TAMU CHEN 304 - L4 (8 pages)

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L4

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L4

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8
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Texas A&M University
Course:
Chen 304 - Chem Engr Fluid Ops
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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 Then the principle of mass conservation tells us that d r r t dV S t n r r t v r t vs r t dS dt V t 36 Using the Leibniz rule equation 33 the left hand side can be written as d r r r t dV r t dV n r r t vs r t dS dt V t t V t S t 37 Combining equations 36 and 37 we get r r t dV n r r t v r t dS t V t S t 38 We can express the right hand side of equation 38 as a volume integral using Gauss s divergence theorem see equation 3a in page 10 36 n r r t v r t dS S t r r t v r t dV 39 V t Introducing this into equation 38 we get r r t r r t v r t dV 0 t V t 40 As this is true for any arbitrary volume V t we conclude that the integrand must be exactly equal to zero at every point i e r r t r r t v r t 0 t 41 which is the differential form of the mass conservation equation This is also known as the Continuity equation From now on we will use a simpler notation and simply refer to r r t as r and so on Equations 36 and 41 then read as d dt r dV V t n r v v dS S 42 S t r rv 0 t 43 Equation 43 can be rewritten as r v r r v t v r v x 44 r r r vy vz x y z 37 r v r vx v v r x r z x y z The left hand side of equation 44 is commonly written compactly as Dr r v r r v Dt t Dr So Dt 45 For incompressible substances r constant everywhere and we get from equation 45 that v 0 46 Equation 46 is simply a special case of equation 45 and is limited to incompressible fluids We can employ either equation 42 or 45 to ensure that mass is conserved in our analysis of any flow problem Which of these two forms of the mass conservation equation is employed depends on the problem We typically use equation 42 when we are trying to get approximate solution for the macroscopic flow characteristics we will see examples of this approach when we study macroscopic balances The differential form given in equation 45 is used when we seek detailed solutions we will see examples of this approach when we study microscopic balances Conservation equation for mass differential element derivation Consider a differential element of fluid in this body of fluid as shown by rectangular box in the figure below 38 dz z y dx x dy The edges are aligned with the coordinate axes in a coordinate system in some Euclidean frame The lengths of the sides are dx dy dz The coordinates of the circled point are x y z We note that for any given surface area dA dA vn the mass flux through it is given by rvn where r is the density of the fluid and vn is the fluid velocity normal to the surface The velocity of the fluid in the x y and z direction are u v and w respectively The mass entering the shaded face of our differential element is rw z dxdy and exiting through the opposite face is rw z dz dxdy Analogous expressions can be derived for the other four faces and the mass balance over the entire element becomes r dxdydz ru x dx ru x dydz r v y dy r v y dxdz r w z dz r w z dxdy 0 t Dividing by dxdydz and taking the limit as dx dy and dz go to zero yields 39 r ru r v r w 0 t x y z This can be written compactly as r r v 0 t which is the same as equation 43 derived earlier Thus …

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