DIFFERENTIAL FORMULATION OF THE BASIC LAWS

Transcription

1 CHAPER DIFFERENIAL FORMULAION OF HE BASIC LAWS. Introdction Differential fmlation of basic las: Conseration of mass Conseration of momentm Conseration of energ. Flo Generation (i) Fced conection. Motion is drien b mechanical means. (ii) Free (natral) conection. Motion is drien b natral fces..3 Laminar s. rblent Flo Laminar flo: no flctations in elocit, ressre, temeratre, rblent flo: random flctations in elocit, ressre, temeratre, ransition from laminar to trblent flo: Determined b the Renolds nmber: Flo oer a flat late: Re V / 500,000 Flo throgh tbes: = Re t t t D 300 laminar trblent t t Fig...4 Conseration of Mass: he Continit Eqation.4. Cartesian Codinates

2 he rincile of conseration of mass is alied to an element ddd Rate of mass added to element - Rate of mass remoed from element = Rate of mass change ithin element m (m ) d d d m m (a) (b) Fig.. m (m ) d (.) Eressing each term in terms of elocit comonents gies continit eqation t 0 (.a) his eqation can be eressed in different fms: 0 t D V 0 V 0 t (.b) (.c) (.d) F constant densit (incomressible flid): V 0 (.3).4. Clindrical Codinates

3 t r r r r 0 3 (.4) r.4.3 Sherical Codinates t r r r sin 0 r (.5) r sin r sin.5 Conseration of Momentm: he Naier-Stokes Eqations of Motion.5. Cartesian Codinates Alication of Neton s la of motion to the element shon in Fig..5, gies F m a ( ) (a) d d d Alication of (a) in the -direction, gies F ( m) a Each term in (b) is eressed in terms of flo field ariables: densit, ressre, and elocit comonents: (b) Fig..5 Mass of the element: m ddd (c) Acceleration of the element a : d D a (d) dt t Sbstitting (c) and (d) into (b) F F Fces: (i) Bod fce bod D ddd g ddd (e) (g) (ii) Srface fce

4 4 Smming all the -comonent fces shon in Fig..6 gies F srface ddd (h) Combining the aboe eqations -direction: D g (.6a) B analog: -direction: D g (.6b) -direction: D g (.6c) IMPORAN HE NORMAL AND SHEARING SRESSES ARE EXPRESSED IN ERMS OF VELOSICY AND PRESSURE. HIS IS VALID FOR NEWOINAN FLUIDS. (See eqations.7a-.7f). HE RESULING EQUAIONS ARE KNOWN AS HE NAVIER-SOKES EQUAIONS OF MOION SPECIAL SIMPLIFIED CASES: (i) Constant iscosit DV g V V (.9) 3 (.9) is alid f: () continm, () Netonian flid, and (3) constant iscosit (ii) Constant iscosit and densit DV g V (.0) (.0) is alid f: () continm, () Netonian flid, (3) constant iscosit and (4) constant densit. he three comonent of (.0) are : g (.0) t

5 5 - - g t (.0) g t (.0).5. Clindrical Codinates he three eqations cresonding to (.0) in clindrical codinates are (.r), (. ), and (.)..5.3 Sherical Codinates he three eqations cresonding to (.0) in sherical codinates are (.r), (. ), and (. )..6 Conseration of Energ: he Energ Eqation.6. Fmlation he rincile of conseration of energ is alied to an element ddd A Rate of change of internal and kinetic energ of element B Net rate of internal and kinetic energ transt b conection C Net rate of heat addition b condction _ d d d Fig..7 D Net rate of k done b element on srrondin gs (.4) he ariables,,,,, and are sed to eress each term in (.4). Assmtions: () continm, () Netonian flid, and (3) negligible nclear, electromagnetic and radiation energ transfer. Detailed fmlation of the terms A, B, C and D is gien in Aendi A he folloing is the reslting eqation D D c k (.5) (.5) is referred to as the energ eqation is the coefficient of thermal eansion, defined as

6 6 (.6) he dissiation fnction is associated ith energ dissiation de to friction. It is imtant in high seed flo and f er iscos flids. In Cartesian codinates is gien b 3 (.7).6. Simlified Fm of the Energ Eqation Cartesian Codinates (i) Incomressible flid. Eqation (.5) becomes k D c (.8) (ii) Incomressible constant condctiit flid. Eqation (.8) is simlified frther if the condctiit k is assmed constant k D c (.9a) k t c (.9b) (iii) Ideal gas. (.5) becomes D k D c (.) V k D c (.3) Clindrical Codinates. he cresonding energ eqation in clindrical codinate is gien in (.4) Sherical Codinates. he cresonding energ eqation in clindrical codinate is gien in (.6)

