abj1 1.System, Surroundings, and Their Interaction 2.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? Lecture 5.0: Convection Flux and Reynolds Transport Theorem

abj2 Motivation for The Reynolds Transport Theorem (RTT) 1.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 2.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

abj3  N in any moving/deforming volume V(t)  Time rate of change of N of V(t)  Convection flux of N through a surface A  RTT Very Brief Summary of Important Points and Equations [1]

abj4 (Physical) System Universe / Isolated System Surroundings Interaction  Mechanical interaction (force)  Thermal interaction (energy and energy transfer)  Electrical, Chemical, etc. 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)

abj5 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. IV IM IS 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.

abj6 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 ? Convection Flux of N Through S

abj7 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/convected with (the mass of) water through the cross section S of a pipe? [Energy / Time] What is the time rate of any property N being transported/convected with the mass flow through a surface S? [ N / Time] Hot water Alaska pipeline From

abj8 Nomenclature The flow of mass through the moving surface element over a period of dt Local fluid/mass velocity relative to a reference frame (RF) A Surface element  Local value of the fields Local surface velocity relative to RF Local relative velocity of fluid wrt surface Extensive property Intensive property of N

abj9 A The flow of mass through the moving surface element over a period of dt Surface element Distance of fluid travelling over = Volume flowrate Volume outflow Mass flowrate N flowrate dl dl cos  Volume element  Local value of the fields

abj10 Q3: Convection Flux of N Through S Net Convection Efflux of N Through S A Surface element  Inside Outside Convection Flux of N Through S Open surface Surface element  Closed surface Nothing but sum all over the closed surface. Net Convection Efflux of N Through S

abj11 Volume, Mass, and N Convection Flux/Flowrate Through S A Surface element  Inside Outside Mass Flowrate Volume Flowrate N Flowrate

abj12 Sign (+ / -) of Volume/Mass Flowrate A Surface element  Inside Outside Volume/Mass outflow is positive: Volume/Mass inflow is negative: A Surface element  Inside Outside NOTE: The sign of N-flowrate depends also on the sign of . If  is a vector component, it can be positive or negative.

abj13 Net Convection Efflux Through A Closed Surface S Closed surface S  Flow Mass Flowrate Volume Flowrate If there is a net rate of outflow, If there is a net rate of inflow,

abj14 Special Case: Uniform Properties Over The Surface A Surface element  Inside Outside If is uniform over A: If are uniform over A:

abj15 A Surface element  Inside Outside If is uniform - but is not - over A: If is uniform – but are not - over A:

abj16 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 is , find the mass flowrate through the same surface S.  The area-averaged velocity is defined by, find over the same surface S. x y z w y = + a y = - a Flow S

abj17 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 ?

abj18 The Total Amount of Property N in A Volume V(t) at time t: Consider an infinitesimal volume dV at any time t : An infinitesimal volume dV [Volume] Mass in an infinitesimal volume dV = dm =  dV [Mass] N contained in an infinitesimal volume dV = dN =  dm =  dV [N] N contained in a finite volume V at time t is then the sum of all dN ’s corresponding to all dV ’s in V V ( t ) can be any volume, material or control, depending upon the choice of the domain of integration. Since N V (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, V(t), S (t) x y z dV, dm =  dV, dN =  dm=  dV Evaluated at Fixed Time t Q1: Property N in A Volume V(t) for A Given Field Dimension [N]

abj19 Q2: Time Rate of Change of  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 t = t +  t V(t), S(t) V(t+  t), S(t+  t)

abj20 Example: Evaluation of Property N in A Volume V(t) IntensiveExtensiveMass IntegralVolume Integral Time Rate of Change in V(t) Property N Mass 1 Linear Momentum Angular Momentum Energy e Entropy s

abj21 Q4:Reynolds Transport Theorem (RTT): What is the relation between the time rates of change of N in the coincident MV and CV ?

abj22 Motivation for The Reynolds Transport Theorem (RTT) 1.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 2.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

abj23 The Reynolds Transport Theorem (RTT) Problem Formulation and Notation t = t MV(t), MS(t) CV(t), CS(t), Coincident MV and CV at time t t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III Due to the motion/deformation of both volumes, MV and CV at a later time t+dt. 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 instant t+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. III – Identified MV moving out. I – New MV moving in.

abj24 The Reynolds Transport Theorem (RTT) Problem Formulation and Notation t = t MV(t), MS(t) CV(t), CS(t), Coincident MV and CV at time t t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III MV and CV at a later time t+dt. III – Identified MV moving out. I – New MV moving in. Obviously Q3:

abj25 The Reynolds Transport Theorem (RTT) Derivation t = t MV(t), MS(t) CV(t), CS(t), t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III III – Identified MV moving out. I – New MV moving in. For simplicity, we evaluate the difference

abj26 The Reynolds Transport Theorem (RTT) t = t MV(t), MS(t) CV(t), CS(t), t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III III – Identified MV moving out. I – New MV moving in. Reynolds Transport Theorem (RTT) Unsteady/Temporal TermNet Convection Efflux Term

abj27 Note on RTT t = t MV(t), MS(t) CV(t), CS(t), t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III III – Identified MV moving out. I – New MV moving in. 1.Instantaneously coincide MV(t) and CV(t). [Coincident MV(t) and CV(t) ] 2.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). ] 3. As demonstrated in the RTT and the diagram (Region I, II, and III), differ by the amount of the net convection efflux of N through CS(t). 4.is the local relative velocity of fluid wrt the moving CS(t).

abj28 Interpretation of RTT t = t MV(t), MS(t) CV(t), CS(t), t = t + dt MV(t+ dt), MS(t+d t) CV (t+d t), CS (t+d t) I II III III – Identified MV moving out. I – New MV moving in. 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.)

abj29 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. The Evaluation of The Unsteady Term Reynolds Transport Theorem (RTT) Unsteady Term

abj30 1.When the whole volume integral, i.e, the total amount of N CV, is not a function of time, regardless of the stationarity of the CV or the steadiness of  and . 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). 1] Example of when the unsteady term vanishes tt + dt x y

abj31 2] Example of when the unsteady term vanishes 1. CV is stationary and non-deforming 2.  and  are steady.

abj32 3] Example of the evaluation of the unsteady term when some fields are uniform over the CV 1. CV is stationary and non-deforming 2.  is uniform over CV. 2.  and  are uniform over CV.

abj33 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 Reynolds Transport Theorem (RTT) Convection Flux Term 1. CV is stationary and non-deforming ( A is stationary and non-deforming)

abj34 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  ) 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. 1.Use the RTT and the coincident stationary and non-deforming control volume CV that includes only the fluid stream in the diffuser (as shown above) to find the time rate of change of 1.Kinetic energy (scalar field) 2. x -linear momentum (component of a vector field) of the coincident material volume MV(t). 2.Given that V 2 < V 1, is the kinetic energy of the coincident material volume MV(t) increasing or decreasing? 3.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) ?

abj35 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 P x and P y of the material volume MV(t) that instantaneously coincides with the stationary and non-deforming control volume CV shown below vanishes; 2. if not, state also - whether they are positive or negative, and - whether there should be the corresponding net force ( F x and F y ) acting on the MV/CV, and - whether the corresponding net force is positive or negative.

abj36 x y V1V1 V 2 = V 1 (a) (yes/no) If not, positive or negative Net F x on CV? (yes/no) If yes, F x positive or negative (b) (yes/no) If not, positive or negative Net F y on CV? (yes/no) If yes, F y positive or negative V1V1 V 2 > V 1 V1V1 V 2 = V 1 V1V1  V1V1