Thermal Conductivity of Homogeneous Materials DETERMINATION BY AN UNSTEADY -STATE METHOD K. 0. BEATTY, JR., A. A. ARMSTRONG, JR., AND E. M. SCHOENBORN North Carolina State College, Raleigh, N . C.
A new, rapid, and rigorous method is described for direct 70" F. and to have had one face suddenly raised to determination of thermal conductivity of materials such the thermal conducand maintained at 220" F. tivity of a material is imas plastics from data taken under unsteady-state condiwhile the opposite face p o r t a n t in m a n y cases tions. The method is based on mathematical solution was maintained a t 70" F. of a boundary value problem, and the outline of the solueven though the material is not being specifically used tion is included. Results are given of application of the The curves are based on a thickness of 2 inches and a either as a heat transfer procedure to several plastic laminates. A copper block thermal diffusivity of 0.005 equipped with a thermocouple is sandwiched between two medium or as an insulator. This is true with plastics, layers of the sample and the resulting sandwich placed square foot per hour. T h e effect of thickness on this for example, where physical between platen surfaces heated to and held at some known and chemical thermal statime lag is shown in Figure constant temperature. The temperature of the copper is bility, pleasantness of feel, 2 where the midpoint temthen taken at a series of time intervals. The slope of the ease of molding, and uniperature divided by t h e straight line obtained by plotting these data on semimean of the end temperalogarithmic paper permits calculation of the thermal formity of cure are all tures is plotted against time conductivity by a simple formula. No assumptions are affected b y t h e c o n d u c for several different thicktivity. involved in analysis of the data other than constancy of nesses. If the thermal conthermal properties over the small temperature change Appropriate methods for while measurements are being made. The method may measurement of this propductivity is independent of temperature, it is apparent be combined with measurements of the thermal diffusiverty depend on the order of ity to calculate heat capacity. The speed of the procedure that the value of the ordimagnitude of the conducis such that all three thermal properties, conductivity, nate should be unity when tivity and on the dimendiffusivity, and heat capacity, can be determined in less steady state is established. sions of the material to be than 1 hour for a 0.5-inch thick plastic specimen. A typical approach to t h e tested. Regarding - the first of these, all substances may solution of these problems be classified either as good is the well-accepted American Society for Testing Materials procedure A.S.T.M. Designaconductors or poor conductors in a fairly sharp manner. The good conductors are materials such as metals, which contain free tion C 177-45T, commonly known as the guarded hot-plate method. electrons and show values of k, varying from about 240 B.t.u./ (hr.)(sq. ft.)(' F./ft.) for silver down to 20, or somewhat less, The use of unsteady-state methods represents an entirely diffor lead and nickel. Nonmetallic materials, on the other hand, ferent approach to the solution since it eliminates some of the inherent difficulties of steady-state methods. Several unsteadyrarely have k values greater than 1 and for the most part are less than 0.5. Physical dimensions of the materials to be tested are state methods (1, 4, 6) have been used with metallic conductors not so easily grouped. It is generally true, however, that good with reasonable success. The use of an unsteady-state method co^nductors are relatively homogeneous and thin specimens may with poor conductors has not been reported in so far as the aube taken as representative. Poor conductors, of which plasthors are aware. A type of unsteady-state procedure has been tics and the common insulation materials are typical, are freproposed by Fitch (3) but since the heat capacity of the sample quently heterogeneous and require thicker samples. is neglected, the analysis is essentially by steady-state proThis paper describes a new method for measuring thermal cedure. conductivity based on an analysis of data obtained during unPRINCIPLE OF PROPOSED METHOD steady-state heat transfer. The method is well adapted to the class of the poor conductors of which plastic laminates may be If a material of high thermal conductivity and known heat taken as a good example. capacity is placed in thermal contact with one face of a slab As ordinarily conducted, the determination of thermal conwhile the other face is heated, the rise in temperature of t h e ductivities of poor conductors presents two principal problems. conductor is a measure of the heat transmitted through the slab. One of these