Heat Transfer Enhancement in Short Corrugated Mini-Tubes

Abstract

Heat transfer phenomena are studied with standing waves inside the tubes for static and moving sinusoidal corrugated walls. The past studies have been done on big-size (dimensions in m) and micro-sized circular tubes (dimensions in μm). We are focusing on intermediate size tubes (dimensions in mm). Numerical simulations, using finite volume commercial software, were performed to study the effects of spatial wavelengths on heat transfer enhancement and associated pressure drop. We imposed 5, 10, 15 and 20 3D sinusoidal radial sine waves along the length of the tube. Heat transfer characteristics of static corrugated wavy walls were calculated for various imposed Reynolds numbers (1 < Re < 120) and amplitude of the wave was varied from 1 to 20 % of the diameter of the tube. For static wall case, upon increasing the number of sine waves, the Nusselt number starts to decrease; the associated pressure drop and friction factor increases very rapidly at the highest values of amplitude. On the other hand, in comparison to the static corrugated wall tube, the pressure drop is reduced by 20–80 % and heat transfer is enhanced by 35–70 % for highest amplitude when frequencies in the range 0 < f < 60 Hz are imposed on tube wall to make the corrugated tube moving in transverse direction.

Appendix

Local tube radius is given by:

$$ R\left( x \right) = R_{0} + A\sin \left( {\frac{2\pi \, x}{\lambda }} \right) $$

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The surface between 0 and x is given by:

$$ S(x) = 2\pi \int\limits_{0}^{x} {\left( {R_{0} + A\sin \left( {\frac{2\pi \,x}{\lambda }} \right)\sqrt {1 + \frac{{4A^{2} \pi^{2} \cos \left( {\frac{2\pi \,x}{\lambda }} \right)^{2} }}{{\lambda^{2} }}} } \right)} dx $$

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The result is given by the following expression:

$$ S\left( x \right) = \frac{1}{4A\pi \lambda }\left( {\frac{{8A^{4} \pi^{3} + 2A^{2} \pi \lambda^{2} + \sqrt {A^{2} } \lambda^{2} \sqrt {4A^{2} \pi^{2} + \lambda^{2} } Log\left[ {2\sqrt {A^{2} } \pi + \sqrt {4A^{2} \pi^{2} + \lambda^{2} } } \right]}}{{\sqrt {1 + \frac{{4A^{2} \pi^{2} }}{{\lambda^{2} }}} }}} \right. + $$

$$ \left( {\sqrt {2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)} \left( {4A\pi R_{0} \sqrt {2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)} } \right.} \right. $$

$$ EllipticE\left[ {\frac{2\pi x}{\lambda },\frac{{4A^{2} \pi^{2} }}{{4A^{2} \pi^{2} + \lambda^{2} }}} \right] - \sqrt {\frac{{2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)}}{{4A^{2} \pi^{2} + \lambda^{2} }}} $$

$$ \left( {2A^{2} \pi \cos \left( {\frac{2\pi x}{\lambda }} \right)} \right.\sqrt {2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)} + \sqrt {A^{2} } \lambda^{2} $$

$$ {\raise0.7ex\hbox{${\left. {\left. {\left. {Log\left[ {2\sqrt {A^{2} } \pi \cos \left( {\frac{2\pi x}{\lambda }} \right) + \sqrt {2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)} } \right]} \right)} \right)} \right)}$} \!\mathord{\left/ {\vphantom {{\left. {\left. {\left. {Log\left[ {2\sqrt {A^{2} } \pi \cos \left( {\frac{2\pi x}{\lambda }} \right) + \sqrt {2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)} } \right]} \right)} \right)} \right)} {}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${}$}} $$

$$ \left. {\left( {\sqrt {\frac{{2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)}}{{\lambda^{2} }}} \sqrt {\frac{{2A^{2} \pi^{2} + \lambda^{2} + 2A^{2} \pi^{2} \cos \left( {\frac{4\pi x}{\lambda }} \right)}}{{4A^{2} \pi^{2} + \lambda^{2} }}} } \right)} \right) $$

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