Systems with Continuous Flow

Systems with Continuous Flow (control-volume, steady or unsteady) describe mass and energy exchange across control surfaces. Key ideas: conservation of mass (continuity) for steady flow implies constant mass flow rate through any cross-section. The steady-flow energy equation (SFEE) relates heat transfer, shaft work, and changes in specific enthalpy, kinetic and potential energy of the flowing fluid. For many engineering devices (nozzles, turbines, compressors, pumps, heat exchangers, throttles) simplifying assumptions (steady state, adiabatic or isothermal, negligible potential energy, ideal gas or incompressible liquid) reduce SFEE to practical design relations. Isentropic relations and ideal-gas constitutive relations are used for compressible devices; incompressible liquid devices use specific-volume approximations for pump work.

Governing FormulaKey relations (SI units): - Continuity (steady): m_dot = rho * A * V - Steady-flow energy equation (general): Q_dot - W_dot_s = m_dot*(h_out + (V_out^2)/2 + g z_out - h_in - (V_in^2)/2 - g z_in) + dE_cv/dt For steady-state and negligible dE_cv/dt: Q_dot - W_dot_s = m_dot*(h_out - h_in + (V_out^2 - V_in^2)/2 + g(z_out - z_in)) - Ideal gas: h = c_p * T, c_p - c_v = R, p = rho R T - Isentropic ideal-gas relations: T2/T1 = (P2/P1)^{(k-1)/k}, p2/p1 = (T2/T1)^{k/(k-1)} - Nozzle (adiabatic, no shaft work): h1 + V1^2/2 = h2 + V2^2/2 => c_p(T1 - T2) = (V2^2 - V1^2)/2 - Throttle (steady, adiabatic, no work, irreversible): h1 = h2 (isenthalpic) - Pump work for incompressible fluid (per unit mass): w_p ≈ (P2 - P1)/rho, W_dot = m_dot * (P2 - P1)/rho - Critical (choked) pressure ratio for isentropic ideal gas: (p*/p0) = (2/(k+1))^{k/(k-1)} (Assume g = 9.81 m/s^2 when needed.)

Knowledge Check

10 Questions

1.For a steady, adiabatic flow through a device with no shaft work and negligible kinetic and potential energy changes, which property of an ideal gas flow remains unchanged between inlet and outlet?

2.Air (rho = 1.20 kg/m^3) flows steadily through a duct of cross-sectional area 0.05 m^2 at velocity 20 m/s. What is the mass flow rate?

3.A steam turbine (ideal, adiabatic) has a mass flow of 2.0 kg/s. Steam enters at 800 K and leaves at 500 K. Treat steam as an ideal gas with c_p = 1005 J/(kg·K). Neglect kinetic and potential energy changes. What is the turbine power output (approx.)?

4.An ideal (isentropic) compressor ingests air at T1 = 300 K, p1 = 100 kPa and compresses it to p2 = 800 kPa. For air use k = 1.4 and c_p = 1005 J/(kg·K). If mass flow is 1.0 kg/s, approximate the ideal (minimum) compressor power required.

5.An adiabatic nozzle accelerates an ideal gas (c_p = 1005 J/(kg·K)). The inlet temperature is 600 K and inlet velocity is 50 m/s; the exit temperature is 300 K. Neglect potential energy and shaft work. What is the approximate exit velocity?

6.A throttling (Joule–Thomson) valve causes a steady, adiabatic, irreversible pressure drop for a generic real fluid with no work interactions. Which of the following properties remains essentially unchanged across the throttle?

7.Air (k = 1.4) issues from a large reservoir at stagnation pressure p0 = 500 kPa. The downstream/back pressure is p_b = 260 kPa. The critical (choked) pressure ratio for air is (p*/p0) = (2/(k+1))^{k/(k-1)} ≈ 0.528. Is the flow through a converging nozzle choked?

8.A counterflow heat exchanger must heat 5.0 kg/s of water (c_p = 4184 J/(kg·K)) from 300 K to 350 K. Assuming negligible potential and kinetic changes, what rate of heat transfer is required (Q_dot)?

9.A pump raises the pressure of water (rho = 1000 kg/m^3) from 0.1 MPa to 2.1 MPa at a mass flow rate of 0.1 kg/s. Neglecting losses and kinetic/potential energy changes, what is the pump power required (approx.)?

10.In a steady duct flow at constant cross-sectional area, pressure is held constant while the fluid temperature increases. For an ideal gas, what must happen to the flow velocity if mass flow rate is to remain constant?