Keywords

1 Introduction

By avoiding the blending of cooling air and hot gas coming from the combustor, the internal cooling method, which includes three main forms, i.e., impinging jet cooling, rib turbulators cooling [1], and pin-fins cooling, provides a more reliable turbine's aerodynamic performance. The rib turbulators, which can enhance local turbulence due to flow separation and confluence in turbine blade serpentine channels, are the subject of research in this paper. Pham et al. [2] studied the influence of rib profiles on the HTEI and the obtained results showed that the HTEI changed significantly when changing the rib profiles. Another study by Dinh et al. [3] investigated a clearance height of 20% compared to the height of the ribs yields the highest HTEI. For cooling channel, the different cooling gases produce different heat transfer characteristics. Shi et al. [4] studied experimentally the differences between steam and air used in a cooling channel with different angled ribs. They found that when steam is used as a coolant of gas turbine internal cooling passage, heat transfer is enhanced significantly.

Besides ribs, pin–fin is another type of structure that promotes turbulence in the cooling cascade of the channel. Do et al. [5] studied the effect of endwall on enlarging the size of high heat transfer region and reducing the low heat transfer regions in channel with pin-fins. Won et al. [6] studied the distribution of heat transfer coefficient in a cooling channel with multiple rows of pin-fins. Based on the results of Shi et al. [4] of the channel with 90º ribs, this work investigated the effects of trenches on the heat transfer characteristics of channels.

2 Numerical Methodology

2.1 Description of Geometry

The computational domain was constructed based on the experimental model of Shi et al. [4] as shown in Fig. 1. The testing sample of Shi et al. [4] consists of three channels: the development, the heated, and the output, where 12 ribs are attached to the heated channel. The height and sample width of the tested channel are W = 80 mm and Hs = 20 mm, respectively, and the top surface of the channel is specified as symmetry to simplify the channel. Hence, the real height of the total channel is H = 40 mm. The cross-section of the computed domain is a rectangle with the aspect ratio AR = W/H = 80/40 = 2. The equivalent diameter of the channel, Dh = 2HW/(H + W) is 53.3 mm. The direction of the flow corresponds to the direction of the x-axis. The detailed dimensions of the design model are given in Table 1. The total length of the calculated domain is Lt = 600 mm. Table 1 describes the design variables of the tested channel.

Fig. 1
An illustration of a rectangle of L t long. It has 3 vertical sections for L d, L, and L o, where L has 12 vertical lines with an angle of alpha. The illustration indicates the flow directions, along with geometric dimensions of the sample.

Schematic diagram of geometric configurations

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Table 1 Variables of geometric configurations
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The number of trenches varies from three to five with in-line and staggered arrangements. The trenches were distributed evenly on the y-direction of ribs in the channel, spread from the middle of the channel for the opposition. The number of trenches will vary from 3 to 5 trenches a rib. Figure 2 illustrates specifically the overview of ribs with trenches of in-line and staggered arrangements.

Fig. 2
2 illustrations of the staggered and in-line arrangements for trenches. In in-line design, the ribs have an in-line arrangement or maintain the same arrangement throughout, while the staggered design has alternative arrangement.

Sample of design arrangements for trenches

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2.2 Data Reduction

The Reynolds number (Re) of the channel is defined as:

$$Re = \frac{{\rho uD_{h} }}{\mu }{ }$$
(1)

where u is the inlet velocity of the channel; ρ is the air density, Dh and μ are the channel hydraulic diameter and dynamic viscosity of the coolant, respectively.

The local heat transfer coefficient (h) of the heated surfaces is determined by:

$$h=\frac{q}{{T}_{w}-{T}_{b}}$$
(2)

where q is the heat flux of the heated wall, Tw is wall temperature, and Tb is the bulk temperature of the fluid.

The local Nusselt number (Nu) is used to represent the local heat transfer capacity, which can be written as:

$$Nu = \frac{{hD_{h} }}{\lambda }$$
(3)

where λ is the thermal conductivity of coolant fluid.

According to the correlation formula of Dittus-Boelter [7], the Nusselt number (Nu0) and friction factor for fully developed turbulent flow in a smooth channel of a rectangular cross-section is defined as:

$$Nu_{0} = 0.023.Re^{0.8} Pr^{0.4}$$
(4)
$${\text{f}}_{0} = \left[ {1.58\ln \left( {\text{Re}} \right) - 3.28} \right]^{ - 2}$$
(5)

The friction factor which represents the pressure loss of the channel is defined as:

$$f = \frac{{\Delta PD_{h} }}{{2\rho \Delta Lu^{2} }}$$
(6)

The heat transfer efficiency index (η) is defined as:

$$\eta = \left( {\frac{Nu}{{Nu_{0} }}} \right)/{ }\left( {\frac{f}{{f_{o} }}} \right)^{1/3}$$
(7)

2.3 Numerical Analysis

In the present study, the governing differential equations are solved numerically to study the flow characteristics and heat transfer in the channel with pin-fins using the commercial software Ansys CFX® 19.1 [8]. The SST turbulence model is used for all the simulations throughout the work. This work uses the structured grid divided in ANSYS-ICEM® software. The elements adjacent to the wall and the pin-fins are specified with a height of 5 × 10–6 (m) to satisfy a y+ value less than 1 for the entire Reynolds number range, to fit the criteria of the turbulence model. The structural grid shown in Fig. 3 is created using hexahedral elements. The test grid will be used for Reynolds 53,648, and the number of grids will vary from 1.0 million to 11.5 million.

Fig. 3
A model presents a rectangular surface with elevated edges. It has a grid with hexahedral elements arranged in parallel with each other on the surface of the structure.

Sample of mesh structure

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The boundary conditions adopted here are used the same as the Ref. [4] as shown in Fig. 4. The fluid was set as steam (IAPWS-97). The walls both in the entrance and in the outlet section are specified as adiabatic. A uniform heat flux of 5000 W/m2 is allocated at the ribbed surfaces and the two sides of smooth walls. The reference pressure of the computational domain and the inlet temperature of steam flow was 0.3 MPa and 447 K, respectively. The outlet was subject to static pressure of 0 Pa. The mass flow rate was given at the inlet according to the corresponding Reynolds numbers ranging from 9098 to 53,648. A turbulence intensity of 5% was given at the inlet. The walls of the channels were applied as the no-slip boundary condition.

Fig. 4
A model presents a rectangular surface with elevated edges. It has the following labels. Inlet, heated walls, adiabatic walls, and outlet.

Boundary conditions

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3 Results and Discussion

3.1 Grid Independence Test and Validation of Numerical Results

A grid independence test has been conducted by using the SST k-ω model at Re = 53,648 with mesh numbers ranging from 1 to 11.5 million. The grid convergence index (GCI) based on the Richardson extrapolation method was analyzed to evaluate the convergence of the grid system used in this study. Table 2 lists the results of the GCI analysis for the channel with ribs. In Fig. 5, the Nusselt number increase when decreasing the normalized grid spacing. It is shown that the Nusselt number is converging to the value of extrapolation by Richardson’s method. Grid 2 was selected for further computation.

Fig. 5
A graph of Nusselt versus normalized grid spacing. It illustrates 2 plots for mesh and Richardson extrapolate. Mesh extends between (0.005, 201) and (0.01, 196) with a decreasing trend. The Richardson Extrapolate has a value of 202. All values are approximate.

Comparison of investigated mesh with Richardson extrapolation

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In order to validate the SST k-ω model, a comparison of numerical and experimental results of averaged Nusselt number distribution on the heated walls for three Reynolds numbers (9098, 26,587, 53,648) is shown in Fig. 6, where the boundary conditions are also in accordance with the experimental investigations [4]. The averaged Nu numbers on the heated walls for Re = 9098, 26,587, and 53,648 are 0.03%, 3.09%, and 8.09% higher than that of experimental data. This is partly because the uncertainty of 8.5% exists in the experimental system. Therefore, the numerical deviations are acceptable.

Fig. 6
A graph of Nusselt number versus Reynolds number. It presents 2 plots for C F D present and experimental data. C F D increases from (9098, 48) to (53648, 210). Experimental data increases between (9098, 48) to (53648, 190). Values are approximate.

Comparison of experimental and numerical data with Re = 53,648

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Table 2 Results of analysis of grid convergence
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3.2 Effect of Ribs with Trenches on Convective Heat Transfer

Figure 8 shows the comparison of Nu, and η with the cases of 3, 4, and 5 trenches with both arrangements. It is discovered that the staggered arrangements induce a higher Nusselt number for all cases compared to the in-line arrangements but accompany by a higher friction factor. The Nusselt number is increased by about 20.1%, 41.9%, and 36.5% for three Re of 9098, 26,587, and 53,648 for the case of 3 trenches with the staggered arrangement and it is increased by 7.9%, 13.7%, and 12.9% for in-line arrangement. Overall, the HTEI of staggered designs is increased by 20.8%, 31.9%, and 26.0% and that of in-line designs show an increase by 11.3%, 16.4%, and 15.8% for all Reynolds number.

Fig. 8
2 graphs, and b, present Nusselt number, and H T E I, respectively. In A, all 7 plots follow an increasing trend throughout. In B, 4 and 5 stags follow an increasing trend, 3 stag, and 3, 4, and 5 in-lines follow an increasing to decreasing trend, and original follows a horizontal trend.

Comparisons of Nu and η with all cases

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4 Conclusion

The results show that the trenches open up some extra passages for the flow to destruct the stagnant vortices in front of and behind the ribs and increase the heat transfer characteristics in the heated walls and the ribs. Besides, the staggered arrangement induces a higher Nusselt number than the in-line cases and hence, creates a higher friction factor. However, the increase in Nusselt number is higher than the increase in friction factor and the overall HTEI is increased. The HTEI of staggered designs is increased by 20.8%, 31.9%, and 26.0% compared to the original design for Re = 9098, 26,587, and 53,648, respectively. Besides, the in-line arrangement comes with a maximum increase of 13.7% in the Nusselt number for Re = 26,587. It combines with a decrease of 9.3% in friction factor to form an increase of 16.4% in HTEI.