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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, mass flow rate is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
of a substance which passes per
unit of time A unit of time is any particular time interval, used as a standard way of measuring or expressing duration. The base unit of time in the International System of Units (SI) and by extension most of the Western world, is the second, defined as ab ...
. Its
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
is
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
per second in SI units, and slug per second or
pound Pound or Pounds may refer to: Units * Pound (currency), a unit of currency * Pound sterling, the official currency of the United Kingdom * Pound (mass), a unit of mass * Pound (force), a unit of force * Rail pound, in rail profile Symbols * Po ...
per second in US customary units. The common symbol is \dot (''ṁ'', pronounced "m-dot"), although sometimes ''μ'' (
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
lowercase mu) is used. Sometimes, mass flow rate is termed '' mass flux'' or ''mass current'', see for example ''Schaum's Outline of Fluid Mechanics''. In this article, the (more intuitive) definition is used. Mass flow rate is defined by the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
: \dot = \lim_ \frac = \frac i.e., the flow of mass through a surface per unit time . The overdot on the 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.


Alternative equations

Mass flow rate can also be calculated by: :\dot m = \rho \cdot \dot V = \rho \cdot \mathbf \cdot \mathbf = \mathbf_ \cdot \mathbf where: *''\dot V'' or Q =
Volume flow rate In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
, *''ρ'' = mass
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the fluid, *v =
Flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
of the mass elements, *A = cross-sectional vector area/surface, * jm = 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 In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, on ...
: :\dot m = \iint_A \rho \mathbf \cdot \mathbf = \iint_A \mathbf_ \cdot \mathbf The
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
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 A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. ...
, 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, \mathbf. The relation is \mathbf = A \mathbf. 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: :\dot m = \rho v A \cos\theta where ''θ'' is the angle between the unit normal \mathbf and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor \cos\theta , as ''θ'' increases less mass passes through. All mass which passes in tangential directions to the area, that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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: :\dot m = \rho v A \cos(\pi/2) = 0 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, ''vs'', with the following relationship: :\dot m_s = v_s \cdot \rho = \dot m/A The quantity can be used in particle Reynolds number or mass transfer coefficient calculation for fixed and fluidized bed systems.


Usage

In the elementary form of the continuity equation for mass, in hydrodynamics: : \rho_1 \mathbf_1 \cdot \mathbf_1 = \rho_2 \mathbf_2 \cdot \mathbf_2 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 mphasis as in the original/ref> 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. Mass flow rate can be used to calculate the energy flow rate of a fluid: :\dot=\dote where: * e = unit mass energy of a system Energy flow rate has SI units of kilojoule per second or
kilowatt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after Jame ...
.


Analogous quantities

In hydrodynamics, mass flow rate is the rate of flow of mass. In electricity, the rate of flow of charge is
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
.


See also

* Continuity equation *
Fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
* Mass flow controller * Mass flow meter * Mass flux * Orifice plate *
Standard cubic centimetres per minute Standard cubic centimeters per minute (SCCM) is a unit used to quantify the flow rate of a fluid. 1 SCCM is identical to 1  cm³STP/min. Another expression of it would be Nml/min. These standard conditions vary according to different reg ...
* Thermal mass flow meter * Volumetric flow rate


References

{{DEFAULTSORT:Mass Flow Rate Fluid dynamics Temporal rates Mass