RU2129264C1 - Procedure of precise determination of steady-state rheological characteristics of various fluid media - Google Patents

Abstract

FIELD: oil excavation, petrochemical, food, biochemical industries, cosmetics, paint and varnish industry. SUBSTANCE: steady-state values of stresses τ and rates of displacement

of various fluid media are measured in agreement with procedure in wide range of rates of displacement. Steady-state rheological characteristics of various fluid media τ_*,μ_* are determined by values measured during experiment and with the aid of generalized rheological model

, where i=1; 0,; -1; n is index of nonlinearity. Elastic plastic materials are described with i=1; n>0, pseudoplastic materials are explained with i=1, n<0, dilatant systems are presented with I= -1, n<0, Newton's media are described with I=0. Exhaustive search of values n is conducted and coefficients μ_*,τ_*, of model resulting in minimum of sum of squares of deviations of calculated and experimental values of stresses are found. Usage of given rheological model allows approximation of rheograms of various non-Newton systems both displaying and not displaying plastic properties obtained on various types of viscosimeters to be conducted. EFFECT: expanded application field. 5 tbl

Description

 The invention relates to the fields of oil production, drilling of oil and gas wells, petrochemistry, biochemistry, food, cosmetic, paint and varnish industry.

 To date, there are more than 20 rheological models that describe the relationship between steady-state stress values and shear rates. The variety of rheological models contributes to the emergence of a large number of works devoted to hydrodynamic calculations of various flows. Rheograms are approximated, as a rule, by means of models assumed to be consistent and the choice of the one that gives the smallest error. Most researchers and engineers, given the complexity of determining model parameters, use the simplest Shvedov-Bingham and Ostwald models, which lead to a significant approximation error. Naturally, from these positions, the most optimal is the use of a single rheological model that describes the flow curve with a high degree of accuracy.

 In addition to the statistical error of studies on viscometers, researchers are faced with the methodological error laid down in the basis of this research method. One of them is the methodological error caused by the non-Newtonian properties of the studied liquids, and arising from the fact that the accepted values of the shear rates on the walls of the viscometers do not correspond to the real ones. Usually, the shear rate on the wall is calculated from the expressions corresponding to the Newtonian medium. Therefore, neglect of the methodological error under consideration can distort the idea of the non-Newtonian system under study.

 The purpose of this invention is the creation of a simple way to accurately determine the rheological characteristics of various types of viscometers, taking into account the statistical error and the methodological error caused by non-Newtonian properties, for various emulsions, suspensions, sols, gels, pastes, oils, solutions and polymer melts.

для самых различных текучих сред в широком диапазоне скоростей сдвига:

или

при i= 1, n>0 описываются вязкопластики:

n<0 описываются псевдопластики: τ_*= τ_∞ (τ → τ_∞ при γ → ∞)

при i=0 описываются ньютоновские жидкости: μ_*= μ,
при i=-1, n<0 описываются дилатантные системы:

где τ_o, η_p, μ_o - начальное напряжение сдвига, пластическая вязкость, вязкость при малых скоростях сдвига,
μ - ньютоновская вязкость,
μ_e - эффективная вязкость,
τ_∞ - напряжение, к которому стремится кривая течения при бесконечной скорости сдвига; фиктивный параметр, свидетельствующий о постоянном разрушении структуры в области ламинарного режима,

скорость сдвига, к которой стремится кривая течения при приложении напряжения, стремящегося к бесконечности, является фиктивным параметром.The goal is achieved using a rheological model that allows you to describe with a high degree of accuracy the relationship between the steady-state values of stresses τ and shear rates

for a wide variety of fluids over a wide range of shear rates:

or

for i = 1, n> 0, viscoplastics are described:

n <0, pseudoplastics are described: τ _* = τ _∞ (τ → τ _∞ as γ → ∞)

for i = 0, Newtonian fluids are described: μ _* = μ,
for i = -1, n <0, dilatant systems are described:

where τ _o , η _p , μ _o - the initial shear stress, plastic viscosity, viscosity at low shear rates,
μ - Newtonian viscosity
μ _e is the effective viscosity
τ _∞ is the stress to which the flow curve tends at an infinite shear rate; a fictitious parameter indicating a constant destruction of the structure in the region of the laminar regime,

the shear rate to which the flow curve tends when applying a voltage tending to infinity is a dummy parameter.

