This is just here as a test because I lose it

Term information

PMID

10631014

definition

Population balance equations (PBEs) have been introduced in several branches of modern science, mainly in branches with particulate entities. This includes topics like crystallization, liquid-liquid extraction, gas-liquid dispersions, liquid-liquid reactions, communition, aerosol engineering, biology (where the separate entities are cells), polymerization, etc. PBEs define how populations of separate entities develop in specific properties over time. Therefore, they belong to a subcategory of equations known as partial differential equations. source: http://en.wikipedia.org/wiki/Population_balance_equation

exact synonym

PBE

PBE equation

has enhanced presentationMathML

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mo>&InvisibleTimes;</mo> <mi>t</mi> </mrow> </mfrac> <mo>&InvisibleTimes;</mo> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>x</mi> </msub> <mo>&InvisibleTimes;</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>&InvisibleTimes;</mo> <mi>d</mi> <mo>&InvisibleTimes;</mo> <msub> <mi>V</mi> <mi>x</mi> </msub> <mo>&InvisibleTimes;</mo> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>r</mi> </msub> <mo>&InvisibleTimes;</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>&InvisibleTimes;</mo> <mi>d</mi> <mo>&InvisibleTimes;</mo> <msub> <mi>V</mi> <mi>r</mi> </msub> </mrow> <mspace width="0.167em"/> <mo>&InvisibleTimes;</mo> <mrow> <mi>f</mi> <mo>&ApplyFunction;</mo> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>t</mi> </mfenced> </mrow> </mrow> <mo>=</mo> <mrow> <mrow> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>x</mi> </msub> <mo>&InvisibleTimes;</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>&InvisibleTimes;</mo> <mi>d</mi> <mo>&InvisibleTimes;</mo> <msub> <mi>V</mi> <mi>x</mi> </msub> <mo>&InvisibleTimes;</mo> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>r</mi> </msub> <mo>&InvisibleTimes;</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>&InvisibleTimes;</mo> <mi>d</mi> <mo>&InvisibleTimes;</mo> <msub> <mi>V</mi> <mi>r</mi> </msub> </mrow> <mspace width="0.167em"/> <mo>&InvisibleTimes;</mo> <mrow> <mi>h</mi> <mo>&InvisibleTimes;</mo> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>Y</mi> <mi>t</mi> </mfenced> </mrow> </mrow> </mrow> </math>

has latex math

\frac{d}{dt} \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,f(x,r,t) = \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,h(x,r,Y,t)

has raw presentationMathML

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>x</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mi>d</mi> <msub> <mi>V</mi> <mi>x</mi> </msub> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>r</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mi>d</mi> <msub> <mi>V</mi> <mi>r</mi> </msub> <mspace width="0.167em"/> <mi>f</mi> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>t</mi> </mfenced> <mo>=</mo> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>x</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mi>d</mi> <msub> <mi>V</mi> <mi>x</mi> </msub> <msub> <mo>&Integral;</mo> <mrow> <msub> <mi>&Omega;</mi> <mi>r</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mi>d</mi> <msub> <mi>V</mi> <mi>r</mi> </msub> <mspace width="0.167em"/> <mi>h</mi> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>Y</mi> <mi>t</mi> </mfenced> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{d}{dt} \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,f(x,r,t) = \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,h(x,r,Y,t) \]</annotation> </semantics> </math>

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