Numerical heat transfer during Herschel–Bulkley fluid natural convection by CVFEM

Numerical heat transfer during Herschel–Bulkley fluid natural convection by CVFEM

Diego A. VascoCarlos Salinas Nelson Moraga Roberto Lemus-Mondaca 

Departamento de Ingeniería Mecánica. Universidad de Santiago de Chile. Av. Lib. Bernardo O’Higgins 3363, Santiago, Chile

Departamento de Ingeniería Mecánica. Universidad del Bío-Bío. Av. Collao 1202, Concepción, Chile

Departamento de Ingeniería Mecánica. Universidad de La Serena. Benavente 980, La Serena, Chile

Departamento de Ciencia de los Alimentos y Tecnología Química, Universidad de Chile. Santos Dumont 964, Independencia, Santiago, Chile

Corresponding Author Email:
19 November 2017
9 April 2018
30 June 2018
| Citation



Numerical prediction of heat transfer by natural convection of a Herschel-Bulkley non-Newtonian fluid inside a square cavity has been computationally analyzed. Unsteady 2D fluid mechanics and heat transfer were described in terms of the non-linear coupled continuity, momentum and heat equations. These equations were solved by the control volume finite element method (CVFEM) with Gauss-Seidel/System Over-Relaxation coupling algorithm. The effect of the Ra, Pr, Bn and the rheological behavior index (n) on the non-Newtonian fluid thermal and momentum behavior were studied. The non-Newtonian fluid flow was described by the rheological model of Herschel-Bulkley. Results for the streamlines and isotherms along the enclosure walls are presented. It was found that the effect of the Pr and Bn is more important when the Ra is lower (103). In addition, the behavior index had a significant effect on the CPU time for the different studied cases.


free convection, heat transfer, non-Newtonian fluid, CVFEM

1. Introduction
2. Physical Situation
3. Mathematical Model
4. CVFEM Implementation
5. Scaling Analysis
6. Results and Discussions
7. Conclusions

[1] Barnes H. (1999). The yield stress-a review-everything flows? Journal of Non-Newtonian Fluid Mechanics 81(1-2): 133-178.

[2] Zhu H, Kim Y, De Kee D. (2005). Non-Newtonian fluids with a yield stress. Journal of Non-Newtonian Fluid Mechanics 129(3): 177-181.

[3] Tabilo-Munizaga G, Barbosa-Cánovas G. (2005). Rheology for the food industry. Journal of Food Engineering 67(1-2): 147-156.

[4] Gratão A, Silveira Jr.V, Telis-Romero J. (2007). Laminar flow of soursop juice through concentric annuli: Friction factors and rheology. Journal of Food Engineering 78(4): 1343-1354.

[5] Trigilio-Tavares D, Alcantara M, Tadini C, Telis-Romero J. (2007). Rheological properties of frozen concentrated orange juice (FCOJ) as a function of concentration and subzero temperatures. International Journal of Food Properties 10(4): 829-839.

[6] Telis-Romero J, Telis V, Yamashita F. (1999). Friction factors and rheological properties of orange juice. Journal of Food Engineering 40(1-2): 101-106.

[7] Vandresen S, Quadri M, de Souza J, Hotza D. (2009). Temperature effect on the rheological behavior of carrot juices. Journal of Food Engineering 92(3): 269-274.

[8] Leong W, Hollands K, Brunger A. (1998). On a physically-realizable benchmark problem in internal natural convection. International Journal of Heat and Mass Transfer 41(23): 3817-3828.  

[9] Leong W, Hollands K, Brunger A. (1999). Experimental Nusselt numbers for a cubical-cavity benchmark problem in natural convection. International Journal of Heat and Mass Transfer 4(11): 1979-1989.

[10] Fusegi T, Hyun J, Kuwahara K, Farouk B. (1991). A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure. International Journal of Heat and Mass Transfer 34(6): 1543-1557.

[11] Wen-Hann ST, Reui-Kuo LR-K. (2011). Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection. International Journal of Heat and Mass Transfer 54(1-3): 447-467.

[12] O’Donovan E, Tanner R. (1984). Numerical study of the Bingham squeeze film problem. Journal of Non-Newtonian Fluid Mechanics 15(1): 75-83.

