Prof. P.-T. BrunDepartment of Chemical and Biological EngineeringCBE 341 – Mass, Momentum, and Energy TransportFall 2017Lecture 3 09/17/171. Problems of practical interest cannot always be solved exactly by analytical methods2. We wish to conduct as few experiments as possible…3. …while working with a suitably scaled model of the geometry of interest4. expose dimensionless groups (or parameters) which can be used to correlate the experimental data, 5. which allows us to use the correlations across geometric scales.6. examine the exact equations of motion critically. By recasting these equations in a dimensionless form, one can recognize the relative importance of various terms in the model equations.Does not require as a starting point the differential equations describing the evolution of various variabless⇢V2Inertial stress Gravitational stress⇢gLViscous stress Interfacial stress µV/L/LDimensionless Groups of Significance in Fluid Mechanics The Reynolds number, Re, is a ratio of the inertial and viscous stresses, inertialviscousVLReμ= = The Froude number, Fr is taken to be the inertial and gravitational stresses.2inertialgravitationalVFrgL= =In some problems such as bubbles rising in a liquid, one refers to an Eötvos number, ,Eo which is a ratio of the buoyancy stress to interfacial stress. ( )2buoyancy.interfacialgLEo= = So, Eo appears to be very similar to Bond number. Here ( ) is the density difference. The interfacial stress is also referred to as capillary stress. The Weber number, ,We is a ratio of the inertial and interfacial stresses. 2inertialinterfacialV LWe = = The capillary number, ,Cais a ratio of the viscous and capillary (interfacial) stresses. viscouscapillaryVCaμ= = The Bond number, ,Bo is a ratio of the gravitational and interfacial stresses. 2gravitationalinterfacialgLBo=
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