Prediction of local shear stress and heat transfer between internal rotating cylinder and longitudinal cavities on stationary cylinder with various shapes

International Journal of Thermal Sciences 138 (2019) 512–520

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Prediction of local shear stress and heat transfer between internal rotating cylinder and longitudinal cavities on stationary cylinder with various shapes

T

A. Nouri-Borujerdi∗, M.E. Nakhchi School of Mechanical Engineering, Sharif University of Technology, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: Simulation of annular flow Turbulent flow Heat transfer Cavities with trapezoidal cross section

A numerical analysis has been performed to simulate flow structure, heat transfer, and pressure drop of turbulent flow in an annulus with a few longitudinal cavities on the outer stationary cylinder. The cross sections of cavities are rectangular, closed and open trapezoidal shapes. This kind of annular flow is applicable to industries applications such as electrical generators where heat generates in the cavities containing wires, heating of axial compressor rotor drams, rotating heat pipes for cooling of superconducting machines or motor rotor. The governing equations of turbulent flow are solved by using Renormalization group (RNG) k–ε model for Reynolds and Taylor numbers in the range of 5 × 103 < Rea < 6.5 × 10 4 and 160 < Ta < 1900 respectively. The angle between the sides and the base of the trapezoid cavity is in the range of 70∘ < β < 135∘ . The results show that the pressure drop is dependent on the cavity angle and reaches a maximum value at β = 91∘ , then declines. Furthermore, Sharp increase in heat transfer coefficient belongs to the corners of the cavity where are located in front of the fluid rotational flow. Furthermore, the averaged Nusselt number is dependent on both the effective Reynolds number and the aspect ratio but the Reynolds number is more effective. The present results are validated with available experimental data in the literature for rectangular cavities.

1. Introduction Fluid flow between two cylinders in the presence of rotational speed and axial velocity is called Taylor-Couette-Poiseuille flow. This type of flow is very popular in rotating heat pipes, gas cooled nuclear reactors, chemical mixers and electrical motors [1–3]. Heat transfer enhancement in rotating devices is one of the most important design factors. Therefore, to prevent overheating and failure the winding wire insulation of electric motors and generators cooling must be done well. Dirker and Meyer [4] conducted a comparative study of literature involving convection heat transfer in annulus. They concluded that more research is needed in the area of convective heat transfer correlations in concentric annulus, as little agreement is found among existing correlations. Gnielinski [5] developed a correlation for turbulent heat transfer coefficient on the basis of a large number of experimental data from the literature. In his research, a proven correlation for heat transfer in circular tubes was extended by factors that take into consideration the effect of the diameter ratio of the annulus and the different boundary conditions for heating or cooling. Lopez et al. [6] considered instabilities driven by the combination of rotation and thermal gradients. This instability determines the dynamics of complex geophysical, astrophysics and industrial flows. Ali and Weidman [7] ∗

performed a detailed linear stability analysis of such flows using axial periodicity and reported on the influence of the Prandtl number and on the stability boundaries. Their results showed a good agreement with the results of Snyder et al. [8] and, to a lesser extent with the results of Sorour and Coney [9]. Ali and Weidman attributed the discrepancies to the limitations of linear stability theory and the infinite-cylinder idealization to capture the experimental details. A similar linear stability analysis by Yoshikawa et al. [10] reported good agreement between numerical results and related results of Lepiller et al. [11]. Nonlinear simulations for small temperature gradients were provided by Ball and Farouk [12] who quantified the heat transfer across the system. Zhao et al. [13] numerically investigated the fully developed Taylor-Couette flow of a drilling fluid between two rotating cylinders. They concluded that the flow field is highly affected by the shear-thinning behavior of the drilling fluid between two cylinders. The effect of the inner cylinder movement on the heat transfer and melting of PCM of a double pipe heat exchanger was numerically investigated by Pahamli et al. [14]. The results show that inner pipe downward movement increases the convection heat transfer which reduces melting time up to 64%. Recently, cylindrical channels with longitudinal cavities on the surface are widely used for heat transfer enhancement due to higher fluid mixing [15–18]. The cavities disturb the incoming boundary layer

Corresponding author. E-mail addresses: [email protected] (A. Nouri-Borujerdi), [email protected] (M.E. Nakhchi).

https://doi.org/10.1016/j.ijthermalsci.2019.01.016 Received 31 March 2018; Received in revised form 18 November 2018; Accepted 14 January 2019 1290-0729/ © 2019 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 138 (2019) 512–520

