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Archimedes number

The Archimedes number (Ar) is a dimensionless quantity in fluid mechanics that quantifies the relative influence of buoyancy forces arising from density differences to viscous forces in fluid flows.^[1] It is defined by the formula
\mathrm{Ar} = \frac{g l^3 \Delta \rho}{\nu^2 \rho},
where g is the acceleration due to gravity, l is a characteristic length scale, \Delta \rho is the density difference between the fluid and the perturbing body or phase, \nu is the kinematic viscosity of the fluid, and \rho is the fluid density.^[1] An equivalent form for particulate systems is
\mathrm{Ar} = \frac{g d_p^3 \rho_f (\rho_p - \rho_f)}{\mu^2},
with d_p as particle diameter, \rho_f and \rho_p as fluid and particle densities, respectively, and \mu as dynamic viscosity.^[2] Physically, the Archimedes number represents the balance between gravitational (buoyant) forces driving motion due to density gradients and the resisting viscous friction within the fluid.^[3] When Ar ≪ 1, viscous forces dominate, leading to laminar or creeping flows where buoyancy has minimal effect, such as in highly viscous media or small-scale systems.^[4] Conversely, when Ar ≫ 1, buoyancy forces prevail, promoting inertial motion, particle settling, or the onset of turbulence in free convection scenarios.^[1] A critical value of Ar often marks transitions, such as from stable laminar to unstable turbulent free-convective flows.^[1] The Archimedes number finds broad applications in analyzing natural and mixed convection, where it parameterizes the competition between free (buoyancy-driven) and forced convection to predict flow regimes and temperature profiles.^[4] In multiphase flows, it aids in modeling particle re-suspension, settling velocities, and minimum stirring requirements, with low Ar values (e.g., Ar < 40) indicating conditions where particles remain suspended only under sufficient agitation.^[2] It is also essential in chemical engineering for designing fluidized and spouted beds, determining minimum fluidization velocities, and classifying particle behavior in gas-solid systems based on density-driven dynamics.^[3]

Definition and Formulation

Definition

The Archimedes number (Ar) is a dimensionless parameter in fluid mechanics that quantifies the influence of density differences on fluid motion within viscous flows. It serves as a measure of the relative strength of buoyancy forces arising from gravitational effects compared to viscous forces resisting the motion.^[2] This parameter is particularly relevant in scenarios where density variations drive flow behavior, such as in systems involving suspended particles or temperature-induced density gradients.^[3] Named after Archimedes of Syracuse (c. 287–212 BC), the ancient Greek mathematician and physicist renowned for his foundational work on buoyancy principles,^[5] it finds application in general contexts like multiphase flows, where interactions between fluids and dispersed phases are governed by buoyancy-viscous balances, and natural convection, where density differences induce fluid circulation without external forcing.^[2]^[3]

Mathematical Expression

The Archimedes number Ar is defined by the formula

Ar = \frac{g L^3 (\rho - \rho_l)}{\nu^2 \rho_l},

where g is the gravitational acceleration, L is the characteristic length, \rho is the density of the denser phase, \rho_l is the density of the reference fluid, and \nu is the kinematic viscosity of the fluid.^[6]^[7] An equivalent expression uses the dynamic viscosity \mu of the fluid:

Ar = \frac{g L^3 \rho_l (\rho - \rho_l)}{\mu^2}.

^[6] This form arises from substituting \mu = \nu \rho_l.^[7] In this formulation, the characteristic length L is typically selected as the particle diameter in particulate suspensions or the system height in natural convection scenarios, while the density difference \rho - \rho_l represents the buoyancy-driving contrast between phases.^[6]^[7] To verify its dimensionless nature, consider dimensional analysis: the numerator g L^3 (\rho - \rho_l) has dimensions [L T^{-2}] [L^3] [M L^{-3}] = [M L T^{-2}], while the denominator \nu^2 \rho_l has ([L^2 T^{-1}]^2) [M L^{-3}] = [M L T^{-2}], yielding a ratio of 1 (dimensionless).^[6]^[7]

