AUTOMATIC VISCOSITY CALCULATION

Clicking the following link, you may download a free spread-sheet Excel to automatically calculate the viscosity of a gas mixture containing up to 7 components

Viscosity is calculated by using empirical formulas indicated in the right column of this page and introduced by Carr and Reichenberg.

The file is freely available to everybody wants to use it, but don't include any warranty from Be.T.A. Strumentazione : we will appreciate every indication to keep this service more effective. Please send your comments to info@beta-strumentazione.it .

EFFECTS AND COMPENSATION OF THE VISCOSITY IN DILUTION PROCESS.

The viscosity of the mixtures involved in a dilution (mixture to be diluted and diluting gas) plays an important role in dilution process using capillaries.

Considering the formula to calculate the gas flow through a capillary tube having length L and a circular hole having radius R,

R⁴ x ΔP

Q = ---------------

8 x η x L

neglecting the dimensional parameters it’s evident a proportional effect of the pressure ΔP applied at capillary ends and a reciprocal effect due to the viscosity η of the gas flowing through the capillary.

Similarly, expressing the dilution rate as the ratio between the gas to be diluted flow and the total diluted flow, it’s possible to see the effects of the ratios η₁ / η₀ and ΔP₀ / ΔP₁, where the indexes 1 and 0 are referred to the mixture to be diluted and to the diluting gas.

K dil. = ------------------------------------------

ΔP₀ η₁

N + (30-N) x ------- x -------

ΔP₁ η₀

This formula is referred to the particular case of a gas divider with 30 nearly-identical capillaries and is used to compensate the viscosity difference of two gases entering in a dilutor by applying pressures inversely proportional to viscosities.

Viscosity values for various components (at standard pressure and temperature) are available in literature for many chemical components, but mixtures viscosity must be calculated.

The rigorous method to calculate a mixture viscosity is too complex and is based upon the molecular collision analysis. Hopefully many scientist did setup simplified calculation procedures, in which the result accuracy is generally growing with complexity : the easier is due to Carr and the more complex is due to Reichenberg.

Both those methods are described in the following text and a link is available to get an automatic calculation in case of mixtures containing up to 7 components in a defined set.

Our customers may ask to include in this set additional components when they need.

SIMPLIFIED CARR METHOD

η_m = [ ∑_i (x_i η_i √M_i )] / [∑_i (x_i √M_i )]

i is between 1 and n (components index in the mixture) and η_m is the mixture viscosity.

x_i = molar fraction of the component i in the mixture

M_i = molecular weight of the component i

The result of this method is particularly approximated when the involved molecules are polar.

METHOD SIMPLIFIED (...comparing it with the rigorous) OF REICHENBERG

The basic formula is :

η_m = ∑ K_i x (1 + 2 A_i + B_i )

^{i =
1}

K_i = y_i η_i / ( y_i + η_i D_i )

D_i = ∑ y_k H_{i k} [ 3 + 2 M_k / M_i ]

^k=1≠i

_(i-1)≠0

A_i = ∑ H_{i j} K_j

^{j = 1}

_{n n}

B_i = ∑ ( ∑ H_{i j} H_{i k} K_j K_k )

^{j=1≠i
k=1≠i}

Calculated parameters :

where the problem data are :

i = index of the component in the mixture ( i = 1...n)

y_i = molar fraction of i_th component in the mixture

T = capillaries and gas temperature

P = pressure of the gas (inlet or outlet of capillary ?)

and the data available in literature tables are :

η_i = viscosity of i_th component

μ_i= polar moment if i_th component

Tc_i = critical temperature of i_th component

Pc_i = critical pressure of i_th component

Mi = molar weight of i_th component

Tr_i =T / Tc_i

Pr_i = P / Pc_i

μ_Ri = 52,46 μ_i Pc_i / Tc_i = polar moment w.out dim.

F_Ri = [ Tr_i^3,5 +(10 μ_Ri)⁷ ] / {Tr_i^3,5 [1+(10 μ_Ri)⁷]}

U_i = { [ 1+0,36 Tr_i (Tr_i -1)]^1/6 F_Ri } / Tr_i^1/2

C_i = M_i^1/4 / ( η_i U_i )^1/2

Tr_{i
j} = T / (Tc_i Tc_j)^1/2

μ_{Ri j} = ( μ_Ri μ_Rj )^1/2

F_{Ri j} = [Tr_{i j}^3,5 ( 10 μ_{Ri j} )⁷] / {Tr_{i j}^3,5 [ 1+ ( 10 μ_{Ri j} )⁷]}

U_{i j} = { [ 1+0,36 Tr_{i j} (Tr_{i j} -1)]^1/6 F_{Ri
j} } / Tr_{i j}^1/2

H_ij = [ M_i x M_j / 32 (M_i + M_j)³]^1/2 x (C_i + C_j)² x U_ij

φ_i / Z_i

Y_i = --------------

∑_i (φ_i / Z_i )

Z_i = compressibility factor (is a property of real gases : for ideal gases his value is = 1)

φ_i = volume fraction of i_th component in the mixture