equipment.widget.helical module

Helically coiled tubes have many applications in engineering. Examples of it use include compact heat exchanger, steam generators, distillation, or chemical reactors…

In curved pipes, the more rapidly flowing central parts of the flow are forced outwards by centrifugal action, while the slower parts along the wall are forced inwards where the pressusre is less, and a secondary flow takes place at right angles to the main flow. If the curvature is significant, the axial velocity distribution is entirely altered by the secondary flow, and a considerable increase in resistance an heat transfer is observed.

The heat transfer and pressure losses depend on the Dean number \(De = Re (d_i/D_c)^{0.5}\)

There is many literature about the flow in curved pipes, here is implemented some of more relevant correlation for friction factor and nusselt number.

equipment.widget.helical.translate(context: str, sourceText: str, disambiguation: str = None, n: int = -1) → str

equipment.widget.helical.Rec_Schmidt(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Schmidt (1967)

\[Re_c = 2300 \left(1+8.6\left(\frac{di}{Dc}\right)^{0.45}\right)\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[3] Schmidt, E.F.; Wärmeübergand und Druckverlust in Rohrschlangen. Chemie Ingenieur Technik 39(13) (1967) 781-789

equipment.widget.helical.Rec_Ito(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Ito (1959)

\[Re_c = 2x10^4 \left(\frac{di}{Dc}\right)^{0.32}\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[4] Ito, H.; Friction Factors for Turbulent Flow in Curved Pipes. J. Basic Eng. 81 (1959) 123-134

equipment.widget.helical.Rec_Kubair(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Kubair-Kuloor (1966)

\[Re_c = 1.273x10^4 \left(\frac{di}{Dc}\right)^{0.2}\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[5] Kubair, V., Kuloor, N.R.; Heat Transfer to Newtonian Fluids in Coiled Pipes in Laminar Flow. Int. J. Heat Mass Transfer 9 (1966) 63-75

equipment.widget.helical.Rec_Srinivasan(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Srinivasan (1968) as shown in [1]. Recomended method by [2].

\[Re_c = 2100 \left(1 + 12\sqrt{\frac{d_i}{D_c}}\right)\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[6] Srinivasan, P.S., Nandapurkar, S.S., Holland, F.A.; Pressure Drop and Heat Transfer in Coils. Chem. Eng. 218 (1968) 113-119

[1] El-Genk, M.S., Timothy, M.S.; A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes. Heat Transfer Eng. 38(5) (2017) 447-474

[2] ; Perry’s Chemical Engineers’ Handbook 9th Edition. McGraw-Hill (2019)

equipment.widget.helical.Rec_Kutateladze(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Kutateladze (1966).

\[Re_c = 2300 + 10500 \left(\frac{d_i}{D_c}\right)^{0.3}\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[7] Kutateladze, S.S., Borishanskii, V.M. ; A Concise Encyclopedia of Heat Transfer. Pergamon Press (1966)

equipment.widget.helical.Rec_SethStahel(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Seth-Stahel (1969).

\[Re_c = 1900 \left(1 + 8 \sqrt{\frac{d_i}{D_c}}\right)\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[20] Seth, K.K., Stahel, E.P.; Heat Transfer from Helical Coils Immersed in Agitated Vessels. Ind. Eng. Chem. 61(6) (1969) 39-49

equipment.widget.helical.Rec_Cioncolini(di, Dc)

Calculates critical Reynolds to define transition between laminar and

turbulent flow using using the correlation of Cioncolini-Santini (2006).

\[Re_c = 30,000 \left(\frac{d_i}{D_c}\right)^{0.47}\]

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Recfloat

Critical reynolds number, [-]

References

[26] Ciencolini, A., Santini, L.; An experimental investigation regarding the laminar to turbulent flow transition in helically coiled pipes. Exp. Thermal Fluid Sci. 30 (2006) 367-380

equipment.widget.helical.f_Schmidt(Re, di, Dc)

Calculate friction factor for internal flow of a helical coil using

the correlation of Schmidt (1967)

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[3] Schmidt, E.F.; Wärmeübergand und Druckverlust in Rohrschlangen. Chemie Ingenieur Technik 39(13) (1967) 781-789

equipment.widget.helical.f_MoriNakayama(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Mori-Nakayama (1965).

\[\frac{f_c}{f_s}=\left(\frac{0.108De^{0.5}}{1-3.253 De^{-0.5}}\right)\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[9] Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (1st Report, Laminar Region). Int. J. Heat Mass Transfer 8(1) (1965) 67-82

[10] Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (2nd Report, Turbulent Region). Int. J. Heat Mass Transfer 10(1) (1967) 37-59

equipment.widget.helical.f_Ju(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil using

the method of Ju et al. (2001).

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[12] Ju, H., Huang, Z., Xu, Y., Duan, B, Yu, Y.; Hydraulic Performance of Small Bending Radius Helical Coil-Pipe. J. Nuclear Sci. Eng. 38(10) (2001) 826-831

equipment.widget.helical.f_MishraGupta(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil using

the method of Mishra-Gupta (1979).

\[\frac{f_c}{f_s} = 1 - \left(1-\left(\frac{11.6}{De}\right)^{0.45} \right)^{\frac{1}{0.45}}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[13] Mishra, P., Gupta, S.N.; Momentum Transfer in Curved Pipes. 1. Newtonian Fluids. Ind. Eng. Chem. Process Des. Dev. 18(1) (1979) 130-137

equipment.widget.helical.f_Prasad(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil using

the method of Prasad et al. (1989).

For laminar flow use a modified Write correlation:

\[\frac{f}{f_s} = \frac{1}{1-\left(1-\left(\frac{B}{De}\right)^{0.45} \right)^{\frac{1}{0.45}}}\]

For turbulent flow use a modified Ito correlation:

\[\frac{f}{f_s} = 1 + 0.18\left(Re \left(\frac{d_i}{D_c}\right)^2\right)^{0.25}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[18] Prasad, B.V.S.S.S., Das, D.H., Prabhaker, A.K.; Pressure Drop, Heat Transfer and Performance of a Helically Coiled Tubular Exchanger. Heat Recovery Systems & CHP 9(3) (1989) 249-256

equipment.widget.helical.f_Ali(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil using

the method of Ali (2001).

Define four flow regime fitted with the correlation:

\[f = Eu\frac{d}{L}=\alpha\left(\frac{d_i}{D_{eq}}\right)^{0.15}Re^{\beta}\]

with the equivalent diameter of coil:

\[D_{eq} = \sqrt{\frac{p^2+\left(\pi D\right)^2}{pi}}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[21] Ali, S.; Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Research 28 (2001) 295-310

equipment.widget.helical.f_ElGenkSchriener(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil using

the method of ElGenk-Schriener (2017).

\[\frac{f_c}{f_s} = 1 + 0.00325 De_m\]

where modified Dean number is defined as:

\[De_m = De^{0.86} \delta^{0.09} \left(\frac{d_i}{D_c}\right)^{-0.38}\]

δ is the curvature defined as:

\[\delta = \frac{d_i/D_c}{1+4\pi^2 \tan^2 \alpha}\]

α is the helix angle:

\[\alpha = \tan^{-1}{\frac{p}{\pi D}}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[1] El-Genk, M.S., Timothy, M.S.; A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes. Heat Transfer Eng. 38(5) (2017) 447-474

equipment.widget.helical.f_Srinivasan(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil using

the method of Srinivasan (1968) as explain in [49].

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[6] Srinivasan, P.S., Nandapurkar, S.S., Holland, F.A.; Pressure Drop and Heat Transfer in Coils. Chem. Eng. 218 (1968) 113-119

[49] Ghobadi, M., Muzychka, Y.S.; A Review of Heat Transfer and Pressure Drop Correlations for Laminar Flow in Curved Circular Ducts. Heat Transfer Eng. 37(10) (2016) 815-839

equipment.widget.helical.f_Ito(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil using

the method of Ito (1969).

For laminar flow:

\[\frac{f_c}{f_s} = 0.1033 De^{0.5} \left(\left(1+\frac{1.729}{De}\right) ^{0.5} - \frac{1.315}{De^{0.5}}\right)^{-3}\]

For turbulent flow:

\[f_c = 4 \left(0.029 sqrt{\frac{d_i}{D_c}} + 0.304 Re^{-0.25}\right)\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[23] Ito, H.; Laminar Flow in Curved Pipes. Z. Angew. Math. Mech. 11 (1969) 653-663

[4] Ito, H.; Friction Factors for Turbulent Flow in Curved Pipes. J. Basic Eng. 81 (1959) 123-134

equipment.widget.helical.f_RaoSadasivudu(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil using

the method of Rao-Sadasivudu (1974) as show in Ali [21].

Several correlation for different Reynolds number range

\[f_c = 1.55 \exp{14.12 \frac{d_i}{D_c}} Re^{-1}, Re < 1200\]

\[f_c = 1.55 \exp{14.12 \frac{d_i}{D_c}} Re^{-0.64}, 1200 < Re < Re_c\]

\[f_c = 0.0382 \exp{11.17 \frac{d_i}{D_c}} Re^{-0.2}, Re_c < Re < 27000\]

\[f_c = 0.01065 \frac{d_i^{0.94}}{D_c^{0.1}} Re^{-0.2}, Re > 27000\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[69] Rao, M.V.R., Sadasividu, D.; Pressure drop studies in helical coils. Indian J. Tech. 12 (1974) 473-474

[21] Ali, S.; Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Research 28 (2001) 295-310

equipment.widget.helical.f_laminar_White(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of White (1929).

\[\frac{f_c}{f_s} = 1 - \left(1-\left(\frac{11.6}{De}\right)^{0.45} \right)^{\frac{1}{0.45}}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[8] White, C.M.; Streamline Flow through Curved Pipes. Proc. R .Soc. London A 123 (1929) 645-663

equipment.widget.helical.f_laminar_Hart(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Hart (1988).

\[\frac{f_c}{f_s} = 1 + 0.09 \frac{De^{1.5}}{70+De}\]

Recomended method in [2] for friction factor in laminar flow.

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[11] Hart, J., Ellenberger, J., Hamersma, P.J.; Single- and Two-Phase Flow Through Helically Coiled Tubes. Chem. Eng. Sci. 43(4) (1988) 775-783

[2] ; Perry’s Chemical Engineers’ Handbook 9th Edition. McGraw-Hill (2019)

equipment.widget.helical.f_laminar_ManlapazChurchill(Re, di, Dc, p)

Calculates friction factor in laminar regimen for internal flow of a

helical coil using the method of Manlapaz-Churchill (1980).

\[\frac{f_c}{f_{s,L}} = \left[\left(1 - \frac{0.18}{\left(1+\left(\frac{35}{De}\right)^2\right)^{1/2}}\right)^m + \left(1+\frac{d_i}{3 D_c}\right)^2 \frac{De}{88.33}\right]^{1/2}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[17] Manlapaz, R.L., Churchill, S.W.; Fully Developed Laminar Flow in a Helically Coiled Tube of Finite Pitch. Chem. Eng. Communications 7 (1980) 57-78

equipment.widget.helical.f_laminar_LiuMasliyah(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Liu-Masliyah (1993).

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[19] Liu, S., Masliyah, J.H.; Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 251 (1993) 315-353

equipment.widget.helical.f_laminar_TarbellSamuels(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Tarbell-Samuels (1973).

\[\frac{f_c}{f_s} = 1 + \left( 8.279e^{-4} + \frac{7.964e{-3}}{d_i/D_c}\right) Re - 2.096e-7 Re^2\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[24] Tarbell, J.M., Samuels, M.R.; Momentum and Heat Transfer in Helical Coils. Chem. Eng. J. 5(2) (1973) 117-127

equipment.widget.helical.f_laminar_Adler(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Adler (1934).

\[\frac{f_c}{f_s} = 0.1064 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[27] Adler, M.; Strömung in gekrümmten Rohren. Z. Angew. Math. Mech. 14(5) 257-275

equipment.widget.helical.f_laminar_Barua(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Barua (1963).

\[\frac{f_c}{f_s} = 0.509 + 0.0918 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[28] Barua, S.N.; On Secondary Flow in Stationary Curved Pipes. Quart. J. Mech. Appl. Math. 16(1) (1963) 61-77

equipment.widget.helical.f_laminar_PimentaCampos(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Pimenta-Campos (2012)

\[\frac{f_c}{f_s} = 1 + \frac{0.028 De^{1.68}}{70+De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[29] Pimenta, T.A., Campos, J.B.L.M.; Friction losses of Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Exp. Thermal Fluid Sci. 36 (2012) 194-204

equipment.widget.helical.f_laminar_Yanase(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Yanase et al. (1989)

\[\frac{f_c}{f_s} = 0.557 + 0.0938 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[30] Yanase, S., Goto, N., Yamamoto, K.; Dual solutions of the flow through a curved tube. Fluid Dyn. Research 5 (1989) 191-201

equipment.widget.helical.f_laminar_Dennis(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Dennis (1980)

\[\frac{f_c}{f_s} = 0.388 + 0.1015 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[31] Dennis, S.C.R.; Calculation of the steady flow through a curved tube using a new finite-difference method. J. Fluid Mech. 99(3) (1980) 449-467

equipment.widget.helical.f_laminar_vanDyke(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Van Dyke (1978)

\[\frac{f_c}{f_s} = 0.47136 De^{0.25}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[32] Van Dyke, M.; Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86(1) 129-145

equipment.widget.helical.f_laminar_CollinsDennis(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Collins-Dennis (1975)

\[\frac{f_c}{f_s} = 0.38036 + 0.1028 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[33] Collins, W.M., Dennis, S.C.R.; The Steady Motion of a Viscous Fluid in a Curved Tube. Q. J. Mech. Appl. Math. 28(2) (1975) 133-156

equipment.widget.helical.f_laminar_Dean(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Dean (1928)

\[\frac{f_c}{f_s} = 0.38036 + 0.1028 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

Notes

Correlation only valid for De < 20

References

[34] Dean, W.R.; The stream-line motion of fluid in a curved pipe (Second paper). London Edinburgh Dublin Phil. Mag. J. Sci. Serie 7 5(30) (1928) 673-695

equipment.widget.helical.f_laminar_Abushammala(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Abushammala et al. (2019)

\[f_c = f_s + A B e^{-C}\]

with:

\[A = p_1 D \left(\frac{D}{Re}\right)^{p_2}\]

\[B = \left(\frac{D_c}{2 d_i} + \frac{2 d_i}{D_c}\right)^{p_3}\]

\[C = p_4 D \frac{p}{d_i} \left(\frac{D_c}{2 d_i}\right)^{-p_5}\]

\[D = \left(\left(\frac{D_c}{2 d_i}\right)^{-p6} \left(1 + \left( \frac{p/d_i}{2 \pi D_c/2/d_i}\right)^2\right)\right)^{-p_7}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

Notes

Correlation only valid for De < 20

References

[35] Abushammala, O., Hreiz, R., Lamaître, C., Favre, E.; Laminar flow friction factor in highly curved helical pipes: Numerical investigation, predictive correlation and experimental validation using a 3D-printed model. Chem. Eng. Sci. 207(7) (2019) 1030-1039

equipment.widget.helical.f_laminar_Gupta(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Gupta et al. (2011)

\[f_c = f_s \left(1 + a Gn^b\right)\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[58] Gupta, R., Wanchoo, R.K., Jafar Ali, T.R.M.; Laminar Flow in Helical Coils: A Parametric Study. Ind. Eng. Chem. Res. 50(2) (2011) 1150-1157

equipment.widget.helical.f_laminar_Hasson(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Hasson (1955) as shown in [21].

\[\frac{f_c}{f_s} = 0.556 + 0.0969 \sqrt{De}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[59] Hasson, D.; Streamline flow resistance in coils. Res. Corresp. 1 S1 (1955).

[21] Ali, S.; Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Research 28 (2001) 295-310

equipment.widget.helical.f_turbulent_Czop(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Czop (1994).

\[f_c = \frac{0.096}{De^{-0.1517}}\]

The paper give this correlation for single phase flow. Give too correlations for two phase flow.

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[14] Czop, V., Barbier, D., Dong, S.; Pressure drop, void fraction and shear stress measurements in an adiabatic two-phase flow in a coiled tube. Nuclear Eng. Design 149 (1994) 323-333

equipment.widget.helical.f_turbulent_Guo(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Guo et al. (2001).

\[f_c = 2.552 Re^{-0.15} \left(\frac{d_i}{D_c}\right)^{0.51}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[22] Guo, L., Feng, Z, Chen, X.; An experimental investigation of the frictional pressure drop of steam–water two-phase flow in helical coils. Int. J. Heat Mass Transfer 44(14) (2001): 2601-2610

equipment.widget.helical.f_turbulent_MandalNigam(Re, di, Dc)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Mandal-Nigam (2009).

\[\frac{f_c}{f_s} = 1 + 0.03{De}^{0.27}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

ffloat

Friction factor, [-]

References

[25] Mandal, M. M., Nigam, K.D.P.; Experimental Study on Pressure Drop and Heat Transfer of Turbulent Flow in Tube in Tube Helical Heat Exchanger. Ind. Eng. Chem. Res. 48(20) (2009) 9318-9324

equipment.widget.helical.f_turbulent_Zhao(Re, do, Dc, eD)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Zhao et al. (2016).

\[\frac{1}{f_c^{0.5}} = 0.923 \ln \left(\frac{0.104\epsilon}{f_c} \left(\frac{d_o}{D_c}\right)^{0.5} + \frac{1.142}{f_c^{1.5}Re} \left(\frac{d_o}{D_c}\right)^{0.5}\right)\]

Parameters:

Refloat

Reynolds number, [-]

dofloat

External diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

eDfloat

Relative roughness, [-]

Returns:

ffloat

Friction factor, [-]

References

[60] Zhao, H., Li, X., Wu, X.; New friction factor equations developed for turbulent flow in rough helical tubes. Int. J. Heat Mass Transfer 95 (2016) 525-534

equipment.widget.helical.f_turbulent_Das(Re, di, Dc, p, eD)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Das (1993).

\[f_c - f_{cs} = 17.5782 Re^{-0.3137} \left(\frac{d_i}{D_c}\right) ^{0.3621} \left(\frac{\epsilon}{D_c}\right)^{0.6885}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Internal diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

eDfloat

Relative roughness, [-]

Returns:

ffloat

Friction factor, [-]

References

[61] Das, S.K.; Water Flow Through Helical Coils in Turbulent Condition. Can. J. Chem. Eng. 71 (1993) 971-973

equipment.widget.helical.f_turbulent_Zheng(Re, di, Dc, p)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Zheng et al. (2023).

\[f_c = \frac{0.0791}{Re^{0.25} + \frac{81858}{Re^{1.54}} \left(\frac{d_i}{D_{cm}}\right)^{0.48}\]

D_cm is defined as:

\[D_{cm} = D_c \left(1+\tan{\alpha}\right)\]

α is the helix angle:

\[\alpha = \tan^{-1}{\frac{p}{\pi D_c}}\]

Parameters:

Refloat

Reynolds number, [-]

difloat

Internal diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

ffloat

Friction factor, [-]

References

[64] Zheng, X., Lu, X., Gao, Y., Jin, D., Hu, Y., Hu, Y., Mao, Y.; Experimental study on friction pressure drop and circumferential heat transfer characteristics in helical tubes. Front. Energy Res. 11 (2023) 1204850.

equipment.widget.helical.Nu_Schmidt(Re, Pr, di, Dc)

Calculates Nusselt number for internal flow of a helical coil using the

correlation of Schmidt (1967)

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[3] Schmidt, E.F.; Wärmeübergand und Druckverlust in Rohrschlangen. Chemie Ingenieur Technik 39(13) (1967) 781-789

equipment.widget.helical.Nu_MoriNakayama(Re, Pr, di, Dc, simple=False)

Calculates Nusselt number for internal flow of a helical coil in

laminar flow using the method of Mori-Nakayama (1965).

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[9] Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (1st Report, Laminar Region). Int. J. Heat Mass Transfer 8(1) (1965) 67-82

[10] Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (2nd Report, Turbulent Region). Int. J. Heat Mass Transfer 10(1) (1967) 37-59

[46] Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (3rd Report, Theoretical Analysis under the Condition of Uniform Wall Temperature and Practical Formulae). Int. J. Heat Mass Transfer 10(5) (1967) 681-695

equipment.widget.helical.Nu_XinEbadian(Re, Pr, di, Dc)

Calculates Nusselt number for internal flow of a helical coil using the

correlation of Xin-Ebadian (1997)

For laminar flow:

\[Nu = \left(2.153 + 0.318 \left(Re \frac{d_i}{D_c}\right)^{0.643}\right) Pr^{0.177}\]

For turbulent flow:

\[Nu = 0.00619 Re^{0.92} Pr^{0.4} \left(1 + 3.455 \frac{d_i}{D_c}\right)\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[15] Xin, R.C., Ebadian, M.A. ; The Effects of Prandtl Numbers on Local and Average Convective Heat Transfer Characteristics in Helical Pipes. J. Heat Transfer 119(3) (1997) 467-73

equipment.widget.helical.Nu_SebanMcLaughlin(Re, Pr, di, Dc)

Calculates Nusselt number for internal flow of a helical coil using the

correlation of Seban-McLaughlin (1963)

For laminar flow:

\[Nu = 1.04 \left(\frac{Re}{1-\left(1-\left(1-\frac{11.6}{De}\right) ^{0.45}\right)^{1/0.45}}\right)^{1/3} Pr^{1/3}\]

For turbulent flow:

\[Nu = 0.023 Re^{0.85} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.1}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[16] Seban R.A., McLaughlin, E.F.; Heat Transfer in Tube Coils with Laminar and Turbulent Flow. Int. J. Heat Mass Transfer 6() (1963) 387-395

equipment.widget.helical.Nu_Prasad(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil using

the method of Prasad et al. (1989).

For laminar flow:

\[Nu = 0.25 \left(\frac{f}{8} Re^2\right)^{1/3} Pr^{1/3}\]

For turbulent flow:

\[Nu = \frac{f}{8} Re\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[18] Prasad, B.V.S.S.S., Das, D.H., Prabhaker, A.K.; Pressure Drop, Heat Transfer and Performance of a Helically Coiled Tubular Exchanger. Heat Recovery Systems & CHP 9(3) (1989) 249-256

equipment.widget.helical.Nu_PawarSunnapwar(Re, Pr, di, Dc)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil using the method of Pawar-Sunnapwar (2013).

For laminar flow:

\[Nu = 0.02198 Re^{0.9314} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.391}\]

For turbulent flow:

\[Nu = 0.0472 De^{0.8346} Pr^{0.4}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[42] Pawar, S.S., Sunnapwar, V.K.; Studies on convective heat transfer through helical coils. Heat Mass Transfer 49(12) (2013) 1741-1754

[45] Pawar, S.S., Sunnapwar, V.K.; Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow. Exp. Thermal Fluid Sci. 44 (2013) 792-804

equipment.widget.helical.Nu_ElGenkSchriener(Re, Pr, di, Dc, p)

Calculates nusselt number for internal flow of a helical coil using

the method of ElGenk-Schriener (2017).

For fluids with Pr < 15:

\[Nu_c = 3.66 + 0.014 Re_m^{0.86} Pr^{0.4}\]

For fluids with Pr > 15:

\[Nu_c = 3.66 + 0.02 Re_m^{0.7} Pr^{0.4}\]

using a modified Reynolds number:

\[Re_m = Re \left(1+3.4 \delta\right)\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

Nufloat

Nusselt number, [-]

References

[1] El-Genk, M.S., Timothy, M.S.; A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes. Heat Transfer Eng. 38(5) (2017) 447-474

equipment.widget.helical.Nu_Zhou(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil using

the method of Zhou et al. (2020).

For laminar regimen:

\[Nu = 0.0254 f Re^{1.197} Pr^{0.159}\]

For turbulent regimen:

\[Nu = 0.013 Re^{0.93} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.177}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[65] Zhao, H., Li, X., Wu, Y., Wu, X.; Friction factor and Nusselt number correlations for forcedconvection in helical tubes. Int. J. Heat Mass Transfer 155 (2020) 119759

equipment.widget.helical.Nu_laminar_KalbSeader(Re, Pr, di, Dc, boundary=0)

Calculates nusselt number for internal flow of a helical coil in

laminar flow using the method of Kalb-Seader (1972).

At constant heat flux boundary condition:

For Pr < 0.05:

\[Nu = 3.31 De^{0.115} Pr^{0.0108}\]

For Pr > 0.7

\[Nu = 0.913 De^{0.476} Pr^{0.2}\]

At uniform wall temperature boundary condition:

\[Nu = 0.836 De^{0.5} Pr^{0.1}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

boundaryint

Index of boundary condition

0 - Constant heat flux 1 - Uniform wall temperature

Returns:

Nufloat

Nusselt number, [-]

References

[36] Kalb, C.E., Seader, J.D.; Heat and Mass Transfer Phenomena for Viscous Flow in Curved Circular Tubes. Int. J. Heat Mass TRansfer 15() (1972) 801-817

[62] Kalb, C.E., Seader, J.D.; Fully Developed Viscous-Flow Heat Transfer in Curved Circular Tubes with Uniform Wall Temperature. AIChE J. 20(2) (1974) 340-346

equipment.widget.helical.Nu_laminar_Dravid(Re, Pr, di, Dc)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Dravid et al. (1971).

\[Nu = \left(0.76 + 0.65 De^{0.5}\right) Pr^{0.175}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[37] Dravid, A.N., Smith, K.A., Merrill, E.W., Brian, P.L.T.; Effect of Secondary Fluid Motion on Laminar Flow Heat Transfer in Helically Coiled Tubes. AIChE J. 17(5) (1971) 1114-1122

equipment.widget.helical.Nu_laminar_JanssenHoogendoorn(Re, Pr, di, Dc, f)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Janssen-Hoogendoorn (1978).

\[Nu = 0.6166 \left(f Re^2\right)^{0.26} Pr^{1/6}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

ffloat

Friction factor, [-]

Returns:

Nufloat

Nusselt number, [-]

References

[38] Janssen, L.A.M., Hoogendoorn, C.J.; Laminar Convective Heat Transfer in Helical Coiled Tubes. Int. J. Heat Mass Transfer 21(9) (1978) 1197-1206

equipment.widget.helical.Nu_laminar_ManlapazChurchill(Re, Pr, di, Dc, p)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Manlapaz-Churchill (1981).

\[Nu = \left(\left(3.657 + \frac{4.343}{\left(1+\frac{957}{Pr He^2} \right)^2}\right)^3 + 1.158 \left(\frac{He}{1+\frac{0.477}{Pr}}\right) ^{1.5}\right)^{1/3}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat, optional

Pitch for twist of 2π radians (360º), [m]

Returns:

Nufloat

Nusselt number, [-]

References

[39] Manlapaz, R.L., Churchill, S.W.; Fully Developed Laminar Convection From a Helical Coil. Chem. Eng. Commun. 9 (1981) 185-200

equipment.widget.helical.Nu_laminar_Salimpour(Re, Pr, di, Dc, p)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Salimpour (2009).

\[Nu = 0.152De^{0.431}Pr^{1.06}\left(\frac{b}{2 \pi D_c}\right)^{-0.277}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat, optional

Pitch for twist of 2π radians (360º), [m]

Returns:

Nufloat

Nusselt number, [-]

References

[40] Salimpour, M.R.; Heat transfer coefficients of shell and coiled tube heat exchangers. Exp. Thermal Fluid Sci. 33(2) (2009) 203-207

equipment.widget.helical.Nu_laminar_PimentaCampos(Re, Pr, di, Dc)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Pimenta-Campos (2013).

\[Nu = \left(0.5 De^{0.481} - 0.465\right) Pr^{0.367}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[41] Pimenta, T.A., Campos, J.B.L.M.; Heat transfer coefficients from Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Int. J. Heat Mass Transfer 58 (2013) 676-690

equipment.widget.helical.Nu_laminar_Hardik(Re, Pr, di, Dc)

Calculates nusselt number for internal flow at constant heat flux

boundary condition of a helical coil in laminar flow using the method of Hardik et al. (2015).

\[Nu = 0.0456 Re^{0.8} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.16}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[43] Hardik, B.K., Baburajan, P.K., Prabhu, S.V.; Local heat transfer coefficient in helical coils with single phase flow. Int. J. Heat Mass Transf. 89 (2015) 522-538

equipment.widget.helical.Nu_laminar_Acharya(Re, Pr, di, Dc, AA=False)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Acharya et al. (2001)

For Pr > 1:

\[Nu = 0.67 Re^{0.5} Pr^{0.21} \left(\frac{d_i}{D_c}\right)^{0.13}\]

For Pr ≤ 1:

\[Nu = 0.69 Re^{0.5} Pr^{0.43} \left(\frac{d_i}{D_c}\right)^{0.13}\]

