Diaphragms for Pressure Actuators

• Introduction
• The Model
• The Selection
• Postscript
• Further reading
• Related case studies

Introduction

A barometer is a pressure actuator. Changes in atmospheric pressure, acting on one side of a diaphragm, cause it to deflect. The deflection is transmitted through mechanical linkage or electromagnetic sensor to a read-out. Similar diaphragms form the active component of altimeters, pressure gauges, and gas-flow controls for diving equipment. Their design requirements are summarized in the table: the diaphragm must be sensitive, remain elastic and, to be accurate, have low hysteresis or damping. What is the best material of which to make diaphragms?

Figure 12.1 A diaphragm for a pressure sensor. It must deflect under pressure, without creep or permanent deformation.

Table 12.1 The design requirements

The Model

Figure 12.1 shows a diaphragm of radius a and thickness t. A pressure difference Δp = p₁ − p₂ acts across it. We wish to maximize the deflection of the center of the diaphragm, subject to the constraint that it remains elastic. The stresses throughout the material must be less than the yield or fracture stress, σ_f.

The deflection δ of a diaphragm caused by Δ_p depends on whether its edges are clamped or free. It is:

(M12.1)

with C₁ = 3 / 16(clamped edges)

or C₁ ≈ 9 / 8 (free edges).

Here E is Young's modulus, and ν is Poisson's ratio. The maximum stress in the diaphragm is

(M12.2)

with C₂ = 1 / 2 (clamped edges)

or C₂ ≈ 3 / 2 (free edges).

This stress must not exceed the yield or failure stress, σ_f, which we identify with the modulus of rupture, σ_MOR (the surface stress, in bending, at which yielding or fracture occurs).

The radius of the diaphragm is generally determined by the design; the thickness t is free. Eliminating t between the two equations gives

(M12.3)

The material properties are grouped in the last brackets. The quantity 1 − ν² is close to 1 for all solids. Therefore, the best material for the diaphragm is that with the largest value of

(M12.4)

The Selection

Figure 12.2 shows the chart we need: the modulus of rupture, σ_MOR, plotted against Young's modulus, E. The solid line shows the index M, positioned to leave a subset of materials exposed. They are listed in Table 12.2.

Figure 12.2 The modulus of rupture, σ_MOR, plotted against Young's modulus, E.

The remaining concerns are those of damping and cost. Figure 12.2 shows a chart of these two quantities, with a 'protective' selection box, the purpose of which is simply to eliminate materials which have high damping or are very expensive. Metals make good diaphragms, notably spring steel, and high-strength titanium alloys. Certain polymers are possible — nylon, polypropylene and PTFE — but they have higher damping, and may creep. So do elastomers: both natural and artificial rubbers. Table 12.2 lists the candidates.

Figure 12.3 The damping coefficient, η, plotted against cost/kg, C_m.

MATERIAL		COMMENT
High-C steels	0.3 – 0.37	The standard choice. Low loss coefficient gives rapid response
Glasses	up to 0.5	Possible if protected from damage
Copper-beryllium alloys	0.28 – 0.31	Good performance combined with corrosion resistance
Bronzes	0.1 – 0.15
Nickel-silvers	0.12 – 0.14
High-strength wrought Al alloys	0.12 – 0.15	Good performance and corrosion resistance
Al-SiC MMCs	0.1 – 0.14
Ti-Alloys	0.2 – 0.3	As good as steel, corrosion resistant, but expensive
Polymers	0.1 – 0.3	Excellent M value, giving large elastic deflection. High loss coefficient limits response time; creep compromises precision
Elastomers	2 – 10

Table 12.2 Materials for diaphragms

Postscript

As always, application of the primary design criterion (large δ without failure) leads to a subset of materials to which further criteria must be applied. Elastomers have the best values of M, but they have high damping, are easily punctured, and may be permeable to certain gases or liquids. If corrosive liquids (sea water, cleaning fluids) are likely to contact the diaphragm, then stainless steel or bronze may be preferable to a high-carbon steel, even though they have smaller values of M. This can be overcome by design: crimping the diaphragm or shaping it like a bellows magnifies deflection without increasing the stress, but adds to the manufacturing cost.

Further Reading

Young, WC (1989) 'Roark's Formulas for Stress and Strain', 6th Edition, McGraw-Hill, NY, USA.

Related Case Studies

1. Materials for Springs

2. Elastic Hinges and Couplings

3. Materials for Static Seals

4. Knife Edges and Pivots

5. Materials for Shaker Tables