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### C4-Isothermal Equations of Change

Course: PGE 322K, Spring 2012
School: University of Texas
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Word Count: 2046

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of Equations Change for Isothermal Systems The equation of continuity (or mass balance equation) Mass influx (inward arrows) Mass balance over a fixed control volume xy z y z u x x u x x x t y z x t xz u y u y y y y y x y u z z u z z z t ux u y uz . u x t y z For incompressible fluids: .u 0 Mass outflux (outward arrows) x z...

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of Equations Change for Isothermal Systems The equation of continuity (or mass balance equation) Mass influx (inward arrows) Mass balance over a fixed control volume xy z y z u x x u x x x t y z x t xz u y u y y y y y x y u z z u z z z t ux u y uz . u x t y z For incompressible fluids: .u 0 Mass outflux (outward arrows) x z Equations of Change for Isothermal Systems The equation of motion Derivation of the equation of motion using momentum principle The balance for x-momentum over a fixed control volume xyz with ux yz xx x xx x x xz yx yx y t xy zx z zx z z xyz g x ix ui u x p ix ix y y i x, y , z xx yx zx u x gx t y z x 1 Equations of Change for Isothermal Systems The equation of motion in a general form u j , z i uiu j p ij ij g j t i x, y ij p gj ui u j j i x , y , z i i x , y , z i For Newtonian flow u j u 2 i ij .u ij xi x j 3 Equations of Change for Isothermal Systems Simplification of the equation of motion 1. Constant density and viscosity : Navier-Stokes equation For incompressible fluids: .u 0 i x, y, z ij i 2u j i i x, y, z 2u j 2 u , z ii 2u j j i x , y .u 0 u uiu j u j i ui ui ij i x , y , z i i x, y , z u j u j i ui ui i i x, y , z i x, y,z u j u j u j i ui ui i ui i i x, y , z i x, y,z i x, y , z 2 Equations of Change for Isothermal Systems Simplification of the equation of motion 1. Constant density and viscosity : Navier-Stokes equation For incompressible fluids: .u 0 i x, y , z ij i 2u j u uiu j ui ij i x , y , z i i x, y, z Equations of Change for Isothermal Systems Simplification of the equation of motion 1. Constant density and viscosity : Navier-Stokes equation The general equation of motion is reduced to u j t u j p 2 u j g j i j u j p 2 u j g j i j u i x, y , z i u j ui t i x , y , z 3 Equations of Change for Isothermal Systems Simplification of the equation of motion 2. Navier-Stokes equation with negligible acceleration term Stokes flow equation: for very slow flow u j u j p 2 ui u j g j j i t x , y , z i j x, y , z 0 p 2u j g j 0 j j x, y , z This equation is important in lubrication theory, particle motion in suspension, and flow in porous media Equations of Change for Isothermal Systems Simplification of the equation of motion 3. Negligible viscous forces in general equation of motion Euler equation: u j t uiu j iij p g j j i x , y , z i i x, y , z 0 With the help of the equation of continuity, Euler equation may be rewritten as u j t u j p ui gj i j i x, y , z 4 Equations of Change for Isothermal Systems Simplification of the equation of motion 3. Negligible viscous forces in general equation of motion (cont.) Euler equation with flow under steady-state conditions u j p ui gj i j i x, y , z From this equation, one can derive Bernoulli equation for steady-state streamline flow B 12 1 u dp gh 2 C D C D B is constant along the streamline in general For example: 1 2 pC 1 2 p D uC uD 2 2 BC BD Equations of Change for Isothermal Systems Simplification of the equation of motion 4. Small variation of density and constant viscosity Boussinesq approximation for temperature-dependent density = (T) - For small variation of with respect to T, the continuity equation has the small variation of respect to T, the continuity equation has the incompressible form .u 0 - and the