Lecture 5.0: Convection Flux and Reynolds Transport Theorem

1. Lecture 5.0: Convection Flux and Reynolds Transport Theorem • System, Surroundings, and Their Interaction • Classification of Systems: Identified Volume, Identified Mass, and Isolated System Questions of Interest: Given the identified volume (IV) of interest and the relevant fields Q1: : How much is N contained in a moving/deforming volume ? Q2: : How much is the time rate of change of N in a moving/deforming volume ? Q3: : Convection Flux of N Through A Surface S: At what rate is N being transported/convected through a moving/deforming surface ? Q4: RTT: What is the relation between the time rates of change of N in the coincident MV and CV?

2. Motivation for The Reynolds Transport Theorem (RTT) • Physical laws (in the form we are familiar with) are applied to an identified mass (MV). They can be written in generic form in terms of the time rate of change of property N of an MV as • However, in fluid flow applications, we are often interested in what happens in a region in space, i.e., in an identified volume or CV. Hence, we want to know the time rate of change of property N of a CV • Thus, in order to apply the physical laws from the point of view of a CV instead, we need to find the relation

3. Very Brief Summary of Important Points and Equations [1] • N in any moving/deforming volume V(t) • Time rate of change ofNof V(t) • Convection flux of N through a surface A • RTT

4. Universe / Isolated System (Physical) System Fundamental Concept: System-Surroundings-Interaction • The very first task in any one problem: • Identify the system • Identify the surroundings • Identify the interactions between the system and its surroundings, e.g., • Mechanics - Force (identify all the forces on the system by its surroundings) • Thermodynamics - Energy and Energy Transfer (identify all forms of energy and energy transfer between the system and its surroundings) Surroundings • Interaction • Mechanical interaction (force) • Thermal interaction (energy and energy transfer) • Electrical, Chemical, etc.

5. IV IM IS Classification of Systems:Identified Volume, Identified Mass, and Isolated System • Identified Volume/Region (IV / IR) • [Control Volume (CV), Open system] • An identified region/volume of interest. • There can be exchange of mass and energy with its surroundings. • Identified Mass (IM) • [Material Volume (MV), Control Mass (CM), Closed system] • A special case of an IV. • It is an IV that always contains the same identified mass. • Thus, there can be exchange of energy with its surroundings, but • not mass. • Isolated System (IS) • A special case of an MV, hence of an IV. • It is an MV that does not exchange energy with its surroundings. • In other words, it is an IV that does not exchange both mass and energy with its surroundings.

6. Convection Flux of N Through S Given a surface S of interest and the relevant fields Q3: : Convection Flux of N Through A Surface S: At what rate is N being transported/convected through a moving/deforming surface ?

7. Hot water Alaska pipeline From http://www.hickerphoto.com/alaska-oil-pipeline-6765-pictures.htm Convection Flux of N Through A Surface S • MOTIVATION for The Expression and Quantification of Flux / Flowrate • What is the volume flowrate of water through the cross section S of a pipe? [Volume / Time] • What is the mass flowrate of water through the cross section S of a pipe? [Mass / Time] • What is the time rate of thermal energy being transported/convectedwith (the mass of) water through the cross section S of a pipe? [Energy / Time] • What is the time rate of any property N beingtransported/convectedwith the mass flow through a surface S? [ N / Time]

8. Local value of the fields q A Surface element • Local fluid/mass velocity relative to a reference frame (RF) • Local surface velocity relative to RF • Local relative velocity of fluid wrt surface The flow of mass through the moving surface elementover a period ofdt • Extensive property • Intensive property of N Nomenclature

9. Local value of the fields dl cosq q Volume element dl A Surface element • Distance of fluid travelling over = • Volume outflow The flow of mass through the moving surface elementover a period ofdt • Volume flowrate • Mass flowrate • N flowrate

10. q q Outside A Surface element Surface element Inside Closed surface Convection Flux of N Through S Net Convection Efflux of N Through S Nothing but sum all over the closed surface. Q3: Convection Flux of N Through SNet Convection Efflux of N Through S Open surface

