Transport processes. 7. Semester Chemical Engineering Civil Engineering

Transcription

1 Transport processes 7. Semester Chemical Engineering Civil Engineering

2 1. Elementary Flui Dynamics 2. Flui Kinematics 3. Finite Control Volume Analysis 4. Differential Analysis of Flui Flow 5. Viscous Flow an Turbulence 6. Turbulent Bounary Layer Flow 7. Principles of Heat Transfer 8. Internal Forces Convection 9. Unsteay Heat Transfer 10. Boiling an Conensation 11. Mass Transfer 12. Porous Meia Flow 13. Non-Newtonian Flow Course plan

3 Toay's lecture Finite Control Volume Analysis Control volume an tem escription Conservation equations Reynols transport theorem

4 Reynols Transport Theorem Learning goal for toay: At the exam you shoul be able to explain the meaning an use of this equation: DB ρb V + ρbvn A

5 Control volume

6 Mass balance Conservation of mass (for a fixe non-eforming ) Mass which enters tem bounaries Mass flow in 1 min mout m Mass which exits tem bounaries Change of the tem mass (accumulation) m m m + m m 0 in out in out m cst E cst Mass flow out Mass flow in m cst E cst Mass flow out m m + m out 2 3 m m m m + m in out

7 Intermezzo - Average velocity Conservation of mass m m m 0 in out ρ AV ρ AV Note: it is implie that V 1 an V 2 is the average velocity Mass velocity (mass flux) G V kg s m 2 ρ / - Often use to avoi any ambiguity Flow rate (volume flow rate) V VA m 3 / s Again V is the average velocity Plug flow Top-Hat velocity profile

8 Example Given a mass flowrate at (1), calculate Mass Flow rates at (2) an (3) m m 2 m 1, m 3 2 The average velocity at (1) an (3) Mass flow rate at (1) 1 ( m 2) m m V, V 0.9V ρa1 ρa3 D 3 ρ A1 D 2 1 G m Vρ A

9 Control volume an tem representation Fixe, non-eforme Fixe/moving, non-eforme Deforming

10 Reynols Transport Theorem Describe flui motion using both tem an control volume concepts System: Fixe quantity of matter Control volume: Volume in space t δt t t+δt

11 Reynols Transport Theorem General parameter: Extensive property B mb Intensive property, per mass basis Fx. If we are setting up a law for mass then: B m, b 1 Fx. If we are setting up a law for kinetic energy then: 1 1 B mv, b V Fx. If we are setting up a law for momentum then: B mv, b V

12 Reynols Transport Theorem is the total amount of a flui parameter in a tem B can be written as: B B lim δ V bi( ρδ i V ) 0 i ρb V Volume of an small flui particle Time rate of change of : B ρb V B Time rate of change of : B ρb V B

13 Example Fire extinguisher B m ρv B m 0 ρv < 0

14 Reynols Transport Theorem Reynols transport equation DB ρb V ρ AV b ρ AVb

15 Example Fire extinguisher #2 Dm ρv + ρ2av 2 2 Constant mass! (Dm /0) ρv ρ AV Thus, the rate change of mass insie the fire extinguisher equals the mass flow out of the fire extinguisher!

16 Reynols Transport Theorem General expression: DB ρb V + ρbvn A

17 Reynols transport theorem Steay flow DB ρb V + ρbvn A

18 Conservation of mass Time rate change of the tem 0 DB ρb V + ρbvn A DM ρv 0 + ρ A Vn

19 Example Air flow steaily, Calculate the air flow DB ρb V + ρbvn A DM ρv + ρvn A 0 ρ AV ρ AV Assuming air as an ieal gas P ρ RT V 1 PTV PT ft / s 1ft/s 1km/t

20 Example Filling of a bathtub, estimate the time rate of change of the water level DB ρb V + ρbvn A DM ρv 0 + ρ A Vn ρ AV A + ρ airvol W V W m watervol water m air m water For the water volume only watervol ρv ρv m water ρh 2 ft 5 ft 10 ft ( )( ) 2 watervol h ρ

21 Momentum equation Newtons secon law DB ρb V + ρbvn A D Change of the tems momentum VρV F Reynols transport equation The sum of forces which act on a tem D ρvv ρvv + VVn ρ A Combination: the linear momentum equation ρvv + ρ A VVn F

22 Example Determine the force on a pipe ben ρvv + ρ A VVn F ρvv VVn ρ + A F Vn V ρ A F + P A + P A y Ay ( )( ) ( )( ) + V m + V + m F + PA + PA Ay Ay ( ) F P + P A mv 1 2 2

23 Example Determine the trust of a jet engine V ρ Vn A F + P A P A Trust ( )( ) ( )( ) + V m + V + m F + PA PA Trust Trust ( ) F PA + PA m V V ρvv + ρ A VVn F Cons. of mass Ieal gas Trust ( ) F PA + P A ρ AV V V Trust P F PA + P A AV V V ( ) RT1

24 First law of thermoynamics 1 law of thermoynamics DB ρb V + ρbvn A D Reynols Transport equation D ρ ( Q in Q out ) + ( W in W out ) e V eρv eρv + eρ A Vn Store energy for a flui particle: 2 V e u + + gz 2 Combine The energy equation Vn e V + eρ A Q + W ρ (,, ) net in net in

25 Flow work Pressure work over a istance Summing up, for the power transfer ue to pressure for all flui particles on the control surface: W PVn A pressure Than the energy equation takes the form: rearranging gives: eρ V + eρvn A Q net, in + W net, in + PVn A 2 P V eρv + u gz ρ A Q net in + W ρ 2 Vn, net, in

26 Enthalpy internal energy, u, an P/ρ both epens on temperature, pressure an specific volume For convenience we efine a quantity, enthalpy as: hu+pv [J/kg]

27 Example Calculate the work output per unit mass from a steam turbine eρv 2 V + h + + gz ρ A Q, 2 Vn 2 2 V2 V 1 m h2 h1+ W 2 shaft net in + W net, in Superheate steam Water/vapor mixture W V V w h h m shaft, in 2 1 shaft, out 2 1+

28 Exercises Exercises, solutions, questions on the course homepage