Модель Шульмана предназначалась автором исключительно для жидкостей, обладающих начальным напряжением сдвига τ_o. В отличие от этого, с помощью уравнения (1) описываются псевдопластичные и дилатантные системы, не обладающие начальным напряжением сдвига, что безусловно расширяет область применения модели (1). Одним из недостатков модели Шульмана является то, что в ней присутствуют различные размерности, определяемые разными степенями m и n, и при m ≠ n параметр η_p теряет свой физический смысл. Кроме этого, определение 4 параметров модели Шульмана приводит на порядок к большему времени счета по сравнению с определением 3 коэффициентов в модели (1).It should be emphasized that this model differs from the Shulman rheological model (Shulman 3. P. Convective heat and mass transfer of rheologically complex liquids. M: Energy, 1975 - 351 p.):

The Schulman model was intended by the author exclusively for liquids having an initial shear stress τ _o . In contrast, using equation (1), pseudoplastic and dilatant systems are described that do not have an initial shear stress, which certainly expands the scope of application of model (1). One of the drawbacks of the Shulman model is that it contains different dimensions determined by different degrees m and n, and for m ≠ n the parameter η _p loses its physical meaning. In addition, the determination of 4 parameters of the Shulman model leads to an order of magnitude longer calculation time compared to the determination of 3 coefficients in model (1).

Как показала практика исследований более 40 реограмм, снимая установившиеся значения

с помощью предлагаемой модели можно с высокой степенью точности описывать реологическое поведение самых различных неньютоновских систем, как обладающих, так и не обладающих пластическими свойствами. При этом обнаружено, что большинство текучих сред лучше описываются как псевдопластики. Средняя погрешность аппроксимации не более 1.5-2.5%, что гораздо ниже по сравнению с другими реологическими моделями /табл. 1-2 /. Таким образом, используя данную реологическую модель, появляется возможность проводить аппроксимацию реограмм и интегрирование уравнений движения на основе единой функциональной зависимости.The method for determining the parameters of the model (1) is as follows. First, the degree of nonlinearity n is set at which equation (1) is reduced to a straight line that can be easily processed using the least squares method to take into account the statistical error of experiments and determine τ $\genfrac{}{}{0ex}{}{\text{1 / n}}{\text{*}}$ , μ $\genfrac{}{}{0ex}{}{\text{1 / n}}{\text{*}}$ , and, accordingly, τ _* , μ _* . Then n is enumerated and those coefficient values are selected that minimize the expression:

As the practice of research of more than 40 rheograms showed, removing steady-state values

Using the proposed model, it is possible to describe with a high degree of accuracy the rheological behavior of the most diverse non-Newtonian systems, both possessing and not possessing plastic properties. It was found that most fluids are better described as pseudoplastics. The average approximation error is not more than 1.5-2.5%, which is much lower compared to other rheological models / table. 1-2 /. Thus, using this rheological model, it becomes possible to approximate rheograms and integrate the equations of motion on the basis of a single functional dependence.

Данное выражение можно привести к виду:

где Q - расход жидкости,
τ_ст - напряжение на стенке трубы,

скорость сдвига на стенке
ΔP - перепад давления,
L,R - длина и радиус капилляра.To account for the methodological error caused by non-Newtonian properties, the Muni-Rabinovich relation is widely known (for example: Leonov E.G., Isaev V.I. Hydroaeromechanics in drilling: Textbook for universities. - M.: Nedra, 1987 - 304 p.) with reference to capillary viscometers:

This expression can be reduced to the form:

where Q is the fluid flow rate,
τ _article - stress on the pipe wall,

wall shear rate
ΔP - pressure drop,
L, R is the length and radius of the capillary.

подходящей функцией Q/(πR³) = f(τ_ст) и подставляя в уравнения (2) , можно определить истинные значения скоростей сдвига на стенке или методическую погрешность, вызванную неньютоновскими свойствами. По значениям напряжений и истинных скоростей сдвига на стенке строится истинная кривая течения. Недостатками этого способа являются применение только для капиллярных вискозиметров и неустановленный вид функции Q/(πR³) = f(τ_ст).
Задворных В.Н. (Реодинамика нелинейно-вязкопластичных буровых растворов в кольцевом пространстве глубокой скважины. - Дисс...канд. техн. наук. М. МИНГ им. Губкина, 1987 - 171 с.) предлагает сразу учитывать этот тип погрешности, определяя коэффициенты реологической модели Шульмана в уравнениях движения для вискозиметров методом перебора. Однако данный способ существует чисто теоретически - перебор 4 коэффициентов модели, сопряженный с численным подсчетом интеграла, приводит либо к неоправданно завышенному времени счета, либо к неточностям определения коэффициентов.Approximating the parameters

