Enthalpy and the Steady Flow Energy Equation
Enthalpy derived rather than declared — flow work always travels with internal energy, so the two are grouped once and never parted.
Skip to the animationMass crossing a boundary carries internal energy plus the flow work needed to push it across, and because those two always travel together they are grouped as enthalpy — which is why every steady-flow device is analysed in h rather than u.
Why the closed-system law is not enough
ΔU = Q − W was derived for a fixed mass. A turbine, nozzle, pump or boiler has mass streaming through it continuously, so the balance needs extra terms — and identifying them is the whole of this topic.
Flow work, and where enthalpy comes from
To push a unit mass of volume v across a boundary against pressure p takes work p·v. It is not energy stored in the fluid; it is the price of admission, charged at the inlet and refunded at the outlet. A closed system never pays it, because nothing crosses.
Since u and p·v accompany every stream, they are grouped once as enthalpy, h = u + p·v. Enthalpy is not a new form of energy and a closed system has little use for it — it exists because of open systems, and for no other reason. This is also why steam tables are tabulated in h.
The steady flow energy equation
Steady means nothing inside the control volume changes with time: mass in equals mass out, and energy in equals energy out.
Q̇ − Ẇ = ṁ[(h₂ − h₁) + (V₂² − V₁²)/2 + g(z₂ − z₁)]. The three bracketed terms are what the stream carries — enthalpy, kinetic energy and potential energy.
Every device is this with terms deleted
| Device | What survives | Result |
|---|---|---|
| Turbine | Adiabatic, ΔKE small | Ẇ = ṁ(h₁ − h₂) |
| Compressor / pump | Same, work negative | Ẇ = ṁ(h₁ − h₂) < 0 |
| Nozzle | No work, no heat | V₂ = √(2(h₁ − h₂)) |
| Diffuser | The nozzle backwards | Velocity converts to enthalpy |
| Throttle | No work, no heat, ΔKE small | h₁ = h₂ — isenthalpic |
| Heat exchanger | No work | Q̇ = ṁ(h₂ − h₁) |
The throttle row is the one to remember: an expansion valve is isenthalpic, and it is the pressure-dropping element of every refrigeration cycle.
Why power plants are worked in enthalpy
At constant pressure Q = ΔH, so a boiler's heat duty is an enthalpy difference and a turbine's work output is another. Since almost every component of a plant is a steady-flow device, working in h means every quantity comes straight off the tables.
When steady flow is the wrong assumption
Filling a tank, starting a turbine or discharging a gas bottle are all unsteady: energy accumulates inside the control volume, and the transient term returns.
A consequence that surprises almost everyone: charging a rigid bottle from a supply line raises its temperature, with no heat added at all. The incoming flow work is deposited into the contents, and it follows directly from the transient form.
The numbers you will be asked for
- Enthalpy
h = u + p·v
Internal energy plus flow work, grouped because they always travel together.
- Steady flow energy equation
Q̇ − Ẇ = ṁ[Δh + ΔV²/2 + gΔz]
Per unit time, for a control volume at steady state.
- Turbine work
Ẇ = ṁ(h₁ − h₂)
Adiabatic, with velocity and elevation changes neglected.
- Nozzle exit velocity
V₂ = √(2(h₁ − h₂))
The enthalpy drop becomes kinetic energy.
- Throttling
h₁ = h₂
No work, no heat, negligible ΔKE — the expansion valve.
Advantages and disadvantages
Advantages
- One equation covers turbines, nozzles, pumps, throttles and exchangers.
- Enthalpy removes the need to track flow work separately.
- Steam tables are tabulated in h, so plant analysis needs no extra conversion.
- At constant pressure the heat added is simply ΔH.
Disadvantages
- Only valid at steady state, which excludes filling, emptying and start-up.
- Enthalpy is often mistaken for a form of energy the fluid contains.
- Kinetic and potential terms are usually dropped, and occasionally should not be.
- It gives no information about irreversibility — that needs the second law.
Watch it work
Check yourself
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