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= exp (αp ⋅ Δp)
with αp in [1/MPa = MPa-1], at the pressure difference Δp = p - pref [MPa]
Typical values of αp for liquids are in between +0.005 and +0.05 MPa-1
(or 5 ⋅ 10-3 MPa-1 ≤ αp ≤ 50 ⋅ 10-3 MPa-1). For water at T < +32 °C, the value of αp is negative, since viscosity decreases with increasing pressure as explained above; for T > +32 °C, αp is
positive, i. e. viscosity increases now with increasing pressure.
In order to estimate viscosity values at pressures at which no measuring values are available, proceed as follows:
1 Select pref (i. e. a pressure, at which an η-value is available), usually is selected:pref = 0.1 MPa (= 1 bar).
2 Calculate the shift factor ap for another available η(p)-value (using Equation 3.11).
3 Calculate the coefficient αp (using Equation 3.12).
4 Calculate the shift factor ap for the desired η(p)-value (using Equation 3.12).
5 Result: Calculate the desired η(p)-value (using Equation 3.11).
3.2.1.1.2Example 1: Calculation of viscosity/pressure coefficient and shift factor of a mineral oil, and determination of viscosity values at further pressures
From a mineral oil is known: η1 = 0.300 Pas (at p1 = 0.1 MPa = 1 bar), and η2 = 2.22 Pas
(at p2 = 100 MPa = 1000 bar). Desired is the viscosity value η3 at p3 = 75 MPa (= 750 bar).
1 Here is selected: pref = p1 = 0.1 MPa
2 ap is calculated for p2 (as ap2): ap2 = η2 (p2) / η1 (pref) = 2.22 Pas / 0.300 Pas = 7.40
3 αp is calculated, with Δp = Δp21 = p2 – pref:ln (ap) = αp ⋅ Δp, and thus: αp = ln (ap) / Δpαp = ln (7.40) / (100 – 0.1) MPa = 2.00 / 99.9 MPa = 0.02 MPa-1 (= 20 ⋅ 10-3 MPa-1)
4 ap is calculated for p3 (as ap3), with Δp = Δp31 = p3 – pref:ap3 = exp (αp ⋅ Δp31) = exp [0.02 MPa-1 ⋅ (75 - 0.1) MPa] = e1.5 = 4.48
5 Result: η3 (p3) = ap3 ⋅ η1 (pref) = 4.48 ⋅ 0.300 Pas = 1.34 Pas
| Table 3.6: Pressure-dependent viscosity values, see the example of Chapter 3.6 | |||||||
| p [MPa] | 0.1 | 1 | 10 | 25 | 50 | 75 | 100 |
| η [Pas] | 0.300 | 0.306 | 0.366 | 0.495 | 0.813 | 1.34 | 2.22 |
The already mentioned and some further pressure-dependent viscosity values are presented in Table 3.6.
3.2.1.1.3Example 2: Calculation of the viscosity/pressure coefficient of a crude oil
From a crude oil is known: η1 = 36 mPas (at p1 = 0.1 MPa = 1 bar),
and η2 = 45 mPas (at p2 = 10 MPa = 100 bar). Desired is the αp-coefficient.
1 pref = p1 = 0.1 MPa
2 ap = η2 (p2) / ηref (p1)= 45/36 = 1.25
3 αp = ln (ap )/ Δp = ln (1.25) / (10 – 0.1) MPa = 0.0225 MPa-1 (= 22.5 ⋅ 10-3 MPa-1)
3.7References
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