Problems with the storage of bulk solids in bins and silos can be avoided if they are designed with respect to the flow properties of the bulk solid which has to be stored. The following essay considers the basic rules for the design of silos.
1 Stresses in silos
Figure 1 shows silos and the pressures and stresses, respectively, acting in the silos. While the pressure (for fluids we will use the word pressure") would increase linearly downwards if the silo would have been filled with a fluid (a), the course of the vertical stress (for bulk solids we will use the word stress") in a silo filled with a bulk solid is rather different (b,c): In the latter case in the vertical (cylindrical) section of the silo the vertical stress increases in a degressive way. If the height to diameter ratio of the silo is sufficiently large (usually: > 3), a constant vertical stress is attained. This means that the vertical stress will not increase further even if the filling height is much larger. The reason for this course are the shear stresses acting between the bulk solid and the silo walls even if the bulk solid is at rest. Due to the shear stresses, the silo walls carry a part of the weight of the bulk solid. A method for the calculation of the stress course in the vertical section was derived by Janssen already in 1895 . This method is the basis for most present standards for the calculation of the load on silo walls for structural silo design [2-4].
Figure 1: a. pressure in a silo filled with a fluid (imaginary); b. vertical stress after filling the silo with a bulk solid; c. vertical stress after the discharge of some bulk solid
The stresses acting in a hopper are different from those in the vertical section. Just after filling an empty silo, the so called filling stress state (also: active stress state, figure 1b) prevails, where the vertical stress in the hopper decreases less in the upper part of the hopper and then more near the imaginary hopper apex. As soon as some bulk solid is discharged for the first time after filling, the stresses in the hopper change and the so-called emptying stress state (also : passive stress state) prevails, figure 1c. When flowing downwards in the hopper, the bulk solid is compressed in the horizontal direction so that the walls of the hopper carry a larger part of the weight of the bulk solid and, hence, the vertical stress in the lower part of the hopper is clearly smaller than after filling. In the emptying stress state the vertical stresses in the lower part of the hopper are nearly proportional to the distance to the imaginary hopper tip or, in other words, the stresses are proportional to the local hopper diameter. This linear course of stress is called the radial stress field . In principle, in the vertical section of the silo the stresses remain unchanged at discharge.
2 Flow Profiles: Mass Flow and Funnel Flow
Two different modes of flow can be observed if a bulk solid is discharged from a silo: mass flow and funnel flow (figure 2a). In case of mass flow, the whole contents of the silo are in motion at discharge. Mass flow is only possible, if the hopper walls are sufficiently steep and/or smooth, and the bulk solid is discharged across the whole outlet opening. If a hopper wall is too flat or too rough, funnel flow will appear. In case of funnel flow (figure 2b), only that bulk solid is in motion first, which is placed in the area more or less above the outlet. The bulk solid adjacent to the hopper walls remains at rest and is called dead" or stagnant" zone. This bulk solid can be discharged only when the silo is emptied completely. The dead zones can reach the surface of the bulk solid filling so that funnel flow becomes obviously when observing the surface. It is possible as well that the dead zones are located only in the lower part of the silo so that funnel flow cannot be recognised by observing the surface of the silo filling.
Figure 2a: Mass flow
Figure 2b: Funnel flow
3 Flow Problems
Typical problems which occur at the storage of bulk solids are:
Figure 3a: Arching
Figure 3b: Ratholing
Figure 3c: Segregation
In a funnel flow silo, all problems mentioned above can occur generally, while in case of mass flow only arching has to be considered: segregation, ratholing, irregular flow and flooding of the bulk solid do not appear in a well designed mass flow silo. The residence time distribution of a mass flow silo is narrow, because it acts as a first in - first out" system (see figure 2a).
Two steps are necessary for the design of mass flow silos: The calculation of the required hopper slope which ensures mass flow, and the determination of the minimum outlet size to prevent arching.
4 Silo design
The flow behaviour of a bulk solid is defined by several well-defined parameters [2,5-8,21]. In general, these are the bulk density rb, the effective angle of internal friction je (a measure for the internal friction of the bulk solid at stationary flow), the unconfined yield strength sc, and the wall friction angle jx. For mass flow design, the wall friction angle jx is the most important parameter, whereby the unconfined yield strength sc is the most important parameter regarding arching. The wall friction angle jx is defined as the friction angle between the surface of the silo wall and the corresponding bulk solid. The unconfined yield strength sc is the compressive strength of a bulk solid. It has to be taken into account that all these parameters are dependent on the stress level, represented by the consolidation stress s1 [2,5-8,21].
The parameters mentioned are measured in dependency on the consolidation stress with shear testers [2,5-8,21], e.g. with the Jenike shear tester or a ring shear tester. The hopper slope required for mass flow and the minimum outlet size to prevent arching can be calculated with the measured values using Jenikes' theory . This method showed its validity in many cases in more than 35 years.
