficient (B.t.u.)/(hour-sq. footh F:) = notation of Bessel function J J o ( z )= Bessel function of first kind and zero order of z J l ( z )= Bessel function of first kind, first order of 5 = thermal conductivity (B.t.u.-foot)/(hour-sq. foot-’ I?.) I; L I D = length to diameter ratio, dimensionless = resistivity rat,io = k / ( h rqn),dimensionless rii
C,
position ratio = r / r m , dimensionless radial dist’ance from center of cylinder, feet radial distance from center of cylinder to outer surface of specimen, feet; numerically equal to radius of cylinder, feet = temperature a t point T and a t time e, O F. = initial uniform temaerature of cylinder, F. = ambient temperature during heating or cooling of specimen, O F. = temperature difference ratio = ( t o - t ) / ( t , - t i ) , dimensionless = thermal diffusivity k/(pC,), (sq. feetlhour) = symbol denoting partial differentiation = heating or cooIing time, hours = v positive root of xJ1(x) = ( h ~ m / k ) J(x) , = density of specimen (lb.)/(cu. foot) = = =
rnl
t ti to
a 6
e
XV
N OIMEN C LATUKE
=
Vol. 46, No. 12
P
LITER4TURE CITED
Am. SOC.Testing Materials, Philadelphia, Method C 17i-4.5, Ball, C. O., Bull. Natl. Research Council, 7, no^ 37 (1923-24). Jacob, Max, “Heat Transfer,” Val. I, pp. 276, 276, Sew York. John Wiley & Sons, 1949. AToAdams, W.H., “Heat Transmission,” 2nd ed., pp. 36, 200, New York, JIcGraw-Hill Book Co., 1942. Manufacturing Chemists’ Association, Washington, D. C.,
“Technical Data on Plastics,” pp, 73, 112, 1952. Olson, F. C. W., and Schulte, 0. T., IND. ENG.CWEM., 34, 874 (1942). RECEITED for reviexy February 27, 1954,
ACCEPTED August 16, 1954
Thermal Conductivities of Liquid Silicon Compounds A. C. JENKINS Linde Air Products Company, Dicision of Union Carbide k Carbon Corp., Tonazcanda, S. Y .
A. J. REID Carbide & Carbon Chemicals Co., Diuision of Union Carbide k Carbon Corp., South Charleston, W . Vu.
T
HE direct measurement of the thermal conductivity of
liquids is difficult. However, extensive and precise measureinents have been made by Bates and associates who have determined nhernial conductivities of water ( 1), mixtures of water and glycerol ( 2 ) , chlorinated hydrocarbons ( b ) , and alcohols and glycols (3). Data on the thermal conductivities of a number of silicon compounds were required as a help in making plant design calculations. For this purpose a method of making the measurements rapidly with a precision of 5% or bett,er )vas sought. An apparatus and method were developed to fill these requirements. APPARATUS AND XIETIIOD
The apparatus shown in Figure 1 makes use of t v o glass vessels. The inner one is maintained a t a fixed temperature (50” C.) by circulation of water through it from a constant temperature bath of large volume. This inner vessel is suspended in a second glass container which holds the 60-inl. saniple. The hottom of the outer glass vessel (containing the sample) is ground flat, arid cenirnt,ed to it are two 3-inch diameter borosilicate glass disks, each */r-inch thick. The Burface of the upper disk i E grooved to hold a copper-constantan thermocouple called t,he intermediate thermocouple. The lower disk ie cemented to the copper bottom of a metal container suspended in a larger galvanized iron container partially filled with ice and water, with a slab of ice in cont’act with the copper.
There is a constant temperature differential of 50” C. between the upper surface of the liquid sample and the bottom of the lower glass disk. Since the thermal conductivity of the glass is constant a t constant t,emperature, the teniperat,ure of the upper surface of the upper glass disk is a function of the thermal conductivity of the particular liquid being investigat’ed. Care niuqt be exercised to ensure a constant liquid sample thickness and to avoid disturbing the saiiiple during measurements. X st,eady state is usually reached in about 2 hours. The apparatus was calibrated by determining the electromotive force values of the intermediate therniocouple for several liquids of kn0T-n t,hernial conductivity. The values used are those determined by Bates (1, 3). The same value for ethylene g1;Vcol has recently been reported by Van der Held and associates (6)> whose results on water also agree closely with those of Bates.
Calibrating Liquids Water Aqueous solution of ethvlene glycol, 32.1y0 by weight Aqueous solution of eth$lene glycol. 61 . S % b y weight Ethylene glycol, 9 9 . 9 % Butanol
Thermal Conductivity. Cal./(Sec.) (Sq Cm.) (‘ C.,’Cm.) 16.02 x 10-4 11.61 X 1 0 - 6 iJ.04 X 10-1 6 . 6 1 X 10-* 3 . M X 10-4
The calibration curve for t,he particular apparatus used is shown in Figure 2. The thermal conductivity may he expressed as a function of the electromotive force of the intermediate thermocouple by the equation y = a9
.2 c
