Heat Transfer Performance of a New Fan-Shaped Pin

Proceedings of ASME Turbo Expo 2013: Turbine Technical Conference and Exposition GT2013 June 3-7, 2013, San Antonio, Texas, USA

GT2013-94193

HEAT TRANSFER PERFORMANCE OF A NEW FAN-SHAPED PIN-FIN IN INTERNAL COOLING CHANNEL Mi-Ae Moon Dept. of Mechanical Engineering, Inha University Incheon, Republic of Korea [email protected]

Kwang-Yong Kim Dept. of Mechanical Engineering, Inha University Incheon, Republic of Korea [email protected]

ABSTRACT Prediction of convective heat transfer around a pin-fin of novel fan-shape has been performed with Reynolds-averaged Navier-Stokes (RANS) analysis in comparison with a circular pin-fin. The low-Reynolds number shear stress transport (SST) model has been selected as the turbulence closure model by comparing the performance with those of the standard k-ε and k-ω models. The fan-shaped pin-fin has shown remarkably improved heat transfer performance compared to the circular pin-fin over the whole range of Reynolds number (Re=5,000-100,000). A parametric study with two geometric parameters of the fan-shaped pin-fin, the lateral reduction angle of the fan-shaped pin-fin and radius of rear part of pin-fin has been performed to find their effects on heat transfer and friction loss.

R Radius of pin-fin Re Reynolds number T Local temperature Uin Average axial velocity at the inlet W Width of cooling channel θ Lateral reduction angle of the fan-shaped pin-fin β Weighting factor in objective function ν Fluid viscosity ρ Fluid density τw Shear stress at the wall Subscript 1 Front part of the fan-shaped pin-fin 2 Rear part of the fan-shaped pin-fin p Pin-fin INTRODUCTION Increase in inlet temperature of turbine stage in a gas turbine leads to improvement in the efficiency of the system. And, a high inlet temperature causes a high heat load on the turbine blades. In order to keep an acceptable temperature of a gas turbine blade, internal cooling channels with various turbulators such as pin-fins, ribs, dimples, and protrusions are used to reduce the heat load on the turbine blade. Especially, pin-fin arrays are widely used to cool an internal cooling channel. The pin-fins have larger surface area per unit volume than the other turbulators. And, the pin-fins induce the complicated three-dimensional turbulent flow and the disturbance of boundary layer, which results in the enhancement of heat transfer in the turbine blade.

NOMENCLATURE A Heat transfer surface AR Aspect ratio (=W/H) Dh Hydraulic diameter (=2HW/(H+W)) f Friction factor f0 Reference friction factor H Height of cooling channel and pin-fin k Fluid thermal conductivity Nu Nusselt number

Nu Δp Pr q0

Area-averaged Nusselt number Pressure drop in a channel Prandtl number Wall heat flux

1

Copyright © 2013 by ASME

There have been many experimental and numerical studies on the heat transfer performance of pin-fins.. Lawson et al. [1] observed that the maximum local Nusselt number occurred near the front part of a pin-fin fin because of the flow impingement. And, their results show that the local Nusselt number decreases in the wake region which is located downstream of the pin-fin. fin. Chyu [2] measured spanwise averaged Nusselt number through in-line line and staggered pin pinfin arrays. He concluded that the high heat transfer occurs as a result of wake impingement in front of pin pin-fins and flow acceleration between two spanwise neighborin neighboring pin-fins. Through these experimental data, the maximum Nusselt number was found in the second row for an in in-line array and the third row for a staggered array. Metzger et al. [3] also observed that peak Nusselt numbers appeared in the second and third rows. Heat transfer performance of staggered longer pin-fins was studied by Peng [4]. He reported that the longer pin-fins fins showed higher heat transfer than the short pin pin-fins in low aspect ratio channel. Metzger and Haley [5] experimentally investigated the effect of streamwise spacing of pin-fins fins on the heat transfer performance. They found that the heat transfer performance is enhanced as the streamwise spacing between pin-fins fins increases. Simoneau and Van Fossen [6] measured the turbulence intensities in rectangular ctangular channels with staggered pin-fins. The effect of pin-fin height on the friction factor in a channel was conducted by Damerow et al. [7]. Their results showed ed that the height of pin pin-fins does not affect the friction coefficient. Numerous studies have been performed to investigate the heat transfer performance and flow characteristics of the various pin-fin geometries. Jeng and Tzeng [[8] experimentally studied the pressure drop and heat transfer of the square pin pinfin array in the rectangular channel el for various longitudinal and transverse pitches. When the longitudinal and transverse pitches have specific values, the heat transfer of the staggered square pin-fin array was higher than others. Uzol and Camci [9] show the results of heat transfer, pressure ssure loss loss, and wake flow field for two-row row staggered elliptical arrays. While the pressure loss for these elliptical pin-fins was lower than the circular pin-fins, fins, Nusselt number on the end wall for the circular pin-fin arrays was higher than those of the elliptical pin-fin fin arrays. The heat transfer and pressure loss characteristics of staggered circular stepped pin pin-fin arrays were experimentally investigated by Goldstein and Chen [[10]. The circular stepped pin-fin fin arrays show showed the better performance than the circular pin-fin fin arrays because the step stepinduced secondary flow promotes the generation of horseshoe vortices which induce the heat transfer enhancement. The flow structures around single turbulator of different geometries (square, circular, triangular,