Include too correlations for alternate axis coil geometric configuration

For Pr > 1:

\[Nu = 0.7 Re^{0.5} Pr^{0.3} \left(\frac{d_i}{D_c}\right)^{0.18}\]

For Pr ≤ 1:

\[Nu = 0.7 Re^{0.5} Pr^{0.375} \left(\frac{d_i}{D_c}\right)^{0.18}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

AAboolean

Set alternalte axis configuraiton for helical coil

Returns:

Nufloat

Nusselt number, [-]

References

[53] Acharya, N., Sen, M., Chang, H.-C.; Analysis of heat transfer enhancement in coiled-tube heat exchangers. Int. J. Heat Mass Transfer 44(17) (2001) 3189-3199

equipment.widget.helical.Nu_laminar_AkiyamaCheng(Re, Pr, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Akiyama-Cheng (1971)

\[\frac{Nu}{Nu_o} = 0.181 Q \left(1 - 0.839Q^{-1} + 35.4Q^{-2} - 207Q^{-3} + 419Q^{-4}\right)\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[54] Akiyama, M., Chen g, K.C.; Boundary Vorticity Method for Lamniar Forced Convection Heat Transfer in Curved Pipes. Int. J. Heat Mass Transfer 14(10) (1971) 1659-1675

equipment.widget.helical.Nu_laminar_Moawed(Re, do, Dc, p)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Moawed (2011)

\[Nu = 0.0345 Re^{0.48} \left(\frac{D_c}{d_o}\right)^{0.914} \left(\frac{p}{d_o}\right)^{0.281}\]

Parameters:

Refloat

Reynolds number, [-]

dofloat

Outer diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat, optional

Pitch for twist of 2π radians (360º), [m]

This correlation use external diameter of pipe, without Prandtl dependence

Returns:

Nufloat

Nusselt number, [-]

References

[55] Moawed, M.; Experimental study of forced convection from helical coiled tubes with different parameters. Energy Conv. Management 52(2) (2011) 1150-1156

equipment.widget.helical.Nu_laminar_NaphonWongwises(Re, Pr, di, Dc)

Calculates friction factor for internal flow of a helical coil with a

spiral configuration in laminar flow using the method of Naphon-Wongwises (2002)

\[Nu = 27.358 De^{0.287} Pr^{-0.949}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[63] Naphon, P., Wongwises, S.; An Experimental Study on the In-Tube Convective Heat Transfer Coefficients in a Spiral Coil Heat Exchanger. Int. Comm. Heat Mass Transfer 29(6) (2002) 797-809

equipment.widget.helical.Nu_laminar_Rainieri(Re, Pr, di, Dc, corrugated)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Rainieri et al. (2013)

\[Nu = 1.168 De^{0.47} Pr^{0.16}\]

For corrugated helical pipe:

\[Nu = 0.0191 De^{1.36} Pr^{0.2}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

corrugatedfloat, optional

Use correlation for corrugated wall tube

Returns:

Nufloat

Nusselt number, [-]

References

[56] Rainieri, S., Bozzoli, F., Cattani, L., Pagliarini, G.; Compound convective heat transfer enhancement in helically coiled wall corrugated tubes. Int. J. Heat Mass Transfer 59 (2013) 353-362

equipment.widget.helical.Nu_laminar_Ayuob(Re, Pr, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Ayuob et al. (2022)

\[Nu = 0.1868 M^{0.6958} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.1703}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[67] Ayuob, S., Mahmood, M., Ahmad, N., Waqas, A., Saeed, H., Sajid, M.B.; Development and validation of Nusselt number correlations for a helical coil based energy storage integrated with solar water heating system. J. Energy Storage 55 (2022) 105777

equipment.widget.helical.Nu_laminar_BergBonilla(Re, Pr, di, Dc)

Calculates friction factor for internal flow of a helical coil in

laminar flow using the method of Berg-Bonilla (1950)

\[Nu = \left(0.0000229 + 0.000636 \frac{d_i}{D_c}\right) Re^{1.29} Pr\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[68] Berg, R.R., Bonilla, C.F.; Heating of fluids in coils. NY Academic Sciences 13 (1950) 12-18

equipment.widget.helical.Nu_turbulent_MandalNigam(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Mandal-Nigam (2009).

\[Nu = 0.55 De^{0.637} Pr^{0.4}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[25] Mandal, M. M., Nigam, K.D.P.; Experimental Study on Pressure Drop and Heat Transfer of Turbulent Flow in Tube in Tube Helical Heat Exchanger. Ind. Eng. Chem. Res. 48(20) (2009) 9318-9324

equipment.widget.helical.Nu_turbulent_RogersMayhew(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Rogers-Mayhew (1964).

\[Nu = 0.023 Re^{0.85} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.1}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[44] Rogers, G.F.C., Mayhew, Y.R.; Heat Transfer and Pressure Loss in Helically Coiled Tubes with Turbulent Flow. Int. J. Heat Mass Transfer 7(11) (1964) 1207-1216

equipment.widget.helical.Nu_turbulent_Shchukin(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Shchukin (1969) as show in [1]

For math:Re (d_i/D_c)^2 < 20

\[Nu = 0.0316 Re^{0.8} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.05}\]

For math:Re (d_i/D_c)^2 > 20

\[Nu = 0.0266 Re^{0.85} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{0.15}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[44] Rogers, G.F.C., Mayhew, Y.R.; Heat Transfer and Pressure Loss in Helically Coiled Tubes with Turbulent Flow. Int. J. Heat Mass Transfer 7(11) (1964) 1207-1216

[1] El-Genk, M.S., Timothy, M.S.; A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes. Heat Transfer Eng. 38(5) (2017) 447-474

equipment.widget.helical.Nu_turbulent_Guo(Re, Pr)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Guo (1998).

\[Nu = 0.023 Re^{0.58} Pr^{0.4}\]

This correlation don’t include any helical coil geometrical parameters dependence.

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

Returns:

Nufloat

Nusselt number, [-]

References

[48] Guo, L., Chen, X., Feng, Z., Bai, B.; Transie:nt convective heat transfer in a helical coiled tube with pulsatile fully developed turbulent flow. Int. J. Heat Mass Transfer 41() (1998) 2867-2875

equipment.widget.helical.Nu_turbulent_Jayakumar(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Jayakumar et al. (2008)

\[Nu = 0.025 De^{0.9112} Pr^{0.4}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[50] Jayakumar, J.S., Mahajani, S.M., Mandal, J.C., Vijayan, P.K., Bhoi, R.; Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chem. Eng. Res. Design 86(3) (2008) 221-232

equipment.widget.helical.Nu_turbulent_Yildiz(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Yildiz et al. (1997)

\[Nu = 0.0551 De^{0.864} Pr^{0.4}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[51] Yildiz, C., Biçer, Y., Pehlivan, D.; Heat Transfer and Pressure Drop in a Heat Exchanger with a Helical Pipe Containing Inside Springs. Energy Convers. Management 38(6) (1997) 619-624

equipment.widget.helical.Nu_turbulent_Wu(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Wu et al. (2025)

\[Nu = 0.023 Re^{0.759} Pr^{0.4} \left(\frac{d_i}{D_c}\right)^{-0.079}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[52] Wu, Z., Li, K., Zhang, K., Tian, W.; Single-phase flow heat transfer characteristics in helical coils with large coil diameters. Appl. Thermal Eng. 266 (2025) 125776

equipment.widget.helical.Nu_turbulent_JhaRajaRao(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Jha et al. (1967)

\[\frac{Nu}{Nu_s} = 1 + 3.46 \frac{d_i}{D_c}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[57] Jha, R.K., Raja Rao, M.; Heat Transfer Through Coiled Tubes in Agitated Vessels. Int. J. Heat Mass Transfer 10(3) (1967) 395-397

equipment.widget.helical.Nu_turbulent_Jeschke(Re, Pr, di, Dc)

Calculates nusselt number for internal flow of a helical coil in

turbulent flow using the method of Jeschke (1925) as shown in [66]

\[Nu = \left(0.039+0.138 \frac{d_i}{D_c}\right) \left(Re Pr\right)^{0.76}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

Returns:

Nufloat

Nusselt number, [-]

References

[66] Jeschke, H.; Wärmeübergang und Druckverlust in Rohrschlangen. VDI Z. 69 (1925) 24-28

[65] Zhao, H., Li, X., Wu, Y., Wu, X.; Friction factor and Nusselt number correlations for forcedconvection in helical tubes. Int. J. Heat Mass Transfer 155 (2020) 119759

equipment.widget.helical.Nu_turbulent_Zheng(Re, Pr, di, Dc, p)

Calculates friction factor for internal flow of a helical coil in

turbulent flow using the method of Zheng et al. (2023).

\[f_c = \frac{0.0791}{Re^{0.25} + \frac{81858}{Re^{1.54}} \left(\frac{d_i}{D_{cm}}\right)^{0.48}\]

D_cm is defined as:

\[D_{cm} = D_c \left(1+\tan{\alpha}\right)\]

α is the helix angle:

\[\alpha = \tan^{-1}{\frac{p}{\pi D_c}}\]

Parameters:

Refloat

Reynolds number, [-]

Prfloat

Prandtl number, [-]

difloat

Internal diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat

Pitch for twist of 2π radians (360º), [m]

Returns:

Nufloat

Nusselt number, [-]

References

[64] Zheng, X., Lu, X., Gao, Y., Jin, D., Hu, Y., Hu, Y., Mao, Y.; Experimental study on friction pressure drop and circumferential heat transfer characteristics in helical tubes. Front. Energy Res. 11 (2023) 1204850.

class equipment.widget.helical.HelicalCoil(**kwargs)

Bases: CallableEntity

Helical coil tube used as anhancing heat transfer equipment.

Parameters:

difloat

Inner diameter of the pipe, [m]

Dcfloat

Diameter of the helix, [m]

pfloat, optional

Pitch for twist of 2π radians (360º), [m]

boundaryint, optional

Set boundary condition in correlation with several implmented

0 - Constant heat flux 1 - Uniform wall temperature

MoriSimpleboolean, optional

Use Simple correlation for Mori-Nakayama nusselt number correlation

AAboolean, optional

Use alternate axis configuration for Acharya nusselt number correlation

corrugatedboolean, optional

Use Rainieri correlation for corrugated wall

eDfloat, optional

Relative roughness of pipe, used in Zhao correlation for turbulent friction factor, [-]

Attributes:

ReCritical

Calculate critical Reynolds number to define transition of regimen

isCalculable

Check if all input are defined

Methods

Nu(Re, Pr)	Calculate nusselt number
calculo()	Definition of twisted tape inserts for annuli sections
f(Re)	Calculate friction factor

inputChanged
valueChanged

TEXT_REYNOLDS_CRITICAL = ('Ito (1959)', 'Schmidt (1967)', 'Kubair-Kuloor (1966)', 'Srinivasan (1968)', 'Kutateladze (1966)', 'Seth-Stahel (1969)', 'Cioncolini-Santini (2006)')

TEXT_LAMINAR_FRICTION = ('Schmidt (1967)', 'White (1929)', 'Mori-Nakayama (1965)', 'Hart (1988)', 'Ju (2001)', 'Mishra-Gupta (1979)', 'Manlapaz-Churchill (1980)', 'Prasad (1989)', 'Liu-Masliyah (1993)', 'Ali (2001)', 'Ito (1969)', 'Tarbell-Samuels (1973)', 'Pimenta-Campos (2012)', 'Adler (1934)', 'Barua (1963)', 'Yanase (1989)', 'Dennis (1980)', 'van Dyke (1978)', 'Collins-Dennis (1975)', 'Dean (1928)', 'Abushammala (2019)', 'ElGenk-Schriener (2017)', 'Srinivasan (1968)', 'Gupta (2011)', 'Hasson (1955)', 'Rao-Sadasivudu (1974)')

TEXT_TURBULENT_FRICTION = ('Schmidt (1967)', 'Mori-Nakayama (1965)', 'Ju (2001)', 'Mishra-Gupta (1979)', 'Czop (1994)', 'Prasad (1989)', 'Ali (2001)', 'Guo (2001)', 'Mandal-Nigam (2009)', 'ElGenk-Schriener (2017)', 'Srinivasan (1968)', 'Ito (1959)', 'Zhao (2016)', 'Das (1993)', 'Zheng (2023)', 'Rao-Sadasivudu (1974)')

TEXT_LAMINAR_HEAT = ('Schmidt (1967)', 'Xin-Ebadian (1997)', 'Mori-Nakayama (1965)', 'Seban-McLaughlin (1963)', 'Prasad (1989)', 'Kalb-Seader (1972)', 'Dravid (1971)', 'Janssen-Hoogendoorn (1978)', 'Manlapaz-Churchill (1981)', 'Salimpour (2009)', 'Pimenta-Campos (2013)', 'Pawar-Sunnapwar (2013)', 'Hardik (2015)', 'ElGenk-Schriener (2017)', 'Acharya (2001)', 'Akiyama-Cheng (1971)', 'Moawed (2011)', 'Rainieri (2013)', 'Naphon-Wongwises (2002)', 'Zhou (2020)', 'Ayuob (2022)', 'Berg-Bonilla (1950)')

TEXT_TURBULENT_HEAT = ('Schmidt (1967)', 'Xin-Ebadian (1997)', 'Mori-Nakayama (1965)', 'Seban-McLaughlin (1963)', 'Prasad (1989)', 'Mandal-Nigam (2009)', 'Rogers-Mayhew (1964)', 'Pawar-Sunnapwar (2013)', 'ElGenk-Schriener (2017)', 'Shchukin (1969)', 'Guo (1998)', 'Jayakumar (2008)', 'Yildiz (1997)', 'Wu (2025)', 'Jha (1967)', 'Zheng (2023)', 'Zhou (2020)', 'Jeschke (1925)')

TEXT_BOUNDARY = ('Constant heat flux', 'Uniform wall temperature')

status = 0

msg = ''

kw = {'AA': False, 'Dc': 0, 'MoriSimple': False, 'boundary': 0, 'corrugated': False, 'di': 0, 'eD': 0, 'methodFrictionLaminar': 0, 'methodFrictionTurbulent': 0, 'methodHeatLaminar': 0, 'methodHeatTurbulent': 0, 'methodReCritic': 0, 'p': 0}

valueChanged

pyqtSignal(*types, name: str = …, revision: int = …, arguments: Sequence = …) -> PYQT_SIGNAL

types is normally a sequence of individual types. Each type is either a type object or a string that is the name of a C++ type. Alternatively each type could itself be a sequence of types each describing a different overloaded signal. name is the optional C++ name of the signal. If it is not specified then the name of the class attribute that is bound to the signal is used. revision is the optional revision of the signal that is exported to QML. If it is not specified then 0 is used. arguments is the optional sequence of the names of the signal’s arguments.

