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Heat transfer from a slot air jet impinging on a circular cylinder
Journal of Food Engineering 63 (2004) 393–401 www.elsevier.com/locate/jfoodeng
Heat transfer from a slot air jet impinging on a circular cylinder E.E.M. Olsson b
a,b,*
, L.M. Ahrne a, A.C. Tr€ ag ardh
b
a SIK-The Swedish Institute for Food and Biotechnology, P.O. Box 5401, SE-402 29 G€oteborg, Sweden Department of Food Technology, Engineering and Nutrition, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
Received 22 March 2003; accepted 16 August 2003
Abstract Heat transfer from a slot air jet impinging on a cylinder shaped food product placed on a solid surface in a semi-confined area was investigated using computational fluid dynamics (CFD). Simulations of a cylinder in cross flow with the k–e, k–x and SST models in CFX 5.5 were compared with measurements in the literature. The SST model predicts the heat transfer better than the other models and is therefore used in this study. The distribution of the local Nusselt numbers around the cylinder for various Reynolds numbers (23,000–100,000), jet-to-cylinder distances, H =d (2–8), and cylinder curvature, d=D (0.29–1.14) was determined. The results show that the local Nusselt numbers varies around the surface of the cylinder and that the average Nusselt number and the stagnation point Nusselt number increases with increasing Reynolds numbers and surface curvature but has little dependency on the jet-to-cylinder distance. The result is Num ¼ 0:14Re0:65 ðH =dÞ0:077 ðd=DÞ0:32 and Nus ¼ 0:46Re0:59 ðH =dÞ0:026 ðd=DÞ0:32 . Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Jet impingement heat transfer; Cooling; Nusselt number; CFD; SST; Cylinder; Food
1. Introduction Heat and mass transfer are involved in many processes. In the food industry, cooling pre-cooked food is necessary to avoid quality degeneration. Various cooling methods are used today and many of them are time and energy consuming. Slow cooling may induce growth of thermophilic and many mesophilic micro-organisms. Cooling using impinging jets creates a high rate of heat transfer which allows the products to be rapidly cooled off to prevent this growth and the quality of food and the rate of production is also increased. Impingement cooling is a rapid cooling method where impinging jets of air with high velocity and low temperature orthogonally strike the surface of the product. Impinging jets are widely used to increase heat and mass transfer in different areas, such as cooling electronics, drying paper and cooling systems in gas turbines. The high velocity of the air enhances the transport of momentum, heat and mass.
*
Corresponding author. Tel.: +46-31-3355686; fax: +46-31-833782. E-mail address: [email protected] (E.E.M. Olsson).
0260-8774/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2003.08.009
In the food industry, impingement systems are used for heating (Marcroft & Karwe, 1999), drying (Braud, Moreira, & Castell-Perez, 2001; Moreira, 2001), cooling (Hu & Sun, 2001) and freezing (Soto & B orquez, 2001). Impinging jets have been used in other areas over half a century but only about a decade in the food industry. The technique is still under development. There are several reviews and surveys on impingement heat transfer in the literature (Downs & James, 1987; Jambunathan, Lai, Moss, & Button, 1992). Impingement in the food industry is reviewed by Ovadia and Walker (1998). Downs and James (1987) reviewed the characteristics of heat transfer from round and slot impinging jets, where they summarised the impact of geometric and temperature effects, interference and cross flow, turbulence levels, incidence and surface curvature. Jambunathan et al. (1992) reviewed experimental data on heat transfer rates in the case of circular jets impinging on a flat surface. Jets with a Reynolds number in the range of 5000-124,000 and a jet-to-cylinder distance of 1.2–16 were considered. The effects of Reynolds number, jet-to-surface distance, nozzle geometry, jet orientation and shape have been most widely studied. The heat transfer increases with higher Reynolds number, Nu / Rem (Jambunathan et al., 1992, among others)
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Nomenclature cp d D d=D Fr h H H =d k k Nu P Pr q Re ReD SST t T T0 u U u0
specific heat (J/(kg °C)) width of jet (m) diameter of cylinder (m) surface curvature (jet width to cylinder diameter ratio) Fr€ ossling number (Fr ¼ Nu=Re0:5 ) heat transfer coefficient (W/(m2 °C)) jet distance (m) jet-to-cylinder distance thermal heat conductivity (W/(m °C)) turbulent kinetic energy (m2 /s2 ) Nusselt number (Nu ¼ hd=k) pressure (Pa) Prandtl number (Pr ¼ lcp =k) heat flux (W/m2 ) Reynolds number (Re ¼ vdq=l) Reynolds number based on cylinder diameter (ReD ¼ vDq=l) shear stress transport time (s) temperature (°C) fluctuating temperature (°C) velocity (m/s) averaged velocity (m/s) fluctuating velocity (m/s)
and the jet-to-surface or jet-to-cylinder distance (H =d) is found to have a heat transfer maximum. This optimal distance coincides with the length of the potential core, which are 6–7 jet diameters or 4–7 slot widths (Downs & James, 1987). There are numerous studies of impingement heat transfer; both numerical and experimental reports are found in the literature. Single jets of air or water are most common and most have been done on a jet impinging on a flat surface (Baughn & Shimizu, 1989; Cooper, Jackson, Launder, & Liau, 1993; Craft, Graham, & Launder, 1993; Lee & Lee, 1999; Yan, Baughn, & Mesbah, 1992). Studies have also been done on a jet impinging upon an object, such as a curved surface (Chan et al., 2002; Lee, Chung, & Kim, 1999), a cylinder (Gori & Bossi, 2000; Kang & Greif, 1992; McDaniel & Webb, 2000; Tawfek, 1999) and a cylindrical pedestal (Baughn, Mesbah, & Yan, 1993). Many studies involve a cylinder in cross flow, which is a similar flow and heat transfer situation (Baughn & Saniei, 1991; Kondjoyan & Boisson, 1997; Tanabe, Kashiwada, Hayashi, & Iwata, 1993). However no studies were found on a jet impinging on a cylinder placed on a solid surface. Computational fluid dynamics (CFD) is a numerical technique being used more frequently as computer
u v x y yþ
0:5
friction velocity (u ¼ ðsw =qÞ ) velocity (m/s) distance from the stagnation point (m) distance from the wall (m) dimensionless distance between the wall and the first node (y þ ¼ u y=m)
Greek symbols e turbulent dissipation rate (m2 /s3 ) h angle on the cylinder (°) l dynamic viscosity (Pa s) m kinematic viscosity (m2 /s) q density (kg/m3 ) s shear stress (N/m2 ) x specific dissipation (s1 ) Subscripts/superscripts m mean value m exponent in Nu ¼ C Rem s stagnation point sim simulated turb turbulent w wall 1 ambient 0 at the surface
power increases. The use of CFD in the food industry is reviewed by Scott and Richardson (1997) and Xia and Sun (2002). Scott and Richardson (1997) reviewed the use of CFD in food processing applications. Xia and Sun (2002) describe different applications in the food industry, list available commercial codes and describe how to make a CFD analyse. Both reviews treat the advantages and drawbacks of CFD and outline future applications of CFD in food systems. CFD has been used for modelling flow and heat and mass transfer in the food industry in various areas such as modelling and validation of a forced convection oven (Verboven, Scheerlinck, De Baerdemaeker, & Nicola€ı, 2000), airblast chilling (Hu & Sun, 2001) and predicting airflow in and through entrances to cold storage rooms to be able to control the temperature (Foster, Barrett, James, & Swain, 2002). In this study, simulations using CFD were made to investigate the heat transfer characteristics of a slot air jet impinging on a solid cylindrical food product. The objective is to determine the distribution of the local Nusselt numbers on the surface of the cylindrical food product for different Reynolds numbers, jet-to-cylinder distances and surface curvatures, and assess the prediction performance of models for heat transfer inherent in the commercial CFD-code.
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2. Materials and methods
2.2. Governing equations
2.1. Simulations
The equations describing the fluid flow and the heat transfer from the impinging jet to the solid cylinder are transport equations of momentum and energy, which are developed from conservation laws of physics. The fluid flow is described by conservation of mass (the continuity equation), momentum (Navier–Stokes equations) and energy (the temperature equation for the fluid). The velocities and temperatures are time-averaged and divided into a mean and a fluctuating value, ui ¼ Ui þ u0i and T ¼ T þ T 0 . Together with the boundary conditions, they form the governing equations for incompressible flow with negligible external and viscous forces:
Steady state simulations in two dimensions of the heat transfer (local Nusselt numbers) from a slot air jet impinging on a solid cylindrical food product in a semiconfined domain were made using the CFD software CFX 5.5 (ANSYS). The Nusselt number describes the dimensionless heat transfer; it is a function of the heat transfer coefficient, the thermal conductivity, and a characteristic length, which in this study is the jet width (d). The computational domain, including the boundary conditions for the simulations is shown in Fig. 1. The infinite cylindrical food has a diameter of 35 mm and the slot jet a width of 30 mm. The impinging jet is assumed to be fully turbulent when exiting the slot pipe; a 1/7th power velocity profile is used. The Reynolds number is based on the average jet velocity and the width of the jet. The solid walls in the domain are assumed to be adiabatic and the pressure in the domain is 0.1 MPa. The temperature of the inflow is 2 °C and the surface of the cylinder has an initial temperature of 35 °C. The mesh consists of tetrahedral control volumes with 25 layers of cumulative prisms near the cylinder. The simulations were made on a Compaq Pentium III 800 MHz with 384 MB RAM and a Sun Ultra 10 workstation with 512 MB RAM. The distribution of local Nusselt numbers around the cylinder is determined for different Reynolds numbers (23,000, 50,000, 70,000 and 100,000), jet-to-cylinder ratios, H =d (2, 4, 6 and 8), and cylinder curvature, d=D (0.29, 0.57, 0.86, 1.0 and 1.14). The diameter of the cylinder is always the same, while the jet width is changed. The simulations involve only heat transfer to the surface of the cylindrical food product, not inside the product.
Wall Inlet d/2
Opening
Symmetry plane
H
D
Solid food
Wall
x Fig. 1. The computational domain.
oUj ¼0 oxj oUi oðUi Uj Þ oP o þq ¼ þ sij þ sturb ; ij oxj oxi oxj ot oUi oUj sij ¼ l ¼ qu0i u0j þ ; sturb ij oxj oxi
ð1Þ
q
oT oðUj T Þ o þ qcp ¼ ðqj þ qturb Þ; j ot oxj oxj lcp oT qj ¼ ; qturb ¼ qcp u0j T 0 j Pr oxj
ð2Þ
qcp
ð3Þ
It would be impossible to solve these equations analytically because of non-linearity and the stochastic nature of turbulence. The extra terms that appear due to averaging the velocity and temperature are the Reynolds stresses and the turbulent heat flux. Modelling these is known as the closure problem of turbulence. An analogy is assumed to exist between viscous stresses and Reynolds stresses on the mean flow. It is assumed that the Reynolds stresses are related to the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modelled as the product of a turbulent velocity and a turbulent length scale. In twoequation turbulence models, such as the k–e and the k–x models, a transport equation is modelled for each of the two properties. The turbulent velocity scale is estimated from the turbulent kinetic energy, and the turbulent length scale is computed from two properties of the fluid, usually the turbulent kinetic energy, k, and the dissipation rate, e, or the specific dissipation, x. The turbulent kinetic energy, the dissipation rate and the specific dissipation are given by their transport equation. 2.2.1. The SST turbulence model The shear stress transport (SST) model of Menter (1994) combines the advantages of the k–x and the k–e model. The pros and cons of the k–x and k–e models are well known (Menter, 1994). The SST model blends between the k–x model near the surface and the k–e model
E.E.M. Olsson et al. / Journal of Food Engineering 63 (2004) 393–401
at the boundary layer edge and outside the boundary layer. In the SST model the definition of the eddy viscosity is modified to account for the transport of the principal turbulent shear stress. The formulation of the turbulent shear stress in two-equation models leads to an overprediction in adverse pressure gradient flows (Menter, 1994). The near-wall formulation is important since it determines the accuracy of the wall shear stress, the wall heat transfer predictions and has an influence on the development of the boundary layer, including the point of separation. The SST model in CFX 5.5 has automatic wall treatment. A low-Reynolds number model that treats the viscous sublayer is used if the grid near the wall is fine, and a wall function is used beyond that.
3. Results and discussion
1.4 1.2 1
Nu/Re 0.5
396
0.8 0.6
SST
0.4
k-w
0.2
k-e Baughn & Saniei
θ =0° θ
(1991)
Knaabel et al. (1982)
0 0
45
90
135
180
θ Fig. 2. Heat transfer distribution on a cylinder in cross flow, ReD ¼ 34,000 and H=d ¼ 20. Simulations with the SST, k–x and k–e models, compared to literature.
3.1. Validation of the simulations 300
θ =0° θ
250
40000 cells (4000 in BL) 40000 cells (6800 in BL) 40000 cells (9600 in BL)
200
Nu
For validation, simulations of a cylinder in cross flow using the k–e, k–x and SST models were compared to measurements of a cylinder in cross flow reported in the literature (Baughn & Saniei, 1991; Knaabel, McKillop, & Baughn, 1982). The SST and k–x models were used with automatic wall treatment (y þ < 1) as well as with scalable wall functions (9 < y þ < 25). The Reynolds number was 34,000 (based on the cylinder diameter), the turbulence intensity less than 0.5% and the distance to the jet, H =d, was 20. The k–x and the SST models with a low-Re model predict the heat transfer well on the upper part of the cylinder (Fig. 2). They also give a good prediction of the separation point. The k–e model fails to predict the heat transfer correctly, especially in the stagnation region and the separation point is also predicted too late. The k–x and the SST models with scalable wall functions also performed poorly (not shown), which is supported by the result of Kondjoyan and Boisson (1997). The heat transfer around the cylinder is measured only to h ¼ 150°. The simulations predict that the heat transfer will continue to increase on the back of the cylinder. While the heat transfer in the recirculation area is probably not properly predicted, owing to the use of the Boussinesq assumption, which is valid only in isotropic flows, and that the k–e model is known to overpredict the turbulent diffusivity in rotating flows. Johnson and King (1985) demonstrated that reducing the eddy viscosity in the wake region in adverse pressure gradient flows improves the model. Thus the SST model is the best model available and will be further used in this study. If not otherwise stated, the simulations have y þ < 2 (most of the cases y þ < 1). The k–e model is used outside the boundary layer in the SST model, and that model probably overpredicts the turbulent diffusion in
240000 cells (6800 in BL)
150 100 50 0 0
45
90
135
180
θ Fig. 3. Grid independence. The original mesh has a total mesh size of 40,000 cells of which 6800 in the boundary layer. This is compared to a finer general mesh and a finer and coarser mesh near the surface.
