Granular Discharge Rate (Beverloo)
Calculate the mass discharge rate of a granular material through a bottom orifice by the Beverloo equation, W = C·ρ·√g·(D₀ − k·d)^2.5, from the discharge coefficient C (~0.58), the bulk density ρ (kg/m³), the orifice diameter D₀ (m), the particle diameter d (m) and the shape factor k (~1.4). The empirical Beverloo equation describes a fascinating behavior distinct from liquids: the grain discharge rate through an orifice does NOT depend on the product height above it (unlike a liquid, whose flow grows with head). This is due to the Janssen arching effect — bottom pressure saturates, so flow depends essentially on orifice size, not the amount of product above. That is why an hourglass keeps time steadily: sand flows at the same rate whether the top bulb is full or nearly empty. Flow is proportional to (D₀ − k·d)^2.5 — note the 2.5 exponent (not 2, of area) and the k·d term, an effective 'empty annulus' near the orifice edge where grains do not pass. Beverloo is fundamental in designing silos, hoppers, feeders and dosers in grain, cement, pharmaceutical and mining industries. Enter the coefficient, density, orifice diameter, particle diameter and shape factor.
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Vazão de descarga granular (Beverloo)
A equação de Beverloo, W = C·ρ·√g·(D₀ − k·d)^2,5, calcula a vazão mássica de descarga de um material granular por um orifício de fundo, e descreve um comportamento fascinante e distinto dos líquidos: a vazão de descarga de grãos por um orifício não depende da altura de produto acima dele. Diferente de um líquido, cuja vazão por um furo cresce com a carga hidráulica (Torricelli), a vazão de grãos é constante, governada essencialmente pelo tamanho do orifício. A causa é o efeito de arqueamento de Janssen: como a pressão no fundo satura, a quantidade de produto acima é irrelevante para a vazão. É exatamente por isso que uma ampulheta marca o tempo de forma constante — a areia escoa à mesma taxa esteja o bulbo superior cheio ou quase vazio. A vazão é proporcional a (D₀ − k·d)^2,5: repare no expoente 2,5 (e não 2, que seria o da área do orifício) e no termo k·d, que representa uma 'coroa vazia' efetiva (empty annulus) junto à borda do orifício, por onde os grãos não conseguem passar — daí o orifício 'útil' ser um pouco menor que o real. Beverloo é fundamental no projeto de silos, moegas, dosadores e alimentadores nas indústrias de grãos, cimento, farmacêutica e de mineração, onde controlar a taxa de descarga de sólidos a granel é essencial. Informe o coeficiente de descarga, a densidade aparente, o diâmetro do orifício, o diâmetro das partículas e o fator de forma.
Related Tools
Silo Emptying Time
Calculate the time to empty a silo by gravity discharge, t = M ÷ W, from the stored product mass M (kg) and the mass discharge rate W (kg/s). Since the discharge rate of a granular material through an orifice is practically CONSTANT (independent of the product height above, by the Janssen effect and per the Beverloo equation), the emptying time is simply total mass divided by rate — a direct relation, unlike a liquid's emptying, which slows as the level falls. This time is an important operational parameter in silo, hopper and storage-unit design and operation: it sets the dispatch capacity (how fast a truck, rail car or ship is loaded), sizes the downstream conveying systems (belts, bucket elevators, screws) that must match the discharge rate, and frames shift logistics and vehicle queues at grain terminals. The discharge rate W can be estimated by the Beverloo equation from the outlet diameter, closing the calculation: larger outlets discharge faster (W ∝ D₀^2.5), reducing emptying time. Enter the stored mass and the discharge rate.
Silo Asymptotic Pressure
Calculate the asymptotic (saturation) vertical pressure of a deep silo, p_∞ = (γ·D) ÷ (4·μ·K), from the product unit weight γ, diameter D, product-wall friction coefficient μ and lateral pressure ratio K. This is the LIMIT value the Janssen vertical pressure tends to at great depth — the maximum bottom pressure a silo can reach, however tall the stored product. It is Janssen's most striking result: while in a liquid pressure would grow without limit with height (p = γ·h), in granular product WALL FRICTION absorbs all added weight beyond a certain depth, making bottom pressure SATURATE. So a 30 m silo of grain may have a bottom pressure equal to only a few metres of product column. This asymptotic pressure is fundamental in design: it sets the maximum bottom and wall load the structure must bear, regardless of height, and explains why silos can be built slender and tall with relatively modest foundations. Note it is proportional to diameter and inversely proportional to friction — wide, smooth-walled silos generate higher pressures. Enter the unit weight, diameter, friction coefficient and lateral pressure ratio.
Silo Discharge Overpressure
Calculate the DISCHARGE horizontal pressure in a silo, p_d = C_d · p_h, from the discharge overpressure coefficient C_d and the static horizontal pressure p_h (from Janssen for the full silo at rest). One of the most important phenomena — historically responsible for many silo failures — is that wall pressures during DISCHARGE are SIGNIFICANTLY HIGHER than static pressures with the silo merely full. When the product starts flowing toward the outlet, flow zones and dynamic arches form, and stress redistribution generates pressure peaks (overpressures) on the wall, especially at the transition from the cylindrical body to the hopper. The overpressure coefficient C_d (typically 1.3-2.0 or more, per the code, flow type — mass or funnel — and geometry) amplifies the static pressure to cover these dynamic peaks. Silo design codes (such as EN 1991-4 / Eurocode and ANSI) prescribe these factors precisely because designing a silo only for static loads, ignoring discharge overpressure, is a classic cause of structural collapse. Enter the overpressure coefficient and the static horizontal pressure.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.