A. Nouri-Borujerdi, M.E. Nakhchi

Nomenclature A a b c Cp D e f h k l L

N Nu p Pr q// r R Re T Ta z

Δ ε μ ν ρ τ Ω

surface area, m2 cavity pitch, m cavity depth, m constant, slot width, m specific heat, J/kg.K cylinder diameter, m air gap between two cylinders, R2-R1, m friction factor heat transfer coefficient, W/m2.K thermal conductivity, turbulent kinetic energy periphery length, m channel length, m number of slot Nusselt number, hDh / k Pressure, Pa Prandtl number, μCp/ k heat flux, W/m2 radial direction Radius, m Reynolds number, uDh / ν temperature, C Taylor number, (R1 Ω) Dh / νair axial direction

difference dissipation energy viscosity, Pa.s kinematic viscosity, m2/s density, kg/m3 shear stress, Pa rotational speed

Subscripts 1,2 a c eff h in s t w

inner, outer axial cavity effective hydraulic inlet smooth turbulence wall

Superscripts / e ]

fluctuation average over cross section average over entire channel

Greek symbols

β

cavity angle

and therefore the heat dissipation from the near wall fluid is increased [19]. Hayase et al. [20] performed a numerical study on heat transfer in annular channels with cavities on the inner and outer surfaces. They concluded that the heat transfer is enhanced up to 20% in the case of cavity on the inner surface and up to 10% in the case of cavity on the outer one. Sommerer and Lauriat [21] carried out a numerical study on heat transfer in an annulus with rectangular cavities which were embedded on the outer surface. They showed that the number of cavities has important effect on heat transfer enhancement inside channels. Fénot et al. [22] experimentally investigated the effect of the entrance region on the heat transfer of the annular channels with rotating inner cylinder including longitudinal cavities. They found that heat transfer is maximum at the leading edge of the cavities. In other experimental studies, Toghraie et al. [23] studied the effect of cavity height on the heat transfer and pressure drop of nanofluids inside the channels. The equations were discretized and numerically solved by using finite volume method. The results show that by increasing the height of the internal cavities, the convective heat transfer and friction factor can be significantly enhanced inside microchannels. Bilen et al. [24] presented an experimental investigation on the effect of slot aspect ratio on heat transfer and pressure drop of a fully developed turbulent air flow in different channels with cavity. Jeng et al. [25] conducted an experimental study on mounting four axial cavities on the rotating inner cylinder in the presence of axial air flow. They observed that the heat transfer increases by a factor of 1.4. Lancial et al. [26] numerically and experimentally investigated the temperature distribution and heat transfer on the rotor with cavity for laminar and turbulence air flows. It was observed that a hot spot is located near the downstream end of the rotor notch. They also conclude that the heat transfer coefficient on the pole face and in the notch region decreases with increasing z due to the growth of thermal boundary layers. Nouri-Borujerdi and Nakhchi [27–29] experimentally investigated the effect of mounting cavities on the outer stationary cylinder on heat transfer and pressure drop of annular flow between two cylinders. They obtained correlations for

Nusselt number and pressure drop as functions of the axial Reynolds number, Taylor number, number of cavities and their aspect ratio. They showed that the heat transfer to pressure drop ratio is a maximum at cavity aspect ratio ofb/ c = 1.42 . Most studies that focused on heat transfer and pressure drop inside annular channels were limited to rectangular cavities and only few studies were conducted on other cavity geometry shapes. Skullong et al. [30] studied the heat transfer characteristics in a solar air heater channel using wavy cavities. It was reported that the cavity pitch ratio has a significant effect on heat transfer enhancement of the solar air heaters. Abou-Ziyan et al. [31] presented an experimental investigation of convective heat transfer in an annular channel by mounting interrupted helical fins on a rotating inner cylinder. They concluded that the shape of the cavity has significant effect on heat transfer enhancement inside rotating annular channels. The effect of mounting longitudinal cavities with triangular cross section on the outer stationary surface on fluid flow inside rotating annular channels was discussed by Zhu et al. [32]. They found that, when the cavity height is smaller than the boundary layer thickness, the torque is the same as that of the smooth surfaces. The main goal of this article is to numerically investigate the heat transfer enhancement and pressure drop in annulus with different aspect ratios and geometry shapes of cavity in the presence of inner cylinder rotation and axial flow. Pressure drop and Nusselt number for various effective Reynolds numbers and cavity angles are studied. Thermal performance analysis has been utilized to obtain optimal design and physical parameters, Furthermore, streamlines and isothermal lines are also investigated. 2. Specification of the system In this section, the geometry, dimensions and aspect ratio of cavity on the stationary outer cylinder and smooth rotating inner cylinder are considered for flow in annular channels. The cross section of the 513

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A. Nouri-Borujerdi, M.E. Nakhchi

cavities is rectangular, closed and open trapezoidal shape with N = 10 cavities on the outer cylinder periphery of the annular channel (see Fig. 1). The inner and outer radii of the annulus are R1 = 44.1 mm and R2 = 47.5 mm respectively with length of L = 191 mm and the air gap width of e = 3.4 mm . The width of the cavity mouth is c = 5 mm with depth in the range of 2.5mm < b < 12.5mm . The cavity angle varies in the range of 70∘ < β < 135∘ . The flow is assumed to be steady state fully turbulent incompressible flow with the Reynolds number in the range of 5 × 103 < Rea < 6.5 × 10 4 , Taylor number of 160 < Ta < 1900 and the effective Reynolds number of 5 × 103 < Reeff < 6.64 × 10 4 . The stationary outer surface is kept at constant temperature of Tw2 = 40∘C while the rotating inner surface is at. Tw1 = 90∘C . The inlet air temperature is Tin = 25∘C . The axial Reynolds number, Taylor number as well as the effective Reynolds number are defined respectively as:

Rea =

uin Dh , νair

Ta =

uin2

(R1 Ω) Dh , νair

Reeff =

(1)

ϕ = [Vz , Vr , Vθ, T − Tw2]T

ϕ (0, r , θ) = 0 where,

ϕ = [Vz − uin , Vr , Vθ, T − Tin]T ,

σε

(11) (12)

P (z = 0) (13)

= Pi

ϕ = [Vz , Vr , Vθ, T ]T

(14)

and the periodic boundary condition in the tangential direction will be given as:

(3)

(5)

=

∂ϕ (z , r , θ) ∂θ

∂ϕ [z , r , θ + (a + c )/ R2] , ∂θ

(15)

θ denotes peripheral direction and superscript “T ” of matrix ϕ is an operator which flips the matrix ϕ over its diagonal. The local Nusselt number at any point on the inner or outer surface of the flow channel is obtained as: ∂T

where, the Reynolds shear stress and the turbulent viscosity are

k2 ε

ϕ (z , R2 , θ) = 0 where,

ϕ (z , r , θ) = ϕ [z , r , θ + (a + c )/ R2]

μ ∂T ⎫ ∂ ∂ ⎧⎡ μ (ρui T ) = + t⎤ ⎢ Pr Pr ∂x j ∂x i ⎨ t⎥ ⎦ ∂x i ⎬ ⎭ ⎩⎣

σk = 1,

(10)

∂ϕ (L, r , θ) = 0 where, ∂z

(4)

Prt = 0.85,

ϕ (z , R1, θ) = 0 where, ϕ = [Vz , Vr , Vθ − R1 Ω, T − Tw1]T

(2)

∂uj ⎤ ⎫ ∂p ∂ ∂ ⎧ ⎡ ∂ui ∂ (uj ρui ) = − [−ρui′uj′] + μ + + x ∂x i ∂x j ∂x j ⎨ ⎢ ∂x i ⎥ ∂x j ∂ ⎦⎬ ⎩ ⎣ j ⎭

μt = ρCμ

Cμ = 0.0845,

in the cylindrical coordinate z = x1, r = x2 and θ = x3 and their corresponding velocities are respectively Vz = u1, Vr = u2 and Vθ = u3. The boundary conditions of the fluid flow for a generic parameter of ϕ = [Vz , Vr , Vθ, T ]T are specified as follows:

The conservative governing equations of mass, momentum and energy for a three -dimensional steady state fully turbulent incompressible flow in an annulus are respectively as:

∂uj ⎤ ∂u 2 2 ∂u − ρkδij − μt k δij − ρui′uj′ = μt ⎡ i + ⎢ ∂x j ∂x i ⎥ 3 3 ∂xk ⎣ ⎦

C2ε = 1.68,

= 1.3

3. Governing equations and solution technique

∂ (ρui ) = 0 ∂x i

(9)

C1ε = 1.42,

Ω) 2 ,

4b (c + b cot β ) + 2(R22 − R12)(a + c )/ R2 4(A1 + A2 ) = (a + c ) R1/ R2 + a + c + 2b (1 + cos β )/sin β l1 + l2

