APPLICATION NOTE NO. 10
Compressibility Compensation of Sea-Bird Conductivity Sensors
Revised May 2013

Print version

Sea-Bird conductivity sensors provide precise characterization of deep ocean water masses. To achieve the accuracy of which the sensors are capable, an accounting for the effect of hydrostatic loading (pressure) on the conductivity cell is necessary. Conductivity calibration certificates show an equation containing the appropriate pressure-dependent correction term, which has been derived from mechanical principles and confirmed by field observations. The form of the equation varies somewhat, as shown below:

SBE 4, 9, 9plus, 16, 19, 21, 25, 25plus, 26, 26plus, and 53 BPR

SBE 16plus, 16plus-IM, 16plus V2, 16plus-IM V2, 19plus, 19plus V2, 37, 45, 49, and 52-MP

where

• a, b, c, d, m, and CPcor are the calibration coefficients used for older sensors (prior to January 1995); Sea-Bird continues to calculate and print these coefficients on the calibration sheets for use with old software, but recommends use of the g, h, i, j, CTcor, CPcor form of the equation for most accurate results
• g, h, i, j, CTcor, and CPcor are the calibration coefficients used for newer sensors.
  Note: The SBE 26, 26plus, and 53 BPR use the SBE 4 conductivity sensor, so both sets of calibration coefficients are reported on the calibration sheet. Seasoft for Waves for DOS, which can be used with the SBE 26 only, only supports use of the a, b, c, d, CTcor, and CPcor coefficients. The current processing software for these instruments, Seasoft for Waves for Windows, only supports use of the g, h, i, j, CTcor, CPcor coefficients.
• CPcor is the correction term for pressure effects on conductivity
• slope and offset are correction coefficients used to make corrections for sensor drift between calibrations; set to 1.0 and 0 respectively on initial calibration by Sea-Bird (see Application Note 31 for details on calculating slope and offset)
• f is the instrument frequency (kHz) for all instruments except the SBE 52-MP.
  For the SBE 52-MP, f = instrument frequency (kHz) * (1.0 + WBOTC * t) ^0.5 / 1000.00
• t is the water temperature (°C).
• p is the water pressure (decibars).

Sea-Bird CTD data acquisition, display, and post-processing software Seasoft for Waves (for SBE 26, 26plus, and 53 only) and Seasoft (for all other instruments) automatically implement these equations.

Conductivity cells do not measure the actual conductance (the desired property), but rather the conductance of a specific geometry of water. The ratio of the cell's length to its cross-sectional area (cell constant) is used to relate the measured conductance to the actual conductance. Under pressure, the conductivity cell's length and diameter are reduced, leading to a lower indicated conductivity. The magnitude of the effect is not insignificant, reaching 0.0028 S/m at 6800 dbars.

The compressibility of the borosilicate glass used in the conductivity cell (and all other homogeneous noncrystalline materials) can be characterized by E (Young's modulus) and ν (Poisson's ratio). For the Sea-Bird conductivity cell, E = 9.1 x 10⁶ psi, ν = 0.2, and the ratio of indicated conductivity divided by true conductivity is:

1 + s

where

• s = ( CPcor ) (p)
  Typical value for CPcor is - 9.57 x 10^-8 for pressure in decibars    or     - 6.60 x 10^-8 for pressure in psi

Note:  This equation, and the mathematical derivations below, deals only with the pressure correction term, and does not address the temperature correction term.

For a cube under hydrostatic load:

ΔL / L = s = - p ( 1 - 2  ν) / E

where

• p is the hydrostatic pressure
• E is Young's modulus
• ν is Poisson's ratio
• ΔL/L and s are strain (change in length per unit length)

Since this relationship is linear in the forces and displacements, the relationship for strain also applies for the length, radius, and wall thickness of a cylinder.

To compute the effect on conductivity, note that R₀ = ρ L / A , where R₀ is resistance of the material at 0 pressure; ρ is volume resistivity, L is length, and A is cross-sectional area. For the conductivity cell:

A = p r ²

where

• r is the cell radius

Under pressure, the new length is L (1 + s ) and the new radius is r (1 + s ). If R_p is the cell resistance under pressure:

R_p = ρ L (1 + s) / (p r ² [1 + s ] ²) = ρ L / p r ² (1 + s ) = R₀ / (1 + s )

Since conductivity is 1/R:

C_p = C₀ (1 + s )      and       C₀ = C_p / (1 + s ) = C_p / (1 + [CPcor] [p])

where

• C₀ is conductivity at 0 pressure
• C_p is the conductivity measured at pressure

A less rigorous determination may be made using the bulk modulus of the material. For small displacements in a cube:

ΔV/V = 3 ΔL/L = - 3p (1 - 2ν) / E    or    ΔV/V = - p / K

where

• ΔV/V is the change in volume per volume or volume strain
• K is the bulk modulus. K is related to E and ν by K = E/3(1 - 2ν).

In this case, ΔL/L = - p / 3K.

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