the recirculation area. This affects the heat transfer in the boundary layer. A test of grid independence of the simulations was done and is shown in Fig. 3. The mesh in the simulations has in total 40,000 cells, of which 6800 is in the boundary layer close to the wall. A refined mesh with 6 times more cells (240,000 cells) gave the same result. Refining the mesh close to the cylinder gave different results, but for P 6800 cells the result was constant. 3.2. Jet flow characteristics The air flow in the computational domain (reflected in the centre), is shown in Fig. 4. The flow from a jet impinging on a circular cylinder (placed on a solid surface) can be divided into the following regions: free jet,
E.E.M. Olsson et al. / Journal of Food Engineering 63 (2004) 393–401
397
Fig. 4. Streamlines of the velocity in the computational domain (reflected in the centre).
stagnation point, cylinder flow, wake recirculation and wall jet. When a free jet impinges in a stagnant fluid, a potential core is formed where the jet centreline velocity is the same as the exit velocity and there is a rapid increase in static pressure as the jet approaches the cylinder. Beyond the stagnation point, the flow accelerates and follows the curvature of the cylinder until the separation point, where the main flow becomes a wall jet. Below the separation point, the flow is reversed because of the separation of the flow, and the interaction with the solid surface under the cylinder. The appearance of the wake is different from that of a free cylinder. The mean velocity field, the turbulent diffusion and the thickness of the boundary layer determine the heat transfer. For the velocity field near the cylinder, see Fig. 5. High velocity creates a thin boundary layer, and a thin boundary layer has a high rate of heat transfer. A turbulent boundary layer increases the rate of heat transfer as compared to a laminar boundary layer. The flow in the stagnation point has a high degree of turbulence, which causes high heat transfer in the stagnation region. The heat transfer decreases over the curved surface; the turbulence level is probably reduced by the vicinity of the wall. The heat transfer is low in the separation region. The recirculation creates an increase in velocity and heat transfer. The flow on the back at the cylinder (h ¼ 180°) is zero indicating zero heat transfer. 3.3. Heat transfer rates 3.3.1. Nusselt number distribution on the cylinder The Nusselt number is a dimensionless heat transfer coefficient. The heat transfer coefficient is a relation between the heat flux and the temperature difference between the surface of the cylinder and the ambient temperature. The heat transfer coefficient has a complex dependence on many variables, such as the fluid properties, the system geometry and the flow velocity.
Fig. 5. The velocity field near the cylinder.
The distribution of the local Nusselt numbers around the surface of the cylinder for various Reynolds numbers is shown for a jet-to-cylinder distance of 4 (see Fig. 6). The jet width and cylinder diameter is constant. The Nusselt number is high on the top of the cylinder, the highest rate is found a few degrees away from the stagnation point, because of the behaviour of the turbulence model. (The development of the so called V2Fmodel (Behnia, Parneix, & Durbin, 1998) was aimed as a remedy for this erroneous behaviour.) The Nusselt number drops near the separation point at 60°–70°. There is a small increase just after the separation due to a small increase in the velocity field (see Fig. 5). There is a recirculation zone behind the cylinder resulting from interaction between the cylinder and the wall. This causes an increase in heat transfer and a maximum at about h ¼ 130°. The heat transfer declines to zero under the cylinder. Compared to the validation case, the predicted Nusselt numbers in the wake are expected to be overpredicted.
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450
250
400 350
Re=50000
200
Re=100000
200
d/D=0.57 d/D=1.0
150
d/D=1.14
Nu
250
d/D=0.29 d/D=0.86
Re=70000
300
Nu
θ =0° θ
Re=23000
θ =0° θ
100
150 100
50
50 0
0 0
45
90
135
180
Fig. 6. Heat transfer around the cylinder for different Reynolds numbers, H=d ¼ 4 and d=D ¼ 0:86.
300
θ =0°
H/d=2
θ
250
H/d=4 H/d=6
Nu
200
H/d=8
150 100 50 0 0
45
90
0
45
90
135
180
θ
θ
135
180
θ Fig. 7. Heat transfer around the cylinder for different jet-to-cylinder distances (H =d), Re ¼ 50,000 and d=D ¼ 0:86.
The heat transfer around the cylinder increases with increasing Reynolds numbers. The heat transfer at the Reynolds numbers investigated does not seem to depend significantly on the jet-to-cylinder distance (H =d), see Fig. 7, in contrast to what is reported in the literature. Several authors report a maximum of heat transfer when H =d ¼ 5–8 (Chan et al., 2002; Gori & Bossi, 2000; Lee et al., 1999; McDaniel & Webb, 2000). These studies involve both round and slot jets impinging on a cylinder or a convex surface in a broad range of Reynolds number (600–87,000). Downs and James (1987) reported 6–7 jet diameters or 4–7 slot widths in their review. A maximum is also reported for a jet impinging on a flat plate (Baughn & Shimizu, 1989; Yan et al., 1992). In general the heat transfer maximum decreases for lower Reynolds numbers. Evidently, the performance of the turbulence model was not successful here. The jet turbulence level is also influencing the heat transfer, higher
Fig. 8. Heat transfer around the cylinder for different surface curvatures (d=D), Re ¼ 23,000 and H =d ¼ 2.
turbulence intensity gives higher heat transfer (Kondjoyan & Boisson, 1997) but its effect is not studied here. The effect of the curvature of the surface (ratio of jet width to cylinder diameter) was also investigated. It was found that the Nusselt number distribution around the cylinder and in the stagnation point is affected by the surface curvature. The Nusselt number increases with increasing surface curvature, see Fig. 8, which is supported by results of Lee, Chung, and Kim (1997). Note that the Nusselt number and the Reynolds number are based on the jet width (d), which is varied. The diameter of the cylinder and the height of the cylinder are kept constant (D ¼ 35 mm and H ¼ 60 mm). The same Reynolds number indicates the same mass flow but not the same velocity. The velocity increases with decreasing jet width. 3.3.2. Stagnation point heat transfer and average heat transfer A stagnation point is created on top of the cylinder (h ¼ 0°). The predicted heat transfer is very high in the stagnation point, and declining to the separation point. The Nusselt number in the stagnation point increases with the Reynolds number, i.e. with increasing velocity and mass flow (see Fig. 9). The fitted curve corresponds to Nus ¼ 0:40Re0:59 . For a round jet impinging on a flat plate, the exponent is found to be 0.5–0.58 depending on the jet distance (Cooper et al., 1993), m ¼ 0.64 for 30,000 < Re < 67,000 (San, Huang, & Shu, 1997) and m ¼ 0.56 for 5000 < Re < 30,000 (Lee & Lee, 1999). The average heat transfer around the product determines the heat transfer needed in the process to cool the product. The rate of heat transfer to the product is limited by the rate of heat transfer that can be conducted inside the product, i.e. the Biot number. In this study the average Nusselt number was found to be Num ¼ 0:12Re0:65 , which is higher dependency on the
E.E.M. Olsson et al. / Journal of Food Engineering 63 (2004) 393–401
399
250
400 350
200
300
Num.,sim
Nus
250 0.59
200
Nus = 0.40Re
150
100
150 100
50
50 0
0 0
20000
40000
60000
80000
0
100000
Reynolds number than in the stagnation point. Gori and Bossi (2000) found that Num ¼ 0:13Re0:69 for Re < 20,000 and McDaniel and Webb (2000) found that m ¼ 0.48– 0.69 for different types of nozzles and 600 < Re < 80,000. The relationship between Nu and Re is dependent on many factors including the definition of Nusselt and Reynolds numbers, for instance the turbulence level, the surface curvature and the geometry of the nozzle. 3.3.3. Correlations of the Nusselt number The simulations indicate that the Nusselt number varies with Reynolds number, radial angle around the circumference and the surface curvature––and not with the jet-to-cylinder distance. The variation in the Nusselt number in the stagnation point and the averaged Nusselt number was examined. The averaged Nusselt number is obtained by integrating the local Nusselt numbers over the radial surface: Z 1 p Num ¼ Nuh dh ð4Þ p 0 The averaged Nusselt number (Num ) and the Nusselt number in the stagnation point (Nus ) are expressed as:
y
z
150
0.65
(H/d)
200
-0.077
(d/D)
250
0.32
Fig. 10. Correlation of the averaged Nusselt number 23,000 < Re < 100,000, 2 < H=d < 8 and 0:28 < d=D < 1:14.
for
400 350 300 250
Nu s,sim
Fig. 9. Nusselt number in the stagnation point as a function of Reynolds number, H =d ¼ 4 and d=D ¼ 0:86.
Nus ¼ Cs Rex ðH =dÞ ðd=DÞ
100
0.14Re
Re
Num ¼ CRea ðH =dÞb ðd=DÞc
50
200 150 100 50 0 0
100
200 0.59
0.46Re
-0.026
(H/d)
300
400
0.32
(d/D)
Fig. 11. Correlation of the stagnation point Nusselt number for 23,000 < Re < 100,000, 2 < H=d < 8 and 0:28 < d=D < 1:14.
tance (H =d) is small on the average Nusselt number and stagnation point Nusselt number.
4. Conclusions
ð5Þ ð6Þ
The data from the simulations are put in an equation system and solved by least squares techniques. The result is: C ¼ 0:14, a ¼ 0:65, b ¼ 0:077, c ¼ 0:32, Cs ¼ 0:46, x ¼ 0:59, y ¼ 0:026 and z ¼ 0:32. The correlation is good (see Figs. 10 and 11). The regression coefficient (R2 ) is 0.99 for the average Nusselt number and 0.95 for the Nusselt number in the stagnation point. The exponent of the Reynolds number is higher for the average Nusselt number than in the stagnation point. It can be noticed that the effect of the jet-to-cylinder dis-
The heat transfer from a slot jet impinging on a circular cylinder placed on a solid surface has been studied using CFD simulations with the shear stress transport model (SST). Validation of the two-equation models in CFX 5.5 showed that the SST model was the best model available. The simulations predicted the heat transfer well on the upper part of the cylinder, but less correct in the non-isotropic region in the wake. For industrial design of thermal processes it is important to predict the heat transfer to the product. The heat transfer in the stagnation point gives the highest rate of heat transfer the product is exposed to and the
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average heat transfer gives the average heat or cold needed in the process. Heat transfer from an impinging jet in the range of Reynolds number of 23,000–100,000 was investigated and the distribution of Nusselt number shows that the rate of heat transfer is non-uniform on the surface of the cylinder. The heat transfer is higher on the top of the product and in the wake, lower in the separation point and the back on the cylinder. The places subjected to more heat transfer than the rest of the product will become more dehydrated. The local Nusselt numbers, the average Nusselt number and the Nusselt number in the stagnation point increase with increasing Reynolds numbers and surface curvature but have a low dependency on the jet-to-cylinder distance. The results are for the average heat 0:077 0:32 transfer, Num ¼ 0:14Re0:65 ðH =dÞ ðd=DÞ and for 0:026 the stagnation point, Nus ¼ 0:46Re0:59 ðH =dÞ 0:32 ðd=DÞ . The average Nusselt number has a higher dependency on the Reynolds number than the heat transfer in the stagnation point.
Acknowledgements This work was financed by the Swedish Knowledge Foundation, Daloon AB in Vadstena, Sweden, and Ircon AB in V€ anersborg, Sweden.
References ANSYS. CFX 5.5 user manual. Baughn, J., Mesbah, M., & Yan, X. (1993). Measurements of local heat transfer for an impinging jet on a cylindrical pedestal. American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD, 239, 57–62. Baughn, J., & Shimizu, S. (1989). Heat transfer measurements from a surface with uniform heat flux and an impinging jet. Journal of Heat Transfer, 111, 1096–1098. Baughn, J. W., & Saniei, N. (1991). The effect of the thermal boundary condition on heat transfer from a cylinder in crossflow. Journal of Heat Transfer, 113, 1020–1023. Behnia, M., Parneix, S., & Durbin, P. A. (1998). Prediction of heat transfer in an axisymmetric turbulent jet impinging on a flat plate. International Journal of Heat and Mass Transfer, 41, 1845–1855. Braud, L. M., Moreira, R. G., & Castell-Perez, M. E. (2001). Mathematical modelling of impingement drying of corn tortillas. Journal of Food Engineering, 50, 121–128. Chan, T. L., Leung, C. W., Jambunathan, K., Ashforth-Frost, S., Zhou, Y., & Liu, M. H. (2002). Heat transfer characteristics of a slot jet impinging on a semi-circular convex surface. International Journal of Heat and Mass Transfer, 45, 993–1006. Cooper, D., Jackson, D. C., Launder, B. E., & Liau, G. X. (1993). Impinging jet studies for turbulence model assessment––I. Flowfield experiments. International Journal of Heat and Mass Transfer, 36(10), 2675–2684. Craft, T. J., Graham, L. J. W., & Launder, B. E. (1993). Impinging jet studies for turbulence model assessment––II. An examination of the performance of four turbulence models. International Journal of Heat and Mass Transfer, 36(10), 2685–2697.
Downs, S. J., & James, E. H. (1987). Jet impingement heat transfer––a literature survey. American Society of Mechanical Engineers (Paper), 87-HT-35, 1–11. Foster, A. M., Barrett, R., James, S. J., & Swain, M. J. (2002). Measurement and prediction of air movement through doorways in refrigerated rooms. International Journal of Refrigeration, 25, 1102–1109. Gori, F., & Bossi, L. (2000). On the cooling effect of an air jet along the surface of a cylinder. International Communications in Heat and Mass Transfer, 27(5), 667–676. Hu, Z., & Sun, D.-W. (2001). Effect of fluctuation in inlet temperature on CFD simulation of air-blast chilling process. Journal of Food Engineering, 48, 311–316. Jambunathan, K., Lai, E., Moss, M. A., & Button, B. L. (1992). A review of heat transfer data for single circular jet impingement. International Journal of Heat and Fluid Flow, 13(2), 106–115. Johnson, D. A., & King, L. S. (1985). Mathematically simply turbulence closure model for attached and separated turbulent boundary layers. AIAA Journal, 23(11), 1684–1692. Kang, S. H., & Greif, R. (1992). Flow and heat transfer to a circular cylinder with a hot impinging air jet. International Journal of Heat and Mass Transfer, 35(9), 2173–2183. Knaabel, J. S., McKillop, A. A., & Baughn, J. W. (1982). Heat transfer to air from a yawed cylinder. International Journal of Heat and Mass Transfer, 25, 409–418. Kondjoyan, A., & Boisson, H. C. (1997). Comparison of calculated and experimental heat transfer coefficients at the surface of circular cylinders placed in a turbulent cross-flow of air. Journal of Food Engineering, 34, 123–143. Lee, D. H., Chung, Y. S., & Kim, M. G. (1997). Turbulent flow and heat transfer measurements on a curved surface with a fully developed round impinging jet. International Journal of Heat and Fluid Flow, 18, 160–169. Lee, D. H., Chung, Y. S., & Kim, M. G. (1999). Technical note. Turbulent heat transfer from a convex hemispherical surface to a round impinging jet. International Journal of Heat and Mass Transfer, 42, 1147–1156. Lee, J., & Lee, S.-J. (1999). Stagnation region heat transfer of a turbulent axisymmetric jet impingement. Experimental Heat Transfer, 12(2), 137–156. Marcroft, H. E., & Karwe, M. V. (1999). Flow field in a hot air jet impingement oven––part I: A single impinging jet. Journal of Food Processing Preservation, 23, 217–233. McDaniel, C. S., & Webb, B. W. (2000). Slot jet impinging heat transfer from circular cylinders. International Journal of Heat and Mass Transfer, 43, 1975–1985. Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8), 1598–1604. Moreira, R. G. (2001). Impingement drying of foods using hot air and superheated steam. Journal of Food Engineering, 49, 291–295. Ovadia, D. Z., & Walker, C. E. (1998). Impingement in food processing. Food Technology, 52(4), 46–50. San, J.-Y., Huang, C.-H., & Shu, M.-H. (1997). Impingement cooling of a confined circular air jet. International Journal of Heat and Mass Transfer, 40(6), 1355–1364. Scott, G., & Richardson, P. (1997). The application of computational fluid dynamics in the food industry. Trends in Food Science and Technology, 8, 119–124. Soto, V., & B orquez, R. (2001). Impingement jet freezing of biomaterials. Food Control, 12, 515–522. Tanabe, S., Kashiwada, Y., Hayashi, O., & Iwata, H. (1993). Effects of wall boundary layer on local heat transfer from a circular cylinder in cross flow. Heat Transfer––Japanese Research, 22(5), 506–517. Tawfek, A. A. (1999). Heat transfer due to a round jet impinging normal to a circular cylinder. Heat and Mass Transfer, 35, 327–333. Verboven, P., Scheerlinck, N., De Baerdemaeker, J., & Nicola€ı, B. M. (2000). Computational fluid dynamics modelling and validation of
E.E.M. Olsson et al. / Journal of Food Engineering 63 (2004) 393–401 the temperature distribution in a force convection oven. Journal of Food Engineering, 43, 61–73. Xia, B., & Sun, D.-W. (2002). Applications of computational fluid dynamics (CFD) in the food industry: A review. Computers and Electronics in Agriculture, 34, 5–24.
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Yan, X., Baughn, J. W., & Mesbah, M. (1992). The effect of Reynolds number on the heat transfer distribution from a flat plate to an impinging jet. American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD, 226, 1–7.
Heat transfer from a slot air jet impinging on a circular cylinder E.E.M. Olsson b
a,b,*
, L.M. Ahrne a, A.C. Tr€ ag ardh
b
a SIK-The Swedish Institute for Food and Biotechnology, P.O. Box 5401, SE-402 29 G€oteborg, Sweden Department of Food Technology, Engineering and Nutrition, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
Received 22 March 2003; accepted 16 August 2003
Abstract Heat transfer from a slot air jet impinging on a cylinder shaped food product placed on a solid surface in a semi-confined area was investigated using computational fluid dynamics (CFD). Simulations of a cylinder in cross flow with the k–e, k–x and SST models in CFX 5.5 were compared with measurements in the literature. The SST model predicts the heat transfer better than the other models and is therefore used in this study. The distribution of the local Nusselt numbers around the cylinder for various Reynolds numbers (23,000–100,000), jet-to-cylinder distances, H =d (2–8), and cylinder curvature, d=D (0.29–1.14) was determined. The results show that the local Nusselt numbers varies around the surface of the cylinder and that the average Nusselt number and the stagnation point Nusselt number increases with increasing Reynolds numbers and surface curvature but has little dependency on the jet-to-cylinder distance. The result is Num ¼ 0:14Re0:65 ðH =dÞ0:077 ðd=DÞ0:32 and Nus ¼ 0:46Re0:59 ðH =dÞ0:026 ðd=DÞ0:32 . Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Jet impingement heat transfer; Cooling; Nusselt number; CFD; SST; Cylinder; Food
1. Introduction Heat and mass transfer are involved in many processes. In the food industry, cooling pre-cooked food is necessary to avoid quality degeneration. Various cooling methods are used today and many of them are time and energy consuming. Slow cooling may induce growth of thermophilic and many mesophilic micro-organisms. Cooling using impinging jets creates a high rate of heat transfer which allows the products to be rapidly cooled off to prevent this growth and the quality of food and the rate of production is also increased. Impingement cooling is a rapid cooling method where impinging jets of air with high velocity and low temperature orthogonally strike the surface of the product. Impinging jets are widely used to increase heat and mass transfer in different areas, such as cooling electronics, drying paper and cooling systems in gas turbines. The high velocity of the air enhances the transport of momentum, heat and mass.
*
Corresponding author. Tel.: +46-31-3355686; fax: +46-31-833782. E-mail address: [email protected] (E.E.M. Olsson).
0260-8774/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2003.08.009
In the food industry, impingement systems are used for heating (Marcroft & Karwe, 1999), drying (Braud, Moreira, & Castell-Perez, 2001; Moreira, 2001), cooling (Hu & Sun, 2001) and freezing (Soto & B orquez, 2001). Impinging jets have been used in other areas over half a century but only about a decade in the food industry. The technique is still under development. There are several reviews and surveys on impingement heat transfer in the literature (Downs & James, 1987; Jambunathan, Lai, Moss, & Button, 1992). Impingement in the food industry is reviewed by Ovadia and Walker (1998). Downs and James (1987) reviewed the characteristics of heat transfer from round and slot impinging jets, where they summarised the impact of geometric and temperature effects, interference and cross flow, turbulence levels, incidence and surface curvature. Jambunathan et al. (1992) reviewed experimental data on heat transfer rates in the case of circular jets impinging on a flat surface. Jets with a Reynolds number in the range of 5000-124,000 and a jet-to-cylinder distance of 1.2–16 were considered. The effects of Reynolds number, jet-to-surface distance, nozzle geometry, jet orientation and shape have been most widely studied. The heat transfer increases with higher Reynolds number, Nu / Rem (Jambunathan et al., 1992, among others)
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Nomenclature cp d D d=D Fr h H H =d k k Nu P Pr q Re ReD SST t T T0 u U u0
specific heat (J/(kg °C)) width of jet (m) diameter of cylinder (m) surface curvature (jet width to cylinder diameter ratio) Fr€ ossling number (Fr ¼ Nu=Re0:5 ) heat transfer coefficient (W/(m2 °C)) jet distance (m) jet-to-cylinder distance thermal heat conductivity (W/(m °C)) turbulent kinetic energy (m2 /s2 ) Nusselt number (Nu ¼ hd=k) pressure (Pa) Prandtl number (Pr ¼ lcp =k) heat flux (W/m2 ) Reynolds number (Re ¼ vdq=l) Reynolds number based on cylinder diameter (ReD ¼ vDq=l) shear stress transport time (s) temperature (°C) fluctuating temperature (°C) velocity (m/s) averaged velocity (m/s) fluctuating velocity (m/s)
and the jet-to-surface or jet-to-cylinder distance (H =d) is found to have a heat transfer maximum. This optimal distance coincides with the length of the potential core, which are 6–7 jet diameters or 4–7 slot widths (Downs & James, 1987). There are numerous studies of impingement heat transfer; both numerical and experimental reports are found in the literature. Single jets of air or water are most common and most have been done on a jet impinging on a flat surface (Baughn & Shimizu, 1989; Cooper, Jackson, Launder, & Liau, 1993; Craft, Graham, & Launder, 1993; Lee & Lee, 1999; Yan, Baughn, & Mesbah, 1992). Studies have also been done on a jet impinging upon an object, such as a curved surface (Chan et al., 2002; Lee, Chung, & Kim, 1999), a cylinder (Gori & Bossi, 2000; Kang & Greif, 1992; McDaniel & Webb, 2000; Tawfek, 1999) and a cylindrical pedestal (Baughn, Mesbah, & Yan, 1993). Many studies involve a cylinder in cross flow, which is a similar flow and heat transfer situation (Baughn & Saniei, 1991; Kondjoyan & Boisson, 1997; Tanabe, Kashiwada, Hayashi, & Iwata, 1993). However no studies were found on a jet impinging on a cylinder placed on a solid surface. Computational fluid dynamics (CFD) is a numerical technique being used more frequently as computer
u v x y yþ
0:5
friction velocity (u ¼ ðsw =qÞ ) velocity (m/s) distance from the stagnation point (m) distance from the wall (m) dimensionless distance between the wall and the first node (y þ ¼ u y=m)
Greek symbols e turbulent dissipation rate (m2 /s3 ) h angle on the cylinder (°) l dynamic viscosity (Pa s) m kinematic viscosity (m2 /s) q density (kg/m3 ) s shear stress (N/m2 ) x specific dissipation (s1 ) Subscripts/superscripts m mean value m exponent in Nu ¼ C Rem s stagnation point sim simulated turb turbulent w wall 1 ambient 0 at the surface
power increases. The use of CFD in the food industry is reviewed by Scott and Richardson (1997) and Xia and Sun (2002). Scott and Richardson (1997) reviewed the use of CFD in food processing applications. Xia and Sun (2002) describe different applications in the food industry, list available commercial codes and describe how to make a CFD analyse. Both reviews treat the advantages and drawbacks of CFD and outline future applications of CFD in food systems. CFD has been used for modelling flow and heat and mass transfer in the food industry in various areas such as modelling and validation of a forced convection oven (Verboven, Scheerlinck, De Baerdemaeker, & Nicola€ı, 2000), airblast chilling (Hu & Sun, 2001) and predicting airflow in and through entrances to cold storage rooms to be able to control the temperature (Foster, Barrett, James, & Swain, 2002). In this study, simulations using CFD were made to investigate the heat transfer characteristics of a slot air jet impinging on a solid cylindrical food product. The objective is to determine the distribution of the local Nusselt numbers on the surface of the cylindrical food product for different Reynolds numbers, jet-to-cylinder distances and surface curvatures, and assess the prediction performance of models for heat transfer inherent in the commercial CFD-code.
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395
2. Materials and methods
2.2. Governing equations
2.1. Simulations
The equations describing the fluid flow and the heat transfer from the impinging jet to the solid cylinder are transport equations of momentum and energy, which are developed from conservation laws of physics. The fluid flow is described by conservation of mass (the continuity equation), momentum (Navier–Stokes equations) and energy (the temperature equation for the fluid). The velocities and temperatures are time-averaged and divided into a mean and a fluctuating value, ui ¼ Ui þ u0i and T ¼ T þ T 0 . Together with the boundary conditions, they form the governing equations for incompressible flow with negligible external and viscous forces:
Steady state simulations in two dimensions of the heat transfer (local Nusselt numbers) from a slot air jet impinging on a solid cylindrical food product in a semiconfined domain were made using the CFD software CFX 5.5 (ANSYS). The Nusselt number describes the dimensionless heat transfer; it is a function of the heat transfer coefficient, the thermal conductivity, and a characteristic length, which in this study is the jet width (d). The computational domain, including the boundary conditions for the simulations is shown in Fig. 1. The infinite cylindrical food has a diameter of 35 mm and the slot jet a width of 30 mm. The impinging jet is assumed to be fully turbulent when exiting the slot pipe; a 1/7th power velocity profile is used. The Reynolds number is based on the average jet velocity and the width of the jet. The solid walls in the domain are assumed to be adiabatic and the pressure in the domain is 0.1 MPa. The temperature of the inflow is 2 °C and the surface of the cylinder has an initial temperature of 35 °C. The mesh consists of tetrahedral control volumes with 25 layers of cumulative prisms near the cylinder. The simulations were made on a Compaq Pentium III 800 MHz with 384 MB RAM and a Sun Ultra 10 workstation with 512 MB RAM. The distribution of local Nusselt numbers around the cylinder is determined for different Reynolds numbers (23,000, 50,000, 70,000 and 100,000), jet-to-cylinder ratios, H =d (2, 4, 6 and 8), and cylinder curvature, d=D (0.29, 0.57, 0.86, 1.0 and 1.14). The diameter of the cylinder is always the same, while the jet width is changed. The simulations involve only heat transfer to the surface of the cylindrical food product, not inside the product.
Wall Inlet d/2
Opening
Symmetry plane
H
D
Solid food
Wall
x Fig. 1. The computational domain.
oUj ¼0 oxj oUi oðUi Uj Þ oP o þq ¼ þ sij þ sturb ; ij oxj oxi oxj ot oUi oUj sij ¼ l ¼ qu0i u0j þ ; sturb ij oxj oxi
ð1Þ
q
oT oðUj T Þ o þ qcp ¼ ðqj þ qturb Þ; j ot oxj oxj lcp oT qj ¼ ; qturb ¼ qcp u0j T 0 j Pr oxj
ð2Þ
qcp
ð3Þ
It would be impossible to solve these equations analytically because of non-linearity and the stochastic nature of turbulence. The extra terms that appear due to averaging the velocity and temperature are the Reynolds stresses and the turbulent heat flux. Modelling these is known as the closure problem of turbulence. An analogy is assumed to exist between viscous stresses and Reynolds stresses on the mean flow. It is assumed that the Reynolds stresses are related to the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modelled as the product of a turbulent velocity and a turbulent length scale. In twoequation turbulence models, such as the k–e and the k–x models, a transport equation is modelled for each of the two properties. The turbulent velocity scale is estimated from the turbulent kinetic energy, and the turbulent length scale is computed from two properties of the fluid, usually the turbulent kinetic energy, k, and the dissipation rate, e, or the specific dissipation, x. The turbulent kinetic energy, the dissipation rate and the specific dissipation are given by their transport equation. 2.2.1. The SST turbulence model The shear stress transport (SST) model of Menter (1994) combines the advantages of the k–x and the k–e model. The pros and cons of the k–x and k–e models are well known (Menter, 1994). The SST model blends between the k–x model near the surface and the k–e model
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at the boundary layer edge and outside the boundary layer. In the SST model the definition of the eddy viscosity is modified to account for the transport of the principal turbulent shear stress. The formulation of the turbulent shear stress in two-equation models leads to an overprediction in adverse pressure gradient flows (Menter, 1994). The near-wall formulation is important since it determines the accuracy of the wall shear stress, the wall heat transfer predictions and has an influence on the development of the boundary layer, including the point of separation. The SST model in CFX 5.5 has automatic wall treatment. A low-Reynolds number model that treats the viscous sublayer is used if the grid near the wall is fine, and a wall function is used beyond that.
3. Results and discussion
1.4 1.2 1
Nu/Re 0.5
396
0.8 0.6
SST
0.4
k-w
0.2
k-e Baughn & Saniei
θ =0° θ
(1991)
Knaabel et al. (1982)
0 0
45
90
135
180
θ Fig. 2. Heat transfer distribution on a cylinder in cross flow, ReD ¼ 34,000 and H=d ¼ 20. Simulations with the SST, k–x and k–e models, compared to literature.
3.1. Validation of the simulations 300
θ =0° θ
250
40000 cells (4000 in BL) 40000 cells (6800 in BL) 40000 cells (9600 in BL)
200
Nu
For validation, simulations of a cylinder in cross flow using the k–e, k–x and SST models were compared to measurements of a cylinder in cross flow reported in the literature (Baughn & Saniei, 1991; Knaabel, McKillop, & Baughn, 1982). The SST and k–x models were used with automatic wall treatment (y þ < 1) as well as with scalable wall functions (9 < y þ < 25). The Reynolds number was 34,000 (based on the cylinder diameter), the turbulence intensity less than 0.5% and the distance to the jet, H =d, was 20. The k–x and the SST models with a low-Re model predict the heat transfer well on the upper part of the cylinder (Fig. 2). They also give a good prediction of the separation point. The k–e model fails to predict the heat transfer correctly, especially in the stagnation region and the separation point is also predicted too late. The k–x and the SST models with scalable wall functions also performed poorly (not shown), which is supported by the result of Kondjoyan and Boisson (1997). The heat transfer around the cylinder is measured only to h ¼ 150°. The simulations predict that the heat transfer will continue to increase on the back of the cylinder. While the heat transfer in the recirculation area is probably not properly predicted, owing to the use of the Boussinesq assumption, which is valid only in isotropic flows, and that the k–e model is known to overpredict the turbulent diffusivity in rotating flows. Johnson and King (1985) demonstrated that reducing the eddy viscosity in the wake region in adverse pressure gradient flows improves the model. Thus the SST model is the best model available and will be further used in this study. If not otherwise stated, the simulations have y þ < 2 (most of the cases y þ < 1). The k–e model is used outside the boundary layer in the SST model, and that model probably overpredicts the turbulent diffusion in
240000 cells (6800 in BL)
150 100 50 0 0
45
90
135
180
θ Fig. 3. Grid independence. The original mesh has a total mesh size of 40,000 cells of which 6800 in the boundary layer. This is compared to a finer general mesh and a finer and coarser mesh near the surface.
the recirculation area. This affects the heat transfer in the boundary layer. A test of grid independence of the simulations was done and is shown in Fig. 3. The mesh in the simulations has in total 40,000 cells, of which 6800 is in the boundary layer close to the wall. A refined mesh with 6 times more cells (240,000 cells) gave the same result. Refining the mesh close to the cylinder gave different results, but for P 6800 cells the result was constant. 3.2. Jet flow characteristics The air flow in the computational domain (reflected in the centre), is shown in Fig. 4. The flow from a jet impinging on a circular cylinder (placed on a solid surface) can be divided into the following regions: free jet,
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397
Fig. 4. Streamlines of the velocity in the computational domain (reflected in the centre).
stagnation point, cylinder flow, wake recirculation and wall jet. When a free jet impinges in a stagnant fluid, a potential core is formed where the jet centreline velocity is the same as the exit velocity and there is a rapid increase in static pressure as the jet approaches the cylinder. Beyond the stagnation point, the flow accelerates and follows the curvature of the cylinder until the separation point, where the main flow becomes a wall jet. Below the separation point, the flow is reversed because of the separation of the flow, and the interaction with the solid surface under the cylinder. The appearance of the wake is different from that of a free cylinder. The mean velocity field, the turbulent diffusion and the thickness of the boundary layer determine the heat transfer. For the velocity field near the cylinder, see Fig. 5. High velocity creates a thin boundary layer, and a thin boundary layer has a high rate of heat transfer. A turbulent boundary layer increases the rate of heat transfer as compared to a laminar boundary layer. The flow in the stagnation point has a high degree of turbulence, which causes high heat transfer in the stagnation region. The heat transfer decreases over the curved surface; the turbulence level is probably reduced by the vicinity of the wall. The heat transfer is low in the separation region. The recirculation creates an increase in velocity and heat transfer. The flow on the back at the cylinder (h ¼ 180°) is zero indicating zero heat transfer. 3.3. Heat transfer rates 3.3.1. Nusselt number distribution on the cylinder The Nusselt number is a dimensionless heat transfer coefficient. The heat transfer coefficient is a relation between the heat flux and the temperature difference between the surface of the cylinder and the ambient temperature. The heat transfer coefficient has a complex dependence on many variables, such as the fluid properties, the system geometry and the flow velocity.
Fig. 5. The velocity field near the cylinder.
The distribution of the local Nusselt numbers around the surface of the cylinder for various Reynolds numbers is shown for a jet-to-cylinder distance of 4 (see Fig. 6). The jet width and cylinder diameter is constant. The Nusselt number is high on the top of the cylinder, the highest rate is found a few degrees away from the stagnation point, because of the behaviour of the turbulence model. (The development of the so called V2Fmodel (Behnia, Parneix, & Durbin, 1998) was aimed as a remedy for this erroneous behaviour.) The Nusselt number drops near the separation point at 60°–70°. There is a small increase just after the separation due to a small increase in the velocity field (see Fig. 5). There is a recirculation zone behind the cylinder resulting from interaction between the cylinder and the wall. This causes an increase in heat transfer and a maximum at about h ¼ 130°. The heat transfer declines to zero under the cylinder. Compared to the validation case, the predicted Nusselt numbers in the wake are expected to be overpredicted.
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450
250
400 350
Re=50000
200
Re=100000
200
d/D=0.57 d/D=1.0
150
d/D=1.14
Nu
250
d/D=0.29 d/D=0.86
Re=70000
300
Nu
θ =0° θ
Re=23000
θ =0° θ
100
150 100
50
50 0
0 0
45
90
135
180
Fig. 6. Heat transfer around the cylinder for different Reynolds numbers, H=d ¼ 4 and d=D ¼ 0:86.
300
θ =0°
H/d=2
θ
250
H/d=4 H/d=6
Nu
200
H/d=8
150 100 50 0 0
45
90
0
45
90
135
180
θ
θ
135
180
θ Fig. 7. Heat transfer around the cylinder for different jet-to-cylinder distances (H =d), Re ¼ 50,000 and d=D ¼ 0:86.
