Calculating Flow Through Tiny Orifices

Accurately calculating the flow of gas or liquid through a tiny orifice is difficult because the shape of the leading edge of the orifice greatly affects the actual flow. For instance, the effective diameter of a sharp-edged orifice is 0.65 of the actual diameter. On the other hand, when an orifice has a leading-edge radius matching the orifice diameter, effective diameter equals the orifice diameter. Thus, the so-called orifice coefficient can vary between 0.65 and 1.0, depending on the radius of the leading-edge.

This is significant because flow through an orifice is proportional to the diameter squared. Thus, the flow may be reduced to as little as 0.65², or 44%, of theoretical full flow, depending on the shape of the leading edge.

Typically, it is almost impossible to measure the leading- edge radius of a tiny orifice with normally available equipment. The best way to determine the diameter required to produce a particular restriction is to first estimate the diameter. Then test an orifice from a range of orifices which have consistent leading-edge shapes, for the given fluid and flow conditions. From the test results, derive a flow constant that applies to the particular flow conditions, and adjust the original estimate using the appropriate gas or liquid flow equations.

Note that the constant (which accounts for the orifice coefficient) only applies for a particular set of flow conditions, fluid, and units. For example, if pressure is measured at different points on a gas-flow test rig, a different constant might have to be established for each because pressure varies throughout a moving gas in an irregular channel.

Liquid flow. For liquids where the pressure at the orifice is known in units of length (head):

Q = k_hD²√h

where Q = the mass flow rate in units of mass/time, D = orifice diameter in units of length, h = head in units of length, and k_h = the flow constant.

For liquids where units of pressure are used:

Q = k_PD²√P

where Q = mass flow rate in units of mass/time, D = orifice diameter in units of length, p = pressure in units of mass/length², and k_P = the flow constant. Note that the constant’s units, although not relevant, differ for units of pressure and units of head.

Gas flow. Calculations for gas flow depend on whether flow is subsonic or supersonic. Supersonic flow is independent of downstream conditions because pressure waves cannot travel upstream faster than the speed of sound. For supersonic gas flow:

where Q = mass flow in units of mass/time, D = orifice diameter in units of length, p₁ = upstream pressure in units of mass/length², T₁ = upstream absolute temperature, and k_s = the flow constant.

A wide variety of formulas apply to subsonic gas flow. For example:

where n = ratio of specific heats at constant pressure and constant volume, P₂ = downstream pressure in units of mass/length², and k_g = the flow constant. Rather than use this or other unwieldy equations — and make the necessary assumptions about system conditions that may affect accuracy — the simple solution is to use the supersonic equation and allow for a reduction in flow when making the initial estimate. In all these cases establish a constant as shown in the table.

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