Mass Flow rate | |
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Common symbols | |

SI unit | kg/s |

Dimension |

In physics and engineering, **mass flow rate** is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is (* ṁ*, pronounced "m-dot"), although sometimes

Sometimes, mass flow rate is termed * mass flux * or *mass current*, see for example *Schaum's Outline of Fluid Mechanics*.^{ [1] } In this article, the (more intuitive) definition is used.

Mass flow rate is defined by the limit:^{ [2] }^{ [3] }

i.e., the flow of mass m through a surface per unit time t.

The overdot on the m is Newton's notation for a time derivative. Since mass is a scalar quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity. The change in mass is the amount that flows *after* crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow.

Mass flow rate can also be calculated by:

where:

*or***Q**= Volume flow rate,*ρ*= mass density of the fluid,**v**= Flow velocity of the mass elements,**A**= cross-sectional vector area/surface,**j**_{m}= mass flux.

The above equation is only true for a flat, plane area. In general, including cases where the area is curved, the equation becomes a surface integral:

The area required to calculate the mass flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface, e.g. for substances passing through a filter or a membrane, the real surface is the (generally curved) surface area of the filter, macroscopically - ignoring the area spanned by the holes in the filter/membrane. The spaces would be cross-sectional areas. For liquids passing through a pipe, the area is the cross-section of the pipe, at the section considered. The vector area is a combination of the magnitude of the area through which the mass passes through, *A*, and a unit vector normal to the area, . The relation is .

The reason for the dot product is as follows. The only mass flowing *through* the cross-section is the amount normal to the area, i.e. parallel to the unit normal. This amount is:

where *θ* is the angle between the unit normal and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor , as *θ* increases less mass passes through. All mass which passes in tangential directions to the area, that is perpendicular to the unit normal, *doesn't* actually pass *through* the area, so the mass passing through the area is zero. This occurs when *θ* = *π*/2:

These results are equivalent to the equation containing the dot product. Sometimes these equations are used to define the mass flow rate.

Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. It is related with superficial velocity, *v _{s}*, with the following relationship:

^{ [4] }

The quantity can be used in particle Reynolds number or mass transfer coefficient calculation for fixed and fluidized bed systems.

In the elementary form of the continuity equation for mass, in hydrodynamics:^{ [5] }

In elementary classical mechanics, mass flow rate is encountered when dealing with objects of variable mass, such as a rocket ejecting spent fuel. Often, descriptions of such objects erroneously^{ [6] } invoke Newton's second law **F** =d(*m***v**)/d*t* by treating both the mass *m* and the velocity **v** as time-dependent and then applying the derivative product rule. A correct description of such an object requires the application of Newton's second law to the entire, constant-mass system consisting of both the object and its ejected mass.^{ [6] }

Mass flow rate can be used to calculate the energy flow rate of a fluid:^{ [7] }

where:

- = unit mass energy of a system

Energy flow rate has SI units of kilojoule per second or kilowatt.

**Continuum mechanics** is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In the theory of Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

**Flux** describes any effect that appears to pass or travel through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).

In fluid dynamics, **Stokes' law** is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

In fluid dynamics, the **Euler equations** are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.

A **continuity equation** or **transport equation** is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

In physics and engineering, in particular fluid dynamics, the **volumetric flow rate** is the volume of fluid which passes per unit time; usually it is represented by the symbol Q. It contrasts with mass flow rate, which is the other main type of fluid flow rate. In most contexts a mention of *rate of fluid flow* is likely to refer to the volumetric rate. In hydrometry, the volumetric flow rate is known as *discharge*.

In fluid mechanics or more generally continuum mechanics, **incompressible flow** refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies **incompressibility** is that the divergence of the flow velocity is zero.

In physics, **circulation** is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.

**Stokes flow**, also named **creeping flow** or **creeping motion**, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

In electromagnetism, **current density** is the amount of charge per unit time that flows through a unit area of a chosen cross section. The **current density vector** is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

In fluid mechanics, **potential vorticity** (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

In physics and engineering, **mass flux** is the rate of mass flow. Its SI units are kg m^{−2} s^{−1}. The common symbols are *j*, *J*, *q*, *Q*, *φ*, or Φ, sometimes with subscript *m* to indicate mass is the flowing quantity. Mass flux can also refer to an alternate form of flux in Fick's law that includes the molecular mass, or in Darcy's law that includes the mass density.

In fluid mechanics and mathematics, a **capillary surface** is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

In fluid dynamics, **Airy wave theory** gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

**Blade element momentum theory** is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

In physics, the first law of thermodynamics is an expression of the conservation of total energy of a system. The increase of the energy of a system is equal to the sum of work done on the system and the heat added to that system:

- ↑ Fluid Mechanics, M. Potter, D.C. Wiggart, Schaum's Outlines, McGraw Hill (USA), 2008, ISBN 978-0-07-148781-8
- ↑ "Mass Flow Rate Fluids Flow Equation - Engineers Edge".
- ↑ "Mass Flow Rate".
- ↑ Lindeburg M. R. Chemical Engineering Reference Manual for the PE Exam. – Professional Publications (CA), 2013.
- ↑ Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
- 1 2 Halliday; Resnick (1977).
*Physics*. Vol. 1. p. 199. ISBN 978-0-471-03710-1.It is important to note that we

[Emphasis as in the original]*cannot*derive a general expression for Newton's second law for variable mass systems by treating the mass in**F**=*d***P**/*dt*=*d*(*M***v**) as a*variable*. [...] We*can*use**F**=*d***P**/*dt*to analyze variable mass systems*only*if we apply it to an*entire system of constant mass*having parts among which there is an interchange of mass. - ↑ Çengel, Yunus A. (2002).
*Thermodynamics : an engineering approach*. Boles, Michael A. (4th ed.). Boston: McGraw-Hill. ISBN 0-07-238332-1. OCLC 45791449.

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