equation of motion is slightly different from Navier-Stokes equation u j t u j p , z ui i j 2u j T g j i x, y is the density evaluated at average temperature 5 Equations of Change for Isothermal Systems Three components of the equation of motion Momentum influx (inward arrows) X-component u x ui u x p ix ix g x t i x , y , z i ix y z Y-component u u p g i y iy iy y t i x , y , z i iy x u y Momentum outflux (outward arrows) Z-component x y z u z u u p g iz iz z iz t i x , y , z i iy Equations of Change for Isothermal Systems A methodology for solving flow problems Statement of flow problems 1. Define the domain boundary. 2. Choose a coordinate system such that there are more vanishing terms (i.e. coordinate system such that there are more vanishing terms (i components of velocity, velocity gradients, and forces). 3. State assumptions to simplify the problem (i.e. set zero more terms). 4. List non-zero components of velocity, velocity gradient, and forces. 5. Apply directly the equations of continuity and motions for each phase. 6. Identify initial and boundary conditions, and then solve the velocity equations. 7. Determine other quantities related to velocity profiles. 6 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Flow of falling film The equation of continuity (see Table B.4) u u u u y x z z 0 x t y z z uz is uniform in z-direction The equation of motion (see Table B.5) u x u u u ux x u y x uz x t x y z yx zx p xx x x y z gx u y u y u y u y ux uy uz t x y z yy zy p xy gy y z y x yz u z u u u p u x z u y z u z z xz zz g z t x y z z z x y Equations Change of for Isothermal Systems Applications of the equations of continuity and motion Flow of falling film The equation of continuity (see Table B.4) uz 0 z The equation of motion (see Table B.5) p p g x g sin x x p 0 y p 0 xz g cos z x 0 7 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion The x-component equation gives p gx sin f ( y , z ) S ince p / y 0 and p x 0 f ( y , z ) patm We have p ( x ) gx sin patm and d xz g cos cos dx Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Flow in tube at steady state The equation of continuity (see Table B.4) 1 1 rur u u z 0 t r r r z The equation of motion (see Table B.6) ur 1 u u p u u u2 1 2u 2u 2 u ur r r uz r rur 2 2r 2r 2 gr t r r r z z r r r r r r u 1 u u u u u u 1 p 1 2u 2u 2 u ur uz r ru 2 2 2 2 r g t r r r z r z r r r r r 2 1 uz u u p u u 2u 1 u ur z z uz z ruz 2 2z 2z gz z r z z r r r r r t 8 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Flow in tube at steady state The equation of continuity (see Table B.4) uz 0 z The equation of motion (see Table B.6) p r 1 p 0 r p 1 0 ruz gz z r r r 0 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Rotational vortex at steady state The equation of continuity (see Table B.4) 1 1 ru r u u z 0 r r r t z z0 z R The equation of motion (see Table B.6) r 2 1 ur u u 2u 2 u p u u u2 1 u ur r r uz r rur 2 2r 2r 2 gr r r r z r r z r r r r t u 1 u u u u u u 2u 1 p 1 2u 2 u ur uz r ru 2 2 2 2 r g t r r r r z r z r r r r uz u u u u ur z z uz z t r z r 1 p 1 2u 2u ruz 2 2z 2z z z r r r r gz 9 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Rotational vortex at steady state The equation of continuity (see Table B.4) 1 u 0 r The equation of motion (see Table B.6) u 2 p r r 1 0 ru r r r p 0 gz z z0 z R r Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Rotational vortex at steady state Integrating the -component equation gives u C 1 C1r 2 r 2 Applying boundary conditions u rR R and u r 0 C1 2 finite C2 0 u r 10 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Rotational vortex at steady state Integrating r- and z-component equations u 2 p 1 2 r p 2 r 2 f1 ( , z ) r r 2 p and g p gz f 2 ( r , ) z Since p ( r , z ) , choosing f1 gz C and f 2 2 r 2 C g ives p gz 1 2r 2 C 2 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Rotational vortex at steady state Applying boundary