By Christophe Morel
This ebook develops the theoretical foundations of disperse two-phase flows, that are characterised via the life of bubbles, droplets or strong debris finely dispersed in a provider fluid, that are a liquid or a fuel. Chapters make clear many tricky topics, together with modeling of the interfacial zone focus. simple wisdom of the themes handled during this publication is vital to practitioners of Computational Fluid Dynamics for two-phase flows in numerous business and environmental settings.
The writer presents a whole derivation of the fundamental equations, by way of extra complex matters like turbulence equations for the 2 stages (continuous and disperse) and multi-size particulate circulation modeling. in addition to theoretical fabric, readers will realize chapters fascinated with closure kinfolk and numerical concerns. Many actual types are offered, protecting key topics together with warmth and mass transfers among levels, interfacial forces and fluid debris coalescence and breakup, among others.
This e-book is very compatible for college kids within the topic region, yet can also be an invaluable reference textual content for extra complex scientists and engineers.
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Extra info for Mathematical Modeling of Disperse Two-Phase Flows
8 2 Microscopic Formulation of Two-Phase Flows In Eq. 8), dðxÞ represents the Dirac delta generalized function, which is the derivative of the Heaviside generalized function (Appendix B). From Eqs. 8), the following topological equation for phase k can be deduced: @vk þ vI Á rvk ¼ 0 @t k ¼ 1; 2 ð2:9Þ From Eqs. 8)1, the following useful relations can also be deduced: Ànk Á rvk ¼ dðFÞjrFj dI , rvk ¼ Ànk dI ð2:10Þ In the above equation, dI is a generalized function having the different interfaces as its support.
N X N X Sj ! dj mj sj wj j¼1 ¼ N X dj Dj þ j¼1 N X I _ d sc À m dj j¼1 qc Tc Á nd dS Sj ð2:78Þ Introducing the following simpliﬁed notations: dd m ¼ N X j¼1 dj m j dd mw ¼ N X j¼1 dj mj wj dd mE ¼ N X j¼1 dj mj Ej ... 78) can be rewritten into the following ones: @ ðdd mÞ þ r Á ðdd mwÞ ¼ dd @t I _ d dS m S @ ðdd mwÞ þ r Á ðdd mwwÞ ¼ dd mg þ dd @t I I _ d vc dS þ dd m S rc Á nd dS S I @ v2c _ ðdd mEÞ þ r Á ðdd mEwÞ ¼ dd mw Á g þ dd md ec þ dS @t 2 S I I À dd qc Á nd dS þ dd rc Á vc Á nd dS S @ ðdd msÞ þ r Á ðdd mswÞ ¼ dd Dd þ dd @t ð2:80Þ S I q _ d sc À c Á nd dS m Tc S References Aris R (1962) Vectors, tensors and the basic equations of fluid mechanics.
19). 23). However, in practical applications, Fs dI is often assumed to be given by the sole interfacial tension force: Fs dI ﬃ ðrs r À rnrs Á nÞdI ð2:62Þ The total energy balance equation for the mixture is given by: ! ! @ v2 v2 q eþ v þrÁ q eþ @t 2 2 ¼ Àr Á q À r Á ðpvÞ þ r Á s Á v þ qv Á g þ EI ! 33). 8 ð2:64Þ Balance Equations for a Whole Fluid Particle We end this chapter by deriving the equations for a whole fluid particle. This particle can be a bubble or a droplet. The equations derived in this section will be useful for the derivation of the hybrid two-fluid model (Chap.
Mathematical Modeling of Disperse Two-Phase Flows by Christophe Morel