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On the optimization of 3D-flow and heat transfer by using the Ice Formation Method: Vane endwall heat transfer
International Journal of Heat and Mass Transfer 88 (2015) 957–964
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the optimization of 3D-flow and heat transfer by using the Ice Formation Method: Vane endwall heat transfer Sven Winkler ⇑, Kristian Haase, Rostyslav Lyulinetskyy, Sven Olaf Neumann, Bernhard Weigand Institute of Aerospace Thermodynamics, University of Stuttgart, 70569 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 22 December 2014 Received in revised form 15 April 2015 Accepted 15 April 2015 Available online 29 May 2015 Keywords: Vane endwall contouring Ice Formation Method Entropy production Heat transfer
a b s t r a c t Technical flow devices are usually optimized by using numerical algorithms that analyze a multitude of design variants for the flow restraining geometry until an optimum solution is found. In this paper, we present an alternative approach: the Ice Formation Method (IFM). We show the application of this method to a turbine vane endwall in order to optimize the endwall with respect to heat transfer. On the basis of a flat endwall, we first created ice-contoured endwalls in our water channel test facility. We then analyzed these contours using numerical simulations with air to approve them for the working medium of gas turbines. Results show that the IFM reliably creates contours with reduced heat transfer coefficients (HTC). In this vein, we created one contour that reduces the average HTC by 5.2% compared to the baseline. Analyzing this contour revealed that it adapts to the flow field by breaking up the near endwall vortices and thus reducing secondary flows. We found that this reduction in HTC results from the natural character of the IFM to reduce entropy production rates due to heat conduction. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The optimization of fluid flow devices with highly threedimensional flow and heat transfer characteristics is a demanding task. Classical optimization methods usually involve a timeconsuming investigation of many different geometries, while one needs to ensure that these geometries cover the entire optimization space so that the optimized geometry is a global optimum. The Ice Formation Method (IFM) is a faster way to create an optimum geometry. The latter simply develops as an ice layer on a cooled baseline geometry in convective water flow. Due to its natural character, this method is free of restrictions to the optimization space and hence ensures that the created geometry is a global optimum. LaFleur [1,2] investigated the Ice Formation Method for a flat plate Couette flow. He could show that, under certain conditions, the resulting steady-state ice layers represent geometries with minimum energy dissipation. Lyulinetskyy [3] showed that this minimum energy dissipation is linked to reduced entropy production rates. This means that the ice layers cause either reduced pressure loss, reduced heat transfer, or a combination thereof, whereas the ice formation always reduces the predominant loss source the
⇑ Corresponding author. Tel.: +49 711 685 62392. E-mail address: [email protected] (S. Winkler). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.045 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
most. Hence, the IFM can be used to optimize flow geometries in order to reduce these two technically important parameters. Carlson [4] used the Ice Formation Method as an optimization tool to reduce the drag coefficient of a cylinder in cross flow. He found contours with ellipsoidal shape, which delay flow detachment and thus reduce drag. LaFleur and Langston [5] first applied the Ice Formation Method to a cylinder/endwall junction and found geometries that reduce the drag coefficient by about 18% compared to the baseline geometry. Later on, LaFleur et al. [6] found similar drag coefficient reductions for a second vane endwall. They created the optimized geometries by melting an initially grown ice layer using air flow with engine realistic Reynolds numbers. Steinbrueck et al. [7] and Zehner et al. [8] used the Ice Formation Method to optimize the separating web of a 180 bend as it is used for turbine blade cooling channels. As opposed to the study mentioned before, they created the ice layers in a water flow channel. Compared to the uncontoured separating web, their icecontoured geometry reduces pressure loss by about 15%. Few researchers have applied the Ice Formation Method to create contours with reduced heat transfer. In [9], the IFM was used to redesign the endwall of the second vane of the Energy Efficient Engine (E3). Numerical predictions of the heat transfer coefficient (HTC) for the resulting ice contour revealed that this geometry reduces the endwall Stanton number by about 24% compared to the original E3 endwall geometry. However, compared to a flat endwall, this contour features an increase in average heat transfer of 10%.
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Nomenclature
c Latin C C ax Cl f HTC k K _ m nþ 1 p P Pr q00 ReC S S_ T Tu u; v ; w uex V x; y; z
true chord (m) axial chord (m) model constant (–) Darcy friction factor, ¼ 8sW =qV 2 (–) convective heat transfer coefficient (W m2 K1 ) thermal conductivity (W m1 K1 ) turbulent kinetic energy (m2 s2 ) mass flow rate (kg s1 ) near wall distance of first grid cells (–) static pressure (Pa) pitch (m), power (W) Prandtl number, ¼ m=a (–) specific heat flux (W m2 ) Reynolds number, ¼ uex C=m1 (–) span (m) entropy production rate (J K1 s1 ) static temperature (K) turbulence intensity (–) velocity components (m s1 ) velocity magnitude at passage exit cross section (m s1 ) velocity magnitude (m s1 ) spatial directions (m)
Greek
a b
thermal diffusivity (m2 s1 ) angle between camera and laser ( )
In the present study, we applied the IFM to the endwall of a stator vane in order to create endwall contours with reduced heat transfer. The main focus of this investigation is to find out how we can understand the reduced heat transfer rates by analyzing in detail the entropy production rates of the flow. The next section describes how we created the ice contours in our experimental facility and the set-up we used for simulating and analyzing these contours numerically. After that, the results section shows heat transfer and entropy production rates for the ice contours. Heat transfer results are first presented for all ice contours. We then analyze the contour with the maximum heat transfer reduction of about 5% in detail and indicate the underlying changes in the flow field. In the same way, we subsequently present entropy production rates for all ice contours and then analyze these more closely for the contour with the lowest HTC. This analysis shows that the IFM reduces entropy production rates due to heat conduction and hence causes the observed reduction in heat transfer.
d f
HC k k2
l m q
sW x
flow turning angle ( ) lateral displacement of laser beam (m) normalized width of plane perpendicular to outflow direction (–) non-dimensional temperature ratio, ¼ ðT F T CI Þ= ðT 1 T F Þ (–) wavelength (m) second eigenvalue (–) dynamic viscosity (kg m1 s1 ) kinematic viscosity (m2 s1 ) density (kg m3 ) wall shear stress (kg m s2 ) dissipation rate (s1 )
Subscripts/Superscripts bulk bulk CI copper inlay Diss dissipation EW endwall F freezing Heat heat conduction in inlet out outlet turb turbulent — mean 0 fluctuating 1 free stream
section comprises an inlet channel, a three-passage linear cascade with the investigated vane profile, and an outlet channel. The inlet channel is 0:5 m long and features a honeycomb structure for flow straightening. It creates an estimated turbulence intensity of Tu 1% to 4%. The length of the outlet channel is 1:2 m, which is long enough to avoid upstream effects of the turning at its end. Fig. 2 shows the linear cascade in detail (left). It has a width of 187:2 mm at the inlet and a height of 140 mm. The cascade holds two full vanes and two half vanes giving a total of three vane passages. Table 1 lists the vane profiles geometrical parameters. The vanes extend over the entire channel height resulting in S=C ax ¼ 2:0 and S=P ¼ 2:24. These values ensure no interaction
2. Methodology The methodology we used comprises three steps: experiments in water flow, numerical simulations for air and results evaluation. First, we present the experimental set-up, which we used for creating the endwall ice contours. Secondly, we explain the numerical setup for the simulations with the experimental ice contours. Finally, the last subsection describes the methods we used to evaluate the results of the numerical simulations. 2.1. Experimental set-up Fig. 1 shows the experimental facility we used to create the icecontoured endwalls. It is a water cycle consisting of a reservoir with 1:3 m3 of water, piping, a Wilo IL 150/190–5,5/4 inline water pump to drive the flow, and a plexiglass test section. This test
Fig. 1. Experimental facility.
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Fig. 2. Linear cascade (left); laser triangulation: height measurement using angle b between laser and camera and displacement d (right).
between the secondary flows of the upper and lower endwall, which was confirmed by numerical analysis. In order to guarantee a fully turbulent boundary layer at the vane passage inlet, we installed a 1:4 mm trip wire 1.5 axial chords upstream of the vanes. Downstream of the vanes, adjustable tailboards are mounted, which we used to adjust periodicity. Using CFD, exact angles for the tailboards were determined in such a way that the pressure distributions around the two inner vanes are almost identical and hence the mid vane passage is in good approximation periodic. The cascade’s lower endwall has a coolable copper inlay that runs from 80 mm in front of the vanes to 80 mm behind them. It is flush mounted with the channels lower endwall. By passing cooling fluid through its meander-shaped channels, the copper inlay can be cooled down to 23 C. The high thermal conductivity of the copper thereby ensures a homogeneous temperature distribution. In the experiments, we used a cooling unit (van der Heijden type 634/R102) and pumped a mixture of 40% Antifrogen N (90%–95% monoethylene glycol) and 60% water through the copper inlay to cool it below the freezing temperature of water. We then set up a constant water flow through the test section. This causes an ice layer to develop on the cooled inlay. At the beginning of the experiment, the arising ice layer interacts with the surrounding flow. On the one hand, the cooling removes latent heat from the phase interface and the ice layer grows. On the other hand, the flow induces latent heat to the phase interface, causing the ice to change its phase back to water. This results in a free boundary which is shaped by the interaction between flow and heat transfer from the ice. After approximately 2 h, these processes reach steadystate. In this state, the water/ice interface is in thermal equilibrium and isothermal at 0 C and hence the shape of the ice layer remains unchanged. Since the Ice Formation Method is a natural process, the steady-state ice layer is assumed to have minimum local entropy production rates [10]. Therefore, it represents a naturally optimized and energetically advantageous flow geometry. In steady state, the shape of the generated ice layer depends only on two control parameters: the Reynolds number ReC and the non-dimensional temperature ratio HC . The Reynolds number is defined as
Table 1 Geometric parameters of vane profile. True chord C Axial chord C ax Flow turning angle c Pitch P Span S
84:7 mm 70 mm 62° 62:4 mm 140 mm
ReC ¼
uex C
m1
;
ð1Þ
where C is the vane profile’s true chord, uex the mean velocity magnitude at the exit cross section of the vane passage without ice formation and m1 the kinematic viscosity of the incident water flow. We determined the velocity uex from the continuity equation using the incident flows mean velocity. The latter was measured with a standard Venturi nozzle (DIN EN ISO 5167–3). To measure the pressure difference in the nozzle, we used calibrated relative pressure sensors (Newport Omega PR41-V-400 mbar-LIN 0.1%) with an accuracy of 40 Pa, which results in a maximum measurement error of 1.5%. The maximum standard deviation for the pressure measurements amounts to 5.7% due to vibration of the water pump. For the kinematic viscosity, we measured the temperature (see below for measurement accuracy) of the incident flow and determined m1 from tabulated fluid properties of water. Over the course of each experiment, we were able to keep the Reynolds number within a maximum deviation of 1.7%. The non-dimensional temperature ratio, which is the normalized ratio of heat fluxes in the solid and the liquid phase, is defined as
HC ¼
T F T CI ; T1 TF
ð2Þ
where T F ; T CI , and T 1 denote the freezing temperature of water, the temperature of the copper inlay, and the water temperature of the flow, respectively. To determine T CI , we measured the temperature inside the copper inlay at three positions: the front, the middle, and the back. Due to its high thermal conductivity, the copper inlay was nearly isothermal and we averaged the three measured temperatures. For T 1 , we measured the temperature of the water flow at the beginning and the end of the test section. We measured all temperatures using an Agilent 34970A data acquisition unit and Newport Omega K-type thermocouples. The latter we calibrated so that their accuracy was 0:1 K. Maximum errors of the temperature measurements were 1.4% for the temperature of the copper inlay and 8.9% for the water temperature, since this temperature is only around 1 C. Maximum standard deviations amount to 0.5% and 0.9%, respectively. Over the course of each experiment, the temperature ratio could be held within a maximum deviation of 10%. To make the experimentally produced ice layers available for the numerical simulations, we digitized them with a laser triangulation system. Fig. 2 (right) shows a schematic of this system. It uses a diode laser (k ¼ 635 nm and P ¼ 23:1 mW) and a Sony XCHR-50 camera. The camera records the lateral displacement d of
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the beam, which the laser projects onto the ice layer. From the displacement, the system then determines the local ice layer height by using trigonometry and the known angle b between laser and camera. The generated data have a resolution of 1430 426 pixels with a distance of 0:2 mm between the pixels. The height measurement has an accuracy of 0:2 mm. 2.2. Numerical simulations In this study, the experimental ice layers were created in water flow. In order to transfer these results to the application, we conducted numerical simulations using the experimentally created ice layers as endwall contours and air as fluid. For each ice layer, we used the same Reynolds number that we used in the experiment to create it. This allowed us to investigate the endwall contours for the working medium used in gas turbines. Fig. 3 shows the solution domain and the boundary conditions for the numerical simulations. The domain represents one periodic segment of the experimental facility’s vane row. It comprises one full vane with, respectively, half a vane passage next to the vanes pressure and suction side. The outflow region is inclined compared to the inflow region by the flow turning angle of the vane profile. The inlet of the domain is four axial chords upstream of the vane passage, the outlet four axial chords downstream of it. At the inlet and outlet, the height of the solution domain is 140 mm and thus equal to the height of the test section from the experiments. The upper endwall is flat throughout the entire domain. In the inflow and outflow region, the lower endwall is also flat and hence the same as for the baseline. In the vane passage, ranging from the vane’s leading to trailing edge, the digitized ice layer constitutes the passage’s lower endwall. Note that in the experiments, the ice layer also extends up- and downstream of this vane passage. However, the ice layer in these regions cannot evolve free of restrictions, since it is influenced by the sudden start and end of the ice at the cooled copper inlay. Therefore, we only used the ice layer within the vane passage as endwall contour and replaced the ice layer up and downstream of the vane passage by linear transitions which have a length of 80 mm. As hydrodynamic boundary conditions, we prescribed a constant mass flow rate at the inlet, which we determined from the Reynolds number of the experiment with the respective ice layer. The outlet was defined as pressure outlet with the constant static
pressure pout . As thermal boundary conditions, we used constant temperatures for the lower endwall (T EW ) and the inflow (T in ). The upper endwall and the vane were adiabatic. In the y-direction (see Fig. 3 for the coordinate system) we used periodic boundaries for all flow variables. To discretize the solution domain, we created hybrid grids with TM the commercial grid generator Centaur [11], version 9.0.2. Near solid walls, we used structured prismatic layers to discretize the entire boundary layer with a dimensionless wall distance of nþ 1 1 for the first grid cells at the walls, with n being the direction normal to the wall. For the remainder of the domain, we used unstructured tetrahedra. Cells were clustered in regions where we expected high gradients, especially at the vanes’ leading and trailing edges. The final grids had a total of approximately six million cells. Fig. 4 (left) shows such a grid near the vane endwall junction. To ensure the solution being grid independent, we determined the Grid Convergence Index (GCI) according to Roache [12]. Based on the grid size we used for our simulations (5:8M cells), we coarsened (3:3M cells) and refined (9:7M cells) the grid, both with a refinement ratio of 1.2 in each spatial direction. Fig. 5 shows for all three grids the distribution of the Darcy friction factor f and the HTC in a plane which is located 0.5 axial chords downstream of the trailing edge and perpendicular to the outflow direction (see Fig. 4 (right) for position). This region features high gradients of the flow variables due to the wake flow and is hence crucial for determining grid independency. The friction factor for this plane is virtually the same for all three grids, with an average relative error of 0.40% for the used grid. For the HTC, results for the used grid are in good agreement with those of the fine grid, the average GCI being 0.47%. Furthermore, for the used grid, we obtained a GCI of 1.60% for average endwall heat transfer and less than 1% for mean flow parameters, such as vane passage exit velocity or pressure drop over the passage. We performed all numerical simulations with the commercial code ANSYSÒ Fluent 12.1, which uses a finite volume formulation of the three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations. For spatial discretization of the equations’ derivatives, we employed second order schemes and simulated the flow compressible and steady-state. To model turbulence, we used the SST turbulence model in a low-Reynolds-number formulation [13]. As mentioned above, we used air with a Prandtl number of Pr ¼ 0:7 for the simulations.
Fig. 3. Numerical solution domain with boundary conditions.
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Fig. 4. Grid at vane and endwall (left); plane in wake flow of vane used for grid independency study (right).
2.3. Evaluation methods
and the other one due to fluctuating velocity gradients
We evaluated the numerical simulations with respect to heat transfer rates at the lower endwall and entropy production rates in the flow. The former indicates the capability of the ice-contoured endwalls to reduce external heat transfer, the latter shows that these reductions are linked to reduced entropy production.
S_ Diss;turb ¼
2.3.1. Heat transfer To demonstrate the capability of the Ice Formation Method for creating contours with reduced heat transfer, we evaluated the simulations with the experimentally created ice layers with respect to endwall heat transfer. We analyzed results in terms of the heat transfer coefficient
HTC ¼
q00 ; DT
ð3Þ
with q00 being the convective heat flux at the lower endwall in W m2 , and DT ¼ T EW T bulk the corresponding temperature difference between the lower endwall and the local bulk temperature. 2.3.2. Entropy production To analyze entropy production rates for the ice layers, we calculated the four source terms in the Reynolds-averaged entropy transport equation as presented in [14]. The first two source terms describe entropy production by viscous dissipation; one due to mean velocity gradients
S_ Diss ¼
" ( 2 2 2 ) 2 du dv dw du dv þ 2 þ þ þ dx dz dy dx T dy 2 2 # du dw dv dw þ þ þ þ dz dx dz dy
l
ð4Þ
" ( 0 2 0 2 ) 0 2 2 du0 dv dw du dv 0 2 þ þ þ þ T dx dy dz dy dx # 0 2 2 du dw0 dv 0 dw0 : ð5Þ þ þ þ þ dz dx dz dy
l
The last two source terms describe entropy production by heat conduction due to finite temperature gradients, again caused by mean temperature gradients
k S_ Heat ¼ 2 T
" 2 2 # 2 dT dT dT þ þ dx dy dz
ð6Þ
and fluctuating temperature gradients
S_ Heat;turb
2 3 2 0 2 0 2 k 4 dT 0 dT dT 5: ¼ 2 þ þ dx dy dz T
ð7Þ
In the equations above, the primes indicate fluctuating variables. The variables u; v , and w describe the velocities with respect to the three spatial coordinates x; y, and z. Furthermore, T is the static temperature and l and k describe the dynamic viscosity and the thermal conductivity of the fluid, respectively. To calculate Eqs. (4) and (6), we directly used the mean velocity and temperature gradients from the numerical simulations. The fluctuating velocity and temperature gradients, which appear as new unknowns due to the time averaging of the equations, however, cannot be obtained from the numerical simulations. Hence, the turbulent entropy production terms, Eqs. (5) and (7), need to be modelled in accordance with the used turbulence model. Following Kock and Herwig [14], we modelled these terms as follows. The turbulent dissipation term, Eq. (5), can be related to
Fig. 5. Darcy friction factor and HTC for all three grids in a plane 0:5C ax downstream of the vane profile (coarse: 3:3M; used grid: 5:8M; fine: 9:7M); see Fig. 4 for definition of coordinate f.
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the mean static Temperature T and the turbulence variables K and x as
qxK ; S_ Diss;turb ¼ C l T
ð8Þ
where K is the turbulence kinetic energy and x its dissipation rate. C l is a model constant (C l ¼ 0:09). Assuming a constant turbulent Prandtl number and an eddy-diffusivity approach according to Boussinesq for modelling the turbulent heat flux, i.e. u0i T 0 ¼ aturb @T=@xi , the turbulent heat conduction term, Eq. (7), can be modelled by using its mean flow counterpart, Eq. (6), as
aturb k S_ Heat;turb ¼ a T2
" 2 2 2 # dT dT dT þ þ ; dx dy dz
ð9Þ
where aturb =a is the ratio of turbulent to laminar thermal diffusivity. 3. Results and discussion In our experimental test facility, we created nine different endwall ice contours using three Reynolds numbers (ReC ¼ 34; 000; 49; 900, and 71,400) and three non-dimensional temperature ratios (HC ¼ 6:5; 8:5, and 12.2). We then digitized these ice layers by laser triangulation and used them in numerical simulations with air. Additionally, we simulated the baseline case of a flat lower endwall for reference. We present all results with respect to this baseline. Note that the design Reynolds number of the vane profile in an engine application is ReC ¼ 200; 000, however, the test facility only allows for a maximum Reynolds number of ReC ¼ 71; 400.
Fig. 6. Difference of area-averaged HTC for ice contours with respect to baseline.
We first present heat transfer results, beginning with global parameters for all contours and then give a detailed heat transfer analysis for one selected contour. Afterwards, we show results of the entropy production terms for the ice contours of the mid Reynolds number of ReC ¼ 49; 900 with the three temperature ratios. Again, we first present global results for all ice contours and then analyze one selected contour in detail.
3.1. Heat transfer results For all ice contours, Fig. 6 shows the numerically predicted difference of area-averaged endwall heat transfer with respect to the flat endwall baseline. Endwall heat transfer is reduced for all ice contours compared to the baseline. The ice-contoured endwalls reduce the HTC between 2% and 5%. These reductions result from the fact that the IFM creates contours adapted to the vane/endwall flow field. The shape of the ice contour thereby is a result of two competing parameters: the non-dimensional temperature ratio HC and the Reynolds number ReC . The former determines the growth rate of the ice layer on the endwall due to heat being removed from the ice layer, the latter determines the melting rate due to heat being added by convective heat transfer from the flow to the ice layer. Both processes contribute to the contouring of the endwall ice layer and generate flow-adapted contours that reduce secondary flows and hence convective heat transfer. For the smallest Reynolds number, however, the flow produces too low heat transfer rates to effectively contour the growing ice. Hence, the HTC for ReC ¼ 34; 000 decreases with increasing HC , since more ice grows on the endwall than can be contoured by the heat transfer caused by the flow. For the highest Reynolds number and the lowest HC , the flow causes high heat transfer rates, but ice-growth is only moderate. Thus, the high heat transfer rates melt more of the ice than necessary for an effective contouring and HTC reduction is low. This leads to the conclusion that there must be an optimum combination of the two parameters, which results in an optimally shaped ice contour with minimum endwall heat transfer. In the present study, we found this optimum for ReC ¼ 49; 900 and HC ¼ 12:2. The ice contour at these parameters reduces the average HTC by 5.2% compared to the baseline. Fig. 7 shows this contour. It features low height levels around the leading edge since at this position the boundary layer rolls up into a horseshoe vortex causing high heat transfer rates and thus low ice thickness. In the rear part of the vane passage, the flow is mainly governed by the passage vortex, which evolves from the horseshoe vortex. The ice contour adapts to this vortex by forming thin ice layers at the pressure side, where the vortex induces high heat transfer and thick ice layers at the suction side, where the
Fig. 7. Height levels (left) and isometric view (right) of ice contour generated at ReC ¼ 49; 900 and HC ¼ 12:2.
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Fig. 8. Local heat transfer distribution for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2 (left) and flat endwall baseline (right).
vortex causes low heat transfer. This also means that the IFM is a method for heat transfer measurement, since it indicates high heat transfer by low ice thickness and low heat transfer by high ice thickness. Fig. 8 shows the local HTC distribution for the ice contour (left) and for the baseline (right). Except for two small regions at the rear pressure side and directly downstream of the trailing edge, the ice contour significantly reduces endwall heat transfer compared to the baseline. Especially at the rear suction side, where the HTC is locally about 35% lower than the baseline HTC, and downstream of the trailing edge, heat transfer is lower for the ice contour. Note that, although the contouring is restricted to the vane passage, it also clearly influences heat transfer downstream of it. The reduced HTC for the ice contour results from the fact that the IFM alters the flow near the endwall and therewith the vortical structures close to it. Fig. 9 illustrates that by showing the vortex cores near the endwall for both the ice contour and the baseline. We extracted them as iso-surfaces of the k2 -eigenvalue according to the method of Jeong et al. [15]. The ice contour causes a break-up of the vortical structures which weakens the vortices by diffusing them. This also causes a shift of the vortex cores away from the endwall. Both effects lead to the reduced endwall heat transfer for the ice contour as shown in Fig. 8. Due to the vortex break-up, vortex cores for the contoured endwall do not extend as far downstream as for the baseline, explaining the regions with reduced heat transfer downstream of the vane passage. 3.2. Entropy production results Although the Ice Formation Method creates endwall contours with reduced heat transfer, the IFMs goal function is the reduction
Table 2 Entropy production terms for ice contours and difference with respect to baseline, in 104 W K1 .
HC 6.5
Dissipation Turb. diss. Conduction S_ Diss S_ Diss;turb S_ Heat
3.122 (0.081) 8.5 3.179 (0.138) 12.2 3.188 (0.147) Baseline 3.041
3.191 (0.321) 3.115 (0.245) 3.282 (0.412) 2.870
98.518 (6:526) 100.283 (4:761) 96.444 (8:600) 105.044
Turb. cond. S_
Total
58.044 (0:043) 58.851 (0.764) 60.894 (2.807) 58.087
162.875 (7:344) 165.427 (4:792) 163.808 (6:411) 170.219
Heat;turb
of entropy production rates. This has been proven by LaFleur [2] for a Couette flow. Hence, Table 2 shows the entropy production terms (Eqs. (4), (6), (8), and (9)) for the ice contours and their difference with respect to the baseline. We averaged these terms over the entire volume of the solution domain. For brevity, we only show results for ReC ¼ 49; 900; results for the other Reynolds numbers are similar. For all HC , the total entropy production is lower than for the baseline. Hence, the IFM is a method to reliably create flow optimized geometries by reducing entropy production. Furthermore, as a natural optimization process, the IFM always tries to diminish the biggest source of loss. We characterize these losses by entropy production. In our vane flow, heat transfer is the predominant physical effect and hence the magnitude of entropy production due to heat conduction clearly exceeds that of dissipation. The highest contribution to entropy production thereby results from entropy production due to mean temperature gradients, S_ Heat (see Eq. (6)). Hence, this term is also reduced the most by the IFM for the ice contours in Table 2.
Fig. 9. Iso-surfaces of k2 -eigenvalue for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2 (left) and flat endwall baseline (right).
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ice contours were created for combinations of the two control parameters Reynolds number and temperature ratio. All ice contours reduced the average HTC compared to the flat endwall baseline. For the contour with the highest HTC reduction, we showed that the IFM generates contours adapted to the vane/endwall flow field, which reduce secondary flows and hence convective heat transfer. An analysis of entropy production rates revealed that the heat transfer reductions result from reduced entropy production rates due to heat conduction, with these two parameters being closely linked to each other. Conflict of interest Fig. 10. Local, cross-section averaged distribution of entropy production rates due to heat conduction for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2; dashed lines indicate leading and trailing edge of vane.
These results are in very good agreement with the heat transfer results from above. To show this, we compare heat transfer reductions at ReC ¼ 49; 900 (see Fig. 6) to the reductions in S_ Heat (Table 2, fourth column). The lower S_ Heat , the lower is the heat transfer for the ice contour. Thus, the highest heat transfer reduction for the contour at ReC ¼ 49; 900 and HC ¼ 12:2 results from the fact that this contour reduces S_ Heat the most. Moreover, this entropy production term changes from one contour to another by about the same ratio as the reduction in heat transfer. For example, from the ice contour at HC ¼ 6:5 to the one at HC ¼ 8:5, heat transfer reduction changes by a factor of 0.68, while the reduction in entropy production changes by a factor of 0.83. Hence, the heat transfer reductions achieved by the IFM directly result from the methods natural character to reduce entropy production for the main source of loss, in this case entropy production due to heat conduction. Fig. 10 shows the local, cross-section averaged entropy production rates due to mean (S_ Heat ) and turbulent (S_ Heat;turb ) temperature gradients along the computational domain for the contour at ReC ¼ 49; 900 and HC ¼ 12:2. Values are again normalized with the baseline. Entropy production rates due to mean temperature gradients increase towards the vanes leading edge because the flow encounters the beginning of the contouring. Between leading and trailing edge, these rates significantly decrease caused by the ice contour in this region. Downstream of the trailing edge, they remain at a level of about 20% below the value of the baseline due to the fact that the ice contouring diminishes the vortical structures in this region. Entropy production rates due to turbulent temperature gradients are increased in this region. However, the absolute values of these rates are lower than the ones for the production rates by mean temperature gradients and, hence, have only minor influence on the total heat transfer. This local distribution again shows that the IFM significantly lowers entropy production rates by heat conduction and, therefore, causes reduced heat transfer in the region of the contouring and even downstream of it. 4. Conclusions In the present paper, we report about the application of the Ice Formation Method to a turbine vane endwall in order to create novel endwall contours with reduced heat transfer. Nine different
None declared. Acknowledgements We gratefully acknowledge the financial support of the DFG (German Research Foundation) and the FVV (Research Association for Combustion Engines e.V.). References [1] R.S. LaFleur, Evolution theory for optimal control of a couette iceform design model, Int. J. Heat Mass Transfer 35 (10) (1992) 2617–2629. [2] R.S. LaFleur, Example of the couette iceform design model: flat plate iceformation, Int. J. Heat Mass Transfer 35 (10) (1992) 2631–2642. [3] R. Lyulinetskyy, Analytical study of entropy production rates for a natural optimization method (ice formation method) for the case of a channel flow (Study thesis), University of Stuttgart, Institute of Space Systems, Stuttgart, Germany, 2013. [4] F.M. Carlson, An investigation of the solidification of a flowing liquid on a circular cylinder in crossflow and its effects on the drag coefficient (PhD Thesis), University of Connecticut, Storrs, CT, USA, 1975. [5] R.S. LaFleur, L.S. Langston, Drag reduction of a cylinder/endwall junction using the iceformation method, J. Fluids Eng. 115 (1) (1993) 26–32. [6] R.S. LaFleur, Second vane total pressure loss due to endwall iceform contouring, in: Proceedings of the ASME Turbo Expo 2008: Power for Land, Sea, and Air, 2008, pp. 1097–1107, GT2008-50439. [7] H. Steinbrueck, S. Zehner, B. Weigand, S. Neumann, J. Gier, Optimization of the web geometry of a 180-degree bend using an experimentally determined ice layer as starting contour: Part 2 – numerical simulation and optimization, in: Proceedings of the 12th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, 2008, ISROMAC12-2008-20053. [8] S. Zehner, H. Steinbrueck, B. Weigand, S.O. Neumann, J. Gier, Optimization of the web geometry of a 180-degree bend using an experimentally determined ice layer as starting contour: Part 1 – experimental results, in: Proceedings of the 12th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, 2008, ISROMAC12-2008-20049. [9] R.S. LaFleur, T.S. Whitten, J.A. Araujo, Second vane endwall heat transfer reduction by iceform contouring, in: Proceedings of the International Gas Turbine and Aeroengine Congress, ASME, Indianapolis, IN, 1999, 99-GT-422. [10] S. Zehner, Experimentelle Untersuchungen von Turbomaschinenkomponenten mit der Eisformationsmethode (Dissertation), University of Stuttgart, Stuttgart, Germany, 2010. [11] CentaurSoft, Centaur Manual, Centaur Software, Austin, latest ed., 2013. see also www.centaursoft.com. [12] P.J. Roache, Perspective: a method for uniform reporting of grid refinement studies, J. Fluids Eng. 116 (3) (1994) 405–413. [13] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (8) (1994) 1598–1605. [14] F. Kock, H. Herwig, Entropy production calculation for turbulent shear flows and their implementations in CFD codes, Int. J. Heat Fluid Flow 26 (4) (2005) 672–680. [15] J. Jeong, F. Hussain, On the identification of a vortex, J. Fluid Mech. 285 (1) (1995) 69–94.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the optimization of 3D-flow and heat transfer by using the Ice Formation Method: Vane endwall heat transfer Sven Winkler ⇑, Kristian Haase, Rostyslav Lyulinetskyy, Sven Olaf Neumann, Bernhard Weigand Institute of Aerospace Thermodynamics, University of Stuttgart, 70569 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 22 December 2014 Received in revised form 15 April 2015 Accepted 15 April 2015 Available online 29 May 2015 Keywords: Vane endwall contouring Ice Formation Method Entropy production Heat transfer
a b s t r a c t Technical flow devices are usually optimized by using numerical algorithms that analyze a multitude of design variants for the flow restraining geometry until an optimum solution is found. In this paper, we present an alternative approach: the Ice Formation Method (IFM). We show the application of this method to a turbine vane endwall in order to optimize the endwall with respect to heat transfer. On the basis of a flat endwall, we first created ice-contoured endwalls in our water channel test facility. We then analyzed these contours using numerical simulations with air to approve them for the working medium of gas turbines. Results show that the IFM reliably creates contours with reduced heat transfer coefficients (HTC). In this vein, we created one contour that reduces the average HTC by 5.2% compared to the baseline. Analyzing this contour revealed that it adapts to the flow field by breaking up the near endwall vortices and thus reducing secondary flows. We found that this reduction in HTC results from the natural character of the IFM to reduce entropy production rates due to heat conduction. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The optimization of fluid flow devices with highly threedimensional flow and heat transfer characteristics is a demanding task. Classical optimization methods usually involve a timeconsuming investigation of many different geometries, while one needs to ensure that these geometries cover the entire optimization space so that the optimized geometry is a global optimum. The Ice Formation Method (IFM) is a faster way to create an optimum geometry. The latter simply develops as an ice layer on a cooled baseline geometry in convective water flow. Due to its natural character, this method is free of restrictions to the optimization space and hence ensures that the created geometry is a global optimum. LaFleur [1,2] investigated the Ice Formation Method for a flat plate Couette flow. He could show that, under certain conditions, the resulting steady-state ice layers represent geometries with minimum energy dissipation. Lyulinetskyy [3] showed that this minimum energy dissipation is linked to reduced entropy production rates. This means that the ice layers cause either reduced pressure loss, reduced heat transfer, or a combination thereof, whereas the ice formation always reduces the predominant loss source the
⇑ Corresponding author. Tel.: +49 711 685 62392. E-mail address: [email protected] (S. Winkler). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.045 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
most. Hence, the IFM can be used to optimize flow geometries in order to reduce these two technically important parameters. Carlson [4] used the Ice Formation Method as an optimization tool to reduce the drag coefficient of a cylinder in cross flow. He found contours with ellipsoidal shape, which delay flow detachment and thus reduce drag. LaFleur and Langston [5] first applied the Ice Formation Method to a cylinder/endwall junction and found geometries that reduce the drag coefficient by about 18% compared to the baseline geometry. Later on, LaFleur et al. [6] found similar drag coefficient reductions for a second vane endwall. They created the optimized geometries by melting an initially grown ice layer using air flow with engine realistic Reynolds numbers. Steinbrueck et al. [7] and Zehner et al. [8] used the Ice Formation Method to optimize the separating web of a 180 bend as it is used for turbine blade cooling channels. As opposed to the study mentioned before, they created the ice layers in a water flow channel. Compared to the uncontoured separating web, their icecontoured geometry reduces pressure loss by about 15%. Few researchers have applied the Ice Formation Method to create contours with reduced heat transfer. In [9], the IFM was used to redesign the endwall of the second vane of the Energy Efficient Engine (E3). Numerical predictions of the heat transfer coefficient (HTC) for the resulting ice contour revealed that this geometry reduces the endwall Stanton number by about 24% compared to the original E3 endwall geometry. However, compared to a flat endwall, this contour features an increase in average heat transfer of 10%.
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Nomenclature
c Latin C C ax Cl f HTC k K _ m nþ 1 p P Pr q00 ReC S S_ T Tu u; v ; w uex V x; y; z
true chord (m) axial chord (m) model constant (–) Darcy friction factor, ¼ 8sW =qV 2 (–) convective heat transfer coefficient (W m2 K1 ) thermal conductivity (W m1 K1 ) turbulent kinetic energy (m2 s2 ) mass flow rate (kg s1 ) near wall distance of first grid cells (–) static pressure (Pa) pitch (m), power (W) Prandtl number, ¼ m=a (–) specific heat flux (W m2 ) Reynolds number, ¼ uex C=m1 (–) span (m) entropy production rate (J K1 s1 ) static temperature (K) turbulence intensity (–) velocity components (m s1 ) velocity magnitude at passage exit cross section (m s1 ) velocity magnitude (m s1 ) spatial directions (m)
Greek
a b
thermal diffusivity (m2 s1 ) angle between camera and laser ( )
In the present study, we applied the IFM to the endwall of a stator vane in order to create endwall contours with reduced heat transfer. The main focus of this investigation is to find out how we can understand the reduced heat transfer rates by analyzing in detail the entropy production rates of the flow. The next section describes how we created the ice contours in our experimental facility and the set-up we used for simulating and analyzing these contours numerically. After that, the results section shows heat transfer and entropy production rates for the ice contours. Heat transfer results are first presented for all ice contours. We then analyze the contour with the maximum heat transfer reduction of about 5% in detail and indicate the underlying changes in the flow field. In the same way, we subsequently present entropy production rates for all ice contours and then analyze these more closely for the contour with the lowest HTC. This analysis shows that the IFM reduces entropy production rates due to heat conduction and hence causes the observed reduction in heat transfer.
