This is not surprising, since the diffusion of particles is not only associated with an momentum transfer, but also with a energy transfer in terms of heat. a slower molecule takes up part of the momentum of the faster molecule. This viscosity is then multiplied by the viscosity ratio (from Fig. The viscosity of ideal gases is mainly based on the momentum transfer due to diffusion between the fluid layers. In using these equations, it is important either to measure the density or to ensure that the z -factor calculation has included the effect of N2, CO2, and H2S using the method of Wichert and Aziz. For this purpose, a correlation between the particle flux diffusing perpendicular to the flow and the temperature must be found. Only the temperature as a variable quantity influences the viscosity. ViscosiTy of gases marcia l. huber and allan h. harvey The following table gives the viscosity of some common gases as a function of temperature . Faster layers thus transfer part of their momentum by diffusion into slower layers. The gas molecules themselves, however, are by no means at rest on the microscopic level. This derivation exploited the fact that the product of particle density and mass of a single particle equals the density \(\rho\) of the gas. Viscosity Converting Chart ; Kinematic viscosity can be converted from SSU to Centistokes with. This website uses cookies. We observe the gas flow on a microscopic level and move along with a fluid layer. 5 (or Eqs. The use of the ideal gas law and the Redlich-Kwong compressibility factor for gas density calculation, and use of the Sutherland Formula for gas viscosity calculation are addressed in this video presentation. If \(\dot n_\text{A}\) denotes the particle flux, i.e. [4] developed a useful analytical method that gives a good estimate of gas viscosity for most natural gases. By the transport of momentum onto the layer underneath, this layer starts to move, etc. Just as the compressibility of natural gas is much greater than that of oil, water, or rock, the viscosity of natural gas is usually several orders of magnitude smaller than oil or water. We now look at a layer at any height \(y\). [27] Momentum transport in gases is generally mediated by discrete molecular collisions, and in liquids by attractive forces which bind molecules close together. For the definition of viscosity one can imagine a fluid between two plates. Let us now consider a directed flow in which all particles move in the same direction with the (mean) velocity \(\overline{v}\). The details are in Table 2. These forces act similar to frictional forces, so that the individual fluid layers try to slow each other down. From this point of view, the flow is no longer directed, but completely random with a mean molecular velocity denoted by \(\overline{v_\text{T}}\). Equation (\ref{na}) can also be put directly into the formula (\ref{eta}) for the viscosity, resulting in the following relationship: \begin{align}&\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda \\[5px]&\eta= 2~ \frac{1}{6} n \cdot \overline{v_\text{T}} \cdot m \cdot \lambda \\[5px]&\eta= \frac{1}{3} \underbrace{n \cdot m}_{\rho} \cdot \lambda \cdot \overline{v_\text{T}} \\[5px]&\boxed{\eta= \frac{1}{3} \cdot \rho \cdot \lambda \cdot \overline{v_\text{T}}} \\[5px]\end{align}. The mean velocity \(v_{x}(y+\lambda)\) of the gas molecules in the layer above at the distance \(\lambda\) can be determined using the velocity gradient \(\frac{\text{d}v}{\text{d}y}\): \begin{align}&v_{x}(y+\lambda)= v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \\[5px]\end{align}. The Since Sutherland's formula is an empirical fit of measured data, the following table of reference data is needed. Gas viscosity is only weakly dependent on pressure near atmospheric pressure. The motive force … Last but not least, the mean free path \(\lambda\) can be expressed by the particle density \(n\) and the diameter of the gas molecules \(d\) (for the derivation of this formula see article Mean free path & collision frequency): \begin{align}& \boxed{\lambda = \frac{1}{\sqrt{2}~n ~\pi d^2}} \\[5px]\end{align}. If one compares this formula with Newton’s law of fluid friction (\ref{tt}), it is apparent that the expression \(2~ \dot n_\text{A} \cdot m \cdot \lambda\) obviously corresponds to the viscosity \(\eta\): \begin{align}&\dot p_\text{A} = – \eta \cdot \frac{\text{d} v}{\text{d} y} \\[5px]&\dot p_\text{A}=- \underbrace{2~ \dot n_\text{A} \cdot m \cdot \lambda}_{\eta} \cdot \frac{\text{d}v}{\text{d}y} \\[5px]\label{eta}&\boxed{\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda} ~~~\text{viscosity of ideal gases}\\[5px]\end{align}. The timings can be used along with a formula to estimate the kinematic viscosity value of the fluid in Centistokes (cSt). The following formula given by Poiseuille shows the dependence of the viscosity of a liquid on temperature- In the case of gases, as mentioned earlier the intermolecular cohesion being negligible the viscosity depends mainly on transfer of molecular momentum in a direction at right angles to the direction of motion. Look up the chart of Fig.2 from Real gases, which gives a value of z = 0.91; then. 4 using the pseudoreduced properties calculated above, which gives μg/μga = 1.32. According to the Maxwell-Boltzmann distribution, the mean speed of a gas particle \(\overline{v_\text{T}}\) is linked to the temperature \(T\) of the gas as follows, \begin{align}\label{a}&\boxed{ \overline{v_\text{T}} = \sqrt{\frac{8 k_B T}{\pi m}}} ~~~\text{arithmetic mean speed} \\[5px]\end{align}. Fig. At this point it is also evident that with ideal gases, pressure has no influence on viscosity. Reliable correlation charts are available to estimate gas viscosity. Why does water boil faster at high altitudes? Note that Figs. First, calculate the pseudocritical properties using Kay’s[6] rules. The transport property equations presented use the independent properties temperature and density as input conditions. Next, the ratio of μg/μga is read from Fig. thermal conductivity \(k\) of ideal gases, Derivation of the continuity equation (conservation of mass), Archimedes’ principle of buoyancy (crown of Archimedes). Presented at the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October. [5] The equations of Lee et al. Source code¶ chemics.gas_viscosity.mu_gas (formula, temp, cas=None, full=False) [source] ¶ Viscosity of gas as a function of temperature. 1 – Viscosity of pure hydrocarbons at 1 atm (from Carr et al.[1]). ν SSU > 100. In gases, however, the molecules exert almost no molecular forces on each other. Help with editing, Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. ν Centistokes = 0.226 ν SSU - 195 / ν SSU (4). We therefore consider gas layers that have a distance \(\lambda\) from each other, so that diffusion processes result in a collision inside those layers and thus in a momentum transfer. The parameters in Sutherland’s formula are as follows: To = reference temperature, K. μo = viscosity of the gas at temperature, To. Before understanding the viscosity index, we need to understand a physical property of fluids called viscosity. Note that this formula only applies to laminar flows where the gas layers do not mix macroscopically and diffusion between the layers only occur at the microscopic level. With ideal gases the viscosity is independent of pressure and increases with increasing temperature! Due to the molecular forces between the molecules, a lower fluid layer always tries to slow down the fluid layer above. The number of particles per unit time and area (particle flux \(\dot n_\text{A}\)) passing through an area perpendicular to the flow can thus be calculated with the following formula: \begin{align}&\dot n_\text{A} = \frac{\text{d}N}{\text{d}A \cdot \text{d}t} =n \cdot \overline{v} \\[5px]&\boxed{\dot n_\text{A} =n \cdot \overline{v} } ~~~\text{particle flux of a directed motion}\\[5px]\end{align}. Kay, W. 1936. At this point an interesting analogy to thermal conductivity \(k\) of ideal gases can be seen (to avoid confusion with the mean free path, the thermal conductivity was not denoted by \(\lambda\) but \(k\)): \begin{align}& \boxed{k= \frac{1}{3} \cdot \rho \cdot \lambda \cdot c_v \cdot \overline{v_\text{T}} } \\[5px]\end{align}. These frictional forces between the layers must be compensated if the uppermost layer is to be moved at a constant speed. Created Date: 5/13/2008 3:44:16 PM Lee correlation is the empirical correlation for the gas viscosity published in 1966. J Pet Technol 18 (8): 997–1000. [4] were originally written to give the viscosity in micropoise, but the modified form above gives the viscosity in the more commonly used centipoise. Lee gas viscosity correlation at T=340F in the PVT software at pengtools.com. Natural Gas Viscosity Calculates viscosity for natural gas based on Lee, Gonzales and Eakin Equations. More information. Matter is what everything is made of - from the tires of a car to the water in a pond. it increases with increasing mean particle speed (increasing temperature). 5 – Pseudocritical properties of methane-based natural gases (from Sutton[2]). That's because they are different states of matter with the most prevalent being solids, liquids and gases. SPE-297-G. Sutton, R.P. Let us again consider the already mentioned laminar flow, which this time consists of an ideal gas as fluid. The molecules do not move in any preferred direction. Although the particle density and thus the diffusing particle flow increases proportionally with increasing pressure, the mean free path decreases to the same extent. One can illustrate the situation with a cart and a ball. 3 – Effect of temperature and pressure on viscosity of natural gases (from Carr et al.[1]). An example of such a treatment is Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the Boltzmann equation. For this we move in thoughts with a layer, so that this layer rests relative to us. \begin{align}\dot p_\text{A}(y) &= \dot p_{A}(y-\lambda) ~-~ \dot p_{A}(y+\lambda) \\[5px] &= \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) – \lambda \cdot \frac{\text{d}v}{\text{d}y} \right)- \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]&= \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) ~- \lambda \cdot \frac{\text{d}v}{\text{d}y} ~-~ v_{x}(y) ~- \lambda \cdot \frac{\text{d}v}{\text{d}y}\right) \\[5px]\end{align}, \begin{align}&\boxed{\dot p_\text{A}= – 2~ \dot n_\text{A} \cdot m \cdot \lambda \cdot \frac{\text{d}v}{\text{d}y}}~~~\text{net momentum flux} \\[5px]\end{align}. The theory on this issue is poorly developed. Chem. Results calculated from coefficients in Yaws’ Critical Property Data for Chemical Engineers and Chemists .CAS (Chemical … Fig. The viscosity equation given in the article was the summation of the dilute gas viscosity and residual fluid viscosity. The viscosity of gases therefore generally increases with temperature and not decreases as with liquids! gas flow rate: P = absolute pressure: p gas = gas density: u = air viscosity: u gas = gas viscosity: K = proportionality factor: T = temperature: v = settling velocity: S = separation factor: N = approximate effective turns: h = inlet height: L cylinder = cylinder length: … Calculate Z's for Sour Gases. This equation used in formula (\ref{eta}) finally shows the following relationship between viscosity and temperature: \begin{align}&\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda \\[5px]&\eta= 2~ \frac{1}{6} n \cdot \sqrt{\frac{8 k_B T}{\pi m}} \cdot m \cdot \lambda \\[5px]\label{ac}&\boxed{\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \lambda} \\[5px]\end{align}. A considered fluid layer thus always moves slower than the fluid layer above it. Using this formula in equation (\ref{ac}), the following formula for calculating the viscosity of ideal gases is finally obtained: \begin{align} &\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \lambda \\[5px] &\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B … The cause of viscosity due to frictional forces acting between the fluid layers can be clearly understood for liquids. However, Kay’s rules require a full gas composition. where: Y = 2.4 - 0.2 X μg = gas viscosity, cp ρ =gas density, g/cm 3 p = pressure, psia T = temperature °R Mg = gas molecular weight = 28.967 γg The more viscous a fluid is, the stronger the internal friction forces and the greater the force required to move the top plate. 2 – Viscosity of natural gases at 1 atm (from Carr et al.[1]). At this point an interesting analogy can be drawn to other transport mechanisms like heat transport and mass transport, which are ultimately described in a similar way: With the help of the kinetic theory of gases, the viscosity of ideal gases can be calculated. First, only the layer directly adhering to the upper plate is set in motion. This in turn is determined by the temperature. [3] to calculate the pseudocritical properties for use with the viscosity calculation. SPE-26668-MS. Lee, A.L., Gonzalez, M.H., and Eakin, B.E. This method lends itself for use in computer programs and spreadsheets. This page was last edited on 3 June 2015, at 15:06. What is the relationship between viscosity and thermal conductivity of ideal gases. Sutherland’s formula for calculating the viscosity of a gas at a specified gas temperature is shown below along with an explanation of the parameters in the equation. Ind. [4] are for specific units as noted below and are as follows: For the data from which the correlation was developed, the standard deviation in the calculated gas viscosity was 2.7%, and the maximum deviation was 9%. Kinematic viscosity of fluids like water, mercury, oils SAE 10 and oil no. Due to the random molecular motion (Brownian motion), molecules also diffuse into adjacent fluid layers. 2. In turbulent flows, the momentum exchange through the turbulence is greater and the viscosity is higher. Fig. The latter has no influence on the temperature anyway; after all, the temperature of a gas does not depend on whether the gas is at rest or moving. Thus, the particle flux is high and so is the momentum transfer. 2 or determined from the gas-mixture composition with Eq. The Sutherland formula canbe used to calculate the viscosity of a gas at a specified temperature if the Sutherland constant isavailable for the gas… Both effects cancel each other out. For a given particle density \(n\) (number of particles per unit volume), the following number of particles \(\text{d}N\) will be found this volume element: \begin{align}&\text{d}N = n \cdot \text{d}V =n \cdot \text{d}A \cdot \overline{v} \cdot \text{d}t \\[5px]\end{align}. [14] For each gas, sum mole fraction x sqrt (Molecular weight) in denominator. Note that dv / dr < 0 because r is measured from the center of the tube. This results in an increasing force, which is necessary to maintain the macroscopic flow (movement of the plate)! Finally, in a state of equilibrium, there is no net momentum flow, which is transferred to the fluid layers, so that these finally move at constant but different speeds according to the linear velocity profile. Viscosity and Temperature. Thus the particles obviously flow through the volume element \(\text{d}V\): \begin{align}&\text{d}V =\text{d}A \cdot \text{d}l = \text{d}A \cdot \overline{v} \cdot \text{d}t \\[5px]\end{align}. The driving force fd on the cylinder is. A force \(F\) can generally be determined from the change in momentum per unit time: \begin{align}&F= \frac{\text{d} p}{\text{d} t} = \dot p ~~~~~\Rightarrow~~~~~\boxed{\text{force = momentum flow rate}}\\[5px]\end{align}. In 1893, he developed an empirical-theoretical relationship between the temperature and viscosity of an ideal gas. Using this formula in equation (\ref{ac}), the following formula for calculating the viscosity of ideal gases is finally obtained: \begin{align}&\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \lambda \\[5px]&\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \frac{1}{\sqrt{2}~n ~\pi d^2} \\[5px]&\boxed{\eta= \sqrt{\frac{4 k_B m~T}{9\pi^3~d^4}}} \\[5px]&\boxed{\eta \sim \sqrt{T}} \\[5px]\end{align}. The fast molecules are slowed down by this diffused gas molecule and the layer slows down. Empirical correlations are usually used depending on temperature and pressure. For the temperature provided above a gas with reference viscosity μ0= centiPoise, The net momentum flux \(\dot p_\text{A}(y)\) in the layer at the height \(y\) is finally the sum of both momentum fluxes. This momentum transport (which ultimately corresponds to a force between the layers) is directed in the direction of decreasing velocity, so to speak, i.e. Calculation of the Viscosity of Gas Mixtures Author: Firmin Joseph Krieger Subject: A presentation of two semiempirical general equations for the viscosity of a mixture of n gaseous components. The notation P = 0 indicates that … (15) η = η 0 + Δ η. Zero-density limit viscosity can take the atmospheric pressure (ideal gas) viscosity. This represents a 2.5% error from the experimentally determined value of 0.0172 cp. Data Gas Gravity . \begin{align}&\dot p_{A}(y+\lambda) = \dot n_\text{A} \cdot \overbrace{m \cdot v_{x}(y+\lambda)}^{\text{momentum of one molecule}} = \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]&\dot p_{A}(y-\lambda) = \dot n_\text{A} \cdot m \cdot v_{x}(y-\lambda) = \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) – \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]\end{align}. the number of particles per unit time and unit area that diffuse from the upper layer or lower layer into the middle layer, then the respective momentum fluxes can be determined with the following formulas. Mol % English Metric . The area-related particle flow \(\dot n_\text{A}\), which diffuses in from a layer above or below, depends in turn on how strongly the gas molecules move due to the random diffusion motion (Brownian motion). [7], [8] split the natural gas viscosity into the sum of the zero-density limit viscosity ( η0) and the residual viscosity (Δ η ). Using (\ref{a}) in equation (\ref{na}) results in the following diffusing particle flux as a function of temperature: \begin{align}&\dot n_\text{A} = \frac{1}{6} n \cdot \underbrace{\sqrt{\frac{8 k_B T}{\pi m}}}_{\overline{v_\text{T}}} \\[5px]\end{align}. ν = η ρ kinematic viscosity [ν] = m² s. The kinematic viscosity puts the viscosity of a fluid in relation to its density and is a measure of how “pasty” a fluid is. 1966. against the velocity gradient. The diffusion of gas molecules between the layers of a laminar flow leads to a momentum transfer on which the viscosity of gases is mainly based! 1. The particle flux directed perpendicular to the main flow (bulk motion) is thus only one-sixth as large: \begin{align}\label{na}&\boxed{\dot n_\text{A} = \frac{1}{6} n \cdot \overline{v_\text{T}} } ~~~\text{particle flux of a random motion}\\[5px]\end{align}. It would not be correct, then, to use the methods of Sutton[2] or Piper et al. In this equation \(m\) denotes the mass of a gas molecule and \(k_\text{B}\) is the Boltzmann constant. Result Unfortunately, your browser is … The value of the dynamic viscosity coefficient is found to be a constant with pressure but the value depends on the temperature of the gas. where . Fig. Unless otherwise noted, the viscosity values refer to a pressure of 100 kPa (1 bar) . Note that the particle flux is identical for both layers if we assume an incompressible gas flow where the particle density is the same at every point in the flow. Note that the speed \(\overline{v_\text{T}}\) represents the mean speed relative to the moving fluid layers and does not include the superposition of the macroscopic flow motion. The viscosity of an (ideal) gas is therefore only dependent on the mass of a gas particle, the mean free path and the particle flux. 1 through 4) that are the most widely used for estimating the viscosity of natural gas from the pseudoreduced critical temperature and pressure. Carr et al. More information about this in the privacy policy. The equations of Lee et al. The most commonly used unit of viscosity is the centi-poise, which is related to other units as follows: 1 c p = 0.01 poise = 0.000672 lbm/ft-s = 0.001 Pa-s Natural gas viscosity is usually expected to increase both with pressure and temperature.
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