Foundations — Pressure-fed cycle — simplest, used in upper stages

This page assumes nothing. Every letter, symbol, and word that the parent note throws at you is built here from the ground up, in an order where each idea leans on the one before it.

1. Pressure — the star of the whole show

Every symbol on this topic is a pressure of some kind, so we must nail down what pressure actually is before anything else.

Why does the topic need this? Because a fluid only moves when pressure is uneven: liquid always flows from where it is squeezed hard to where it is squeezed less. That single rule — high pressure → low pressure — is the engine of everything here.

Look at the figure: the same force $F$ on a small area gives a big pressure (sharp push), and on a big area gives a small pressure (soft push). A drawing pin works because your thumb's force lands on a tiny point.

Reveal check:

2. The named pressures: $p_c$, $p_{\text{tank}}$, and the drops $\Delta p$

Now that "pressure" means something, here are the specific pressures the topic labels.

Why the topic needs $\Delta p$: every obstacle between tank and chamber eats some pressure. The tank must start with enough extra to survive all these tolls and still arrive above $p_c$. Picture it as a marble rolling downhill losing a bit of height at each bump.

The figure is the pressure "staircase": start high at the tank on the left, step down at each loss, and you must still land above the chamber line on the right for liquid to flow in.

3. Density $\rho$ and the weight of liquid ($\rho gh$)

The parent note's Bernoulli line uses $\rho$, $g$, $h$, and $v$. Let's earn each.

Put them together and $\rho gh$ is a pressure — the extra squeeze at the bottom of a liquid column caused by the weight of liquid above it. This is why a deep swimming pool presses harder on your ears at the bottom.

Why the topic needs it: in the pressure budget, the weight of the propellant column can help push liquid down (adding pressure) or hinder it under acceleration. It's usually small compared to $p_c$, but it belongs in the honest accounting.

This trade between static pressure and motion is exactly what Bernoulli's equation describes, which the parent uses to build its pressure budget.

4. Material strength: $\sigma$, $t$, $r$ — why tanks get heavy

The mass-penalty section uses hoop stress. These three symbols carry it.

The figure cuts a spherical tank in half. The gas inside pushes the two halves apart across the circle of area $\pi r^2$; the thin ring of wall (area $2\pi rt$) holds them together. Balancing "push apart = wall holds" gives the hoop-stress relation: $\sigma =\frac{pr}{2t}$

Why the topic needs it: this is the reason pressure-fed engines run at low chamber pressure. Higher pressure $p$ means you need thicker walls $t$, which means more metal, which means a heavier tank. The penalty is baked into these three letters.

5. The ideal gas law: $pV=nRT$

The pressurant-sizing example uses this. Every symbol:

Why the topic needs it: the Ideal gas law tells you how much helium mass you must carry to fill the emptying tank at the right pressure. Squeeze more gas into a space (bigger $n$) and pressure rises; that is the whole logic of a pressurant bottle.

6. How it all fits — the prerequisite map

Read it top to bottom: pressure is the seed everything grows from. It branches into the pressure budget (with help from Bernoulli, density, speed) and into hoop stress (with help from material strength), and both branches meet at "why low chamber pressure", which is the heart of why the pressure-fed cycle lives on upper stages.

7. One tiny numerical sanity check

Equipment checklist

Cover the right side and test yourself before moving on: