# Calculating leakage in an all steel piston cylinder arrangement

I wish to calculate air leakage in a piston cylinder arrangement. One side is pressurized and the other is at atmospheric pressure. All steel construction. No piston rings or lubrication. The piston oscillates at about 50 to 80 times per sec, amplitude 5 mm, and is used as an indentation marking tool.

I know the piston diameter, cylinder diameter, piston length, air pressure and roughness. I need to calculate the leakage in liters per minute to determine: If the clearance is increased, how much increase in piston length is required for the same leakage? Less clearance gives more air efficiency, but is difficult to manufacture.

• Are there any piston rings and/or lubrication? It seems to me that your answer will be depend on the application—high-speed or low-speed, automotive or hydraulic, etc.
– Air
May 13, 2015 at 20:25
• No piston rings or lubrication. The piston oscillates at about 50 to 80 times per sec, amplitude 5 mm. used as a indentation marking tool. less clearence gives more air efficiency, but difficult to manufacture. May 14, 2015 at 10:54

If the piston is in (slight) clearance inside the cylinder, it's a bit like calculating the flow rate through a capillary, with the difference in diameter being the capillary diameter and the contact length between the piston and the cylinder the capillary length. There are various online calculators to help you with the actual calculation, such as this one. If you're after the theory, it's based on Bernoulli's principle.

EDIT

The length doesn't actually come into account in Bernoulli's equation, only the pressure and flow rate, if you neglect the difference in altitude. If you want to take account of the length, then you also need to look at the friction and things start to get messy. The most common approach is generally the Darcy-Weisbach equation. It's expressed as pressure loss, but you can rearrange the equation to work out the fluid velocity and therefore the flow rate. Careful which friction factor you use, the Darcy factor is 4 times the so-called Fanning friction factor. For laminar flow, it is generally approximated as $\frac{64}{R_e}$ where $R_e$ is the Reynolds number.

The air properties are also dependent on pressure and temperature, but assuming dry air at ambient temperature at standard atmospheric pressure (not exactly true, but it won't be far off) and a pressure drop of 1 bar between the piston chamber and the atmosphere, I get a flow rate of ~20.4 mL/min with a 1mm length and ~1.7mL/min with a 12mm length. I checked a posteriori that the assumption of laminar flow was valid.

Darcy-Weisbach equation

Pressure loss form:

$\Delta P = f_D * \frac{L}{D} * \frac{\rho u^2}{2}$

Re-arrange to give flow rate, given that $Q = A * u$

$Q = A * \sqrt{\frac{2}{\rho} * \frac{\Delta P * D}{f_D * L}}$

Friction factor

Assume flow is laminar (need to check a posteriori):

$f_D = \frac{64}{R_e} = 64 * \frac{\nu A}{QD}$

which gives (using $\nu = \mu / \rho$):

$Q = \frac{A * \Delta P * D^2}{L} * \frac{1}{32 \mu}$

Numerical application

D = 0.04e-3;    % [m], diameter
L = 1e-3;       % [m], length
dP = 1e5;       % [Pa], pressure difference
mu = 1.846e-5;  % [kg/(m*s)], dynamic viscosity of dry air at 1atm and 300K
rho = 1.2922;   % [kg/m^3], air density at standard conditions for P and T
A = pi*D^2/4;   % [m^2], cross area
Q = A*dP*D^2/(L*32*mu);     % [m^3/s], flow rate
Re = Q*D/(A*(mu/rho));    % [-], Reynolds number


This gives a flow rate of ~20.4 ml/min and a Reynolds number of ~758, which is less than 2000, so the assumption of laminar flow is valid.

Change the length to 12mm, and you get a flow rate of ~1.7 ml/min and a Reynolds number of ~63.

• Thank you. your reply seems to point almost near to required solution. My requirement is that, if I increase length of piston, the increased friction will reduce flow rate. How to calculate that ? My piston is dia 16 mm, clearance is 0.04 mm. If piston length increased from 1 mm to say 12 mm ? Please suggest online calc. May 14, 2015 at 11:04
• See my edit, there is no "magic" solution, things get complicated quite quickly. May 14, 2015 at 13:50
• Thank you for this additional explanation. I searched for Darcy–Weisbach equation calculators. and found some. I will read some more and ask. Thank you so much, I am an electronics engineer and not quite close to pneumatics but must do the work :-) May 15, 2015 at 15:29
• No worries. I have added the detailed steps I took to get the results. If it answers your question, feel free to accept the answer :-) May 15, 2015 at 16:07
• Yes, indeed. I mentioned Darcy-Weisbach equation as an additional tool/equation to take account of friction. Nov 13, 2017 at 9:46