Viscous dissipation of a power law fluid in axial flow between isothermal eccentric cylinders

Kolitawong, Chanyut; Kananai, Nantarath; Giacomin, A. Jeffrey; Nontakaew, Udomkiat

doi:10.1016/j.jnnfm.2010.11.004

Cited by 17 publications

(8 citation statements)

References 12 publications

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“…For this case, Eq . simplifies to,

0 = k {\frac{1}{h} \frac{\partial}{\partial ξ} \frac{1}{h} \frac{\partial T}{\partial ξ} + \frac{1}{h} \frac{\partial}{\partial θ} \frac{1}{h} \frac{\partial T}{\partial θ}} - {τ_{ξ ζ} \frac{1}{h} \frac{\partial v_{ζ}}{\partial ξ} + τ_{θ ζ} \frac{1}{h} \frac{\partial v_{ζ}}{\partial θ}}

The shear stress can be written in terms of the velocity gradients such that :

τ_{ξ ζ} = - η \frac{1}{h} \frac{\partial v_{ζ}}{\partial ξ},

τ_{θ ζ} = - η \frac{1}{h} \frac{\partial v_{ζ}}{\partial θ}

where, η is defined in Eq . ; v ζ , the velocity profile for the flow between the eccentric cylinders.…”

Section: Discussionmentioning

confidence: 99%

“…For small dimensionless eccentricity, ε ≪ 1, the scale factor simplifies to as:

h = \frac{a}{X} = \frac{R_{0} true(1 - ɛ \cos β true)}{{(1 - ɛ^{2})}^{\frac{1}{2}}}

where,

X = \frac{1 + ɛ \cos θ}{ɛ} = \frac{1 - ɛ^{2}}{ɛ true(1 - ɛ \cos β true)},

and R 0 is the average radius,

R_{0} = \frac{R_{1} + R_{2}}{2}

and β is measured from the x ‐axis as shown in Fig. .…”

Section: Methodsmentioning

confidence: 99%

“…Kolitawong and Giacomin derived the following dimensionless axial velocity profile between eccentric cylinders for power‐law liquids in bipolar cylindrical coordinates,

\begin{matrix} v_{ζ p} (κ, β) = - \frac{n}{n + 1} \frac{d (1 - ɛ \cos β)}{2 R_{2}}^{1 + \frac{1}{n}} {| κ |}^{1 + \frac{1}{n}} - 1 ɛ < \frac{1}{5} \end{matrix}

where, n is defined in Eq . ; d = R 2 – R 1 , the annular gap; R 2 and R 1 , the outer and inner radii of the annulus, respectively.…”

Section: Methodsmentioning

confidence: 99%

“…Later, Kolitawong and Giacomin mapped the eccentric cylinder cross section in Fig. into bipolar cylindrical coordinates in Fig.…”

Section: Introductionmentioning

confidence: 99%

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Viscous dissipation in plastic pipe extrusion

2013

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Modeling the non-Newtonian flow of polymer melts in single-screw extrusion generally requires numerical methods. This study analyzes the viscous dissipation of the melt-conveying zone, which is mainly responsible for the axial melt temperature increase, in single-screw extruders for both one-and two-dimensional stationary, fully developed flows of a power-law fluid. Rewriting the flow equations and applying the theory of similarity revealed three independent parameters that influence the physics of the fluid flow: the dimensionless pressure gradient P p;z , the power-law exponent n, and the screw-pitch ratio t=D b. Based on these parameters, we carried out a comprehensive numerical parametric study evaluating viscous dissi-pation and flow rate. Here, we present four heuristic models that predict the viscous dissipation of a power-law fluid in the melt-conveying zone of single-screw extruders. For one-dimensional and two-dimensional flows, we developed models for both a given pressure gradient and a given throughput. The approximation equations obtained allow fast and stable prediction without the need for numerical simulations of viscous dissipation. The accuracy of the heuristic models developed was validated in an error analysis, which showed that our approaches provide excellent approximations of the numerical results.

