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An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow

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Abstract

Forced convection heat transfer in a non-Newtonian fluid flow inside a pipe whose external surface is subjected to non-axisymmetric heat loads is investigated analytically. Fully developed laminar velocity distributions obtained by a power-law fluid rheology model are used, and viscous dissipation is taken into account. The effect of axial heat conduction is considered negligible. The physical properties are assumed to be constant. We consider that the smooth change in the velocity distribution inside the pipe is piecewise constant. The theoretical analysis of the heat transfer is performed by using an integral transform technique – Vodicka’s method. An important feature of this approach is that it permits an arbitrary distribution of the surrounding medium temperature and an arbitrary velocity distribution of the fluid. This technique is verified by a comparison with the existing results. The effects of the Brinkman number and rheological properties on the distribution of the local Nusselt number are shown.

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Abbreviations

A ilm , B ilm :

constants in Eqs.(28) and (39)

Br :

Brinkman number = \(\kappa u^{{1 + \nu}}_{\rm max} R^{1- \nu} /[\lambda (T_{0{\rm b}} - T_{\rm s})]\)

c :

specific heat (J kg −1 K −1)

G m :

function defined by Eq. (31)

h :

outside heat transfer coefficient (W m −2 K −1)

\(\widehat{h}\) :

local heat transfer coefficient (W m −2 K −1)

H :

dimensionless outside heat transfer coefficient = hR/λ

J m():

Bessel function of the first kind of order m

n :

number of partitions

Nu :

local Nusselt number = \(2\widehat{h}R/\lambda\)

P m :

function defined by Eq. (24)

Q :

heat generation term = Br[(1 + ν)/ν]ν+1 η (1+ν)/ν

r :

radial coordinate

R :

pipe radius (m)

T :

fluid temperature (K)

u :

axial fluid velocity (m/s), Eq. (2)

u m :

mean axial fluid velocity (m/s)

u max :

maximum fluid velocity = (1 + 3ν)u m/(1 + ν)

U :

dimensionless velocity = u/u m

W m :

function defined by Eq. (27)

x :

axial coordinate

X ilm :

Eigenfunctions corresponding to the lth eigenvalue for the ith region, Eqs. (28) and (39)

Y m():

Bessel function of the second kind of order m

\(\phi\) :

circumferential coordinate

Φ lm :

function corresponding to the lth eigenvalue defined by Eqs.(37) and (39)

γ lm :

lth eigenvalues

η :

dimensionless radial coordinate = r/R

κ :

power-law model parameter (Pa s ν)

λ :

thermal conductivity (W m −1 K −1)

ν :

power-law model index

θ :

dimensionless fluid temperature = (T − T s)/(T 0b − T s)

ρ :

fluid density (kg m −3)

τ rx :

shear stress (Pa), Eq. (1)

ξ :

dimensionless axial coordinate =λ x/(u m R 2 ρ c)

0:

inlet

0b:

bulk quantity at inlet

∞:

surrounding

B:

bulk

i :

region number

s:

representative

w:

wall

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Authors and Affiliations

  1. Department of Mechanical Engineering, Miyagi National College of Technology, Medeshimasiote, Natori, 9811239, Japan

    Ryoichi Chiba

  2. Department of Mechanical Engineering, Ishinomaki Sensyu University, Minamisakai, Ishinomaki, 9868580, Japan

    Masaaki Izumi

  3. Department of Mechanical Engineering, Iwate University, 4-3-5 Ueda, Morioka, 0208551, Japan

    Yoshihiro Sugano

Authors
  1. Ryoichi Chiba
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  2. Masaaki Izumi
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  3. Yoshihiro Sugano
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Correspondence to Ryoichi Chiba.

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Chiba, R., Izumi, M. & Sugano, Y. An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow. Arch Appl Mech 78, 61–74 (2008). https://doi.org/10.1007/s00419-007-0141-1

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