∂ ∂ ⎧ ⎡ μt ∂ε ⎫ ε + μ⎤ (ρui ε) = + [C1ε Gk − ρC2ε ε ] ⎢ ⎥ ⎬ ∂x i ∂x j ⎨ ∂ σ x k ε j ⎦ ⎩⎣ ⎭

Where Gk = −ρui′uj′ ∂x is the production of kinetic energy. The constant i coefficients are chosen as:

+ 0.5(R1 in which, ueff = and uin is the axial inlet velocity and Ω is the rotational speed of the inner surface.Dh .is the hydraulic diameter of the annulus cross section and is obtained as follows:

Dh =

(8)

∂uj

ueff Dh νair

∂k ⎫ ∂ ∂ ⎧ ⎡ μt (ρui k ) = + μ⎤ + Gk − ρε ⎥ ⎢ ∂ ∂x i ∂x j ⎨ σ k ⎦ xj ⎬ ⎭ ⎩⎣

Nuj =

hj Dh k

=

Dh ⎡− ∂n ⎤ j⎦ ⎣ , Twj − T¯

j = 1, 2 (16)

The above relationship is used at any point on the inner or outer surface of the cylinder like GH or ABCDEF in Fig. (2) as a part of channel. j = 1 or 2 indicates the inner or outer cylinder. nj is a perpendicular vector to the surface. The averaged Nusselt number over the surface of ABCDEF or GH in Fig. (2) is obtained at any level of axial direction as:

(6)

(7)

Turbulent kinetic energy and energy dissipation are

Fig. 1. Part of an annular channel with dimensions and coordinate system. (a) rectangular cavity, β = 90∘, (b) closed trapezoidal cavity, β < 90∘, (c) open trapezoidal cavity β > 90∘ . 514

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A. Nouri-Borujerdi, M.E. Nakhchi

Fig. 2. Grid distribution in computational domain at a. cross section of a longitudinal cavity on outer cylinder surface. lj

hj Dh

Nuj =

k

∂T

Dh ∫ − ∂n dx =

0

j

(Twj − T¯ ) l j

j = 1, 2

,

(17)

where, l j measures the distance along the cylinder perimeter in which l1 = (a + c ) R1/ R2 and l2 = a + c + 2b (1 + cos β )/sin β are the periphery length of ABCDEF and GH respectively. T¯ is the averaged fluid temperature at any cross section area of the flow channel and is given by:

Fig. 3. Average Nusselt number of the entire outer cylinder surface vs. mesh number for a rectangular cavity (β = 90∘, b/ c = 0.5) .

points for our values of interest have reached a steady solution and (iii) the domain has imbalances of less than 1%. The RMS is the square root of the arithmetic mean of the squares of the error values. In the present work, we applied the criterion of 10−7 for summation of residuals in convergence monitoring of the solution results. Fig. 3 shows the averaged Nusselt number of the outer surface against the grid points for a rectangular shaped cavity, (β = 90∘, b/ c = 0.5). To verify the mesh independency study, the number of grid points is kept constant at 60 and 200 in the tangential and axial directions respectively but those in the radial direction changed to 16, 20, 24, 28, 32, 36 and 40 for different Reynolds numbers in the range of 7395 < Rea < 64600 . The analysis of the numerical results shows that almost nothing changes in the average Nusselt number if the radial grid number is larger than 32. Therefore, in order to minimize the computational time, the number of grid points in the radial direction is considered to be 32. In a similar manner, the number of grid points in the tangential and axial directions is considered to be 55 and 121 respectively. In these figures, the total grid points in the space between two cylinder is 32 × 55 × 121 = 212960 and the required grid points in the cavities is taken to be 24 × 18 × 121 = 52272 or in sum is identical to 265232 grid points. The smallest grid size located adjacent to the wall in the r-, θ− and z-directions are 5.04 × 10−5m , 7.43 × 10−5m and 6.42 × 10−5m , respectively.

