abj1 Lecture 6.0 : Finite Control Volume Formulation of Physical Laws and C-Mass 1.Finite Control Volume Formulation of Physical Laws 2.Conservation of Mass  Special Case 1:Incompressible Flows (Steady and Uniform Density Field)  Special Case 2:Steady Density Field (e.g., Compressible Flows) 3.Conservation of Volume for Incompressible Liquids  An illustration of the wider use of the Reynolds Transport Theorem to convert from the material volume MV viewpoint to control volume CV viewpoint

abj2 Very Brief Summary of Important Points and Equations [1]  C-Mass:  C-Volume: (for incompressible liquids) Physical Laws RTT  Finite CV formulation of physical laws:

abj3 Finite Control Volume Formulation of Physical Laws Physical Laws [1]: RTT [2]: MV(t), MS(t) CV(t), CS(t) Efflux through CS Increase in CV Finite CV Formulation of PL, [1] = [2]: Physical Laws RTT

abj4 Finite Control Volume Formulation of Physical Laws MV(t), MS(t) CV(t), CS(t) Efflux through CS Increase in CV Mass Entropy Energy Linear Momentum Angular Momentum

abj5 Conservation of Mass (C-Mass) MV(t), MS(t) CV(t), CS(t) Efflux through CS Increase in CV C-Mass

abj6 Special Case 1: Incompressible Flow (Steady and Uniform Density Field) 1.CV is stationary and non-deforming 2.Incompressible flow (steady and uniform density field) C-Mass Net mass efflux through CS = 0 Net volume efflux through CS = 0 C-Mass Unsteady Term:

abj7 Special Case 2: Steady Density Field (e.g., Compressible Flow) 1.CV is stationary and non-deforming 2.Steady density field (need not be uniform) C-Mass Net mass efflux through CS = 0 C-Mass Unsteady Term: Note that – unlike the previous incompressible flow case - in this case the density at various parts of (CV and) CS may not be equal. Hence, we have only the net mass efflux – and not necessarily net volume efflux - vanish. Examples are in the case of steady (density), compressible flows where the density field is steady but not uniform.

abj8 Conservation of Volume for Incompressible Liquids An illustration of The Wider Use of The Reynolds Transport Theorem to convert from the material volume MV viewpoint to control volume CV viewpoint

abj9 Conservation of Volume for Incompressible Liquids Illustration of The Wider Use of The Reynolds Transport Theorem  The RTT and Finite CV Formulation of Physical Law are not applicable only to main “physical laws.”  They are applicable to any material volume MV so long as we have the relation for the time rate of change of the property N of the material volume  The Conservation of Volume for incompressible liquids below illustrates this point. Physical Laws RTT

abj10 Conservation of Volume (for incompressible liquids)  For incompressible liquids (not incompressible flow), its volume as we follow the material volume does not change with time. Thus, we have Physical Laws: RTT: Thus, we can formulate the conservation of volume for incompressible liquids as C-Volume: (for incompressible liquids)

abj11 Special Case :Incompressible Liquids + Stationary and Non-Deforming CV 1.Stationary and non-deforming CV 2.Incompressible liquids C-Volume: (for incompressible liquids) Unsteady Term: C-Volume: (for incompressible liquids) Net volume efflux through CS = 0 Note that neither steady nor uniform density field assumption is necessary so long as 1.CV is stationary and non-deforming 2.It is an incompressible liquids. That is, if we follow any one material volume (even though the density field inside the coincident CV may not be steady or uniform), its volume does not change.