Physical Significance

Force Interpretation

The Archimedes number quantifies the relative importance of buoyancy-driven forces to viscous forces in fluid systems involving density variations. Specifically, it represents the ratio of the characteristic buoyancy force, which scales with the term g (\rho_p - \rho_f) L^3 where g is gravitational acceleration, \rho_p and \rho_f are the densities of the particle or denser phase and the fluid, respectively, and L is the characteristic length, to the characteristic viscous force, which scales with \mu^2 / \rho_f where \mu is the dynamic viscosity of the fluid.^[8]^[4] This scaling arises from dimensional analysis in multiphase flows, where buoyancy arises from gravitational effects on density differences and viscous effects resist motion through internal friction.^[2] High values of the Archimedes number indicate regimes where buoyancy forces dominate, promoting flows driven primarily by gravitational settling or convection, such as rapid particle descent in low-viscosity fluids.^[9] In contrast, low values signify viscous-dominated conditions, where creeping flows prevail and buoyancy-induced motions are significantly damped by friction, as seen in highly viscous media or small-scale systems.^[2] These implications help delineate transitions between flow behaviors, with buoyancy enhancing instability and disorder while viscosity stabilizes the system.^[8] The buoyancy component of the Archimedes number is fundamentally connected to Archimedes' principle, which asserts that an immersed object experiences an upward force equal to the weight of the displaced fluid, thereby providing the density-difference mechanism that drives the gravitational term in the number.^[10]^[11] This principle underpins the physical balance captured by the number without invoking historical context, emphasizing the role of displaced fluid mass in generating the effective gravitational force.

Role in Dimensionless Analysis

The Archimedes number arises in dimensional analysis through the application of the Buckingham π theorem to problems involving buoyancy-driven flows, particularly in multiphase systems where gravitational effects dominate particle or bubble motion. Consider a scenario such as particle settling or fluidization, where the relevant dimensional variables include gravitational acceleration g, a characteristic length scale L (e.g., particle diameter), fluid density \rho, particle density \rho_l, and fluid viscosity \mu. With five variables and three fundamental dimensions (mass, length, time), the theorem yields two independent dimensionless π groups; one of these is the Archimedes number, which combines the variables to capture the balance between buoyancy forces—arising from density differences under gravity—and viscous forces resisting motion. This grouping ensures that the Archimedes number encapsulates the ratio of these forces without inertial contributions, distinguishing it from other π terms like the Reynolds number. In practice, deriving the groups involves selecting repeating variables (typically \rho, \mu, and L) to nondimensionalize the remaining parameters, leading to the Archimedes number as the key group for gravitational-viscous interactions. In ensuring dynamic similarity for scaling fluid problems, the Archimedes number must be matched between a model and its prototype to preserve the relative influence of buoyancy and viscosity, enabling valid predictions from experiments or simulations in multiphase flows such as fluidized beds. For instance, in computational fluid dynamics-dem discrete element method (CFD-DEM) simulations of fluidized beds, geometric, kinematic, and dynamic similarity require equal Archimedes and Reynolds numbers to replicate particle motion accurately across scales.^[12] Failure to match the Archimedes number can distort flow regimes, as it governs transitions from viscous-dominated to inertia-influenced behaviors under gravity.^[13] Compared to general dimensionless groups in multiphase flows, the Archimedes number is particularly specific to gravitational effects through buoyancy, whereas the Reynolds number emphasizes inertial-to-viscous ratios and the Froude number highlights inertial-to-gravitational balances; this specificity makes it indispensable for analyzing settling, rising, or suspension phenomena where density-driven convection prevails over other forces.^[14]