(a) Computational domain

(b) Configuration of the fan fan-shaped pin-fin Fig. 1 Schematic of computational domain and pin pin-fin shape and rhomboidal) were experimentally investigated by Armellini et al. [11]. The separation structures downstream of various turbulators depended on the shape of heat transfer me larger and stronger as the promoters, and these became blockage effect increased (passing from triangular to rhomboidal, circular, and square pin-fins). Velocity pin-fins were distributions for the elliptical and circular pin experimentally conducted by Uzol and Camci [12]. The streamline and velocity distributions in the wake region of the fins depended on the geometries of the pin-fins. Weilin pin-fins and Abel [13] experimentally studied on the in-line and fins and showed the staggered arrays staggered square pin-fins in-line having a higher heat transfer performance than the in arrays. shaped pin pin-fin is proposed in this A new design of fan-shaped work, and analysis of heat transfer performance and parametric study for the fan-shaped pin-fin has been three-dimensional RANS numerically performed using three fan-shaped equations. The heat transfer performance of the fan pin-fin has been evaluated in comparison with that of a circular pin-fin that has been widely used as a turbulence promoter in the gas turbine internal cooling channel.

2


Table 1 : Cases in the parametric study for various fan-shaped pin-fin fin configurations Case 1 2 3 4 5 6 7 8 (Reference) 9

(a)

R2 1.00 1.00Rp 1.00 1.00Rp 1.00 1.00Rp 1.00 1.00Rp 1.00 1.00Rp 0.90 0.90Rp 1.10 1.10Rp 1.20 1.20Rp 1.30 1.30Rp

θ 40.0° 50.0° 60.0° 70.0° 80.0° 60.0° 60.0° 60.0° 60.0°

SHAPED PIN PIN-FIN GEOMETRY OF FAN-SHAPED Fig. 1 shows the computational domain and the new shape of pin-fin that was designed to enhance the heat transfer performance on the cooling surface. Aspect ratio ((AR=W/H) of the cooling channel is 2.05 with the height of the channel ((H) of 28.7mm, which is same as that used by Simoneau and Van ]. The hydraulic diameter of the rectangular Fossen [14]. channel, Dh is 38.55mm and the length of cooling channel is pin-fin, of which diameter (Rp) is 203.2mm. A single circular pin diameter ratio of 3.01) 3.01), is installed in the 9.53mm (length-to-diameter walls and pincooling channel. The distance between the side wall fin is 0.50W and the streamwise distance from the inlet section to the pin-fin (Lp) is 3.62Dh. The radius of front part of the fanfan-shaped pin-fin (R1) is Rp, and the radius of rear part of fan shaped pin-fin (R2) is 1.20Rp. The lateral reduction angle of 0.0°. the fan-shaped pin-fin (θ) is 60.0

(b) high Re k-ε model

TURBULENCE MODELING has to be calculated The unknown Reynolds stress tensor ha by an appropriate turbulence model for closure to solve the RANS equations for steady incompressible flow. In this study, three different turbulence models such as k-ε, k-ω, and low-Re SST models were tested.. The k-ε model was proposed in the standard form by Launder and Spalding [15]. The standard kω model was developed by Wilcox [[16], which is sensitive to stream turbulence properties. In contrast, k-ε model inlet free-stream stream turbulence. Mentor [17] is insensitive to the free-stream developed the shear stress transport (SST) turbulence model, which combines the advantages of the k-ε and k-ω models with a blending function. The model works by solving a based model ((k-ω) at the near wall turbulence/frequency-based region and a k-ε model for the bulk flow. A blending function ensures a smooth transition between the two models. Bardina et al. [18]] showed that the SST model captures the flow separation under an adverse pressure gradient more effectively than other eddy viscosity models. Thus, it predicts more

