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Analysis of forced convection of Phan–Thien–Tanner fluid in slits and tubes of constant wall temperature with viscous dissipation

  • S. Z. DaghighiEmail author
  • M. Norouzi
Technical Paper
  • 19 Downloads

Abstract

In the present work, analytical solutions are presented for thermal convection of the linear Phan–Thien–Tanner fluid (LPTT) in slits and tubes of constant wall temperature by taking account of the viscous dissipation term. Unlike the similar previous studies in which the advection term was neglected in the heat transfer equation, it is considered in this investigation. A continuous relation between the Nusselt number and the Brinkman number is obtained. Expressions for the temperature distribution are derived in closed form and in terms of a Frobenius series for the slit and tube flows, respectively. Based on these solutions, the effects of fluid elasticity and Brinkman number on thermal convection of LPTT fluid flows are studied in detail. It is shown that at negative Brinkman numbers (fluid cooling), increasing the Deborah number leads to a decrease in the Nusselt number, but an increase in the centerline temperature. Nonetheless, this trend is opposite for positive Brinkman numbers (fluid heating), i.e., an increase in the Nusselt number and a decrease in the centerline temperature. Also, there is a Brinkman number beyond which the Nusselt number is smaller than zero, meaning that there is weak heat convection in the flow. Also, the results confirm that the extensibility parameter affects the temperature profile in the same way as the Deborah number.

Keywords

Viscoelastic fluid Phan–Thien–Tanner model Forced convection Constant wall temperature tube and slit Viscous dissipation 

List of symbols

\( A \)

Area of cross section, \( {\text{m}}^{2} \)

\( {\text{Br}} \)

Brinkman number, \( {\text{Br}} = \eta U^{2} /k(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} ) \)

\( c_{\text{p}} \)

Specific heat, J kg−1 K−1

\( d_{\text{h}} \)

Hydraulic diameter, \( d_{\text{h}} = 2R \) for tube flow and \( d_{\text{h}} = 4H \) for slit flow, \( {\text{m}} \)

\( {\mathbf{D}} \)

Deformation rate tensor, Eq. (8)

De

Deborah number, defined as \( {\text{De}} = \,\lambda U/R \) for tube case and \( {\text{De}} = \,\lambda U/H \) for slit case

F

Dimensionless function

\( h \)

Heat transfer coefficient, W m−2 K−1

\( H \)

Half of the distance between two parallel plates, \( {\text{m}} \)

J

Equals 0 for slit and 1 for tube

\( k \)

Conductivity coefficient, W m−1 K−1

\( K \)

Equals 1.5 and 2 for slit and tube, respectively

\( {\text{Nu}} \)

Nusselt number, \( {\text{Nu}} = hd_{\text{h}} /k \)

\( p \)

Pressure, \( {\text{Pa}} \)

\( P \)

Perimeter of cross section, \( {\text{m}} \)

\( R \)

Tube radius, \( {\text{m}} \)

\( T \)

Fluid temperature, \( {\text{K}} \)

\( u \)

Axial velocity, \( {\text{ms}}^{ - 1} \)

U

Mean velocity, \( {\text{ms}}^{ - 1} \)

\( {\mathbf{V}} \)

Velocity vector

\( y \)

Radial (tube) or transverse (slit) direction, \( {\text{m}} \)

\( z \)

Axial direction, \( {\text{m}} \)

Greek symbols

\( \alpha \)

Equals 1 for pure entropy elasticity and 0 for pure energy elasticity

\( \alpha_{1} \)

\( \alpha_{1} { = }1.5_{{}} U_{\text{N}} /U \) for slit and \( \alpha_{1} { = }2\,U_{\text{N}} /U \) for tube

\( \delta \)

Constant, Eq. (12)

\( \varepsilon \)

Extensibility coefficient

\( \eta \)

Constant viscosity coefficient, \( {\text{Pa s}} \)

\( \lambda \)

Relaxation time, \( {\text{s}} \)

\( \rho \)

Density, Kg m−3

\( \mathop {\varvec{\uptau}}\limits_{{}}^{\nabla } \)

Upper-convected derivative of stress tensor, Eq. (9)

\( \phi \)

Viscous dissipation, Eq. (15)

Subscripts

m

Mean value

max

Maximum value

N

Newtonian case

w

Wall

Superscripts

T

Transpose operator

~

Dimensional parameter

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringShahrood University of TechnologyShahroodIran

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