inputChanged

pyqtSignal(*types, name: str = …, revision: int = …, arguments: Sequence = …) -> PYQT_SIGNAL

types is normally a sequence of individual types. Each type is either a type object or a string that is the name of a C++ type. Alternatively each type could itself be a sequence of types each describing a different overloaded signal. name is the optional C++ name of the signal. If it is not specified then the name of the class attribute that is bound to the signal is used. revision is the optional revision of the signal that is exported to QML. If it is not specified then 0 is used. arguments is the optional sequence of the names of the signal’s arguments.

property isCalculable

Check if all input are defined

calculo()

Definition of twisted tape inserts for annuli sections

property ReCritical

Calculate critical Reynolds number to define transition of regimen flow from laminar to turbulent

Nu(Re, Pr)

Calculate nusselt number

f(Re)

Calculate friction factor

class equipment.widget.helical.UI_Helical(parent=None)

Bases: ToolGui

Helical coil dialog

Methods

loadUI()	Add widget
setVisibleMod()	Enable widget with special parameters for selected method

title = 'Use helical coil'

loadUI()

Add widget

setVisibleMod()

Enable widget with special parameters for selected method

class equipment.widget.helical.Dialog(parent=None)

Bases: QDialog

Component list config dialog

__init__(parent=None)

References

[1] (1,2,3,4,5,6)

El-Genk, M.S., Timothy, M.S.; A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes. Heat Transfer Eng. 38(5) (2017) 447-474

[2] (1,2,3,4)

; Perry’s Chemical Engineers’ Handbook 9th Edition. McGraw-Hill (2019)

[3] (1,2,3)

Schmidt, E.F.; Wärmeübergand und Druckverlust in Rohrschlangen. Chemie Ingenieur Technik 39(13) (1967) 781-789

[4] (1,2)

Ito, H.; Friction Factors for Turbulent Flow in Curved Pipes. J. Basic Eng. 81 (1959) 123-134

[5]

Kubair, V., Kuloor, N.R.; Heat Transfer to Newtonian Fluids in Coiled Pipes in Laminar Flow. Int. J. Heat Mass Transfer 9 (1966) 63-75

[6] (1,2)

Srinivasan, P.S., Nandapurkar, S.S., Holland, F.A.; Pressure Drop and Heat Transfer in Coils. Chem. Eng. 218 (1968) 113-119

[7]

Kutateladze, S.S., Borishanskii, V.M. ; A Concise Encyclopedia of Heat Transfer. Pergamon Press (1966)

[8]

White, C.M.; Streamline Flow through Curved Pipes. Proc. R .Soc. London A 123 (1929) 645-663

[9] (1,2)

Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (1st Report, Laminar Region). Int. J. Heat Mass Transfer 8(1) (1965) 67-82

[10] (1,2)

Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (2nd Report, Turbulent Region). Int. J. Heat Mass Transfer 10(1) (1967) 37-59

[11]

Hart, J., Ellenberger, J., Hamersma, P.J.; Single- and Two-Phase Flow Through Helically Coiled Tubes. Chem. Eng. Sci. 43(4) (1988) 775-783

[12]

Ju, H., Huang, Z., Xu, Y., Duan, B, Yu, Y.; Hydraulic Performance of Small Bending Radius Helical Coil-Pipe. J. Nuclear Sci. Eng. 38(10) (2001) 826-831

[13]

Mishra, P., Gupta, S.N.; Momentum Transfer in Curved Pipes. 1. Newtonian Fluids. Ind. Eng. Chem. Process Des. Dev. 18(1) (1979) 130-137

[14]

Czop, V., Barbier, D., Dong, S.; Pressure drop, void fraction and shear stress measurements in an adiabatic two-phase flow in a coiled tube. Nuclear Eng. Design 149 (1994) 323-333

[15]

Xin, R.C., Ebadian, M.A. ; The Effects of Prandtl Numbers on Local and Average Convective Heat Transfer Characteristics in Helical Pipes. J. Heat Transfer 119(3) (1997) 467-73

[16]

Seban R.A., McLaughlin, E.F.; Heat Transfer in Tube Coils with Laminar and Turbulent Flow. Int. J. Heat Mass Transfer 6() (1963) 387-395

[17]

Manlapaz, R.L., Churchill, S.W.; Fully Developed Laminar Flow in a Helically Coiled Tube of Finite Pitch. Chem. Eng. Communications 7 (1980) 57-78

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Prasad, B.V.S.S.S., Das, D.H., Prabhaker, A.K.; Pressure Drop, Heat Transfer and Performance of a Helically Coiled Tubular Exchanger. Heat Recovery Systems & CHP 9(3) (1989) 249-256

[19]

Liu, S., Masliyah, J.H.; Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 251 (1993) 315-353

[20]

Seth, K.K., Stahel, E.P.; Heat Transfer from Helical Coils Immersed in Agitated Vessels. Ind. Eng. Chem. 61(6) (1969) 39-49

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Ali, S.; Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Research 28 (2001) 295-310

[22]

Guo, L., Feng, Z, Chen, X.; An experimental investigation of the frictional pressure drop of steam–water two-phase flow in helical coils. Int. J. Heat Mass Transfer 44(14) (2001): 2601-2610

[23]

Ito, H.; Laminar Flow in Curved Pipes. Z. Angew. Math. Mech. 11 (1969) 653-663

[24]

Tarbell, J.M., Samuels, M.R.; Momentum and Heat Transfer in Helical Coils. Chem. Eng. J. 5(2) (1973) 117-127

[25] (1,2)

Mandal, M. M., Nigam, K.D.P.; Experimental Study on Pressure Drop and Heat Transfer of Turbulent Flow in Tube in Tube Helical Heat Exchanger. Ind. Eng. Chem. Res. 48(20) (2009) 9318-9324

[26]

Ciencolini, A., Santini, L.; An experimental investigation regarding the laminar to turbulent flow transition in helically coiled pipes. Exp. Thermal Fluid Sci. 30 (2006) 367-380

[27]

Adler, M.; Strömung in gekrümmten Rohren. Z. Angew. Math. Mech. 14(5) 257-275

[28]

Barua, S.N.; On Secondary Flow in Stationary Curved Pipes. Quart. J. Mech. Appl. Math. 16(1) (1963) 61-77

[29]

Pimenta, T.A., Campos, J.B.L.M.; Friction losses of Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Exp. Thermal Fluid Sci. 36 (2012) 194-204

[30]

Yanase, S., Goto, N., Yamamoto, K.; Dual solutions of the flow through a curved tube. Fluid Dyn. Research 5 (1989) 191-201

[31]

Dennis, S.C.R.; Calculation of the steady flow through a curved tube using a new finite-difference method. J. Fluid Mech. 99(3) (1980) 449-467

[32]

Van Dyke, M.; Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86(1) 129-145

[33]

Collins, W.M., Dennis, S.C.R.; The Steady Motion of a Viscous Fluid in a Curved Tube. Q. J. Mech. Appl. Math. 28(2) (1975) 133-156

[34]

