Basic equations for suspension flows with interphase mass-transfer

In the case of suspension flows, the rate of interphase momentum transfer M(k) and that of interphase energy transfer E(k), which were expressed as a sum of infinite discontinuities by Ishii, have been reduced to the sum of several terms which have concise physical significance. M(k) is composed of the following terms: (i) the momentum carried by the interphase mass transfer; (ii) the interphase drag force due to the relative motion between phases; (iii) the interphase force produced by the concentration gradient of the dispersed phase in a pressure field. And E(k) is composed of the following four terms, that is, the energy carried by the interphase mass transfer, the work produced by the interphase forces of the second and third parts above, and the heat transfer between phases. It is concluded from the results that (i) the term, (-alpha-k-nabla-p), which is related to the pressure gradient in the momentum equation, can be derived from the basic conservation laws without introducing the "shared-pressure presumption"; (ii) the mean velocity of the action point of the interphase drag is the mean velocity of the interface displacement, upsilonBAR-i. It is approximately equal to the mean velocity of the dispersed phase, upsilonBAR-d. Hence the work terms produced by the drag forces are f(dc) . upsilonBAR-d, and f(cd) . upsilonBAR-d, respectively, with upsilonBAR-i not being replaced by the mean velocity of the continuous phase, upsilonBAR-c; (iii) by analogy, the terms of the momentum transfer due to phase change are upsilonBAR-d-GAMMA-c, and upsilonBAR-d-GAMMA-d, respectively; (iv) since the transformation between explicit heat and latent heat occurs in the process of phase change, the algebraic sum of the heat transfer between phases is not equal to zero. Q(ic) and Q(id) are composed of the explicit heat and latent heat, so that the sum Q(ic) + Q(id)) is equal to zero.