Heat and Mass Transfer

, Volume 52, Issue 3, pp 635–655 | Cite as

Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions

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Abstract

Closed form approximate solutions to nonlinear heat (mass) diffusion equation with power-law nonlinearity of the thermal (mass) diffusivity have been developed by the integral-balance method avoiding the commonly used linearization by the Kirchhoff transformation. The main improvement of the solution is based on the double-integration technique and a new approach to the space derivative. Solutions to Dirichlet and Neumann boundary condition problems have been developed and benchmarked against exact numerical and approximate analytical solutions available in the literature.

Keywords

Penetration Depth Series Solution Flux Boundary Condition Transient Heat Conduction Pointwise Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

Thermal diffusivity (m2/s)

a0

Thermal diffusivity of the linear problem (m = 0) (m2/s)

b

Coefficient in Eq. (24b) which should be defined trough the initial condition \(\delta \left( {t = 0} \right) = 0\)

Cp

Specific heat capacity (J/kg)

EL(n, m, t)

Squared-error function in accordance with the Langford criterion (Eq. 35)

EMT

Square-error function in accordance with the method of Mitchell and Myers [32] for the case of fixed temperature BC problem

EMq(HBIM) and EMq(DIM)

Square-error functions in accordance with the method of Mitchell and Myers [32] for the case of fixed temperature and fixed flux BC problems, respectively

eMT(DIM)

Squared-error sub-function in accordance with the method of Mitchell and Myers and related to DIM solution of the fixed temperature BC problem

k

Thermal conductivity (W/mK)

k0

Thermal conductivity of the linear problem (m = 0) (W/mK)

m

Dimensionless parameter of the nonlinearity

n

Dimensionless exponent of the parabolic profile

nmT

Dimensionless exponent of the parabolic profile for the fixed temperature BC problem defined by matching HBIM and DIM solutions

nmq

Dimensionless exponent of the parabolic profile for the fixed flux BC problem defined by matching HBIM and DIM solutions

q0

Heat flux surface density (W/m2)

T

Temperature (K)

Ta

Approximate temperature (K)

Ts

Surface temperature (at x = 0) (K)

T0

Initial temperature of the medium (K)

t

Time (s)

x

Space coordinate (m)

u

Dimensionless temperature (fixed temperature boundary condition problem)

ua

Approximate dimensionless temperature

U(ξ, t)

Dimensionless approximate profile (fixed flux BC problem) expressed through the Zener’s coordinate

V(ξ, t)

Dimensionless approximate profile (fixed temperature BC problem) expressed through the Zener’s coordinate

w

Kirchhoff transforms defined and used in Eq. (2)

Greek symbols

δ

Thermal penetration depth (m)

δq(HBIM)

Thermal penetration depth in case of fixed flux BC and HBIM solution (m)

δq(DIM)

Thermal penetration depth in case of fixed flux BC and DIM solution (m)

δT(HBIM)

Thermal penetration depth in case of fixed temperature BC and HBIM solution (m)

δT(DIM)

Thermal penetration depth in case of fixed temperature BC and DIM solution (m)

ΦT(ξ, t)

Error function of the heat conduction equation expressed through the Zener’s coordinate (Eq. 46a) and the fixed temperature BC problem

Φq(ξ, t)

Error function of the heat conduction equation expressed through the Zener’s coordinate (Eq. 48) and the fixed flux BC problem

φ(ua(x, t))

Error function defined by Eq. 31

φT(ua(x, t))

Error function defined by Eq. 31 for the case of fixed temperature BC problem

φq(θa(x, t))

Error function defined by Eq. 31 for the case of fixed temperature BC problem

\(\eta = x/\sqrt {at}\)

Boltzmann similarity variable (dimensionless)

λ = Tm

A transform used in Eq. (3)

ρ

Density (kg/m3)

θ

Dimensionless temperature in the fixed flux boundary condition problem

θa

Dimensionless approximate temperature (fixed flux boundary condition problem)

τ = t/m

A transform used in Eq. (3)

ξ = x/δ

Dimensionless Zener’s coordinate

Abbreviations

BC

Boundary condition

DIM

Double integration method

HBIM

Heat-balance integral method

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Chemical EngineeringUniversity of Chemical Technology and MetallurgySofiaBulgaria

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