I thought of an injector nozzle for a diesel engine with an alternate combustion chamber format (somewhat like a Heron combustion chamber), and would now like to get an estimate of the size of the opening for it.

The injector nozzle is basically a tube with a pintle inside it; the pintle has a flange at its end, outside of the tube and in the combustion chamber, which extends past the radius of the tube. When the injector is spraying, the fuel flows between the tube wall and pintle and is deflected outwards by the flange into the combustion chamber, forming a planar 'disk' of fuel between the piston and the top of the cylinder. The problem is to calculate an estimate of the opening $l$ between the flange and the butt end of the tube from a simplified description of the intended spray front.

Stylized illustration of the nozzle

I'll be using cylindrical coordinates $r,\theta,z$.

Design parameters:

  • $Q_0$, the mass flow rate in the tube
  • $r_0$, the flange radius (and thus of the nozzle)
  • $r_c$, the radius of the combustion chamber
  • $t_c$, the time allotted for fuel injection at max RPM.

Problem domain:

  • $S(r)=2\pi(r-r_0) l$ (spray front)
  • $V(r)=\pi(r^2-r_0^2) l$ (volume occupied)


  • The fuel occupies the whole opening of the nozzle, doesn't diverge vertically and distributes itself uniformly ($\frac{\partial\rho}{\partial r} = \frac{\partial\rho}{\partial\theta} =\frac{\partial\rho}{\partial z} = 0$)
  • $Q_0$ is the flow not just in the nozzle, but also at the spray front ($Q_0=\iint_S \mathbf j\cdot d\mathbf S$)


$$m(t_c)=\pi\rho(r_c^2-r_0^2) l=Q_0 t_c\\ \implies l=\frac{Q_0 t_c}{\pi\rho(r_c^2-r_0^2)}$$

I get $l$ = 1.34×10-5 cm = 134 nm for an application in a high-speed two-stroke ($r_c$ = 2.2cm, $r_0$ = 1mm, $t_c$ = 2×10-4s (more or less the time taken to traverse 10°, my estimate of the injection duration, at 8000 rpm), $Q_0$ = .8 g/s), which is absurdly tiny.

Even though the order of magnitude of the result comes mainly from the injection duration, I don't trust my results and think that it comes from the assumptions I made. On top of that, I don't know how to take into account the end of injection prior to combustion, so I assumed the injector stops injecting at combustion.

Is there anything wrong with the analysis or calculation that I did?

  • $\begingroup$ How did you determine your flow rate? $\endgroup$
    – Dan
    Dec 5, 2016 at 4:45
  • $\begingroup$ Took a sample BSFC, converted it to g/s per injector. Could be too small... $\endgroup$
    – setun-90
    Dec 5, 2016 at 7:44

1 Answer 1


Revisiting the problem anew, it turns out that the units were interacting in a way I wasn't taking into account.

When I asked this question, I had input $r_c$ and $r_0$ in cm. However, $\rho$ was in kg/m3, and $Q_0$ was in g/s; additionally, my $Q_0$ was too large (an Audi 3.3L V8 TDI achieves 0.014 g/s per injector).

Surely enough, putting $r_c$ and $r_0$ into meters, then putting $\rho$ in g/m3 and finally correcting $Q_0$ gave me $l$ = 2.9342×10-6m ≈ 2.93 μm, which is more reasonable given that $2\pi r_0 l$ = 9.2181×10-8 m2 ≈ 9.22 μm2, comparable to the injection area per injector for a PSA HDi DW10D with 8×110-micrometer nozzles of 7.6 μm2.

If there's a lesson to be learned here, it's to double check not just your dimensions, but your units... especially the ones indirectly related to the result.

  • $\begingroup$ A good reason to make sure the units balance on both sides of the equation. $\endgroup$
    – Solar Mike
    Sep 30, 2017 at 17:18

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