is the measurement of small heat fluxes and may If the heated face is maintained a t a constant temperature, TO, be solved by any of several techniques. The second problem the rate of transmission of heat through the slab to the conductor is that of establishing steady state. The only solution to this will continuously vary both because of the decreasing temis to control the temperatures on each side of the sample and then perature difference across the slab and because of the changing wait a sufficient length of time to permit steady state t o be temperatures within the slab itself. By appropriate mathematiapproximated. cal analysis these variations may be taken into account and The time required to approximate steady-state conditions the thermal conductivity obtained directly from the data. within any specified degree of precision depends on the thermal Consider a slab of plastic of thickness, L, in the direction, 2, diffusivity and the thickness of the specimen. I n Figure I, and of unlimited size in the ?J and z directions whose thermal the temperature distribution through a slab is shown for several conductivity is to be measured. The face z = L of the plastic time intervals. The slab is assumed to have been initially at is placed in intimate thermal contact with a similar semi-infinite K N O W L E D G E of
1527
.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1528
Vol. 42, No. 8
A tan A = R
(8)
Equation 7 may be simplified further in form by substituting for An according to Equation 9 X,
=
Z/KR
(9)
By comparing Equations 8 and 9 it is apparent’that the fnctor, Z, is itself a function of R. Substitution of Equation 9 in 7 and slight rearrangement gives
6%
-
-
0.4
Distance
-
-
-
-
0.8 1.2 1.6 2 .o . in I n c h e s f r o m H e a t e d S u r f a c e
’
Figure 1. Calculated Temperature Distribution in a 2-inch Thick Slab Heated from One Side
slab of copper of thickness L‘. The exposed face-i.e., the face not in contact with the plastic-of the copper slab is assumed to be perfectly insulated. The plastic and copper are both initially a t the uniform temperature, to. At the instant 6’ = 0, the face x = 0 of the plastic is suddenly raised to the constant temperature, To, and maintained a t this temperature for the duration of the test. It is assumed that because of its high thermal conductivity, the temperature of the copper may be taken to be uniform throughout and equal to the temperature of the plastic surface z = L a t the copper-plastic interface. Under these conditions, the appropriate equations and boundary conditions are
t(x,O) =
t o for all values of
t(0,O) = T ofor 8
>
z
(2)
0
(3)
8 > 0 Equation 4 may be rearranged to yield ( F ) ( E ) + G at =oatx=li
Consideration of the terms in Equation 10 reveals that as 8 increases, the terms after the first become less and less significant. If only the first term is considered and the natural logarithm of both sides of Equalion 10 is taken, Equation I1 results. In To
To
-t - to
=
In Equation 11 only the first term of the infinite series is considered. The significance of t,his neglect of other t,erms may be seen from a st.udy of Equations 7 and 8. Consideration of Equation 8 shows that the positive roots, An, are alternately in t’he first and third quadrants. Consequently t’he value of sin An in 13quation 7 is alterna,tely positive and negative. Since the other coefficients are positive, the series is an alkrnating one. I t may be shown further that the series meets the two requirements for convergence of such an alternating series-namely, that the limiting value of the last term as n approaches infinity is zero, and that any term ( n 1) is less in absolute value than the preceding term n of the series. I t follovr~sby B well-known t,heorem that the absolute error made in the sum by stopping a t the first term is equal or less than the absolute value of the second t’erm. Csing thermal propert,ies of the plmtic materials employed in these experiments] calculation s h o w t.hat for a 0.5-inrh specimen the error in considering only
+
where
The prime on the
at’
w
term in Equation 4 may be dropped in
Equation 5 by virtue of the assumption that a t x = L,t = t’. The solution of Equation 1 under the boundary conditions stated may be made by any of several procedures for solving the Laplace equation. The details of the solution although interesting are not important to the use of the method. A discussion of the method of solution for this problem, together with one form of the resulting solution, is available in the treatise on heat conduction problems by Carslaw and Jaeger ( g ) . One general solution of the equation may be written in the form ~To - t To - to
2
2(A,Z
+ R2)e
- n = l x,,(x,z+
where An, for n = I, 2, 3 transcendental equation
R2
- - eah: L2
+ R)
X,x sin
(7)
. . , . ., etc., are the positive roots of the
Time
Figure 2.
