Foundations — Turbopump design — centrifugal pump, axial turbine stages, NPSH

This page assumes you have seen nothing. We build every letter, ratio, and squiggle the parent note leans on, in an order where each piece rests on the one before it. Link back to the parent any time: the full topic note.

1. The absolute basics of "how much stuff moves"

Mass flow rate — $\dot{m}$

The picture. Imagine a pipe. Draw an invisible slice across it. Count the kilograms that cross that slice in one second — that number is $\dot{m}$. The little dot on top always means "per second" (a rate).

Why the topic needs it. A rocket engine is a flow machine — propellant is constantly streaming through. Every power and force we compute is "per second," so we must know how much mass streams per second.

Density — $\rho$

The picture. A $1\text{m}\times 1\text{m}\times 1\text{m}$ box. Fill it with the liquid and weigh it. Water gives $1000\text{kg}$, so ${\rho }_{\text{water}}=1000$. Liquid hydrogen (LH₂) is fluffy: only $71\text{kg}$ in that same box.

Why the topic needs it. LH₂ being so light is exactly why its pressure numbers turn into gigantic "column heights" later — keep this fact in your pocket.

2. Circles and spinning — turning rotation into speed

Angular speed — $\omega$

What is a radian? Walk around a circle. One full loop is $2\pi \approx 6.28$ radians. So a radian is just a way to measure "how much of a turn" using the number $\pi$ instead of degrees.

RPM to $\omega$

Engineers often quote spin as RPM (revolutions per minute — full turns each minute). To convert:

$\omega =\frac{2\pi N}{60}$

Blade speed — $u=\omega r$

The picture. See figure below: the same $\omega$ gives a small speed near the axis ($u_1=\omega r_1$) and a large speed at the rim ($u_2=\omega r_2$). The subscript just labels which radius: $1$ = inlet (near axis, the "eye"), $2$ = outlet (the rim).

Why the topic needs it. The whole pump works by dragging liquid up to a fast rim speed $u_2$. Since head grows like $u_2^2$, a bigger rim or faster spin is the single biggest lever in the design.

3. Splitting a velocity into useful pieces

When liquid leaves the impeller rim it moves in some direction — partly outward, partly sideways (around the circle). We must separate these two.

The picture. Stand at the rim and draw the fluid's velocity arrow. Drop it onto two directions: one pointing straight out (radial), one pointing sideways along the circle (tangential). The sideways one is $c_u$.

• $c_{u1}$ = whirl at the inlet. Design choice: often $c_{u1}=0$ (liquid fed straight in, no pre-spin).
• $c_{u2}$ = whirl at the outlet (rim) — this is the whirl the impeller adds.

4. Energy per kilogram, and "head"

Torque — $T$

Power — $P=T\omega$

Gravity — $g$

$g\approx 9.81{\text{m/s}}^2$ is the acceleration gravity gives falling things. It appears because we choose to measure energy as an equivalent height of liquid (next).

Head — $H$

Pressure — $p$, and rise $\Delta p$

$\Delta p$ (the triangle $\Delta$ means "change in") is the pressure the pump adds: outlet minus inlet.

5. Gas-side extra symbols (for the turbine)

The turbine is spun by hot gas, and gas needs a few more letters.

• $c_p$ — specific heat at constant pressure: joules needed to warm one kilogram of gas by one degree, at fixed pressure. Units $\text{J/(kg⋅K)}$. Big $c_p$ = gas stores lots of thermal energy to hand over.
• $T_0$ — stagnation (total) temperature: the temperature the gas would have if brought smoothly to rest. Think "total thermal energy on tap." Units: kelvin, $\text{K}$.
• $\gamma$ — heat capacity ratio ("gamma"): a pure number (~1.2–1.4) describing how a gas heats up when squeezed. It controls how much energy expansion can release.
• ${\pi }_t$ — turbine pressure ratio: inlet pressure divided by outlet pressure across the turbine. Bigger ratio = more room for the gas to expand and give up energy.
• $\eta$ — efficiency ("eta"): fraction of ideal energy actually captured, between $0$ and $1$. Subscripts say which device: ${\eta }_t$ turbine, ${\eta }_{\text{pump}}$ pump, ${\eta }_{\text{mech}}$ shaft bearings/seals.

6. The boiling-safety symbols

• $p_v$ — vapor pressure: the pressure below which the liquid boils at its current temperature. LOX and LH₂ sit close to this already — dangerous.
• $z$ — static height: how high the tank sits above the pump. Higher tank = gravity gives extra push (helps).
• $h_{\text{loss}}$ — friction head loss: metres of "push" eaten by rubbing against pipe walls on the way down (hurts).
• NPSH — Net Positive Suction Head: the safety margin, in metres, between the pressure the liquid actually has at the pump inlet and the pressure at which it would boil.

The prerequisite map

Each root symbol feeds one of the three pillars — pump head, turbine power, NPSH — which together are the turbopump. Related cycle ideas live in Gas Generator Cycle and Staged Combustion Cycle; the pump/turbine share the same physics as the Euler Turbomachinery Equation, and the inlet-boiling logic rests on Bernoulli's Principle.

Equipment checklist

Cover the answers and test yourself — you are ready for the topic when every line comes instantly.