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The theory of a vibrating-rod densimeter

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Abstract

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.

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Abbreviations

A, B, C, D :

constants in equation (60)

A j , B j :

constants in equation (18)

a +j , a −j :

wavenumbers given by equation (19)

C f :

drag coefficient defined in equation (64)

C /0 f , C /1 f :

components of C f in series expansion in powers of ε

c :

speed of sound

D b :

drag force of fluid b

D 0 :

coefficient of internal damping

E :

extensional modulus

\(F,\tilde F\) :

force per unit length

F +j , F −j :

constants in equation (24)

f, g :

functions of σ defined in equations (56)

G :

modulus of rigidity

I :

second moment of area

K :

constant in equation (90)

k, k′ :

constants defined in equations (9)

L :

half-length of oscillator

Ma :

Mach number

m a :

mass per unit length of fluid a

m b :

added mass per unit length of fluid b

m s :

mass per unit length of solid

n j :

eigenvalue defined in equation (17)

P :

power (energy per cycle)

P a , P b :

power in fluids a and b

p :

pressure

R :

radius of rod or outer radius of tube

R c :

radius of container

R i :

inner radius of tube

r :

radial coordinate

T :

tension

T visc :

temperature rise due to heat generation by viscous dissipation

t :

time

v r , v θ :

radial and angular velocity components

y :

lateral displacement

z :

axial coordinate

α :

dimensionless tension

β a :

dimensionless mass of fluid a

β b :

dimensionless added mass of fluid b

β′ b :

dimensionless drag of fluid b

γ :

dimensionless parameter associated with λ

Δ0 :

dimensionless coefficient of internal damping

\(\Delta \tilde \omega \) :

dimensionless half-width of resonance curve

\(\Delta \tilde \omega _r \) :

dimensionless frequency difference defined in equation (87)

δ :

spatial resolution of amplitude

δR, δμ, δρ, δρ s , δω:

increments in R, μ, ρ, ρ s , ω

ε :

dimensionless amplitude of oscillation

ζ :

dimensionless axial coordinate

η :

ratio of \(\tilde \omega _0 \) to \(\tilde \omega _r \)

η a , η b :

ratios of \(\tilde \omega _0 \) to \(\tilde \omega _r \) for fluids a and b

θ :

angular coordinate

λ:

parameter arising from distortion of initially plane cross-sections

λ f :

thermal conductivity of fluid

Λ:

dimensionless parameter associated with λ

μ :

viscosity of fluid

μ a , μ b :

viscosity of fluids a and b

ξ :

dimensionless displacement

ξ j :

jth component of ξ

ρ :

density of fluid

ρ a , ρ b :

density of fluids a and b

ρ s :

density of tube or rod material

ρ′ :

density of fluid calculated on assumption that σ* → ∞

σ :

dimensionless radial coordinate

σ*:

dimensionless radius of container

\(\tau ,\tilde \tau \) :

dimensionless times

τ rr τrr, τrθ :

radial normal and shear stress components

φ :

spatial component of ξ defined in equation (13)

φ j :

jth component of φ

Φ:

dimensionless streamfunction

Φ0, Φ1 :

components of Φ in series expansion in powers of ε

χ:

phase angle

χ r :

phase difference

χ ra , χ rb :

phase difference for fluids a and b

Ψ:

streamfunction

Ψ j :

jth component defined in equation (22)

Ω:

dimensionless frequency (based on ρ)

Ω a , Ω b :

dimensionless frequency in fluids a and b

Ω s :

dimensionless frequency (based on ρ s )

ω :

angular frequency

ω 0 :

resonant frequency in absence of fluid and internal damping

ω r :

resonant frequency in absence of internal fluid

ω ra , ω rb :

resonant frequencies in fluids a and b

\(\tilde \omega \) :

dimensionless frequency

\(\tilde \omega _r \) :

dimensionless frequency when β a vanishes

\(\tilde \omega _{ra} ,\tilde \omega _{rb} \) :

dimensionless frequencies when β a vanishes in fluids a and b

\(\tilde \omega _0 \) :

dimensionless resonant frequency when β a , βb, β′b and Δ0 vanish

\(\tilde \omega _1 \) :

dimensionless resonant frequency when β a , βb and β′ b vanish

\(\tilde \omega _2 \) :

dimensionless resonant frequency when β b and β′ b vanish

\(\tilde \omega _ + ,\tilde \omega _ - \) :

dimensionless frequencies at which amplitude is half that at resonance

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Authors and Affiliations

  1. Department of Chemical Engineering and Chemical Technology, Imperial College, Prince Consort Road, SW7 2BY, London, England, UK

    T. Retsina, S. M. Richardson & W. A. Wakeham

Authors
  1. T. Retsina
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  2. S. M. Richardson
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  3. W. A. Wakeham
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Retsina, T., Richardson, S.M. & Wakeham, W.A. The theory of a vibrating-rod densimeter. Appl. Sci. Res. 43, 127–158 (1986). https://doi.org/10.1007/BF00386040

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