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Constructal Design of Elliptical Conduits for Cooling of Gas Turbine Blades with External Thermal Barrier Coating


Gas turbines (GTs) are thermal machines used to transform the energy released in combustion with a hydrocarbon into mechanical power, in order to drive a machine or generate thrust in aircraft. The critical issue in the GT design are the parts exposed to extreme mechanical and thermal conditions, e.g., the first row of turbine blades. The GT thermal efficiency is limited by the maximum temperature the blade materials can withstand without softening or creeping. Currently, the maximum operating temperature is above the softening point of the blade material thanks to techniques of ceramic coatings of low thermal conductivity, called Thermal Barrier Coating (TBC), and techniques of blade cooling. The internal cooling of blades involves conduits inside them for air that comes from a bleed in an intermediate compressor stage. The air bleeding is around 3 to 5% of the main GT flow. This air and the heat flow that it receives are not used to generate power, so it is necessary to optimize the cooling techniques in order to control the temperature using the least amount of air and minimum heat flux evacuated, for holding the GT overall efficiency high. The present work studies the internal cooling of Elemental Gas Turbine Blade (EGTB) with a fixed thickness of the TBC and the optimization of the conduit shape and position over a cross section in 2D. The optimization is carried out by exhaustive searching method based on the Constructal Theory. The optimization of the position, size, and aspect ratio of EGTBs was done for two types of standard elliptical conduits of different geometries, uniformly distributed. Two different objective functions are analyzed: minimum maximum temperature on the metal and maximum heat evacuation efficiency. The outcome of this work establishes that the use of elliptical conduits of aspect ratio 2:5 leads to improvement in the thermal performance of cooled blades. As compared with circular conduits of the same area, elliptical conduits allow transfer of a greater amount of heat; with a correct design, they enable a lower maximum temperature on the metal. Besides, the constructal designs obtained in this study for the minimum maximum relative temperature \(\tilde{T}_{\rm{max}}\) or maximum heat evacuation efficiency ξ were not identical.

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A :

area, dm2

D h :

hydraulic diameter

f :

Darcy friction coefficient

H :

elemental blade height, dm

h :

convection coefficient, W/(m2 · K)

H 0 :

vertical semi-axis of the ellipse 0, dm

H 1 :

vertical semi-axis of the ellipse 1, dm

H 2 :

wall thickness of ellipse 1, dm

k :

thermal conductivity coefficient, W/(m·K)

L :

elemental blade width, dm

L 0 :

horizontal semi-axis of the ellipse 0, dm

L 1 :

horizontal semi-axis of ellipse 1, dm

Nu :

Nusselt number

ε :


μ :

dynamic viscosity, kg/m·s

ν :

cinematic viscosity, m2/s

ξ :

heat evacuation efficiency

ρ :

density, kg/m3

P :


Pe :

Peclet number


Prandtl number

q :

total heat, W

R :

thermal resistance, m2·K/W


Reynolds number

S :

heat source, W/m3

T :

temperature, K

V :

velocity, m/s

W :

transversal dimension, dm

x :

abscissa coordinate, dm

y :

ordinates coordinate, dm

σ :

Stefan-Boltzmann constant, W/(K4m2)

ϕ :

material ratio

ϕ 0 :

1/4 ellipse 0 dimensionless area

ϕ 1 :

1/2 ellipse 1 dimensionless area


non dimensional parameter


ellipse 0


ellipse 1






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Correspondence to C. Bosc, L. A. O. Rocha, F. R. Centeno or F. Gutierrez.

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Bosc, C., Rocha, L.A.O., Centeno, F.R. et al. Constructal Design of Elliptical Conduits for Cooling of Gas Turbine Blades with External Thermal Barrier Coating. J. Engin. Thermophys. 28, 507–528 (2019).

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