Mathematical Theorems
Appendix A Mathematical Theorems The mathematical theorems needed in order to derive the governing model equations are defined in this appendix. A.1 Transport Theorem for a Single Phase Region The transport theorem is employed deriving the conservation equations in continuum mechanics. The mathematical statement is sometimes attributed to, or named in honor of, the German Mathematician Gottfried Wilhelm Leibnitz (1646–1716) and the British fluid dynamics engineer Osborne Reynolds (1842–1912) due to their work and con- tributions related to the theorem. Hence it follows that the transport theorem, or alternate forms of the theorem, may be named the Leibnitz theorem in mathematics and Reynolds transport theorem in mechanics. In a customary interpretation the Reynolds transport theorem provides the link between the system and control volume representations, while the Leibnitz’s theorem is a three dimensional version of the integral rule for differentiation of an integral. There are several notations used for the transport theorem and there are numerous forms and corollaries. A.1.1 Leibnitz’s Rule The Leibnitz’s integral rule gives a formula for differentiation of an integral whose limits are functions of the differential variable [7, 8, 22, 23, 45, 55, 79, 94, 99]. The formula is also known as differentiation under the integral sign. H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1361 © Springer International Publishing Switzerland 2014 1362 Appendix A: Mathematical Theorems b(t) b(t) d ∂f (t, x) db da f (t, x) dx = dx + f (t, b) − f (t, a) (A.1) dt ∂t dt dt a(t) a(t) The first term on the RHS gives the change in the integral because the function itself is changing with time, the second term accounts for the gain in area as the upper limit is moved in the positive axis direction, and the third term accounts for the loss in area as the lower limit is moved.
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