Department of Chemical Engineering University of Tennessee Prof. David Keffer. Course Lecture Notes SIXTEEN

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1 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow Deamen of Chemical Enee Uniesi of Tennessee Pof. Daid Keffe Couse Lecue Noes SIXTEEN SECTION.6 DIFFERENTIL EQUTIONS OF CONTINUITY SECTION.7 DIFFERENTIL EQUTIONS OF MOTION Geankolis ae We hae soled ssems wih maeial and mechanical ene balances on macoscoic conol olumes. We hae also used shell balances, which use a conol olume wih a diffeenial dimension one diecion, o oba eloci and sess ofiles ha diecion. To adance ou sud, we now need o conside a conol olume wih diffeenial elemens all hee saial dimensions.. Goals: Deie eneal diffeenial equaions of conui (mass balance). Deie eneal diffeenial equaion of chane (momenum balance). Use hese enealied equaions o sole a aicula oblem b kee onl ems elean o he oblem while discad unnecessa ems. B. Tes of deiaies: 1. aial ime deiaie chane densi a a fied o (,,) wih ime We ae a canoe on a ie, addl wih jus enouh effo so ha we don moe a all. Then we look down and see a he same o he ie. (Fied osiion, diffeen aicles (elemens) of fluid.). oal ime deiaie d d d d d d d d d d chane densi of he fluid while we moe aound wih some eloci -1

2 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow d d d d T d d We ae a canoe on a ie, addl aound, mabe coss-seam, mabe useam, wih a ien eloci,, a eco. Then we look down and see a diffeen os ie and diffeen aicles of he fluid. (Diffeen osiion, diffeen aicles (elemens) of fluid.). Subsanial ime deiaie (a subse of oal ime deiaies) D D chane densi of he fluid while mo wih he fluid eloci [ ] T We ae a canoe on a ie, allow ouseles o moe wih he ie (no addl). Then we look down and see a diffeen o ie bu fo he same aicle of he fluid. (Diffeen osiion, same aicles (elemens) of fluid.) C. Lea leba and Veco Calculus Oeaions: 1. Gadien of a scala T The adien of a scala is a 1 eco of he deiaie of ha scala all saial dimensions.. dieence of a eco -

3 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow The dieence of a eco is a scala, eesen he do oduc of he adien oeao and he eco.. dieence of a mai The dieence of a mai is a 1 eco, eesen he do oduc of he adien oeao and he mai. 4. Lalacian of a scala This is a scala. The Lalacian is he sequenial oeaion of fis he adien oeao and hen he dieence oeao. Check ou some ules of eco aleba and eco calculus equaions (.6-4 o equaion.6-16), Geankolis, ae D. Deie he eneal diffeenial equaion of conui accumulaion - ou eneaion - consumion Defe diffeenial olume elemen as shown on ae167, Geankolis. Defe he fie ems he mass balance. Thee is no eneaion o consumion of he fluid. en con 0 acc V -

4 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow -4 ou Pu hese fie ems mass balance: Diide b diffeenial olume: Reaane o a fom econiable as he defiion of a deiaie: Take limis as diffeenial elemens aoach 0 and al he defiion of he deiaie: Now conside he law fo he deiaie and he same fo and D

5 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow D This is he conui equaion. Fo he case of an comessible fluid (consan densi) a sead o unsead sae: 0 This is a mass balance. I ma no look like i bu i is. Jus o back houh he deiaion and see ha his is noh bu an eession of accumulaion - ou eneaion - consumion when hee is no eneaion o consumion and when he fluid is comessible. This equaion does no assume sead sae, een houh hee is no ime deiaie he equaion. This is a fis ode aial diffeenial equaion PDE) Eamle.6-1. ae 168 The conui equaion can also be eessed sheical and cldical coodaes, which ae useful if ou hae a ssem which nauall lends iself o ha ssem, as a cicula ie lends iself o cldical coodaes. In cldical coodaes: is sill ue bu cosθ, sθ, and so he conui equaion becomes: 1 In sheical coodaes: is sill ue bu ( ) 1( ) ( ) θ θ sθcosφ, sθsφ, and cosθ SO: -5

