7.1 Turbines
For an adiabatic turbine with small KE/PE changes, the shaft work rate is approximately:
Ẇ_s ≈ ṁ (h₁ − h₂)
Here, the drop in enthalpy includes the reduction in internal energy and flow energy; both contribute to shaft work output.
1) Introduction and intuition
In open systems (control volumes) like turbines, compressors, nozzles, pumps, and heat exchangers, mass crosses the system boundary. To push a fluid element across a boundary against the surrounding pressure, some work is required. This “push” is the essence of flow work. When we express it per unit mass, we often call it flow energy and denote it by the product p v (pressure × specific volume).
Intuitively: every kilogram of fluid entering or leaving a device carries not only its internal energy, but also the “energy of having been pushed in” against the environment at pressure p. That energy bookkeeping leads to the appearance of the pv term in open-system analyses.
2) Definitions: flow work and flow energy
- Flow work (boundary flow work): The work required to push fluid into or out of a control volume against the static pressure at the boundary.
- Per unit mass form (flow energy): The quantity of flow work associated with one unit mass of fluid crossing the boundary equals p v (units of energy per mass).
- Total rate form: For a stream with mass flow rate ṁ, the rate of flow work is p V̇ = p (ṁ v) = ṁ (p v).
Here, p is the thermodynamic pressure at the boundary section and v is the specific volume of the stream there; both are properties of state at that section.
3) Derivation from control volume perspective
Consider a steady stream crossing a control surface with local pressure p and cross-sectional area A. Over a small distance dx, the boundary exerts a force pA on the stream, and the displacement is dx. The incremental work done on the stream segment is:
δW_flow = pA dx = p dV
Dividing by the mass of that segment, with dV = v dm, yields the per-mass increment:
δw_flow = p v (dm/m) → per unit mass: w_flow = p v
Thus, each kilogram crossing that section carries an amount of “flow energy” equal to p v. In rate form for a steady stream, the power associated with flow work is p V̇.
4) Is flow energy a property?
This is the subtle point:
- Work in general is a path function (not a property). “Flow work” as an interaction (energy crossing the boundary due to pushing) is a kind of work and hence path-dependent when considered as a process quantity.
- However, the quantity p v evaluated at a state is constructed from properties p and v. Therefore, p v for a given state is determined solely by that state.
Conclusion: The interaction “flow work” is not a property (it’s work), but the per-unit-mass state quantity we call “flow energy,” p v, is a state-dependent quantity because it is a product of properties. In practice, we avoid confusion by combining u and p v into the enthalpy h = u + p v, which is unequivocally a property.
5) Connection to enthalpy
For open systems, the natural energy bookkeeping quantity is the enthalpy:
h = u + p v
- u (specific internal energy) accounts for microscopic energy modes inside the fluid.
- p v accounts for the energy needed to push the fluid into/out of the control volume.
Because h is a property, it greatly simplifies the steady-flow energy equation (SFEE), allows tabulation (steam tables, refrigerant tables), and corresponds directly to measurable device performance.
6) Steady-flow energy equation (SFEE) and the pv term
For a single-inlet, single-outlet steady device, neglecting potential/kinetic energy for brevity and using the engineering sign convention (heat to system positive, work by system positive):
Q̇ − Ẇ = ṁ (h₂ − h₁)
Expanded, with h = u + p v:
Q̇ − Ẇ = ṁ [(u₂ − u₁) + (p₂ v₂ − p₁ v₁)]
The term (p v) naturally appears as part of enthalpy differences, embodying the effect of flow work without having to track boundary forces and displacements explicitly.
7) How flow energy appears in common devices
7.2 Compressors and pumps
For an adiabatic compressor (small KE/PE changes), shaft power input is:
Ẇ_s ≈ ṁ (h₂ − h₁)
Raising pressure increases p v and u such that h rises; this enthalpy rise is supplied by shaft work (and heat if present).
7.3 Nozzles and diffusers
With adiabatic flow and negligible shaft work:
h₁ + V₁²/2 ≈ h₂ + V₂²/2
Enthalpy converts to kinetic energy (nozzles) or vice versa (diffusers). The p v portion within h participates in this conversion.
7.4 Throttling valves
Throttling is approximately adiabatic with no shaft work and negligible KE/PE changes:
h₂ ≈ h₁
Despite large pressure drops (and changes in p v), enthalpy remains constant. Internal energy typically increases to offset the p v decrease, which manifests as temperature changes depending on fluid and conditions.
7.5 Heat exchangers
Usually modeled as adiabatic overall (no heat to surroundings) but with heat transfer between streams. Enthalpy changes describe how energy is carried into and out of the device; p v is embedded within h.
8) Worked examples
Example 1: Checking units of pv
Pressure p [Pa] = N/m², specific volume v [m³/kg]. Product p v → (N/m²)(m³/kg) = (N·m)/kg = J/kg. Thus, pv has the units of specific energy. This confirms that pv per unit mass is an energy-like quantity.
Example 2: Flow energy magnitude for air
Air at p = 200 kPa and v ≈ 0.8 m³/kg (approx. at moderate T): pv = (200,000 Pa)(0.8 m³/kg) = 160,000 J/kg = 160 kJ/kg. This is a significant portion of stream energy compared to typical enthalpy levels for gases.
Example 3: Turbine enthalpy drop and work
A steam turbine operates adiabatically with ṁ = 5 kg/s. Inlet h₁ = 3400 kJ/kg, outlet h₂ = 2600 kJ/kg. Shaft power ≈ ṁ (h₁ − h₂) = 5 × (800) = 4000 kW. The 800 kJ/kg drop includes both internal energy and pv contributions (combined as h).
Example 4: Throttling valve (h = const.)
A refrigerant is throttled from 1.2 MPa to 0.3 MPa, adiabatically, no shaft work. Enthalpy remains essentially constant: h₂ ≈ h₁. Although p v changes significantly, u adjusts so that u₂ + p₂ v₂ = u₁ + p₁ v₁ (i.e., h₂ = h₁). Temperature may drop, illustrating that p v alone is not conserved; h is the right property to track.
9) Common misconceptions and clarifications
- “Flow work is a property.” Not exactly. Work is not a property; it is path-dependent. But the per-mass quantity p v evaluated at a state depends only on that state because p and v are properties. To avoid confusion, use enthalpy h = u + p v, which is a property.
- “pv is always small.” Not true. For gases, pv can be large (comparable to u). For liquids (nearly incompressible), v is small so pv may be modest, but not always negligible depending on pressure levels.
- “Tracking flow work separately is necessary.” In most steady-flow analyses, it is cleaner to track h directly. Separating u and pv is useful for teaching and for special derivations.
- “pv is conserved across devices.” No. Neither u nor pv is conserved individually. Device performance relates to changes in h (plus KE/PE and heat/work terms per SFEE).
10) Summary checklist
- Meaning: Flow work is the work needed to push fluid across a boundary; per unit mass it is p v (flow energy).
- Property question: Flow work is not a property (work is path-dependent), but p v at a state depends only on that state. In practice, use enthalpy h = u + p v (a property).
- Why it matters: Open-system energy analysis becomes simple with h, embedding the pv contribution automatically.
- Devices: Turbines, compressors, nozzles, valves, and heat exchangers are naturally described with enthalpy changes (and KE/PE where relevant).
- Units: pv has units of J/kg (specific energy), consistent with its interpretation.