TAMU CHEN 304 - L4 (8 pages)

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L4



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L4

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Pages:
8
School:
Texas A&M University
Course:
Chen 304 - Chem Engr Fluid Ops
Chem Engr Fluid Ops Documents

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Mathematical toolkit Leibniz Formula for Differentiating Integrals In transport problems we frequently deal with differentiating expressions such as G t f x t dV 32 V t where a field f is changing with time and the closed region of interest V is also changing with time A generalization of Leibniz s rule in 1 D to multi dimensional space takes the form dG d f f x t dV dV n vS f dS dt dt V t V t t S t 33 where the 2nd term accounts for the change in V Here vS denotes the velocity of the surface which can be a function of position and time and n is unit outward normal see figure below When the volume V is invariant with time the 2nd term clearly drops out n vs V t S t If we have a vector quantity b defined as b a x t dV 34 V t then db d a a x t dV dV n v S a ds dt dt V t V t t S t 34 35 CBE341 Lecture 4 9 20 Mass conservation equation Conservation equations There are two major parts to any transport model The first is the conservation equation which is a mathematical statement of established physical laws concerning mass momentum and energy The second is the constitutive equation describing the manner in which the material under investigation responds to stimuli We will begin our analysis by writing down the mass conservation principle in a mathematical form Conservation equation for mass Conservation of mass is simply rate of mass accumulation rate of mass in rate of mass out Let us first consider equations describing conservation of scalar quantities A control volume is any closed region in space selected for its usefulness in formulating a desired balance equation v n vs V t S t A control volume is assumed to have a volume V t and surface area S t where t is time n denotes unit outward normal vector v r t denotes the fluid velocity field and r is the position vector vS r t defined only on the surface S t denotes the velocity of the control surface itself 35 Let r r t be the density The mass flux at any point relative to the same frame in which r is defined is given by r r t v r t The mass flux at a point on the surface S t relative to the surface is given by r r t v r t vs r t It then follows that the net rate at which mass enters the control volume is given by n r r t v r t v r t dS s S t The total mass inside the control volume at any given time is given by r r t dV V t Therefore the rate of accumulation of mass is given by d r r t dV dt V t



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