∫ ρ Cp u TdA T¯ =

A

∫ ρ Cp u dA

,

j = 1, 2 (18)

A

where, A = 4b (c + b cot β ) + 2(R22 − R12)(a + c )/ R2 is the cross sectional area of the flow channel. ρ and Cp are fluid density and specific heat at constant pressure. In a similar manner, the averaged Nusselt number over the entire surface of the inner or outer cylinder along the flow channel yields. L lj

Nuj =

hj Dh k

∂T

Dh ∫ ∫ − ∂n dx dz =

0 0

j

(Twj − T ) l j L

,

j = 1, 2 (19)

where, 0 ≤ z ≤ L measures the distance along the channel. The average friction factor inside the channel can be defined as:

f=

Δp 1 ρuin 2 2

(20)

To solve the conservative governing equations of (3–5) and (8, 9), a staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite difference method. For the second-order spatial derivatives, the second upwind scheme of Shirvan et al. [33] is used. The numerical method is validated against available analytical solutions. Fig. 2 shows a non-uniform hexahedral mesh generation within a part of the flow cross-section. The mesh size is fine next to the surface due to the rapid changes in velocity and temperature gradients and is coarse far from the surface with mesh size increment of 1.04. The SIMPLE algorithm suggested by Patankar and Spalding [34] is employed for velocity-pressure coupling. The results of the (RNG) k- ε turbulent model were compared with (i) Standard k- ε model, (ii) Standard k-omega model and (iii) SST k-omega model. Among them, the (RNG) k-ε model gives the best agreement with the experimental results. The RNG model which is a mathematical technique that can be used to derive a turbulence model similar to the k- ε results in a modified form of the ε equation which attempts to account for the different scales of motion through changes to the production term. Therefore, in this work, the renormalization group (RNG) k- ε model has been used as a turbulent model.

4. Results and discussion 4.1. Code validation In order to investigate the accuracy of the present work, the numerical results of the friction factor of the entire surface of the outer cylinder with cavities are compared with the experimental data of Nouri-Borujerdi and Nakhchi [27] with rectangular cavity shape. Table 1 reports the numerical results of the average friction factor of a rectangular slot surface (β = 90∘) . The results are validated with the experimental data of Nouri-Borujerdi and Nakhchi [27]. Table 1 shows a good agreement between the numerical and experimental results. The corresponding errors for b/ c = 1 and 1.5 are in the range of 2.38 < Error < 6.25 and 0 < Error < 3.65 percent respectively. Table 2 presents this comparison for two different cavity aspect ratios of b/ c = 1 and 1.5 at Ta = 1.44 × 10 4 and for different effective Reynolds number in the range of 6211 < Reeff < 9367. The table shows a good agreement between the numerical and experimental results and the corresponding errors for the two cases of b/ c = 1 and1.5are in the range of 1.03 < Error < 2.94 and1.75 < Error < 3.94 percent respectively. In this table, the Taylor number is defined based on the

3.1. Convergence and mesh independence study In numerical simulations, the following conditions are required to ensure that the results are valid: (i) residual RMS error values have reduced to an acceptable value (typically 10−4 or 10−5 ), (ii) monitor 515

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A. Nouri-Borujerdi, M.E. Nakhchi

definition of ref. [27] when the Taylor number is defined as Ta = (Rm Ω e / νair )2e / Rm in which Rm = (R1 + R2)/2 and e = R2 − R1.

constant value when the air flow becomes hydraulically fully developed far from the inlet. It is found that the wall shear stress of the inner surface is higher than that of the outer surface. The total shear stress on the inner rotating cylinder is generally higher in comparison with the outer stationary one. This is due to the rotational speed of the inner cylinder (w = R1 Ω) which has a significant effect on the shear stress of the τrθ component. By moving forward from the inlet to the outlet through the air gap, the shear stress of the both walls decreases due to the pressure loss. Because of the insignificant momentum change, the wall shear stress on the both surfaces is balance by the pressure force. Fig. 7 presents the numerical results of pressure drop across the channel versus the effective Reynolds number with rectangular cavity (β = 90∘) for different aspect ratios at Ta = 1422 . The pressure drop increases with both the Reynolds number as well as the aspect ratio. The experimental data of Nouri-Borujerdi and Nakhchi [27] is added for comparison. When b/c = 2, the peak pressure drop reaches 338.6 Pa at Reeff = 3 × 10 4 . Fig. 8 illustrates the numerical results of the pressure drop across the channel versus the cavity angle at Ta = 900 and b/c = 0.5 for different effective Reynolds numbers. It is shown that the pressure drop increases with increasing the effective Reynolds number due to higher perturbations and mixing of the air flow inside the cavitiy and the air gap. Furthermore, the pressure drop is a function of the angle and aspect ratio of the cavity and reaches a peak value at about β = 91∘ , then is reduced with the open mouth. The reason is that the air flow can move easily in the open trapezoidal shape (β > 90∘) and faces with less resistance in comparison with the closed one (β < 91∘). Table 3 shows the effect of the cavity aspect ratio on the channel pressure drop against the Taylor number. The results show that pressure drop enhances with increasing the aspect ratio and is highest at b/ c = 2.5. This is because of the high depth of the cavity that the fluid cannot spin easily due to the rotation of the inner cylinder, compared with the short depth. In addition, the fluid has an axial motion and in turn causes more pressure losses than the rest.