Dean, W.R.; The stream-line motion of fluid in a curved pipe (Second paper). London Edinburgh Dublin Phil. Mag. J. Sci. Serie 7 5(30) (1928) 673-695

[35]

Abushammala, O., Hreiz, R., Lamaître, C., Favre, E.; Laminar flow friction factor in highly curved helical pipes: Numerical investigation, predictive correlation and experimental validation using a 3D-printed model. Chem. Eng. Sci. 207(7) (2019) 1030-1039

[36]

Kalb, C.E., Seader, J.D.; Heat and Mass Transfer Phenomena for Viscous Flow in Curved Circular Tubes. Int. J. Heat Mass TRansfer 15() (1972) 801-817

[37]

Dravid, A.N., Smith, K.A., Merrill, E.W., Brian, P.L.T.; Effect of Secondary Fluid Motion on Laminar Flow Heat Transfer in Helically Coiled Tubes. AIChE J. 17(5) (1971) 1114-1122

[38]

Janssen, L.A.M., Hoogendoorn, C.J.; Laminar Convective Heat Transfer in Helical Coiled Tubes. Int. J. Heat Mass Transfer 21(9) (1978) 1197-1206

[39]

Manlapaz, R.L., Churchill, S.W.; Fully Developed Laminar Convection From a Helical Coil. Chem. Eng. Commun. 9 (1981) 185-200

[40]

Salimpour, M.R.; Heat transfer coefficients of shell and coiled tube heat exchangers. Exp. Thermal Fluid Sci. 33(2) (2009) 203-207

[41]

Pimenta, T.A., Campos, J.B.L.M.; Heat transfer coefficients from Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Int. J. Heat Mass Transfer 58 (2013) 676-690

[42]

Pawar, S.S., Sunnapwar, V.K.; Studies on convective heat transfer through helical coils. Heat Mass Transfer 49(12) (2013) 1741-1754

[43]

Hardik, B.K., Baburajan, P.K., Prabhu, S.V.; Local heat transfer coefficient in helical coils with single phase flow. Int. J. Heat Mass Transf. 89 (2015) 522-538

[44] (1,2)

Rogers, G.F.C., Mayhew, Y.R.; Heat Transfer and Pressure Loss in Helically Coiled Tubes with Turbulent Flow. Int. J. Heat Mass Transfer 7(11) (1964) 1207-1216

[45]

Pawar, S.S., Sunnapwar, V.K.; Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow. Exp. Thermal Fluid Sci. 44 (2013) 792-804

[46]

Mori, Y., Nakayama, W.; Study on Forced Convective Heat Transfer in Curved Pipes (3rd Report, Theoretical Analysis under the Condition of Uniform Wall Temperature and Practical Formulae). Int. J. Heat Mass Transfer 10(5) (1967) 681-695

[47]

Shchukin, V.K.; Correlation of Experimental Data on Heat Transfer in Curved Pipes. Teploenergetika 16(2) (1969) 72-76

[48]

Guo, L., Chen, X., Feng, Z., Bai, B.; Transie:nt convective heat transfer in a helical coiled tube with pulsatile fully developed turbulent flow. Int. J. Heat Mass Transfer 41() (1998) 2867-2875

[49] (1,2)

Ghobadi, M., Muzychka, Y.S.; A Review of Heat Transfer and Pressure Drop Correlations for Laminar Flow in Curved Circular Ducts. Heat Transfer Eng. 37(10) (2016) 815-839

[50]

Jayakumar, J.S., Mahajani, S.M., Mandal, J.C., Vijayan, P.K., Bhoi, R.; Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chem. Eng. Res. Design 86(3) (2008) 221-232

[51]

Yildiz, C., Biçer, Y., Pehlivan, D.; Heat Transfer and Pressure Drop in a Heat Exchanger with a Helical Pipe Containing Inside Springs. Energy Convers. Management 38(6) (1997) 619-624

[52]

Wu, Z., Li, K., Zhang, K., Tian, W.; Single-phase flow heat transfer characteristics in helical coils with large coil diameters. Appl. Thermal Eng. 266 (2025) 125776

[53]

Acharya, N., Sen, M., Chang, H.-C.; Analysis of heat transfer enhancement in coiled-tube heat exchangers. Int. J. Heat Mass Transfer 44(17) (2001) 3189-3199

[54]

Akiyama, M., Chen g, K.C.; Boundary Vorticity Method for Lamniar Forced Convection Heat Transfer in Curved Pipes. Int. J. Heat Mass Transfer 14(10) (1971) 1659-1675

[55]

Moawed, M.; Experimental study of forced convection from helical coiled tubes with different parameters. Energy Conv. Management 52(2) (2011) 1150-1156

[56]

Rainieri, S., Bozzoli, F., Cattani, L., Pagliarini, G.; Compound convective heat transfer enhancement in helically coiled wall corrugated tubes. Int. J. Heat Mass Transfer 59 (2013) 353-362

[57]

Jha, R.K., Raja Rao, M.; Heat Transfer Through Coiled Tubes in Agitated Vessels. Int. J. Heat Mass Transfer 10(3) (1967) 395-397

[58]

Gupta, R., Wanchoo, R.K., Jafar Ali, T.R.M.; Laminar Flow in Helical Coils: A Parametric Study. Ind. Eng. Chem. Res. 50(2) (2011) 1150-1157

[59]

Hasson, D.; Streamline flow resistance in coils. Res. Corresp. 1 S1 (1955).

[60]

Zhao, H., Li, X., Wu, X.; New friction factor equations developed for turbulent flow in rough helical tubes. Int. J. Heat Mass Transfer 95 (2016) 525-534

[61]

Das, S.K.; Water Flow Through Helical Coils in Turbulent Condition. Can. J. Chem. Eng. 71 (1993) 971-973

[62]

Kalb, C.E., Seader, J.D.; Fully Developed Viscous-Flow Heat Transfer in Curved Circular Tubes with Uniform Wall Temperature. AIChE J. 20(2) (1974) 340-346

[63]

Naphon, P., Wongwises, S.; An Experimental Study on the In-Tube Convective Heat Transfer Coefficients in a Spiral Coil Heat Exchanger. Int. Comm. Heat Mass Transfer 29(6) (2002) 797-809

[64] (1,2)

Zheng, X., Lu, X., Gao, Y., Jin, D., Hu, Y., Hu, Y., Mao, Y.; Experimental study on friction pressure drop and circumferential heat transfer characteristics in helical tubes. Front. Energy Res. 11 (2023) 1204850.

[65] (1,2)

Zhao, H., Li, X., Wu, Y., Wu, X.; Friction factor and Nusselt number correlations for forcedconvection in helical tubes. Int. J. Heat Mass Transfer 155 (2020) 119759

[66] (1,2)

Jeschke, H.; Wärmeübergang und Druckverlust in Rohrschlangen. VDI Z. 69 (1925) 24-28

[67]

Ayuob, S., Mahmood, M., Ahmad, N., Waqas, A., Saeed, H., Sajid, M.B.; Development and validation of Nusselt number correlations for a helical coil based energy storage integrated with solar water heating system. J. Energy Storage 55 (2022) 105777

[68]

Berg, R.R., Bonilla, C.F.; Heating of fluids in coils. NY Academic Sciences 13 (1950) 12-18

[69]

Rao, M.V.R., Sadasividu, D.; Pressure drop studies in helical coils. Indian J. Tech. 12 (1974) 473-474