d f
HC k k2
l m q
sW x
flow turning angle ( ) lateral displacement of laser beam (m) normalized width of plane perpendicular to outflow direction (–) non-dimensional temperature ratio, ¼ ðT F T CI Þ= ðT 1 T F Þ (–) wavelength (m) second eigenvalue (–) dynamic viscosity (kg m1 s1 ) kinematic viscosity (m2 s1 ) density (kg m3 ) wall shear stress (kg m s2 ) dissipation rate (s1 )
Subscripts/Superscripts bulk bulk CI copper inlay Diss dissipation EW endwall F freezing Heat heat conduction in inlet out outlet turb turbulent — mean 0 fluctuating 1 free stream
section comprises an inlet channel, a three-passage linear cascade with the investigated vane profile, and an outlet channel. The inlet channel is 0:5 m long and features a honeycomb structure for flow straightening. It creates an estimated turbulence intensity of Tu 1% to 4%. The length of the outlet channel is 1:2 m, which is long enough to avoid upstream effects of the turning at its end. Fig. 2 shows the linear cascade in detail (left). It has a width of 187:2 mm at the inlet and a height of 140 mm. The cascade holds two full vanes and two half vanes giving a total of three vane passages. Table 1 lists the vane profiles geometrical parameters. The vanes extend over the entire channel height resulting in S=C ax ¼ 2:0 and S=P ¼ 2:24. These values ensure no interaction
2. Methodology The methodology we used comprises three steps: experiments in water flow, numerical simulations for air and results evaluation. First, we present the experimental set-up, which we used for creating the endwall ice contours. Secondly, we explain the numerical setup for the simulations with the experimental ice contours. Finally, the last subsection describes the methods we used to evaluate the results of the numerical simulations. 2.1. Experimental set-up Fig. 1 shows the experimental facility we used to create the icecontoured endwalls. It is a water cycle consisting of a reservoir with 1:3 m3 of water, piping, a Wilo IL 150/190–5,5/4 inline water pump to drive the flow, and a plexiglass test section. This test
Fig. 1. Experimental facility.
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Fig. 2. Linear cascade (left); laser triangulation: height measurement using angle b between laser and camera and displacement d (right).
between the secondary flows of the upper and lower endwall, which was confirmed by numerical analysis. In order to guarantee a fully turbulent boundary layer at the vane passage inlet, we installed a 1:4 mm trip wire 1.5 axial chords upstream of the vanes. Downstream of the vanes, adjustable tailboards are mounted, which we used to adjust periodicity. Using CFD, exact angles for the tailboards were determined in such a way that the pressure distributions around the two inner vanes are almost identical and hence the mid vane passage is in good approximation periodic. The cascade’s lower endwall has a coolable copper inlay that runs from 80 mm in front of the vanes to 80 mm behind them. It is flush mounted with the channels lower endwall. By passing cooling fluid through its meander-shaped channels, the copper inlay can be cooled down to 23 C. The high thermal conductivity of the copper thereby ensures a homogeneous temperature distribution. In the experiments, we used a cooling unit (van der Heijden type 634/R102) and pumped a mixture of 40% Antifrogen N (90%–95% monoethylene glycol) and 60% water through the copper inlay to cool it below the freezing temperature of water. We then set up a constant water flow through the test section. This causes an ice layer to develop on the cooled inlay. At the beginning of the experiment, the arising ice layer interacts with the surrounding flow. On the one hand, the cooling removes latent heat from the phase interface and the ice layer grows. On the other hand, the flow induces latent heat to the phase interface, causing the ice to change its phase back to water. This results in a free boundary which is shaped by the interaction between flow and heat transfer from the ice. After approximately 2 h, these processes reach steadystate. In this state, the water/ice interface is in thermal equilibrium and isothermal at 0 C and hence the shape of the ice layer remains unchanged. Since the Ice Formation Method is a natural process, the steady-state ice layer is assumed to have minimum local entropy production rates [10]. Therefore, it represents a naturally optimized and energetically advantageous flow geometry. In steady state, the shape of the generated ice layer depends only on two control parameters: the Reynolds number ReC and the non-dimensional temperature ratio HC . The Reynolds number is defined as
Table 1 Geometric parameters of vane profile. True chord C Axial chord C ax Flow turning angle c Pitch P Span S
84:7 mm 70 mm 62° 62:4 mm 140 mm
ReC ¼
uex C
m1
;
ð1Þ
where C is the vane profile’s true chord, uex the mean velocity magnitude at the exit cross section of the vane passage without ice formation and m1 the kinematic viscosity of the incident water flow. We determined the velocity uex from the continuity equation using the incident flows mean velocity. The latter was measured with a standard Venturi nozzle (DIN EN ISO 5167–3). To measure the pressure difference in the nozzle, we used calibrated relative pressure sensors (Newport Omega PR41-V-400 mbar-LIN 0.1%) with an accuracy of 40 Pa, which results in a maximum measurement error of 1.5%. The maximum standard deviation for the pressure measurements amounts to 5.7% due to vibration of the water pump. For the kinematic viscosity, we measured the temperature (see below for measurement accuracy) of the incident flow and determined m1 from tabulated fluid properties of water. Over the course of each experiment, we were able to keep the Reynolds number within a maximum deviation of 1.7%. The non-dimensional temperature ratio, which is the normalized ratio of heat fluxes in the solid and the liquid phase, is defined as
HC ¼
T F T CI ; T1 TF
ð2Þ
where T F ; T CI , and T 1 denote the freezing temperature of water, the temperature of the copper inlay, and the water temperature of the flow, respectively. To determine T CI , we measured the temperature inside the copper inlay at three positions: the front, the middle, and the back. Due to its high thermal conductivity, the copper inlay was nearly isothermal and we averaged the three measured temperatures. For T 1 , we measured the temperature of the water flow at the beginning and the end of the test section. We measured all temperatures using an Agilent 34970A data acquisition unit and Newport Omega K-type thermocouples. The latter we calibrated so that their accuracy was 0:1 K. Maximum errors of the temperature measurements were 1.4% for the temperature of the copper inlay and 8.9% for the water temperature, since this temperature is only around 1 C. Maximum standard deviations amount to 0.5% and 0.9%, respectively. Over the course of each experiment, the temperature ratio could be held within a maximum deviation of 10%. To make the experimentally produced ice layers available for the numerical simulations, we digitized them with a laser triangulation system. Fig. 2 (right) shows a schematic of this system. It uses a diode laser (k ¼ 635 nm and P ¼ 23:1 mW) and a Sony XCHR-50 camera. The camera records the lateral displacement d of
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the beam, which the laser projects onto the ice layer. From the displacement, the system then determines the local ice layer height by using trigonometry and the known angle b between laser and camera. The generated data have a resolution of 1430 426 pixels with a distance of 0:2 mm between the pixels. The height measurement has an accuracy of 0:2 mm. 2.2. Numerical simulations In this study, the experimental ice layers were created in water flow. In order to transfer these results to the application, we conducted numerical simulations using the experimentally created ice layers as endwall contours and air as fluid. For each ice layer, we used the same Reynolds number that we used in the experiment to create it. This allowed us to investigate the endwall contours for the working medium used in gas turbines. Fig. 3 shows the solution domain and the boundary conditions for the numerical simulations. The domain represents one periodic segment of the experimental facility’s vane row. It comprises one full vane with, respectively, half a vane passage next to the vanes pressure and suction side. The outflow region is inclined compared to the inflow region by the flow turning angle of the vane profile. The inlet of the domain is four axial chords upstream of the vane passage, the outlet four axial chords downstream of it. At the inlet and outlet, the height of the solution domain is 140 mm and thus equal to the height of the test section from the experiments. The upper endwall is flat throughout the entire domain. In the inflow and outflow region, the lower endwall is also flat and hence the same as for the baseline. In the vane passage, ranging from the vane’s leading to trailing edge, the digitized ice layer constitutes the passage’s lower endwall. Note that in the experiments, the ice layer also extends up- and downstream of this vane passage. However, the ice layer in these regions cannot evolve free of restrictions, since it is influenced by the sudden start and end of the ice at the cooled copper inlay. Therefore, we only used the ice layer within the vane passage as endwall contour and replaced the ice layer up and downstream of the vane passage by linear transitions which have a length of 80 mm. As hydrodynamic boundary conditions, we prescribed a constant mass flow rate at the inlet, which we determined from the Reynolds number of the experiment with the respective ice layer. The outlet was defined as pressure outlet with the constant static
