ME336 TBCH7

7 CONTROL VOLUME ANALYSIS

7.1 IntroductionControl volume analysis is a method used for fluid flow problems.

Conservation Laws:

  • Conservation of mass: In a moving fluid, mass is neither created nor destroyed.

  • Conservation of momentum: Time rate of change of linear momentum equals sum of body and surface forces acting on the fluid.

  • Conservation of energy: Time rate of change in internal plus kinetic energy equals rate of heat added plus rate of work done by forces.

  • Conservation of angular momentum: Time rate of change equals the sum of torques acting on the fluid.

7.2 Basic Concepts: System and Control VolumeSystem: Fixed, identifiable quantity of matter with a boundary surface.

  • Mass cannot cross the system boundary, but energy and work can.Control Volume (CV): Region in space through which fluid flows, defined by a control surface.

  • Applies conservation laws to analyze fluid flow.

  • Ports: Control surface sections where mass can enter or exit.Types of control volumes:

  • Fixed CV: Stationary with a constant volume/shape.

  • Moving CV: Moves with the fluid.

  • The choice of a CV is critical for convenient analysis.

7.3 System and Control Volume AnalysisBoth approaches yield the same results independently of chosen methods.Example: Analyzing a gas compression using system vs. control volume at different stages.

  • Mass conservation can be analyzed using mass flow rate across ports versus changes within the system.

7.4 Reynolds Transport Theorem for a SystemDerives the time rate of change of extensive properties in a system using:

  • Volume Integral: Accounts for changes within the system.

  • Surface Integral: Accounts for fluid entering/exiting.

7.5 Reynolds Transport Theorem for a Control VolumeFor a control volume coinciding with a system, can express in terms of:

  • CV (time rate of change of property within CV) and CS (mass transport across surface).

  • Fundamental for developing conservation equations (mass, momentum, energy, angular momentum).

7.6 Control Volume AnalysisSimplifies obtaining global fluid properties rather than detailed solutions.Derives equations for conservation laws applied in integral form:

  • Mass Balance: [ \frac{dM_{CV}}{dt} = \dot{M}{in} - \dot{M}{out} ]Accounts for mass entering and exiting the control volume.

  • Momentum Balance: [ \frac{d(P_{CV})}{dt} = \dot{P}{in} + F{net} ]Relates forces acting, accumulation of momentum.

  • Energy Balance: [ \frac{dE_{CV}}{dt} = \dot{E}{in} - \dot{E}{out} + W ]Involves work done and heat transfer rates.

  • Angular Momentum Balance: [ \frac{d(L_{CV})}{dt} = \tau_{net} + \dot{L}_{transport} ]Relevant for rotating machinery and flow analysis around rotational axes.

7.6.1 Mass BalanceDerived from the Reynolds transport theorem, leading to:Rate of change of mass in CV = Mass flow into CV - Mass flow out of CV.

7.6.2 Momentum BalanceDerived as:Rate of accumulation of momentum = Net inflow of momentum + Forces acting on CV.

7.6.3 Energy BalanceExpresses the energy change in a CV:Accumulation of energy = Energy entering - Energy exiting + Work done on the fluid.

7.6.4 Angular Momentum BalanceEssential for devices with rotating parts, expressing:Rate of change of angular momentum = Net torque + Transport of angular momentum.

7.7 SummarySystem vs. Control Volume:

  • System: Fixed, identifiable volume with mass interactions.

  • CV: Passage defined for fluid movement, allowing analysis of mass, momentum, and energy.

  • The application of Reynolds transport theorem links both analyses for fluid dynamics.

  • Amongst conservation principles, interdependent behaviors of fluids can be assessed effectively using methods aligned with the above principles.