(c) k-ω and low-Re Re SST models Fig. 2 Examples of grid system for different turbulence models

Fig. 3 Grid dependency test

3


(a) Experimental data [12]

(b) Low-Re SST model

(c) k-ε model

(d) k-ω model

Fig. 4 Validation of numerical solutions for local velocity distribution the y+ values for the next-to-wall nodes of the order of 1, which is placed near y = 0.001H. For the boundary condition conditions, the uniform velocity profile at the inlet section was assumed and the pressure was specified at the outlet. The numerical analysis was conducted over Reynolds number range from 5,000 to 100,000. The bulk temperature is calculated at the inlet to the chan channel. The turbulence level is specified as 1.0% and a uniform heat flux is adopted on the heated surfaces including the surface of pin pinfin. The no-slip slip condition is applied on the whole surfaces. The incompressible governing equations were solved through numerical merical analysis considering the small temperature rise and subsequently small density variation of approximately 1.79% 1.79%. The root mean square (RMS) relative residual values of all flow parameters were set to 1.0E 1.0E-6, and energy and mass imbalances of less than han 0.003 were adopted as the convergence criteria in the computation domain. The solver finished a single simulation in 4 hrs with approximately 500 iterations using an Intel Core i7 3.41 GHz CPU.

wall turbulence that plays a vital role in accurately the near-wall the accurate prediction of turbulent heat transfer. Garg and Ameri [19]] also found that the SST model resolved the equation turbulence passage vortex better than other two-equation models (e.g., the k-ε and k-ω models)) in the case of turbomachinery heat transfer applications, thus yielding better comparison with experimental data. SOLUTION METHODOLOGY dimensional RANS analyses of the In present study, three-dimensional fluid flow and convective heat transfer were performed using CFX 11.0 [20], which employs an unstructured grid ANSYS-CFX system. ANSYS ICEM 11.0 has been used to construct a hexahedral grid for the analysis. Fig. 2 shows an example of the computational grids for different turbulence models. The ated at the wall region to resolve the high grids are concentrated type grids are adopted around the fan fanvelocity gradient. O-type shaped pin-fin. wall turbulence, which determines The treatment of near-wall the accuracy of the wall shear stress and heat transfer ole in turbulence modeling. The prediction, plays a key role computational grid should be consistent with the turbulence modes used in the analysis. For example, in order to use the empirical wall function (high-Re model) near the wall, the aced in the log log-law first grid points near the wall should be placed low-Reynolds region, relatively far from the wall, but if the low number turbulence model is applied, the first grid points ld be located very close to the wall. Therefore, the should different grid systems were adopted to the different turbulence dels as shown in Fig. 2. The grid system to use the wall models wall region consists of about function formulation in the near-wall 2.5 million nodes and the first grid points are placed near y = law region, y+ > 30. The 0.05H to locate the points in the log-law Re SST models is other grid system to adopt the k-ω and low-Re made of about 3.8 million nodes. This grid system produces

RESULTS AND DISCUSSION To find the optimal number of grids, the grid grid-dependency test was performed for the Nusselt number distribution on the heated surface with the fan-shaped shaped pin pin-fin with low-Re SST turbulence model as shown in Fig. 3. From these results, the optimal number of grids for the reference geometry was determined to be about 3.8 million nodes. The grid systems for the various geometries tested in this study were constructed with approximately same grid spacing with this optimum grid system for the reference shape. Th Therefore, the numbers of grids in the range from 3.5 million to 4.2 million were used depending on the geometry. Present numerical solutions for the reference shape (Case 8 in Table 1) have been validated in comparison with experimental data of Simoneau and Van Fossen [14] and Uzol and Camci [12] for the same conditions. Fig. 4 shows the velocity distributions on a specified plane, and Fig. 5 presents the area-averaged averaged Nusselt number variations in the Reynolds

4


(a) Circular pin pin-fin Fig. 5 Validation of numerical solutions for variation of area-averaged averaged Nussselt number with Reynolds number [14]

