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ABSTRACT
Taking the strongly nonlinear energy conservation equation and the transient diffusion equations as examples, the differences of solving procedure and the computation time are compared and discussed between the fast cosine transformation (FCT) and the matrix multiplication transformation (MMT) in the application of spectral methods. To compare the solving procedures for the energy equation, the temperature distribution in the fin with variable thermal conductivity, variable heat transfer coefficient, and internal heat generation are considered. To compare the computation time, the transient diffusion equations in 1D and 2D enclosures, which contain the Poisson operator and the source term, are analyzed. All variables are approximated by the Chebyshev polynomials and all equations are discretized by the Chebyshev–Gauss–Lobatto (CGL) collocation points in space. The comparisons indicate that the FCT can be adopted to solve the strongly nonlinear equation directly and it can provide the same accuracy as MMT does. The comparison of computation time indicates that the MMT is faster than the FCT when the grid number is small or moderate (less than 200), while, if the grid number is large (greater than 200), the FCT is more faster than the MMT.
Nomenclature
| A | = | dimensionless parameter of thermal conductivity |
| a | = | adjustment parameter |
| B | = | matrix defined in Eqs. (8) & (11) |
| Ci | = | dimensionless internal heat generation parameters |
| ci | = | internal heat generation parameters |
| = | identified coefficients | |
| D | = | derivative matrix |
| F | = | matrix defined in Eqs. (8) & (11) |
| h | = | Convective heat transfer coefficient, W/(m2 · K) |
| i, j, k | = | index counters |
| L | = | length of fin, m |
| N | = | grid number |
| Ncc | = | dimensionless convection-conduction parameter |
| Nrc | = | dimensionless radiation-conduction parameter |
| q | = | internal heat generation, W/m3 |
| T | = | temperature, K |
| t | = | time, s |
| u | = | normal variable defined in Eq. (27) |
| X | = | dimensionless Cartesian coordinate in horizontal direction |
| x | = | Cartesian coordinate in horizontal direction, m |
| y | = | Cartesian coordinate in vertical direction, m |
| α | = | linear coefficient defined in thermal conductivity |
| δ | = | half thickness of fin, m |
| ϵ | = | emissivity of wall, error |
| λ | = | thermal conductivity, W/(m · K) |
| ρ | = | source term defined in Eq. (23) |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8 W/(m2 · K4) |
| Θ | = | dimensionless temperature |
| Subscripts | = | |
| a | = | of convection sink |
| b | = | of fin-base |
| bench | = | of benchmark result |
| FCT | = | of fast cosine transformation result |
| exact | = | of exact solution |
| i, j, k | = | of index counters |
| max | = | of maximum value |
| s | = | of radiation sink |
| Superscripts | = | |
| n | = | of the nth time step value |
| β | = | exponential index defined in convective heat transfer coefficient |
| * | = | of the last iterative value |
| ´ | = | matrices after exchange |
| + | = | direction from physical to spectral spaces |
| − | = | direction from spectral to physical spaces |
| (1), (2) | = | first and second order |
Nomenclature
| A | = | dimensionless parameter of thermal conductivity |
| a | = | adjustment parameter |
| B | = | matrix defined in Eqs. (8) & (11) |
| Ci | = | dimensionless internal heat generation parameters |
| ci | = | internal heat generation parameters |
| = | identified coefficients | |
| D | = | derivative matrix |
| F | = | matrix defined in Eqs. (8) & (11) |
| h | = | Convective heat transfer coefficient, W/(m2 · K) |
| i, j, k | = | index counters |
| L | = | length of fin, m |
| N | = | grid number |
| Ncc | = | dimensionless convection-conduction parameter |
| Nrc | = | dimensionless radiation-conduction parameter |
| q | = | internal heat generation, W/m3 |
| T | = | temperature, K |
| t | = | time, s |
| u | = | normal variable defined in Eq. (27) |
| X | = | dimensionless Cartesian coordinate in horizontal direction |
| x | = | Cartesian coordinate in horizontal direction, m |
| y | = | Cartesian coordinate in vertical direction, m |
| α | = | linear coefficient defined in thermal conductivity |
| δ | = | half thickness of fin, m |
| ϵ | = | emissivity of wall, error |
| λ | = | thermal conductivity, W/(m · K) |
| ρ | = | source term defined in Eq. (23) |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8 W/(m2 · K4) |
| Θ | = | dimensionless temperature |
| Subscripts | = | |
| a | = | of convection sink |
| b | = | of fin-base |
| bench | = | of benchmark result |
| FCT | = | of fast cosine transformation result |
| exact | = | of exact solution |
| i, j, k | = | of index counters |
| max | = | of maximum value |
| s | = | of radiation sink |
| Superscripts | = | |
| n | = | of the nth time step value |
| β | = | exponential index defined in convective heat transfer coefficient |
| * | = | of the last iterative value |
| ´ | = | matrices after exchange |
| + | = | direction from physical to spectral spaces |
| − | = | direction from spectral to physical spaces |
| (1), (2) | = | first and second order |