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Conjugate natural convection between horizontal eccentric cylinders

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Abstract

This study involved the numerical investigation of conjugate natural convection between two horizontal eccentric cylinders. Both cylinders were considered to be isothermal with only the inner cylinder having a finite wall thickness. The momentum and energy equations were discretized using finite volume method and solved by employing SIMPLER algorithm. Numerical results were presented for various solid–fluid conductivity ratios (KR) and various values of eccentricities in different thickness of inner cylinder wall and also for different angular positions of inner cylinder. From the results, it was observed that in an eccentric case, and for KR < 10, an increase in thickness of inner cylinder wall resulted in a decrease in the average equivalent conductivity coefficient (\(\overline{{K_{eq} }}\)); however, a KR > 10 value caused an increase in \(\overline{{K_{eq} }}\). It was also concluded that in any angular position of inner cylinder, the value of \(\overline{{K_{eq} }}\) increased with increase in the eccentricity.

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Abbreviations

a :

Location of the positive pole of the bipolar coordinate system on the y-axis, r o sinhη o

D :

Dimensionless parameter, 2(r o  − r ii )

e :

The distance between centers of two cylinders (m)

E :

Dimensionless eccentricity, e/(r o  − r oi )

E 1 :

Dimensionless eccentricity, e/(r o  − r ii )

F :

Bouancy force or convective mass flux at face of control volume

g :

Gravitational accelaration (ms−2)

h :

Coordinate transformation scale factor

i :

Index of the numerical grid in ξ and φ directions

k :

Auxiliary index for averaging properties of interface nodes

H :

Dimensionless coordinate transformation factor, h/D

KR :

Solid–fluid conductivity ratio (Ks/Kf)

K eq :

Local equivalent conductivity coefficient

\(\overline{{K_{eq} }}\) :

Average equivalent conductivity coefficient

L :

r o  − r oi

L* :

r o  − r ii

N :

Annulus radius ratio, r oi /r o

n :

Number of bipolar grid nodes in η-direction

p :

Pressure of fluid (Pa)

P :

Dimensionless pressure of fluid

q :

Number of cylindrical grid nodes in r-direction

r o :

Radius of outer cylinder (m)

r ii :

Inner radius of inner cylinder (m)

r oi :

Outer radius of inner cylinder (m)

r′:

Distance from the center of the inner cylinder to the outer cylinder (m)

R :

Dimensionless component of cylindrical coordinate, R = r/r ii

R* :

Dimensionless distance in radii direction

RR :

Radius ratio of outer cylinder to inner cylinder, r o /r ii

S φ :

Source term

T :

Temprature at any point (K)

w :

Velocity in ξ direction (ms−1)

v :

Velocity in η direction (ms−1)

u ref :

Reference velocity (ms−1)

V :

Dimensionless η-velocity component

W :

Dimensionless ξ-velocity component

Ra :

Rayleigh number

Pr:

Prandtl number

α :

Thermal diffusivity of fluid or the angle measured from top with the origin of Cartesian coordinate

β :

Volumetric coefficient of thermal expansion

Γ:

Diffusion coefficient

η :

The first transverse bipolar coordinate

η i :

Value of η on the outer surface of inner cylinder, Cosh−1[(N(1 + E 2) + 1 − E 2)/(2 N·E)]

η o :

Value of η on the surface of outer cylinder, Cosh−1[(N(1 − E 2) + 1 + E 2)/(2E)]

θ :

Dimensionless temperature

μ :

Dynamic viscosity of fluid (N s m−2)

ν :

Kinematic viscosity (m2 s−1)

ξ :

The second transverse bipolar coordinate

ρ :

Fluid density (kg m−3)

φ:

The azimuth angle in cylindrical coordinate

ψ:

Stream function

c :

Cold

f :

Fluid

h :

Hot

i :

Inner

o :

Outer

s :

Solid

N, E :

Evaluated in north and east of control volume, respectively

W, S :

Evaluated in west and south of control volume, respectively

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Authors and Affiliations

  1. Department of Mechanical Engineering, Yazd University, P.O.B. 89195-741, Yazd, Iran

    Davood Nasiri, Ali Akbar Dehghan & Mohammad Reza Hadian

  2. Department of Civil Engineering, Yazd University, P.O.B. 89195-741, Yazd, Iran

    Mohammad Reza Hadian

Authors
  1. Davood Nasiri
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  2. Ali Akbar Dehghan
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  3. Mohammad Reza Hadian
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Correspondence to Ali Akbar Dehghan.

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Nasiri, D., Dehghan, A.A. & Hadian, M.R. Conjugate natural convection between horizontal eccentric cylinders. Heat Mass Transfer 53, 799–811 (2017). https://doi.org/10.1007/s00231-016-1862-x

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