Intuition The one core ideaWhen fluid flows through a machine (pipe, nozzle, turbine), energy doesn't just sit in a box — it is carried across the boundary by the moving mass, and the surroundings must push that mass in and out. Every symbol below exists to bookkeep one piece of that story, and they all combine into a single sentence: energy carried in + heat added = energy carried out + work extracted.
This page assumes you have seen none of the notation in the parent note. We build each symbol from a picture, in an order where each one leans only on the ones before it.
Definition System, boundary, surroundingsA system is the stuff we choose to study. The boundary is the imaginary skin we draw around it. Everything else is the surroundings . Heat and work cross the boundary; we watch what happens inside .
There are two ways to draw the boundary, and the whole topic is about the difference.
Definition Closed system vs open system (control volume)
A closed system is a fixed lump of matter — the boundary moves with the matter, so no mass ever crosses it. Think of gas sealed in a balloon.
An open system (a control volume , CV) is a fixed region of space — the boundary (the control surface , CS) stays put while matter streams through it. Think of a fixed window drawn around a nozzle: air rushes in one face and out the other.
Why the topic needs this: the reliable energy law we own is written for closed systems (fixed matter). But a nozzle is an open system. The parent note's whole trick is to follow a closed lump through the open region, then translate. You cannot understand the derivation without holding both pictures at once. See Closed-system first law of thermodynamics for the closed version we start from.
m δ m m δ m little bit of m tiny travelling parcel of fluid we track from inlet to outlet. The little δ
Definition Mass flow ratem ˙ dot on any symbol means "per second" (rate). So ==m ˙ each second . Picture a turnstile at the pipe mouth counting kilograms as they pass.
Intuition Why the dot mattersIn steady flow the same m ˙ Conservation of mass — continuity equation . The dot lets us talk about a continuous stream instead of one frozen lump.
p p outward per unit area on a surface (units: pascals = N/m2
Definition Specific volumev v one kilogram of the fluid (units: m3 opposite of density: thin gas → big v v v
Every travelling parcel hauls a "backpack" of energy in three pockets. We measure each per kilogram (that's what "specific" means) so parcels of any size compare fairly.
Definition Internal energyu u inside the fluid as molecular jiggling and bonds (J/kg). Hotter fluid = faster molecules = more u
Definition Kinetic energy2 1 V 2 V speed of the flow (m/s) — how fast the whole stream moves, not the molecular jiggling. Its energy per kilogram is 2 1 V 2 2 1 m V 2 m
V v big V = fast-moving stream; big v = thin, spread-out fluid. Read the case carefully — capital V v
Definition Potential energyg z z g ≈ 9.81 2 z g z name it before we drop it.
Here is the punchline that the whole topic is built around.
u p v u and pays the flow-work toll p v always travel as a pair at a boundary. Rather than write them separately forever, we glue them into one bundle and give it a name.
h h = u + p v Enthalpy = "internal energy plus the flow work needed to push the fluid across a boundary." Picture a parcel's backpack (u plus the pre-paid ticket (p v
q q temperature difference (J/kg), heat added counts positive . Picture a flame under the pipe warming the stream. The rate form is Q ˙
w s w s useful work carried across by a spinning shaft or blade (J/kg), work out counts positive (the "work-out" convention). Picture a turbine blade the flow pushes on the way through (work out, w s > 0 w s < 0 W ˙ s
Common mistake Merging shaft work with flow workThey are different . Flow work p v pass through the ports — already hidden inside h w s extra work via a moving blade. Counting p v
"Steady" means nothing at a fixed point changes with time . The stream may speed up along the pipe, but at any chosen spot the readings hold constant. Symbolically ∂ / ∂ t ( ) = 0 ∂ / ∂ t
Definition Stagnation state andh 0 , T 0 rest (speed → 0) without adding heat. Its kinetic energy pocket empties into the enthalpy pocket, so enthalpy rises to the stagnation enthalpy h 0 = h + 2 1 V 2 stagnation temperature T 0 Stagnation properties & isentropic relations and Speed of sound and Mach number .
c p h = c p T c p one kilogram of the gas by one kelvin at constant pressure (J/kg·K). For a calorically perfect gas enthalpy is simply proportional to temperature:
h = c p T . Why this helps: it converts the abstract "enthalpy" into a thermometer reading. Combined with the stopped-flow idea it gives the master link T 0 = T + 2 c p V 2
Pressure p and specific volume v
Kinetic energy half V squared
Steady-Flow Energy Equation
Mass flow rate m dot and steady flow
Heat q and shaft work w_s
Read it bottom-right: pressure and volume make flow work ; flow work plus internal energy make enthalpy ; enthalpy plus the moving/lifting energies, fed into the closed-system law under steady flow, become the SFEE ; specialise it and out drops the stagnation relation.
Test yourself — cover the right side and answer before revealing.
What does a dot (as in m ˙ "Per second" — a rate of flow, e.g. kilograms crossing a face each second.
Difference between a closed and an open system? Closed = fixed lump of matter, boundary moves with it (no mass crosses); open = fixed region of space, mass streams through the boundary.
What is specific volume v Volume occupied by one kilogram of fluid (m3
Distinguish V v Capital V v 3
Where does flow work p v Force p A L v = A L p v
What is enthalpy and why is it defined? h = u + p v
Sign convention for q w s q > 0 w s > 0 out (turbine positive, compressor negative).
What are the three pockets in a parcel's energy backpack? Internal u 2 1 V 2 g z
What does stagnation enthalpy h 0 The enthalpy the flow would have if brought adiabatically to rest: h 0 = h + 2 1 V 2
For a perfect gas, how does enthalpy relate to temperature? h = c p T
