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ABSTRACT
In this paper a novel shape design method is introduced for the numerical solution of inverse heat convection problems (IHCPs) of nanofluids. The proposed method is a novel extension of the ball-spine algorithm (BSA) inverse method, which is recently adapted for inverse heat transfer problems (IHTPs). Here it is shown how, by a novel physical-sense remedy, the method is applicable to IHCPs as well. In this respect, after validation of the proposed method, two types of IHCPs are introduced and numerically solved. In both types of introduced inverse problems, the objective is shape optimization of a duct containing such steady-state incompressible laminar nanofluid flow that it satisfies a prescribed heat flux distribution along the walls of the geometry. The results show merits and robustness of the BSA application in capturing the target geometry corresponding to a given heat flux distribution in forced heat convection problems with a low computational cost.
Nomenclature
| C | = | overall under/over relaxation factor |
| Cp | = | specific heat, J/(kg K) |
| = | unit vector in radial direction | |
| it | = | iteration |
| k | = | thermal conductivity, W/(m K) |
| kf | = | fluid thermal conductivity, W/(m K) |
| ks | = | solid thermal conductivity, W/(m K) |
| M | = | number of boundary nodes |
| N | = | dimensionless normal vector of the surface |
| p | = | pressure, N/m2 |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| q | = | heat flux, W/m2 |
| qw | = | wall heat flux, W/m2 |
| = | displacement vector | |
| Re | = | Reynolds number |
| Res | = | residual |
| s | = | coordinate along the surface of solid body, m |
| S | = | circumference of solid body, m |
| T | = | temperature, K |
| Tin | = | inlet temperature, K |
| Tw | = | wall temperature, K |
| x | = | horizontal Cartesian coordinate, m |
| X | = | horizontal dimensionless Cartesian coordinate |
| y | = | vertical Cartesian coordinate, m |
| Y | = | vertical dimensionless Cartesian coordinate |
| u | = | horizontal velocity component, m/s |
| U | = | dimensionless horizontal velocity component |
| V | = | vertical velocity component, m/s |
| V | = | vertical dimensionless velocity component |
| Win | = | inlet width |
| α | = | thermal diffusivity, m2/s |
| ρ | = | density, kg/m3 |
| φ | = | solid volume fraction of the nanofluid |
| ν | = | kinematic viscosity, m2/s |
| θ | = | dimensionless temperature |
| θh | = | heat source dimensionless temperature |
| θin | = | inlet dimensionless temperature |
| θw | = | wall dimensionless temperature |
| | | | = | absolute value |
| Superscript | = | |
| Cur | = | current |
| I.G | = | initial guess |
| N | = | current iteration |
| n+1 | = | next iteration |
| Tar | = | target |
| Subscript | = | |
| f | = | fluid |
| h | = | heat source |
| i | = | indices for specified boundary nodes |
| in | = | inlet |
| nf | = | nanofluid |
| s | = | solid particles |
| w | = | wall |
Nomenclature
| C | = | overall under/over relaxation factor |
| Cp | = | specific heat, J/(kg K) |
| = | unit vector in radial direction | |
| it | = | iteration |
| k | = | thermal conductivity, W/(m K) |
| kf | = | fluid thermal conductivity, W/(m K) |
| ks | = | solid thermal conductivity, W/(m K) |
| M | = | number of boundary nodes |
| N | = | dimensionless normal vector of the surface |
| p | = | pressure, N/m2 |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| q | = | heat flux, W/m2 |
| qw | = | wall heat flux, W/m2 |
| = | displacement vector | |
| Re | = | Reynolds number |
| Res | = | residual |
| s | = | coordinate along the surface of solid body, m |
| S | = | circumference of solid body, m |
| T | = | temperature, K |
| Tin | = | inlet temperature, K |
| Tw | = | wall temperature, K |
| x | = | horizontal Cartesian coordinate, m |
| X | = | horizontal dimensionless Cartesian coordinate |
| y | = | vertical Cartesian coordinate, m |
| Y | = | vertical dimensionless Cartesian coordinate |
| u | = | horizontal velocity component, m/s |
| U | = | dimensionless horizontal velocity component |
| V | = | vertical velocity component, m/s |
| V | = | vertical dimensionless velocity component |
| Win | = | inlet width |
| α | = | thermal diffusivity, m2/s |
| ρ | = | density, kg/m3 |
| φ | = | solid volume fraction of the nanofluid |
| ν | = | kinematic viscosity, m2/s |
| θ | = | dimensionless temperature |
| θh | = | heat source dimensionless temperature |
| θin | = | inlet dimensionless temperature |
| θw | = | wall dimensionless temperature |
| | | | = | absolute value |
| Superscript | = | |
| Cur | = | current |
| I.G | = | initial guess |
| N | = | current iteration |
| n+1 | = | next iteration |
| Tar | = | target |
| Subscript | = | |
| f | = | fluid |
| h | = | heat source |
| i | = | indices for specified boundary nodes |
| in | = | inlet |
| nf | = | nanofluid |
| s | = | solid particles |
| w | = | wall |