(b) Fan-shaped shaped pin pin-fin (Case 8)

Fig. 8 Nusselt number ratio contours on the heated surface at Re = 5,000

Fig. 6 Variations of area-averaged averaged Nusselt number with Reynolds number for circular and fan-shaped shaped pin pin-fins

(a) Circular pin pin-fin (a) Re = 5,000

(b) Fan-shaped shaped pin pin-fin (Case 8)

(b) Re = 100,000 Fig. 9 Nusselt number ratio contours on the heated surface at Re = 100,000

Fig. 7 Distributions of Nusselt number ratio for circular and fan-shaped pin-fins

5


(a) Circular pin-fin (a) Re = 5,000

(b) Fan-shaped pin-fin (Case 8) (b) Re = 100,000 Fig. 10 Vorticity (|ωx|) distributions on the x-z plane (y/W = 0.50) at Re = 5,000

Fig. 12 Results of parametric study for Nu

(a) Re = 5,000

(a) Circular pin-fin

(b) Re = =100,000 Fig. 13 Results of parametric study for f/f0

(b) Fan-shaped pin-fin (Case 8)

temperature averaged over the specified surfaces in the channel that is same as that of the experiment [14]. As shown in the results of comparison (Figs. 4 and 5), the computational Re SST turbulence model shows best results using low-Re presented in Fig. 2. The value oof wall temperature used in the averaged Nusselt number shown in Fig. 5 is the area-averaged agreement with the experimental data. The test cases of the fin for parametric study are listed in Table 1. fan-shaped pin-fin condition as the All the results were obtained under the same condi experiment [14].

Fig. 11 Distributions of entropy generation due to the heat transfer on the heated surface at Re = 5,000 number range from 5,000 to 100,000, which was presented in previous work [21]. These figures compare the results calculated by three different turbulence models, i.e., k-ε, Re SST models, with the grid systems standard k-ω, and low-Re

6


Fig. 15 Results of parametric study for S H

(a) Re = 5,000

transfer on the heated surface are shown in Fig. 111. The entropy generation induced by the heat transfer ((SH) is related to the available work in the total system from the viewpoint of thermodynamics. SH is defined as follows [[22]: 

 

 

 

 =     +   +    

(1)

The decrease in entropy generation due to heat transfer causes the energy conservation in the gas turbine. In Fig. 111, it is found that the fan-shaped shaped pin pin-fin is more useful for reduction of entropy production in the flow field as well as for enhancement of heat transfer compared to the circular ppin-fin. Results of the parametric study are presented in Figs. 112 and 13 for Nusselt number and friction factor, respectively, at Reynolds numbers 5,000 and 100,000. The friction factor factors are defined as follows:

(b) Re =100,000 Fig. 14 Results of parametric study for η averaged Nusselt number Fig. 6 presents variation of area-averaged shaped pin pin-fin in with Reynolds number for the fan-shaped fin of radius Rp [14]. comparison with that for the circular pin-fin averaged Nusselt number for the fan fan-shaped The value of area-averaged fin shows the high level of heat transfer rate over the pin-fin eynolds number. As the Reynolds number whole range of Reynolds increases, the heat transfer improvement for the fan-shaped pin-fin reaches 22.8% at Re = 80,000 as compared to that for the circular pin-fin [14]. streamwise averaged Nusselt number The cross-streamwise distributions on the heated surface for the circular [14] and fan-shaped pin- fins are presented in Fig. 7 at Re = 5,000 and 100,000. The Nusselt number has relatively high around the fins due to the horseshoe vortices. The maximum value of pin-fins pin-fin. The fanthe Nusselt number occurs just in front of the pin fin shows higher heat transfer around and shaped pin-fin fin than the circular pin pin-fin. The local downstream of the pin-fin Nusselt number ratio distributions on the heated surface are shown in Figs. 8 and 9. As previously mentioned, the high pin-fin Nusselt number region is found around front part of pin ke. At both the Reynolds numbers, high Nu and in the wake. fin persists longer in the wake region of the fan-shaped pin-fin fin [14] giving the higher overall heat than the circular pin-fin transfer performance. The vorticity (|ωx|) distributions on x-z plane ((y/W = 0.50) are shaped pin pin-fin, the shown in Fig. 10. In case of the fan-shaped strength of the vortical motion is shown to be clearly enhanced, which promotes heat transfer performance fin. The asymmetry of vorticity downstream of the pin-fin. distribution is caused by the enhanced horseshoe vortices in front of the fan-shaped pin-fin. The local entropy generation distributions due to heat



 = 

(2)



0.316 ⁄  = 

1.8  

  .



