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Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs
Accepted Manuscript Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs Tabish Alam, Man-Hoe Kim PII: DOI: Reference:
S1359-4311(17)31394-7 http://dx.doi.org/10.1016/j.applthermaleng.2017.07.181 ATE 10849
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
1 March 2017 7 June 2017 25 July 2017
Please cite this article as: T. Alam, M-H. Kim, Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng. 2017.07.181
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Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs Tabish Alam, Man-Hoe Kim* School of Mechanical Engineering, Kyungpook National University, Daegu 41566, South Korea Abstract Application of protrusion rib roughnesses on the absorber plate of solar air heater (SAH) duct can effectively enhance the heat transfer rate irrespective of pressure drop penalty. This paper presents the numerical investigation of SAH duct, roughened with conical protrusion ribs. Effect of relative ribs pitch (6≤p/e≤12) and relative ribs height (0.020≤e/D≤0.044) on Nusselt number and friction factor have been studied for the range of Reynolds number from 4000 to 16000. Thermal efficiency of roughened duct have been determined using useful energy gain to air and heat losses to environment. The maximum thermal efficiency (η) and efficiency enhancement factor (EEF) are found as 69.8% and 1.346, respectively. Correlations of friction factor and Nusselt number have also been developed as function of Reynolds number and roughness parameters of conical ribs. Keywords: Heat transfer enhancement, Efficiency enhancement factor, Conical rib, Solar air heater
*
Corresponding Author. Tel.: +82-53-950-5576; Fax: +82-53-950-6550. E-mail addresses: [email protected] (M. H. Kim), [email protected] (T.Alam)
1
Nomenclature
I ka kg m N Nu Nus P P/e Pr
Area of absorber plate, m2 Specific heat of air at constant pressure, J/Kg.K Duct height, mm Rib height, mm Relative rib height Flat efficiency factor, Heat removal factor Friction factor of roughened duct Friction factor of smooth duct Convective heat transfer coefficient, W/m2.K Solar insolation, W/m2 Thermal conductivity of air, W/m/K Thermal conductivity of glass, W/m/K Mass flow rate, Kg/s Number of glass cover Nusselt number of roughened duct Nusselt number of smooth duct Pitch of ribs, mm Relative pitch ratio Prandtl number
Qu
Useful heat gain, J
Ap Cp D e e/D F’ Fo f fs h
Re Tg
Reynolds number Mean Temperature of glass, K
Ti To Tpm ti tg tg Ub Ue
Inlet temperature of air, K Outlet temperature of air, K Average plate mean temperature, K Thickness of insulation, mm Thickness of glass cover, mm Height of collector edge, mm Back heat loss coefficient, W/m2.K Edge heat loss coefficient, W/m2.K
Ut Ul εg εp ηth ρ σ τ µ (τα)
Top heat loss coefficient, W/m2.K Overall heat loss coefficient, W/m2.K Emissivity of glass cover Emissivity of absorber plate Thermal efficiency of collector Density of air, kg/m3 Stefan-Boltzmann Constant, W/m2.K4 Trasnmissivity of glass cover Dynamic viscosity of air, N.s/m2 Transmittance-absorbent product of glass cover
1. Introduction In the past few years, we are facing the problems of energy crisis due to higher prices and exhaustive nature of fossil fuels such as crude oil, coal, natural gas and so on. High usage of fossil fuels causes to threat the environmental asset in term of wildlife, land disturbance, pollution and human health. Renewable energy can minimize our dependency on fossil fuels, thereby, renewable energies are getting importance in the recent years because energy can renewed and will never run out. Renewable energy is ecofriendly and results in little to no effect to the environment. Out of many renewable energies, solar energy is considered to be clean source of energy and available on every part of the world. Solar energy is exploited in many application, included heating purposes and generation of electricity. Solar air heater (SAH) is 2
one of the most economical and elementary device which employ to supply the heated air to drying the crops, industrial purposes, heating the building and space. Absorber plate of SAH converts solar insolation into thermal energy of air when passing through underside of absorber plate when it exposed to solar radiation, however, convective heat transfer coefficient is inherent low causes to low performance of SAH. The low convective heat transfer coefficient between absorber plate and flowing air is observed due to presence of viscous laminar sublayer on absorber plate. Artificial roughness is considered to be good technique to enhance the convective heat transfer coefficient. Artificial roughness on absorber plate disturb viscous laminar sublayer causes to increase heat transfer rate[1]. Artificial roughness not only enhance the heat transfer rate but also increase pressure drop which is inadmissible. Artificial roughness, provided in the form of transverse small diameter wires, was first investigated in SAH and shown remarkable heat transfer enhancement is observed in comparison to conventional smooth solar air heater duct [1,2]. Small roughness wires alter the flow pattern and creates the turbulence in the vicinity of heated surface without disturbing the core flow. Various arrangements of wires, namely; V-shaped [3], W-shaped [4,5], multi V-shaped[6,7], broken angled ribs[8,9], V-ribs with gap[10,11], multi V-ribs with gap [12,13]have been investigated.
Combinations of ribs arrangements such as transvers and inclined ribs [14],
chamfered ribs and grooves [15,16], rectangular ribs and groove [17] have also been studied. The grooves were created by machining process and require great attentions. Various researchers investigated the spherical/protrusion ribs out of many artificial roughness, it does not show much more pressure penalty as compared to other roughness [18], thereby, spherical/protrusion rib roughness is getting importance. In this regards, Saini and Verma [19] investigated the spherical ribs on absorber plate. The spherical ribs were created by pressurized the indentation–producing device on absorber plate. Effect of roughness parameters, i.e. relative ribs height (0.0189-0.0379) and pitch (8-12) on heat transfer and friction characteristics were studied. Considerable heat
3
transfer rate was reported as a results of spherical ribs provided on absorber plate. Bhushan and Singh [20] introduced new parameters of spherical ribs roughness, namely; relative short way length (18.75-37.50), relative long way length (25.0-37.50) and relative print diameter (0.1470.3670). It was concluded that Nusselt number and friction number were enhanced upto 3.2 and 2.2 times with respect to smooth duct in the ranges of parameters investigated. Effect of arc arrangement of spherical ribs were studied and angle of arcs were varied from 45° to 75°[21,22]. Angle of arc affected the heat transfer and maximum heat transfer was found at 60° angle of arc. The correlations of friction factor and Nusselt number were also developed as function spherical ribs parameters. Yadav et al. [23] also investigated the similar pattern of spherical ribs as investigated by Sethi [21,22], although, dimension of ducts were different. The application of protrusion roughness in SAH may be attractive because protrusion does not add extra mass on absorber plate and manufacturing of this roughness does not require special treatment. These roughness can be created by punching the thin absorber plate using indentation device. The literature survey shows that protrusions ribs of hemispherical/spherical shape have been investigated previously which are very common shape. In order to increase heat transfer further, conical shape protrusion ribs have been investigated in this work with the aim of sharp apex corner of cone would contribute the turbulence in the flow resulting high heat transfer rate from absorber plate. In this regards, Numerical study has been conducted to investigate the effect of relative ribs pitch (p/e) and relative ribs height (e/D) of conical shape protrusions ribs attached to absorber plate on friction factor and Nusselt number. Using the numerical data, friction factor and Nusselt number correlations have also been developed.
4
2. Detail of computational model In the present work, numerical simulations of SAH have been carried out using finite volume method. The effect of protrusion ribs of conical shape on absorber plate have been investigated. The details of geometry, computational flow domain, mesh generation, boundary conditions, solution method, selection of turbulence model and results validation have been discussed in following sub-sections. 2.1 Description of geometry and parameters Height, length and width of computational flow domain of SAH are considered as 25 mm, 1000 mm and 300 mm, respectively as per ASHARE’s recommendation [24]. Although, actual solar air heater consisted inlet plenum and outlet plenum along with test section which are omitted in present numerical studies to make the model simple [25–27]. Only test section has been considered in the present numerical studies. One broad wall of model is considered as absorber plate on which protrusion ribs of conical shape have been attached. Absorber plate has been exposed to uniform heat flux of 1000 W/m2 which is done to provide insolation, falling on absorber plate. Three sides of the duct were smooth and insulated. Air enter in the duct at inlet section, extract heat from absorber plate when air comes in contact with absorber plate and then air comes out from the duct at outlet section. The computational flow domain of the solar air heater has been showing in Fig. 1. The ranges of relative ribs height (e/D) and relative ribs pitch (p/e) of conical protrusion ribs have been chosen as 0.020-0.044 and 6-12, respectively, which are similar to previous studies, conducted for spherical ribs [19]. All the simulations have been carried out at four values
5
of Reynolds number (Re) in the range of 4000-16000. However, relative base diameter of ribs (d/e) are fixed for all simulations. Ranges of ribs parameter have been listed in Table 1.
Fig. 1. Computation flow domain Table 1. Parameter ranges
S. No.
Parameter
Range
1
Relative rib height ratio (e/D)
0.020-0.044
2
Relative rib pitch ratio (p/e)
6-12
3
Reynolds number (Re)
4000-16000
2.2 Computational flow domain and mesh generation 3-Dimensional flow domains have been created using design modular of Ansys 16.0. Only half flow domain has been exploited in all simulation due to symmetrical body of flow domain and symmetry boundary condition has been applied on the mid-section (z=0) of the duct. 6
In order to discretize the flow domain, unstructured mesh has been created for all simulations. Very fine prism mesh near the absorber and rib walls have been created to resolve the boundary layer in the vicinity of wall bounded domain. Non-dimensionless wall distance (y+) for absorber wall and ribs bounded flow have been estimated less then unity. Initially, grid independent test has been conducted for different number of elements and nodes in the ranging from 13.75×106 to 17.17×106 and from 3.77×106 to5.09×106, respectively for conical protrusion ribs having relative ribs height of 0.0289 and relative ribs pitch ratio (p/e) of 10 at Reynolds number (Re) of 12000. Relative percentage variation in Nusselt number have been observed as 1.63%, 0.42% and 0.37% when number of elements increases from 13.75×106 to 15.03×106 , 15.03×106 to 16.01×106 and 16.01×106 to 17.17×106, respectively. Table 2 shows the effect of variation elements number on the percentage variation of Nusselt number and friction factor. Based on the results obtained, number of elements of 16×106 has been considered for further simulation. Fig. 2. shows the meshed flow domain of the model.
Fig. 2. Partial view of meshed domain 7
Table 2. Detail of grid independent test S. No.
Number of elements
Percentage variation in Nusselt Number
Percentage variation in friction factor
1
13.75×106
--
--
2
15.03×10
6
1.63%
2.36%
3
16.01×106
0.42%
0.54%
4
17.17×106
0.37%
0.81%
2.3 Boundary Conditions After generating the mesh, similar boundary conditions have been imposed on the flow domain as found on actual solar air heater. In order to save computer memory and simulation times, only half flow domains have been exploited in the present work and symmetric wall condition have been applied at mid plane (z =0) on the duct. Top side of domain has been considered as absorber plate on which conical ribs have been created and heat flux of 1000 W/m2, equivalent to typical insolation has been imposed on the absorber plate. Sides and bottom plane are considered to be smooth and insulated and no slip boundary condition has been imposed. Pressure outlet and velocity inlet conditions have been assigned on the outlet and inlet section of the domain, respectively. Based on the desired Reynolds number, values of inlet velocities have been determined at uniform temperature of 300° K. Due to small rise in temperature of air, all physical properties are considered to be fixed which are determined on 300° K temperature and listed in Table 3.
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Table 3. Thermo-physical properties of air at 300° K
Property Name
Value
Density (ρ)
1.117 kg/m-3
Prandlt Number (Pr)
0.71
Specific heat (Cp)
1007 J/kg.°K
Viscosity (µ)
1.857×10-8 Ns/m2
2.4 Solution Methods In present simulations, commercial code of Ansys Fluent 16.0 has been used to analyze the fluid flow and heat transfer of 3-dimensional computational domain using finite volume method. Using the boundary conditions, the governing equations in the incompressible and steady state have been solved. The transport equations of model are given in Appendix A. Velocity and pressure field have coupled using SIMPLE (semi-implicit method for pressure linked equations) algorithm. Second order upwind scheme has been exploited to special discretization [28,29]. Convergence criteria for energy equation has been taken in the account of 10-6 order, however, criteria for velocity component and momentum are considered as 10-4 [30,31]. 2.5 Selection of turbulence model and result validations In author’s previous work [31], different turbulence models were used for smooth duct and based on the comparison of CFD results with the result obtained from correlations of Nusselt number (Gnielinski and Dittus-Boelter correlation[32]) and friction factor (Blasius Equation 9
[33]), RNG k-ε turbulence model was selected. The correlations of Nusselt number and friction factor have been given below. Gnielinski correlation,
Nu =
(f/8)(Re - 1000)Pr for 3000 < Re < 10000 1 + 12.7(f/8) 1/2(Pr 2/3 - 1)
(1)
where, Darcy factor, f = {0.79 ln (Re ) - 1.64 }
-2
Dittus-Boelter correlation, Nu = 0.023 Re 0.8 Pr
0.4
for Re > 10000
(2)
Blasius Equation, f s = 0.0085 Re
-0.25
(3)
In order to select the turbulence model in the present work, simulations of hemispherical protrusion ribs created on absorber plate have been carried out using five turbulence model, namely; standard k-ε, RNG k-ε, Realizable k-ε, Standard k-ω and SST k-ω. In this context, hemispherical protrusion ribs of relative ribs height of 0.0289 and ribs pitch of 10 have been created on the absorber plate which is very similar to the experimental work, conducted by Verma and Saini [19]. In order to check the accuracy of the different models, predicted values of Nusselt number and friction factor obtained at various Reynolds number (Re = 3054, 6119, 9701 and 11138) from different turbulence models have been compared with the experimental data [19]. Comparison of Nusselt number and friction have been presented in Fig. 3 and Fig. 4, respectively.
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Fig. 3. Comparison of Nusselt number predicted by different turbulence models
Fig. 4. Comparison of friction factor predicted by different turbulence models
11
These graphs clearly shows that out of different turbulence models, SST model predict Nusselt number and friction factor very close to experimental data [19]. Minimum average absolute standard deviation error for friction factor and Nusselt number has been observed as 10.1% and 9.4%, respectively in case of SST turbulence model, however, Realizable k-ε turbulence model and Standard k-ε turbulence model predict the maximum average absolute standard deviation for friction factor and Nusselt number, respectively. It has been clear from this analyses that results, predicted by SST k-ε turbulence model, are in good agreement with results obtained from experimental work. Further, present work exploited SST k-ω turbulence model for all simulations which was also recommended by other investigators [27,34]. 3. Results and discussion For the protrusions conical ribs on absorber plate, predicted data of Nusselt number and friction factor have been obtained for the Reynolds number in the range of 4000 – 16000. The effect of shape of ribs, Reynolds number, relative ribs height and relative ribs pitch have been discussed below. 3.1 Effect of rib shapes Nusselt number of roughened duct equipped with conical protrusion ribs and spherical ribs have been compared with Nusselt number of smooth duct (using Gnielinski correlation (Eq. 1) and Dittus-Boelter correlation (Eq. 2)) as a function of Reynolds number for relative ribs height of 0.0289 and relative ribs pitch of 10, as shown in Fig. 5. It is clearly seen that Nusselt number of both the roughened ducts are much higher than the Nusselt number of smooth duct. This is due to fact that higher turbulence in the flow provide higher Nusselt number in case of roughened duct. Addition to this, Nusselt number provided by conical protrusion ribs are observed higher in comparison to Nusselt number, obtained from spherical ribs. In this context, velocity vector around the protrusion conical ribs and spherical ribs have been shown on the mid plane (z = 0) of the duct in the direction of flow for Reynolds number of 12000, as shown in Fig. 12
6. Sharp edge of conical ribs are attributes to high turbulence in the flow along with strong reattachment point on surface. Strong re- attachments on the surface disturb sub-laminar layer and high turbulence help to mix the flow near surface to core flow. However, re-attachment point has also been observed in case of spherical ribs but intensity of reattachment point is too week to enhance the heat transfer rate. The variation of friction factor of smooth duct (using Blasius Equation (eq. 3)) and roughened ducts with conical protrusion ribs and spherical ribs as a function of Reynolds number have been presented in Fig. 7. As the value of Reynolds number increase, friction factor of smooth duct and both
the roughened ducts decrease due to
deterioration of laminar-sub layer [25] and higher friction factor have been found for protrusion conical ribs because sharp edge of cone causes to higher pressure drop in the flow [35]. Average enhancement in Nusselt number and friction factor of conical ribs with respect to spherical ribs are found as 1.3 and 1.07, respectively.
Fig. 5. Variation of Nusselt number as a function of Reynolds number 13
Fig. 6. Velocity vector around conical and spherical ribs
14
Fig. 7. Variation of friction factor as a function of Reynolds number 3.2 Effect of Reynolds number (Re) Similar trend of Nusselt number with respect to Reynolds number have been observed at other parameters of conical ribs as shown in Fig. 5, which was expected. An increment in Nusselt number as a function of Reynolds number can be explained on the basis of turbulence kinetic energy. In this context, contours of turbulence kinetic energy at various Reynolds number have been shown on the mid plane (z = 0) of the duct along the flow direction (Fig. 8). Intensity of turbulence kinetic energy increases with increase in Reynolds number and region of high turbulence kinetic energy have been found downstream of the rib at all Reynolds number. High turbulence kinetic energy region past to ribs have been found due to large velocity gradient in the flow [29]. Region of high turbulence kinetic energy decreases with increase in Reynolds number, however, scale of turbulence kinetic energy increases rapidly. The maximum turbulence kinetic energy have been found as a function of Reynolds number and the value of maximum turbulence 15
kinetic energy increase from 0.26 J/kg to 1.74 J/kg with the values of Reynolds number, changes from 4000 to 16000.
Fig.8. Turbulence kinetic energy contours for different Reynolds Number
3.3 Effect of relative ribs height (e/D) In order to show the effect of relative ribs height, variation of Nusselt number as function of relative ribs height have been plotted in Fig. 9 for different values of Reynolds number and fixed values of relative ribs pitch (p/e=10). It is observed from these plots that Nusselt Number increase linearly with increase in the values of relative ribs height for given relative ribs pitch and all values of Reynolds number. Relative ribs height of 0.044 provide maximum friction 16
factor and Nusselt number, and relative rib height of 0.020 provide minimum friction factor and Nusselt number. It is fact that strong turbulence increases due to local fluid velocity profile which is mixed vigorously in between the ribs, when relative ribs height increase. These results are inline with results of obtained in case of rectangular ribs [36] and equilateral triangular ribs [29]. Variation of friction factor as function of relative ribs height have also been plotted in Fig. 10 for different values of Reynolds number and fixed value of relative pitch (p/e=10). It has been clearly seen from these plots that friction factor increase when relative ribs height increase from 0.020 to 0.044. This is due to fact that higher relative roughness height contributed more disruption to flow in comparison to disruption offered by low roughness height. Flow disruption due to ribs lead to the dissipation of dynamic pressure of the fluid causes to higher values of friction factor, which is also reported in other investigation[29].
Fig. 9. Variation of Nusselt number as a function of relative ribs height
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Fig. 10. Variation of friction factor as a function of relative ribs height Enhancement factor in friction factor and Nusselt number of roughened duct with respect to friction factor and Nusselt number of smooth duct have also been determined for different relative ribs height and fixed value of relative ribs pitch of 10. The maximum values of enhancement factor for friction factor and Nusselt number at various relative ribs height have been listed in Table 4 for the range of Reynolds considered. Maximum enhancement factor in friction factor and Nusselt number with increase in the values of relative ribs height. Table 4. Maximum values of enhancement factor at different relative ribs height S. No.
Parameter
Nu/Nus
f/fs
1
e/D = 0.0200
2.18
5.05
2
e/D = 0.0289
2.30
5.43
3
e/D = 0.0360
2.40
6.18
4
e/D = 0.0440
2.49
7.39
18
3.4 Effect of relative ribs pitch (p/e) Effect of the relative ribs pitch on Nusselt number has been presented for all values of Reynolds number by keeping fixed value of relative ribs height (e/D=0.0289), as shown in Fig. 11. The values of Nusselt number increase with increasing the value of relative ribs pitch from 6 to 10, beyond the relative ribs pitch of 10, the values of Nusselt number begin to decrease for all values of Reynolds number. Relative pitch value of 6 offers the lowest values of Nusselt number and relative ribs pitch value of 10 offers the highest values of Nusselt number. In the present study, 10 relative ribs pitch has been found as optimum for the range of Reynolds number considered. These results are in line with results of spherical ribs[19]. Higher Nusselt number for relative pitch of 10 may due to the lateral movement of air when air passes over the absorber plate. Lateral movement of air causes to reduce the re-circulation zone and hence better heat transfer area is achieved [25]. In this context, velocity vector around the ribs on the plane, parallel to absorber plate have been presented in Fig. 13 for different values of relative ribs pitch and fixed values of Reynolds number (Re=12000). It can been seen from the velocity vector that as lateral movement of air increase with increasing the relative ribs pitch from 6 to 12, however, increasing values of relative ribs pitch offers low level of turbulence as number of ribs is decreased. So, relative pitch of 10 attributed the desirable conditions which gives the highest Nusselt number. Fig. 12 shows the effect of relative ribs pitch on friction factor for different values of Reynolds number and fixed values of relative ribs height of 10. The values of friction factor decrease continuously with increasing the relative pitch ratio, which is due to fact that higher values of relative pitch attributed to low resistance offer to flow. The minimum and maximum values of friction factor were observed at relative pitch of 12 and 6, respectively.
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Fig. 11. Variation of Nusselt number as a function of relative ribs pitch
Fig. 12. Variation of friction factor as a function of relative ribs pitch 20
Fig. 13. Velocity vectors for different relative pitch at parallel plane to absorber plate (at Re = 12000) Enhancement factor in friction factor and Nusselt number of roughened duct with respect to friction factor and Nusselt number of smooth duct have also been determined. Table 5. has been made to show the maximum values of enhancement factor in friction factor and Nusselt number for different values of relative pitch ratio and fixed value of relative rib height of 0.0289. Maximum values of enhancement factor in Nusselt number is observed at relative pitch of 6, either side of relative pitch of 10, it decrease. Although, maximum value of enhancement factor in friction factor decrease with increment of relative ribs pitch.
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Table 5. Maximum values of enhancement factor at different relative ribs height S. No.
Parameter
Nu/Nus
f/fs
1
p/e = 6
2.02
5.89
2
p/e = 8
2.21
5.71
3
p/e = 10
2.30
5.43
4
p/e = 12
2.13
4.68
4. Thermal efficiency (η) Determination of thermal efficiency of roughened duct is similar to the efficiency of smooth duct. It has been noticed in previous discussion that heat transfer rate of roughened duct is found higher than the heat transfer rate in smooth duct. So it is necessary to determine the thermal efficiency of roughened duct for different roughness parameters. In this context, thermal efficiency have been calculated which is based on insolation fall on absorber plate, heat energy gain to air and thermal losses to environment. The following heat energy (Qu) gain to air is given by Bondi et al. [37]; Q u = Fo Ap (I (ia ) - U l (To - Ti ))
(4)
Where, Fo is heat removal factor and F’ is collector fin efficiency, which are given below: mC P é ì F'APU l ü ù Fo = êexp í ý - 1ú APU l êë î mC P þ ûú
U and F ' = æçç1 + l ö÷÷ è
-1
h ø
Total heat loss coefficient (Ul) is sum of bottom loss, edge loss and top loss coefficient as given by Duffie and Beckmann [38] in following manner; Ul = Ub +U e +Ut
(5)
22
Where, back loss coefficient (Ub) and edge loss coefficient (Ue) are expressed as; Ub =
k i and (L+ W ) t e k i Ue = ti LWt i
Calculation of top loss coefficient is very complex. It can be calculated using graphical method and iterative procedures. However, some correlations available to calculate it, very common correlations is given below as proposed by Akhtar and Mullick [39]: é ê 2 2 1 ê σ (T pm + Tg )(T pm + Tg ) æç k a Nu1 =ê + ç L Ut ê æ 1 ö g è ç + 1 - 1÷ ê çε ÷ è p εg ø ë
-1
ù ú öú ÷ ú + σε g (Tg2 + Ta2 )(Tg + Ta ) + hw ÷ øú ú û
[
]
-1
+
tg kg
(6)
Finally, thermal efficiency (η) is given below:
η th =
(T - Ti ) ü Qu ì = Fo í(ai ) - U l o ý I IA p î þ
(7)
Plots of thermal efficiency (η) with respect to Reynolds number (Re) at different relative ribs height have been shown in Fig. 14. For comparison purposes, thermal efficiency of smooth duct has also been plotted. It can be seen clearly from these plots that thermal efficiency increase with increase in Reynolds number for all values of relative ribs height, however, rate of increment of thermal efficiency of roughened duct with respect to Reynolds are higher than the rate of increment in case of smooth duct. Maximum and minimum thermal efficiency are found at relative roughness height of 0.044 and 0.020, respectively, for all range of Reynolds number. This trend of the variation of thermal efficiency is related to the similar variation of Nusselt number. The enhancement in thermal efficiency has also been determined in terms of efficiency enhancement factor (EEF) which is defined as the ratio of the thermal efficiency of roughened
23
duct with smooth duct. Efficiency enhancement factor (EEF) at different relative roughness height has been listed in 6.
Fig. 14. Thermal efficiency (η) as function of Reynolds number at different roughness height (e/D) Table 6. Efficiency enhancement factor (EEF) at different relative roughness height
Reynolds Number
Efficiency enhancement factor (EEF) e/D=0.020 e/D=0.0289 e/D=0.036 e/D=0.044
Re=4000
1.239
1.286
1.323
1.346
Re=8000
1.185
1.205
1.219
1.231
Re=12000
1.132
1.145
1.152
1.159
Re=16000
1.100
1.108
1.114
1.118
Similarly, plots of thermal efficiency (η) with respect to Reynolds number (Re) at different relative pitch have been shown in Fig. 15. It can be seen that maximum thermal 24
efficiency have been found for relative rib pitch of 10, and either this value of relative rib pitch of 10, thermal efficiency begin to decrease for all values of Reynolds number. Trend of thermal efficiency is same as the trend of Nusselt number with respect of relative rib pitch. Maximum thermal efficiency are found at relative ribs pitch of 10 and minimum thermal efficiency are found for relative ribs pitch of 6 for all Reynolds number. Efficiency enhancement factor (EEF) at different relative ribs pitch has been listed in Table 7.
Fig. 15. Thermal efficiency (η) as function of Reynolds number at different roughness pitch (p/e) Table 7. Efficiency enhancement factor (EEF) at different relative rib pitch Reynolds Number
Efficiency enhancement factor (EEF) p/e=6
p/e=8
p/e=10
p/e=12
Re=4000
1.204
1.275
1.286
1.252
Re=8000
1.166
1.200
1.205
1.190
Re=12000
1.122
1.142
1.145
1.134
25
Re=16000
1.094
1.106
1.108
1.101
5. Correlations Development It has been shown in the previous sections that Nusselt number and friction factor are found to be strong functions of relative ribs height, relative ribs pitch and Reynolds number. The functional relationships of Nusselt number and friction factor in terms of roughens parameters and operating parameter can be stated as: Nu = F1 (Re, e D , p e )
(8)
f = F2 (Re, e D , p e )
(9)
Similar procedure has been adopted to develop the statistical correlations as used in previous published papers [40,41]. Forced convection heat transfer always results in power law relationship between Nusselt number and Reynolds Number in the following form, which is written as;
Nu = A0 Re n
(10)
In order to get the relationship between Nusselt number and Reynolds number, plots of ln(Nu) verses ln(Re) have been shown for different relative ribs height and fixed value of relative ribs pitch, as shown in Fig. 16. The relation between ln(Nu) and ln(Re) are found to be linear. So, logarithm form of the eq (10) can be written the following form: lnNu = ln A0 + n ln Re
(11)
A linear curve has been generated using all data point of roughened duct, fitted with least square method as shown in Fig. 17 and equation of the curve has been given below: lnNu = A1 + 0.984lnRe
(12)
where, A1 is antilog of A0 and ‘n’ is exponent. 26
By taking antilog,
Nu = A0 ( Re) 0.984
or
Nu = Ao Re 0.984
Coefficient A0(=Nu/Ren) is a function of relative ribs height and relative ribs pitch.
Fig. 16. Variation of ln(Nu) verses ln(Re) for different relative ribs height
27
(13)
Fig. 17. Plot of ln(Nu) verses ln(Re) In order to develop the relation between Nusselt number and relative ribs height, variation of ln(A0) with respect to ln(e/D) have been examined which shows the linear relationship between ln(A0) and ln(e/D). So, linear relationship between ln(A0) and ln(Re) can be written as follows; ln(A0 ) = ln(B0 ) + m ln(e / D)
(14)
A composite straight line has been prepared using all the data point as shown in Fig. 17 and following equation has been established. ln(A0 ) = ln(B0 ) + 0.280 ln(e / D)
(15)
By taking antilog, (16)
A 0 = B0 (e / D) 0.280
or
Re
0.984
Nu = B0 (e / D ) 0.280
(17)
28
where, coefficient, B0= æçç
è Re
0.984
ö Nu ÷ 0.280 ÷ (e / D ) ø
is function of relative ribs pitch.
Fig. 18. Plot of ln(A0) verses ln(e/D) Further, log-log plots of coefficient, B0 and relative ribs pitch have been examined. The relation between ln(B0) and ln(p/e) has been found in the form polynomial of second order in the following form. ln(B 0 ) = ln(C1 ) + o ln( p / e) + p{ln( p / e )}
(18)
2
A composite polynomial curve of second order has been plotted as shown in Fig. 19. A regression analysis is to fit a curve is given below. ln(B0 ) = ln(C1 ) + 4.085 ln( p / e) + (- 0.922 ){ln( p / e )}
(19)
2
By taking antilog
B0 = C0 ( p / e) 4.085 exp[ -0.922{ln( p / e)} ] 2
(20)
Where, C0 is antilog of C1
29
Or
Nu Re
0.984
(e / D )
0.280
( p / e)
4.085
exp[ -0.922{ln( p / e)} ] 2
= C0
(21)
Rearranging the eq. (21)
Nu = C0 Re 0.984(e / D )
0.280
( p / e) 4.085 exp[ -0.922{ln( p / e)} ] 2
(22)
where, C0 is the constant with the value of C0=2.29×10-4. Therefore, final Nusselt number correlation can be written as;
Nu = 2.29 ´ 10-4 Re 0.984(e / D)
0.280
( p / e) 4.085 exp[-0.922{ln( p / e)} ] 2
(23)
Fig. 19. Plot of ln(B0) verses ln(p/e) The values of Nusselt number predicted by correlation (Eq. 23) and CFD simulations have been compared, as shown in Fig. 20. It can be seen that all data points of Nusselt number
30
fall within the deviation limit of ±10%. Average absolute standard deviation in Nusselt, predicted by correlation, has been found as 2.78%. Similarly, correlation of friction factor as function of Reynolds number, relative ribs pitch and relative ribs height has been developed which is written below:
f = 2.19 ´ 104 Re -0.352(e / D)
5.839
exp[0.739{ln(e / D)} ]( p / e)1.860 exp[-0.523{ln( p / e)} ] (24) 2
2
These correlations are only valid for the parameter values in the ranges of 0.02≤e/D≤0.44 and 6≤p/e≤12.
Fig. 20. Comparison of Nusselt number predicted by correlation and CFD simulations The values of friction factor predicted by above correlation have been plotted against the values of friction factor obtained from CFD simulations, as shown in Fig. 21. All the data point fall within the maximum deviation limit of ±10% and average absolute standard deviation of
31
friction factor obtained from correlation has been observed as 5.25%. Therefore, it can concluded that correlations of Nusselt number and friction factor are able to predict their respective values with reasonable accuracy.
Fig. 21. Comparison of friction factor predicted by correlation and CFD simulations 6. Conclusions Numerical simulations have been conducted to predict the thermal hydraulic performance of SAH duct, roughened with conical protrusion ribs. The effect of roughness parameters on heat transfer, friction factor, and thermal efficiency is obtained. The major findings from the study are as below: 1. Providing the conical protrusion ribs results in considerable heat transfer enhancement in comparison to spherical ribs. Average enhancement in Nusselt 32
number and friction factor of conical protrusion ribs over spherical ribs have been observed as 1.30 and 1.07, respectively for relative ribs pitch of 10 and relative ribs height of 0.0298. 2. Maximum Nusselt number and friction factor occurred at relative pitch of 10 and 6, respectively. 3. Maximum enhancement in Nusselt number and friction factor of roughened duct with respect to smooth duct have been observed as 2.49 and 7.29, respectively for relative ribs pitch of 10 and relative ribs height of 0.044. 4. Maximum thermal efficiency and efficiency enhancement factor have been found as 69.8% and 1.346, respectively for relative roughness height of 0.044 and relative rib pitch of 10. 5. Nusselt number and friction factor correlations have been developed as function of roughness and operating parameters. Nusselt number and friction factor correlations predict the data with average absolute standard deviation of 2.78% and 5.25%, respectively. Acknowledgements This research was supported by the Technology Innovation Program (Grant No. 10070117) by the Ministry of Trade, Industry & Energy, Korea. Appendix A. The transports equations of SST k-ω model are given below. ¶ æç ¶k ¶ ¶ Gk ( rk ) + ( rkui ) = ¶x j çè ¶x j ¶xi ¶t
ö ÷ + Gk - Yk + S k ÷ ø
¶ æç ¶w ö÷ ¶ ¶ Gk + Gw - Yw + Sw ( rw) + ( rwui ) = ¶x j çè ¶x j ÷ø ¶xi ¶t
Where 33
Gk is effective diffusivity of k; Gw is effective diffusivity of ɓ; Gk is turbulent kinetic energy due to mean velocity gradient; Gɓ is generation of ɓ; Yk dissipation of k due to turbulence; Yɓ dissipation of ω due to turbulence; References 1. B.N. Prasad, J.S. Saini, Effect of artificial roughness on heat transfer and friction factor in a solar air heater, Solar Energy. 41 (1988) 555–560. 2. B.N. Prasad, A.K. Behura, L. Prasad, Optimizarion of thermo hydraulic performance in three sides artificially roughened solar air heater, Solar Energy. 105 (2015) 313–319. 3. A.M. Ebrahim-Momin, J.S. Saini, S.C. Solanki, Heat transfer and friction in solar air heater duct with V-shaped rib roughness on absorber plate, Int. J. Heat Mass Transf. 45 (2002) 3383–3396. 4. A. Lanjewar, J.L. Bhagoria, R.M. Sarviya, Heat transfer and friction in solar air heater duct with W-shaped rib roughness on absorber plate, Energy. 36 (2011) 4531–4541. 5. A.M. Lanjewar, J.L. Bhagoria, R.M. Sarviya, Performance analysis of W-shaped rib roughened solar air heater, Journal of Renewable and Sustainable Energy. 3 (2011) 43110. 6. V.S. Hans, R.P. Saini, J.S. Saini, Heat transfer and friction factor correlations for a solar air heater duct roughened artificially with multiple v-ribs, Solar Energy. 84 (2010) 898– 911. 7. D. Jin, M. Zhang, P. Wang, S. Xu, Numerical investigation of heat transfer and fluid flow in a solar air heater duct with multi V-shaped ribs on the absorber plate, Energy. 89 (2015) 178–190. 8. K.R. Aharwal, B.K. Gandhi, J.S. Saini, Heat transfer and friction characteristics of solar air heater ducts having integral inclined discrete ribs on absorber plate, International Journal of Heat and Mass Transfer. 52 (2009) 5970–5977. 9. K.R. Aharwal, B.K. Gandhi, J.S. Saini, Experimental investigation on heat-transfer enhancement due to a gap in an inclined continuous rib arrangement in a rectangular duct of solar air heater, Renewable Energy. 33 (2008) 585–596. 10. S. Singh, S. Chander, J.S. Saini, Heat transfer and friction factor of discrete V-down rib roughened solar air heater ducts, Journal of Renewable and Sustainable Energy. 3 (2011) 13108. 11. S. Singh, S. Chander, J.S. Saini, Investigations on thermo-hydraulic performance due to flow-attack-angle in V-down rib with gap in a rectangular duct of solar air heater, Applied. Energy. 97 (2012) 907–912. 12. A. Kumar, R.P. Saini, J.S. Saini, Experimental investigation on heat transfer and fluid flow characteristics of air flow in a rectangular duct with Multi v-shaped rib with gap roughness on the heated plate, Solar Energy. 86 (2012) 1733–1749. 34
13. A. Kumar, R.P. Saini, J.S. Saini, Development of correlations for Nusselt number and friction factor for solar air heater with roughened duct having multi v-shaped with gap rib as artificial roughness, Renewable Energy. 58 (2013) 151–163. 14. Varun, R.P. Saini, S.K. Singal, Investigation of thermal performance of solar air heater having roughness elements as a combination of inclined and transverse ribs on the absorber plate, Renewable Energy. 33 (2008) 1398–1405. 15. A. Layek, J.S. Saini, S.C. Solanki, Heat transfer and friction characteristics for artificially roughened ducts with compound turbulators, nternational Journal of Heat and Mass Transfer. 50 (2007) 4845–4854. 16. A. Layek, J.S. Saini, S.C. Solanki, Effect of chamfering on heat transfer and friction characteristics of solar air heater having absorber plate roughened with compound turbulators, Renewble. Energy. 34 (2009) 1292–1298. 17. A.R. Juarker, J.S. Saini, B.K. Ghandi, Heat transfer and friction characteristics of rectangular solar air heater duct using rib-grooved artificial roughness, Solar Energy. 80 (2006) 895–907. 18. S.K. Sharma, V.R. Kalamkar, Thermo-hydraulic performance analysis of solar air heaters having artificial roughness–A review, Renewable and Sustainable Energy Reviews. 41 (2015) 413–435. 19. R.P. Saini, J. Verma, Heat transfer and friction factor correlations for a duct having dimple-shape artificial roughness for solar air heaters, Energy. 33 (2008) 1277–1287. 20. B. Bhushan, R. Singh, Nusselt number and friction factor correlations for solar air heater duct having artificially roughened absorber plate, Solar Energy. 85 (2011) 1109–1118. 21. M. Sethi, N.S. Thakur, Varun, Heat transfer and friction characteristics of dimple-shaped roughness element arranged in angular fashion (arc) on the absorber plate of solar air heater, Journal of Renewable and Sustainable Energy. 4 (2012) 23112. 22. M. Sethi, Varun, N.S. Thakur, Correlations for solar air heater duct with dimpled shape roughness elements on absorber plate, Solar Energy. 86 (2012) 2852–2861. 23. S. Yadav, M. Kaushal, Varun, Siddhartha, Nusselt number and friction factor correlations for solar air heater duct having protrusions as roughness elements on absorber plate, Experimental Thermal and Fluid Science. 44 (2013) 34–41. 24. ASHRAE, Standard, 93–97, Method of Testing to Determine the Thermal Performance of Solar, Refrigeration and Air Conditioning Engineering, New York, 1977. 25. S. Singh, B. Singh, V.S. Hans, R.S. Gill, CFD (computational fluid dynamics) investigation on Nusselt number and friction factor of solar air heater duct roughened with non-uniform cross-section transverse rib, Energy. 84 (2015) 509–517. 26. S. Kumar, R.P. Saini, CFD based performance analysis of a solar air heater duct provided with artificial roughness, Renew. Energy. 34 (2009) 1285–1291. 27. A. Boulemtafes-Boukadoum, A. Benzaoui, CFD based Analysis of Heat Transfer Enhancement in Solar Air Heater Provided with Transverse Rectangular Ribs, Energy Procedia. 50 (2014) 761–772. 28. P. Promvonge, S. Sripattanapipat, S. Kwankaomeng, Laminar periodic flow and heat transfer in square channel with 45° inline baffles on two opposite walls, International Journal of Thermal Science 49 (2010) 963–975. 29. A.S. Yadav, J.L. Bhagoria, A CFD based thermo-hydraulic performance analysis of an artificially roughened solar air heater having equilateral triangular sectioned rib roughness on the absorber plate, International Journal of Heat and Mass Transfer 70 35
(2014) 1016–1039. 30. A. Bekele, M. Mishra, S. Dutta, Heat transfer augmentation in solar air heater using delta-shaped obstacles mounted on the absorber plate, International Journal Sustainable. Energy. 32 (2013) 53–69. 31. T. Alam, M. Kim, Numerical study on thermal hydraulic performance improvement in solar air heater duct with semi ellipse shaped obstacles, Energy. 112 (2016) 588–598. 32. F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 6th ed., Jefferson city, 2007. 33. R.K. Kakac, S., Shah, W. Aung, Hand Book of Single-Phase Convective Heat Transfer., Wiley, New York, 1987. 34. A. Chaube, P.K. Sahoo, S.C. Solanki, Analysis of heat transfer augmentation and flow characteristics due to rib roughness over absorber plate of a solar air heater, Renew. Energy. 31 (2006) 317–331. 35. E.M. Sparrow, L.M. Hossfeld, Effect of rounding of protruding edges on heat transfer and pressure drop in a duct, International Journal of Heat and Mass Transfer. 27 (1984) 1715–1723. 36. A.S. Yadav, J.L. Bhagoria, A numerical investigation of square sectioned transverse rib roughened solar air heater, Inernational Journal of Thermal Science 79 (2014) 111–131. 37. P. Bondi, L. Cicala, G. Farina, Performance analysis of solar air heaters of conventional design, Solar Energy. 41 (1988) 101–107. 38. D.E. Klein, J.A. Duffie, W.A. Beckman, A design procedure for solar air heating systems, Solar Energy. 18 (1977) 509–512. 39. S.A. Klein, Calculation of fiat-plate collector loss coefficients, Solar Energy. 17 (1975) 79–80. 40. R.P. Saini, J.S. Saini, Heat transfer and friction factor correlations for artificially roughened ducts with expanded metal mesh as roughness element, International Journal of Heat and Mass Transfer. 40 (1997) 973–986. 41. T. Alam, R.P. Saini, J.S. Saini, Effect of circularity of perforation holes in V-shaped blockages on heat transfer and friction characteristics of rectangular solar air heater duct, Energy Conversion and Management. 86 (2014) 952–963. Highlights · · · ·
Investigate CFD analysis of SAH duct, roughened with conical protrusion ribs. Provide thermohydraulic performance for SAH duct with conical protrusions ribs. Obtain maximum thermal efficiency of 70% and efficiency enhancement factor of 1.35. Provide Nusselt number and friction factor correlations as function of Re number.
36
S1359-4311(17)31394-7 http://dx.doi.org/10.1016/j.applthermaleng.2017.07.181 ATE 10849
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
1 March 2017 7 June 2017 25 July 2017
Please cite this article as: T. Alam, M-H. Kim, Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng. 2017.07.181
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Heat transfer enhancement in solar air heater duct with conical protrusion roughness ribs Tabish Alam, Man-Hoe Kim* School of Mechanical Engineering, Kyungpook National University, Daegu 41566, South Korea Abstract Application of protrusion rib roughnesses on the absorber plate of solar air heater (SAH) duct can effectively enhance the heat transfer rate irrespective of pressure drop penalty. This paper presents the numerical investigation of SAH duct, roughened with conical protrusion ribs. Effect of relative ribs pitch (6≤p/e≤12) and relative ribs height (0.020≤e/D≤0.044) on Nusselt number and friction factor have been studied for the range of Reynolds number from 4000 to 16000. Thermal efficiency of roughened duct have been determined using useful energy gain to air and heat losses to environment. The maximum thermal efficiency (η) and efficiency enhancement factor (EEF) are found as 69.8% and 1.346, respectively. Correlations of friction factor and Nusselt number have also been developed as function of Reynolds number and roughness parameters of conical ribs. Keywords: Heat transfer enhancement, Efficiency enhancement factor, Conical rib, Solar air heater
*
Corresponding Author. Tel.: +82-53-950-5576; Fax: +82-53-950-6550. E-mail addresses: [email protected] (M. H. Kim), [email protected] (T.Alam)
1
Nomenclature
I ka kg m N Nu Nus P P/e Pr
Area of absorber plate, m2 Specific heat of air at constant pressure, J/Kg.K Duct height, mm Rib height, mm Relative rib height Flat efficiency factor, Heat removal factor Friction factor of roughened duct Friction factor of smooth duct Convective heat transfer coefficient, W/m2.K Solar insolation, W/m2 Thermal conductivity of air, W/m/K Thermal conductivity of glass, W/m/K Mass flow rate, Kg/s Number of glass cover Nusselt number of roughened duct Nusselt number of smooth duct Pitch of ribs, mm Relative pitch ratio Prandtl number
Qu
Useful heat gain, J
Ap Cp D e e/D F’ Fo f fs h
Re Tg
Reynolds number Mean Temperature of glass, K
Ti To Tpm ti tg tg Ub Ue
Inlet temperature of air, K Outlet temperature of air, K Average plate mean temperature, K Thickness of insulation, mm Thickness of glass cover, mm Height of collector edge, mm Back heat loss coefficient, W/m2.K Edge heat loss coefficient, W/m2.K
Ut Ul εg εp ηth ρ σ τ µ (τα)
Top heat loss coefficient, W/m2.K Overall heat loss coefficient, W/m2.K Emissivity of glass cover Emissivity of absorber plate Thermal efficiency of collector Density of air, kg/m3 Stefan-Boltzmann Constant, W/m2.K4 Trasnmissivity of glass cover Dynamic viscosity of air, N.s/m2 Transmittance-absorbent product of glass cover
1. Introduction In the past few years, we are facing the problems of energy crisis due to higher prices and exhaustive nature of fossil fuels such as crude oil, coal, natural gas and so on. High usage of fossil fuels causes to threat the environmental asset in term of wildlife, land disturbance, pollution and human health. Renewable energy can minimize our dependency on fossil fuels, thereby, renewable energies are getting importance in the recent years because energy can renewed and will never run out. Renewable energy is ecofriendly and results in little to no effect to the environment. Out of many renewable energies, solar energy is considered to be clean source of energy and available on every part of the world. Solar energy is exploited in many application, included heating purposes and generation of electricity. Solar air heater (SAH) is 2
one of the most economical and elementary device which employ to supply the heated air to drying the crops, industrial purposes, heating the building and space. Absorber plate of SAH converts solar insolation into thermal energy of air when passing through underside of absorber plate when it exposed to solar radiation, however, convective heat transfer coefficient is inherent low causes to low performance of SAH. The low convective heat transfer coefficient between absorber plate and flowing air is observed due to presence of viscous laminar sublayer on absorber plate. Artificial roughness is considered to be good technique to enhance the convective heat transfer coefficient. Artificial roughness on absorber plate disturb viscous laminar sublayer causes to increase heat transfer rate[1]. Artificial roughness not only enhance the heat transfer rate but also increase pressure drop which is inadmissible. Artificial roughness, provided in the form of transverse small diameter wires, was first investigated in SAH and shown remarkable heat transfer enhancement is observed in comparison to conventional smooth solar air heater duct [1,2]. Small roughness wires alter the flow pattern and creates the turbulence in the vicinity of heated surface without disturbing the core flow. Various arrangements of wires, namely; V-shaped [3], W-shaped [4,5], multi V-shaped[6,7], broken angled ribs[8,9], V-ribs with gap[10,11], multi V-ribs with gap [12,13]have been investigated.
Combinations of ribs arrangements such as transvers and inclined ribs [14],
chamfered ribs and grooves [15,16], rectangular ribs and groove [17] have also been studied. The grooves were created by machining process and require great attentions. Various researchers investigated the spherical/protrusion ribs out of many artificial roughness, it does not show much more pressure penalty as compared to other roughness [18], thereby, spherical/protrusion rib roughness is getting importance. In this regards, Saini and Verma [19] investigated the spherical ribs on absorber plate. The spherical ribs were created by pressurized the indentation–producing device on absorber plate. Effect of roughness parameters, i.e. relative ribs height (0.0189-0.0379) and pitch (8-12) on heat transfer and friction characteristics were studied. Considerable heat
3
transfer rate was reported as a results of spherical ribs provided on absorber plate. Bhushan and Singh [20] introduced new parameters of spherical ribs roughness, namely; relative short way length (18.75-37.50), relative long way length (25.0-37.50) and relative print diameter (0.1470.3670). It was concluded that Nusselt number and friction number were enhanced upto 3.2 and 2.2 times with respect to smooth duct in the ranges of parameters investigated. Effect of arc arrangement of spherical ribs were studied and angle of arcs were varied from 45° to 75°[21,22]. Angle of arc affected the heat transfer and maximum heat transfer was found at 60° angle of arc. The correlations of friction factor and Nusselt number were also developed as function spherical ribs parameters. Yadav et al. [23] also investigated the similar pattern of spherical ribs as investigated by Sethi [21,22], although, dimension of ducts were different. The application of protrusion roughness in SAH may be attractive because protrusion does not add extra mass on absorber plate and manufacturing of this roughness does not require special treatment. These roughness can be created by punching the thin absorber plate using indentation device. The literature survey shows that protrusions ribs of hemispherical/spherical shape have been investigated previously which are very common shape. In order to increase heat transfer further, conical shape protrusion ribs have been investigated in this work with the aim of sharp apex corner of cone would contribute the turbulence in the flow resulting high heat transfer rate from absorber plate. In this regards, Numerical study has been conducted to investigate the effect of relative ribs pitch (p/e) and relative ribs height (e/D) of conical shape protrusions ribs attached to absorber plate on friction factor and Nusselt number. Using the numerical data, friction factor and Nusselt number correlations have also been developed.
4
2. Detail of computational model In the present work, numerical simulations of SAH have been carried out using finite volume method. The effect of protrusion ribs of conical shape on absorber plate have been investigated. The details of geometry, computational flow domain, mesh generation, boundary conditions, solution method, selection of turbulence model and results validation have been discussed in following sub-sections. 2.1 Description of geometry and parameters Height, length and width of computational flow domain of SAH are considered as 25 mm, 1000 mm and 300 mm, respectively as per ASHARE’s recommendation [24]. Although, actual solar air heater consisted inlet plenum and outlet plenum along with test section which are omitted in present numerical studies to make the model simple [25–27]. Only test section has been considered in the present numerical studies. One broad wall of model is considered as absorber plate on which protrusion ribs of conical shape have been attached. Absorber plate has been exposed to uniform heat flux of 1000 W/m2 which is done to provide insolation, falling on absorber plate. Three sides of the duct were smooth and insulated. Air enter in the duct at inlet section, extract heat from absorber plate when air comes in contact with absorber plate and then air comes out from the duct at outlet section. The computational flow domain of the solar air heater has been showing in Fig. 1. The ranges of relative ribs height (e/D) and relative ribs pitch (p/e) of conical protrusion ribs have been chosen as 0.020-0.044 and 6-12, respectively, which are similar to previous studies, conducted for spherical ribs [19]. All the simulations have been carried out at four values
5
of Reynolds number (Re) in the range of 4000-16000. However, relative base diameter of ribs (d/e) are fixed for all simulations. Ranges of ribs parameter have been listed in Table 1.
Fig. 1. Computation flow domain Table 1. Parameter ranges
S. No.
Parameter
Range
1
Relative rib height ratio (e/D)
0.020-0.044
2
Relative rib pitch ratio (p/e)
6-12
3
Reynolds number (Re)
4000-16000
2.2 Computational flow domain and mesh generation 3-Dimensional flow domains have been created using design modular of Ansys 16.0. Only half flow domain has been exploited in all simulation due to symmetrical body of flow domain and symmetry boundary condition has been applied on the mid-section (z=0) of the duct. 6
In order to discretize the flow domain, unstructured mesh has been created for all simulations. Very fine prism mesh near the absorber and rib walls have been created to resolve the boundary layer in the vicinity of wall bounded domain. Non-dimensionless wall distance (y+) for absorber wall and ribs bounded flow have been estimated less then unity. Initially, grid independent test has been conducted for different number of elements and nodes in the ranging from 13.75×106 to 17.17×106 and from 3.77×106 to5.09×106, respectively for conical protrusion ribs having relative ribs height of 0.0289 and relative ribs pitch ratio (p/e) of 10 at Reynolds number (Re) of 12000. Relative percentage variation in Nusselt number have been observed as 1.63%, 0.42% and 0.37% when number of elements increases from 13.75×106 to 15.03×106 , 15.03×106 to 16.01×106 and 16.01×106 to 17.17×106, respectively. Table 2 shows the effect of variation elements number on the percentage variation of Nusselt number and friction factor. Based on the results obtained, number of elements of 16×106 has been considered for further simulation. Fig. 2. shows the meshed flow domain of the model.
Fig. 2. Partial view of meshed domain 7
Table 2. Detail of grid independent test S. No.
Number of elements
Percentage variation in Nusselt Number
Percentage variation in friction factor
1
13.75×106
--
--
2
15.03×10
6
1.63%
2.36%
3
16.01×106
0.42%
0.54%
4
17.17×106
0.37%
0.81%
2.3 Boundary Conditions After generating the mesh, similar boundary conditions have been imposed on the flow domain as found on actual solar air heater. In order to save computer memory and simulation times, only half flow domains have been exploited in the present work and symmetric wall condition have been applied at mid plane (z =0) on the duct. Top side of domain has been considered as absorber plate on which conical ribs have been created and heat flux of 1000 W/m2, equivalent to typical insolation has been imposed on the absorber plate. Sides and bottom plane are considered to be smooth and insulated and no slip boundary condition has been imposed. Pressure outlet and velocity inlet conditions have been assigned on the outlet and inlet section of the domain, respectively. Based on the desired Reynolds number, values of inlet velocities have been determined at uniform temperature of 300° K. Due to small rise in temperature of air, all physical properties are considered to be fixed which are determined on 300° K temperature and listed in Table 3.
8
Table 3. Thermo-physical properties of air at 300° K
Property Name
Value
Density (ρ)
1.117 kg/m-3
Prandlt Number (Pr)
0.71
Specific heat (Cp)
1007 J/kg.°K
Viscosity (µ)
1.857×10-8 Ns/m2
2.4 Solution Methods In present simulations, commercial code of Ansys Fluent 16.0 has been used to analyze the fluid flow and heat transfer of 3-dimensional computational domain using finite volume method. Using the boundary conditions, the governing equations in the incompressible and steady state have been solved. The transport equations of model are given in Appendix A. Velocity and pressure field have coupled using SIMPLE (semi-implicit method for pressure linked equations) algorithm. Second order upwind scheme has been exploited to special discretization [28,29]. Convergence criteria for energy equation has been taken in the account of 10-6 order, however, criteria for velocity component and momentum are considered as 10-4 [30,31]. 2.5 Selection of turbulence model and result validations In author’s previous work [31], different turbulence models were used for smooth duct and based on the comparison of CFD results with the result obtained from correlations of Nusselt number (Gnielinski and Dittus-Boelter correlation[32]) and friction factor (Blasius Equation 9
[33]), RNG k-ε turbulence model was selected. The correlations of Nusselt number and friction factor have been given below. Gnielinski correlation,
Nu =
(f/8)(Re - 1000)Pr for 3000 < Re < 10000 1 + 12.7(f/8) 1/2(Pr 2/3 - 1)
(1)
where, Darcy factor, f = {0.79 ln (Re ) - 1.64 }
-2
Dittus-Boelter correlation, Nu = 0.023 Re 0.8 Pr
0.4
for Re > 10000
(2)
Blasius Equation, f s = 0.0085 Re
-0.25
(3)
In order to select the turbulence model in the present work, simulations of hemispherical protrusion ribs created on absorber plate have been carried out using five turbulence model, namely; standard k-ε, RNG k-ε, Realizable k-ε, Standard k-ω and SST k-ω. In this context, hemispherical protrusion ribs of relative ribs height of 0.0289 and ribs pitch of 10 have been created on the absorber plate which is very similar to the experimental work, conducted by Verma and Saini [19]. In order to check the accuracy of the different models, predicted values of Nusselt number and friction factor obtained at various Reynolds number (Re = 3054, 6119, 9701 and 11138) from different turbulence models have been compared with the experimental data [19]. Comparison of Nusselt number and friction have been presented in Fig. 3 and Fig. 4, respectively.
10
Fig. 3. Comparison of Nusselt number predicted by different turbulence models
Fig. 4. Comparison of friction factor predicted by different turbulence models
11
These graphs clearly shows that out of different turbulence models, SST model predict Nusselt number and friction factor very close to experimental data [19]. Minimum average absolute standard deviation error for friction factor and Nusselt number has been observed as 10.1% and 9.4%, respectively in case of SST turbulence model, however, Realizable k-ε turbulence model and Standard k-ε turbulence model predict the maximum average absolute standard deviation for friction factor and Nusselt number, respectively. It has been clear from this analyses that results, predicted by SST k-ε turbulence model, are in good agreement with results obtained from experimental work. Further, present work exploited SST k-ω turbulence model for all simulations which was also recommended by other investigators [27,34]. 3. Results and discussion For the protrusions conical ribs on absorber plate, predicted data of Nusselt number and friction factor have been obtained for the Reynolds number in the range of 4000 – 16000. The effect of shape of ribs, Reynolds number, relative ribs height and relative ribs pitch have been discussed below. 3.1 Effect of rib shapes Nusselt number of roughened duct equipped with conical protrusion ribs and spherical ribs have been compared with Nusselt number of smooth duct (using Gnielinski correlation (Eq. 1) and Dittus-Boelter correlation (Eq. 2)) as a function of Reynolds number for relative ribs height of 0.0289 and relative ribs pitch of 10, as shown in Fig. 5. It is clearly seen that Nusselt number of both the roughened ducts are much higher than the Nusselt number of smooth duct. This is due to fact that higher turbulence in the flow provide higher Nusselt number in case of roughened duct. Addition to this, Nusselt number provided by conical protrusion ribs are observed higher in comparison to Nusselt number, obtained from spherical ribs. In this context, velocity vector around the protrusion conical ribs and spherical ribs have been shown on the mid plane (z = 0) of the duct in the direction of flow for Reynolds number of 12000, as shown in Fig. 12
6. Sharp edge of conical ribs are attributes to high turbulence in the flow along with strong reattachment point on surface. Strong re- attachments on the surface disturb sub-laminar layer and high turbulence help to mix the flow near surface to core flow. However, re-attachment point has also been observed in case of spherical ribs but intensity of reattachment point is too week to enhance the heat transfer rate. The variation of friction factor of smooth duct (using Blasius Equation (eq. 3)) and roughened ducts with conical protrusion ribs and spherical ribs as a function of Reynolds number have been presented in Fig. 7. As the value of Reynolds number increase, friction factor of smooth duct and both
the roughened ducts decrease due to
deterioration of laminar-sub layer [25] and higher friction factor have been found for protrusion conical ribs because sharp edge of cone causes to higher pressure drop in the flow [35]. Average enhancement in Nusselt number and friction factor of conical ribs with respect to spherical ribs are found as 1.3 and 1.07, respectively.
Fig. 5. Variation of Nusselt number as a function of Reynolds number 13
Fig. 6. Velocity vector around conical and spherical ribs
14
Fig. 7. Variation of friction factor as a function of Reynolds number 3.2 Effect of Reynolds number (Re) Similar trend of Nusselt number with respect to Reynolds number have been observed at other parameters of conical ribs as shown in Fig. 5, which was expected. An increment in Nusselt number as a function of Reynolds number can be explained on the basis of turbulence kinetic energy. In this context, contours of turbulence kinetic energy at various Reynolds number have been shown on the mid plane (z = 0) of the duct along the flow direction (Fig. 8). Intensity of turbulence kinetic energy increases with increase in Reynolds number and region of high turbulence kinetic energy have been found downstream of the rib at all Reynolds number. High turbulence kinetic energy region past to ribs have been found due to large velocity gradient in the flow [29]. Region of high turbulence kinetic energy decreases with increase in Reynolds number, however, scale of turbulence kinetic energy increases rapidly. The maximum turbulence kinetic energy have been found as a function of Reynolds number and the value of maximum turbulence 15
kinetic energy increase from 0.26 J/kg to 1.74 J/kg with the values of Reynolds number, changes from 4000 to 16000.
Fig.8. Turbulence kinetic energy contours for different Reynolds Number
3.3 Effect of relative ribs height (e/D) In order to show the effect of relative ribs height, variation of Nusselt number as function of relative ribs height have been plotted in Fig. 9 for different values of Reynolds number and fixed values of relative ribs pitch (p/e=10). It is observed from these plots that Nusselt Number increase linearly with increase in the values of relative ribs height for given relative ribs pitch and all values of Reynolds number. Relative ribs height of 0.044 provide maximum friction 16
factor and Nusselt number, and relative rib height of 0.020 provide minimum friction factor and Nusselt number. It is fact that strong turbulence increases due to local fluid velocity profile which is mixed vigorously in between the ribs, when relative ribs height increase. These results are inline with results of obtained in case of rectangular ribs [36] and equilateral triangular ribs [29]. Variation of friction factor as function of relative ribs height have also been plotted in Fig. 10 for different values of Reynolds number and fixed value of relative pitch (p/e=10). It has been clearly seen from these plots that friction factor increase when relative ribs height increase from 0.020 to 0.044. This is due to fact that higher relative roughness height contributed more disruption to flow in comparison to disruption offered by low roughness height. Flow disruption due to ribs lead to the dissipation of dynamic pressure of the fluid causes to higher values of friction factor, which is also reported in other investigation[29].
Fig. 9. Variation of Nusselt number as a function of relative ribs height
17
Fig. 10. Variation of friction factor as a function of relative ribs height Enhancement factor in friction factor and Nusselt number of roughened duct with respect to friction factor and Nusselt number of smooth duct have also been determined for different relative ribs height and fixed value of relative ribs pitch of 10. The maximum values of enhancement factor for friction factor and Nusselt number at various relative ribs height have been listed in Table 4 for the range of Reynolds considered. Maximum enhancement factor in friction factor and Nusselt number with increase in the values of relative ribs height. Table 4. Maximum values of enhancement factor at different relative ribs height S. No.
Parameter
Nu/Nus
f/fs
1
e/D = 0.0200
2.18
5.05
2
e/D = 0.0289
2.30
5.43
3
e/D = 0.0360
2.40
6.18
4
e/D = 0.0440
2.49
7.39
18
3.4 Effect of relative ribs pitch (p/e) Effect of the relative ribs pitch on Nusselt number has been presented for all values of Reynolds number by keeping fixed value of relative ribs height (e/D=0.0289), as shown in Fig. 11. The values of Nusselt number increase with increasing the value of relative ribs pitch from 6 to 10, beyond the relative ribs pitch of 10, the values of Nusselt number begin to decrease for all values of Reynolds number. Relative pitch value of 6 offers the lowest values of Nusselt number and relative ribs pitch value of 10 offers the highest values of Nusselt number. In the present study, 10 relative ribs pitch has been found as optimum for the range of Reynolds number considered. These results are in line with results of spherical ribs[19]. Higher Nusselt number for relative pitch of 10 may due to the lateral movement of air when air passes over the absorber plate. Lateral movement of air causes to reduce the re-circulation zone and hence better heat transfer area is achieved [25]. In this context, velocity vector around the ribs on the plane, parallel to absorber plate have been presented in Fig. 13 for different values of relative ribs pitch and fixed values of Reynolds number (Re=12000). It can been seen from the velocity vector that as lateral movement of air increase with increasing the relative ribs pitch from 6 to 12, however, increasing values of relative ribs pitch offers low level of turbulence as number of ribs is decreased. So, relative pitch of 10 attributed the desirable conditions which gives the highest Nusselt number. Fig. 12 shows the effect of relative ribs pitch on friction factor for different values of Reynolds number and fixed values of relative ribs height of 10. The values of friction factor decrease continuously with increasing the relative pitch ratio, which is due to fact that higher values of relative pitch attributed to low resistance offer to flow. The minimum and maximum values of friction factor were observed at relative pitch of 12 and 6, respectively.
19
Fig. 11. Variation of Nusselt number as a function of relative ribs pitch
Fig. 12. Variation of friction factor as a function of relative ribs pitch 20
Fig. 13. Velocity vectors for different relative pitch at parallel plane to absorber plate (at Re = 12000) Enhancement factor in friction factor and Nusselt number of roughened duct with respect to friction factor and Nusselt number of smooth duct have also been determined. Table 5. has been made to show the maximum values of enhancement factor in friction factor and Nusselt number for different values of relative pitch ratio and fixed value of relative rib height of 0.0289. Maximum values of enhancement factor in Nusselt number is observed at relative pitch of 6, either side of relative pitch of 10, it decrease. Although, maximum value of enhancement factor in friction factor decrease with increment of relative ribs pitch.
21
Table 5. Maximum values of enhancement factor at different relative ribs height S. No.
Parameter
Nu/Nus
f/fs
1
p/e = 6
2.02
5.89
2
p/e = 8
2.21
5.71
3
p/e = 10
2.30
5.43
4
p/e = 12
2.13
4.68
4. Thermal efficiency (η) Determination of thermal efficiency of roughened duct is similar to the efficiency of smooth duct. It has been noticed in previous discussion that heat transfer rate of roughened duct is found higher than the heat transfer rate in smooth duct. So it is necessary to determine the thermal efficiency of roughened duct for different roughness parameters. In this context, thermal efficiency have been calculated which is based on insolation fall on absorber plate, heat energy gain to air and thermal losses to environment. The following heat energy (Qu) gain to air is given by Bondi et al. [37]; Q u = Fo Ap (I (ia ) - U l (To - Ti ))
(4)
Where, Fo is heat removal factor and F’ is collector fin efficiency, which are given below: mC P é ì F'APU l ü ù Fo = êexp í ý - 1ú APU l êë î mC P þ ûú
U and F ' = æçç1 + l ö÷÷ è
-1
h ø
Total heat loss coefficient (Ul) is sum of bottom loss, edge loss and top loss coefficient as given by Duffie and Beckmann [38] in following manner; Ul = Ub +U e +Ut
(5)
22
Where, back loss coefficient (Ub) and edge loss coefficient (Ue) are expressed as; Ub =
k i and (L+ W ) t e k i Ue = ti LWt i
Calculation of top loss coefficient is very complex. It can be calculated using graphical method and iterative procedures. However, some correlations available to calculate it, very common correlations is given below as proposed by Akhtar and Mullick [39]: é ê 2 2 1 ê σ (T pm + Tg )(T pm + Tg ) æç k a Nu1 =ê + ç L Ut ê æ 1 ö g è ç + 1 - 1÷ ê çε ÷ è p εg ø ë
-1
ù ú öú ÷ ú + σε g (Tg2 + Ta2 )(Tg + Ta ) + hw ÷ øú ú û
[
]
-1
+
tg kg
(6)
Finally, thermal efficiency (η) is given below:
η th =
(T - Ti ) ü Qu ì = Fo í(ai ) - U l o ý I IA p î þ
(7)
Plots of thermal efficiency (η) with respect to Reynolds number (Re) at different relative ribs height have been shown in Fig. 14. For comparison purposes, thermal efficiency of smooth duct has also been plotted. It can be seen clearly from these plots that thermal efficiency increase with increase in Reynolds number for all values of relative ribs height, however, rate of increment of thermal efficiency of roughened duct with respect to Reynolds are higher than the rate of increment in case of smooth duct. Maximum and minimum thermal efficiency are found at relative roughness height of 0.044 and 0.020, respectively, for all range of Reynolds number. This trend of the variation of thermal efficiency is related to the similar variation of Nusselt number. The enhancement in thermal efficiency has also been determined in terms of efficiency enhancement factor (EEF) which is defined as the ratio of the thermal efficiency of roughened
23
duct with smooth duct. Efficiency enhancement factor (EEF) at different relative roughness height has been listed in 6.
Fig. 14. Thermal efficiency (η) as function of Reynolds number at different roughness height (e/D) Table 6. Efficiency enhancement factor (EEF) at different relative roughness height
Reynolds Number
Efficiency enhancement factor (EEF) e/D=0.020 e/D=0.0289 e/D=0.036 e/D=0.044
Re=4000
1.239
1.286
1.323
1.346
Re=8000
1.185
1.205
1.219
1.231
Re=12000
1.132
1.145
1.152
1.159
Re=16000
1.100
1.108
1.114
1.118
Similarly, plots of thermal efficiency (η) with respect to Reynolds number (Re) at different relative pitch have been shown in Fig. 15. It can be seen that maximum thermal 24
efficiency have been found for relative rib pitch of 10, and either this value of relative rib pitch of 10, thermal efficiency begin to decrease for all values of Reynolds number. Trend of thermal efficiency is same as the trend of Nusselt number with respect of relative rib pitch. Maximum thermal efficiency are found at relative ribs pitch of 10 and minimum thermal efficiency are found for relative ribs pitch of 6 for all Reynolds number. Efficiency enhancement factor (EEF) at different relative ribs pitch has been listed in Table 7.
Fig. 15. Thermal efficiency (η) as function of Reynolds number at different roughness pitch (p/e) Table 7. Efficiency enhancement factor (EEF) at different relative rib pitch Reynolds Number
Efficiency enhancement factor (EEF) p/e=6
p/e=8
p/e=10
p/e=12
Re=4000
1.204
1.275
1.286
1.252
Re=8000
1.166
1.200
1.205
1.190
Re=12000
1.122
1.142
1.145
1.134
25
Re=16000
1.094
1.106
1.108
1.101
5. Correlations Development It has been shown in the previous sections that Nusselt number and friction factor are found to be strong functions of relative ribs height, relative ribs pitch and Reynolds number. The functional relationships of Nusselt number and friction factor in terms of roughens parameters and operating parameter can be stated as: Nu = F1 (Re, e D , p e )
(8)
f = F2 (Re, e D , p e )
(9)
Similar procedure has been adopted to develop the statistical correlations as used in previous published papers [40,41]. Forced convection heat transfer always results in power law relationship between Nusselt number and Reynolds Number in the following form, which is written as;
Nu = A0 Re n
(10)
In order to get the relationship between Nusselt number and Reynolds number, plots of ln(Nu) verses ln(Re) have been shown for different relative ribs height and fixed value of relative ribs pitch, as shown in Fig. 16. The relation between ln(Nu) and ln(Re) are found to be linear. So, logarithm form of the eq (10) can be written the following form: lnNu = ln A0 + n ln Re
(11)
A linear curve has been generated using all data point of roughened duct, fitted with least square method as shown in Fig. 17 and equation of the curve has been given below: lnNu = A1 + 0.984lnRe
(12)
where, A1 is antilog of A0 and ‘n’ is exponent. 26
By taking antilog,
Nu = A0 ( Re) 0.984
or
Nu = Ao Re 0.984
Coefficient A0(=Nu/Ren) is a function of relative ribs height and relative ribs pitch.
Fig. 16. Variation of ln(Nu) verses ln(Re) for different relative ribs height
27
(13)
Fig. 17. Plot of ln(Nu) verses ln(Re) In order to develop the relation between Nusselt number and relative ribs height, variation of ln(A0) with respect to ln(e/D) have been examined which shows the linear relationship between ln(A0) and ln(e/D). So, linear relationship between ln(A0) and ln(Re) can be written as follows; ln(A0 ) = ln(B0 ) + m ln(e / D)
(14)
A composite straight line has been prepared using all the data point as shown in Fig. 17 and following equation has been established. ln(A0 ) = ln(B0 ) + 0.280 ln(e / D)
(15)
By taking antilog, (16)
A 0 = B0 (e / D) 0.280
or
Re
0.984
Nu = B0 (e / D ) 0.280
(17)
28
where, coefficient, B0= æçç
è Re
0.984
ö Nu ÷ 0.280 ÷ (e / D ) ø
is function of relative ribs pitch.
Fig. 18. Plot of ln(A0) verses ln(e/D) Further, log-log plots of coefficient, B0 and relative ribs pitch have been examined. The relation between ln(B0) and ln(p/e) has been found in the form polynomial of second order in the following form. ln(B 0 ) = ln(C1 ) + o ln( p / e) + p{ln( p / e )}
(18)
2
A composite polynomial curve of second order has been plotted as shown in Fig. 19. A regression analysis is to fit a curve is given below. ln(B0 ) = ln(C1 ) + 4.085 ln( p / e) + (- 0.922 ){ln( p / e )}
(19)
2
By taking antilog
B0 = C0 ( p / e) 4.085 exp[ -0.922{ln( p / e)} ] 2
(20)
Where, C0 is antilog of C1
29
Or
Nu Re
0.984
(e / D )
0.280
( p / e)
4.085
exp[ -0.922{ln( p / e)} ] 2
= C0
(21)
Rearranging the eq. (21)
Nu = C0 Re 0.984(e / D )
0.280
( p / e) 4.085 exp[ -0.922{ln( p / e)} ] 2
(22)
where, C0 is the constant with the value of C0=2.29×10-4. Therefore, final Nusselt number correlation can be written as;
Nu = 2.29 ´ 10-4 Re 0.984(e / D)
0.280
( p / e) 4.085 exp[-0.922{ln( p / e)} ] 2
(23)
Fig. 19. Plot of ln(B0) verses ln(p/e) The values of Nusselt number predicted by correlation (Eq. 23) and CFD simulations have been compared, as shown in Fig. 20. It can be seen that all data points of Nusselt number
30
fall within the deviation limit of ±10%. Average absolute standard deviation in Nusselt, predicted by correlation, has been found as 2.78%. Similarly, correlation of friction factor as function of Reynolds number, relative ribs pitch and relative ribs height has been developed which is written below:
f = 2.19 ´ 104 Re -0.352(e / D)
5.839
exp[0.739{ln(e / D)} ]( p / e)1.860 exp[-0.523{ln( p / e)} ] (24) 2
2
These correlations are only valid for the parameter values in the ranges of 0.02≤e/D≤0.44 and 6≤p/e≤12.
Fig. 20. Comparison of Nusselt number predicted by correlation and CFD simulations The values of friction factor predicted by above correlation have been plotted against the values of friction factor obtained from CFD simulations, as shown in Fig. 21. All the data point fall within the maximum deviation limit of ±10% and average absolute standard deviation of
31
friction factor obtained from correlation has been observed as 5.25%. Therefore, it can concluded that correlations of Nusselt number and friction factor are able to predict their respective values with reasonable accuracy.
Fig. 21. Comparison of friction factor predicted by correlation and CFD simulations 6. Conclusions Numerical simulations have been conducted to predict the thermal hydraulic performance of SAH duct, roughened with conical protrusion ribs. The effect of roughness parameters on heat transfer, friction factor, and thermal efficiency is obtained. The major findings from the study are as below: 1. Providing the conical protrusion ribs results in considerable heat transfer enhancement in comparison to spherical ribs. Average enhancement in Nusselt 32
number and friction factor of conical protrusion ribs over spherical ribs have been observed as 1.30 and 1.07, respectively for relative ribs pitch of 10 and relative ribs height of 0.0298. 2. Maximum Nusselt number and friction factor occurred at relative pitch of 10 and 6, respectively. 3. Maximum enhancement in Nusselt number and friction factor of roughened duct with respect to smooth duct have been observed as 2.49 and 7.29, respectively for relative ribs pitch of 10 and relative ribs height of 0.044. 4. Maximum thermal efficiency and efficiency enhancement factor have been found as 69.8% and 1.346, respectively for relative roughness height of 0.044 and relative rib pitch of 10. 5. Nusselt number and friction factor correlations have been developed as function of roughness and operating parameters. Nusselt number and friction factor correlations predict the data with average absolute standard deviation of 2.78% and 5.25%, respectively. Acknowledgements This research was supported by the Technology Innovation Program (Grant No. 10070117) by the Ministry of Trade, Industry & Energy, Korea. Appendix A. The transports equations of SST k-ω model are given below. ¶ æç ¶k ¶ ¶ Gk ( rk ) + ( rkui ) = ¶x j çè ¶x j ¶xi ¶t
ö ÷ + Gk - Yk + S k ÷ ø
¶ æç ¶w ö÷ ¶ ¶ Gk + Gw - Yw + Sw ( rw) + ( rwui ) = ¶x j çè ¶x j ÷ø ¶xi ¶t
Where 33
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Investigate CFD analysis of SAH duct, roughened with conical protrusion ribs. Provide thermohydraulic performance for SAH duct with conical protrusions ribs. Obtain maximum thermal efficiency of 70% and efficiency enhancement factor of 1.35. Provide Nusselt number and friction factor correlations as function of Re number.
36