11. q Outside A Surface element Inside Volume Flowrate Mass Flowrate N Flowrate Volume, Mass, and N Convection Flux/Flowrate Through S

12. Volume/Mass outflow is positive: Outside q A A Surface element Volume/Mass inflow is negative: Inside Outside q Surface element Inside NOTE:The sign of N-flowrate depends also on the sign of h. If his a vector component, it can be positive or negative. Sign (+ / -) of Volume/Mass Flowrate

13. q If there is a net rate of outflow, If there is a net rate of inflow, Flow Volume Flowrate Mass Flowrate Net Convection Efflux Through A Closed Surface S Closed surface S

14. q Outside A Surface element Inside If is uniform over A: If are uniform over A: If are uniform over A: Special Case: Uniform Properties Over The Surface

15. q Outside A Surface element Inside If is uniform - but is not - over A: If is uniform – but are not - over A:

16. y y = + a w x Flow y = - a S z Example: Evaluate the flux by using the elemental area element Problem: The velocity field is given by • Find the volume flowrate Q through the cross sectional surface S. • If the density of fluid isr, find the mass flowrate through the same surface S. • The area-averaged velocity is defined by , find over the same surface S.

17. Given the identified volume (IV) of interest and the relevant fields Q1: : How much is N contained in a moving/deforming volume ? Q2: : How much is the time rate of change of N in a moving/deforming volume ?

18. y x z Q1: Property N in A Volume V(t) for A Given Field Evaluated at Fixed Time t dV, dm = rdV, dN = h dm= h rdV V(t), S (t) Dimension [N] • The Total Amount of Property N in A Volume V(t)at timet: • Consider an infinitesimal volume dV at any timet: • An infinitesimal volume dV[Volume] • Mass in an infinitesimal volume dV=dm = r dV [Mass] • Ncontained in an infinitesimal volume dV=dN = hdm = hrdV [N] • N contained in a finite volume V at timetis then the sum of all dN’s corresponding to all dV’sinV • V (t) can be any volume, material or control, depending upon the choice of the domain of integration. • Since NV(t)depends upon , , and the domain V (t), • After the volume integration (with domain variable with time t), is a function of t alone, . in the same field, if the MV(t) and CV(t) coincide,

19. Q2: Time Rate of Change of V(t), S(t) V(t+d t), S(t+d t) • After the function is found, the time rate of change of N within the volume V(t)as we follow the volume can be found from the time derivative t = t+d t t = t

20. Example: Evaluation of Property N in A Volume V(t)

21. Q4: Reynolds Transport Theorem (RTT): What is the relation between the time rates of change of N in the coincidentMV and CV?

22. Motivation for The Reynolds Transport Theorem (RTT) • Physical laws (in the form we are familiar with) are applied to an identified mass (MV). They can be written in generic form in terms of the time rate of change of property N of an MV as • However, in fluid flow applications, we are often interested in what happens in a region in space, i.e., in an identified volume or CV. Hence, we want to know the time rate of change of property N of a CV • Thus, in order to apply the physical laws from the point of view of a CV instead, we need to find the relation

23. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t Due to the motion/deformation of both volumes, MV and CV at a later time t+dt. Coincident MV and CV at time t The Reynolds Transport Theorem (RTT) Problem Formulation and Notation III – Identified MV moving out. I – New MV moving in. • MV is a moving/deforming material volume, MV (t). • CV is a moving/deforming identified/control volume, CV (t). • At an instant t : • Coincident MV and CV : At any time t, we can identify the coincident MV and CV. • At a later instantt+dt : • Region III: Part of the identified and interest MV is moving out of the identified CV . • Region I: Part of a new MV – which is not the one of interest at present - is moving into the identified CV.

24. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t MV and CV at a later time t+dt. Coincident MV and CV at time t Obviously Q3: The Reynolds Transport Theorem (RTT) Problem Formulation and Notation III – Identified MV moving out. I – New MV moving in.

