INDUSTRIAL AND ENGINEERING CHEMISTRY
416
with 2 per cent hydrochloric acid for 2 hours a t 100" C. The hydrolyzate was strongly reducing. Hydrolysis:.of Arabinogalactan Chemical Constituents
Three grams of the dried polysaccharide were dissolved
in 150 cc. of 2 per cent sulfuric acid contained in a 250-cc. beaker. The beaker was covered with a watch glass to prevent excessive evaporation and placed in an air bath which was maintained a t 100" C. for 8 hours. The solution was cooled and nearly neutralized with a hot aqueous solution of barium hydroxide. The alkali was added slowly and with constant stirring to prevent local overheating. The neutralization was completed with barium carbonate. The neutralized hydrolyzate was evaporated in vacuo to a thin sirup. The sirup was treated with ethyl alcohol to precipitate a small amount of gum which was always present, filtered, and the alcohol removed by evaporation in vmuo. The hydrolyzate was then treated with methanol containing a small amount of acetic acid and seeded with a crystal of galactose as described by Wise and Peterson. When, after standing in the ice box for 3 days, no crystals separated, the methanol was removed by evaporation in vacuo. five grams of diphenylhydrazine dissolved in the least amount of 95 per cent ethyl alcohol were then added and the solution allowed to stand at room temperature for a week. At the end of this time the crystalline precipitate of mixed diphenylhydrazones was filtered onto a small Buchner funnel and washed several times with ether to remove unchanged
Vol. 23, No. 4
diphenylhydrazine. The hydrazones were then fractionally crystallized from 75 per cent alcohol. After three fractionations the less soluble fraction melted at 186-3.87' C., which was unchanged on recrystallization. The melting point of 1-arabinose diphenylhydrazone prepared from Special Chemicals Company 1-arabinose melted at 185-186' c. under the same conditions. A mixture of the two melted a t 185-186' C. The more soluble fraction melted a t 157-158' C. A sample of d-galactose diphenylhydrazone melted a t 155-156" C. A mixture of the two melted at 156-157" C. Araban was determined by distillation of the arabinogalactan with 12 per cent hydrochloric acid and precipitation of the furfural as the phloroglucide. The quantity indicated was 12.2 per cent. The presence of galactose was further established by treatment of the polysaccharide with nitric acid and isolation of mucic acid. Acknowledgment
The authors wish to thank The Masonite Corporation for furnishing the cyclone condensate and wood chips used in this investigation as well as for the use of their laboratory during a part of this work. Literature Cited (1) Boehm, IND. ENG. CHEN.,32, 493 (1930). (2) Kirkpatrick, Chem. Met. Eng., 84, 342 (1927). (3) Schorger and Smith, J. IND. END. CAEN., 8, 494 (1916) (4) Wise and Peterson, I b i d . , aa, 362 (1930).
Dimensional Analysis Applied to the Thermal Conductivity of Liquids' J. F. Downie Smith2 37 GORHAM. Sr., CANBRIDQE, MASS.
Assuming that thermal conductivity of liquids is a Variables Affecting Thermal Conductivity of function of molecular weight, density, specific heat, s e a r c h one is conLiquids viscosity, gas constant, thermal expansion, and comfronted with the probpressibility, an equation has been derived by dimenlem of the determination of The first thing to do is to sional analysis connecting these variables. This equaa particular p r o p e r t y of a settle which properties affect tion is: fluid or solid. It often hapthe thermal conductivity of kK'/i z'/d pens, also, that this deterliquids. -c'/a p l /X'/a a mh/rt = 4 mination is difficult and reI n 1923, B r i d g m a n (3) By making certain modifications this equation can be quires special technic and exsuggested an equation giving k pensive equipment. I n such thermal conductivity, k , as simplified, and a graph is shown connecting a case it would be extremely a f u n c t i o n of the gas con2 L - ,-C convenient if this property stant, a, the v e l o c i t y of with ( K C ml/2 This graph shows a maximum could be c a l c u l a t e d from sound in the liquid, v , and error in thermal conductivity of 4.5 per cent. known values of other properthe mean distance of separaties which are more readily obtion of centers of the moletainable. If, however, it is known that the property wanted cules, d, assuming an arrangement cubical on the average. is a function of several other properties, it may be a long, The equation is tedious process to find the function. The application of 2av dimensional analysis usually simplifies the problem by reduc(1) k = F ine the number of semrate Quantitiesthat vary. Engineers are, in-generai, not familiar with this subject, and, of course, it is dimensionally correct, so in this article the principle is considered in some detail, The arrangement of the variables in the equation could avoiding proofs of the theorems. The particular problem have been determined by dimensional analysis without having is to derive an equation for thermal conductivity of liquids in the picture portrayed by Bridgman in his derivation provided terms of other properties. that k is a function of a, v, and d. The procedure is as follows:
REQUENTLY in re-
F
(5)
.