in M i n u t e s
Effect of Thickness on Rate of Approach to Steady State in a Slab
1529
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1950
"
m = -k
-
L
(13)
(2.303)(L'p'c')L
or
From Equation 14 it is easily seen that the value of k depends on the slope of the line, the properties of the copper block, and on 2. The value of the factor Z may be determined in three ways. The first method for finding Z involves use of the diffusivity. If the diffusivity is known, the value of the product Z R = X2 may be obtained by rearrangement of Equation 12.
ZR
Figure 3. Correction Factor, 2, as Function
LY
ofZXR
the first term of the series becomes less than 0.1% after 12 minutes. For thinner specimens, the time would be somewhat less. The error introduced by failure t o meet the boundary conditions exactly is more difficult to evaluate precisely. Careful considerations have led t o the conclusion that the error is negligible in its effect on the slope of the curve but does significantly affect the intercept. The error is introduced because of the temporary chilling effect of plastic on the platen surface a t initial contact. Consequently, the plastic surface at z = 0
By combination of Equations 8 and 9 it is possible to determine Z from this value of ZR. A plot of this relationship is given in Figure 3. If the heat capacity of the plastic and, consequently, the value of R are known, 2 may be found directly from Equations 8 and 9. For convenience, a plot of Z as a function of R has been prepared and is given as Figure 4. The third method for finding 2 may be termed the intercept method. I t requires no previous knowledge of the thermal properties of the material and permits calculation of thermal conductivity, heat capacity, and thermal diffusivity from a single set of measurements if the density is known. Reference to Equation 11 shows that the intercept of the straight line (obtained by plotting on semilogarithmic paper) for the condition where x = L is given by Equation 16 Intercept = I =
(sin
dm)
(16)
The intercept is a function of R only since Z is itself a function of R. Thus it is possible to determine the value of R and of Z from the intercept without need of either diffusivity or heat capacity data. A plot showing the relationship of I to R and Z is given in Figure 5. The values from which Figures 3, 4, and 5 were plotted are given in Table I. R
Figure 4. Correction Factor, Z , as Function of Ratio of Heat Capacities, R
is not brought to temperature T Oa t time zero as required by the boundary conditions. The effect is to shift the scale of the time axis somewhat. Theoretical considerations indicate that the use of copper platens would very materially reduce this source of difficulty. I t is apparent from Equation 11 that the curve obtained by To - t plotting T versus 8 on semilogarithmic paper would be a 0
- to
straight line. The slope of this line is given by Equation 12. The subscript on 2 has been dropped in Equation 12 and those following since only one value of 2-viz. Z1-is being considered.
The factor 2.303 arises from the use of graph paper based on logarithms to the base 10 instead of to the base e. At first glance it might appear that the slope of the line as given by Equation 12 depends on the thermal diffusivity, a, rather than on the thermal conductivity, k . Such a conclusion fails to consider the effect of the factor, R. If now, the value of R as given by Equation 6 and for a its equivalent k / p c are substituted in Equation 12, Equation 13 results.
-
To t Figure 5. Intercept of Plot of US. 8 To to as Function of Correction Factor, Z , and of Ratio of Heat Capacities, R This last method suffers somewhat because of the difficulties of precise evaluation of the intercept. However, as may be seen in Figure 5, the values for Z for use in Equation 14 are relatively insensitive to slight variations in the intercept. Consequently, the thermal conductivity may be determined with quite satisfactory precision from relatively inaccurate values of the intercept. Calculation of the heat capacity by determining R
INDUSTRIAL AND ENGINEERING CHEMISTRY
1530
from the intercept is also feasible. For reasonable precision this requires accurate determinations of the intercept value. APPLICATION OF THE METHOD
In an effort to simulate experimentally the boundary conditions as set forth a t the beginning of the previous section, a n arrangement of apparatus was made as shown diagrammatically in Figure 6. A block of copper of high purity and carefully machined surfaces was sandwiched between two pieces of the plastic to be tested. These three layers were in turn placed between the steam-heated steel platens of a small hydraulic press under an effective pressure of approximately 200 pounds per square inch. Glass wool insulation was placed around the sides to prevent edge losses. The temperature of the copper block was taken with a thermocouple placed in a l/ls-inch diameter hole drilled in the edge. Platen temperatures were taken in a similar manner.
I
Vol. 42, No. 8
The measured slopes and intercepts are indicated on the figuie. Dimensions of the plastic and of the copper block together with other pertinent data are given in Table 111. The preceding derivation has been based on a material hoinogeneous in the direction of heat flow. The data given here are on a paper-base phenolic laminate which does not meet this requirement. The justification for this is that the nonhomogeneity is of order of magnitude less than the sample thickness. Practical justification is also offered in that the results on a single 0.5-inch thick piece are in experimental agreement with those obtained using a single 0.125-inch thick piece and using four 0.125-inch pieces. The use of the three methods of calculation can best be shown by illustration using the actual measurements combined with appropriate data. SAIfPLE
CALCULATION USINGDIFFUSIVITY DATA. For sample
a, one layer of 0.125-inch nominal thickness plastic was used.
From Equation 15 using the dimensions and diffusivity given in Table I11 and the measured slope from Figure 8
Sfearn H e a t e d P l a t e n fi"
I
fi'
Therefore, from Figure 3 Z = 0.937 Substituting this value in Equation 14 gives
Sfeam Heated Platen
Figure 6.
Thermal Conductivity Assembly
It was recognized, of course, that the use of a guard ring or similar device would have been preferable to simple insulation. Work is no\?- under way on making a new apparatus t o incorporate this and other features. The use of the plastic sample on both sides of the copper eliminated the necessity for insulating one surface. The use of a hydraulic press for clamping the system together improved the thermal contact between hard faces of the plastic and metal. With softer samples this pressure would be neither necessary nor desirable.
=
0.147 B.t.u./(hr.)(sq. f t . ) ( " F./ft.)
h similar calculation for sample b consisting of one layer of
0.5-inch nominal thicknpss plastic gives
Therefore, 2 = 0.775
=
0.153 B.t.u./(hr.)(sq. ft.)(" F./ft.)
TABLE I. VALUESOF 2, ZR, AND I AS FUNCTIONS OF R CSED 16 PREPARATION OF FIGURES 3, 4, AKD 5 Steam
Heated
Platen I
Figure 7 . Thermal Diffusivity Assembly
The determination of diffusivity may be made with this same appwatus and procedure by omitting the copper block between the plastic specimens. A thermocouple between them measures the midplane temperatures. A diagram of the arrangement is shown in Figure 7. The data are plotted in exactly the same manner as for the conductivity experiment, replacing the copper block temperature by the midplane temperature. In this case the slope of the straight line portion of the curve is a direct measure of the diffusivity by Equation 17. 4L2m2.303 a = ir2
=
-0.932 L2m
CALCULATIOnS AND RESULTS
Typical data are given in Table I1 and plotted in Figure 8. These show results taken on one layer of 0.125 inch thickness and on one layer of 0.5-inch thick paper-base phenolic laminate.
