INDUSTRIAL AND ENGINEERING CHEMISTRY
908
tion 3-namely, those involving A4, As, and A8-should be noted. The coefficients of A4, AS, and A8 in Equation 3 have been empirically determined. The terms L , P I ,and Pa in Equation 3 are likewise empirical. NOMENCLATURE
AH
‘188
=
standard enthalpy change for a chemical reaction a t 298 Kelvin, in calories per gram mole
K = equilibrium constant of a chemical reaction, in the
usual units (atmospheres, etc.) logarithm of K to the base 10 ASOZQS= standard entropy change for a chemical reaction a t 298 Kelvin, in calories/( Kelvin) (gram mole) T = temperature, degrees Kelvin
log K
=
O
Vol. 40, No. 5
LITERATURE CITED
(1) Anderson, J. W., Beyer, G . H., and Watson, K., A-atl. Petroleum N e u s , 36, R476-84 (July 5, 1944). (2) Aston, J. G., IND. ENG.CHEM., 34,514 (1942).
Latimer, J . Am. Chem. Soc., 43, 818 (1921). Nernst, “Grundlagen des neuen T~’armesatzes,”Halle. Verlagsgesellschaft, 19 18. ( 5 ) Parks and Huffman, “Free Energies of Some Organic Compounds,” h’ew York, Reinhold Publishing Corp., 1933. ( 6 ) Sackur, Ann. p h g s i k . , (4) 40,87 (1913). (7) Tetrode, I b i d . (4), 38,434 (1912). (8) Wenner, “Thermochemical Calculations,” New York, McGrawHill Book Co., 1941. (3) (4)
O
RECEIVED October 7, 1946.
efrigerating Capacity of Two-Component Systems GLYCEROL-WATER AND PROPYLENE GLYCOLWATER
’
FRANK T. GUCKER, JR.’, AND GLENN A. MARSH2 Northwestern University, Evanston, I l l . Engineering design of quick-freezing equipment and refrigerating cold plates requires a knowledge of the specific heat capacities of the solutions and two-phase mixtures used for these purposes. The specific heats of the solutions can be measured without difficulty, but any particular two-phase mixture can be studied conveniently only just below its freezing point. A thermodynamic analysis yields an estimatfon of the specific heats of such mushes from the slope of the freezing point curve, the heat of fusion of ice, and the heat of dilution of the solution. The analysis also shows that a t any temperature the specific heat of a two-phase mixture is a linear function of the composition; hence experiments on a few mixtures a t several different temperatures can be correlated to allow a calculation of the specific heat of any mixture a t any
temperature in the experimental range. There is one mixture composition which is the most effective thermal buffer a t any temperature or over any given temperature range. The graphs and tables allow a choice of the most efficient mixture for any particular application. The authors have determined the specific heats of glycerol (2570 to 65%)-water mixtures and of propylene glycol (2070 t6 50yc)-water mixtures, from -25“ to +35” F. The results are accurate to about *l% for the solutions and *S% for the two-phase mixtures. The specific heat of the glycerol solutions is about 90% of that of the corresponding propylene glycol solutions. However, the twophase mixtures are far more effective thermal buffers. Here the best propylene glycol mixtures are only 70% to SOY0 as efficient as the best glycerol mixtures.
S
specific heats of these mixtures t o compare their thermal buffering capacity Tith that of the glycerol-water mixtures. The measurement of the specific heat of one of the solutions a t any temperature is perfectly straightforward, and the results are easily correlated, since the specific heat changes only slightly n-ith temperature and with composition. The case of the twophase mixtures, however, is quite different. Here most of the heat capacity is due t o melting of the ice. The heat capacity changes rapidly with composition and temperature, and may become very large as the last ice in the mixture is melted. Since the two-phase mixtures become too stiff to stir a few degrees below the freezing point, a thermodynamic relation is required, t o allow correlation of a reasonably small number of experiments and calculation of any other results in the experimental range.
OLUTIOKS of water and glycerol, and two-phase mixtures containing ice, are used in the quick-freeze and refrigerating industries. Being nontoxic, they form a bath in which the food can be immersed directly, and thus rapid freezing is assured. Such solutions do not expand appreciably on freezing; hence they are also used in cold plates to fill the space between the refrigerating coils and the inner surface of the plate. Thus they improve heat transfer and provide a thermal buffer capable Qf absorbing considerable heat without appreciable change in temperature. Apparently the heat capacities of glycerol-water mixtures have not been studied below room temperature; this work was undertaken to provide the basic information for engineering design. Tables based on a thermodynamic analysis of the experimental results allow a prediction of the mixture with the highest specific heat, and consequently the greatest thermal buffering power, over any temperature range a t which refrigeration is t o be maintained. The specific heats of aqueous mixtures of the closely related propylene glycol also are of interest. Since no suitable data were found in the literature, the present authors studied the Present address, Indiana University, Bloomington, Ind. Present address. Illinois Institute of Technology, Chicago, Ill.