4.2. Flow structure and isothermal lines

4.4. Heat transfer

Fig. 4 illustrates the streamlines and isothermal lines of air flow inside a cavity of the annulus with rectangle, open and closed trapezoid shape at axial Reynolds number of Rea = 5800, Taylor number of Ta = 1400 and at section of z / L = 0.95. If the rotational speed of the inner cylinder is large enough, the viscous drag of the rotating cylinder would set up a swirling flow inside the cavity. In an open trapezoidal shape (β > 90∘), the fluid temperature is more affected by the fluid within the air gap between two cylinders and more mixing process occurs. As a result more heat transfer rate between the fluid flow and the surface is expected to occur in comparison with the closed trapezoidal or rectangular shape. Fig. 5 plots the isothermal lines of air flow inside the annulus with a rectangular shape for Reeff = 1.26 × 10 4 , Ta = 1085 and β = 90∘ at four axial locations of z / L = 0.05, 0.35, 0.65 and 0.95. It is seen that at the beginning of the channel, the isothermal lines are parallel and very close to each other so that the temperature gradient is the biggest and gradually decreases along the channel. As a result, the heat transfer coefficient at the beginning is very high and gradually approach a constant value if the channel length is long enough corresponding to a fully developed condition.

Fig. 9 presents the local Nusselt number along the surface with cavity at the axial location of z / L = 0.5 and Ta = 164 for three shapes of rectangular, closed and open trapezoidal. If the inner cylinder rotates clockwise, fluid collides to the leading edge of the cavity (point E on Fig. 2) and the local Nusselt number increases sharply. This enhancement can be attributed to the stagnant point while, the Nusselt number at points C and D reach a minimum value due to the immobility of the fluid. The trend of the Nusselt number is in agreement with the numerical results of Bouafia et al. [35] and Lancial et al. [36] in an annular flow with a rectangular cavity. Fig. 10 reports the local Nusselt number along the cavity periphery at the level of z / L = 0.5 for different effective Reynolds numbers in the range of 7395 < Reeff < 64600. The Taylor number, the aspect ratio and the angle of the open trapezoidal are Ta = 164, b/ c = 1 and β = 100∘ respectively. The sharp difference between the results of Figs. 10 and 9 is related to the axial velocity which exists in Fig. 10. It is observed that the Nusselt number remains nearly constant as long as the effective Reynolds number is large enough. If the effective Reynolds number is reduced due to the reduction of the axial velocity component, the rotational velocity component will be predominant and point E plays as a stagnant point. In other words, the local Nusselt number of point E sharply increases. Table 4 indicates the numerical results of the averaged Nusselt number on the outer cylinder at level of z / L = 0.5 against the effective Reynolds number at β = 100∘ for different aspect ratios. The averaged Nusselt number increases with both the effective Reynolds number and aspect ratio but the Reynolds number is more effective. It is expected at very low Reynolds numbers the thermal boundary layer grows very fast and the cavity dimensions to be more effective.

Table 1 Average friction factor of the rectangular cavity and experimental data of Nouri-Borujerdi and Nakhchi [27]. Reeff

f b/ c = 1

6211 6704 7395 7789 8282 8775 9367

b/ c = 1.5

Present work

Exp [27].

Error%

Present work

Exp [27].

Error%

0.084 0.083 0.082 0.081 0.080 0.078 0.075

0.089 0.087 0.084 0.085 0.082 0.080 0.080

5.61 4.60 2.38 4.71 2.44 2.50 6.25

0.094 0.092 0.089 0.086 0.083 0.081 0.079

0.094 0.093 0.089 0.088 0.084 0.084 0.082

0.00 1.08 0.00 2.27 1.19 3.57 3.65

Table 2 Average Nusselt number of the outer cylinder surface with a rectangular cavity (β = 90∘ ) and experimental data of Nouri-Borujerdi and Nakhchi [27]. Reeff

Nu2 b/ c = 1

6211 6704 7395 7789 8282 8775 9367

b/ c = 1.5

Present work

Exp [27].

Error%

Present work

Exp [27].

