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A closed steam boiler in a coffee machine idles at around 124 °C, the steam and water inside at 1.2 barg. When a steam valve is opened, steam rushes out through a steam wand inserted in milk, to aerate and heat it.

What I'm trying to determine is the flow rate of that steam, but all thermodynamic info on steam boilers I'm finding is for industrial situations and too complex for my very basic understanding of thermodynamics.

I'm disregarding any further heating from the boiler when the valve is open and am just assuming an infinitely big boiler with unlimited pre-generated steam

I'm thinking the steam tube cross sectional area is important, as would be the length and type of tube for losses. Can someone put me on the right track as far as formulae go.

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You can use the Hagen-Poiseuille equation to determine the flow rate as a result of a pressure gradient. A caveat in using this equation is that the flow has to be laminar, i.e. inertial effects can be neglected.

It is best used in the form: $$Q=\frac{1}{128}\frac{\pi d^4}{\mu}\frac{\Delta P}{L}$$ from which you can see that the volumetric flow rate $Q$ is dependent (indeed) on the cross sectional area throught the tube diameter $d$, the length $L$ of the tube and the pressure drop $\Delta P$ applied along the tube. Furthermore, the type of fluid is important through the viscosity $\mu$; a more viscous fluid would be more difficult to force through the tube at the same pressure gradient.

This equation already accounts for frictional losses at the tube walls (hence the required pressure drop) but it doesn't take into account any bends in the tubing or the type of tubing. However, I expect their influence to be negligible as long as most of the tubing is straight and rough enough to allow the use of the no-slip condition but not so much to induce flow instabilities.

As mentioned earlier, after determining the flow rate from the information you have, you will need to check that the flow is indeed laminar. If not the equation cannot be used. You can do this by calculating the Reynolds number:

$$\mathrm{Re}=\frac{\rho vd}{\mu}\approx\frac{\rho Q}{\mu d}<1$$

As long as the above condition holds you are fine. If not then the approach suggested by @idkfa is best used.

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You stated that most things you found are too complex for your basic understanding, is it safe to assume that the answer you are seeking may be an educated guess? If so I would go with Bernoulli

$$g z_1 + \frac{c_1^2}{2} = g z_2 + \frac{c_2^2}{2} + \int_1^2 \frac{dp}{\varrho}$$

Assume isothermal state change. Also assume $c_1 = 0$ if you say you regard the boiler as infinitely big. Properties of the steam could be acquired via books or online tables.

Lastly you need the mentioned geometry of the tube.

For more details use the instationary form and add friction losses if you like. I personally would neglect this.

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  • $\begingroup$ Yes, just need to be ball park to see if a concept is worth exploring further. $\endgroup$ – jontyc Jul 5 '16 at 4:32
  • $\begingroup$ Why is there an integration here? $\endgroup$ – Algo Jul 6 '16 at 6:11
  • $\begingroup$ @algo I'd regard the steam as a compressible fluid therefore you need to integrate the density as it is not constant. But it should be $dp$ instead of $p$ thanks for making me aware. $\endgroup$ – idkfa Jul 6 '16 at 7:55
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I don't recommend using the Hagen-Poiseuille equation as the flow is most probably turbulent judging from the given pressure difference. and since the flow rate will be mainly governed by the pressure difference between the boiler and the wand outlet (say ambient) I think @idkfa 's answer is enough.

But, if it you have a situation where the length of the pipe/friction losses are of significance, you can use the Unwin-Babcock equation which is an extension of the Darcy-Weisbach equation to be applicable for steam:

$$ \Delta P = \frac{0.0001306\,QL}{d^3\rho _ {steam}} (1 + \frac{3.6}{d})$$

where $\Delta P$ is the pressure drop in psi, $Q$ is the volumetric flow rate, $L$ is the pipe length, $d$ is the pipe diameter and $\rho _{steam}$ is the density of the steam. However, the equation is not always guaranteed to give accurate result for all cases of steam flow so you should use the Moody chart alongside (giving it the upper hand as a judge of the pressure drop result).

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  • $\begingroup$ The density of steam at inlet or outlet? $\endgroup$ – Salomon Turgman Jul 5 '16 at 23:29
  • $\begingroup$ Oops. Never mind. For flow rate at outlet, outlet density. For inlet, inlet density. $\endgroup$ – Salomon Turgman Jul 5 '16 at 23:48
  • $\begingroup$ @sturgman Well, since the flow is supposedly isothermal (for insulated steam distribution pipes where the equation is originally applicable) the difference in density won't be significant between inlet and outlet, so it doesn't matter (or you may use an average value). $\endgroup$ – Algo Jul 6 '16 at 6:02

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