  # The Definition of Angle of Repose

Many empirical tests are used to assess powder fluidity, including measurement of Hausner ratio, angle of rest (Fig. 12.8) with a larger angle indicating a more cohesive powder, Carr indices (Carr, 1965), flow through a defined aperture, and pressure tests. These tests are not useful in the design of hoppers and silos for reliable flow. However, they can be useful in quality control, where a change in a measured value may indicate a change in the flow behavior of a particular material. The angle of rest increases with moisture content. Mohsenin credits Fowler and Wyatt with developing an equation for calculating the angle of rest of wheat, sand, canaryseed, and other solids: there are many methods for measuring the angle of rest, and each gives slightly different results. The results also depend on the exact methodology of the experimenter. As a result, data from different laboratories are not always comparable. One method is the triaxial shear test, another is the direct shear test. The angle of rest plays a role in several topics of technology and science, including: Here is a list of the different materials and their angle of rest.  All measurements are approximate.

The angle of rest is sometimes used in the design of equipment for the treatment of particulate solids. For example, it can be used to design a suitable hopper or silo to store the material or to size a conveyor belt for transporting the material. It can also be used to determine if a slope (e.g. an uncompacted pile or gravel bank) is likely to collapse or not; The slope of the slope is derived from the angle of rest and represents the steepest slope that a pile of granular material will take. This angle of rest is also crucial for the correct calculation of stability in ships. If the static coefficient of friction of a material is known, the following function can be used to make a good approximation of the angle of rest. This feature is accurate enough for stacks, where the individual objects in the stack are tiny and stacked in a random order.  The angle of rest is another important physical property used to characterize the majority of particulate foods such as seeds, cereals, flours, groats and fruits.

When granular solids are stacked on a flat surface, the sides of the pile are at a certain reproducible angle with the surface horizontally leveled (Fig. 3.8). This angle is called the resting angle of the material. The angle of rest is important for the design of particulate matter treatment, storage and transport systems. If the grains are smooth and rounded, the angle of rest is small. With very thin and sticky materials, the angle of rest is high. Low angle of rest materials are very fluid and can be transported with gravitational force or low energy. Tausig possessed this calmness in a technical way, and his touch was wonderful; But he never brought tears to your eyes.

The angle of rest (Fig. 7) is a parameter commonly used to evaluate the interparticle force.16 The simplest method for determining the angle of rest is the casting angle. A funnel with a wide outlet is placed at a distance of 10 cm above the bench, where a piece of paper is placed just below the funnel. Powder is added while the funnel is closed. The content circulates and accumulates on paper. The diameter of the cone (D) and the two opposite sides (l1 + l2) is measured with rulers. The angle of rest (θ) is calculated from the arc equation cos [D/(l1 + l2)]. The relationship between flow properties and resting angle was established. If the angle of rest is less than 25 degrees, the river is called excellent; On the other hand, if the angle of rest is greater than 40 degrees, the flow is considered low. where Φr is the angle of rest, n is the form factor based on the specific surface, M is the percentage of moisture, Dav is the average diameter of the sieve particles, sg is the density, and a, b, c, d are constants. The orders of magnitude of the coefficients were determined as follows: a = 0.4621, b = 0.0342, c = −0.0898 and d = 0.0978, with an amplitude of the correlation coefficient R = 0.97. These coefficients can be used to calculate the resting angle sizes of different solids.

The angle of rest has long been used to characterize bulk solids.28–30 The angle of rest is a characteristic that refers to the friction of particles or resistance to movement between particles. According to USP, this is the constant three-dimensional angle (relative to the horizontal base) assumed by a pile of cone-shaped material formed by one of the different methods. Due to the high dependence of resting angle measurements on test conditions, the resting angle is not a very robust way to quantify powder flow.31 This method is suitable for fine-grained, non-cohesive materials with individual particle sizes of less than 10 mm. The material is placed in a box with a transparent side to observe the granular test material. It should initially be level and parallel to the bottom of the box. The box is slowly tilted until the material begins to slide in large quantities and the tilt angle is measured. Dry nanoparticle powders are characterized by a knock density test and a resting angle test, which provides a measure of the fluidity of dry powders and is important for powder inhaler formulations. Tapped and untapped (bulk) densities are determined by delineating a small bowl with known volumes, then introducing a small mass of powder into the bowl (bulk density) and tapping it 50 times vertically against a padded table top (knocking density). The mass is divided by the initial and final volumes. These values are also used to determine the Hausner ratio (typed density/bulk density) and Carr index (Ci) (typed density-bulk density)/exploited density × 100%) for each sample.