e).
u
1 Received 9
September 25, 1930. National Research Fellow, Harvard University Engineering School.
k
=
+l(a, v , d ) or +(k, a, v d ) = 0
INDUSTRIAL A N D ENGINEERING CHEMISTRY
April, 1931
417
The quantities and their dimensions may be tabulated as follows : QUANTITY
SYMBOL
Thermal conductivity Gas constant Velocity of sound in liquid Mean distance of separation of molecules
ka
DKMBNSIONS
V
M LLS/ T / TW% L/T
d
1;
From M, From L, From T, From 8 , Solving,
It has been shown (1, 4 ) that any equation of the type
-
Therefore,
~ ( Q QLZ ~Qs, * .Qn) = 0 describing a relation among n different kinds of quantity Q1,Q 2 . . . . .Qn, is always reducible to the form +(TI,
xi, ra,
= o
l + w + z
1+2w+x+y = o -3 2a - x = o -1 - w = o w = -1,x = - l , y = 2 , z = 0 kd2 x = or - = a constant
-
e av'
av
which is Bridgman's complete equation except that the number 2 is not explicitly determined.
. . .. . ~ n - - r ) = 0
where each of the variables T represents a dimensionless product of the form T = ,'IQ Q:, QaC,. . . .Qn",r is the number of independent fundamental units needed in specifying the units of the n kinds of quantity, and 6 is an unknown function to be determined by experiment. This is known as the % theorem." I n the case of thermal conductivity, therefore, n = 4 ( k , a, v, d) and r = 4 ( M , L, T , 0). Thus there appears to be no group involving the four quantities. If, however, we define temperature as the mean energy of the atom, its dimensions are M L 2 / T Zand the gas constant has no dimensions. The new table Is as follows:
Derivation of Equation
Although this equation is theoretically interesting, results obtained from it are not very accurate. I n one instance, that of carbon disulfide, the calculated value is 37 per cent in error, which is perhaps not any closer than one could guess, for after all, outside of water and a few alcohols, thermal .a
0
QUANTITY SYMBOLDIMENSIONS Thermal conductivity k 1/LT Velocity of sound in liquid 0 L/T Mean distance of separation of molecules d I.
I n this case n = 3, and r = 2. Therefore 4 ( T ) = 0, where ?T = k" vu da. It has also been shown (2) that, in general, any one of the exponents can be assigned a t pleasure. Therefore, since we are interested in k , let us assign the value of 1 to : G . Therefore T = k vu da. Apply .. . dimensions: From L, From T, Therefore,
Therefore,
-1+y+z'= 0 -1-y = o y = -1, z = 2 T = kd 2 and 4
(Y)
E
== constant A
k =
AD d2
I
mc Figure 1
conductivities-do not vary greatly for liquids. Since, however, this equation gives approximate results, we might forget the physical picture and introduce some other variable and, by dimensional analysis or some other means, create a modifying factor until fairly accurate values of thermal conductivity are obtainable. This method does not appeal to us, however, so we shall try to solve the problem in another way. Weber (6) suggested the following empirical equation relating thermal conductivity, k , with density, p , specific heat, c, and molecular weight, m:
= 0
(2)
which is Bridgman's equation except t,hat the constant A is not explicitly determined. Its value, however, can be found from experiment, and it would be approximately 2a. Suppose, not having read Bridgman's paper, we had suspected that thermal conductivity depended not only on the factors u, a, and d, but, also on the mass of the molecule, m; then we would have ( k , a, 0, d, m) = 0. Giving temperature separate dimensions, and considering the mass of a molecule has dimensions M , we have n = 5, and T = 4. Therefore, + ( T ) = 0, where T = ka" uZ dY m'. Apply dimensions, and we get:
k = Apc
+
(2)'''
(3)
The results obtained from this formula, however, are often considerably in error. Smith (5) proposed an empirical formula for thermal conductivity of liquids in terms of viscosity, z , specific gravity, p, specific heat, c, and molecular weight of the liquid, m, which gave very satisfactory results.
T h e r m a l Conductivity D a t a Determined by Dimensional Analysis bK'/2 --
SUBSTANCE Water Methyl alcohol Ethyl alcohol Benzene Carbon disulfid e Toluene Acetone n-Octane n-Heptane n-Hexane n-Pentane ??-Xylene
cnr'/ip
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