RTT relates between the system approach with finite control volume approach for a system property:
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1 8//8 ME 3: FLUI MECHANI-I r. A.B.M. Tofiqe Hasan Professor eparmen of Mecanical Enineerin Banlades Universiy of Enineerin & Tecnoloy (BUET, aka Lecre- 8//8 Flid ynamics eacer.be.ac.bd/ofiqeasan/ bd/ofiqeasan/ r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 eynolds Transpor Teorem (TT TT relaes beween e sysem approac wi finie conrol volme approac for a sysem propery: p B sys ρ ρb dv b V n Conservaion of mass M (mass B M (mass, b mass M sys ρ dv V n ρ dv V B = any exensive propery (sc as mass, momenm, enery ec. b = any inensive propery per ni mass (sc as mass per mass, momenm per mass, ec. V n V n General Coniniy Eqaion in ineral form applied o a conrol volme (seady/nseady Coniniy Eqaion in ineral form applied o a conrol volme for Seady flow r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8
2 8//8 Flow ro Variable area passae Seady flow coniniy i eqaion = ρ dv V n V n V n V n ( ( V V No conribion from oer s ( flow A A V A V (Inflow ve, Oflow +ve A A AV AV ( m m ; m in m o V V AV consan Mass flow rae is consan across e A In case of incompressible flow (densiy is consan A V A V ( Q Q ; Q in Q o AV Consan Volme flow rae (discare is consan across e Area and velociy inversely proporional in case of incompressible flow. Wa abo compressible flow??? r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 3 Incompressible seady laminar waer flows develops in a srai pipe of radis. A secion, e velociy profile is niform and is eqal o U. A secion, e velociy is parabolic and axisymmeric wi zero velociy a e pipe wall and a imm velociy a e cenerline. (i elae U and. (ii elae beween averae velociy a secion wi. Solion: General Coniniy Eqaion in ineral form applied o e sown conrol volme = ρ dv V n V n = = V n V n V n V n ( (3 ( V and n are perpendiclar, V n Vncos9 ( r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8
3 8//8 ( V n V n ( ( A secion (, velociy is niform V n A A U m ( ( Inflow is neaive (-ve V n U V n Vncos8 ( V A secion (, velociy is no niform raer parabolic and axisymmeric ( V n V n rdr m ( Oflow is posiive (+ve V n Vncos V r dr r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 5 ( n rdr V r n ( V r r n V ( V n ( r rdr rdr Now Eqn ( comes as: V n V n ( ( ( U U ; Incompressible flow, ρ = cons. r dr r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan
4 8//8 Considerin i averae velociy a secion ( wic ms be niform a eac poin across e secion, en: ( V n V V V r dr r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 7 (Unseady flow A.5 m i, m diameer cylindrical waer ank wose op is open o e amospere is iniially filled wi waer. Now, e discare pl near e boom of e ank is plled o, and a waer wose diameer is. m sreams o (Fi.. Te averae velociy of e is iven by: V m/s were is e ei of waer in e ank measred from e cener of e ole and is e raviaional acceleraion. eermine (i How lon i will ake for e waer level in e ank o drop o.75 m from e boom? (ii ( How lon i will ake o empy pye ank? Solion: General Coniniy Eqaion in ineral form applied o e sown conrol volme ρ dv V n Unseady, d V ρ V r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 8
5 8//8 (Unseady flow Now, ρ dv V n d d V m A V j j No inflow; only o flow ro e ole (+ve Ten, V A V m ank d ank d d ank d d d ank Unseady, dv ρ d ank d r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 9 (Unseady flow Now, inerain from = a wic = o = a wic = ank d d ank ank ank r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 5
6 8//8 (Unseady flow Time reqired for e waer level in e ank o drop o.75 m from e boom: ank s. 7 min Time reqired o empy e waer ank: s 9 min. Time reqiremen is NOT linear (AN UNSTEAY POBLEM r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 r. A.B.M. Tofiqe Hasan (BUET L-3 T-, ep. of ME ME 3: Flid Mecanics-I (Jan. 8 6
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