Term information
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
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>⁢</mo> <msub> <mo>∫</mo> <mrow> <msub> <mi>Ω</mi> <mi>x</mi> </msub> <mo>⁢</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>⁢</mo> <mi>d</mi> <mo>⁢</mo> <msub> <mi>V</mi> <mi>x</mi> </msub> <mo>⁢</mo> <msub> <mo>∫</mo> <mrow> <msub> <mi>Ω</mi> <mi>r</mi> </msub> <mo>⁢</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>⁢</mo> <mi>d</mi> <mo>⁢</mo> <msub> <mi>V</mi> <mi>r</mi> </msub> </mrow> <mspace width="0.167em"/> <mo>⁢</mo> <mrow> <mi>f</mi> <mo>⁡</mo> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>t</mi> </mfenced> </mrow> </mrow> <mo>=</mo> <mrow> <mrow> <msub> <mo>∫</mo> <mrow> <msub> <mi>Ω</mi> <mi>x</mi> </msub> <mo>⁢</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>⁢</mo> <mi>d</mi> <mo>⁢</mo> <msub> <mi>V</mi> <mi>x</mi> </msub> <mo>⁢</mo> <msub> <mo>∫</mo> <mrow> <msub> <mi>Ω</mi> <mi>r</mi> </msub> <mo>⁢</mo> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </msub> <mo>⁢</mo> <mi>d</mi> <mo>⁢</mo> <msub> <mi>V</mi> <mi>r</mi> </msub> </mrow> <mspace width="0.167em"/> <mo>⁢</mo> <mrow> <mi>h</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mi>x</mi> <mi>r</mi> <mi>Y</mi> <mi>t</mi> </mfenced> </mrow> </mrow> </mrow> </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)
<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>∫</mo> <mrow> <msub> <mi>Ω</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>∫</mo> <mrow> <msub> <mi>Ω</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>∫</mo> <mrow> <msub> <mi>Ω</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>∫</mo> <mrow> <msub> <mi>Ω</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>