Publications devoted to the study of convective heat transfer in an air swirling flow in pipes or vortex devices have appeared over the past three decades. High heat-transfer rate in vortex devices is highlighted in [1, 2]. However, the results of these studies are applicable to either long pipes or specific designs of the tested chambers [3].

Heat transfer in vortex chambers as applied to cooled turbine blades or recuperators has not yet been studied sufficiently [4, 5], and there are no publications at all about studies of the proposed designs of vortex chambers at the trailing edges of cooled blades.

Installation of VC at the blade leading edge leads to spiral coolant swirling inducing transverse components of the velocity additionally turbulizing the flow. This fact is confirmed by the conclusions given in [3] since the heat transfer rate rises by 10–30% with increasing turbulence intensity [6–10]. These factors acting together sharply increase the exchange of coolant portions between the near-wall layers and the flow core, thereby enhancing heat-and-mass transfer.

In the vortex chambers (Fig. 1) where the considered coolant flow pattern is implemented, air flows through distribution chamber 1 and inlet holes 2 to a cylindrical chamber from which it is directed via outlet holes to collection header 4 and then is discharged to the atmosphere. The inlet and outlet holes are made tangentially with respect to the chamber and arranged in a staggered pattern along the chamber height.

Fig. 1.
figure 1

Design of the vortex chamber. Air flow during (I) direct or (II) reverse purge. 1—Distribution cavity; 2—inlet holes; 3—chamber; 4—outlet holes; 5—collection header.

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An analysis of the effect of the vortex chambers’ design features on their thermohydraulic characteristics is presented below. Its results will enable us to reduce the time for design and development of cooling blades/vanes and compact heat exchangers with the indicated design on the inner cavity.

THERMAL TESTS OF VORTEX CHAMBERS AND A PROCEDURE FOR PROCESSING THE TEST RESULTS

The experimental investigations into thermohydraulic characteristics of VCs were performed by a calorimetric method in models whose dimensions are given in Table 1. The method of calorimetry in a liquid-metal thermostat makes is possible to find the functional relationship between the design style of the inner cavity in the models and their temperature conditions [11–15].

Table 1.   Geometric characteristics of the studied models of VC
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The models were milled from corrosion-resistant steel 12X18H10T with the subsequent use of argon-arc welding. The assembly quality of individual model elements was controlled by X-ray examination.

During the experiment, coolant supply conditions were varied. At the first stage (with a direct blow), air was supplied via holes 2 with the diameter din and then discharged to the atmosphere via holes 4 with the diameter dout. At the second stage (reverse air blow), the air flow direction was reversed. This allowed us to determine the effect of the coolant supply conditions on the thermohydraulic characteristics of the vortex chamber provided that the overall dimensions of the model are the same.

The cooling air pressure ratio πvc across the vortex camber, i.e., the ratio of the VC inlet pressure рin to the atmospheric pressure р0, was from 1.1 to 2.0 in the experiments, which is specific for operation of compact heat exchangers of gas-turbine units and cooled blades of high-temperature turbines of gas turbine engines [6–10].

The local heat flux q was determined from the thickness of the zinc crust formed on the outer surface of the model immersed in the melt of crystallizing high-purity zinc as the model inner cavity was purged with cooling air [12]. The accuracy of determining such parameters as the heat flux q and the heat transfer coefficient α, is directly related to the uncertainty of the method of calorimetry in a liquid-metal thermostat, which is 5–7% [11–16]. At each set of test conditions, five to seven experiments were carried out. After that, the local heat flux and the local heat transfer coefficient were determined by the developed procedure in the monitored section at leading edge points located at an interval of 1 mm. The accuracy of the experiment was estimated by the agreement between the cooling air temperature increase values calculated by the heat balance equation and measured in the experiments.

Hydraulic and thermal characteristics of the models were plotted as a function of the cooling air pressure ratio πvc across the vortex chamber.

Air temperature at the model inlet was measured with shielded chromel-alumel thermocouples with a 0.15 × 10–3 m-diameter junction installed in front of the coolant inlet to the distribution cavity.

The average heat flux in the calculation cross-section of the model was found by the formula

$$q = \sum\limits_{j = 1}^n {\frac{{{{q}_{j}}}}{n}} ,$$

where qj is the local heat flux at a given point averaged over five experiments; n is the number of zinc crust thickness measurement point along the model height H in the calculation section.