4,000 <  < 10 [24]  ≤ 4,000   ≥ 10 [25]

(3)

Here, f represents presents the Darcy fricrion factor [23]. f0 is the friction factor for the fully developed turbulence flow in a smooth channel, and obtained from Blasius [24] and Tuzson [25] correlations. And, τw, ρ and Uin are shear stress at the wall, fluid density and average axial velocity, respectively. According ding to the change of the two geometric parameters, the values of Nusselt number and friction factor have different tendencies. As the lateral reduction angle of the fan fan-shaped pin-fin (θ)) increases (Cases 22-5), the heat transfer performance becomes maximum at Case 3 ((θ = 60.0°), while the friction factor reaches minimum at Case 4 ((θ = 70.0°) at both low and high Reynolds numbers. As the radius of rear part of pin-fin (R2) increases (Cases 66-9), the heat transfer rate becomes maximum at Case 8 ((R2 =1.20Rp), but the pressure loss monotonically increases. For Nusselt number, all the test cases show better performances than the case of the circular pin-fin. Fig. 14 shows the results of the parametric study for the

7


thermal performance (η). The thermal performance is considering both heat transfer and pressure loss by defining as follows;   ⁄ 

 = (⁄ )⁄ 

[3] Metzger, D. E., Berry, R. A. and Bronson, J. P., 1982, “Developing Heat Transfer in Rectangular Ducts with Staggered Arrays of Short Pin Fins,” ASME Journal of Heat Transfer, Vol. 104, pp. 700-706. [4] Peng, Y., 1984, “Heat Transfer and Friction Loss Characteristics of Pin Fin Cooling Configurations,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 106, pp. 246-251. [5] Metzger, D. E. and Haley, S. W., 1982, “Heat Transfer Experiments and Flow Visualization for Arrays of Short Pin Fins,” Proceedings of ASME Turbo Expo 1982, GT1982-138. [6] Simoneau, R. J. and Van Fossen, G. J., 1984, “Effect of Location in an Array on Heat Transfer to a Short Cylinder in Crossflow,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 106, pp. 42-48. [7] Damerow, W. P., Murtaugh, J. C. and Burgraf, F., 1972, “Experimental and Analytical Investigation of the Coolant Flow Characteristics in Cooled Turbine Airfoils,” NASA CR-120883. [8] Jeng, T. M. and Tzeng, S. C., 2007, “Pressure Drop and Heat Transfer of Square Pin-Fin Arrays in In-Line and Staggered Arrangement,” International Journal of Heat and Mass Transfer, Vol. 50, pp. 2364-2375. [9] Uzol, O. and Camci, C., 2005, “Heat Transfer, Pressure Loss and Flow Field Measurements Downstream of Staggered Two-Row Circular and Elliptical Pin Fin Arrays,” ASME Journal of Heat Transfer, Vol. 127, pp. 458-471. [10] Goldstein, R. J. and Chen, S. B., 1998, “Flow and Mass Transfer Performance in Short Pin-Fin Channels with Different Fin Shapes,” International Journal of Rotating Machinery, Vol. 4, pp. 113-128. [11] Armellini, A., Casarsa, L. and Giannattasio, P., 2010, “Low Reynolds Number Flow in Rectangular Cooling Channels Provided with Low Aspect Ratio Pin Fins,” International Journal of Heat and Fluid Flow, Vol. 31, pp. 689-701. [12] Uzol, O. and Camci, C., 2001, “Elliptical Pin Fins as an Alternative to Circular Pin Fins for Gas Turbine Blade Cooling Applications - Part 2 : Wake Flow Field Measurements and Visualization Using Particle Image Velocimetry,” Proceedings of ASME Turbo Expo 2001, GT2011-0181. [13] Weilin, Q. and Abel, S. H., 2009, “Experimental Study of Saturated Flow Boiling Heat Transfer in an Array of Staggered Micro-Pin-Fins,” International Journal of Heat and Mass Transfer, Vol. 52, pp. 1853-1863. [14] Simoneau, R. J. and Van Fossen, G. J., 1984, “Effect of Location in an Array on Heat Transfer to a Short Cylinder in Crossflow,” ASME Journal of Heat Transfer, Vol. 106, pp. 42-48. [15] Launder, B. E. and Spalding, D. B., 1974, “The Numerical Computation of Turbulent Flow,” Computer Methods in Applied Mechanics and Engineering, Vol. 3, pp. 269-289. [16] Wilcox, D. C., 1988, “Reassessment of the Scale-