6 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow -6 so he conui equaion becomes: φ θ θ θ θ θ φ s 1 s s 1 1 Secion.7 Now do he same analsis fo a momenum balance E. Deie he eneal diffeenial equaion of chane (momenum balance) accumulaion - ou eneaion - consumion Defe diffeenial olume elemen as shown on ae 167. Momenum is a eco. Sce hee is no sic diffeence beween,, and coodaes, we can deie he equaion of chane fo he -comonen of momenum and hen make analoous saemens abou he and comonens. Look onl a -comonen of momenum V acc Momenum can flow and ou b conecion: ou Momenum can flow and ou b molecula diffusion: ou Momenum can be eneaed he diffeenial olume b a bod foce like ai: V en

7 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow -7 Momenum can be eneaed he diffeenial olume b a ne foce ac on he elemen, due o he diffeence essues: en Pu hese fie ems mass balance: Diide b olume, eaane, and ake limis and al he defiion of he deiaie Now, conside aa he ule fo deiaies of a oduc: and he same fo and, i.e. Subsiu hese ules o ou momenum balance, we hae:

8 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow ( ) ( ) ( ) ( ) ( ) ( ) Recall, he subsanial deiaie of he -momenum comonen has he defiion: ( ) ( ) ( ) ( ) ( ) D ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D ( ) bu us he oduc ule fo diffeeniaion aa: ( ) D( ) D D D Fom he mass balance: ( ) so ( ) D( ) D D ( ) D( ) D ( ) -8

9 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow nd he momenum balance becomes: D ( ) D This is he equaion of moion he -diecion. I is a momenum balance. The - and -comonens of he momenum ae obaed a ecisel analoous manne. The look like: D D which eco noaion looks like: D Geankolis,.7-1 This momenum balance is ue fo an conuous medium. B. Equaions fo sesses us Newon s Law of iscosi (ae 17-17) 1. Shea sess comonens fo Newonian fluids ecanula coodaes See Geankolis ae 17, eqns (.7-14 o.7-0). Shea sess comonens fo Newonian fluids cldical coodaes See Geankolis ae 17-17, eqns (.7-1 o.7-7). Shea sess comonens fo Newonian fluids sheical coodaes See Geankolis ae 17, eqns (.7-8 o.7-4) Fo he qualiaie elanaion of he basis of hese equaions, ou mus o o n Inoducion o Fluid Dnamics b Sanle Middleman, Wile, New Yok, 1998, ae Eame ecanula case: oba 1-D esuls fom -D case b look a equaion.7-17 when he comonen of he eloci is eo. -9

10 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow -10 Subsiue he defiions of he Newonian sess o he equaion of chane. Oba he equaion of chane fo a Newonian fluid. Eamle ien fo comonen ecanula coodae ssem. (.7-5) D C. Naie-Sokes equaions (Equaion of chane fo comessible Newonian fluid) (ae ) When he densi is consan (comessible) and he iscosi is consan (isohemal condiions), he equaions of (.7-10 o.7-1) combed wih Newon s law equaions (equaions.7-14 o.7-4) become he Naie-Sokes equaions. 1. Equaions of chane fo comessible Newonian fluid ecanula coodaes Sa wih equaion (.7-5), eneal equaion of chane fo Newonian fluid and assume consan densi: D D D

11 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow -11 D D D [ ] D Fom he mass balance, we know ha fo an comessible fluid, he dieence of he eloci is eo. D This is he equaion of chane fo an comessible Newonian fluid he -diecion, us ecanula coodaes. The and equaions look like: D

12 D. Keffe - ChE 40: Hea Tansfe and Fluid Flow D and he hee equaions can be eessed eco noaion as D. Equaions of chane fo comessible Newonian fluids cldical coodaes See Geankolis ae 174, eqns (.7-40 o.7-4). Equaions of chane fo comessible Newonian fluids sheical coodaes See Geankolis ae , eqns (.7-4 o.7-46) -1

Control Volume Derivation

Control Volume Derivation School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass