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Heat transfer from a non-isothermal rotating rough disk subjected to forced flow
International Communications in Heat and Mass Transfer 110 (2020) 104395
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Heat transfer from a non-isothermal rotating rough disk subjected to forced flow
T
Muhammad Usmana, , Ahmer Mehmooda, Bernhard Weigandb ⁎
a b
Department of Mathematics & Statistics, International Islamic University Islamabad, 44000, Pakistan Institute of Aerospace Thermodynamics, University of Stuttgart, Germany
ARTICLE INFO
ABSTRACT
Keywords: Sinusoidal rotating disk Three-dimensional boundary-layer Non-similar flow Heat transfer enhancement Non-isothermal disk Forced flow
This communication aims to report heat transfer characteristics from a non-isothermal wavy disk rotating in a forced flow. The primary objective of this kind of studies is to seek a suitable mechanism for the quick removal of heat energy from a rotating disk surface. The achievement of an efficient mechanism which ensures a further higher rate of heat transfer is a fundamental goal in such studies. In this regard active and passive techniques are of fundamental importance whereby the current study utilizes a combination of both. There are three key elements, namely, the nonuniform disk temperature; surface roughness, and a radial free flow, which are responsible for the augmentation of the heat transfer. The analysis has been carried out for a variety of fluids depicting the increased rate of heat transfer due to a variation of these different elements. In doing so, the local and global heat transfer rates are manipulated in order to obtain a clear picture of the heat transfer process at a corrugated disk. The considered rough non-isothermal disk (with two sinusoids) rotating in the uniform stream of air (Pr = 0.71) leads to a significant (about 263%) enhancement in the overall heat transfer rate compared to that of a flat free rotating disk having surface temperature as quatratic function of the radial coordinate. Moreover, some threshold values (which correspond to seizing the heat transfer process) of the used power-index of the disk temperature (Tw − T∞ = c0rn∗) are also identified which are observed to vary slightly due to the surface irregularities and the relative motion of fluid and disk.
1. Introduction Rotating disks can be found in turbomachinery [1], rotating disk electrode, rotating machinery, rotating disk contractors [2], rotating heat exchangers [3,4] and several other technical applications. The occurrence of flow and heat transfer in all these units requires the modeling of rotating disk systems which can be classified into rotating cavities, rotor-stator systems, and free disks. The free rotating disk serves as a first prototype being a preliminary stage, with the simplest model, to analyze the different technical applications. Numerous theoretical and experimental works have been devoted to free rotating disk flow concerning to its important role in all aforementioned mechanical rotating devices after the pioneering study of von-Kármán [5]. Instead of making a review on the large number of literature available on this subject related to free rotating disk flow, the reader is referred to a quite recent monograph by Shevchuk [6,7] which not only includes a detailed review but a rigorous analysis of convective heat transfer in rotating disk systems under different situations including uniform or non-uniform temperature distribution on the disk surface. The impact of an outer forced flow orthogonal to a rotating disk is highly useful for cooling or heating the disk surface. Particularly, rotor
⁎
end surfaces in a gas turbine are cooled through the application of such flows. Fundamental work regarding the analysis of these flows on a rotating flat disk along with heat transfer phenomenon can be found in [8–11]. More recently, Wiesche and Helcig [12] showed theoretical as well as experimental investigation concerning convective heat transfer from a rotating disk subjected to a free stream of air. They pointed out the advantageous aspects (especially on heat transfer enhancement) of a forced flow on a rotating disk in different situations. Despite having huge literature related to the flat disk case with (or without) a forced flow, there are some vital areas to which less attention had been given in the past. One is the consideration of surface irregularities on the rotating disk surface. In Sec. 3 it will be realized that the surface topography plays an important role towards the enhancement of heat transfer characteristics. In 1989, Le Palec [13] explored the influence of a bumpy surface on the heat transfer phenomenon from an isothermal rotating disk placed in a quiescent air. For mathematical simplicity he introduced the surface irregularities in the shape of periodic sinusoids which resulted in a 15% heat transfer increase in comparison to a flat disk. A bit later, Le Palec et al. [14] presented a theoretical and experimental study
Corresponding author. E-mail addresses: [email protected] (M. Usman), [email protected] (B. Weigand).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104395
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International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
confirming such a significant heat transfer enhancement owing to a corrugated disk surface subjected to uniform wall heat flux. In [13,14] Le Palec and his co-authors did not investigate the influence of surface corrugation on flow properties. Yoon et al. [15,16] explored the flow variation due to disk surface undulations. Recently, Mehmood et al. [17] pointed out that the wavy disk surface not only augments the heat transfer rate but also increases the torque and mass flow rate. The increased torque increases the overall power of rotating machinery while the increased mass flow rate contributes towards the heat transfer augmentation. It is already known that a significant heat transfer enhancement can be achieved owing to the application of forced flow on a flat rotating disk (see for instance [8–12]). Therefore, provoking from the idea of [13,14] of surface corrugation and employing forced flow, the present work combines these two phenomena (i.e. corrugated disk in forced flow) along with the consideration of non-uniform temperature distribution on the disk surface. The non-uniform disk temperature is considered in the form of a power-law function of its radius which also includes the isothermal disk temperature condition as one special case. Theoretically, any non-constant smooth function of the radial variable can be considered. But the power-law form, particularly chosen here, bears a practical reason. The temperature distribution on the walls of rotating cavities in a gas turbine, actually, does also follow a power-law form. Effects of temperature distribution (power-law) exponent and forced flow on the convective transport process are observed and discussed in detail. The contribution of surface undulations towards flow and heat transfer augmentation has also been noted in comparison with the flat disk case. The flow induced by the rotation of a wavy disk without free flow addressed by Le Palec [13] and the axisymmetric stagnation point flow on a resting or a rotating disk studied by Mabuchi et al. [9] are also recovered as special cases of this study.
Fig. 1. Configuration of wavy disk.
Accordingly u = (ux, uy, uθ) denotes the related velocity vector. The kinematic viscosity and thermal diffusivity are respectively denoted by ν and α. The fluid temperature is denoted by T and the ambient temperature is T∞. The disk temperature is Tw and is taken to be non-uniform, following a power-law distribution of the form Tw = T∞ + c0rn∗, where n∗ is a real number and is called the power-index of the disk temperature, and c0 is a constant having suitable dimensions. The flow and thermal boundary conditions are summarized as
ux = u y = u r = 0, Tw = T + c0 r n , u x = ar , u = 0, T = T ,
uref
one (two) reference velocities present in this problem, see for instance Eq. (6), a sum of the two can be regarded as an appropriate reference velocity. Thus the variables x and y are normalized as
The physical model consists of a free rotating disk with a sinusoidal wavy surface subjected to a steady radial forced flow in an inertial frame of reference. The waviness of the disk surface is described by a smooth periodic function of the form
r s (r ) = a0 cos (2N r ); r = b
=
(1)
y
+
ux dr =0 r dx
ux
ux
u u + uy x y
ux u r
dr = dx
+ y
= (f f g
T T + uy = x y
Pr
(4)
=
x uref
(7) 2 1/2
uref x
(x , y ) , g( , ) = r
uref x
(x , y ) , r
(, )=
T Tw
T T
(8)
+
2
g2 +
1 3 dR + ff 2 2R d
+
2
a
f f)
(9)
dR 1 3 dR f g + + fg Rd 2 2R d 1 3 dR + f 2 2R d
dR Rd
dR n f Rd
= (f g
= (f
g f)
f)
(10) (11)
whereby satisfying the continuity Eq. (2) identically. In the transformed system of Eqs. (9)–(11), R = r/λ denotes the dimensionless radial coordinate and Pr = α/ν is the Prandtl number. These partial differential Eqs. (9) and (10) reduce to the classical self-similar form derived by von Kármán [5] by setting a0/λ = 0 & a/ω = 0 and to those investigated by Le Palec [13] for a0/λ ≠ 0, a/ω = 0, & n∗ = 0 with the consideration of Eq. (11). The associated boundary conditions is also normalized due to Eqs. (7) and (8) and takes the form
2T
y2
2
(3)
Energy equation:
ux
;
dR 2 f R d
f
2u
y2
y
The utilization of these relations ((7) and (8)) in Eqs. (3)–(6) yield the following set of non-similar governing equations
2u
x y2
=
f( , )=
Momentum equations:
u 2 dr u = ux x r dx x
,
where uref = ΛR and Λ = (a + ω ) . The corresponding dimensionless dependent variables in terms of stream functions and dimensionless temperature are obtained as
(2)
ux u + uy x x y
x
2
where a0 denotes the amplitude of the wavy surface, N denotes the number of waves fitted onto the disk radius b, and r denotes the radial coordinate. The ratio of wave amplitude a0 to the wavelength λ is kept small enough so that the boundary-layer assumptions remain valid. This allows one to reduce the Navier-Stokes, and energy equations to momentum and thermal boundary-layer equations. In view of these assumptions the governing boundary-layer equations of the problem under investigation. Continuity equation:
ux + x
(6)
The flow under consideration is of non-similar nature because of the presence of appropriate characteristic lengths in every spatial direction. Therefore, the curvilinear variable x is normalized by the wavelength of the surface undulations and the wall normal variable y is normalized by the boundary-layer thickness. Similar to other boundary-layer flows the boundary-layer thickness varies as ~ x . Since there are more than
2. Transport equations and auxiliary data
uy
at y = 0 as y
(5)
In Eqs. (1)–(5) an orthogonal curvilinear coordinate system (x, y, θ) is used (see Fig. 1) where x, y, and θ describe the variables in longitudinal; normal, and azimuthal directions to the wavy surface, respectively. 2
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
Table 1 Comparison of numerical data of heat transfer rate for non-isothermal flat disk. a/ω
0
0.5 1 10
Pr
n∗ = − 2
0.1 1 10 100 0.71 1 0.71 1 0.71 1
n∗ = 0
n∗ = 2
n∗ = 4
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0766 0.3963 1.1341 2.6871 0.4194 0.4815 0.4897 0.5536 0.6381 0.7270
0.0766 0.3963 1.1341 2.6871 0.4194 0.4815 0.4897 0.5536 0.6381 0.7270
0.1417 0.6159 1.6206 3.7422 0.6049 0.6910 0.6905 0.7849 0.9037 1.0250
0.1417 0.6159 1.6206 3.7422 0.6049 0.6910 0.6905 0.7849 0.9037 1.0250
0.1977 0.7557 1.9460 4.4421 0.7305 0.8324 0.8283 0.9395 1.0808 1.2233
0.1977 0.7557 1.9460 4.4421 0.7305 0.8324 0.8283 0.9395 1.0808 1.2233
where the bar refers to the quantities used in [6,7]. Clearly, Table 1 shows an excellent agreement of the present results with those reported in [6,7]. Moreover, the present results have been obtained after several runs by taking different step sizes to ensure the grid independence. Thus, the reported data is sure to be accurate and grid independent. 3. Heat transfer characteristics The consideration of a non-uniform disk temperature and the presence of an external flow significantly influence the heat transfer characteristics. Based upon the practical implementations of intentionally roughened surfaces in state-of-the-art heat transferring devices, operating under extreme conditions, our primary focus is to calculate a net gain in the heat transfer rate and to identify those situations where it is achieved. The local rate of the dimensionless heat transfer is usually expressed as local Nusselt number and is defined as
Fig. 2. Spatial variation of Nusselt number for different values of n∗ at fixed a/ ω.
Re
1/2
Nu =
R 2
1= = 0,
1 = 0,
at
=0
at
=
(12)
The non-similar partial differential Eqs. (9)–(11) along with the boundary conditions (12) have been solved numerically by applying the Keller-box method, see [18] programmed in MATLAB. The classical von Kármán [5] self-similar (ε = 0) solution serves as an initial solution to start the solution process. Validity of the code is justified by attaining the results of the isothermal wavy disk case studied by Le Palec [13]. Moreover, the current procedure is also validated by comparing the results for the case of a non-isothermal flat disk rotating in uniform axial flow studied by Shevchuk [6,7] (see Table 1). The current variables, defined in (7) & (8), follow the following relationship with those of [6,7]:
=
g (1 + k 2)1/2 f 1+k h , = , = 1 + k, 1+k f (1 + k 2)1/2 g h
=
1+k ; k = a/ (1 + k 2)1/2
(13)
where Re = is the rotational Reynolds number. The spatial variation of the heat transfer rate (for n∗ ≠ 0) is illustrated in Fig. 2 depicting a wavy nature of the heat transfer rate undergoing two cycles per wavelength (wavelength of Nu ≅ λ/2) with an decreasing amplitude. Such an exhibition of local Nusselt number with increased frequency had already been reported by Le Palec [13] for an isothermal free rotating wavy disk (n∗ = 0). Clearly (see Fig. 2) the higher values of temperature distribution exponent (n∗ > 0) correspond to an increase local heat transfer rate and its decreasing values (n∗ < 0) result in lowering the local heat transfer rate in comparison to an isothermal disk (n∗ = 0). A similar trend of the local heat transfer rate is also shown in Fig. 3 when the relative velocity is varied (i.e. a/ω ≠ 0). This Figure illustrates that an increase in relative velocity ratio leads to an increase in the associated local heat transfer rate. This is an obvious fact which has already been observed for a flat disk too. The periodicity of the local Nusselt number is again twice in comparison to the surface topology. This establishes the fact that the curvature of the local Nusselt number curves is entirely due to the surface topography and is not altered by the additional features such as the variable disk temperature or the external forced flow etc. A zero value of relative velocity ratio refers to a flow induced purely by the rotation of a sinusoidal disk, whereas its infinite value corresponds to the axi-symmetric stagnation point flow over a resting wavy disk. An important fact which can be seen in these Figures (Figs. 2 & 3) is that the surface irregularities of the wavy disk reduce the local Nusselt number in comparison to a smooth disk. This behavior is a little bit deceiving and had already been reported for an isothermal sinusoidal-shaped free rotating disk (i.e. n∗ = 0 & a/ω = 0) by Le Palec [13]. From such a behavior one should not simply assume that the surface undulations cause to reduce the heat transfer rate. In Figs. 2 and 3 Nu has been plotted against the flat radial coordinate which also serves as longitudinal variable for a flat disk but not for a wavy disk. For a wavy disk the longitudinal
Fig. 3. Spatial variation of Nusselt number for different values of a/ω at fixed n∗.
f=g=f =g a f = , g = 0,
( , 0)
3
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
Table 3 Dependence of threshold values of n∗ (for which heat transfer process is seized from the disk surface) on surface waviness with N = 2 and Pr = 0.71. a0/λ
a/ω
0.0 1/16 1/10
Table 2 Values of mean Nusselt number Nup for different temperature distribution exponent and relative velocity ratio with N = 2 and Pr = 0.71. a0/λ
n∗ = − 1
n∗ = 0
n∗ = 1
n∗ = 2
n∗ = 4
0.0
0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10
0.1893 0.1928 0.1979 0.3132 0.3190 0.3272 0.3763 0.3832 0.3930 0.4174 0.4251 0.4358 0.4392 0.4473 0.4588
0.3259 0.3318 0.3403 0.4858 0.4947 0.5073 0.5765 0.5870 0.6019 0.6368 0.6484 0.6648 0.6690 0.6812 0.6987
0.4319 0.4397 0.4510 0.6065 0.6175 0.6332 0.7143 0.7273 0.7457 0.7872 0.8014 0.8217 0.8262 0.8412 0.8627
0.5185 0.5279 0.5413 0.7006 0.7133 0.7315 0.8211 0.8360 0.8571 0.9033 0.9196 0.9428 0.9476 0.9647 0.9893
0.6555 0.6674 0.6842 0.8461 0.8614 0.8832 0.9851 1.0028 1.0281 1.0812 1.1007 1.1284 1.1332 1.1536 1.1829
0.5 1 2 ∞
0.5
1
2
∞
−2 -2 −1.9996
−2 -2 −2.0001
−2 −2.0001 −2.0004
-2 −2.0001 −2.0004
-2 −2.0001 −2.0004
variable is x which runs along the disk texture radially outwards. Therefore, a comparison between the two situations (flat and wavy disks) can't be made from here because the curvilinear radius of a wavy disk is larger than the radius of corresponding flat disk. To ensure a comparison of the two situations it is necessary to bring the two data on equal scaling; either the flat disk area can be regarded as reference and the results of the wavy disk should be reduced accordingly or the wavy disk area can be taken as reference where the results of the flat disk should be reduced accordingly. For this purpose we take the flat disk surface area as reference and thus have to increase the Nusselt number of the wavy disk case by the area ratio S 2 , as was also done by Le Palec [13]. Before b doing so, it is important to calculate the average Nusselt number in order to get a clear picture. The average heat transfer rate from a non-isothermal wavy disk is measured by means of the average Nusselt number (Nu ) which is acquired by integrating Eq. (13) over the whole disk surface and then dividing by the disk surface area, as
Fig. 4. Mean Nusselt number Nu (curve 1) and Nup (curve 2) under the influence of a non-uniform temperature distribution and the amplitude-to-wavelength ratio.
a/ω
0
NuRe
1/2
=
1 S
Re
1/2
Nu dS
S
(14)
where S refers to the wavy disk area. As mentioned above, this average Nusselt number Nu is corrected by an area ratio S 2 to get the upgraded b average Nusselt number, ready to compare with the flat disk results. This can be expressed by
Nup = Nu
S b2
(15)
Notice that as the surface undulations are flattened out (i.e. a0/λ →
Fig. 5. Heat transfer rate (Nup Re
1/2 )
plotted against number of sinusoids (N) for Pr = 0.71.
4
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
0) the surface area S → πb2 (of the corresponding flat disk) and Nup Nu , the Nusselt number of the corresponding flat disk. These calculations (Eqs. (14) & (15)) are shown in Fig. 4 where three values of the temperature distribution exponent n∗ are considered and a same pattern for the overall heat transfer rate is observed for each considered value of n∗. In Fig. 4, curve 1 shows the calculation of Eq. (14) (i.e. without correction) and it clearly decreases with increasing values of the surface roughness ratio a0/λ. However, the mean Nusselt number obtained from Eq. (15) (i.e. with correction) shown as curve 2 in Fig. 4 reflects that heat transfer is enhanced with increasing roughness ratio of the disk. Apart from this it can also been noted in Fig. 4 that the temperature distribution exponent n∗ has also a substantial influence on the heat transfer enhancement and higher values yield considerable improvement in the exchange of heat energy from a hot wavy disk to the cold surrounding fluid. For instance, the considered rough non-isothermal disk rotating in a uniform stream of air (Pr = 0.71, a/ω = ∞ & a0/λ = 0.1) leads to a significant (about 263%) enhancement of the overall heat transfer rate compared to that of a flat free rotating isothermal disk when the wavy disk (with two sinusoids) temperature is quatratic function of its radial distance (see Table 2). Whereby an isothermal rough rotating disk in a forced flow gives an enhancement of about 114% when compared to flat free rotating isothermal disk. A value of 3.46 (for any a0/λ) in the heat transfer enhancement is observed when n∗ is varied from −1 to 4 for the situation of flow induced by pure rotation of sinusoidal disk (i.e. a/ω = 0). For a relative velocity ratio a/ω = 0.5 & ∞, a rise of 2.70 & 2.58 (respectively) times in average heat transfer rate can be observed. Obviously, the larger values of a relative velocity ratio correspond to frail the influence of the temperature distribution exponent. Interestingly, the values of the temperature exponent n∗ find no association with the surface waviness but it enhances the heat transfer rate at the same time whether the disk is flat or rough. The average heat transfer is also found to depend on the number of sinusoids fitted to the disk radius 0 ≤ r ≤ b as depicted in Fig. 5. It is observed that whatever the flow situation is (for any a/ω) and whether the disk is isothermal or non-isothermal (for any n∗) the mean Nusselt number increases slowly with an increase in number of cycles N up to N = 4; and after that it becomes almost constant. According to Shevchuk [6], the heat transfer process from a free rotating flat disk seizes (i.e. Nup = 0 ) at a value n∗ = − 2. This value does also remain invariant for a flat rotating disk in the presence of a radial pressure gradient or with an alteration of the Prandtl number. But interestingly, this threshold value (which corresponds to the zero heat transfer rate from the disk surface) is found to depend (although slightly) upon the waviness of the disk surface; and the radial pressure gradient. However, in the case of a flat disk the threshold values of n∗ does not depend on the pressure gradient. This is, in fact, caused by the waviness of the disk surface which not only effects the threshold value of n∗ by itself but also switches on its dependence upon pressure gradient. These variations are further reduced when the surface roughness ratio is smaller than 0.1. Furthermore, the relative motion of the wavy disk and the forced flow also alter this value (n∗ = − 2). From Table 3 it is evident that for smaller values of a0/λ the temperature exponent n∗ retains its flat disk value (n∗ = − 2) at which the disk behaves as an insulated disk.
disk rotating inside a uniform fluid stream has been examined numerically. The numerical results reveal that the power-law exponent of the non-isothermal temperature distribution plays a key role in enhancing the heat transfer rate in the presence of surface roughness. The temperature exponent is not much a dominant parameter with increasing the Prandtl numbers similar to the flat disk case. A double periodicity in the distribution of the Nusselt number has also been exhibited for the non-isothermal disk situation in the presence or absence of a radial pressure gradient. In the presence of radial pressure gradient a heat transfer enhancement of 263% is observed for the wavy disk (with two sinusoids) when the temperature is a quatratic function of the radial coordinate for Pr = 0.71. The mean Nusselt number increases due to the surface roughness ratio. This enhancement in heat transfer rate is observed to be a strong function of the amplitude-to-wavelength ratio; the relative velocity ratio, and the temperature exponent. Moreover some threshold values (which correspond to seizing the heat transfer process) of n∗ are also identified which are observed to vary slightly due to surface irregularities and relative motion of fluid and disk. References [1] J.M. Owen, Fluid flow and heat transfer in rotating disc systems, in: D.E. Metzger (Ed.), Proceedings of the 1982 International Center for Heat and Mass Transfer Symposium on Heat and Mass Transfer in Rotating Machinery, Hemisphere, Washington, DC, 1983. [2] W.J. Yang, Gas-liquid mass transfer in rotating performance disc contractors, Lett. Heat Mass Transf. 9 (1982) 119–129. [3] E.W. Eisele, W. Leidenfrost, A.E. Muthunayagam, Studies of heat transfer from rotating heat exchangers, in: T.F. Irvine, W.E. Ibele, J.P. Hartnett, R.J. Goldstein (Eds.), Progress in Heat and Mass Transfer, 2 Pergamon Press, Oxford, 1969, p. 483498. [4] S. Mochizuki, W.J. Yang, Heat transfer and friction loss in laminar radial flows through rotating annular disks, Trans. Am. Soc. Mech. Eng. Ser. C J. Heat Transf. 103 (1981) 212–217. [5] T. von Kármán, Über laminare und turbulente Reibung, Z. Angew. Math. Mech. 1 (1921) 233–252. [6] I.V. Shevchuk, Convective Heat and Mass Transfer in Rotating Disk Systems, Springer, Berlin, 2009. [7] I.V. Shevchuk, Modelling of Convective Heat and Mass Transfer in Rotating Flows, Springer International Publishing Switzerland, 2016. [8] C.L. Tien, J. Tsuji, Heat transfer by laminar forced flow against a non-isothermal rotating disk, Int. J. Heat Mass Transf. 7 (1963) 247–252. [9] I. Mabuchi, T. Tanaka, M. Kumada, Studies on convective heat transfer from a rotating disk (3rd report, heat and mass transfer in a laminar flow about a rotating disk with suction or injection in the axial stream), Bull. JSME 11 (1968) 875–884. [10] I. Mabuchi, T. Tanaka, Y. Sakakibara, Studies of convective heat transfer from a rotating disk (5th report, experiment on the laminar heat transfer from a rotating isothermal disk in a uniform forced stream), Bull. JSME 14 (1971) 581–589. [11] D.M. Hannah, Forced flow against a rotating disc, Br. Aero. Res. Comm. Rep. Memo 2772 (1947). [12] S. aus der Wiesche, C. Helcig, Convective Heat Transfer from Rotating Disks Subjected to Streams of Air, Springer Cham Heidelberg, New York Dordrecht London, 2016. [13] G. Le Palec, Numerical study of convective heat transfer over a rotating rough disk with uniform wall temperature, Int. Com. Heat Mass Transf. 16 (1989) 107–113. [14] G. Le Palec, P. Nardin, D. Rondot, Study of laminar heat transfer over a sinusoidalshaped rotating disk, Int. J. Heat Mass Transf. 33 (1990) 1183–1192. [15] M.S. Yoon, J.M. Hyun, J.S. Park, Flow and heat transfer over a rotating disk with surface roughness, Int. J. Heat Fluid Flow 28 (2007) 262–267. [16] M.S. Yoon, J.S. Park, J.M. Hyun, Magnetohydrodynamics flow over a rapidly rotating axisymmetric wavy disk, Fluid Dyn. Res. 43 (2011) 1–16. [17] A. Mehmood, M. Usman, B. Weigand, Heat and mass transfer phenomena due to rotating non-isothermal wavy disk, Int. J. Heat Mass Transf. 129 (2019) 96–102. [18] T. Cebeci, A.M.O. Smith, Analysis of Turbulent Boundary Layers, Academic Press Inc, 1974.