25. III – Identified MV moving out. I – New MV moving in. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t The Reynolds Transport Theorem (RTT) Derivation For simplicity, we evaluate the difference

26. III – Identified MV moving out. I – New MV moving in. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t Reynolds Transport Theorem (RTT) Unsteady/Temporal Term Net Convection Efflux Term The Reynolds Transport Theorem (RTT)

27. III – Identified MV moving out. I – New MV moving in. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t Note on RTT • Instantaneously coincide MV(t) and CV(t). [Coincident MV(t) and CV(t)] • In the form given in the previous slide, it is applicable to moving/deforming CV(t). [CV is a function of time; hence, CV(t).] • As demonstrated in the RTT and the diagram (Region I, II, and III), • differ by the amount of the net convection efflux ofNthroughCS(t). • is the local relative velocity of fluid wrt the movingCS(t).

28. III – Identified MV moving out. I – New MV moving in. III II I MV(t), MS(t) CV(t), CS(t), MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) t = t + dt t = t Reynolds Transport Theorem (RTT) Increase in MV = Increase in CV + Efflux Through CS = Increase in CV + [Outflow – Inflow] (See the diagram and Region I, II, III for better understanding.) Interpretation of RTT

29. Reynolds Transport Theorem (RTT) The Evaluation of The Unsteady Term Unsteady Term • In principle, in order to evaluate the unsteady term , we must • first find the volume integral , then • later take time derivative . • In other words, the order of differentiation and integration is important.

30. y x t t + dt 1] Example of when the unsteady term vanishes • When the whole volume integral , i.e, the total amount of NCV, is not a function of time, regardless of the stationarity of the CV or the steadiness of h and r. • A container filled with water is moving. • In this case, even though • the CV is moving, CV(t), • the density field as described by the coordinate system fixed to earth is not steady (at one time, one point has the density of water, the next instant the point has the density of air), • but since , (total mass in the container remains constant with respect to time).

31. 2] Example of when the unsteady term vanishes • CV is stationary and non-deforming • r and h are steady.

32. 1. CV is stationary and non-deforming 2. h is uniform over CV. 2. r and h are uniform over CV. 3] Example of the evaluation of the unsteady term when some fields are uniform over the CV

33. Reynolds Transport Theorem (RTT) The Evaluations of The Convection Efflux Term 1] Example of the evaluation of the convection flux term when some fields are uniform over the surface A of interest Convection Flux Term 1.CVis stationary and non-deforming (A is stationary and non-deforming)

34. Example 2: Finding The Time Rate of Change of Property N of an MV By The Use of A Coincident CV and The RTT Problem: Flow Through A Diffuser An incompressible flow of water (density r) with steady velocity field passes through a conical diffuser at the volume flowrate Q. Assume that the velocity is axial and uniform at each cross section. • Use the RTT and the coincident stationary and non-deforming control volumeCV that includes only the fluid stream in the diffuser (as shown above) to find the time rate of change of • Kinetic energy (scalar field) • x-linear momentum (component of a vector field) of the coincident material volumeMV(t). • Given that V2< V1, is the kinetic energy of the coincident material volumeMV(t) increasing or decreasing? • According to Newton’s second law, should there be any net force in the x direction acting on the MV(t) , or equivalently CV(t) ?

35. Example 3: Finding The Time Rate of Change of Property N of an MV By The Use of A Coincident CV and The RTT Problem: Given that the velocity field is steady and the flow is incompressible 1. state whether or not the time rate of change of the linear momenta Px and Py of the material volume MV(t) that instantaneously coincides with the stationary and non-deforming control volume CVshown below vanishes; 2. if not, state also - whether they are positive or negative, and - whether there should be the corresponding net force (Fx and Fy ) acting on the MV/CV, and - whether the corresponding net force is positive or negative.

36. V2 = V1 V2 > V1 V1 V2 = V1 V1 V1 V1 V2 = V1 • (yes/no) If not, positive or negative • Net Fx on CV? (yes/no) If yes, Fxpositive or negative • (b) (yes/no) If not, positive or negative • Net Fy on CV? (yes/no) If yes, Fypositive or negative q V2 = V1 V1 y x