with the appropriate function Q / (πR ³ ) = f (τ _st ) and substituting into equations (2), one can determine the true values of shear rates on the wall or the methodical error caused by non-Newtonian properties. Using the values of stresses and true shear rates, a true flow curve is constructed on the wall. The disadvantages of this method are the use only for capillary viscometers and the unknown function type Q / (πR ³ ) = f (τ _article ).
Zadvornykh V.N. (Reodynamics of nonlinear-viscous-plastic drilling fluids in the annular space of a deep well. - Diss ... candidate of technical sciences. M. MING named after Gubkin, 1987 - 171 pp.) Suggests immediately taking into account this type of error by determining the coefficients of the Shulman rheological model in equations of motion for viscometers by brute force. However, this method exists purely theoretically - enumeration of 4 coefficients of the model, coupled with a numerical calculation of the integral, leads either to unreasonably high calculation time, or to inaccuracies in determining the coefficients.

 In an early work (VA Iktisanov. An accurate description of the rheological characteristics of non-Newtonian systems with and without plastic properties. M., VNIIOENG, Geology, geophysics and oil field development, N9, 1995, pp. 51-70) a method for accounting for this type of error, also associated with the numerical calculation of the integral and with the use of iterations, but without enumerating the coefficients of the rheological model. Unfortunately, this method is cumbersome for practical use.

 As a result, studies on viscometers are currently being conducted without taking into account the methodological error caused by non-Newtonian properties, which leads to a distortion of the results.

Далее из выражения (7) подставляем Q/(πR³) в уравнение (5), дифференцируем и получаем следующее уравнение для истинной скорости сдвига на стенке:

Таким образом, используя коэффициенты модели, полученные для уравнения (7) и экспериментальные значения τ_ст, определяем по выражению (8) истинные

Обладая значениями

получаем истинную кривую течения, которая впоследствии аппроксимируется той же самой моделью (1).Let us solve the problem of determining the methodological error caused by non-Newtonian properties for capillary viscometers, having a single rheological model (1). First, the approximation is made according to the above algorithm for the shear rate on the wall for Newtonian fluid 4Q / (πR ³ ) and stress τ _st :

Next, from expression (7) we substitute Q / (πR ³ ) into equation (5), differentiate and obtain the following equation for the true shear rate on the wall:

Thus, using the coefficients of the model obtained for equation (7) and the experimental values of τ _st , we determine from expression (8) the true

Possessing values

we obtain the true flow curve, which is subsequently approximated by the same model (1).

Решим задачу учета методической погрешности для ротационных вискозиметров. Для течения Куэтта при вращении внешнего цилиндра с угловой скоростью w₂ можно вывести следующее уравнение:

для вязкопластиков: τ_z= τ_o при τ₁> τ_o> τ₁δ², τ_z= τ₁δ² при τ_o< τ₁δ²,
для псевдопластиков и дилатантных: τ_z= τ₁δ².
Повторяя те же выкладки, но для условия, когда внешний цилиндр неподвижный (w₂ = 0), а внутренний вращается с угловой скоростью w₁ в направлении, противоположном w₂, получаем:

для вязкопластиков: τ_z= τ_o при τ₂/δ²> τ_o> τ₂, τ_z= τ₂ при τ_o< τ₂,
для псевдопластиков и дилатантных: τ_z= τ₂,
где τ₁, τ₂ - напряжения на внутреннем и внешних цилиндрах,
R₁, R₂ - радиусы внутреннего и внешнего цилиндров,
δ = R₁/R₂ - относительный кольцевой зазор.The methodological error in determining the shear rate caused by non-Newtonian properties, σ for capillary viscometers, is quite simple to determine by taking the ratio of equation (8) to expression (7):

