Abstract
This study presents a numerical solution to convective heat transfer in laminar flow in the thermal entrance region of a rectangular duct with two indented sides. The flow is considered to be hydrodynamically fully developed and thermally developing laminar flow of incompressible, Newtonian fluids with constant thermal properties. The ducts are subjected to a constant wall temperature. An algebraic technique is used to discretize the solution domain and a boundary-fitted coordinate system is numerically developed. The governing equations in the boundary-fitted coordinates are solved by the control volume-based finite difference method. Distribution of the bulk temperature and the Nusselt number along the direction of flow is calculated and presented graphically. Also calculated is the thermal entrance length of the rectangular ducts with two indented sides. The parameters, such as the friction factor times the Reynolds number, and the Nusselt number for the fully developed flow and thermally developing flow are obtained.
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Abbreviations
- a :
-
half-width of duct [m]
- A :
-
cross-sectional area [m2]
- b :
-
half-height of duct [m]
- C p :
-
specific heat [kJ kg-1 K-1]
- D h :
-
hydraulic diameter [m]
- f :
-
skin friction factor =\(\frac{{D_h (dp/dz)}}{{1/2\rho w_m^2 }}\)
- h z :
-
local heat transfer coefficient [Wm−2 K−1]
- h T :
-
asymptotic heat transfer coefficient [Wm−2 K−1]
- J :
-
Jacobian matrix of transformation, Eq. (20)
- k :
-
thermal conductivity [Wm−1 K−1]
- L :
-
thermal entrance length [m]
- L * :
-
dimensionless thermal entrance length =L/D h RePr
- L1 :
-
maximum number of grids in Ξ direction
- M1 :
-
maximum number of grids in η direction
- Nu T :
-
asymptotic Nusselt number =h T D h /k
- p :
-
pressure [n m−2]
- P :
-
circumferential length [m]
- Pr :
-
Prandtl number = ν/αT
- Re :
-
Reynolds number =w m D h /ν
- T :
-
temperature [K]
- T b :
-
bulk temperature [K]
- T i :
-
inlet temperature
- T w :
-
circumferential duct wall temperature [K]
- w :
-
velocity [ms−1]
- w m :
-
mean velocity [ms−1]
- W :
-
dimensionless velocity = −\(\frac{{\mu w}}{{D_h^2 (dp/dz)}}\)
- W m :
-
dimensionless mean velocity
- x, y :
-
transveral coordinates [m]
- X,Y :
-
dimensionless transversal coordinates =\(\frac{z}{{D_h RePr}}\)
References
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Communicated by S. N. Atluri, March 29, 1991
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Dong, Z.F., Ebadian, M.A. Convective heat transfer in the entrance region of a rectangular duct with two indented sides. Computational Mechanics 8, 269–278 (1991). https://doi.org/10.1007/BF00577380
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DOI: https://doi.org/10.1007/BF00577380