pressure pout . As thermal boundary conditions, we used constant temperatures for the lower endwall (T EW ) and the inflow (T in ). The upper endwall and the vane were adiabatic. In the y-direction (see Fig. 3 for the coordinate system) we used periodic boundaries for all flow variables. To discretize the solution domain, we created hybrid grids with TM the commercial grid generator Centaur [11], version 9.0.2. Near solid walls, we used structured prismatic layers to discretize the entire boundary layer with a dimensionless wall distance of nþ 1 1 for the first grid cells at the walls, with n being the direction normal to the wall. For the remainder of the domain, we used unstructured tetrahedra. Cells were clustered in regions where we expected high gradients, especially at the vanes’ leading and trailing edges. The final grids had a total of approximately six million cells. Fig. 4 (left) shows such a grid near the vane endwall junction. To ensure the solution being grid independent, we determined the Grid Convergence Index (GCI) according to Roache [12]. Based on the grid size we used for our simulations (5:8M cells), we coarsened (3:3M cells) and refined (9:7M cells) the grid, both with a refinement ratio of 1.2 in each spatial direction. Fig. 5 shows for all three grids the distribution of the Darcy friction factor f and the HTC in a plane which is located 0.5 axial chords downstream of the trailing edge and perpendicular to the outflow direction (see Fig. 4 (right) for position). This region features high gradients of the flow variables due to the wake flow and is hence crucial for determining grid independency. The friction factor for this plane is virtually the same for all three grids, with an average relative error of 0.40% for the used grid. For the HTC, results for the used grid are in good agreement with those of the fine grid, the average GCI being 0.47%. Furthermore, for the used grid, we obtained a GCI of 1.60% for average endwall heat transfer and less than 1% for mean flow parameters, such as vane passage exit velocity or pressure drop over the passage. We performed all numerical simulations with the commercial code ANSYSÒ Fluent 12.1, which uses a finite volume formulation of the three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations. For spatial discretization of the equations’ derivatives, we employed second order schemes and simulated the flow compressible and steady-state. To model turbulence, we used the SST turbulence model in a low-Reynolds-number formulation [13]. As mentioned above, we used air with a Prandtl number of Pr ¼ 0:7 for the simulations.
Fig. 3. Numerical solution domain with boundary conditions.
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Fig. 4. Grid at vane and endwall (left); plane in wake flow of vane used for grid independency study (right).
2.3. Evaluation methods
and the other one due to fluctuating velocity gradients
We evaluated the numerical simulations with respect to heat transfer rates at the lower endwall and entropy production rates in the flow. The former indicates the capability of the ice-contoured endwalls to reduce external heat transfer, the latter shows that these reductions are linked to reduced entropy production.
S_ Diss;turb ¼
2.3.1. Heat transfer To demonstrate the capability of the Ice Formation Method for creating contours with reduced heat transfer, we evaluated the simulations with the experimentally created ice layers with respect to endwall heat transfer. We analyzed results in terms of the heat transfer coefficient
HTC ¼
q00 ; DT
ð3Þ
with q00 being the convective heat flux at the lower endwall in W m2 , and DT ¼ T EW T bulk the corresponding temperature difference between the lower endwall and the local bulk temperature. 2.3.2. Entropy production To analyze entropy production rates for the ice layers, we calculated the four source terms in the Reynolds-averaged entropy transport equation as presented in [14]. The first two source terms describe entropy production by viscous dissipation; one due to mean velocity gradients
S_ Diss ¼
" ( 2 2 2 ) 2 du dv dw du dv þ 2 þ þ þ dx dz dy dx T dy 2 2 # du dw dv dw þ þ þ þ dz dx dz dy
l
ð4Þ
" ( 0 2 0 2 ) 0 2 2 du0 dv dw du dv 0 2 þ þ þ þ T dx dy dz dy dx # 0 2 2 du dw0 dv 0 dw0 : ð5Þ þ þ þ þ dz dx dz dy
l
The last two source terms describe entropy production by heat conduction due to finite temperature gradients, again caused by mean temperature gradients
k S_ Heat ¼ 2 T
" 2 2 # 2 dT dT dT þ þ dx dy dz
ð6Þ
and fluctuating temperature gradients
S_ Heat;turb
2 3 2 0 2 0 2 k 4 dT 0 dT dT 5: ¼ 2 þ þ dx dy dz T
ð7Þ
In the equations above, the primes indicate fluctuating variables. The variables u; v , and w describe the velocities with respect to the three spatial coordinates x; y, and z. Furthermore, T is the static temperature and l and k describe the dynamic viscosity and the thermal conductivity of the fluid, respectively. To calculate Eqs. (4) and (6), we directly used the mean velocity and temperature gradients from the numerical simulations. The fluctuating velocity and temperature gradients, which appear as new unknowns due to the time averaging of the equations, however, cannot be obtained from the numerical simulations. Hence, the turbulent entropy production terms, Eqs. (5) and (7), need to be modelled in accordance with the used turbulence model. Following Kock and Herwig [14], we modelled these terms as follows. The turbulent dissipation term, Eq. (5), can be related to
Fig. 5. Darcy friction factor and HTC for all three grids in a plane 0:5C ax downstream of the vane profile (coarse: 3:3M; used grid: 5:8M; fine: 9:7M); see Fig. 4 for definition of coordinate f.
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the mean static Temperature T and the turbulence variables K and x as
qxK ; S_ Diss;turb ¼ C l T
ð8Þ
where K is the turbulence kinetic energy and x its dissipation rate. C l is a model constant (C l ¼ 0:09). Assuming a constant turbulent Prandtl number and an eddy-diffusivity approach according to Boussinesq for modelling the turbulent heat flux, i.e. u0i T 0 ¼ aturb @T=@xi , the turbulent heat conduction term, Eq. (7), can be modelled by using its mean flow counterpart, Eq. (6), as
aturb k S_ Heat;turb ¼ a T2
" 2 2 2 # dT dT dT þ þ ; dx dy dz
ð9Þ
where aturb =a is the ratio of turbulent to laminar thermal diffusivity. 3. Results and discussion In our experimental test facility, we created nine different endwall ice contours using three Reynolds numbers (ReC ¼ 34; 000; 49; 900, and 71,400) and three non-dimensional temperature ratios (HC ¼ 6:5; 8:5, and 12.2). We then digitized these ice layers by laser triangulation and used them in numerical simulations with air. Additionally, we simulated the baseline case of a flat lower endwall for reference. We present all results with respect to this baseline. Note that the design Reynolds number of the vane profile in an engine application is ReC ¼ 200; 000, however, the test facility only allows for a maximum Reynolds number of ReC ¼ 71; 400.
Fig. 6. Difference of area-averaged HTC for ice contours with respect to baseline.
We first present heat transfer results, beginning with global parameters for all contours and then give a detailed heat transfer analysis for one selected contour. Afterwards, we show results of the entropy production terms for the ice contours of the mid Reynolds number of ReC ¼ 49; 900 with the three temperature ratios. Again, we first present global results for all ice contours and then analyze one selected contour in detail.