4. Conclusion Flow and heat transfer phenomena for a non-isothermal sinusoidal
5
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Heat transfer from a non-isothermal rotating rough disk subjected to forced flow
T
Muhammad Usmana, , Ahmer Mehmooda, Bernhard Weigandb ⁎
a b
Department of Mathematics & Statistics, International Islamic University Islamabad, 44000, Pakistan Institute of Aerospace Thermodynamics, University of Stuttgart, Germany
ARTICLE INFO
ABSTRACT
Keywords: Sinusoidal rotating disk Three-dimensional boundary-layer Non-similar flow Heat transfer enhancement Non-isothermal disk Forced flow
This communication aims to report heat transfer characteristics from a non-isothermal wavy disk rotating in a forced flow. The primary objective of this kind of studies is to seek a suitable mechanism for the quick removal of heat energy from a rotating disk surface. The achievement of an efficient mechanism which ensures a further higher rate of heat transfer is a fundamental goal in such studies. In this regard active and passive techniques are of fundamental importance whereby the current study utilizes a combination of both. There are three key elements, namely, the nonuniform disk temperature; surface roughness, and a radial free flow, which are responsible for the augmentation of the heat transfer. The analysis has been carried out for a variety of fluids depicting the increased rate of heat transfer due to a variation of these different elements. In doing so, the local and global heat transfer rates are manipulated in order to obtain a clear picture of the heat transfer process at a corrugated disk. The considered rough non-isothermal disk (with two sinusoids) rotating in the uniform stream of air (Pr = 0.71) leads to a significant (about 263%) enhancement in the overall heat transfer rate compared to that of a flat free rotating disk having surface temperature as quatratic function of the radial coordinate. Moreover, some threshold values (which correspond to seizing the heat transfer process) of the used power-index of the disk temperature (Tw − T∞ = c0rn∗) are also identified which are observed to vary slightly due to the surface irregularities and the relative motion of fluid and disk.
1. Introduction Rotating disks can be found in turbomachinery [1], rotating disk electrode, rotating machinery, rotating disk contractors [2], rotating heat exchangers [3,4] and several other technical applications. The occurrence of flow and heat transfer in all these units requires the modeling of rotating disk systems which can be classified into rotating cavities, rotor-stator systems, and free disks. The free rotating disk serves as a first prototype being a preliminary stage, with the simplest model, to analyze the different technical applications. Numerous theoretical and experimental works have been devoted to free rotating disk flow concerning to its important role in all aforementioned mechanical rotating devices after the pioneering study of von-Kármán [5]. Instead of making a review on the large number of literature available on this subject related to free rotating disk flow, the reader is referred to a quite recent monograph by Shevchuk [6,7] which not only includes a detailed review but a rigorous analysis of convective heat transfer in rotating disk systems under different situations including uniform or non-uniform temperature distribution on the disk surface. The impact of an outer forced flow orthogonal to a rotating disk is highly useful for cooling or heating the disk surface. Particularly, rotor
⁎
end surfaces in a gas turbine are cooled through the application of such flows. Fundamental work regarding the analysis of these flows on a rotating flat disk along with heat transfer phenomenon can be found in [8–11]. More recently, Wiesche and Helcig [12] showed theoretical as well as experimental investigation concerning convective heat transfer from a rotating disk subjected to a free stream of air. They pointed out the advantageous aspects (especially on heat transfer enhancement) of a forced flow on a rotating disk in different situations. Despite having huge literature related to the flat disk case with (or without) a forced flow, there are some vital areas to which less attention had been given in the past. One is the consideration of surface irregularities on the rotating disk surface. In Sec. 3 it will be realized that the surface topography plays an important role towards the enhancement of heat transfer characteristics. In 1989, Le Palec [13] explored the influence of a bumpy surface on the heat transfer phenomenon from an isothermal rotating disk placed in a quiescent air. For mathematical simplicity he introduced the surface irregularities in the shape of periodic sinusoids which resulted in a 15% heat transfer increase in comparison to a flat disk. A bit later, Le Palec et al. [14] presented a theoretical and experimental study
Corresponding author. E-mail addresses: [email protected] (M. Usman), [email protected] (B. Weigand).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104395
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International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
confirming such a significant heat transfer enhancement owing to a corrugated disk surface subjected to uniform wall heat flux. In [13,14] Le Palec and his co-authors did not investigate the influence of surface corrugation on flow properties. Yoon et al. [15,16] explored the flow variation due to disk surface undulations. Recently, Mehmood et al. [17] pointed out that the wavy disk surface not only augments the heat transfer rate but also increases the torque and mass flow rate. The increased torque increases the overall power of rotating machinery while the increased mass flow rate contributes towards the heat transfer augmentation. It is already known that a significant heat transfer enhancement can be achieved owing to the application of forced flow on a flat rotating disk (see for instance [8–12]). Therefore, provoking from the idea of [13,14] of surface corrugation and employing forced flow, the present work combines these two phenomena (i.e. corrugated disk in forced flow) along with the consideration of non-uniform temperature distribution on the disk surface. The non-uniform disk temperature is considered in the form of a power-law function of its radius which also includes the isothermal disk temperature condition as one special case. Theoretically, any non-constant smooth function of the radial variable can be considered. But the power-law form, particularly chosen here, bears a practical reason. The temperature distribution on the walls of rotating cavities in a gas turbine, actually, does also follow a power-law form. Effects of temperature distribution (power-law) exponent and forced flow on the convective transport process are observed and discussed in detail. The contribution of surface undulations towards flow and heat transfer augmentation has also been noted in comparison with the flat disk case. The flow induced by the rotation of a wavy disk without free flow addressed by Le Palec [13] and the axisymmetric stagnation point flow on a resting or a rotating disk studied by Mabuchi et al. [9] are also recovered as special cases of this study.
Fig. 1. Configuration of wavy disk.
Accordingly u = (ux, uy, uθ) denotes the related velocity vector. The kinematic viscosity and thermal diffusivity are respectively denoted by ν and α. The fluid temperature is denoted by T and the ambient temperature is T∞. The disk temperature is Tw and is taken to be non-uniform, following a power-law distribution of the form Tw = T∞ + c0rn∗, where n∗ is a real number and is called the power-index of the disk temperature, and c0 is a constant having suitable dimensions. The flow and thermal boundary conditions are summarized as
ux = u y = u r = 0, Tw = T + c0 r n , u x = ar , u = 0, T = T ,
uref
one (two) reference velocities present in this problem, see for instance Eq. (6), a sum of the two can be regarded as an appropriate reference velocity. Thus the variables x and y are normalized as
The physical model consists of a free rotating disk with a sinusoidal wavy surface subjected to a steady radial forced flow in an inertial frame of reference. The waviness of the disk surface is described by a smooth periodic function of the form
r s (r ) = a0 cos (2N r ); r = b
=
(1)
y
+
ux dr =0 r dx
ux
ux
u u + uy x y
ux u r
dr = dx
+ y
= (f f g
T T + uy = x y
Pr
(4)
=
x uref
(7) 2 1/2
uref x
(x , y ) , g( , ) = r
uref x
(x , y ) , r
(, )=
T Tw
T T
(8)
+
2
g2 +
1 3 dR + ff 2 2R d
+
2
a
f f)
(9)
dR 1 3 dR f g + + fg Rd 2 2R d 1 3 dR + f 2 2R d
dR Rd
dR n f Rd
= (f g
= (f
g f)
f)
(10) (11)
whereby satisfying the continuity Eq. (2) identically. In the transformed system of Eqs. (9)–(11), R = r/λ denotes the dimensionless radial coordinate and Pr = α/ν is the Prandtl number. These partial differential Eqs. (9) and (10) reduce to the classical self-similar form derived by von Kármán [5] by setting a0/λ = 0 & a/ω = 0 and to those investigated by Le Palec [13] for a0/λ ≠ 0, a/ω = 0, & n∗ = 0 with the consideration of Eq. (11). The associated boundary conditions is also normalized due to Eqs. (7) and (8) and takes the form
2T
y2
2
(3)
Energy equation:
ux
;
dR 2 f R d
f
2u
y2
y
The utilization of these relations ((7) and (8)) in Eqs. (3)–(6) yield the following set of non-similar governing equations
2u
x y2
=
f( , )=
Momentum equations:
u 2 dr u = ux x r dx x
,
where uref = ΛR and Λ = (a + ω ) . The corresponding dimensionless dependent variables in terms of stream functions and dimensionless temperature are obtained as
(2)
ux u + uy x x y
x
2
where a0 denotes the amplitude of the wavy surface, N denotes the number of waves fitted onto the disk radius b, and r denotes the radial coordinate. The ratio of wave amplitude a0 to the wavelength λ is kept small enough so that the boundary-layer assumptions remain valid. This allows one to reduce the Navier-Stokes, and energy equations to momentum and thermal boundary-layer equations. In view of these assumptions the governing boundary-layer equations of the problem under investigation. Continuity equation:
ux + x
(6)
The flow under consideration is of non-similar nature because of the presence of appropriate characteristic lengths in every spatial direction. Therefore, the curvilinear variable x is normalized by the wavelength of the surface undulations and the wall normal variable y is normalized by the boundary-layer thickness. Similar to other boundary-layer flows the boundary-layer thickness varies as ~ x . Since there are more than
2. Transport equations and auxiliary data
uy
at y = 0 as y
(5)
In Eqs. (1)–(5) an orthogonal curvilinear coordinate system (x, y, θ) is used (see Fig. 1) where x, y, and θ describe the variables in longitudinal; normal, and azimuthal directions to the wavy surface, respectively. 2
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
Table 1 Comparison of numerical data of heat transfer rate for non-isothermal flat disk. a/ω
0
0.5 1 10
Pr
n∗ = − 2
0.1 1 10 100 0.71 1 0.71 1 0.71 1
n∗ = 0
n∗ = 2
n∗ = 4
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
Shevchuk [6,7]
Present
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0766 0.3963 1.1341 2.6871 0.4194 0.4815 0.4897 0.5536 0.6381 0.7270
0.0766 0.3963 1.1341 2.6871 0.4194 0.4815 0.4897 0.5536 0.6381 0.7270
0.1417 0.6159 1.6206 3.7422 0.6049 0.6910 0.6905 0.7849 0.9037 1.0250
0.1417 0.6159 1.6206 3.7422 0.6049 0.6910 0.6905 0.7849 0.9037 1.0250
0.1977 0.7557 1.9460 4.4421 0.7305 0.8324 0.8283 0.9395 1.0808 1.2233
0.1977 0.7557 1.9460 4.4421 0.7305 0.8324 0.8283 0.9395 1.0808 1.2233
where the bar refers to the quantities used in [6,7]. Clearly, Table 1 shows an excellent agreement of the present results with those reported in [6,7]. Moreover, the present results have been obtained after several runs by taking different step sizes to ensure the grid independence. Thus, the reported data is sure to be accurate and grid independent. 3. Heat transfer characteristics The consideration of a non-uniform disk temperature and the presence of an external flow significantly influence the heat transfer characteristics. Based upon the practical implementations of intentionally roughened surfaces in state-of-the-art heat transferring devices, operating under extreme conditions, our primary focus is to calculate a net gain in the heat transfer rate and to identify those situations where it is achieved. The local rate of the dimensionless heat transfer is usually expressed as local Nusselt number and is defined as
Fig. 2. Spatial variation of Nusselt number for different values of n∗ at fixed a/ ω.
Re
1/2
Nu =
R 2
1= = 0,
1 = 0,
at
=0
at
=
(12)
The non-similar partial differential Eqs. (9)–(11) along with the boundary conditions (12) have been solved numerically by applying the Keller-box method, see [18] programmed in MATLAB. The classical von Kármán [5] self-similar (ε = 0) solution serves as an initial solution to start the solution process. Validity of the code is justified by attaining the results of the isothermal wavy disk case studied by Le Palec [13]. Moreover, the current procedure is also validated by comparing the results for the case of a non-isothermal flat disk rotating in uniform axial flow studied by Shevchuk [6,7] (see Table 1). The current variables, defined in (7) & (8), follow the following relationship with those of [6,7]:
=
g (1 + k 2)1/2 f 1+k h , = , = 1 + k, 1+k f (1 + k 2)1/2 g h
=
1+k ; k = a/ (1 + k 2)1/2
(13)
where Re = is the rotational Reynolds number. The spatial variation of the heat transfer rate (for n∗ ≠ 0) is illustrated in Fig. 2 depicting a wavy nature of the heat transfer rate undergoing two cycles per wavelength (wavelength of Nu ≅ λ/2) with an decreasing amplitude. Such an exhibition of local Nusselt number with increased frequency had already been reported by Le Palec [13] for an isothermal free rotating wavy disk (n∗ = 0). Clearly (see Fig. 2) the higher values of temperature distribution exponent (n∗ > 0) correspond to an increase local heat transfer rate and its decreasing values (n∗ < 0) result in lowering the local heat transfer rate in comparison to an isothermal disk (n∗ = 0). A similar trend of the local heat transfer rate is also shown in Fig. 3 when the relative velocity is varied (i.e. a/ω ≠ 0). This Figure illustrates that an increase in relative velocity ratio leads to an increase in the associated local heat transfer rate. This is an obvious fact which has already been observed for a flat disk too. The periodicity of the local Nusselt number is again twice in comparison to the surface topology. This establishes the fact that the curvature of the local Nusselt number curves is entirely due to the surface topography and is not altered by the additional features such as the variable disk temperature or the external forced flow etc. A zero value of relative velocity ratio refers to a flow induced purely by the rotation of a sinusoidal disk, whereas its infinite value corresponds to the axi-symmetric stagnation point flow over a resting wavy disk. An important fact which can be seen in these Figures (Figs. 2 & 3) is that the surface irregularities of the wavy disk reduce the local Nusselt number in comparison to a smooth disk. This behavior is a little bit deceiving and had already been reported for an isothermal sinusoidal-shaped free rotating disk (i.e. n∗ = 0 & a/ω = 0) by Le Palec [13]. From such a behavior one should not simply assume that the surface undulations cause to reduce the heat transfer rate. In Figs. 2 and 3 Nu has been plotted against the flat radial coordinate which also serves as longitudinal variable for a flat disk but not for a wavy disk. For a wavy disk the longitudinal
Fig. 3. Spatial variation of Nusselt number for different values of a/ω at fixed n∗.
f=g=f =g a f = , g = 0,
( , 0)
3
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
Table 3 Dependence of threshold values of n∗ (for which heat transfer process is seized from the disk surface) on surface waviness with N = 2 and Pr = 0.71. a0/λ
a/ω
0.0 1/16 1/10
Table 2 Values of mean Nusselt number Nup for different temperature distribution exponent and relative velocity ratio with N = 2 and Pr = 0.71. a0/λ
n∗ = − 1
n∗ = 0
n∗ = 1
n∗ = 2
n∗ = 4
0.0
0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10 0.0 1/16 1/10
0.1893 0.1928 0.1979 0.3132 0.3190 0.3272 0.3763 0.3832 0.3930 0.4174 0.4251 0.4358 0.4392 0.4473 0.4588
0.3259 0.3318 0.3403 0.4858 0.4947 0.5073 0.5765 0.5870 0.6019 0.6368 0.6484 0.6648 0.6690 0.6812 0.6987
0.4319 0.4397 0.4510 0.6065 0.6175 0.6332 0.7143 0.7273 0.7457 0.7872 0.8014 0.8217 0.8262 0.8412 0.8627
0.5185 0.5279 0.5413 0.7006 0.7133 0.7315 0.8211 0.8360 0.8571 0.9033 0.9196 0.9428 0.9476 0.9647 0.9893
0.6555 0.6674 0.6842 0.8461 0.8614 0.8832 0.9851 1.0028 1.0281 1.0812 1.1007 1.1284 1.1332 1.1536 1.1829
0.5 1 2 ∞
0.5
1
2
∞
−2 -2 −1.9996
−2 -2 −2.0001
−2 −2.0001 −2.0004
-2 −2.0001 −2.0004
-2 −2.0001 −2.0004
variable is x which runs along the disk texture radially outwards. Therefore, a comparison between the two situations (flat and wavy disks) can't be made from here because the curvilinear radius of a wavy disk is larger than the radius of corresponding flat disk. To ensure a comparison of the two situations it is necessary to bring the two data on equal scaling; either the flat disk area can be regarded as reference and the results of the wavy disk should be reduced accordingly or the wavy disk area can be taken as reference where the results of the flat disk should be reduced accordingly. For this purpose we take the flat disk surface area as reference and thus have to increase the Nusselt number of the wavy disk case by the area ratio S 2 , as was also done by Le Palec [13]. Before b doing so, it is important to calculate the average Nusselt number in order to get a clear picture. The average heat transfer rate from a non-isothermal wavy disk is measured by means of the average Nusselt number (Nu ) which is acquired by integrating Eq. (13) over the whole disk surface and then dividing by the disk surface area, as
Fig. 4. Mean Nusselt number Nu (curve 1) and Nup (curve 2) under the influence of a non-uniform temperature distribution and the amplitude-to-wavelength ratio.
a/ω
0
NuRe
1/2
=
1 S
Re
1/2
Nu dS
S
(14)
where S refers to the wavy disk area. As mentioned above, this average Nusselt number Nu is corrected by an area ratio S 2 to get the upgraded b average Nusselt number, ready to compare with the flat disk results. This can be expressed by
Nup = Nu
S b2
(15)
Notice that as the surface undulations are flattened out (i.e. a0/λ →
Fig. 5. Heat transfer rate (Nup Re
1/2 )
plotted against number of sinusoids (N) for Pr = 0.71.
4
International Communications in Heat and Mass Transfer 110 (2020) 104395
M. Usman, et al.
0) the surface area S → πb2 (of the corresponding flat disk) and Nup Nu , the Nusselt number of the corresponding flat disk. These calculations (Eqs. (14) & (15)) are shown in Fig. 4 where three values of the temperature distribution exponent n∗ are considered and a same pattern for the overall heat transfer rate is observed for each considered value of n∗. In Fig. 4, curve 1 shows the calculation of Eq. (14) (i.e. without correction) and it clearly decreases with increasing values of the surface roughness ratio a0/λ. However, the mean Nusselt number obtained from Eq. (15) (i.e. with correction) shown as curve 2 in Fig. 4 reflects that heat transfer is enhanced with increasing roughness ratio of the disk. Apart from this it can also been noted in Fig. 4 that the temperature distribution exponent n∗ has also a substantial influence on the heat transfer enhancement and higher values yield considerable improvement in the exchange of heat energy from a hot wavy disk to the cold surrounding fluid. For instance, the considered rough non-isothermal disk rotating in a uniform stream of air (Pr = 0.71, a/ω = ∞ & a0/λ = 0.1) leads to a significant (about 263%) enhancement of the overall heat transfer rate compared to that of a flat free rotating isothermal disk when the wavy disk (with two sinusoids) temperature is quatratic function of its radial distance (see Table 2). Whereby an isothermal rough rotating disk in a forced flow gives an enhancement of about 114% when compared to flat free rotating isothermal disk. A value of 3.46 (for any a0/λ) in the heat transfer enhancement is observed when n∗ is varied from −1 to 4 for the situation of flow induced by pure rotation of sinusoidal disk (i.e. a/ω = 0). For a relative velocity ratio a/ω = 0.5 & ∞, a rise of 2.70 & 2.58 (respectively) times in average heat transfer rate can be observed. Obviously, the larger values of a relative velocity ratio correspond to frail the influence of the temperature distribution exponent. Interestingly, the values of the temperature exponent n∗ find no association with the surface waviness but it enhances the heat transfer rate at the same time whether the disk is flat or rough. The average heat transfer is also found to depend on the number of sinusoids fitted to the disk radius 0 ≤ r ≤ b as depicted in Fig. 5. It is observed that whatever the flow situation is (for any a/ω) and whether the disk is isothermal or non-isothermal (for any n∗) the mean Nusselt number increases slowly with an increase in number of cycles N up to N = 4; and after that it becomes almost constant. According to Shevchuk [6], the heat transfer process from a free rotating flat disk seizes (i.e. Nup = 0 ) at a value n∗ = − 2. This value does also remain invariant for a flat rotating disk in the presence of a radial pressure gradient or with an alteration of the Prandtl number. But interestingly, this threshold value (which corresponds to the zero heat transfer rate from the disk surface) is found to depend (although slightly) upon the waviness of the disk surface; and the radial pressure gradient. However, in the case of a flat disk the threshold values of n∗ does not depend on the pressure gradient. This is, in fact, caused by the waviness of the disk surface which not only effects the threshold value of n∗ by itself but also switches on its dependence upon pressure gradient. These variations are further reduced when the surface roughness ratio is smaller than 0.1. Furthermore, the relative motion of the wavy disk and the forced flow also alter this value (n∗ = − 2). From Table 3 it is evident that for smaller values of a0/λ the temperature exponent n∗ retains its flat disk value (n∗ = − 2) at which the disk behaves as an insulated disk.
disk rotating inside a uniform fluid stream has been examined numerically. The numerical results reveal that the power-law exponent of the non-isothermal temperature distribution plays a key role in enhancing the heat transfer rate in the presence of surface roughness. The temperature exponent is not much a dominant parameter with increasing the Prandtl numbers similar to the flat disk case. A double periodicity in the distribution of the Nusselt number has also been exhibited for the non-isothermal disk situation in the presence or absence of a radial pressure gradient. In the presence of radial pressure gradient a heat transfer enhancement of 263% is observed for the wavy disk (with two sinusoids) when the temperature is a quatratic function of the radial coordinate for Pr = 0.71. The mean Nusselt number increases due to the surface roughness ratio. This enhancement in heat transfer rate is observed to be a strong function of the amplitude-to-wavelength ratio; the relative velocity ratio, and the temperature exponent. Moreover some threshold values (which correspond to seizing the heat transfer process) of n∗ are also identified which are observed to vary slightly due to surface irregularities and relative motion of fluid and disk. References [1] J.M. Owen, Fluid flow and heat transfer in rotating disc systems, in: D.E. Metzger (Ed.), Proceedings of the 1982 International Center for Heat and Mass Transfer Symposium on Heat and Mass Transfer in Rotating Machinery, Hemisphere, Washington, DC, 1983. [2] W.J. Yang, Gas-liquid mass transfer in rotating performance disc contractors, Lett. Heat Mass Transf. 9 (1982) 119–129. [3] E.W. Eisele, W. Leidenfrost, A.E. Muthunayagam, Studies of heat transfer from rotating heat exchangers, in: T.F. Irvine, W.E. Ibele, J.P. Hartnett, R.J. Goldstein (Eds.), Progress in Heat and Mass Transfer, 2 Pergamon Press, Oxford, 1969, p. 483498. [4] S. Mochizuki, W.J. Yang, Heat transfer and friction loss in laminar radial flows through rotating annular disks, Trans. Am. Soc. Mech. Eng. Ser. C J. Heat Transf. 103 (1981) 212–217. [5] T. von Kármán, Über laminare und turbulente Reibung, Z. Angew. Math. Mech. 1 (1921) 233–252. [6] I.V. Shevchuk, Convective Heat and Mass Transfer in Rotating Disk Systems, Springer, Berlin, 2009. [7] I.V. Shevchuk, Modelling of Convective Heat and Mass Transfer in Rotating Flows, Springer International Publishing Switzerland, 2016. [8] C.L. Tien, J. Tsuji, Heat transfer by laminar forced flow against a non-isothermal rotating disk, Int. J. Heat Mass Transf. 7 (1963) 247–252. [9] I. Mabuchi, T. Tanaka, M. Kumada, Studies on convective heat transfer from a rotating disk (3rd report, heat and mass transfer in a laminar flow about a rotating disk with suction or injection in the axial stream), Bull. JSME 11 (1968) 875–884. [10] I. Mabuchi, T. Tanaka, Y. Sakakibara, Studies of convective heat transfer from a rotating disk (5th report, experiment on the laminar heat transfer from a rotating isothermal disk in a uniform forced stream), Bull. JSME 14 (1971) 581–589. [11] D.M. Hannah, Forced flow against a rotating disc, Br. Aero. Res. Comm. Rep. Memo 2772 (1947). [12] S. aus der Wiesche, C. Helcig, Convective Heat Transfer from Rotating Disks Subjected to Streams of Air, Springer Cham Heidelberg, New York Dordrecht London, 2016. [13] G. Le Palec, Numerical study of convective heat transfer over a rotating rough disk with uniform wall temperature, Int. Com. Heat Mass Transf. 16 (1989) 107–113. [14] G. Le Palec, P. Nardin, D. Rondot, Study of laminar heat transfer over a sinusoidalshaped rotating disk, Int. J. Heat Mass Transf. 33 (1990) 1183–1192. [15] M.S. Yoon, J.M. Hyun, J.S. Park, Flow and heat transfer over a rotating disk with surface roughness, Int. J. Heat Fluid Flow 28 (2007) 262–267. [16] M.S. Yoon, J.S. Park, J.M. Hyun, Magnetohydrodynamics flow over a rapidly rotating axisymmetric wavy disk, Fluid Dyn. Res. 43 (2011) 1–16. [17] A. Mehmood, M. Usman, B. Weigand, Heat and mass transfer phenomena due to rotating non-isothermal wavy disk, Int. J. Heat Mass Transf. 129 (2019) 96–102. [18] T. Cebeci, A.M.O. Smith, Analysis of Turbulent Boundary Layers, Academic Press Inc, 1974.
4. Conclusion Flow and heat transfer phenomena for a non-isothermal sinusoidal
5