2.455-mm cells used) and t=0.00282, the absolute turbidity found is 0.3542.

For products which are low in both color and turbidity the graph shown in figure 83 is used. If the scale is such that C and -log T are

0.0030

-log T

00100

FIGURE 83.-Enlarged graph of lower portion of figure 82 for products low in both color and turbidity.

plotted at 50 mm=0.001 unit, it is possible to read C and -log T values accurately to the fourth decimal place.

Values of log T and of R, determined at one thickness, may be converted into corresponding values at another thickness, or -log T into log t. Over a wide range of thickness, the absorbancy is not directly proportional to the depth, b, in the turbid solutions, but is a power function of it, according to the equation

(c) PROCEDURE OF LANDT AND WITTE

Landt and Witte [10], employing a Zeiss-Pulfrich turbidimeter, studied the turbidity in 45-Brix sugar solutions which had been filtered through paper. Only the practical application of their method is given

here.

The procedure in making measurements is similar to that of Sauer [3], as described above, in that the turbid solution is first compared in brightness with an arbitrary glass turbidity standard to give the relative turbidity. From this value is deducted the turbidity value of the water surrounding the sample cell, giving a value called the corrected relative turbidity. This value is divided by the brightness of the calibrated turbidity standard, H, of the instrument relative to the turbidity brightness of the arbitrary standard used, thus giving the value A, or A corr. rel. turb./H. The value A is next multiplied by f(k), which, according to Landt and Witte, is found by the Sauer formula, applying only to the plane parallel cells:

f(k):

kd(√2-1)2.303 10-[1-10-(2-1)]

(94)

Here k is a factor depending upon the extinction coefficient and dis the thickness of layer. For cylindrical beakers, account is taken of the fact that the primary as well as the scattered light suffers a weakening in passage through layers of solution on two sides of the actual Tyndall pencil. These factors are designated, respectively, m and n, and formula 94 becomes for the 26- and 36-mm beakers:

f(k):

=

kd(√2-1)2.303

10-k(d+m+n) [1-10—kd (√2−1)]*

(95)

Values of k and ƒ (k), as calculated for various sizes of plane parallel cells are given in table 42, and for the 26-mm and 36-mm beakers, in table 43.

TABLE 42.-Values of k and f(k) for plane parallel cells of various thicknesses (Landt and Witte)

TABLE 43.—Values of k and f(k) for cylindrical beakers 26 mm and 36 mm in diameter (Landt and Witte)

Next, in order to obtain strict comparison with other plane parallel cells or for calculation to absolute turbidity Af(k) must be multiplied by the factor, D, which is the ratio of the layer thickness of the calibrated standard to the layer thickness of the plane parallel cell. Finally, to reduce the results to terms of absolute turbidity, the whole is multiplied by t, the turbidity of the calibrated standard. The formula is therefore:

Absolute turbidity, T=Af(k)Dt.

3. REFERENCES

[1] R. T. Balch, Ind. Eng. Chem., Anal. Ed. 3, 124 (1931).

[2] H. Sauer, Z. Instrumentenk. 51, 408 (1931).

[3] H. Sauer, Z. tech. Physik. 12, 148 (1931).

[4] F. W. Zerban and L. Sattler, Ind. Eng. Chem., Anal. Ed. 3, 326 (1931). [5] F. W. Zerban and L. Sattler, Ind. Eng. Chem., Anal. Ed. 7, 157 (1935). [6] F. W. Zerban and L. Sattler, Ind. Eng., Chem., Anal. Ed. 8, 168 (1936). [7] F. W. Zerban and L. Sattler, Ind. Eng. Chem., Anal. Ed. 9, 229 (1937). [8] F. W. Zerban and L. Sattler, Ind. Eng. Chem., Anal. Ed. 10, 9 (1939). [9] F. W. Zerban, L. Sattler and I. Lorge, Ind. Eng. Chem., Anal. Ed. 6, 178 (1934).

[10] E. Landt and H. Witte, Z. Ver. deut. Zucker-Ind. 84, 450 (1934). [11] Zeiss Nephelometer, Advertising booklet (Carl Zeiss, Inc., 485 Fifth Ave., New York, N. Y.)

XXI. VISCOSITY OF SUGAR SOLUTIONS

1. THEORETICAL

Viscosity is the property of homogeneous fluids that causes them to offer resistance to flow. It is expressed mathematically by the constant of proportionality between shearing stress and rate of shear. It has the dimensional formula ML-1T-1, and is generally expressed as 7. In the cgs system, the unit of viscosity is the poise. The onehundredth part of this unit (the centipoise) is frequently used in practice. The viscosity of water at 20° C is often taken as 1.005 centipoises. The ratio of viscosity to density is called the kinematic viscosity, and the cgs unit is called the stoke, poises/(g/cm3). The reciprocal of viscosity is fluidity. The cgs unit of fluidity is the rhe. An extensive investigation of the flow in capillary tubes was first undertaken by Poiseuille about 1838. He found that the rate of discharge was directly proportional to the first power of the pressure difference and to the fourth power of the diameter and inversely proportional to the length of the capillary tubes. This relationship, which is known as Poiseuille's law, expressed mathematically, is

7=Viscosity, in poises,

V=cm3 discharge in t (seconds),

Ap difference in pressure between the two ends of the tube (dynes/cubic centimeter),

L-length of the tube in centimeters,

R=radius of the tube in centimeters.

This formula may be derived, assuming a capillary tube of uniform diameter and sufficiently long that kinetic energy and end effects are negligible. Consider a cylindrical volume element of the fluid of length dL, radius r, and with the difference in pressure dp between the ends. The resultant force tending to push this volume element downstream is ridp. This force is resisted by the shearing stresses due to viscosity, assuming the fluid adheres to the walls of the tube. Let S denote the shearing stress at radius r, i. e., the tangential force per unit area exerted upon the cylindrical surface, 2ardL, by the fluid between it and the wall of the tube. The total shearing force acting upstream is 27rSdL. Under steady flow conditions (no acceleration), these two forces must be equal, so that

2πr SdL= r2dp.

When conditions are uniform throughout the length of the tube,

(97)

Let v denote the velocity at radius r. The velocity gradient with respect to increasing values of r is -dv/dr, where dv is the difference in velocities at the radial distances r and r+dr from the axis of the tube. The definition of viscosity, n, expressed mathematically, is

The volume of fluid flowing per unit time is r2dv, and, integrating over the entire tube, using eq 100, gives

which is the usual form of Poiseuille's law. It also may be written

where C R/8LV is a constant for a given capillary viscometer. Where the liquid of density, p, flows by gravity with an effective

hydrostatic head, h, so that Ap=phg, eq 102 gives for the ratio of viscosities of two liquids

(103)

Equations 102 and 103 have been accurately verified experimentally for long capillaries with small rates of flow. The results of measurements over a convenient range of rates of flow with many types of capillary viscometers have been found [1,2, 3] to be represented, within experimental errors, by the relation

where A and B are instrumental constants for a particular viscometer and direction of flow. The constant B has been found to be approximately equal to V/8TL (using cgs units in eq 104), but it should be determined by experiment.

2. CAPILLARY-TUBE VISCOMETERS

(a) OSTWALD VISCOMETER

The Ostwald viscometer (fig. 84-I) is one of the earliest forms. It consists of a glass U-shaped tube, one limb of which contains a smaller bulb discharging into a capillary tube, while the other contains a tube of larger diameter with a larger bulb near the bottom. There are no standard dimensions for this instrument, and since the hydrostatic head causing flow cannot be varied, any considerable change in viscosities must usually be covered by the use of a series of instruments with capillaries of different sizes. With the usual type of Ostwald viscometer, a certain volume of liquid is introduced into the wider limb, frequently by means of a pipette, and drawn up through the capillary to a mark above the smaller bulb. In making a measurement with all Ostwald instruments, the liquid is forced through the capillary tube to a mark above the upper bulb, and the time required for the meniscus to fall by gravity from the upper mark, C, to another mark, D, below the bulb, is measured.

(b) BINGHAM VISCOMETER

The Bingham viscometer (fig. 84-II) is a refinement of the Ostwald capillary tube type. An advantage is the small sample required, 4 ml being a common capacity of the bulb, C, which is emptied or filled during a measured time interval. There are no standard dimensions, the range of viscosities to be measured determining the size of capillary chosen. Any one instrument can be used to measure a wide range of viscosities, since various pressures may be used. The average hydrostatic head may be made negligible, the liquid being forced through the capillary by air pressure, which is kept as constant as possible during a measurement. A trap makes it possible to take readings at increasing temperatures without refilling. The working volume of liquid is from A to H or from E to M. Drainage errors may be avoided by forcing the liquid through the capillary into a dry bulb, in which case the time required for the meniscus to pass from D to B is measured.