3.1. Heat transfer results For all ice contours, Fig. 6 shows the numerically predicted difference of area-averaged endwall heat transfer with respect to the flat endwall baseline. Endwall heat transfer is reduced for all ice contours compared to the baseline. The ice-contoured endwalls reduce the HTC between 2% and 5%. These reductions result from the fact that the IFM creates contours adapted to the vane/endwall flow field. The shape of the ice contour thereby is a result of two competing parameters: the non-dimensional temperature ratio HC and the Reynolds number ReC . The former determines the growth rate of the ice layer on the endwall due to heat being removed from the ice layer, the latter determines the melting rate due to heat being added by convective heat transfer from the flow to the ice layer. Both processes contribute to the contouring of the endwall ice layer and generate flow-adapted contours that reduce secondary flows and hence convective heat transfer. For the smallest Reynolds number, however, the flow produces too low heat transfer rates to effectively contour the growing ice. Hence, the HTC for ReC ¼ 34; 000 decreases with increasing HC , since more ice grows on the endwall than can be contoured by the heat transfer caused by the flow. For the highest Reynolds number and the lowest HC , the flow causes high heat transfer rates, but ice-growth is only moderate. Thus, the high heat transfer rates melt more of the ice than necessary for an effective contouring and HTC reduction is low. This leads to the conclusion that there must be an optimum combination of the two parameters, which results in an optimally shaped ice contour with minimum endwall heat transfer. In the present study, we found this optimum for ReC ¼ 49; 900 and HC ¼ 12:2. The ice contour at these parameters reduces the average HTC by 5.2% compared to the baseline. Fig. 7 shows this contour. It features low height levels around the leading edge since at this position the boundary layer rolls up into a horseshoe vortex causing high heat transfer rates and thus low ice thickness. In the rear part of the vane passage, the flow is mainly governed by the passage vortex, which evolves from the horseshoe vortex. The ice contour adapts to this vortex by forming thin ice layers at the pressure side, where the vortex induces high heat transfer and thick ice layers at the suction side, where the
Fig. 7. Height levels (left) and isometric view (right) of ice contour generated at ReC ¼ 49; 900 and HC ¼ 12:2.
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Fig. 8. Local heat transfer distribution for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2 (left) and flat endwall baseline (right).
vortex causes low heat transfer. This also means that the IFM is a method for heat transfer measurement, since it indicates high heat transfer by low ice thickness and low heat transfer by high ice thickness. Fig. 8 shows the local HTC distribution for the ice contour (left) and for the baseline (right). Except for two small regions at the rear pressure side and directly downstream of the trailing edge, the ice contour significantly reduces endwall heat transfer compared to the baseline. Especially at the rear suction side, where the HTC is locally about 35% lower than the baseline HTC, and downstream of the trailing edge, heat transfer is lower for the ice contour. Note that, although the contouring is restricted to the vane passage, it also clearly influences heat transfer downstream of it. The reduced HTC for the ice contour results from the fact that the IFM alters the flow near the endwall and therewith the vortical structures close to it. Fig. 9 illustrates that by showing the vortex cores near the endwall for both the ice contour and the baseline. We extracted them as iso-surfaces of the k2 -eigenvalue according to the method of Jeong et al. [15]. The ice contour causes a break-up of the vortical structures which weakens the vortices by diffusing them. This also causes a shift of the vortex cores away from the endwall. Both effects lead to the reduced endwall heat transfer for the ice contour as shown in Fig. 8. Due to the vortex break-up, vortex cores for the contoured endwall do not extend as far downstream as for the baseline, explaining the regions with reduced heat transfer downstream of the vane passage. 3.2. Entropy production results Although the Ice Formation Method creates endwall contours with reduced heat transfer, the IFMs goal function is the reduction
Table 2 Entropy production terms for ice contours and difference with respect to baseline, in 104 W K1 .
HC 6.5
Dissipation Turb. diss. Conduction S_ Diss S_ Diss;turb S_ Heat
3.122 (0.081) 8.5 3.179 (0.138) 12.2 3.188 (0.147) Baseline 3.041
3.191 (0.321) 3.115 (0.245) 3.282 (0.412) 2.870
98.518 (6:526) 100.283 (4:761) 96.444 (8:600) 105.044
Turb. cond. S_
Total
58.044 (0:043) 58.851 (0.764) 60.894 (2.807) 58.087
162.875 (7:344) 165.427 (4:792) 163.808 (6:411) 170.219
Heat;turb
of entropy production rates. This has been proven by LaFleur [2] for a Couette flow. Hence, Table 2 shows the entropy production terms (Eqs. (4), (6), (8), and (9)) for the ice contours and their difference with respect to the baseline. We averaged these terms over the entire volume of the solution domain. For brevity, we only show results for ReC ¼ 49; 900; results for the other Reynolds numbers are similar. For all HC , the total entropy production is lower than for the baseline. Hence, the IFM is a method to reliably create flow optimized geometries by reducing entropy production. Furthermore, as a natural optimization process, the IFM always tries to diminish the biggest source of loss. We characterize these losses by entropy production. In our vane flow, heat transfer is the predominant physical effect and hence the magnitude of entropy production due to heat conduction clearly exceeds that of dissipation. The highest contribution to entropy production thereby results from entropy production due to mean temperature gradients, S_ Heat (see Eq. (6)). Hence, this term is also reduced the most by the IFM for the ice contours in Table 2.
Fig. 9. Iso-surfaces of k2 -eigenvalue for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2 (left) and flat endwall baseline (right).
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ice contours were created for combinations of the two control parameters Reynolds number and temperature ratio. All ice contours reduced the average HTC compared to the flat endwall baseline. For the contour with the highest HTC reduction, we showed that the IFM generates contours adapted to the vane/endwall flow field, which reduce secondary flows and hence convective heat transfer. An analysis of entropy production rates revealed that the heat transfer reductions result from reduced entropy production rates due to heat conduction, with these two parameters being closely linked to each other. Conflict of interest Fig. 10. Local, cross-section averaged distribution of entropy production rates due to heat conduction for ice-contoured endwall at ReC ¼ 49; 900; HC ¼ 12:2; dashed lines indicate leading and trailing edge of vane.
These results are in very good agreement with the heat transfer results from above. To show this, we compare heat transfer reductions at ReC ¼ 49; 900 (see Fig. 6) to the reductions in S_ Heat (Table 2, fourth column). The lower S_ Heat , the lower is the heat transfer for the ice contour. Thus, the highest heat transfer reduction for the contour at ReC ¼ 49; 900 and HC ¼ 12:2 results from the fact that this contour reduces S_ Heat the most. Moreover, this entropy production term changes from one contour to another by about the same ratio as the reduction in heat transfer. For example, from the ice contour at HC ¼ 6:5 to the one at HC ¼ 8:5, heat transfer reduction changes by a factor of 0.68, while the reduction in entropy production changes by a factor of 0.83. Hence, the heat transfer reductions achieved by the IFM directly result from the methods natural character to reduce entropy production for the main source of loss, in this case entropy production due to heat conduction. Fig. 10 shows the local, cross-section averaged entropy production rates due to mean (S_ Heat ) and turbulent (S_ Heat;turb ) temperature gradients along the computational domain for the contour at ReC ¼ 49; 900 and HC ¼ 12:2. Values are again normalized with the baseline. Entropy production rates due to mean temperature gradients increase towards the vanes leading edge because the flow encounters the beginning of the contouring. Between leading and trailing edge, these rates significantly decrease caused by the ice contour in this region. Downstream of the trailing edge, they remain at a level of about 20% below the value of the baseline due to the fact that the ice contouring diminishes the vortical structures in this region. Entropy production rates due to turbulent temperature gradients are increased in this region. However, the absolute values of these rates are lower than the ones for the production rates by mean temperature gradients and, hence, have only minor influence on the total heat transfer. This local distribution again shows that the IFM significantly lowers entropy production rates by heat conduction and, therefore, causes reduced heat transfer in the region of the contouring and even downstream of it. 4. Conclusions In the present paper, we report about the application of the Ice Formation Method to a turbine vane endwall in order to create novel endwall contours with reduced heat transfer. Nine different
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