(4)

In comparison with the circular pin-fin, thermal performance of Case 3 shows the maximum improvement by 44.3%. Enhancement of thermal performance of new pin-fin shape is attributed to the facts that the heat transfer is promoted by the enhanced horseshoe vortices while pressure loss is not changed remarkably.  The area-averaged entropy generation (  ) is presented in Fig. 15. Area-averaged entropy generations by the fan-shaped pin-fins (Cases 1-8) are smaller than that by the circular pinfin. Although the heat transfer enhancement for the fan-shaped pin-fins is higher than that for the circular pin-fin, a maximum decrease of 28.8% in the area-averaged entropy generation for Case 4 is observed. CONCLUSIONS The heat transfer coefficient and friction factor of a new fan-shaped pin-fin were evaluated numerically using RANS equations compared with a circular pin-fin at various Reynolds numbers. The low-Re SST turbulence model showed better performance than the k-ε and k-ω models, and the numerical results for the circular pin-fin show good agreements with the experimental data. The fan-shaped pin-fin shows remarkably improved area-averaged Nusselt number over the entire Reynolds number range (5,000-10,000) compared to the circular pin-fin. The improvement is prominent at higher Reynolds numbers, and the heat transfer rate for the fan-shaped pin-fin is increased by 22.8% at Re = 80,000. Effects of two geometric parameters of the fan-shaped pin-fin, the lateral reduction angle of the fan-shaped pin-fin and radius of rear part of pin-fin, on heat transfer and friction loss have been tested in a parametric study, and the results indicate that the heat transfer rate has maxima with the variations of these two parameters but the friction factor has minimum with the variation of the lateral reduction angle and monotonically increases with the rear part radius. ACKNOWLEDGENTS This work was supported by the National Research Foundation of Korea (NRF), grant No. 20090083510, funded by the Korean government (MEST) through Multi-phenomena CFD Engineering Research Center. REFERENCE [1] Lawson, S. A., Thrift, A. A., Thole, K. A. and Kohli, A., 2011, “Heat Transfer from Multiple Row Arrays of Low Aspect Ratio Pin Fins,” International Journal of Heat and Mass Transfer, Vol. 54, pp. 4099-4109. [2] Chyu, M. K., 1990, “Heat Transfer and Pressure Drop for Short Pin-Fin Arrays with Pin-Endwall Fillet,” ASME Journal of Heat Transfer, Vol. 112, pp. 926-932.

8


Determining Equation for Advanced Turbulence Models,” AIAA Journal, Vol. 26, pp. 1299-1310. [17] Menter, F. R., 1994, “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Application,” AIAA Journal, Vol. 32, pp. 1598-1605. [18] Bardina, J. E., Huang, P. G. and Coakley, T. J., 1997, “Turbulence Modeling Validation,” AIAA Journal, Vol. 35, pp. 1997-2121. [19] Garg, V. K. and Ameri, A. A., 2001, “Two-Equation Turbulence Model for Prediction of Heat Transfer on a Transonic Turbine Blade,” International Journal of Heat and Fluid Flow, Vol. 225, pp. 93-602. [20] CFX-11.0 Solver Theory, Ansys inc., 2008.

[21] Moon, M. A. and Kim, K. Y., 2012, “Prediction of Turbulent Heat Transfer around Pin-Fins,” 7th Turbulence, Heat and Mass Transfer, Paper No. J035. [22] Heinz, H. and Tammo, W., 2011, “Second Law Analysis of Momentum and Heat Transfer in Unit Operation,” International Journal of Heat and Mass Transfer, Vol. 54, pp. 1323-1330. [23] White, F. M., 2003, “Fluid Mechanics,” McGraw Hill, 5th Edition. [24] Blasius, P. R. H., 1913, “Das Äehnlichkeitasgesetz bei Reibungsvorgängen in Flussigkeiten,” Forschungsheft, Vol. 131, pp. 1-41. [25] Tuzson, J., 2000, “Centrifugal Pump Design,” John Wiley & Sons, New York, 1th Edition.

9
