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Mass flow rate is density times cross-sectional area times mean normal speed.
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Formula
ṁ in kg/s; ρ in kg/m3; A in m2; v̄ in m/s
| Symbol | Meaning | Unit |
|---|---|---|
| Mass per unit time through the control section | kg/s | |
| Fluid density at the section | kg/m3 | |
| Cross-sectional area | m2 | |
| v̄ | Mean normal velocity | m/s |
\dot m=\rho A\bar vLocal mass flux is ρv; integrating it over a section gives mass flow. For uniform density and an appropriate mean speed this becomes ṁ=ρA v̄. In steady flow without side streams, mass flow is conserved even as density and speed change. For strongly varying profiles or density, use the area integral explicitly.
Local mass flux is ρv; integrating it over a section gives mass flow. For uniform density and an appropriate mean speed this becomes ṁ=ρA v̄. In steady flow without side streams, mass flow is conserved even as density and speed change. For strongly varying profiles or density, use the area integral explicitly.
Mass flow rate is density times cross-sectional area times mean normal speed.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Mass flow rate through a section” and check its unit and magnitude.
Air with ρ=1.2 kg/m3 flows through A=0.05 m2 at 8 m/s: ṁ=0.48 kg/s.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: ṁ in kg/s; ρ in kg/m3; A in m2; v̄ in m/s.
Formula cluster
Mass flow rate is density times cross-sectional area times mean normal speed.
Air with ρ=1.2 kg/m3 flows through A=0.05 m2 at 8 m/s: ṁ=0.48 kg/s.
Write mass balances for ducts, pipes, nozzles and thermal systems.
Here is how to work through a typical Mass flow rate through a section (ṁ = ρA v̄) task step by step:
Task
Work through the numerical example for “Mass flow rate through a section” and check its unit and magnitude.
Solution path
Air with ρ=1.2 kg/m3 flows through A=0.05 m2 at 8 m/s: ṁ=0.48 kg/s.
Mass conservation requires the same mass throughput in steady flow with no side streams. If density falls, speed must rise at the same area to keep ρA v̄ constant. This is typical of compressible flow. Volume flow A v̄ can change. Mass flow is conserved in a steady control volume only when all inlets and outlets are counted and there is no accumulation. The product ρA v̄ assumes suitable mean density and speed at the section. For nonuniform profiles, integrate density times normal velocity over area.
Its unit is kilograms per second. With ρ in kg/m3, A in m2 and speed in m/s, the product is kg/m3·m2·m/s=kg/s. This dimensional check helps distinguish mass flow from volume flow, whose unit m3/s contains no density. Mass flow is conserved in a steady control volume only when all inlets and outlets are counted and there is no accumulation. The product ρA v̄ assumes suitable mean density and speed at the section. For nonuniform profiles, integrate density times normal velocity over area.
Write a control-volume balance: total incoming mass flow minus outgoing mass flow equals the rate of mass accumulation inside. At steady state, accumulation is zero. Use signs consistently; a negative branch flow means its actual direction is opposite to the assumed one. Mass flow is conserved in a steady control volume only when all inlets and outlets are counted and there is no accumulation. The product ρA v̄ assumes suitable mean density and speed at the section. For nonuniform profiles, integrate density times normal velocity over area.
Write the quantity and unit from the variable list next to every value: ṁ in kg/s; ρ in kg/m3; A in m2; v̄ in m/s. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example Air with ρ=1.2 kg/m3 flows through A=0.05 m2 at 8 m/s: ṁ=0.48 kg/s. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Mass flow is conserved in a steady control volume only when all inlets and outlets are counted and there is no accumulation. The product ρA v̄ assumes suitable mean density and speed at the section. For nonuniform profiles, integrate density times normal velocity over area.
The background states this starting condition: Local mass flux is ρv; integrating it over a section gives mass flow. For uniform density and an. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Mass flow is conserved in a steady control volume only when all inlets and outlets are counted and there is no accumulation. The product ρA v̄ assumes suitable mean density and speed at the section. For nonuniform profiles, integrate density times normal velocity over area.
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