**By Gary Wu**Mechanical Engineer

Millipore Corp.

Allen, Tex.

**EDITED BY VICTORIA REITZ**

Most mass-flow controllers are thermal based. That is, they use the thermal properties of a gas to directly measure the mass-flow rate. But for many control situations — such as when downstream pressure is too low, allowable pressure drop is too small, or gas (vapor) properties change — pressurebased controllers which use choked-flow phenomenon are the better choice.

Understanding the basic principles of choked flow at the valve opening lets designers take advantage of some built-in properties.

**CHOKED FLOW BASICS**When gas travels through an orifice the flow velocity at the opening depends on upstream and downstream pressures

*P*

*and*

_{u }*P*

*. Increasing the upstream pressure or decreasing the downstream pressure will increase velocity until it reaches sonic speed. When*

_{d}*P*

*/*

_{d }*P*

*< 0.528, pressure equals absolute pressure and increasing the upstream pressure or decreasing the downstream pressure further will not increase the velocity at the orifice. This point is called choked flow. For choked flow, the flow rate is a linear function of the upstream pressure, which changes the gas density. Downstream pressure changes do not affect the flow velocity and flow rate. Unless a specially shaped nozzle is used, the downstream gas velocity cannot surpass sonic speed. Downstream disturbances cannot travel backwards fast enough to influence the upstream flow. By controlling upstream pressure, the valve can deliver gas with a stable flow rate. A ball in the valve opening, for instance, can be adjusted to control upstream pressure.*

_{u }**CALCULATING RESPONSE**Designers have two options when dealing with choked-flow valves. They can calculate response time for the pressure change from initial pressure

*P*

*to final pressure*

_{i }*P*

*based on the valve design, or design the valve to obtain a certain response time.*

_{f }The response time is related to the dead volume *V*_{0 }inside the valve, the orifice flow rate, and the initial and final pressures *P** _{i }*and

*P*

*.*

_{f}First, transfer the gas in the dead volume V0 to the standard condition (14.7 psia and 70°F) volume *Q** _{i}*.

Some gas will then flow out of or into the valve with a flow rate •*m*, which is a function of *P*. In addition, *P *is a function of time *t*.

Assuming that completely closing the valve decreases the pressure and completely opening the valve raises the pressure, the standard gas volume is

The pressure change in the chamber depends on *Q*. It can be expressed as

Substitute the pressure change into the gas volume

Differentiating the equation gives

or

Integrate the equation to produce

Use the initial conditions, *t *= 0 and *Q *= *Q** _{i}*, to get

*C*=

*lnQ*

*. The equation can be written in terms of time*

_{i}Substituting *Q** _{i }*from the first equation into the response time results in

To find the time needed for the pressure to drop from *P** _{i }*to

*P*

*, the equation can be expressed as*

_{f}If the response time is known, dead volume of the valve is given as

In the case of a pressure increase, the equations become