The borders between funnel and mass flow, which result from the calculations of Jenike , are shown in figure 4a for the wedge shaped hopper and in figure 4b for the conical hopper. In the diagrams the wall friction angle jx is drawn over the hopper slope angle Q measured against the vertical. The effective angle of internal friction je, which is a measure of the internal friction of the bulk solid, is the parameter of the mass flow/funnel flow borderlines. The borderlines separate all pairs of values leading to mass flow from those leading to funnel flow.
Figure 4a: Design diagram for mass flow (wedge-shaped hopper)
Figure 4b: Design diagram for mass flow (conical hopper)
Conditions which lie within the borderline yield mass flow whereas funnel flow is present in case of conditions outside of the borderline. If the wall friction angle jx and the effective angle of internal friction je are known (measured with a shear tester, e.g. with the ring shear tester), the maximum slope angle Q of the hopper wall against the vertical which ensures mass flow can be determined with this diagram. The courses of the borderlines indicate, that the larger the wall friction angle jx is, the steeper (smaller Q) the hopper has to be for mass flow. The wedge shaped hopper allows a somewhat (often 8° to 10°) larger slope angle Q against the vertical with the same material properties. That means that the walls of a wedge shaped mass flow hopper can be flatter than the walls of a conical mass flow hopper [7,12].
When bulk solid is discharged from a mass flow silo, the radial stress field prevails in the hopper as already described in section 1 (see figure 1c). In the hopper (at least beneath a sufficiently large distance from the vertical section) the major principal stress s1 is proportional to the local hopper diameter (figure 5). It decreases to zero towards the imaginary hopper apex. The stress s1 acts as a consolidation stress thus determining the properties of the bulk solid, e.g. the bulk density rb and the unconfined yield strength sc. The unconfined yield strength sc of a bulk solid can be measured for each major principal stress (consolidation stress) s1 (see ). The function sc =f(s1) (figure 6) is called the flow function. Usually, the unconfined yield strength increases with the consolidation stress. If the flow function has been measured, the unconfined yield strength sc can be drawn in figure 5 at each position of the hopper.
Figure 5: Stress conditions in the hopper (emptying)
s1' is the bearing stress acting where an imaginary stable arch of bulk solid is carried by the hopper walls. s1' is proportional to the local hopper diameter such as s1. An arch can only be stable in that are of the hopper where the unconfined yield strength sc is larger than the bearing stress s1' . This is the case beneath the point of intersection of the sc curve with the s1' curve. Above that intersection the unconfined yield strength is smaller than the bearing stress of the arch. In this case, the unconfined yield strength is not large enough to support an arch, i.e. an arch would not be stable at this position. The point of intersection indicates the lowest possible position in the hopper (height h*, figure 5) for an outlet opening large enough to avoid arching. The diameter of this minimum outlet opening is called dcrit. If a smaller outlet opening would be chosen, h* indicates up to where flow promoting devices have to be installed beginning at the outlet.
Figure 6: Flow function and time flow function
Some bulk solids tend to consolidate increasingly with the period of storage at rest (time consolidation [8,21]). It can be found a time flow function sct = f(s1) (figure 6) for each storage time analogously to the flow function. If the time flow function would be drawn in figure 5 then this would yield to a point of intersection of the s1'-curve and the sct-curve, which would be above the already determined point of intersection of the s1c- and s1'-curves. This means that larger outlets are required to prevent arching with increasing storage time at rest.
For practical silo design, equations or diagrams derived by Jenike  are used to determine the stresses s1 and s1' in dependence on the flow properties measured (je, jx, rb) and the silo geometry (Q). With this means the minimum outlet sizes of conical as well as wedge-shaped hoppers can be calculated. Furthermore, the minimum outlet sizes for avoidung ratholing at funnel flow can be determined .
5 Choice of the hopper geometry
Figure 7 [7,9,12] shows some opportunities to design mass flow silos. The calculations of Jenike (see design diagrams, figure 4) refer to conical hoppers (a) and wedge shaped hoppers (b). In case of these basic hopper forms, the maximum slope angles of the walls to achieve mass flow (Qax in case of a conical, Qeb in case of a wedge-shaped hopper) and the outlet dimensions (d, b) to prevent arching can be determined. In case of the wedge shaped hopper it is assumed, that the influence of the vertical end walls can be neglected if the length of the outlet L is at least three times the width b.
Figure 7: Hopper forms 
The variants c and d are advantageous as well to ensure mass flow if the maximum slope angles as indicated in the figure are not exceeded. The pyramidal hopper geometry (e) is disadvantageous because the bulk solid has to flow from the top in the edges of the hopper and in the edges to the outlet. Thus, the bulk solid has to overcome wall friction at two sides what supports the formation of dead zones. If mass flow has to be achieved with such a hopper geometry, the edges have to be rounded on the inside, and the maximum slope angle against the vertical of the edges must not exceed Qax. Because the walls of a pyramidal hopper are always steeper than the adjacent edges, a pyramidal mass flow hopper is steeper than a conical hopper for a specific bulk solid. Variant f is just a transition from a cylindrical section to a square outlet. In this case, the slope of the hopper walls against the vertical must not exceed Qax at any position.