[13] Turan O, Chakraborty N, Pool R. (2010). Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls. Journal of Non-Newtonian Fluid Mechanics 165(15-16): 901-913.

[14] Khalifeh A, Clermont J. (2005). Numerical simulations of non-isothermal three-dimensional flows in an extruder by a finite-volume method. Journal of Non-Newtonian Fluid Mechanics 126(1): 7-22.

[15] Barth W, Carey G. (2006). On a natural-convection benchmark problem in non-Newtonian fluids. Numerical Heat Transfer, Part B: Fundamentals 50(3): 193-216. https://DOI: 10.1080/10407790500509009.

[16] Anwar-Hossain M, Reddy-Gorla R. (2009). Natural convection flow of non-Newtonian power-law fluid from a slotted vertical isothermal surface. International Journal of Numerical Methods for Heat & Fluid Flow 19(7): 835-846.

[17] Lamsaadi M, Naïmi M. (2006). Natural convection in a vertical rectangular cavity filled with a non-Newtonian power law fluid and subjected to a horizontal temperature gradient. Numerical Heat Transfer, Part A: Applications 49(10): 969-990.

[18] Lamsaadi M, Naïmi M, Hasnaoui M. (2006). Natural convection heat transfer in shallow horizontal. Energy Conversion and Management 47(15-16): 2535-2551.

[19] Vola D, Boscardin L, Latché J. (2003). Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results. Journal of Computational Physics 187(2): 441-456.

[20] Siginer D, Valenzuela-Rendon A. (2000). On the laminar free convection and stability of grade fluids in enclosures. International Journal of Heat and Mass Transfer 43(18): 3391–3405.

[21] Kumar A, Bhattacharya M. (1991). Transient temperature and velocities profiles in a canned non-Newtonian liquid food during sterilization in a still-cook retort. International Journal of Heat and Mass Transfer 34(4-5): 1083-1096.

[22] Kim G, Hyun J, Kwak H. (2003). Transient buoyant convection of a power-law non-Newtonian fluid in an enclosure. International Journal of Heat and Mass Transfer 46(19): 3605-3617.

[23] Dean E, Glowinski R, Guidoboni G. (2007). On the numerical simulation of Bingham visco-plastic flow: Old and new results. Journal of Non-Newtonian Fluid Mechanics. 1421-3): 36–62.

[24] Baliga B, Patankar S. (1980). A new finite element formulation for convection diffusion problems. Numerical Heat Transfer 3(4): 393-409.

[25] Voller V. (2009). Basic control volume finite element methods for fluid and solids, IISC Press, Singapore. 

[26] Silva J, De Moura, L. (2001). A control-volume finite-element method (CVFEM) for unsteady, incompressible, viscous fluid flows. Numerical Heat Transfer, Part B: Fundamentals 40(1): 61-82. 

[27] Salinas C, Chavez C, Gatica Y, Ananias R. (2011). Simulation of wood drying stresses using CVFEM. Latin American Applied Research 41(1): 23-30.

[28] Estacio K, Nonato L, Mangiavacchi N, Carey G. (2008). Combining CVFEM and meshless front tracking in Hele–Shaw mold filling simulation. International Jouurnal of Numerical Methods in Fluid. 56: 1217-1223.

[29] Charoensuk J, Vessakosol P. (2010). A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials. Heat and Mass Transfer 46(11-12): 1261-1276. https://DOI 10.1007/s00231-010-0649-8.

[30] Herschel W, Bulkley R, (1926). Konsistenzmessungen von Gummi-Benzollösungen. Colloid Polym Sci. 39: 291–300.

[31] Papanastasiou T. (1987). Flows of materials with yield. Journal of Rheology 31(5): 385-404.

[32] Abdali S, Mitsoulis E, Markatos N. (1992). Entry and exit flows of Bingham fluids. Journal of Rheology 36: 389-407.

[33] Stoer J, Bulirsch R. (1980). Introduction to numerical analysis, Springer, New York, USA.

[34] Vahl Davis G. (1983). Natural convection of air in a square cavity: A bench mark numerical solution. International Journal for Numerical Methods in Fluids 3: 249-264.