R
Z
ZR
I
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 3.00
0.9678 0,9366 0.9076 0.8797 0,8636 0.8286 0.8049 0.7821 0,7607 0.7401 0.6510 0,5799 0,4740
0.0968 0.1873 0.2723 0.3519 0.4268 0.4972 0.5634 0,625; 0.6846 0.7401 0.9763 1.1597 1 4-221
1.0162 1 ,0308 1,0447 1,0380 1.0701 1.0814 1.0910 1,1016
1.llOT
1.1191 1,1537 1,1784 1.2102
TABLE 11. TEMPERATURE-TIME DATAFOR THERMAL COKDUCTIVITY AIEASUREMENT O S PAPER-BBSE PHENOLIC LAMISATF: (Sample a, single layer 0.120 inch thick) Platen Copper Time, Time, Temp., Block Temp., hlin. 8 , Hr. To, F, t , F. 0 0.000 232.2a 82.4 2 0.033 228.4 116.1 4 0.067 228.4 144.1 6 0.100 228.6 165.4 8 0,133 228.6 181.2 10 0.167 228.7 193.3 12 0.200 228.6 202.1 14 0.233 228.5 208.9 16 0.267 228.6 214.0 .4v. = 228.6 a Omitted froin average.
e To to
1.000 0.770 0.577 0.433 0,324 0.242 0.181 0.134 0,0998
TABLE
.
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I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
August 1950
DATAFOR THERMAL CONDUCTIVITY 111. AUXILIARY TESTS
Single Layer, Paper-Baas Laminate Phenolic Plastic Sample a, Sample b, 0 . 5 Inch Thick 0.125 Inch Thick Over-all dimensions, inches 5 x 5 x 0.120 5 X 5 X 0.489 Thickness L, ft. 0.010 0.04075 Weight 65 grams = 0.143 lb. 267.1 grams = 0.589 lb. 82.5 82.5 Densit;. 0 . lb./cu. ft. Heat iapacity as measured b calorimeter c, B.t.u./ 0.366 (lb.)(' k.) 0.366 Thermal diffusivity by direct measurement, sq. ft./hr. 0.00479 0.00479 Ratio, R = 0.194 0.800 w'c' we product, B.t.u./OF. 0.0523 0.216 L p c product, B.t.u./(" F.)(sq. it.) 0 302 1.241
w'
_
I
5
Copper Blocka Over-all
8
dimensions,
-
Time
'
0.3 in Hours
I
I
I
0.4
Figure 8. Thermal Conductivity of Paper-Base PhenoIic Laminate Measured by Unsteady-State Method 0.0933 0.270 L'h'cf&oduct, B.t.u./ 1.556 (" F.)(sq. ft.) * Since in operation plastic specimens were placed on each side of the copper block, the effective thickness and weight are only one half the oyerall values.
SAMPLECALCULATION USING HEATCAPACITYDATA. For sample a,by Equation 6
From Figure 4 Z = 0.936, Hence, k = 0.147.B.t.u./(hr.)(sq. ft.)(" F./ft.) Similar calculation for sample b gives
R = 0.800; 2 = 0.782 Hence, k = 0.151 B.t.u./(hr.)(sq. ft.)(O F./ft.) SAMPLECALCULATION USING INTERCEPT METHOD. For sample a: The measured intercept, I = 1.04 From Figure 5, the corresponding value of R is 0.261 and of Z is 0.918. The calculated value of k is then
=
0.150 B.t.u./(hr.)(sq. ft.)(OF./ft.)
The value of the heat capacity, c, may be calculated from R by rearrangement of Equation 6 c=-
L'p'c'R = (1.556)(0.261) LP
(0.01)(82.5)
= 0.492 B.t.u./(lb.)(
O
F.)
Similar calculations using the intercept of 1.08 obtained with sample b give
k
= 0.142 B.t.u./(hr.)(sq.
ft.)(OF./ft.)
and c = 0.271 B.t.u./(lb.)(" F.)
Although the intercept method ave heat capacity data differing by 30% from that measurefin the calorimeter, the corresponding error in the thermal conductivity is less than one third of this or about 8%. The greatest difficulty in getting the true intercept is associated with determining the zero of the time axis. A change of less than 2 minutes on the time axis would give results with the intercept method which would coincide with those by the diffusivity method.
DISCUSSION OF RESULTS
The results and calculations given above indicate that the proposed transient method may be used to give consistent results using a simple apparatus. The data reported here and i n another paper (6) indicate that values are obtained which are somewhat higher than those obtained with the guarded hot, plate method A.S.T.M. Designation C 177-451'. This may be' attributed to better thermal contact in the present metho& between the hot plates and the plastic. With the hard and not too flat surfaces presented by the plastic laminates, good contact cannot be obtained without the use of pressure. The rapidity with which data can be obtained as compared to steady-state methods should be an important feature for many purposes. As has been pointed out previously, the same apparatus and general procedure may be used for measuring diffusivity. I n consequence, the heat capacity, diffusivity, and thermal conductivity may all be determined in less time than i s usually required for either the thermal conductivity or the heat capacity alone. There are certain difficulties still t o be overcome in the interpretation of data. The first and most important of these is the determination of the temperature to which the measured value of k should be assigned. During the measurement, the temperature distribution is continuously changing, and the equation for the temperature curve does not have a simple form. It would appear that if conductivity varies with temperature, the data obtained should fall on a curve instead of a straight line. The slope of the tangent to this curve at any point would correspond to the thermal conductivity for the particular mean temperature existing at that instant. The fact that each set of data so far obtained lies on a straight line appears t o indicate that the conductivities of the plastics studied do not change significantly over the temperature range used. More mathematical and experimental work is now under way to clarify this point. A second difficulty arises when an attempt is made to use the method with materials made up of layers of materials of different thermal properties, as, for example, a composite insulation board consisting of a layer of paper and a mineral wool filler. T h e problem appears to be soluble for certain simple cases, but it is possible that in complex cases, such as structural walls, the solution would be impractically complex. Other problems arise relative to the effect of deviations from the assumed boundary conditions. Chief among these are instantaneous rise of surface temperature at the platen-sample interface, and slight unavoidable variations during the run i n
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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Vol. 42, No. 8
CONCLUSIONS
thermal conductivity, B.t.u./(hr.)(sq. ft.)(’ F./ft.) thickness of sample, f t . L’ thickness of copper block, ft. To - t m = slope of plot of - versus 0 on semilogarthmic To - t o paper, hr.-I wc 12 = - a dimensionless ratio
1. A practical, rapid, and simple unsteady-state procedure for measuring thermal conductivity of homogeneous materials has been developed requiring only approximate values of diffusivity or heat capacity in addition to the data taken. 2. A procedure for measurement of all thermal properties, diffusivity, heat capacity, and conductivity from a single set of unsteady-state data with no additional thermal data is presented. The results obtained are not of high precision, but the limitations are experimental. More careful attention to experimental procedure will improve these values. 3. The application of the method to substances which are nonhomogeneous because the conductivity changes significantly with temperature should be possible by graphical and probably by analytical means. 4. Substances having nonhomogeneities because of composition differences in the direction of heat flow cannot be handled by the proposed method in its present form.
temperature a t any point in sample a t any time, F. ini$al uniform temperature of sample and copper block, F. t’ = temperature of copper block, F. = t a t z = L T O= ronstant elevated temperature of facr z = 0 for times greater than zero, O F. w = iyeight of sample, lb. w’= weight of copper block, Ib. TC = distance normal t o surface of sample mrasured from heated surface A2 Z = -- , a dimensionless factor R a: = thermal diffusivity k / p c , sq. ft./hr. h = a dimensionless factor, root of A tan A = R p = density of sample, Ib./cu. ft. p ’ = density of copper, Ib./cu. ft. 0 = time, hr.
the temperature a t this interface. The consistency of the data reported would attest to these not being major problems. These inconsistencies between assumptions and practice are present in all unsteady-state heat transfer problems and usually cause no serious error. The matter does, however, deserve careful consideration.
The authors wish t o acknowledge the generous assistance of John Cell of the department of mathematics. Revere Copper and Brass Incorporated furnished the highly purity copper blocks. NOMENCLATURE
= c = c‘ = e =
I
=
L
= = =
w’c”
ACKNOWLEDGMENT
A
Iz
area normal to direction of heat flow, sq. ft. heat capacity of sample, B.t.u./(lb.)(’ F.) heat capacity of copper block, B.t.u./(lb.)(” F.) base of natural logarithms, 2.71828 To - t intercept of plot of versus 0 on semilogarithmic To - t o paper, dimensionless ~
t
to
= =
O
d=,
LITERATCRE CITED
(1) Carslaw, H. S., and Jaeger, J. C., “Conduction of Heat in
Solids.” London, Oxford University Press, 1947.
(2) Ibid., p. 107, Equation 21. (3) Fitch, A. L., Am. Phys. Teacher, 3, S o . 3, 135-6 (1935). (4) Ingersoll, L. R., Zobel, 0. J., and Ingersoll, -4.C., “Heat Con-
duction,” New York, McGraw-Hill Book Co., Inc., 1948.
(5 j Schoenborn, Srmstrong, and Beatty, unpublished data. (6) Worthing and Holliday, “Heat,” New York, John Wiley & Sons, Inc., 1948. RECEIVED December 2, 1949. Presented before the Division of Industrial and Engineering Chemistry at the 117th Meeting of the AMERICAN CHEMICAL SOCIETY,Detroit, 3licli.
Preparation of Tall Oil Esters E. R. JIUELLER, P. L. ENESS, AND E. E. RSCSWEENEY Battelle Memorial InstitzLte, Columbzcs, Ohio T a l l oil was esterified with three different polyhydric alcohols: glycerol, pentaerythritol, and sorbitol. The effect of temperature, catalyst, carbon dioxide flow rate, and amount of alcohol on the rate of esterification and changes in the color, viscosity, and drying time of the ester was studied. Within practical esterification temperatures, catalysts had no appreciable effect upon the esterification rate of any alcohols studied. Ten per cent excess alcohol (over stoichiometric) produced the maximum rate obtain-
able for glycerol and pentaerythritol; acid numbers of 10 were readily obtained. Four and one half equivalents of tall oil per mole of sorbitol yielded esters with acid numbers over 20. For the temperatures investigated (437”, 475 ’, 526:, 560°, and 585’ F.), no advantage in esterification is gained by use of temperatures higher than 526” F. for glycerol, 535” F. for sorbitol, or 560” F. for pentaerythritol. Color, viscosity, and drying time of the esters were not appreciahly affected in any case.
I
oil and the developnient of the best possiblr properties in the ester. It was planned t o study the effect of temperature, catalyst, and carbon dioxide flow on the rate of esterification with the three common polyols: glycerol, pentaerythritol, and sorbitol, as well as changes in the color, viscosity, and drying time of the esters. A 15% excess of glycerol and pentaerythritol, based on acid content,, was used. Although other excesses are often recommended, 15% was selected because a number of commercial users of the material formulate to such compositions. Later, the effect of excesses ranging from 0 to 20% was studied and is discussed below. T i t h sorbitol, 4.5 equivalents of tall oil per mole were used, as earlier runs confirmed the work of Konen ( 9 )that larger amounts of tall oil resulted in extremely high acid values arid rmaller amounts are impractical.
N A research program on the utilization of tall oil in organic
protective coatings, it early became apparent that the most favorable conditions for its esterification would have t o be established. Esterification rate studies on fatty acids have been conducted by several workers (3-4, 6, 9 ) . So far as is known, no extensive investigations on the esterification of rosin have been reported, although esterification procedure6 are given (8). Essentially no information was available on the most suitable esterification procedures for use with tall oil, although some recommendations were found ( 1,6 , 8 , 1 0 , 1 1 ) . ESTERIFICATIOY PROCEDURE
This study was directed toward determining the most suitable and economically practical conditions for the esterification of tall