Error%

21.77 22.93 25.16 27.89 31.74 33.19 33.88

22.43 23.17 25.91 28.67 32.64 34.03 34.64

2.94 1.03 2.89 2.72 2.75 2.47 2.19

23.17 26.67 29.11 31.89 32.23 34.41 35.85

24.12 27.22 29.63 32.56 33.18 34.92 36.40

3.94 2.02 1.75 2.05 2.86 1.46 1.51

4.3. Pressure drop and local shear stress Fig. 6 shows total wall shear stress along the inner smooth and outer surface with cavity. In this figure, the cavity aspect ratio is b/ c = 0.5, Ta = 1250 and β = 90∘. The wall shear stress decreases monotonically in a similar manner to surface heat flux. This is because of the analogy between the energy and momentum equations. Furthermore, the wall shear stress is a maximum at the inlet of the channel and tends to a 516

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A. Nouri-Borujerdi, M.E. Nakhchi

Fig. 4. Streamlines (right) and isothermal lines (left) at section of z / L = 0.95 for three different shapes of cavity at Rea = 5800 and.Ta = 1400 .

for b/ c = 2.5 in comparison with small values of b/ c. Also, the effect of the Taylor numbers on the Nusselt number is very prominent especially at large b/ c. This is due to the rotational speed of the inner cylinder which causes a thin boundary layer and formed next to the surface.

Table 5 presents the averaged Nusselt number of the outer surface at the level of z / L = 0.5 against the Taylor number for different aspect ratios of β = 100∘ and Rea = 5000 . The Taylor number is defined based on Eq. (1). The variation of the averaged Nusselt number is significant

Fig. 5. Isothermal lines at different axial locations along the annular channel for Reeff = 1.26 × 10 4 , Ta = 1085 and.β = 90∘ . 517

International Journal of Thermal Sciences 138 (2019) 512–520

A. Nouri-Borujerdi, M.E. Nakhchi

Table 3 Pressure drop across the annular channel vs. Taylor number for different cavity aspect ratios at β = 100∘ and Rea = 6000 . Ta

508 615 745 983 1149 1366

ΔP [Pa] b/ c = 0.5

b/ c = 1

b/ c = 1.5

b/ c = 2

b/ c = 2.5

44.7 47.3 49.9 51.5 53.8 55.3

47.5 48.4 51.3 55.7 57.1 60.1

51.4 53.1 55.6 59.0 61.9 65.0

53.0 55.2 58.4 62.1 65.7 73.7

59.2 61.8 68.5 75.2 83.4 94.2

Fig. 6. Periphery averaged wall shear stress along the inner and. outer surface with cavity for different effective Reynolds numbers.

Fig. 9. Local Nusselt number along the cavity periphery without axial velocity at the axial location of z / L = 0.5 and Rea = 0, Ta = 164 .

Fig. 7. Pressure drop across the channel with rectangular cavity. (β = 90∘) for different aspect ratios at.Ta = 1422

Fig. 10. Local Nusselt number along the cavity periphery with axial velocity for different effective Reynolds numbers at β = 100∘, Ta = 164, z / L = 0.5 and.b/ c = 1. Fig. 8. Pressure drop across the channel vs. angle of aspect ratio. at Ta = 900 and b/c = 0.5 for different effective Reynolds numbers.

flow is thermally developing in the air flow path which is in agreement with experimental study of Fenot et al. [22]. The results show that the Nusselt number of the open trapezoidal shape is higher than that of the other two shapes. This is mainly due to the better fluid mixing between the air gap and the open trapezoidal cavity. By moving forward from inlet to the outlet, the entrance effect becomes smaller and conformably

Fig. 11 shows the local Nusselt number of the outer cylinder at b/ c = 1 for three different cavity angles accordance with the rectangular, closed and open trapezoidal shapes. It can be observed that the fluid 518

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A. Nouri-Borujerdi, M.E. Nakhchi

decreases with increasing the Reynolds number.

Table 4 Average Nusselt number of the outer cylinder surface with cavity vs. the effective reynolds number for different cavity aspect ratios at Ta = 1250, z / L = 0.5 and β = 100∘ . Reeff

4651 12457 20329 28552 36750 44242 52397 59335 64270

4.5. Thermal performance There are two design parameters for the cavity configuration: one the aspect ratio and the other is angle. In order to optimize these two parameters simultaneously, the overall heat transfer and the total pressure drop across the channel are obtained by varying these two parameters in the range of 5000 < Reeff < 66400. The performance of heat transfer of air flow in between two cylinders is measured by (Nuc 2/ Nus2)/(fc 2 / fs2 ) [27]. Table 6 shows thermal performance of three geometric shapes including rectangular, open and closed trapezoidal. It turns out that the rectangle has a better performance than the rest, because the reaction between the surface and the fluid is facing less resistance. Conversely, the closed trapezoidal has the worst performance.