Engineering Applications

Fluidized Bed Design

In fluidized bed design, the Archimedes number plays a crucial role in predicting the transition from a fixed packed bed to a fluidized state, where the drag force exerted by the upward fluid flow balances the gravitational force on the particles. This dimensionless parameter encapsulates the interplay between particle properties (such as size and density) and fluid properties (viscosity and density), enabling engineers to assess fluidization behavior without extensive experimentation. By evaluating Ar, designers can determine the conditions under which particles begin to suspend and move freely, which is essential for optimizing reactor performance in processes like catalytic cracking and drying.^[15] A key application of the Archimedes number is in estimating the minimum fluidization velocity u_{mf}, the superficial fluid velocity at which fluidization initiates. The widely adopted correlation developed by Wen and Yu provides this estimate as follows:

Re_{p,mf} = \sqrt{33.7^2 + 0.0408 \, Ar} - 33.7

where Re_{p,mf} = \frac{u_{mf} d_p \rho_f}{\mu} is the particle Reynolds number at minimum fluidization, d_p is the particle diameter (or Sauter mean diameter for non-spherical particles), \rho_f is the fluid density, and \mu is the fluid viscosity. Solving for u_{mf} yields:

u_{mf} = \frac{\mu}{d_p \rho_f} \left( \sqrt{33.7^2 + 0.0408 \, Ar} - 33.7 \right)

This empirical relation, derived from pressure drop data across a broad range of particle and fluid systems, assumes a bed voidage at minimum fluidization of approximately 0.4 and is particularly effective for gas-solid systems where the fluid density is much lower than the particle density.^[15] The Archimedes number in this context is influenced primarily by the particle size (where the characteristic length L = d_p) and the density difference between the solid particles and the fluid, as captured in its formulation Ar = \frac{d_p^3 \rho_f (\rho_p - \rho_f) g}{\mu^2}, with \rho_p denoting particle density and g gravity. Larger particles or greater density contrasts increase Ar, leading to higher u_{mf} and requiring more vigorous fluid flow for fluidization, while higher fluid viscosity elevates the threshold by enhancing drag resistance at low velocities. These factors guide the selection of particle sizes (typically 50–1000 μm for Geldart Group B particles) to achieve efficient fluidization in industrial beds.^[15] This correlation assumes spherical particles for accuracy, as deviations in shape can alter drag coefficients and voidage distributions, potentially over- or under-predicting u_{mf} by up to 20–30% for highly irregular shapes. Extensions to non-spherical particles incorporate shape factors, such as sphericity \phi (the ratio of surface areas of a sphere and particle with equal volumes), by modifying the particle diameter in Ar as d_p / \phi or adjusting the constants in the correlation to account for increased drag. Such modifications have been validated in generalized flow regime diagrams that classify fluidization states based on Ar and particle shape.^[15]^[16]

Bubble Column Reactors

In bubble column reactors, the Archimedes number (Ar) plays a crucial role in design by characterizing the buoyancy-driven rise of gas bubbles in liquid media, enabling predictions of key hydrodynamic parameters such as gas holdup (ε_g). Empirical correlations often combine Ar with the Eötvös number (Eo), which accounts for surface tension effects on bubble shape, and the Froude number (Fr), which reflects inertial forces relative to gravity, to estimate ε_g as a function of these dimensionless groups: ε_g = f(Ar, Eo, Fr). For instance, a modified correlation incorporating gas properties achieves predictions within ±15% accuracy for systems with fine pore spargers, where Ar is defined using the column diameter to capture system-scale buoyancy-viscous interactions.^[17] This approach aids in optimizing reactor performance for gas-liquid contacting processes like oxidation or hydrogenation, where accurate holdup estimation ensures efficient mass transfer.^[17] The physical basis of Ar in these systems stems from its representation of the ratio of buoyancy forces, driven by the density difference between gas and liquid phases, to viscous drag, which governs bubble dynamics and stability. High Ar values, typically arising in low-viscosity liquids (e.g., water) with larger bubble sizes (d_b > 5 mm), promote faster bubble rise velocities and enhance coalescence, influencing transitions from the homogeneous bubbly regime—characterized by uniform small bubbles—to the churn-turbulent regime with chaotic large bubbles and higher turbulence.^[18] Conversely, lower Ar (e.g., Ar < 10^4 in viscous media) stabilizes bubbly flows, delaying transitions and allowing operation at higher superficial gas velocities (U_gs up to 0.2 m/s) without excessive mixing.^[19] These regime shifts directly impact design choices, such as selecting reactor heights (H > 5 m for scale-up stability) to maintain desired holdup levels (ε_g ≈ 0.1–0.3) and superficial velocities that balance throughput with energy input.^[17] Recent advancements since 2010 have integrated Ar into computational fluid dynamics (CFD) models to facilitate scale-up from lab to industrial bubble columns (D = 0.15–3 m), addressing challenges in predicting heterogeneous regime hydrodynamics. For example, Eulerian-Eulerian simulations incorporate Ar to scale liquid upflow rates (Q_Lup ∝ D² (gD)^{1/2}) and void fraction gradients, validating against experiments with Ar ranging from 10^9 to 10^14 for air-water systems at U_gs = 0.09–0.25 m/s.^[18] These models, often using population balance equations for bubble size distribution, improve predictions of regime transitions and holdup uniformity, reducing empirical reliance and enabling virtual prototyping for energy-efficient designs.

Spouted Bed Systems

In spouted bed systems, the Archimedes number (Ar) is integral to the design and operation, particularly in predicting the minimum spouting velocity (u_ms), the gas velocity required at the inlet to initiate a stable central spout and particle circulation. Empirical correlations typically express the dimensionless form u_ms / √(g D) as a function of Ar and geometric parameters such as bed diameter (D), inlet diameter (D_0), and bed height (H_0), enabling engineers to scale operations for processes like drying, granulation, and chemical reactions. For conical spouted beds equipped with draft tubes, one widely used correlation is u_ms ∝ Ar^{0.3} (H_0 / D_0)^{0.5} (L_H / D_0)^{1.2} / D_0^{0.3}, where L_H is the height of the entrainment zone, valid for Ar in the range 4000–8000 and applicable to particles with diameters of 0.5–2 mm.^[20] Advanced models, including back-propagation neural networks trained on experimental datasets, further incorporate Ar alongside factors like particle sphericity and gas properties to achieve prediction accuracies exceeding 95% for diverse spouted bed configurations. The mechanism underlying Ar's influence involves quantifying the competition between buoyancy forces driving particle entrainment into the upward gas jet and viscous drag resisting motion in the spout and annular regions. In the spout, high-velocity gas creates a dilute core where particles are lifted, while in the annulus, denser packing leads to downward flow; Ar, defined as Ar = g d_p^3 ρ (ρ_s - ρ) / μ^2, captures how particle size (d_p), density difference (ρ_s - ρ), and fluid viscosity (μ) determine the ease of this cyclic entrainment, promoting stable regimes at higher Ar values. This balance prevents spout collapse or excessive particle carryover, with Ar effectively scaling the inertial-buoyancy effects relative to drag in multiphase flow dynamics.^[21] Design considerations emphasize selecting optimal Ar for stable spouting, typically targeting values above 10^4 for coarse particles to ensure uniform circulation without flooding or defluidization. Particle density and size directly modulate Ar—denser or larger particles increase Ar, lowering the required u_ms and enhancing stability for Geldart Group D materials common in spouted beds—while bed geometry must align to avoid mismatches that elevate energy demands. Experimental validations from 1990s–2010s studies, such as those on open-sided draft tube systems, confirm Ar-based correlations predict u_ms with relative errors below 8% across contactor angles of 28°–45° and aperture ratios of 57%–78%, underscoring their reliability for industrial-scale reactors.^[21]

Particle Settling and Convection

In particle settling processes, the Archimedes number (Ar) plays a crucial role in determining the terminal velocity regimes of solid particles in a fluid, enabling the prediction of settling behavior across laminar, transitional, and turbulent flow conditions. For low Reynolds number (Re) flows, where viscous forces dominate, the terminal Reynolds number is given by Re_t = Ar / 18, extending Stokes' law to account for buoyancy-driven settling; this correlation arises from balancing gravitational and drag forces, with the terminal velocity u_t approximated as u_t = [ρ_f (ρ_p - ρ_f) g d_p^2] / (18 μ), where ρ_p and ρ_f are particle and fluid densities, d_p is particle diameter, g is gravity, and μ is fluid viscosity. ^[22] In intermediate and high Re regimes, more complex explicit correlations express Re_t as a function of Ar alone, such as solutions to quadratic drag relations like Re_t = [-b + √(b² - 4ac)] / (2a) derived from C_d = 24/Re_t + 0.44, allowing direct computation of u_t = Re_t μ / (d_p ρ_f) without iterative methods. ^[23] These relations highlight Ar's role in scaling settling dynamics, as referenced in viscous force balances from earlier interpretations. ^[23] Applications of Ar extend to sedimentation in slurries, where it informs critical deposition velocities in pipe flows at low solid volume fractions (e.g., 0.5–3%), through functional forms linking particle Reynolds number (Re_pc) to Ar for predicting the onset of bed formation. ^[24] For instance, in dilute suspensions, Ar helps model the transition from heterogeneous to homogeneous settling, ensuring slurry transport without excessive energy loss. In the dynamics of free-falling disks, Ar governs the onset of oscillations and path instability; numerical studies identify critical values around Ar ≈ 900, where descent velocity and angle of attack stabilize, with vortex interactions intensifying as Ar increases, influencing applications like sediment fallout simulations. ^[9] Additionally, Ar facilitates the Geldart classification of fluidizable particles, with boundaries such as Ar = 88.5 separating aeratable Group A (smooth fluidization, e.g., catalysts) from Group B (bubbling onset), and Ar = 176,900 marking the Group B-D transition for larger particles prone to slugging. ^[25] ^[22] In mixed and natural convection, Ar parameterizes the relative strength of buoyancy-driven flows against forced convection or inertial effects, often appearing in heat transfer correlations as Nu = f(Ar, Re, Pr) for density-driven systems like enclosures or jets. ^[2] Defined as Ar = Ra_H / (Pr Re_H^2) in thermal contexts (with Ra_H as Rayleigh number based on height H), it quantifies buoyant-to-inertial ratios, aiding predictions of flow regimes in ventilation or heat exchangers where free convection dominates at high Ar. ^[2] Emerging applications in the 2020s leverage Ar in environmental flows, such as sediment transport in rivers, where it normalizes grain-scale transitions (e.g., Ar ≈ 5.5–8.5 for gravel-to-sand bed shifts on Earth and Titan), enabling cross-planetary comparisons of discharge and erosion dynamics. ^[26] In bio-reactors involving particle-fluid interactions, Ar supports modeling of suspension uniformity in fluidized systems, though specific implementations remain tied to broader sedimentation principles. ^[22]

Grashof Number

The Grashof number (Gr) is a dimensionless parameter that quantifies the ratio of buoyancy forces to viscous forces in flows driven by thermal convection, defined as