We solve the problem of accounting for the methodological error for rotational viscometers. For the Couette flow during the rotation of an external cylinder with an angular velocity w ₂ , the following equation can be derived:

for viscoplastics: τ _z = τ _o for τ ₁ > τ _o > τ ₁ δ ² , τ _z = τ ₁ δ ² for τ _o <τ ₁ δ ² ,
for pseudoplastics and dilatant: τ _z = τ ₁ δ ² .
Repeating the same calculations, but for the condition when the outer cylinder is stationary (w ₂ = 0) and the inner cylinder rotates with an angular velocity w ₁ in the direction opposite to w ₂ , we get:

for viscoplastics: τ _z = τ _o for τ ₂ / δ ² > τ _o > τ ₂ , τ _z = τ ₂ for τ _o <τ ₂ ,
for pseudoplastics and dilatant: τ _z = τ ₂ ,
where τ ₁ , τ ₂ - stresses on the inner and outer cylinders,
R ₁ , R ₂ - the radii of the inner and outer cylinders,
δ = R ₁ / R ₂ is the relative annular gap.

Таким образом, для ньютоновской жидкости достаточно построения кривой течения по следующим параметрам:

или

где

скорости сдвига для ньютоновской среды на внутреннем и внешнем цилиндрах,
M - момент сил, L - высота цилиндра.For a Newtonian fluid, when f (τ) = τ / μ, the values of integrals (10) and (11) are equal to:

Thus, for a Newtonian fluid, it suffices to construct a flow curve using the following parameters:

or

Where

shear rates for a Newtonian medium on the inner and outer cylinders,
M is the moment of forces, L is the height of the cylinder.

 The moment of forces removed from the external or internal cylinders is equal in magnitude and is different only in direction. The effective viscosity determined by the values of parameters (13) and (14) is the same. Thus, irrespective of which cylinder rotates and with which of them the moment of forces is taken, the construction of the flow curve can be carried out either according to equations (13) or according to (14). Therefore, it is enough to consider the problem when using stress and shear rate for one of the cylinders, for example, for the inner cylinder according to equation (13).

можно получить следующее выражение, аналогичное уравнению Муни-Рабиновичу для капиллярного вискозиметра:

при условии существования неподвижного ядра для вязкопластиков в зазоре, т.е. при τ₁> τ_o> τ₁δ².
Из уравнения (15) следует, что разность истинных скоростей сдвига равна:
f(τ₁)-f(τ₁δ²) = 2τ₁dw₂/dτ₁= 2τ₁dw₁/dτ₁ (16)
Чтобы решить поставленную задачу учета методической погрешности для ротационных вискозиметров, вначале произведем аппроксимацию параметров τ₁ и 2w₂/(1-δ²) или 2w₁/(1-δ²) реологическим уравнением (1) :

В результате дифференцирования w₂ или w₁ из выражения (17) уравнение (16) можно привести к виду:

В отличие от соотношения (5) для капиллярного вискозиметра в данном случае присутствуют две неизвестные истинные скорости сдвига. Для того, чтобы определить истинную скорость сдвига на внутреннем цилиндре, необходимо знать соотношение истинных скоростей сдвига на внешнем и внутреннем цилиндрах. В первом приближении можно принять, что данное соотношение определяется с помощью коэффициентов модели для уравнения (17):

Впоследствии, определяя по уравнению (18) уточненные скорости сдвига, вновь производится аппроксимация и определяются более точные коэффициенты реологической модели. Далее уточняется соотношение скоростей сдвига (19) для новых коэффициентов модели и все повторяется вновь. При этом в выражении (18) коэффициенты не меняются. Как показала практика, достаточно 3-8 итераций для определения истинных скоростей сдвига и коэффициентов модели, описывающих истинную реограмму.Expressions (13), (14) are derived for the Newtonian medium, however, they are universally used for non-Newtonian systems, which certainly distorts the actual flow curve. For rotational viscometers from equations (10), (11), given that

you can get the following expression, similar to the Muni-Rabinovich equation for a capillary viscometer:

subject to the existence of a fixed core for viscoplastics in the gap, i.e. when τ ₁ > τ _o > τ ₁ δ ² .
From equation (15) it follows that the difference in true shear rates is equal to:
f (τ ₁ ) -f (τ ₁ δ ² ) = 2τ ₁ dw ₂ / dτ ₁ = 2τ ₁ dw ₁ / dτ ₁ (16)
To solve the problem of accounting for the methodological error for rotational viscometers, we first approximate the parameters τ ₁ and 2w ₂ / (1-δ ² ) or 2w ₁ / (1-δ ² ) with the rheological equation (1):

As a result of differentiation of w ₂ or w ₁ from the expression (17), equation (16) can be reduced to the form:

