A - top area of slice [m²]
B - diameter (span) of opening at bottom of hopper [m]
B_crit - critical diameter for flow from hopper [m]
C - constant of integration [-]
E_WS - material-dependent constant in Warren-Spring equation 31 [kPa]
F_WS - material-dependent constant in Warren-Spring equation 31 [kPa]
F - material-dependent constant [-]
G - material-dependent constant [-]
H - function of that replaces m [-]
K - Janssen constant (varies for bin, hopper, author) [-]
K_WS - material-dependent constant in Warren-Spring equation 32 [-]
L_WS - material-dependent constant in Warren-Spring equation 32 [kPa]
M - material-dependent constant [-]
N - material-dependent constant [-]
P - perimeter of slice [m]
R - ratio of A to P [m]

- angle from vertical to hopper wall [rad]
- void fraction of powder bed [-]
- specific bulk gravity [kg/(m²s²]
- arctan(1/ff) [rad]
- coefficient indicating active or passive state [-]
- 3.14159... [-]
- internal friction angle [rad]
_W - powder-wall friction angle [rad]
- angle from horizontal to the normal of a slice [rad]
_B - bulk density [kg/m³]
_S - skeletal (solid) density [kg/m³]
₁ - principal stress in active state [N]
_1,a - principal stress where arch meets wall [N]
₂ - principal stress in passive state [N]
_n - normal stress [N]
_t - tangental stress [N]
_y - vertical stress at y from top of powder bed [N]
_{y, h} - hydraulic stress at y from top of powder bed [N]
= arcsin(sin _W / sin ) [rad]

References

Drescher, A., A.J. Waters, and C.A. Rhoades - "Arching in hoppers: I. Arching theories and bulk material flow properties and II. Arching theories and critical outlet size", Powder Technology 84: 165-183 (1995)

Enstad, G. - "On the theory of arching in mass-flow hoppers", Chem. Eng. Sci. 30: 1273-1283 (1975)

Enstad, G. - "A note on the stresses and dome formation in axially symmetric mass flow hoppers", Chem. Eng. Sci. 32: 339-342 (1977)

Janssen, H.A. - "Versuche uber Getreidedruck in Silozellen", Zeit. Ver. Deutsch. Ing. 39: 1045-1049 (1895)

Jenike, A.W. - "Gravity flow of bulk solids", Utah Univ. Eng. Exp. Stn., Bull. 108 (1961)

Jenike, A.W. - "Storage and flow of solids", Utah Univ. Eng. Exp. Stn., Bull. 123 (1964)

Nedderman, R.M. - Statics and kinematics of granular materials, (Cambridge University Press, 1992)

Shamlou, P.A. - Handling of bulk solids - Theory and practice (Butterworths, London, 1990) Chapter 2, pp 20-50, and Chapter 3, pp 51-79

Walker, D.M. - "An approximate theory for pressures and arching in hoppers", Chem. Eng. Sci. 28: pp 975-997 (1966)