An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates

Prashant K. Jain, Suneet Singh, Rizwan-uddin

Research output: Research - peer-reviewArticle

Abstract

Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r-θ spherical coordinates. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. However, inhomogeneous boundary conditions of the first, second or third kind may be applied at the inner and outer radial boundaries of the concentric layers. It is noted that the proposed solution is "free" from imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of radial eigenvalues on those in the θ-direction. Solution is shown to be relatively simple for the most common spherical geometries-(multilayer) hemisphere and full sphere. An illustrative problem of heat conduction in a three-layer hemisphere is solved. Results along with the isotherms are shown graphically and discussed.

LanguageEnglish (US)
Pages2133-2142
Number of pages10
JournalInternational Journal of Heat and Mass Transfer
Volume53
Issue number9-10
DOIs
StatePublished - Apr 2010

Fingerprint

spherical coordinates
conductive heat transfer
Heat conduction
Multilayers
Boundary conditions
eigenvalues
hemispheres
boundary conditions
Boundary value problems
Isotherms
heat sources
boundary value problems
isotherms
geometry
Geometry
Hot Temperature

Keywords

  • Analytical
  • Conduction
  • Cones and wedges
  • Hemisphere
  • Multilayer
  • Spherical
  • Transient

ASJC Scopus subject areas

  • Mechanical Engineering
  • Condensed Matter Physics
  • Fluid Flow and Transfer Processes

Cite this

An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates. / Jain, Prashant K.; Singh, Suneet; Rizwan-uddin.

In: International Journal of Heat and Mass Transfer, Vol. 53, No. 9-10, 04.2010, p. 2133-2142.

Research output: Research - peer-reviewArticle

@article{18717160467b4d4eab67f027a66d2dab,
title = "An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates",
abstract = "Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r-θ spherical coordinates. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. However, inhomogeneous boundary conditions of the first, second or third kind may be applied at the inner and outer radial boundaries of the concentric layers. It is noted that the proposed solution is {"}free{"} from imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of radial eigenvalues on those in the θ-direction. Solution is shown to be relatively simple for the most common spherical geometries-(multilayer) hemisphere and full sphere. An illustrative problem of heat conduction in a three-layer hemisphere is solved. Results along with the isotherms are shown graphically and discussed.",
keywords = "Analytical, Conduction, Cones and wedges, Hemisphere, Multilayer, Spherical, Transient",
author = "Jain, {Prashant K.} and Suneet Singh and Rizwan-uddin",
year = "2010",
month = "4",
doi = "10.1016/j.ijheatmasstransfer.2009.12.035",
volume = "53",
pages = "2133--2142",
journal = "International Journal of Heat and Mass Transfer",
issn = "0017-9310",
publisher = "Elsevier Limited",
number = "9-10",

}

TY - JOUR

T1 - An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates

AU - Jain,Prashant K.

AU - Singh,Suneet

AU - Rizwan-uddin,

PY - 2010/4

Y1 - 2010/4

N2 - Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r-θ spherical coordinates. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. However, inhomogeneous boundary conditions of the first, second or third kind may be applied at the inner and outer radial boundaries of the concentric layers. It is noted that the proposed solution is "free" from imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of radial eigenvalues on those in the θ-direction. Solution is shown to be relatively simple for the most common spherical geometries-(multilayer) hemisphere and full sphere. An illustrative problem of heat conduction in a three-layer hemisphere is solved. Results along with the isotherms are shown graphically and discussed.

AB - Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r-θ spherical coordinates. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. However, inhomogeneous boundary conditions of the first, second or third kind may be applied at the inner and outer radial boundaries of the concentric layers. It is noted that the proposed solution is "free" from imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of radial eigenvalues on those in the θ-direction. Solution is shown to be relatively simple for the most common spherical geometries-(multilayer) hemisphere and full sphere. An illustrative problem of heat conduction in a three-layer hemisphere is solved. Results along with the isotherms are shown graphically and discussed.

KW - Analytical

KW - Conduction

KW - Cones and wedges

KW - Hemisphere

KW - Multilayer

KW - Spherical

KW - Transient

UR - http://www.scopus.com/inward/record.url?scp=76749115239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76749115239&partnerID=8YFLogxK

U2 - 10.1016/j.ijheatmasstransfer.2009.12.035

DO - 10.1016/j.ijheatmasstransfer.2009.12.035

M3 - Article

VL - 53

SP - 2133

EP - 2142

JO - International Journal of Heat and Mass Transfer

T2 - International Journal of Heat and Mass Transfer

JF - International Journal of Heat and Mass Transfer

SN - 0017-9310

IS - 9-10

ER -