In this case, the heat flux averaged over the outer surface of the vortex chamber can be determined by the formula

$$\bar {q} = \frac{{\sum\limits_{j = 1}^N q }}{N},$$

where N is the number of calculation section on the model outline.

The average heat transfer coefficient αav was calculated by the expression:

$${{{\alpha }}_{{{\text{av}}}}} = \frac{{{{R}_{{{\text{out}}}}}}}{{{{R}_{{{\text{in}}}}}}}\frac{1}{{\left( {\frac{{{{T}_{{\text{w}}}} - {{T}_{{{\text{in}}}}}}}{{\bar {q}}} - \frac{{{{R}_{{{\text{out}}}}}}}{{{{\lambda }_{{\text{w}}}}}}} \right)\ln \frac{{{{R}_{{{\text{out}}}}}}}{{{{R}_{{{\text{in}}}}}}}}},$$

where Rout and Rin are the outer and the inner radius of the leading edge, respectively; Tw and Tin are the wall temperature on the cooler side and the air temperature at the model inlet, respectively, K.

For the investigated models, the wall thermal conductivity λw was 16 W/(m2 K).

To evaluate the heat transfer enhancement level in the investigated models, the dependence of the relative heat transfer coefficient

$$\bar {\alpha } = \frac{{{{\alpha }_{{{\text{av}}}}}}}{{{{\alpha }_{{{\text{sm}}}}}}}$$

on the experimental conditions and the channel hydraulic diameter was determined.

In this expression, αsm is the heat transfer coefficient in a smooth round channel [6, 7].

The Nusselt number for a fully developed turbulent flow in a straight smooth pipe was determined by the Dittus–Boelter correlation derived in 1930 and updated in 1985 to consider the temperature factor (Тw/Тin) in [15, 16]:

$${\text{N}}{{{\text{u}}}_{{{\text{sm}}}}} = \frac{{{{{\alpha }}_{{{\text{sm}}}}}{{d}_{{\text{h}}}}}}{{{{{\lambda }}_{{{\text{air}}}}}}} = 0.023{{\left( {\frac{{{{G}_{{{\text{ph}}}}}{{d}_{{\text{h}}}}}}{{{{F}_{{{\text{eq}}}}}{{{\mu }}_{{{\text{air}}}}}}}} \right)}^{{0.8}}}{\text{P}}{{{\text{r}}}^{{0.4}}}{{\left( {\frac{{{{T}_{{\text{w}}}}}}{{{{T}_{{{\text{in}}}}}}}} \right)}^{{ - 0.55}}},$$

where Gph is the physical air flow through a model immersed in a zinc melt; dh is the hydraulic diameter (for a round pipe, dh = d is the dimeter of inlet (or outlet) holes); \({{F}_{{{\text{eq}}}}} = n \times F\) is the area of the equivalent surface; n and F are the number of inlet (or outlet) holes and the area of one hole; \({{{\lambda }}_{{{\text{air}}}}} = {{f}_{1}}\left( {{{T}_{{{\text{in}}}}}} \right)\) is the air thermal conductivity; μair = f2(Тin) is the air dynamic viscosity; Pr is the Prandtl number (equal to approximately 0.7). The diameter and number of holes are given without subscripts since the inlet and outlet holes during a direct purge become, respectively, outlet and inlet holes during a reverse purge.

Hydraulic characteristics obtained in purging the model in a melt of crystalizing metal (see Fig. 2) demonstrated that the flow capacity of the models depends directly on the number and characteristics of the inlet holes [5].

Fig. 2.
figure 2

Dependence of physical air flow Gph on the pressure ratio πvc for the investigated models (a) M1, M2 and (b) M3 for the direct or reverse purge. The experimental data for the direct purge are shown with dark symbols; those for the reverse purge are shown with bright symbols.

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ANALYSIS OF THE RESULTS FROM EXPERIMENTAL INVESTIGATIONS OF THE VORTEX CHAMBERS

Figure 3 presents the dependence of the ratio of air flows during direct and reverse purge \({{G_{{{\text{ph}}}}^{{{\text{dir}}}}} \mathord{\left/ {\vphantom {{G_{{{\text{ph}}}}^{{{\text{dir}}}}} {G_{{{\text{ph}}}}^{{{\text{rev}}}}}}} \right. \kern-0em} {G_{{{\text{ph}}}}^{{{\text{rev}}}}}}\) on the pressure ratio πvc for the investigated models. An analysis of the flow capacities has demonstrated that the model M3 is characterized by the same air flowrates with a fixed pressure ratio in direct and reverse purges. For the models M1 and M2, the air flow during the direct purge is, respectively, 20 and 40% lower than that during the reverse purge. It is evident that the presented curves correspond to the self-similarity mode for \({{{\pi }}_{{{\text{vc}}}}} \geqslant {\text{1}}.{\text{2}}\) (see Fig. 3).

Fig. 3.
figure 3

Dependences of the ratio of physical air flows in the direct and reverse purges, \({{G_{{{\text{ph}}}}^{{{\text{dir}}}}} \mathord{\left/ {\vphantom {{G_{{{\text{ph}}}}^{{{\text{dir}}}}} {G_{{{\text{ph}}}}^{{{\text{rev}}}}}}} \right. \kern-0em} {G_{{{\text{ph}}}}^{{{\text{rev}}}}}}\), of the studied models on the pressure ratio πvc. See Fig. 2 for designations.

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Distributions of the local heat flux q along the height H of the investigated models (from the air inlet to the air outlet) for the zero section at \({{{\pi }}_{{{\text{vc}}}}} = {\text{1}}.{\text{5}}\) are shown in Fig. 4. It is evident that the average level of heat flux and the nonuniformity of its distribution both become greater along the model height with an increase in the diameter and pitch Sh of the air inlet holes with a simultaneous increase in the overall dimensions (Rin) of the chamber proper. It should be noted that the dependence \(q = f(H)\) is smoother for the reverse purges for all the models (when the coolant is supplied to the chamber via larger diameter holes). This can be explained by a lower impingement velocity of the coolant on the chamber inner surface.

Fig. 4.
figure 4

Distribution of the heat flux density q along the height H of the studied models (a) M1, (b) M2, and (b) M3. Dashed lines show the holes with a diameter din, and dash-dotted lines show the holes with a diameter dout.

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For the model M2, the peak values of q correspond to the areas opposite the air inlet holes (see Fig. 4b); however, for the M3 model, they are shifted by (0.3–0.4) Sh in case of the direct purges or by 0.5 Sh in case of the reverse purges towards the peripheral region of the model (see Fig. 4c). This is explained by a change in the coolant flow direction as it impinges tangentially on the inner surface of the vortex chamber with an increase in the parameter Rout and the hole pitch Sh (the coolant flow trajectory takes a more pronounced spiral shape). Thus, extreme heat fluxes q are observed in the region of the tangential supply of cooling air, which is explained by a lower temperature and a higher swirling velocity of the cooling flow. This fact agrees with the conclusions of [5]. These regularities are valid in the entire investigated range of the experimental conditions.

Figure 5 presents the dependences of \({{{{{\bar {q}}}_{{{\text{dir}}}}}} \mathord{\left/ {\vphantom {{{{{\bar {q}}}_{{{\text{dir}}}}}} {{{{\bar {q}}}_{{{\text{rev}}}}}}}} \right. \kern-0em} {{{{\bar {q}}}_{{{\text{rev}}}}}}}\) for direct or reverse purges of the models (πvc = 1.5) of the relative length \(\bar {l} = {{{{l}_{i}}} \mathord{\left/ {\vphantom {{{{l}_{i}}} l}} \right. \kern-0em} l}\) of the outer counter of the vortex chamber.

Fig. 5.
figure 5

Distribution of the ratio of the average heat fluxes \({{{{{\bar {q}}}_{{{\text{dir}}}}}} \mathord{\left/ {\vphantom {{{{{\bar {q}}}_{{{\text{dir}}}}}} {{{{\bar {q}}}_{{{\text{rev}}}}}}}} \right. \kern-0em} {{{{\bar {q}}}_{{{\text{rev}}}}}}}\) during (a) direct or (b) reverse purges along the outer surface \(\bar {l}\) of the vortex chambers with the fixed value of πvc = 1.5. See Fig. 2 for designations.

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It can be seen that the heat flux averaged over the contour of the leading edge is higher in the direct purges of the models than that in the reverse ones, with the maximum value of the complex \({{{{{\bar {q}}}_{{{\text{dir}}}}}} \mathord{\left/ {\vphantom {{{{{\bar {q}}}_{{{\text{dir}}}}}} {{{{\bar {q}}}_{{{\text{rev}}}}}}}} \right. \kern-0em} {{{{\bar {q}}}_{{{\text{rev}}}}}}} \approx 1.6\) corresponding to the M3 model, while that for the model M2 is approximately 1.0.

An analysis of the curves presented in Fig. 6 suggests that the relative heat transfer coefficient \({\bar {\alpha }}\) is higher in the reverse purges than in the direct ones (by 48% for M1, 36% for M2, and 24% for M3). The maximum cooling efficiency is therefore observed for the models with air supply through holes with a larger dimeter and air removal via smaller diameter holes. However, the thermal problem should not be considered without analyzing the \({{G}_{{{\text{ph}}}}} = f({{{\pi }}_{{{\text{vc}}}}})\) (see Fig. 2) characterizing the flow capacity of the models. For the M3 model, it is advisable that the diameter of the air inlet holes be larger than the diameter of air outlet holes, since its flow capacity is the same in both direct and reverse purges. The flow capacity of the model M1 is 21% higher in the reverse purges than that in the direct purges.

Fig. 6.
figure 6

Dependence of the relative heat transfer coefficient \({\bar {\alpha }}\) on the pressure ratio πvc for the studied models. See Fig. 2 for designations.

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In the reverse purges, the maximum value of \({\bar {\alpha }}\) is observed in the model M1 (42% higher than that in the model M2), while the flow capacity of M1 is 50% lower than that of the model M2.

A lower level of \({\bar {\alpha }}\) for the model М2 (nin = 4, nout = 5) in the direct or reverse purges as compared to that for the model M1 is likely to be caused by a lower swirling degree and more intense heating of the flow as it moves along the model channel height [5].

With an increase in the cooling air pressure ration πvc across the vortex chamber, the relative heat transfer coefficient \({\bar {\alpha }}\) drops for M1 and rises for М2 and М3 (see Fig. 6). This behavior is observed in both direct and reverse purges of the models. At \({{{\pi }}_{{{\text{vc}}}}}\) > 1.4, the self-similarity region is observed, i.e., the value of \({\bar {\alpha }}\) remains constant irrespective of the experimental conditions.

Figure 7 presents Nu vs. Re curves in the logarithmic coordinates for the investigated models for the direct or reverse purges. A comparative analysis of the results (Nu vs. Re dependence) for the studied models and models with a flow structure induced by forced swirl [3] demonstrates that these data coincide almost completely (97–98%). Based on the experimental data, criterial dependences of the Nusselt number on Reynolds number have been derived (Table 2).

Fig. 7.
figure 7

Dependence of the Nusselt number on the Reynolds number for the studied models in the direct or reverse purges. See Fig. 2 for designations.

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Table 2.   Dependences of Nusselt number on the Reynolds number for the studied models in the direct or reverse purges
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CONCLUSIONS

(1) The performed investigations suggest that, with an increase in the number of air inlet holes and a decrease in their diameter, the hydraulic resistance of the model increases with a simultaneous growth in the heat transfer and smaller nonuniformity in the heat transfer distribution over the vortex chamber outer surface.

(2) With an increase in the diameter of the vortex chamber, the coolant flow in it obtains a more pronounced spiral-like pattern and is accompanied by an increase in the relative heat transfer coefficient \({\bar {\alpha }}{\text{.}}\) However, for \(d > 7 \times {{10}^{{-3}}}\) m, the coolant flow is likely to separate from the walls thereby decreasing \({\bar {\alpha }}{\text{.}}\)

(3) A stable swirl of the flow and a uniform air heating along the height of the model are observed with reverse purges and n = 9 holes with a larger diameter at a relative parameter of \({H \mathord{\left/ {\vphantom {H n}} \right. \kern-0em} n} < {\text{4}}.{\text{5}}{\text{.}}\)

(4) An increase in the heat transfer with increasing tangential channels stems from lower heating of the coolant due to its additional supply along the channel’s height and the stabilization of flow swirl.

(5) The results of the experimental studies allow us to cut down the material expenditures and time for development of cooled blades having an inner cavity with vortex chambers, for example, in the region of the leading edge of a blade in an AL-31FP engine.