THEORY O F SPECIFIC HEATS OF TWO-PHASE MIXTURES
When two-phase mixtures are heated, some of the ice melts, and this latent heat is added t o the specific heats of the solution and the ice in the original mixture. The situation may be analyzed thermodynamically as follows: Let 2 2 , ZI,and represent, respectively, the weight fractions of solute, water, and ice in the mixture, h the enthalpy per gram of
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1948
mixture, and t the centigrade temperature. If the mixture is heated infinitesimally, so t h a t the solution remains saturatedt h a t is, in equilibrium with ice-the heat capacity of the mush is:
This equation states that the total heat capacity is made up of two terms: the heat required t o raise the temperature without melting any ice, plus the heat required t o melt enough ice t o bring the mixture into equilibrium a t the higher temperature. The first term is:
Here c8' is the specific h e i t of the solution (the prime being added t o show t h a t it is saturated) and ci is that of ice at the experimental temperature. Furthermore, if the solute forms no solid solution in the ice, the change of enthalpy with concentration is:
(g)t il =
id
(3)
where -1, is the heat absorbed in melting 1 gram of ice at the experimental temperature and il is the heat absorbed per gram of water dissolving in a solution of the constant composition in equilibrium at this temperature. The quantity (& - la) is the partial heat of solution of ice in the saturated solution. The change of 2 1 with t in the saturated solution is obtained from the freezing point diagram. Figure 10, the freezing point diagram for the system glycerol-water up t o 65% glycerol, illustrates the general principles. If at any temperature the horizontal line ad intersects the freezing point curve a t a point c, then ac and (1 - ac) are the weight fractions of glycerol and water, respectively, in the saturated solution. Let the former be represented by y, so that the latter is (1 y). If ab is equal t o 5 2 , the weight fraction of glycerol in the mixture of ice and saturated solution, then:
-
(4)
Hence the change of the concentration of water with temperature is: (5)
Here dyldt is the reciprocal of the slope of the freezing point curve. Since this slope is negative] the sign of the whole term is positive. At any fixed temperature the weight fraction of ice is:
909
Also, it can be evaluated from experimental results as:
A suitable plot of k calculated from a small series of measurements of cm a t different temperatures correlates the experimental results and allows the calculation of cm at any desired temperature and composition. THERMODYNAMIC EVALUATION OF k . An exact evaluation of k from Equation 9 required a knowledge of the heat capacity of ice, its latent heat of fusion, the freezing point-composition curve of the system, the partial heat of solution of water in the saturated solution, and the heat capacity of the saturated solution. All of these values must be known at the particular temperature for which k is t o be calculated. The first two values are properties of the solvent alone. The specific heat of ice a t any centigrade temperature can be calculated with sufficient accuracy from the equation: ci = 0.591
+ 6.68 X 1Ovat + 5.6 X 10-st? cal./" C./gram
(11)
based on the data of Nernst (3) at -lo", -20°, and -60' C. The latent heat of fusion of ice at various centigrade temperatures was obtained from the equation:
- Z;
=
79.78 4-0.5025t
- 0.0157t2 cal./gram
(12)
This was calculated from the equation of Young (8)for the molal heat of fusion of ice. This e'quation is valid near the freezing point but is subject t o increasing uncertainty a t low temperatures, because of the necessary extrapolation of the heat capacity of supercooled water. Lack of thermal data and uncertainty in the freezing pointcomposition curve for the propylene glycol-water system preclude the thermodynamic calculation of k for this system. The information on the glycerol-water system is not complete but allows a fair estimate of k and the specific heats of the two-phase mushes which will illustrate the application of Equation 9. These calculations will be given after a description of the determination of the specific heats of the solutions. CALCULATION OF HEATABSORBED OVER A RANGEOF TEMPERATURE. A knowledge of k over a temperature range allows a graphical integration of the heat absorbed by any particular mixture over a certain temperature range. The same result can be calculated more simply between any lower temperature and 0' C., as follows: Assume t h a t the ice is removed from the solution and each is heated separately over the temperature interval. The necessary amount of heat, obtained by integration of Equation 2, is:
Hence the pure heat capacity term is:
(Z)
= ci
+ - (c: 22
-
Ci)
Y
i.