In contrast to relation (5) for a capillary viscometer, in this case there are two unknown true shear rates. In order to determine the true shear rate on the inner cylinder, it is necessary to know the ratio of the true shear rates on the outer and inner cylinders. In a first approximation, we can assume that this ratio is determined using the model coefficients for equation (17):

Subsequently, determining the specified shear rates by equation (18), the approximation is again performed and more accurate coefficients of the rheological model are determined. Next, the ratio of shear rates (19) for the new model coefficients is specified and everything is repeated again. Moreover, in expression (18), the coefficients do not change. As practice has shown, 3-8 iterations are enough to determine the true shear rates and model coefficients that describe the true rheogram.

 The methodological error caused by non-Newtonian properties is determined by the ratio of the true shear rate from equation (18) to the shear rate for Newtonian media according to equation (17).

 Analyzing equation (18) for rotational viscometers and equation (8) for a capillary viscometer, it can be noted that neglecting the methodological error leads to an overestimated effective viscosity for visco- and pseudoplastics and to an underestimated effective viscosity for dilated systems. For a capillary viscometer, the true shear rate on the wall does not depend on the radius of the capillary, but is only a function of the wall voltage and model coefficients. For rotational viscometers, the considered methodological error, in addition to the model coefficients, depends on the value of the relative annular gap δ. At δ → 1, for rotational viscometers, the methodological error can be neglected. However, in practice, a decrease in δ leads to difficulties, and sometimes the impossibility of studying highly viscous and coarse non-Newtonian systems.

 Using the proposed algorithm for the exact description of non-Newtonian fluids, a user program for IBM PC has been created. Processing of experimental data indicates that the methodological error in the general case is small 1-3% / table. 3, 4 /. However, when studying systems with pronounced non-Newtonian properties on viscometers with a wide annular gap (δ <0.98), the methodological error can be significant and reach 20% / table. 5 / or more.

 Thus, using the proposed rheological model, it is possible to carry out approximations of rheograms and integrate the equations of motion on the basis of a single relationship. Using this method of determining the true rheological characteristics, it becomes possible to conduct research processing on rotational viscometers with various clearance and on capillary viscometers with different diameters for a wide variety of fluids with a high degree of accuracy.

 An increase in the degree of accuracy in describing various non-Newtonian systems, in turn, leads to an increase in the accuracy of hydrodynamic calculations of the flows of these systems in pipes, annular channels, a porous medium, jets, and boundary layers; to more accurate forecasting and creation of new materials with desired properties.

Sources of information:
1. Leonov E.G., Isaev V.I. Hydroaeromechanics in drilling: Textbook. for universities. - M .: Nedra, 1987 - 304 p.

 2. Shulman 3.P. Convective heat and mass transfer of rheologically complex liquids. M .: Energy, 1975 - 351 p.

3. Iktisanov V.A. An accurate description of the rheological characteristics of non-Newtonian systems with and without plastic properties. Moscow, VNIIOENG, Geology, geophysics and oil field development, N9, 1995, p. 51-70
4. Rheodynamics of nonlinear visco-plastic drilling fluids in the annular space of a deep well. - Diss ... cand. tech. sciences. M. MING them. Gubkina, 1987 - 171 p.

Claims (1)

A method for determining the steady-state rheological characteristics of a fluid, including experimentally obtaining its rheological stress-shear curve, followed by a mathematical description of this curve by approximating, measured in the experiment, steady-state stresses and shear rates, characterized in that when approximating the experimental values obtained in the study both rotational and capillary viscometers use a rheological model described by the equation

where n is an indicator of nonlinearity;
τ _* , μ _* - steady-state rheological characteristics;
i = 1; 0; -1,
for i = 1, n> 0, viscoplastics τ _* = τ ₀ , μ _* = η _p , n <0 are described; pseudoplastics τ _* = τ _∞ , μ _* = μ ₀ ;
for i = -1, n <0 - dilatant systems

at i = 0 - Newtonian fluids μ _* = μ, where τ is the shear stress;

shear rate;
τ ₀ is the initial shear stress;
η _p is the plastic viscosity;
μ ₀ - viscosity at low shear rates;
μ is the Newtonian viscosity;
τ _∞ - the stress to which the flow curve tends at an infinite shear rate is a dummy parameter;

the shear rate to which the flow curve tends when applying a voltage tending to infinity is a dummy parameter.

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