\mathrm{Gr} = \frac{g \beta \Delta T L^3}{\nu^2},

where g is the acceleration due to gravity, \beta is the thermal expansion coefficient of the fluid, \Delta T is the temperature difference driving the buoyancy, L is a characteristic length scale, and \nu is the kinematic viscosity.^[27]^[28] The Archimedes number (Ar) shares a structural similarity with the Grashof number as both represent the competition between buoyancy and viscous effects in natural convection, but Ar generalizes to density differences not induced by temperature. Specifically, when density variations arise from thermal effects under the Boussinesq approximation, where the relative density change (\rho - \rho_0)/\rho_0 \approx \beta \Delta T, the Archimedes number approximates the Grashof number such that \mathrm{Ar} \approx \mathrm{Gr}.^[1] Ar is particularly favored in non-thermal contexts, such as particle-laden flows, where temperature gradients are absent.^[1] While the Grashof number specifically addresses buoyancy from temperature-induced density gradients, the Archimedes number applies to compositional density differences, such as those from suspended particles or dissolved solutes in multiphase flows.^[28]^[2] This distinction arises because Gr relies on the thermal expansion coefficient \beta, which is irrelevant in non-thermal buoyancy mechanisms. Both numbers appear in correlations for natural convection heat and mass transfer, particularly in predicting flow regimes and transition to turbulence, but the Archimedes number is preferred in multiphase applications without significant heating, such as sedimentation or fluidization processes.^[27]^[2]

Galileo Number

The Galileo number (Ga) is a dimensionless quantity in fluid mechanics that represents the ratio of gravitational forces to viscous forces in flows dominated by inertia and gravity, without accounting for density differences between phases. It is defined as

\text{Ga} = \frac{g L^3}{\nu^2},

where g is the acceleration due to gravity, L is a characteristic length scale (such as particle diameter), and \nu is the kinematic viscosity of the fluid.^[6] This formulation arises from dimensional analysis of buoyancy-free gravitational flows, emphasizing the competition between body forces driving motion and viscous dissipation.^[29] The Galileo number is mathematically related to the Archimedes number (Ar) through the density ratio, specifically \text{Ga} = \text{Ar} \times \frac{\rho_l}{\rho - \rho_l}, where \rho_l is the fluid density and \rho is the density of the dispersed phase (e.g., particles).^[29] This relation adjusts the Archimedes number's inclusion of buoyancy strength (\Delta \rho = \rho - \rho_l) to isolate pure gravitational effects; in dilute suspensions where \Delta \rho \approx \rho_l (common for particles with densities roughly twice that of the fluid), the two numbers become approximately equal and are sometimes used interchangeably.^[30] While the Archimedes number is primarily applied to buoyant, particle-laden flows such as sedimentation or fluidization, the Galileo number finds use in scenarios involving free-surface flows or inertial gravity waves, where buoyancy modulation by density contrast is negligible.^[31] Historically, in sedimentation studies, the Galileo number has been referred to as a modified form of the Archimedes number to highlight its role in gravity-viscous balances absent explicit buoyancy terms.^[30]

Richardson Number

The Richardson number, denoted as Ri, quantifies the ratio of buoyancy forces to inertial forces in fluid flows and is defined as Ri = \frac{Gr}{Re^2}, where Gr is the Grashof number representing buoyancy relative to viscous forces and Re is the Reynolds number representing inertial relative to viscous forces.^[32] In contexts involving density differences, such as particulate or multiphase flows, a density-based variant emerges where buoyancy is characterized by the Archimedes number Ar, leading to the approximation Ri \approx \frac{Ar}{Re^2}.^[5] This formulation underscores the interplay between buoyancy-driven effects captured by Ar and the flow's inertial scaling via Re. A key aspect of the Richardson number is its role in assessing stratification stability in buoyancy-influenced flows; specifically, when Ri > 0.25, the flow tends toward stable layering where buoyancy dominates and suppresses shear-induced instabilities, as derived from linear stability analysis.^[33] In contrast to the Archimedes number, which primarily serves as an input parameter for the magnitude of buoyancy forces independent of flow velocity, the Richardson number evaluates regime transitions—such as from forced to mixed or free convection—by integrating inertial influences, thereby providing a dynamic measure of flow stability.^[34] In applications to atmospheric and oceanic mixed convection, the Richardson number connects the Archimedes number to turbulence suppression mechanisms, enabling predictions of vertical mixing and diffusivity in stably stratified boundary layers where high Ri values limit turbulent exchange.^[34] This linkage is particularly valuable for modeling transport processes in environments with density gradients, such as ocean thermoclines or atmospheric inversions, where buoyancy effects modulate convective vigor.^[35]

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