conditions zz p r 00 patm 1 p patm g z z0 2 r 2 2 The shape of the air-liquid interface 2 2 z z0 r 2g 11 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Pitot tube p 1 2 p1 1 2 p2 u1 u2 2 2 2 0 2 p2 p1 u1 p2 gh2 with p1 gh1 1/ 2 u1 2 g h2 h1 1/ 2 p2 1 u12 p1 is called 'stagnation presure' 2 1 u12 is called 'dynamic presure' 2 Equations of Change for Isothermal Systems Applications of the equations of continuity and motion Orifice flow Distribution of p-patm at the orifice p 1 2 p atm u atm gh 2 u 2 gh loss of potential energy = gain of kinetic energy w Ac u Ac 2 gh Ac Aorifice for rounded orifice, otherwise 60% Aorifice for sharp-edged orifice 12 Equations of Change for Isothermal Systems Dimensional analysis of the equations of change Non-dimensionalizing Navier-Stokes equation Consider the x-component of the Navier-Stokes equation u x u u u p u x x u y x u z x 2u x g x x y z x t Introducing the following dimensionless variables using characteristic length (l0), characteristic velocity (u0), and characteristic pressure (P0) x x / l0 ; t u0t / l0 ; y y / l0 ; z z / l0 ; P P0 u x u x / u0 ; P 2 u0 Equations of Change for Isothermal Systems Dimensional analysis of the equations of change Non-dimensionalizing Navier-Stokes equation Inserting the dimensionless variables into the x-component Navier-Stokes equation gives 2 2 2 u0 ux u0 ux u0 ux u0 ux ux u y uz l0 / u0 t l0 x l0 y l0 z 2 u0 p u0 2 2 ux g x l0 x l0 ux ux ux ux p 2 l0 g x u ux u y uz 2 x u0 t x y z x u0 l0 13 Equations of Change for Isothermal Systems Dimensional analysis of the equations of change Non-dimensionalizing Navier-Stokes equation (cont.) Or ux ux ux ux p 1 2 1 u x u y u z ux Fr t x y z x Re 1 1 l0 g x 2 Re u0 l0 Fr u0 Re and Fr are called Reynolds and Froude numbers, respectively. The dimensionless equation shows that - Its solutions are independent on the dimensions of flow domain, projection of gravity force on the x-direction, and fluid properties (i.e. viscosity and density). - Its solutions are dependent on dimensionless numbers (i.e. Re and Fr) - It can be simplified for very high or low values of Re and Fr. For example Re ; ux ux ux ux p 1 ux u y uz t x y z x Fr 14
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University of Texas - PGE - 322K
Mechanisms of Energy TransferMechanisms of energy transferEnergy transfer by work p xx yx zx xyp yy zy xz yz p zz Energy influxes (inward arrows)yP(x,y,z)zxHeat sourcesEnergy outfluxes (outward arrows)Mechanisms of Energy TransferMech
University of Texas - PGE - 322K
Shell Energy BalanceEnergy conservation for homogeneous mediaEnergy influxes (inward arrows)yP(x,y,z)zxHeat sourceEnergy outfluxes (outward arrows) Energy balance for an unsteady-state systemRate ofenergy inRate ofenergy outRate of workby f
University of Texas - PGE - 322K
Non-isothermal Equations of changeEnergy conservation for homogeneous mediaTotal energy balance( E K U ) .( E K U )u .q t . p u . .u u.g SKinetic energy balanceEK . uEK . pu p .u . .u : u u.g tInternal energy balance( U ) .( U u ) .q p .u : u S
University of Texas - PGE - 322K
Homework Set #1:Due date: 26/01/12 by 11:00 PMProblem #1:Consider a spherical particle falling freely in a large fluid body. It is found experimentally thatthe terminal settling velocity u of the particle is dependent on: (1) particle diameter D; (2)
University of Texas - PGE - 322K
Homework Set #2:Due date: 02/09/12 by 11:00 amProblem #1:A thin plate of 30x60x0.1 cm3 size is falling vertically in a narrow slit as shown in Fig 1. Theplate thickness is assumed to be negligible as compared to the slit width of 1.5 cm. The slit isf
University of Texas - PGE - 322K
Homework Set #3:Due date: 02/16/12, by 11:00 AMProblem #1:Problem 2B.3 at the end of chapter 2 of BSL, parts (a), (b) and (c) only. Follow the terminologyin the diagram in Figure 2B.3 on p. 63. Note that x = 0 in the center of the slit in this figure.
