Nozzle & Diffuser: Continuity Equation Applications

A nozzle and a diffuser are the two simplest pieces of flow hardware, and they are mirror images of one another. A nozzle is a duct whose cross-section shrinks along the flow; a diffuser is one whose cross-section grows. For a steady, incompressible flow through either, the single-inlet–single-outlet equation A₁V₁ = A₂V₂ fixes the velocity at every section once you know the area there.

In a nozzle the area falls from inlet to outlet, A₂ < A₁, so the velocity must rise: V₂ = (A₁/A₂) V₁, and because A₁/A₂ > 1 the outlet is faster than the inlet. A garden-hose nozzle, a jet engine's exhaust, a fire-fighter's tip — all squeeze the flow through a smaller exit and the fluid leaves faster than it entered. A diffuser does the opposite: A₂ > A₁, so V₂ < V₁, and the flow slows down. The outlet duct of a wind tunnel and the volute of a centrifugal pump both diffuse — they trade speed for a gentler outflow.

Notice what continuity does and does not say. It tells you exactly how the velocity changes with area; it says nothing about why the fluid obligingly speeds up. That 'why' is pressure, and it is the energy equation's job (Module 5), not continuity's. For now treat A₁V₁ = A₂V₂ as a purely kinematic constraint: given the areas and one velocity, the other velocity is fixed.

Real pipe systems branch. A single main may split into two feeds, or several lines may merge into one. Continuity still holds — it must, because mass is conserved at every junction — but now the single-inlet–single-outlet form is replaced by a sum over all the ports meeting at that junction.

For a steady liquid junction the balance is on volume: add the inflowing Q's, subtract the outflowing Q's, set the result to zero, and any one unknown flow follows immediately. The junction itself need not be symmetric or tidy; the balance cares only about totals.

That same balance is continuity's quiet superpower: a steady flow that does not balance is impossible. If two pipes feed a junction at 3 kg/s and 2 kg/s and a single outlet draws 5 kg/s, the books balance (5 in, 5 out). If that outlet drew 4 kg/s instead, the steady state could not exist — mass would have to vanish, or pile up and the flow would not actually be steady. So a mass-flow balance is a built-in sanity check on flow-meter readings, design sketches, and textbook answers: if the numbers do not close, something is wrong upstream of the equation.

So far every example has been steady, so the accumulation term ∂/∂t ∫_CV ρ dV was zero. The moment a flow is unsteady — a tank filling or draining, a pipeline filling during start-up, a balloon inflating — that term wakes up and does the work.

For a tank with a free surface and a fixed cross-sectional area A_tank, the mass inside is ρ A_tank h (with h the depth), so the accumulation term is ρ A_tank dh/dt. Continuity then reads that the rate of mass build-up equals the net mass flowing in. Divide through by the density for a liquid and you get the clean result that the free surface rises or falls at exactly the rate the volume imbalance demands.

If more flows in than out, the level rises; if more flows out, it falls; if they match, the level holds steady even though fluid is continually passing through. This is the same equation as before — only now the control volume's contents are explicitly changing with time. The differential form of the next lesson takes this 'changing contents' idea to its limit, shrinking the control volume all the way down to a point.