Triangular ducts differ from circular tubes in several important aspects. Because the duct periphery is not symmetrical to the flow field, the transition from laminar to turbulent flow follows a more complicated process. Also, because of this asymmetry, additional thermal boundary conditions can exist and thus make the determination of the heat transfer more difficult.

The transition process in isosceles triangular ducts has been investigated experimentally by Eckert and Irvine (1956) and later by Bandopadhayay and Hindwood (1973). Both studies reported that Turbulence first appears in the wide area of the cross section near the triangle base and with increasing Reynolds Number spreads toward the narrow apex region. Figure 1, which shows the flow visualization measurements of Eckert and Irvine (1956) for an isosceles triangular duct with an apex angle of 11.5 degrees, illustrates this transition process. For example, from Figure 1, at a Reynolds number of 2000 (based on the hydraulic diameter), approximately 25% of the duct, measured along the center line, contains laminar flow while the remaining 75% is turbulent. Also seen in the figure is that at a Reynolds number of 800, the flow is half laminar and half turbulent. Even at a Reynolds number of 8000, almost 10% of the "apex" corner flow is laminar.

**Figure 1. Determination of the laminar zone in an isosceles triangular duct. From Eckert, E. R. G. and Irvine, T. F. Jr (1965). Flow in Corners of Passages with Non-Circular Cross Sections, Trans. Am. Soc. Mech. Engrs. pp. 709–718, with permission.**

Such considerations are important since they indicate that the local heat transfer can be relatively poor in the apex region. This in turn can cause local "hot spots" in the corners whose prediction can be critical in the design of heat exchanger passages.

The classical thermal boundary conditions for round tubes are either a constant wall temperature or a constant wall heat flux. Their Nusselt Numbers are indicated by Nu_{T} and Nu_{H}, respectively. Nothing needs to be said about the thermal conditions around the perimeter because of symmetry, i.e., no peripheral wall temperature gradients can exist. For triangular ducts, however, peripheral wall temperature variations can exist for the constant heat flux boundary condition. This in turn leads to three generally accepted thermal boundary conditions which are illustrated in Table 1.

**Table 1. Triangular duct thermal boundary conditions**

Boundary condition | Symbol |

Constant wall temperature in the flow and peripheral directions. |
Nu_{T} |

Constant heat flux in the flow direction and constant peripheral wall temperature (thick wall duct). |
Nu_{H1} |

Constant heat flux in the flow direction and constant peripheral heat flux (thin wall duct). |
Nu_{H2} |

Constant heat flux in the flow direction with peripheral heat conduction in the wall (an actual duct). |
Nu_{H4} |

In the following sections, information is presented on the steady state pressure drop and heat transfer for fully developed constant property flow in triangular ducts for both laminar and turbulent conditions. For information on such topics as hydrodynamic and thermal entrance lengths, further readings are suggested.

The pressure drop characteristics of fully developed duct flows are normally given as the relation between the Reynolds Number and the Friction Factor (see Nomenclature). For fully developed laminar duct flow, it can be shown that the product of these two quantities is a constant depending on the duct geometry. Thus it is sufficient to tabulate the constant for various geometries.

Shaw and London (1978) have presented a most useful compendium of laminar flow and heat transfer data for different ducts including triangular ducts. Table 2, taken from their work, lists the friction factor Reynolds number product for isosceles triangular ducts as a function of the duct apex angle, 2α. Also included in Table 2 are heat transfer data which will be discussed later. Shaw and London (1978) also include calculated information on hydrodynamic entrance lengths and excess entrance pressure drops.

For fully developed turbulent flow, an additional complication arises in the flow field. Because of the nonisotropic Reynolds stress distribution around the peripheral wall, secondary flows occur which complicate an analytical approach to the problem. Nevertheless, both analytical and experimental results are available in the literature which are summarized below.

There have been a number of experimental studies of triangular duct turbulent friction factors. These would of course include both the transition and secondary flow effects mentioned above. In addition, several analysis have been reported only one of which, Malak et al. (1975), included a correlation which accounts for the effect of apex angle on friction factor.

A convenient way to correlate the turbulent friction factor data, both experimental and theoretical, is by a *modified Blasius equation*:

Carlson and Irvine (1961) reported that the constant "C" in Eq. (1) is a weak function of the Reynolds number and depends primarily on the apex angle.

Figure 2 presents the results of ten experiments on isosceles triangular ducts where the constant "C" is plotted against the apex angle. Also shown are the theoretical (numerical) results of Malak et al. (1975) and Usui et al. (1983). The later considered the effects of secondary flows while the former did not. In a separate study Kokorev et al. (1971) estimated that the effects of secondary flows on the friction factor was less than 10% for isosceles triangular ducts.