Nu2 b/ c = 0.5

b/ c = 1

b/ c = 1.5

b/ c = 2

b/ c = 2.5

21.7 40.3 58.4 79.1 93.5 110.6 129.6 142.0 150.6

27.6 49.5 67.2 90.5 107.3 121.3 140.1 151.3 159.4

32.0 55.1 73.3 99.6 118.9 135.4 154.5 170.3 177.0

34.7 61.9 85.1 113.2 132.7 152.0 172.8 187.5 193.1

35.1 66.8 92.0 130.0 151.2 178.5 203.4 218.8 239.7

Table 5 Average Nusselt number of the outer cylinder surface with cavity at z / L = 0.5 vs. Taylor number for different aspect ratios at β = 100∘ and Rea = 5000 . Ta

508 615 745 983 1149 1326

5. Conclusions This paper deals with numerical investigation of annular flow between two concentric cylinders: the inner cylinder is rotating with a smooth surface and the other one is stationary surface with some longitudinal cavities. The geometry of the cavities consists of rectangular, closed and open trapezoidal shapes. The effect of dimensions and geometry of the cavity have been considered on heat transfer and pressure drop in the range of 5 × 103 < Rea < 6.5 × 10 4 and 165 < Ta < 1900. In the open trapezoidal shape, β > 90∘, the fluid temperature is more affected by the fluid between the two cylinders and more mixing occurs. As a result more heat transfer rate is expected to occur in comparison with the closed trapezoidal or rectangular shape. The total shear stress on the rotor is generally higher than that of the outer surface. This is mainly due to the rotational flow near the rotor. Based on the present results, the pressure drop is dependent on the cavity angle and reaches its maximum value at β = 91∘. At any point in the channel, the average air temperature increases by decreasing the Reynolds number. Sharp increase in heat transfer coefficient belongs to the corner of the cavity where is located in front of the fluid rotational speed. Furthermore, the averaged Nusselt number is dependent on both the effective Reynolds number and the aspect ratio but the Reynolds number is more effective.

Nu2 b/ c = 0.5

b/ c = 1

b/ c = 1.5

b/ c = 2

b/ c = 2.5

92.3 93.2 93.9 95.1 96.0 96.4

99.4 101.3 103.1 105.6 108.3 108.2

107.1 109.0 112.8 114.2 116.9 120.3

118.3 124.7 130.5 139.0 144.2 146.9

128.0 138.9 148.5 165.7 176.4 185.1

Acknowledgement This research was financially supported by Vice Research and Development of MAPNA GROUP, under the contract No. 4600000164.

Fig. 11. Local Nusselt number of the outer cylinder along the channel at b/ c = 1.

to that the difference among the local Nusselt numbers decreases. Fig. 12 reports the dimensionless average temperature of air flow along the channel at constant wall temperature of Tw1and Tw2, Taylor number of Ta = 1250, cavity aspect ratio of b/ c = 0.5 with the cavity angle of β = 90∘ for different effective Reynolds numbers. The results show that the average air temperature increase along the channel at each specified effective Reynolds number. Because in a flow channel under a constant wall temperature, the average fluid temperature is ˙ p) in which the overall heat transfer proportional to exp(−πDh Uz / mc coefficient is U ∼ Reneff and the mass flow rare is m˙ ∼ Reeff . Since in a laminal or turbulent flow n < 1, then the fluid temperature decreases along the channel, z, and increases with increasing of the effective Reynolds number. The slope of the temperature is also steeper at low Reynolds numbers. This means that the average air temperature

Fig. 12. Average air temperature along the channel at Ta = 1250 for different effective Reynolds numbers. 519

International Journal of Thermal Sciences 138 (2019) 512–520

A. Nouri-Borujerdi, M.E. Nakhchi

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Table 6 Thermal performance of outer cylinder with different cavity shapes. Slot shape

β∘

b/c

Nuc 2 Nus2

fc 2 fs2

Nuc 2 fc 2 / Nus2 fs2

70 80 90

1.8 1.7 1.7

2.14 2.26 2.37

1.81 1.88 1.64

1.18 1.20 1.44

100 105 110

1.5 1.6 1.2

2.62 2.73 2.45

1.52 1.47 1.47

1.72 1.86 1.66

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