Draft:Mass Flow Rate from Surface Density and Linear Velocity

• Comment: Nothing showing that this is deserving of its own page compared to adding a section on the main mfr page. Moritoriko (talk) 05:50, 3 June 2025 (UTC)

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Overview Mass flow rate with surface density refers to the calculation of mass flow rate (ṁ) when dealing with materials distributed over a surface rather than throughout a volume. This formulation is particularly relevant in industrial processes involving thin layers, coatings, membranes, and surface transport phenomena. Mathematical Formulation General Formula The mass flow rate expressed in terms of surface density is given by: ṁ = ρs × v × A where:

ṁ is the mass flow rate (kg/s) ρs is the surface density (kg/m²) v is the linear velocity (m/s) A is the cross-sectional area or relevant width (m)

Simplified Formula For applications where the surface area is implicit or normalized, the formula simplifies to: ṁ = ρs × v × l where:

l is the length or width of the moving zone, depending on the context (m)

Distinction from Volumetric Density This formulation differs from the conventional mass flow rate formula that uses volumetric density: ṁ = ρ × v × A where ρ represents volumetric density (kg/m³). The key distinction lies in the density parameter:

Surface density (ρs): mass per unit area (kg/m²) Volumetric density (ρ): mass per unit volume (kg/m³)

Applications This formulation is commonly used in: Industrial Processes

Coating and film deposition: Calculating material flow rates in thin film applications Conveyor systems: Determining mass transport rates of materials on moving surfaces Membrane processes: Analyzing mass transfer across permeable surfaces

Material Science

Powder transport: Flow of particulate matter distributed over surfaces Catalytic processes: Mass transfer rates in catalyst beds with surface-active sites

Environmental Engineering

Sediment transport: Movement of particles along surfaces in water bodies Atmospheric particle deposition: Calculation of pollutant deposition rates

Practical Considerations When applying these formulas, several factors must be considered:

Uniformity assumption: The surface density ρs is assumed to be uniform across the area Steady-state conditions: The velocity v is considered constant over time Surface geometry: The effective area A or length l must accurately represent the transport zone

References

General Concepts

Mass flow rate. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Mass\ flow\ rate(https://en.wikipedia.org/wiki/Mass_flow_rate) citeturn3search0

 Defines the mass flow rate $\dot{m}=\displaystyle\frac{dm}{dt}$ and its conventional volumetric formulation $\dot{m}=\rho v A$, offering the standard background on mass flow rate in SI units. citeturn3search0

Area density. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Area\ density(https://en.wikipedia.org/wiki/Area_density) citeturn2search10

 Introduces surface (area) density $\rho_s$ as mass per unit area (kg/m²) and shows its relation to volumetric density via $\rho_s = \rho\,l$. citeturn2search10

Mass flux. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Mass\ flux(https://en.wikipedia.org/wiki/Mass_flux) citeturn2search11

 Describes mass flux $\mathbf{j}_m=\rho\,\mathbf{v}$ (kg·s⁻¹·m⁻²), explaining how mass flow rate through a surface is found via $\dot{m}=\iint_A \rho\,\mathbf{v}\cdot d\mathbf{A}$. citeturn2search11

Find the mass flow rate, given a surface, density and velocity field. (2014). Math StackExchange. Retrieved June 3, 2025, from [1](https://math.stackexchange.com/questions/1294489/find-the-mass-flow-rate-given-a-surface-density-and-velocity-field) citeturn1search0turn5search9

 Demonstrates the general surface‐integral formulation $\dot{m}=\iint_S \delta(x,y,z)\,\mathbf{V}(x,y,z)\cdot d\mathbf{S}$ for non‐volumetric (surface‐distributed) densities. citeturn1search0turn5search9

Distinct Formulations

Mass flow rate equations. NASA Glenn Research Center. (2024). In Beginners’ Guide to Aeronautics. Retrieved June 3, 2025, from [2](https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/mass-flow-rate-equations/) citeturn0search3

 Reviews $\dot{m}=\rho\,v\,A$ for standard volumetric density and highlights “mass current” vs. “mass flux” terminology. citeturn0search3

Surface‐Density–Based Applications

Spin coating. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Spin\ coating(https://en.wikipedia.org/wiki/Spin_coating) citeturn0search10

 Discusses thin‐film deposition by centrifugal spreading; surface density of the film governs the mass flow per unit width during spin‐off. citeturn0search10

Slot‐die coating. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Slot-die\ coating(https://en.wikipedia.org/wiki/Slot-die_coating) citeturn2search12

 Describes how the wet layer thickness is controlled by volumetric pump rate, coating speed, and coating width—implicitly invoking $\dot{m}=\rho_s\,v\,\ell$ when relating mass per unit area to coating velocity and width. citeturn2search12

Sediment transport. Wikipedia contributors. (2025). In Wikipedia. Retrieved June 3, 2025, from Sediment\ transport(https://en.wikipedia.org/wiki/Sediment_transport) citeturn5search4

 Provides formulae for sediment transport rate (mass/time) and notes the use of unit‐width rates ($\,q_s$ in kg·m⁻¹·s⁻¹), reflecting mass flow per surface area in rivers or coastal environments. citeturn5search4

Additional Contexts and Derivations

Thermal dispersion mass flow meters. Sierra Instruments. (2014). “New Developments in Thermal Dispersion Mass Flow Meters.” Retrieved June 3, 2025, from [3](https://www.sierrainstruments.com/userfiles/file/aga-new-developments-thermal-dispersion.pdf) citeturn0search4

 Introduces “mass velocity” $V_s$ (m/s) so that $\dot{m}_s = \rho_s\,V_s$ yields the mass flow rate per unit area inside flow conduits—analogous to $\dot{m}=\rho_s\,v\,A$ when extended across a cross‐section. citeturn0search4

Area density (areal density). In The Materials Science of Thin Films (Arshad’s Class Notes, 2018). Retrieved June 3, 2025, from [4](https://arshadnotes.wordpress.com/wp-content/uploads/2018/02/the_materials_science_of_thin_films.pdf) citeturn2search2

 Explains that surface density (kg/m²) of adatoms on a substrate determines film mass transport in vapor‐deposition processes, providing foundational context for $\rho_s$. citeturn2search2

Lindeburg, M. R. (2013). Chemical Engineering Reference Manual for the PE Exam. Professional Publications.

 Presents introductory mass flux definitions and notes superficial mass flow in porous media ($\dot{m}_s = \rho_s\,v_s$), which underpins the surface‐density formulation $\dot{m}=\rho_s\,v\,A$. citeturn0search0

Notes on Practical Considerations

Uniformity and Steady‐State Assumptions:

 The integration in Math StackExchange’s derivation (Turn 1search0) assumes constant $\delta=\rho_s$ and constant normal velocity $v$, simplifying $\dot{m}=\displaystyle\iint_S \rho_s\,v\,dA$ to $\rho_s\,v\,A$. citeturn1search0

Geometry of Transport Zone:

 Ensuring $A$ (or length $\ell$) accurately represents the physical width of the moving surface is critical; see Slot‐die coating (Turn 2search12) for how coating width $\ell$ enters the simplified form $\dot{m}=\rho_s\,v\,\ell$. citeturn2search12

Distinction from Volumetric Formulation:

 Conventional $\dot{m}=\rho\,v\,A$ (Turn 3search0) uses $\rho$ in kg/m³, whereas $\dot{m}=\rho_s\,v\,A$ replaces volumetric density with surface density (kg/m²) to account for material confined to a thin layer or film. citeturn3search0turn2search10

All retrieval dates are “June 3, 2025.”