(7)
and the heat capacity of the two-phase mixture is:
The symbols have the same significance as in the preceding section. Here ca is the specific heat of the solution which is saturated only a t the lowest temperature. The second integral is obtained analytically from Equation 11 as:
"lo
cidt = -0.591t
At any fixed temperature] t, the expression in brackets is a constant, which we shall call kt; hence the heat capacity of twophase mixtures must be a linear function of 52, increasing from ci when 52 = 0 to
[
z 1
- - - (Z1 - I,)
in the saturated solution. This analysis shows that k can be calculated from the equation: =
c,
1
-Y
[(c: -
Ci) ,
-
-yl dd-ty (11- - Zi)]
(9)
- 3.34
X 1O-Y
-
1.87 X 10-5ta
(14)
Here t is the centigrade temperature. Evaluation of the first term requires a knowledge of the specific heats of the solutions involved. If now the ice is melted, it absorbs a n amount of heat equal t o -&, where -I$ is the heat of fusion per gram of ice at 0' c. Finally, if the resulting water is mixed with the original solution, it absorbs an additional amount of heat equal t o xiAII, where Ab is the integral heat of solution of water t o form the
910
INDUSTRIAL AND ENGINEERING CHEMISTRY
n
VoI. 40, No. 5
determine the freezing point curve. Application of Equation 15 then would give the amount of heat absorbed in heating various mixtures from any low temperature t o 0" C. These values could be plotted against the temperature, and tangents at any desired point would give the specific heat of the mixture at t h a t temperature. The practicrll choice of the better method would depend upon the particular system and the information available in the literature.
EF-
EXPERIMENTAL TECHNIQUE
APPARATLX The calorimeter (Figure 1) consisted of a pint (500-ml.) Dewar flask A , supplied with a t T 5 o-stage reciprocating stirrer, operated a t the rate of 26 strokes per minute. The stirre1 also served as an electric heater. Each stage was made of tm;) concentric disks of thin Bakelite, B , C, and B', C'. Disks B , B , and C' were mounted firmly on KO. 17 B.$S. gage (1.15-mm. diameter) Advance (constantan) supporting wires, D, D, crimped above and below each disk. The disk C' was placed 2 cm. below B' so as to stir the bottom of the flask. Ten small holes were equally spaced around the circumference of each disk, and about 1 meter of No. 37 B.&S. gage (0.11-mm. diameter) bare Advance wire, with a resistance of approximately 40 ohms, was strung bcbtween the disks in each pair to form two star-shaped webs i n series. The top web supported center disk C, and the wires increased the efficiency of the stirring. The ends of the resistance wire were soldered t o wires D, which served as connections to the flexible current and potential leads, E, F. The n ires supporting the stirrer were guided by closely fitting holes drilled in Bakelite disks G, H , fastened to cork stopper I . The full lines show the stirrer a t the top of the stroke, and the broken lines a t the bottom. The heating current was supplied by a battery of storage cells. The current and voltage were measured potentiometrically, and the time by means of an electric stop clock, fixing the energy input to 0.170. Mercury thermometer J, graduated to 0.2" C., was read t o about 0.01" by means of a cathetometer. It was found to read 0.12" C. low a t the ice point and also a t the eutectic temperature of XaCl.H20 (-21.13" C.) reported by Rodebush (7). A corresponding correction was applied at all temperatures. During an experiment the thermometer was read a t I-minute intervals over a long enough rating period (usually 5 minutes) t o establish the temperature drift due to heat leakage and stirring. This amounted to about 0.05 O per minute for the solutions and a s little as 0.01 per minute for the two-phase mixtures. Then th; heating current was turned on and the temperature raised by 1 to 5 " C. A second rating period allowed a n estimation of the temperature change to about 0.04 C. A pre1iminary.experiment Yith w a t , r gave the heat capacity of the calorimeter as 9.7 calories per C. MATERIALS AND SOLUTIONS.The solutions or mixtures were made up to approximately even concentrations with distilled water, cooled below the starting temperature by means of a dry ice-acetone mixture, transferred to the calorimeter, and weighed. After the experiment they were heated to room temperaturc and analyzed. The glycerol used was C.P. (U.S.P.) grade, containing about 57, water. Solutions of the glycerol were analyzed by means of duplicate pycnometer measurements of apparent density a t 25 C. and comparison with the standard values of Bosart and Snoddy (6). This gave the glycerol content of the solution withm about O
P 2I :4 : ;CM,' 6
0.10;.
Figure 1. Calorimeter E n d solution froni the initial one, a t 0' C. The total heat absorbed by the mixture over the temperature range is:
DEGREES F.
