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Research Papers: Heat and Mass Transfer

Heat Transfer From Particles Confined Between Two Parallel Walls

[+] Author and Article Information
Ashok S. Sangani

Department of Biomedical and
Chemical Engineering,
Syracuse University,
Syracuse, NY 13210
e-mail: asangani@syr.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 1, 2018; final manuscript received September 27, 2018; published online November 22, 2018. Editor: Portonovo S. Ayyaswamy.

J. Heat Transfer 141(2), 022001 (Nov 22, 2018) (9 pages) Paper No: HT-18-1493; doi: 10.1115/1.4041802 History: Received August 01, 2018; Revised September 27, 2018

The rate of heat conduction (or mass transfer by diffusion) from a cylindrical or a spherical particle confined between two walls is determined as a function of the position and the radius of the particle. It is shown that the appropriate Green's function can be determined using the method of images even when the resulting series is divergent with the help of Shanks transformation. Asymptotic expansions for small particle radius compared to the distance between the walls are combined with the expressions for the case in which the gap between the particle and one of the walls is small compared to the particle radius to provide formulas that are surprisingly accurate for estimating the rate of heat transfer for the entire range of parameters that include the radius and the position of the particle. Results are also presented for the thermal dipole induced by a spherical or a cylindrical particle placed between two walls with unequal temperatures and these are used to predict the effective thermal conductivity of thin composite films containing spherical or cylindrical particles.

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References

Bergman, T. L. , Incropera, F. P. , DeWitt, D. P. , and Lavine, A. S. , 2011, Fundamentals of Heat and Mass Transfer, Wiley, Hoboken, NJ.
Suryanarayana, N. V. , 2015, Engineering Heat and Mass Transfer, Penram International Publishing, Mumbai, India.
Shanks, D. , 1955, “ Non‐Linear Transformations of Divergent and Slowly Convergent Sequences,” J. Math. Phys., 34(1-4), pp. 1–42. [CrossRef]
Kushch, V. , 2013, Micromechanics of Composites: Multipole Expansion Approach, Butterworth-Heinemann, Oxford, UK.
Sangani, A. S. , and Yao, C. , 1988, “ Transport Processes in Random Arrays of Cylinders. I. Thermal Conduction,” Phys. Fluids, 31(9), pp. 2426–2434. [CrossRef]
Davit, Y. , and Peyla, P. , 2008, “ Intriguing Viscosity Effects in Confined Suspensions: A Numerical Study,” Europhys. Lett., 83 (6), p. 64001. [CrossRef]
Bhattacharya, S. , Bławzdziewicz, J. , and Wajnryb, E. , 2005, “ Hydrodynamic Interactions of Spherical Particles in Suspensions Confined Between Two Planar Walls,” J. Fluid Mech., 541(1), pp. 263–292. [CrossRef]
Sangani, A. S. , Acrivos, A. , and Peyla, P. , 2011, “ Roles of Particle-Wall and Particle-Particle Interactions in Highly Confined Suspensions of Spherical Particles Being Sheared at Low Reynolds Numbers,” Phys. Fluids, 23(8), p. 083302. [CrossRef]
Swan, J. W. , and Brady, J. F. , 2010, “ Particle Motion Between Parallel Walls: Hydrodynamics and Simulation,” Phys. Fluids, 22(10), p. 103301. [CrossRef]
Maxwell, J. C. , 1881, A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, Wotton-under-Edge, Gloucestershire, UK.
Jeffrey, D. J. , 1973, “ Conduction Through a Random Suspension of Spheres,” Proc. R. Soc. Lond. A, 335(1602), pp. 355–367. [CrossRef]
Sangani, A. S. , and Yao, C. , 1988, “ Bulk Thermal Conductivity of Composites With Spherical Inclusions,” J. Appl. Phys., 63(5), pp. 1334–1341. [CrossRef]
Bonnecaze, R. T. , and Brady, J. F. , 1991, “ The Effective Conductivity of Random Suspensions of Spherical Particles,” Proc. R. Soc. Lond. A, 432(1886), pp. 445–465. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

The coefficients c0−c2 in Eq. (18) as functions of s. The open circles represent the computed values and the solid lines, the fits as given by Eqs. (19)(21).

Grahic Jump Location
Fig. 2

The shape factor for a cylinder placed midway between the two walls as a function of a, the radius of the cylinder divided by the half-gap between the walls. The open circles represent the converged numerical results, the dashed line is the asymptote (18) correct to O(a4), the dashed-and-dot line shows the leading approximation (1) for small a, the stars the opposite limit a→1 as given by Eq. (22), and the solid line represents the approximation (24).

Grahic Jump Location
Fig. 3

λ0−λ2 as functions of s. The open circles are determined from the computed values of c0−c2 using Eqs. (26)(28) and the solid lines are the fits given by Eqs. (29)(31).

Grahic Jump Location
Fig. 4

The shape factor as a function of the radius of the cylinder for s=0.2, 0.5, and 0.8. The open circles represent the numerical results and the lines the approximation (24).

Grahic Jump Location
Fig. 5

Shape factor for a sphere of radius ah with its center at a distance h from a wall. The open circles represent the exact numerical results; the dashed line, the small a approximation given by Eq. (33); the stars, the asymptotic result (34) for a→1; and the solid line, the approximation (35) which combines the two asymptotes.

Grahic Jump Location
Fig. 6

The coefficient b0 in Eq. (36) as a function of s. The open circles are the results of computations and the solid line represents (38).

Grahic Jump Location
Fig. 7

Shape factor for a sphere placed between two walls. The open circles represent the exact numerical results and the solid lines, the composite expression (39).

Grahic Jump Location
Fig. 8

Dipole correction as a function of a for α=0,5,10, and ∞: (a) s=0 and (b)s=0.5. The open circles represent the exact numerical results; the dashed lines, the small a approximation (48); and the solid line, the approximation (51).

Grahic Jump Location
Fig. 9

The temperature of the sphere with α=∞ as a function of a and s. The circles represent the numerical results and the lines the small-a approximation (52).

Grahic Jump Location
Fig. 10

The correction to the O(ϕ) coefficient for the effective conductivity as a function of a for α=0, 5, 10, and ∞

Grahic Jump Location
Fig. 11

Dipole correction for a cylinder placed at the mid-plane between the two walls (s=0) as a function a for α=0 (the lowest curve), 5, 10, and ∞ (the two uppermost curves). The open circles are the exact results of numerical computations and the dashed lines are the small a approximation given by Eq. (55). The solid curve represents Eq. (58) for α=∞.

Grahic Jump Location
Fig. 12

The correction to the O(ϕ) term for the effective conductivity as a function of a for α=0 (the lowest curve), 5, 10, and ∞ (the uppermost curve)

Grahic Jump Location
Fig. 13

The normalized temperature of the cylinder as function a for s=0.25 (the lowest curve), 0.5, and 0.7 (the uppermost curve) and α=∞

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