**Figure 2. Experimental results and theoretical predictions of the constant "C" plotted against the apex angle 2α.**

As seen in Figure 2, for apex angles greater than 15 degrees, the experimental data for "C" are greater than the analyses but are less than the analyses for apex angles less than 15 degrees. It is recommended that the solid line which represents the experimental data be used in calculating turbulent friction factors.

The three Nusselt numbers Nu_{T}, Nu_{H1}, and Nu_{H2} defined in Table 1 are listed in Table 2 for isosceles triangular ducts for apex angles from 0 to 180 degrees. It can be seen from the table that the Nusselt numbers for the constant heat flux conditions are quite sensitive to the thermal boundary conditions, i.e., whether the Nu_{H1} or Nu_{H2} condition is specified. For example, for small apex angles, e.g., 7.15 degrees, the two Nusselt numbers differ by a factor of 60. Thus, extreme caution must be exercised in choosing the proper thermal boundary conditions for a particular design calculation.

Actual triangular ducts have thermal boundary conditions that are best represented by Nu_{H4} since there may be significant wall heat conduction. These situations have been investigated by Irvine and Cheng (1993) and further information on the Nu_{H4} boundary condition may be obtained from that reference.

Of all the physical processes involved in triangular duct flow and heat transfer, the least attention has been directed toward the problem of predicting the heat flow under turbulent flow conditions. Based on heat transfer in circular tubes, it is probable that triangular duct heat transfer in turbulent flow is less sensitive to the thermal boundary conditions than is the case for laminar flow. However, this lack of sensitivity has not been investigated experimentally.

Heat transfer experiments have been reported by Eckert and Irvine (1960) on an 11.46 degree isosceles triangular duct under the NuH4 boundary condition. Unfortunately, they did not have fully developed thermal flow and their main contribution was to show that using circular tube correlations with the hydraulic diameter as the characteristic dimension in the Nusselt and Reynolds numbers was not appropriate.

Altemani and Sparrow (1980) performed experiments on an equilateral triangular duct with the NuH1 boundary condition. They also measured thermal entrance lengths. On the basis of their measurements, they proposed that for fully developed flow the *Petukhov-Popov (1972) correlation* for a round tube be used with the friction factor calculated from Eq. 1. This correlation is given by:

Equation (2) is recommended at the present time for triangular ducts along with a healthy skepticism regarding the results.

It is impossible to report in more detail the flow and heat transfer aspects of triangular ducts in the present article. Readers are directed to the appropriate references for further information and topics not considered in this presentation.

C | Constant in Equation 1 (–) |

D_{h} | Hydraulic Diameter = |

f | Friction Factor = |

L | Duct height in Figure 1 (m) |

Pr | Prandtl number = |

Re | Reynolds Number = |

Average velocity (m/s) | |

x | Centerline coordinate in Fig. 1 (m) |

z | Flow direction coordinate (m) |

α Half apex angle (degrees)

κ Thermal diffusivity (m^{2}/s)

ν Kinematic viscosity (m^{2}/s)

ρ Density (kg/m^{3})

#### REFERENCES

Altemani, C. A. C. and Sparrow, E. M. (1980) Turbulent heat transfer and fluid flow in an unsymmetrically heated triangular duct, *J. Heat Trans.*, 102, 590–597.

Bandopadhayay, P. C. and Hindwood, J. B. (1973) On the coexistence of the laminar and turbulent flow in a narrow triangular duct, *J. Fluid Mechs.*, 59, 775–784.

Carlson, L. W. and Irvine, T. F. Jr. (1961) Fully developed pressure drop in triangular shaped ducts, *J. Heat Trans.*, 83, 441–444.

Chegini, H. and Chaturvedi, S. K. (1986) An experimental and analytical investigation of friction factors for fully developed flow in internally finned triangular ducts, *J. Heat Trans.*, 108, 507–512.

Eckert, E. R. G. and Irvine, T. F. Jr. (1956) Flow in corners of passages with non-circular cross sections, *Trans. Am. Soc. Mech. Engrs.*, 78,709–718.

Eckert, E. R. G. and Irvine, T. F. Jr. (1960) Pressure drop and heat transfer in a duct with triangular cross section, *J. Heat Trans.*, 82, 125–135.

Irvine, T. F. Jr. and Cheng J. A. (1993) Conjugated laminar heat transfer for Newtonian and power law fluids in triangular ducts, *J. Energy Heat Mass Trans.*, 15, 107–113.

Kokorev, L. S. et al. (1971) Effect of secondary flows on the velocity distribution and hydraulic drag in turbulent liquid flows in non-circular channels, *Heat Trans. Sov. Res.*, 3, 66–78.

Leonhardt, W. J. and Irvine, T. F. Jr. (1977) Experimental friction factors for fully developed flow of dilute aqueous polyethylene-oxide solutions in smooth wall triangular ducts, *Heat and Mass Transfer Source Book (Fifth All-Union Conference, Minsk)*, 236-250, Scripta Publishing Co. and John Wiley and Sons.

Malak, J. et al. (1975) Pressure losses and heat transfer in non-circular channels with hydraulically smooth walls, *Int. J. Heat and Mass Trans.*, 18, 139–148. DOI: 10.1016/0017-9310(75)90016-2

Schiller, L. (1923) Über den Stromungswiderstand von Rohen Verschiedenen Querschnitts und Rauhigkeitsgrades, *Zeit. Aug. Math und Mech.*, 2–10.

Shah, R. K. and London, A. L. (1978) Laminar flow forced convection in ducts, *Advances in Heat Transfer Supplement 1*, T. F. Irvine Jr. and J. P. Hartaett, eds., Academic Press.

Tung, S. S. and Irvine, T. F. Jr. (1979) Experimental study of the flow of a viscoelastic fluid in a narrow triangular duct, *Studies in Heat Transfer*, 309–329, Hemisphere Pub. Corp.

Usai, H. et al. (1982) Turbulence measurements and mass transfer in fully developed flow in a triangular duct with a narrow apex angle, *Int. J. Heat Mass Trans.*, 25, 615–623. DOI: 10.1016/0017-9310(82)90166-1

Usui, H. et al. (1983) Fully developed turbulent flow in isosceles triangular ducts, *J. Chem. Eng. Japan*, 16, 13–18.

#### References

- Altemani, C. A. C. and Sparrow, E. M. (1980) Turbulent heat transfer and fluid flow in an unsymmetrically heated triangular duct,
*J. Heat Trans.*, 102, 590â€“597. - Bandopadhayay, P. C. and Hindwood, J. B. (1973) On the coexistence of the laminar and turbulent flow in a narrow triangular duct,
*J. Fluid Mechs.*, 59, 775â€“784. DOI: 10.1017/S0022112073001862 - Carlson, L. W. and Irvine, T. F. Jr. (1961) Fully developed pressure drop in triangular shaped ducts,
*J. Heat Trans.*, 83, 441â€“444. - Chegini, H. and Chaturvedi, S. K. (1986) An experimental and analytical investigation of friction factors for fully developed flow in internally finned triangular ducts,
*J. Heat Trans.*, 108, 507â€“512. - Eckert, E. R. G. and Irvine, T. F. Jr. (1956) Flow in corners of passages with non-circular cross sections,
*Trans. Am. Soc. Mech. Engrs.*, 78,709â€“718. - Eckert, E. R. G. and Irvine, T. F. Jr. (1960) Pressure drop and heat transfer in a duct with triangular cross section,
*J. Heat Trans.*, 82, 125â€“135. - Irvine, T. F. Jr. and Cheng J. A. (1993) Conjugated laminar heat transfer for Newtonian and power law fluids in triangular ducts,
*J. Energy Heat Mass Trans.*, 15, 107â€“113. - Kokorev, L. S. et al. (1971) Effect of secondary flows on the velocity distribution and hydraulic drag in turbulent liquid flows in non-circular channels,
*Heat Trans. Sov. Res.*, 3, 66â€“78. - Leonhardt, W. J. and Irvine, T. F. Jr. (1977) Experimental friction factors for fully developed flow of dilute aqueous polyethylene-oxide solutions in smooth wall triangular ducts,
*Heat and Mass Transfer Source Book (Fifth All-Union Conference, Minsk)*, 236-250, Scripta Publishing Co. and John Wiley and Sons. - Malak, J. et al. (1975) Pressure losses and heat transfer in non-circular channels with hydraulically smooth walls,
*Int. J. Heat and Mass Trans.*, 18, 139â€“148. DOI: 10.1016/0017-9310(75)90016-2 - Schiller, L. (1923) Ãœber den Stromungswiderstand von Rohen Verschiedenen Querschnitts und Rauhigkeitsgrades,
*Zeit. Aug. Math und Mech.*, 2â€“10. - Shah, R. K. and London, A. L. (1978) Laminar flow forced convection in ducts,
*Advances in Heat Transfer Supplement 1*, T. F. Irvine Jr. and J. P. Hartaett, eds., Academic Press. - Tung, S. S. and Irvine, T. F. Jr. (1979) Experimental study of the flow of a viscoelastic fluid in a narrow triangular duct,
*Studies in Heat Transfer*, 309â€“329, Hemisphere Pub. Corp. - Usai, H. et al. (1982) Turbulence measurements and mass transfer in fully developed flow in a triangular duct with a narrow apex angle,
*Int. J. Heat Mass Trans.*, 25, 615â€“623. DOI: 10.1016/0017-9310(82)90166-1 - Usui, H. et al. (1983) Fully developed turbulent flow in isosceles triangular ducts,
*J. Chem. Eng. Japan*, 16, 13â€“18. DOI: 10.1252/jcej.16.13