In order to achieve mass flow, variants e and f must have the steepest walls. The conical hopper (a) can be designed more shallow, and the largest slope angles measured against the vertical can be achieved with geometry b,c or d (wedge-shaped hoppers).
In some industries non-symmetrical silos are preferred (e.g. pyramidal hoppers with differently sloped walls). From the view of mass flow design, there is no reason to build such silos. If mass flow has to be achieved, symmetrical hoppers usually require the lowest height for the transition from the silo cross-section to the outlet cross-section to achieve mass flow .
6 Application of the results on the design of silos
In section 4 the silo design procedure due to the theory of Jenike  was described in a shortened way. Further details and information can be given besides the determination of the hopper slope for mass flow and the size of the outlet to prevent arching. Some examples are listed shortly in the following (further examples of silo design: [17,18,22,23]):
The design of silos in order to obtain reliable flow is possible on the basis of measured material properties and calculation methods. Because badly designed silos can yield operational problems and a decrease of the product quality, the geometry of silos should be determined always on the basis of the material properties. The expenses for testing and silo design are small compared to the costs of loss of production, quality problems and retrofits.
 Janssen, H.A.: Ztg. Ver. dt. Ing. 39 (1895), pp. 1045-1049 (in German)
 Martens, P. (Hrsg): Silohandbuch, Verlag Ernst & Sohn, Berlin (1988) (in German)
 DIN 1055 part 6: German code: Lasten in Silozellen, Mai 1987 (available in English)
 Schwedes, J.: Chem.-Ing.-Tech. 56 (1984) 4, pp. 291-298 (in German)
 Schwedes, J.: Fließverhalten von Schüttgütern in Bunkern, Verlag Chemie, Weinheim (1968) (in German)
 Schwedes, J.: Chem.-Ing.-Tech. 48 (1976) 4, pp. 294-300 (in German)
 Jenike, A.W.: Storage and Flow of Solids, Bull. No. 123, Engng. Exp. Station, Univ. Utah, Salt Lake City (1970)
 Schulze, D.: Grundlagen der Schüttgutmechanik, ZDS-Seminar Pulvertechnologie, 1.-3. Sept. 1993 (Zentralfachschule der Deutschen Süßwarenwirtschaft) (in German)
 Woodcock, C.R.; Mason, J.S.: Bulk Solids Handling, Leonard Hill, Glasgow and London (1987)
 Schulze, D.; Schwedes, J.: Storage and Flow of Bulk Solids in Silos and Information for Planning new Installations, VGB-Kraftwerkstechnik (English Issue) 70 (1990) 9, pp. 665-669
 Reisner, W.; v.Eisenhardt-Rothe, M.: Silos und Bunker für die Schüttgutspeicherung, Trans Tech Publications, Clausthal-Zellerfeld (1971) (in German)
 Schulze, D.: Storage, Feeding, Proportioning, in: Chulia, D., Deleuil, M., Pourcelot, Y (Editors): Powder Technology and Pharmaceutical Processes, Handbook of Powder Technology Vol. 9, Elsevier Amsterdam - London - New York - Tokyo 1994, pp. 285-317 (ISBN 0-444-81533-3)
 Schulze, D.: Flowability and Time Consolidation Measurements Using a Ring Shear Tester, Powder Handling & Processing 8 (1996) 3, pp. 221-226
 Schulze, D.; Schwedes, J.: bulk solids handling 12 (1992) 1, pp. 33-39
 Schulze, D.: Silos - Design Variants and Special Types, Bulk Solids Handling 16 (1996) 2, pp. 225-232
 Johanson, J.R.: Bulk Solids Handling 2 (1983) 3, pp. 495-498
 Schulze, D.: Feeders and Flow Promoting Devices, in: Silos - Fundamentals of Theory, Behaviour and Design, edited by C.J. Brown and J. Nielsen, publ. by E & FN Spon, London and New York (1998), pp. 200-220 (ISBN 0-419-21580-8)
 Schwedes, J.; Schulze, D.: Bulk Solids Handling 11 (1991) 1, pp. 47-52
 Schäfer, R.; Schröer, H.; Schwedes, J.: Zement Kalk Gips 75 (1986) 11, pp. 587-592 (in German)
 Schulze, D.; Schwedes, J.: Chem.-Ing.-Techn. 63 (1991) 3, pp. 256-257 (in German)
 Schulze, D.: Flow Behaviour of Powders and Bulk Solids (Fundamentals), https://www.dietmar-schulze.de/info1e.html (click here)
 Schulze, D.; Schwedes, J.; Leonhardt, D.; Kossert, J.: Design of a Silo for the Storage of 10.000 t Sulphur Granules, Bulk Solids Handling 18 (1998) 2, pp. 211-217
 Kwade, A.; Schulze, D.: Proper Silo Design for Food Products - Today Strategies for Avoiding Segregation, Degradation and Hang-ups, PARTEC 98, Preprints "1st Europ. Symp. Process Technology in Pharmaceutical and Nutritional Sciences", 10.-12.3.98, Nuremberg, pp. 157-166