Experimental investigation of laminar flow of viscous oil through a circular tube having integral axial corrugation roughness and fitted with twisted tapes with oblique teeth

Abstract

The experimental friction factor and Nusselt number data for laminar flow of viscous oil through a circular duct having integral axial corrugation roughness and fitted with twisted tapes with oblique teeth have been presented. Predictive friction factor and Nusselt number correlations have also been presented. The thermohydraulic performance has been evaluated. The major findings of this experimental investigation are that the twisted tapes with oblique teeth in combination with integral axial corrugation roughness perform significantly better than the individual enhancement technique acting alone for laminar flow through a circular duct up to a certain value of fin parameter.

Appendix

1.1 Uncertainty analysis

All the quantities that are measured to estimate the Nusselt number and the friction factor are subject to certain uncertainties due to errors in the measurement. These individual uncertainties as well as the combined effect of these are presented here. The analysis is carried out on the basis of the suggestion made by Kline and McClintock [50].

1.2 Analysis

First the analysis for the friction factor is presented. The analysis for the Nusselt number is presented after that.

1.2.1 Friction Factor

$$ f = \frac{1}{2}\left\{ {\frac{{\Delta P}}{{L_{p} }}} \right\}\left\{ {\frac{{\rho D^{3} }}{{\text{Re}^{2} \mu^{2} }}} \right\} $$

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$$ \frac{{\Delta f}}{f} = \frac{1}{f}\left[ {\left\{ {\frac{\partial f}{{\partial \left( {\Delta P} \right)}}\Delta \left( {\Delta P} \right)} \right\}^{2} + \left\{ {\frac{\partial f}{{\partial L_{p} }}\Delta L_{p} } \right\}^{2} + \left\{ {\frac{\partial f}{\partial D}\varDelta D} \right\}^{2} + \left\{ {\frac{\partial f}{{\partial \left( {\text{Re} } \right)}}\Delta \text{Re} } \right\}^{2} } \right]^{0.5} $$

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or,

$$ \frac{{\Delta f}}{f} = \left[ {\left\{ {\frac{{\Delta \left( {\Delta P} \right)}}{{\Delta P}}} \right\}^{2} + \left\{ {\frac{{\Delta L_{p} }}{{L_{p} }}} \right\}^{2} + \left\{ {\frac{{3\Delta D}}{D}} \right\}^{2} + \left\{ {\frac{{2\Delta \text{Re} }}{\text{Re}}} \right\}^{2} } \right]^{0.5} $$

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$$ \Delta P \propto h $$

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$$ \therefore \frac{{\Delta \left( {\Delta P} \right)}}{{\Delta P}} = \frac{{\Delta h}}{h} $$

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$$ \text{Re} = \frac{{4\dot{m}}}{\pi D\mu } $$

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$$ \frac{{\Delta \text{Re} }}{\text{Re}} = \left[ {\left( {\frac{{\Delta \dot{m}}}{{\dot{m}}}} \right)^{2} + \left( {\frac{{\Delta D}}{D}} \right)^{2} } \right]^{0.5} $$

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The uncertainty in friction factor has been calculated from the above equations.

1.2.2 Nusselt number

$$ Nu = \frac{hD}{k} $$

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$$ \frac{{\Delta Nu}}{Nu} = \frac{1}{Nu}\left[ {\left\{ {\frac{\partial }{\partial h}(Nu)\Delta h} \right\}^{2} + \left\{ {\frac{\partial }{\partial D}\left( {Nu} \right)\Delta D} \right\}^{2} + \left\{ {\frac{\partial }{\partial k}\left( {Nu} \right)\Delta k} \right\}^{2} } \right]^{0.5} $$

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or

$$ \frac{{\Delta Nu}}{Nu} = \left\{ {\left( {\frac{{\Delta h}}{h}} \right)^{2} + \left( {\frac{{\Delta D}}{D}} \right)^{2} } \right\}^{0.5} $$

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$$ h = \frac{{q^{\prime\prime}}}{{T_{wi} - T_{b} }} $$

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$$ \frac{{\Delta h}}{h} = \frac{1}{h}\left[ {\left\{ {\frac{\partial h}{{\partial q^{\prime\prime}}}\Delta q^{\prime\prime}} \right\}^{2} + \left\{ {\frac{\partial h}{{\partial T_{wi} }}\Delta T_{wi} } \right\}^{2} + \left\{ {\frac{\partial h}{{\partial T_{b} }}\Delta T_{b} } \right\}^{2} } \right]^{0.5} $$

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$$ \frac{{\Delta h}}{h} = \left[ {\left\{ {\frac{{\Delta q^{\prime\prime}}}{{q^{\prime\prime}}}} \right\}^{2} + \left\{ {\frac{{\Delta T_{wi} }}{{T_{wi} - T_{b} }}} \right\}^{2} + \left\{ {\frac{{\Delta T_{b} }}{{T_{wi} - T_{b} }}} \right\}^{2} } \right]^{0.5} $$

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$$ q^{\prime\prime} = \frac{0.5}{{\pi DL_{h} }}\left[ {\left( {V^{2} /R} \right) + \dot{m}C_{p} \left( {T_{bo} - T_{bi} } \right)} \right] $$

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$$ \frac{{\Delta q^{\prime\prime}}}{{q^{\prime\prime}}} = \frac{1}{{q^{\prime\prime}}}\left[ \begin{gathered} \left\{ {\frac{\partial }{\partial R}\left( {q^{\prime\prime}} \right)\Delta R} \right\}^{2} + \left\{ {\frac{\partial }{\partial V}\left( {q^{\prime\prime}} \right)\Delta V} \right\}^{2} + \left\{ {\frac{\partial }{{\partial \dot{m}}}\left( {q^{\prime\prime}} \right)\Delta \dot{m}} \right\}^{2} + \left\{ {\frac{\partial }{{\partial T_{bo} }}\left( {q^{\prime\prime}} \right)\Delta T_{bo} } \right\}^{2} \hfill \\ + \left\{ {\frac{\partial }{{\partial T_{bi} }}\left( {q^{\prime\prime}} \right)\Delta T_{bi} } \right\}^{2} + \left\{ {\frac{\partial }{\partial D}\left( {q^{\prime\prime}} \right)\Delta D} \right\}^{2} + \left\{ {\frac{\partial }{{\partial L_{h} }}\left( {q^{\prime\prime}} \right)\Delta L_{h} } \right\}^{2} \hfill \\ \end{gathered} \right]^{0.5} $$

$$ \frac{{\Delta q^{\prime\prime}}}{{q^{\prime\prime}}} = \left[ \begin{gathered} \frac{1}{{\left( {1 + \dot{m}C_{p} R\Delta T_{b} /V^{2} } \right)^{2} }}\left( {\frac{{\Delta R}}{R}} \right)^{2} + \frac{4}{{\left( {1 + \dot{m}C_{p} R\Delta T_{b} /V^{2} } \right)^{2} }}\left( {\frac{{\Delta V}}{V}} \right)^{2} \hfill \\ + \frac{1}{{\left( {1 + \frac{{V^{2} }}{{R\dot{m}C_{p} \varDelta T_{b} }}} \right)^{2} }}\left( {\frac{{\Delta \dot{m}}}{{\dot{m}}}} \right)^{2} + \frac{1}{{\left( {1 + \frac{{V^{2} }}{{R\dot{m}C_{p}\Delta T_{b} }}} \right)^{2} }}\left( {\frac{{\Delta T_{bo} }}{{\Delta T_{b} }}} \right)^{2} \hfill \\ + \frac{1}{{\left( {1 + \frac{{V^{2} }}{{R\dot{m}C_{p}\Delta T_{b} }}} \right)^{2} }}\left( {\frac{{\Delta T_{bi} }}{{\Delta T_{b} }}} \right)^{2} + \left( {\frac{{\Delta D}}{D}} \right)^{2} + \left( {\frac{{\Delta L_{h} }}{{L_{h} }}} \right)^{2} \hfill \\ \end{gathered} \right]^{0.5} $$

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where

$$ \Delta T_{b} = T_{bo} - T_{bi} $$

The uncertainty in Nusselt number has been calculated from the above equations.

The accuracies of the measured quantities are given below in the tabular form:

Quantity	Accuracy	Quantity	Accuracy
ΔDh	0.00002 m	ΔL	0.001 m
Δ$\dot{m}$	1.667E−5 kg/s	Δh	0.001 m
ΔT	0.025 °C	ΔV	0.1 V
ΔR	0.0000 Ω