JUNE, 1939
INDUSTRIAL AND ENGINEERING CHEMISTRY
freezer within the necessary time interval before incipient churning starts. The realization of this fact should do much to clarify the confusion that exists concerning the comparative whipping properties of various mixes and the procedure that must be followed to obtain the higher overruns. Since the work here reported was concerned primarily with overrun, no reference has heretofore been made to ice cream quality. Other things being equal, the ice cream that can be drawn from the freezer with the greatest amount of water in the form of ice has the best texture because the ice
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crystals will not grow so much in the hardening room. (This is probably the reason ice creams made from aged mixes are superior.) Therefore, those mixes that can be drawn from the freezer a t the desired overrun with the greatest amount of water in the form of ice present are to be preferred. Moreover, ice cream should be drawn from the freezer while the temperature-overrun equilibrium exists, because a t this point the greatest amount of water possible a t the chosen overrun has been frozen to ice and the best possible texture is therefore to be had.
Enthalw-Concentration Charts from Vapor Pressure Data WILLIAM HALTENBERGER, JR. Villanova College, Villanova, Penna.
T
HE minimum thermal data required for the construction of an enthalpy-concentration chart for a liquid solution are the heats of dilution a t one temperature, and heat capacities over the entire temperature and concentration range to be covered; or else, heat capacities a t one concentration, and heats of dilution over the entire temperature and concentration range (6). Experimental data on either heat capacities or heats of dilution a t higher concentrations and temperatures are scarce; consequently enthalpy-concentration charts have been prepared for only few systems. However, information is available on the equilibrium properties (notably vapor pressures) of many liquid solutions, from which heats of dilution a t higher concentrations and temperatures can be calculated. The enthalpy-concentration chart for the system can then be constructed by combining the calculated heats of dilution with experimental heat capacities a t low concentrations. The relations involved in one of the several possible conversions of this nature are derived and illustrated here. The experimental data required for the construction of an enthalpy-concentration chart by the method under consideration are ( a ) heat capacities of the solution a t any one concentration, over the entire temperature range and ( b ) vapor pressures of the solution over the concentration and temperature range to be covered. The method is applicable to all systems containing a single volatile constituent. The accuracy of the calculated values is discussed a t the end of the paper. All methods based on pure thermodynamics for the calculation of thermal properties from equilibrium properties include the differentiation of the equilibrium data. The differentiation can be greatly simplified if the equilibrium data can be converted to a linear or nearly linear form. The most convenient of the several possible “linear” relations for the present purpose is the Duhring plot, since it has to be constructed anyway for calculations on most processes involving solutions. Brown (3) showed how the Clapeyron relations for a liquid and for a solution containing that liquid as solvent can be combined to give a relation in which the differential term is the slope of the Duhring line. The slopes of the Duhring lines are, by all present evidence, constant a t low concentra-
tions and vary only slightly at higher concentrations. The relation is : dt’ _ - AHAV’T‘
dt - A H ‘ A V T
-
(1)
where at a given concentration,
t’
=
satur:tion or
F.
temp. of solution at pressure P ,
C.
T’ = satpration temp. of solution at pressure P , K or t =
R. saturation temp. of solvent at pressure P , or
C.
O F .
T = saturation temp. of solvent at pressure P, K or R. D = slope of Duhring line AH = latent heat of solvent at temp. t and pressure P AH‘ = latent heat of solvent in solution at temp. t’and pressure P AV = difference between gas and liquid volumes of solvent at temp. t and pressure P AV’ = difference between gas and liquid volumes of solvent at temp. t’ and pressure P
This is an exact relation, applicable to all systems containing a single volatile constituent. To simplify future discussion, it will be assumed that the volatile constituent is water, and English units will be used. This will make all quantities in Equation 1 and future equations, steam table quantities. Equation 1 can be simplified, without introducing a serious error, by neglecting the volume of the liquid phase. Then AV = V,,,, denoted by V hereafter. Furthermore, for ideal gases a t the same pressure, T’V’ -
T V -
(F)’
If 4, a measure of deviation from the ideal gas law, is defined as
Equation 1 can be thrown into a convenient form for calculations : AH‘
=
T[(F) AH
+@I
(3)
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VOL. 31, NO. 6
(8) The heat balance is:
s
AH'd(1
FIGURE1.
where AH' is the latent heat a t any concentration, and d ( l - W,/W) is the number of pounds of water removed a t that concentration. Noting that H , - AH' = RI (Equation 4) and solving for H 6 , we obtain Equation 5. The value of the relative enthalpy, H,, a t concentration W , and temperature t', may be assumed to be zero-that is, the datum from which all other enthalpies are calculated. Since the precision in determining relative enthalpies a t low concentration may be poor by this method, experimental heats of dilution at low concentrations should be available if it is desired to refer the enthalpies to the customary standard state of infinite dilution a t some chosen temperature. When the relative enthalpy a t a given temperature and Concentration is known, the relative enthalpy a t the same concentration but fi different temperature may be calculated by means of the Person-Kirchoff law:
q5 us. t FOR STEAM AT CONSTANT SUPERHEAT TEMPERATURES
Figure 1, based on Keenan's steam tables ( 5 ) )gives values of 4 us. t for steam, for boiling point elevations up to 100' F. For values of t below 100' F., 4 can safely be neglected. AH' has been defined as the latent heat of water in the solution. It is therefore the difference between the heat contents of water in the vapor and the liquid state: AH' = H ,
- W J W ) is the heat input into the solution,
-
- Hi
or
where C,
=
dH = C,dt (10) heat capacity of solution at constant pressure
Application The application of Equations 3, 4, 5, and 10 to the calculation of the 120' F. isotherm on the enthalpy-concentration chart for sodium hydroxide solutions is shown in the example
PI = H , - AH' ( 4) = partial heat content of water in solution, B. t. u./lb.
where H,
=
water heat content of steam at temp. t' and pressure P , B. t. u./lb. steam
The heat quantity appearing on the enthalpy-concentration chart is the relative enthalpy of the solution, H , in B. t. u. per pound of solution, plotted as isotherms vs. concentration. H is referred to arbitrarily chosen conditions of the solvent and the solute. At any given temperature, t', the relative enthalpy of a solution of concentration Wa pounds of solute per pound of solution can be calculated by
where Ha = relative enthalpy of solution at temp. t' and concentration W,, B. t. u./lb. of solution Hb = relative enthalpy of solution at temp. t' and concentration Wb, B. t. u./lb. of solution Equation 5 can be derived as follows: Consider the isothermal evaporation of water from one pound of dilute sohtion. Let W be the number of pounds of solute per pound of solution; consider the initial concentration as W,. If X pounds of. water are removed, the final concentration is: Wb
= Wa/(l -
x)
(6) X BY WEIGHT NaOH
From this, the water removed is:
x
= 1
Writing a material balance,
- Wa/Wb
(7)
FIGURE 2. ENTHALPY-CONCENTRATION CHART FOR SODIUM HYDROXIDE SOLUTIONS Reference conditions: liquid water at 32 F. f o r the solvent, infinite dilution at 68 a F. for the solute I
INDUSTRIAL AND ENGINEERING CHEMISTRY
JUNE, 1939
which follows. Other isotherms are calculated in an identical manner. The enthalpy-concentration chart (Figure 2) shows a comparison for values of H to 50 per cent by weight sodium hydroxide and 160" F. between the calculated values and experimental values referred t o later. CONSTRUCTION OF DUHRINGPLOT. Complete data on the vapor pressures of s6dium hydroxide solutions, as determined by various investigators, are available up to about 60 per cent by weight from 32" to 212" F. (4). A convenient correlation of the original data is a plot of (1' - t ) isotherms as. concentration (Figure 3). The smoothed values of (t' - t ) can be transferred directly to the Duhring plot (Figure 4). The
785
in the divergence between observed and calculated values at higher concentrations in Figure 2. Values of A H , the latent heat of water a t saturation temperature t , and H,,the heat content of superheated steam a t 120" F. and the pressure corresponding to saturation temperature t of water, were read from plots based on the Keenan steam tables (6). Values of the correction, 6, are read from Figure 1 at the 120" F. isotherm and the various values of t. The following table summarizes the calculation of HI a t 120" F. for two concentrations (from Equations 3 and 4) : % NaOH
D 1. F. T. R.
(T'/T)*
c
30 1.0490 95.09 554.78 1.0918 0.0004
% NaOH by Weight
by Weight
50 1.1068 52.77 512.46 1.2796 0.0003
LTk/T)*
++
-HVA H ' H,
30 1.0922 1039.0 1081.8 1113.3 +31.5
50 1.2799 1062.2 1228.3 1114.1 -114.2
CONSTRUCTION OF ENTHALPY-CONCENTRATION CHART. Pure water a t 68" F. was chosen as reference condition for the solute, since reliable experimental heats of dilution down to infinite dilution are available a t that temperature. The reference condition of the solvent is that of the steam tablesnamely, liquid water a t 32" F. As a basis in the application of Equation 5, the.relative enthalpy, Ha, is assumed to be that of a solution of concentration 20 per cent by weight sodium hydroxide a t 120" F. This value for Ha-namely, 73.9 B. t. u. per pound of solution-was obtained by application of Equation 10 to experimental data on heats of dilution (1) between 20 per cent by weight sodium hydroxide and infinite dilution, and experimental heat capacities (2) a t the same concentration. The following table shows the steps required in the calculation of H . The last column gives Bertetti and McCabe's experimental values for H, referred to the same conditions as the calculated values.
FIGURE3. (1' - 1) vs. CONCENTRATION AT CONSTAXT BOILINGTEMPERATURES FOR SODIUM HYDROXIDE SOLUTIONS
heavy line shows the position of the same value of (t' - t ) on the two plots. The correlation should be based on the original data and done with great care. It will be shown later to what extent errors in the Duhring plot affect the calculated values of H. CALCULATION of IT1. For the construction of an isotherm on-the enthalpy-concentration chart, values of HIat one temperature and over the contemplated concentration range are required. Equations 3 and 4 are used for the calculation of R1. Values of t a t various concentrations corresponding to t' = 120 are read from the Duhring plot. The Duhring plot should be large enough so that values of t can be read within 0.1" F. The Duhring lines on Figure 4 are indicated as being straight a t all concentrations. This is undoubtedly incorrect above 30 per cent concentration, but the available data are not exact enough to establish the curvature of the lines. The extent of error introduced by assuming straight lines shows up
0.2 0.3 0.4 0.5
1 0.667 0.500 0.400
0 0.333 0.500
0,600
4-76.0 +31.5 -62.0 -114.2
0 +20.2 +19.0 f10.0
73.9 80.5 109.8 159.7
73.9 80.7 110.1 162.3
FIGURE4. DUHRING PLOTFOR SODIUM HYDROXIDE SOLUTIONS
-
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INDUSTRIAL AND ENGINEERING CHEMISTRY
786
Accuracy
Error in H ,
The ohief source of errors in this method lies in the determination of the Duhring line slopes. The order of magnitude of A H for steam is 1000 B. t. u. From Equation 3 an error of one thousandth in D will introduce an error of about 1B. t. u. in AH’. This error will be numerically transferred to B1.The resultant error in H , the relative enthalpy, is a function of the concentration at which the error occurs. From Equation 5, for an average error of E B. t. u. in AH’ betaween concentrations W4and W,, the error in H is:
w4
Wh
0.1 0.2 0.3
0.2 0.3
0.001 error
0.4 0.5
0.4
in D
1.0 0.50
0.33 0.25
B. t. u.
0.01 error in D 10 5.0
3.3 2.5
These errors become cumulative as Equation 5 is integrated from low to high concentration. This, combined with t h e fact that vapor pressure data are likely t o be less reliable at high than at low concentrations, will tend t o give larger errors at higher concentrations.
Literature Cited
or E (I
- Wb/wa) Be t. U.
The following table shows the order of magnitude of errors in H due to errors of one thousandth and one hundredth in D over a 10 per cent concentration range:
(1) Bertetti, J. W., and McCabe, W. L., IND. ENG.CHEM.,28, 242 (1936). (2) Ibid., 28, 375 (1936). (3) Brown, G.G.,J. Franklin Inst., 219, 406 (1935). (4) Fricke, R., in Landolt-Bornstein physikalisch-chemische Tabellen, Berlin, Julius Springer, 1931, Eg. I1 b, p. 1332; Hsyward and Perman, Ibid., p. 1333. (5) Keenan, J. H.,“Steam Tables”, New York, John Wiley & Sons, 1937. (9) McCabe, mi. L., Truns. Am. Inst. Chem Engrs., 31,129 (1934).
CORRESPONDENCE Study of Liquid Flow SIR: In The Du Pont Magazine [33,No. 3, 3 et seq. (1939)] we find an article by W. T. Collins “Research Employs Plastics.” This publication is primarily devoted to a discussion of the outstanding advantages to be gained by the use of transparent plastics-for example, “Pyra1in”-in the construction of models for hydraulic research work. The possibility of being able to follow visually the flow of a liquid in systems of involved construction should be of great advantage to the engineer engaged in studies of liquid flow. Models such as those described by Collins will permit the detection of extreme cases of turbulence, usually responsible for erosion, cavitation, etc. Frequently the addition of dyes, insoluble in the liquid, will assist in detecting turbulent zones of flow in the system. However, these methods are not sufficiently accurate to permit a study of the type of flow encountered or the detection of turbulence in cases of low rates of flow. These factors, moreover, are not only of importance to the civil engineer, but equally so to the chemical engineer in the construction of stills, rectifiers, piping in general, columns, etc., since maximum efficiency will largely depend on flow characteristics of the liquid. We have found that the phenomenon of birefringence, or stream double refraction of colloidal dispersions consisting of anisometric particles, can be advantageously applied in such studies. It is known that anisometric particles will always tend t o orient with one axis parallel t o their direction of flow, and that
such orientation becomes readily detectable if we place the container through which the liquid flows between crossed Polaroid films or plates and illuminate it with a strong source of diffuse light. Highly diluted water dispersions of natural bentonite have proved especially suited for such work. Other colloidal sols exhibiting stream double refraction-i. e., vanadium pentoxide, ferric oxide, soap solutions, etc.-have also been tested. However, they have certain disadvantages which make their use less attractive. The most important are color, change of the surface tension of the liquid, difficulty of production, and cost. A well prepared one per cent dispersion of bentonite in water obtained by fractionating bentonite in a supercentrifuge and selecting fractions with particles below 50 mp will be practically clear to the eye, have a viscosity and a surface tension close to that of water, and exhibit pronounced birefringence even at extremely low rates of flow for temperatures up to the boiling point of water. A detailed study of liquid flow under different conditions is in progress and will be published later. E. A. HAWSER AND D. R. DEWEY,~ W D MASBACHUSETTS INSTIlUTE
CAMBRIDQE, MABEL Maroh 29, 1939
OF TECENOLOQY