7 .7 Soltions to the emeratre Distribtion he flo field (elocit distribtion) is needed f the determination of the temeratre distribtion. 7 IMPORAN: able. shos that f constant densit and iscosit, continit and momentm (f eqations) gie the soltion to,,, and. hs f this condition the flo field and temeratre fields are ncoled (smallest rectangle). F comressible flid the densit is an added ariable. Energ eqation and the eqation of state roide the fifth and sith reqired eqations. F this case the elocit and temeratre fields are coled and ths mst be soled simltaneosl (largest rectangle in able.). Basic la No. of Eqations ABLE. Unknons Energ Continit Momentm Eqation of State Viscosit relation (, ) Condctiit relation k k(, ) k 3 k.8 he Bossinesq Aroimation Flid motion in free conection is drien b boanc fces. Grait and densit change de to temeratre change gie rise to boanc. Accding to able., continit, momentm, energ and eqation of state mst be soled simltaneosl f the 6 nknons:.,,, and Starting ith the definition of coefficient of thermal eansion, defined as (.6)

8 8 his reslt gies ( ) (.8) Based on the aboe aroimation, the momentm eqation becomes DV g V (.9).9 Bondar Conditions () No-sli condition. At the all, 0 V (,0, 0 (f) (.30a) (,0, (,0, (,0, 0 (.30b) () Free stream condition. Far aa from an object ( ) (,, V (.3) Similarl, nifm temeratre far aa from an object is eressed as (,, (.3) (3) Srface thermal conditions. o common srface thermal conditions are sed in the analsis of conection roblems. he are: (i) Secified temeratre. At the all: (,0, (.33) s (ii) Secified heat fl. Heated cooled srface: (,0, k q o (.34).0 Non-dimensional Fm of the Goerning Eqations: Dnamic and hermal Similarit Parameters Eress the goerning eqations in dimensionless fm to: () identif the goerning arameters () lan eeriments (3) gide in the resentation of eerimental reslts and theetical soltions Dimensional fm: Indeendent ariables:,, and t

9 Unknon ariables are:,,, and. hese ariables deend on the f indeendent ariables. In addition arios qantities affect the soltions. he are, V,, L, g,, s, and Flid roerties c, k,,, and 9 Geometr.0. Dimensionless Variables Deendent and indeendent ariables are made dimensionless as follos: V V, V ( ) V, ( ( s ) g, g, ) g V,,, t t L L L L Using (.35) the goerning eqations are reritten in dimensionless fm. (.35).0. Dimensionless Fm of Continit No arameters aear in (.37) D V 0 (.37).0.3 Dimensionless Fm of the Naier-Stokes Eqations of Motion DV Gr g P V (.38) Re Re Constant (characteristic) qantities combine into to goerning arameters: Re V L V 3 L, Renolds nmber (iscos effec (.39) g s L Gr, Grashof nmber (free conection effec (.40).0.4 Dimensionless Fm of the Energ Eqation Consider to cases: (i) Incomressible, constant condctiit

10 0 D c (.4a) RePr Constant (characteristic) qantities combine into to additional goerning arameters: Re c c c / Pr k k / c V ( s, Prandtl nmber (heat transfer effec (.4), Eckert nmber (dissiation effect high seed, large iscosi (.43) ) (ii) Ideal gas, constant condctiit and iscosit D RePr D c c Re (.4b) No ne arameters aear..0.5 Significance of the Goerning Parameters Dimensionless temeratre soltion: f (,,, t ; Re, Pr, Gr, Ec) (.45) NOE: Si qantities:,, s, V, L, g and fie roerties c, k,,, and, are relaced b f dimensionless arameters: Re, Pr, Gr and Ec. Secial case: negligible free conection and dissiation: o goerning arameters: f (,,, t ; Re, Pr) (.46) Geometricall similar bodies hae the same soltion hen the arameters are the same. Eeriments and crelation of data are eressed in terms of arameters rather than dimensional qantities. Nmerical soltions are eressed in terms of arameters rather than dimensional qantities..0.6 Heat ransfer Coefficient: he Nsselt Nmber Local Nsselt nmber: N = f ( ; Re, Pr, Gr, Ec) (.5) Secial case: negligible boanc and dissiation:

11 N = f ( ; Re, Pr) (.5) Free conection, negligible dissiation N = f ( ; Gr, Pr) (. 53) F the aerage Nsselt nmber, is eliminated in the aboe..9 Scale Analsis A rocedre sed to obtain der of magnitde estimates ithot soling goerning eqations.