University of Texas - PGE - 322K
Homework Set #4:Due date: 03/08/12, by 11:00 amProblem # 1:A Newtonian fluid fills a long cylindrical tube of radius R held vertically as shown. The tubewall is moving upward with velocity V &lt; 0 (V is less than zero if the tube is moving up and z isd
University of Texas - PGE - 322K
Homework Set #5:Due date: 03/22/12, by 11:00 am.Problem #1:Solve problem 10.B.1 in BSL. Only questions a, b, and c.Problem #2:Read the statement of Problem 10.B.4 in BSL, but do NOT answer questions (a) and (b) in thetext. Answer the followings:(a)
University of Texas - PGE - 322K
Homework Set #6:Due date: 04/03/12, by 11:00 am.Problem #1:Solve Problem 3A.1 in Chapter 3 of BSL.Problem #2:The annular space between two coaxial long circular pipes is filled with a viscous, Newtonianfluid. The radii of the inner and outer wetted
University of Texas - PGE - 322K
Homework Set #7:Due date: 04/12/12, by 11:00 am.Problem # 1:The crude oil with temperature-independent physical properties is in fully developed laminarflow between two flat surfaces placed a distance 2B apart. For z &lt; 0 the fluid is uniform at T =T1
University of Texas - PGE - 322K
Please read the following parts in the textbook (Bird et al, 2nd edition)1. Review of mathematics: (to be completed within two weeks from the date thisassignment posted)Appendix A2. Chapter 1 : the following sections (to be completed by 01/28/12)- 1.
University of Texas - PGE - 322K
Reading assignments:1. Chapter 9 :- Sections: 9.1, 9.2, 9.5, 9.6, 9.7, 9.82. Chapter 10 :- Sections: 10.4, 10.7, 10.83. Chapter 11 :- Section: 11.1, 11.4:4. Chapter 12 :- Section: 12.1
University of Texas - PGE - 322K
Solutions for homework set #1:Problem #1:Consider a spherical particle falling freely in a large fluid body. It is found experimentally thatthe terminal settling velocity u of the particle is dependent on: (1) particle diameter D; (2) thebuoyant weigh
University of Texas - PGE - 322K
Solutions for homework set #2:Problem #1:A thin plate of 30x60x0.1 cm3 size is falling vertically in a narrow slit as shown in Fig 1. Theplate thickness is assumed to be negligible as compared to the slit width of 1.5 cm. The slit isfilled with a 50-c
University of Texas - PGE - 322K
Homework Set #3: SOLUTIONSProblem #1:Problem 2B.3 at the end of chapter 2 of BSL, parts (a), (b) and (c) only. Follow the terminologyin the diagram in Figure 2B.3 on p. 63. Note that x = 0 in the center of the slit in this figure.Note also that the an
University of Texas - PGE - 322K
Solutions for homework set #4Problem # 1:cylinder wall movesupward with velocity VzNewtonian fluidinside tubeRa. The last equation you could use is equation 3.6-7 in the example problem, or equation 2.3-16in Chapter 2. Note that everything is ide
University of Texas - PGE - 322K
Solutions for Homework Set #5Problem #1:Solve problem 10.B.1 in BSLAnswer:a. Equation setupDue to the motionless body of fluid, heat conduction is the only mechanism of energy transferaccounting for the temperature distribution in the fluid. Obvious
University of Texas - PGE - 322K
Problem #1:Solve Problem 3A.1 in Chapter 3 of BSL.Answer:Equation 3.6-31 describes the torque required to turn an outer cylinder at an angular velocity o.This equation is the same for the torque required to turn an inner rotating shaft at an angularv
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 3110
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Toledo - ACCT - 4310
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