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“…For this case, Eq . simplifies to,

0 = k {\frac{1}{h} \frac{\partial}{\partial ξ} \frac{1}{h} \frac{\partial T}{\partial ξ} + \frac{1}{h} \frac{\partial}{\partial θ} \frac{1}{h} \frac{\partial T}{\partial θ}} - {τ_{ξ ζ} \frac{1}{h} \frac{\partial v_{ζ}}{\partial ξ} + τ_{θ ζ} \frac{1}{h} \frac{\partial v_{ζ}}{\partial θ}}

The shear stress can be written in terms of the velocity gradients such that :

τ_{ξ ζ} = - η \frac{1}{h} \frac{\partial v_{ζ}}{\partial ξ},

τ_{θ ζ} = - η \frac{1}{h} \frac{\partial v_{ζ}}{\partial θ}

where, η is defined in Eq . ; v ζ , the velocity profile for the flow between the eccentric cylinders.…”

Section: Discussionmentioning

confidence: 99%

“…For small dimensionless eccentricity, ε ≪ 1, the scale factor simplifies to as:

h = \frac{a}{X} = \frac{R_{0} true(1 - ɛ \cos β true)}{{(1 - ɛ^{2})}^{\frac{1}{2}}}

where,

X = \frac{1 + ɛ \cos θ}{ɛ} = \frac{1 - ɛ^{2}}{ɛ true(1 - ɛ \cos β true)},

and R 0 is the average radius,

R_{0} = \frac{R_{1} + R_{2}}{2}

and β is measured from the x ‐axis as shown in Fig. .…”

Section: Methodsmentioning

confidence: 99%

“…Kolitawong and Giacomin derived the following dimensionless axial velocity profile between eccentric cylinders for power‐law liquids in bipolar cylindrical coordinates,

\begin{matrix} v_{ζ p} (κ, β) = - \frac{n}{n + 1} \frac{d (1 - ɛ \cos β)}{2 R_{2}}^{1 + \frac{1}{n}} {| κ |}^{1 + \frac{1}{n}} - 1 ɛ < \frac{1}{5} \end{matrix}

where, n is defined in Eq . ; d = R 2 – R 1 , the annular gap; R 2 and R 1 , the outer and inner radii of the annulus, respectively.…”

Section: Methodsmentioning

confidence: 99%

“…Later, Kolitawong and Giacomin mapped the eccentric cylinder cross section in Fig. into bipolar cylindrical coordinates in Fig.…”

Section: Introductionmentioning

confidence: 99%

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Viscous dissipation in plastic pipe extrusion

2013

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show abstract

“…Feldman et al [9] have studied the problem of the thermal entrance region for the case of an incompressible Newtonian fluid; the dynamic regime is supposed fully developed and the temperature uniform at the entry, different thermal conditions being considered. When viscous dissipation is taken into account as it is encountered in polymers (non-Newtonian fluids) melting manufacturing, annular eccentric geometry is largely present, because the die mandrel is displaced downward to get a uniform annular shape at the exit as possible, this kind of problem have been studied numerically and analytically by several authors (see for instance [10,11]). For an eccentric curved annulus, Nobari et al [12] study numerically the problem for a Newtonian fluid using a second order finite difference method.…”

Section: Introductionmentioning

confidence: 99%

Laminar mixed convection in an eccentric annular horizontal duct for a thermodependent non-Newtonian fluid

Messaoudene¹,

Horimek²,

Nouar³

et al. 2011

International Journal of Heat and Mass Transfer

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Transport Phenomena in Eccentric Cylindrical Coordinates

2017

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Studies in transport phenomena have been limited to a select few coordinate systems. Specifically, Cartesian, cylindrical, spherical, Dijksman toroidal, and bipolar cylindrical coordinates have been the primary focus of transport work. The lack of diverse coordinate systems, for which the equations of change have been worked out, limits the diversity of transport phenomena problem solutions. Here, we introduce eccentric cylindrical coordinates and develop the corresponding equations of change (continuity, motion, and energy). This new coordinate system is unique, distinct from bipolar cylindrical coordinates, and does not contain cylindrical coordinates as a special case. We find eccentric cylindrical coordinates to be more intuitive for solving transport problems than bipolar cylindrical coordinates. Specific applications are given, in the form of novel exact solutions, for problems important to chemical engineers, in momentum, heat and mass transfer. We complete our analysis of eccentric cylindrical coordinates by using the new equations to solve one momentum, one energy, and one mass transport problem exactly. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3563–3581, 2017

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Viscous dissipation of a power law fluid in axial flow between isothermal eccentric cylinders

Cited by 17 publications

References 12 publications

Viscous dissipation in plastic pipe extrusion

Viscous dissipation in plastic pipe extrusion

Laminar mixed convection in an eccentric annular horizontal duct for a thermodependent non-Newtonian fluid

Transport Phenomena in Eccentric Cylindrical Coordinates

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