The heat transfer around the cylinder increases with increasing Reynolds numbers. The heat transfer at the Reynolds numbers investigated does not seem to depend significantly on the jet-to-cylinder distance (H =d), see Fig. 7, in contrast to what is reported in the literature. Several authors report a maximum of heat transfer when H =d ¼ 5–8 (Chan et al., 2002; Gori & Bossi, 2000; Lee et al., 1999; McDaniel & Webb, 2000). These studies involve both round and slot jets impinging on a cylinder or a convex surface in a broad range of Reynolds number (600–87,000). Downs and James (1987) reported 6–7 jet diameters or 4–7 slot widths in their review. A maximum is also reported for a jet impinging on a flat plate (Baughn & Shimizu, 1989; Yan et al., 1992). In general the heat transfer maximum decreases for lower Reynolds numbers. Evidently, the performance of the turbulence model was not successful here. The jet turbulence level is also influencing the heat transfer, higher
Fig. 8. Heat transfer around the cylinder for different surface curvatures (d=D), Re ¼ 23,000 and H =d ¼ 2.
turbulence intensity gives higher heat transfer (Kondjoyan & Boisson, 1997) but its effect is not studied here. The effect of the curvature of the surface (ratio of jet width to cylinder diameter) was also investigated. It was found that the Nusselt number distribution around the cylinder and in the stagnation point is affected by the surface curvature. The Nusselt number increases with increasing surface curvature, see Fig. 8, which is supported by results of Lee, Chung, and Kim (1997). Note that the Nusselt number and the Reynolds number are based on the jet width (d), which is varied. The diameter of the cylinder and the height of the cylinder are kept constant (D ¼ 35 mm and H ¼ 60 mm). The same Reynolds number indicates the same mass flow but not the same velocity. The velocity increases with decreasing jet width. 3.3.2. Stagnation point heat transfer and average heat transfer A stagnation point is created on top of the cylinder (h ¼ 0°). The predicted heat transfer is very high in the stagnation point, and declining to the separation point. The Nusselt number in the stagnation point increases with the Reynolds number, i.e. with increasing velocity and mass flow (see Fig. 9). The fitted curve corresponds to Nus ¼ 0:40Re0:59 . For a round jet impinging on a flat plate, the exponent is found to be 0.5–0.58 depending on the jet distance (Cooper et al., 1993), m ¼ 0.64 for 30,000 < Re < 67,000 (San, Huang, & Shu, 1997) and m ¼ 0.56 for 5000 < Re < 30,000 (Lee & Lee, 1999). The average heat transfer around the product determines the heat transfer needed in the process to cool the product. The rate of heat transfer to the product is limited by the rate of heat transfer that can be conducted inside the product, i.e. the Biot number. In this study the average Nusselt number was found to be Num ¼ 0:12Re0:65 , which is higher dependency on the
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399
250
400 350
200
300
Num.,sim
Nus
250 0.59
200
Nus = 0.40Re
150
100
150 100
50
50 0
0 0
20000
40000
60000
80000
0
100000
Reynolds number than in the stagnation point. Gori and Bossi (2000) found that Num ¼ 0:13Re0:69 for Re < 20,000 and McDaniel and Webb (2000) found that m ¼ 0.48– 0.69 for different types of nozzles and 600 < Re < 80,000. The relationship between Nu and Re is dependent on many factors including the definition of Nusselt and Reynolds numbers, for instance the turbulence level, the surface curvature and the geometry of the nozzle. 3.3.3. Correlations of the Nusselt number The simulations indicate that the Nusselt number varies with Reynolds number, radial angle around the circumference and the surface curvature––and not with the jet-to-cylinder distance. The variation in the Nusselt number in the stagnation point and the averaged Nusselt number was examined. The averaged Nusselt number is obtained by integrating the local Nusselt numbers over the radial surface: Z 1 p Num ¼ Nuh dh ð4Þ p 0 The averaged Nusselt number (Num ) and the Nusselt number in the stagnation point (Nus ) are expressed as:
y
z
150
0.65
(H/d)
200
-0.077
(d/D)
250
0.32
Fig. 10. Correlation of the averaged Nusselt number 23,000 < Re < 100,000, 2 < H=d < 8 and 0:28 < d=D < 1:14.
for
400 350 300 250
Nu s,sim
Fig. 9. Nusselt number in the stagnation point as a function of Reynolds number, H =d ¼ 4 and d=D ¼ 0:86.
Nus ¼ Cs Rex ðH =dÞ ðd=DÞ
100
0.14Re
Re
Num ¼ CRea ðH =dÞb ðd=DÞc
50
200 150 100 50 0 0
100
200 0.59
0.46Re
-0.026
(H/d)
300
400
0.32
(d/D)
Fig. 11. Correlation of the stagnation point Nusselt number for 23,000 < Re < 100,000, 2 < H=d < 8 and 0:28 < d=D < 1:14.
tance (H =d) is small on the average Nusselt number and stagnation point Nusselt number.
4. Conclusions
ð5Þ ð6Þ
The data from the simulations are put in an equation system and solved by least squares techniques. The result is: C ¼ 0:14, a ¼ 0:65, b ¼ 0:077, c ¼ 0:32, Cs ¼ 0:46, x ¼ 0:59, y ¼ 0:026 and z ¼ 0:32. The correlation is good (see Figs. 10 and 11). The regression coefficient (R2 ) is 0.99 for the average Nusselt number and 0.95 for the Nusselt number in the stagnation point. The exponent of the Reynolds number is higher for the average Nusselt number than in the stagnation point. It can be noticed that the effect of the jet-to-cylinder dis-
The heat transfer from a slot jet impinging on a circular cylinder placed on a solid surface has been studied using CFD simulations with the shear stress transport model (SST). Validation of the two-equation models in CFX 5.5 showed that the SST model was the best model available. The simulations predicted the heat transfer well on the upper part of the cylinder, but less correct in the non-isotropic region in the wake. For industrial design of thermal processes it is important to predict the heat transfer to the product. The heat transfer in the stagnation point gives the highest rate of heat transfer the product is exposed to and the
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average heat transfer gives the average heat or cold needed in the process. Heat transfer from an impinging jet in the range of Reynolds number of 23,000–100,000 was investigated and the distribution of Nusselt number shows that the rate of heat transfer is non-uniform on the surface of the cylinder. The heat transfer is higher on the top of the product and in the wake, lower in the separation point and the back on the cylinder. The places subjected to more heat transfer than the rest of the product will become more dehydrated. The local Nusselt numbers, the average Nusselt number and the Nusselt number in the stagnation point increase with increasing Reynolds numbers and surface curvature but have a low dependency on the jet-to-cylinder distance. The results are for the average heat 0:077 0:32 transfer, Num ¼ 0:14Re0:65 ðH =dÞ ðd=DÞ and for 0:026 the stagnation point, Nus ¼ 0:46Re0:59 ðH =dÞ 0:32 ðd=DÞ . The average Nusselt number has a higher dependency on the Reynolds number than the heat transfer in the stagnation point.
Acknowledgements This work was financed by the Swedish Knowledge Foundation, Daloon AB in Vadstena, Sweden, and Ircon AB in V€ anersborg, Sweden.
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