Systems and Control Volumes

A system (control mass) is a fixed amount of matter tracked through time; no mass crosses its boundary, but energy can. A control volume (CV) is a specified region in space through which mass and energy can cross the control surface. Analysis of CVs uses integral conservation laws (mass, momentum, energy) often in the form of the Reynolds Transport Theorem to relate system behavior to control-volume behavior. For steady-flow devices the steady-flow energy equation (SFEE) is commonly used. Sign conventions: work done by the system is positive in many thermodynamics texts (W_by), while engineering machine power produced is often given as positive output for turbines; state which sign convention is used in each problem.

Governing FormulaKey relations: - Reynolds Transport Theorem (RTT): dB_system/dt = d/dt \int_{CV} b\rho \,dV + \int_{CS} b\rho (\mathbf{V}\cdot\mathbf{n}) \,dA, where b is property per unit mass and B is system property. - Conservation of mass (integral): d/dt \int_{CV} \rho \,dV + \int_{CS} \rho (\mathbf{V}\cdot\mathbf{n}) \,dA = 0. For steady one-dimensional flow: \dot m = \rho A V (and A_1 V_1 = A_2 V_2 for incompressible steady flow). - Steady-flow energy equation (neglecting shaft work other than W_s and using enthalpy): \dot Q - \dot W_s = \dot m \left( h_{out} - h_{in} + \tfrac{V_{out}^2 - V_{in}^2}{2} + g(z_{out}-z_{in}) \right). - Moving boundary (quasi-equilibrium) work for closed systems: W = \int_{V1}^{V2} P_{ext} \,dV. - Momentum (steady): \sum F = \dot m (V_{out} - V_{in}) + \sum \text{pressure forces on CV faces} (use sign conventions).

Knowledge Check

10 Questions

1.Which statement correctly distinguishes a control mass (system) from a control volume?

2.In steady incompressible flow a pipe contracts from cross-sectional area A1 = 0.05 m^2 to A2 = 0.02 m^2. If the velocity at section 1 is V1 = 3.0 m/s, what is the velocity at section 2? (Assume incompressible steady flow.)

3.Air with density ρ = 1.20 kg/m^3 flows through a duct of area A = 0.02 m^2 at velocity V = 10 m/s. What is the mass flow rate \dot m?

4.Which statement best expresses the Reynolds Transport Theorem (RTT) in words for a generic extensive property B of a system?

5.A steady adiabatic turbine receives steam at specific enthalpy h_in = 3200 kJ/kg and exhausts at h_out = 2800 kJ/kg. If mass flow rate is \dot m = 2.0 kg/s and kinetic/ potential energy changes are negligible, what is the shaft power produced (assume sign: power produced = \dot m(h_in - h_out))?

6.Which statement correctly describes a throttling (expansion) valve operating between two pressure levels for a liquid?

7.A piston-cylinder assembly contains gas initially at V1 = 0.010 m^3. It is quasi-statically compressed by a constant external pressure P_ext = 200 kPa until the volume is V2 = 0.005 m^3. What is the work done on the gas (magnitude) during compression? (Assume quasi-static with P_ext constant and neglect other work.)

8.A water jet strikes a stationary flat plate normally and is brought to rest. The jet mass flow rate is \dot m = 0.50 kg/s and jet speed V = 20.0 m/s (normal). Neglect weight and buoyancy; what is the steady force exerted by the jet on the plate (magnitude)?

9.A steady adiabatic nozzle (no shaft work, negligible potential energy) has inlet conditions h_in = 300 kJ/kg and V_in = 20.0 m/s. The outlet enthalpy is h_out = 280 kJ/kg. What is the outlet velocity V_out (magnitude)? (Use h in J/kg when combining with kinetic energy.)

10.A control volume initially contains mass m0 = 10.0 kg. Mass enters at 1.0 kg/s and leaves at 0.6 kg/s. After 5.0 s of operation, what is the mass inside the control volume? (Assume uniform mixing and constant inlet/outlet rates.)