| Mass flow rate | |
|---|---|
Common symbols | |
| SI unit | kg/s |
| Dimension | |
Inphysics andengineering,mass flow rate is therate at whichmass of a substance changes overtime. Itsunit iskilogram persecond (kg/s) inSI units, andslug per second orpound per second inUS customary units. The common symbol is (pronounced "m-dot"), although sometimes (Greek lowercasemu) is used.
Sometimes, mass flow rate as defined here is termed "mass flux" or "mass current".[a]Confusingly, "mass flow" is also a term formass flux, the rate of mass flow per unit of area.[2]
Mass flow rate is defined by thelimit[3][4]i.e., the flow of mass through a surface per time.
The overdot on isNewton's notation for atime derivative. Since mass is ascalar quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity. The change in mass is the amount that flowsafter 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 forsteady flow.
Mass flow rate can also be calculated by
where
The above equation is only true for a flat, plane area. In general, including cases where the area is curved, the equation becomes asurface integral:
Thearea 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 afilter or amembrane, 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. Thevector area is a combination of the magnitude of the area through which the mass passes through,, and aunit vector normal to the area,. The relation is.
The reason for thedot product is as follows. The only mass flowingthrough the cross-section is the amount normal to the area, i.e.parallel to the unit normal. This amount is
where 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 increases less mass passes through. All mass which passes in tangential directions to the area, that isperpendicular to the unit normal,doesn't actually passthrough the area, so the mass passing through the area is zero. This occurs when: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 withsuperficial velocity,, with the following relationship:[5]The quantity can be used inparticle Reynolds number ormass transfer coefficient calculation for fixed and fluidized bed systems.
In the elementary form of thecontinuity equation for mass, inhydrodynamics:[6]
In elementary classical mechanics, mass flow rate is encountered when dealing withobjects of variable mass, such as a rocket ejecting spent fuel. Often, descriptions of such objects erroneously[7] invokeNewton's second law by treating both the mass and the velocity 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.[7]
Mass flow rate can be used to calculate the energy flow rate of a fluid:[8]where is the unit mass energy of a system.
Energy flow rate has SI units ofkilojoule per second orkilowatt.
It is important to note that wecannot derive a general expression for Newton's second law for variable mass systems by treating the mass inF =dP/dt =d(Mv) as avariable. [...] Wecan useF =dP/dt to analyze variable mass systemsonly if we apply it to anentire system of constant mass having parts among which there is an interchange of mass.[Emphasis as in the original]
