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ABSTRACT
A horizontal double-pipe heat exchanger with an inverted outer equilateral triangular tube is modeled to numerically investigate the low-temperature thermal energy storage capability of an impure phase change material (PCM). The energy source fluid (hot water) flows through the inner tube and transfers heat to the PCM (heat sink) residing in the annular gap. The results show that the inlet temperature of the heat transfer fluid (HTF) has a significant effect on the melting process compared with the mass flow rate (MFR). The configuration, as well the concentricity/eccentricity of the inner tube has a great influence on the energy storage.
Nomenclature
| A* | = | Darcy coefficient (dimensionless) |
| = | weighted average mixture specific heat, kJ.kg−1.°C−1 | |
| C | = | morphology constant (dimensionless) |
| φm | = | mixture gravitational acceleration, m. s−2 |
| gL | = | volume fraction of liquid (dimensionless) |
| gS | = | volume fraction of solid (dimensionless) |
| fL | = | mass fraction of liquid (dimensionless) |
| fS | = | mass fraction of solid (dimensionless) |
| hL | = | sensible enthalpy of liquid, kJ/m |
| hS | = | sensible enthalpy of solid, kJ/m |
| hm | = | sensible enthalpy for mixture, kJ/m |
| K′ | = | permeability of porous media |
| P | = | hydrodynamic pressure, Pa |
| Pr | = |
|
| Di | = | diameter of inner cylinder, m |
| Do | = | diameter of outer cylinder, m |
| Ra | = |
|
| Ra* | = |
|
| Ste | = |
|
| Stei | = | initial Stefan number (dimensionless) |
| S | = | source term (dimensionless) |
| SΦ | = | source term associated with Φ (dimensionless) |
| T | = | temperature, °C |
| TWALL | = | inner tube wall temperature, °C |
| Ti | = | initial temperature of the solid PCM, °C |
| ΔT | = | temperature difference between the inner tube wall and the solidus temperature of PCM, °C |
| Tref | = | reference temperature, °C |
| t | = | time, min |
| = |
| |
| = | enthalpy at the inner wall (dimensionless) | |
| = | initial enthalpy (dimensionless) | |
| um | = | mixture velocity component along the x direction, ms−1 |
| vm | = | mixture velocity component along the y direction, ms−1 |
| = | mixture velocity components along the X and Y directions, respectively (nondimensional) | |
| U | = | contravariant velocity component in the ξ -direction |
| V | = | contravariant velocity component in the η -direction |
| xξ, xη, yξ, yη | = | metric derivatives |
| NuL | = | local Nusselt number (nondimensional) |
| = | surface-averaged Nusselt number (nondimensional) | |
| Greek Symbols | = | |
| α, β, γ | = | geometric relations between coordinate systems |
| ΔH | = | nodal latent heat, kJ.kg−1 |
| λPCM | = | latent heat of fusion, kJ.kg−1 |
| ΓΦ | = | diffusion coefficient associated with Φ |
| ρL | = | actual density of liquid, kg.m−3 |
| ρS | = | actual density of solid, kg.m−3 |
| μm | = | effective mixture viscosity, kg.m−1.s−1 |
| μl | = | absolute viscosity, kg.m−1.s−1 |
| Φ | = | generalized dependent variable |
| Гeff | = | effective diffusivity, m2s−1 |
| αPCM | = | thermal diffusivity of PCM, m2s−1 |
| β | = | coefficient of thermal expansion, |
| νPCM | = | kinematic viscosity, m2s−1 |
| τ | = | Fourier number, |
| ξ, η | = | axes of nonorthogonal curvilinear coordinate system |
| Subscripts | = | |
| L | = | liquid |
| S | = | solid |
| i | = | inner cylinder |
| o | = | outer cylinder |
| REF | = | reference value |
| w | = | bulk |
| wall | = | bulk |
| Superscripts | = | |
| * | = | nondimensional variables |
Nomenclature
| A* | = | Darcy coefficient (dimensionless) |
| = | weighted average mixture specific heat, kJ.kg−1.°C−1 | |
| C | = | morphology constant (dimensionless) |
| φm | = | mixture gravitational acceleration, m. s−2 |
| gL | = | volume fraction of liquid (dimensionless) |
| gS | = | volume fraction of solid (dimensionless) |
| fL | = | mass fraction of liquid (dimensionless) |
| fS | = | mass fraction of solid (dimensionless) |
| hL | = | sensible enthalpy of liquid, kJ/m |
| hS | = | sensible enthalpy of solid, kJ/m |
| hm | = | sensible enthalpy for mixture, kJ/m |
| K′ | = | permeability of porous media |
| P | = | hydrodynamic pressure, Pa |
| Pr | = |
|
| Di | = | diameter of inner cylinder, m |
| Do | = | diameter of outer cylinder, m |
| Ra | = |
|
| Ra* | = |
|
| Ste | = |
|
| Stei | = | initial Stefan number (dimensionless) |
| S | = | source term (dimensionless) |
| SΦ | = | source term associated with Φ (dimensionless) |
| T | = | temperature, °C |
| TWALL | = | inner tube wall temperature, °C |
| Ti | = | initial temperature of the solid PCM, °C |
| ΔT | = | temperature difference between the inner tube wall and the solidus temperature of PCM, °C |
| Tref | = | reference temperature, °C |
| t | = | time, min |
| = |
| |
| = | enthalpy at the inner wall (dimensionless) | |
| = | initial enthalpy (dimensionless) | |
| um | = | mixture velocity component along the x direction, ms−1 |
| vm | = | mixture velocity component along the y direction, ms−1 |
| = | mixture velocity components along the X and Y directions, respectively (nondimensional) | |
| U | = | contravariant velocity component in the ξ -direction |
| V | = | contravariant velocity component in the η -direction |
| xξ, xη, yξ, yη | = | metric derivatives |
| NuL | = | local Nusselt number (nondimensional) |
| = | surface-averaged Nusselt number (nondimensional) | |
| Greek Symbols | = | |
| α, β, γ | = | geometric relations between coordinate systems |
| ΔH | = | nodal latent heat, kJ.kg−1 |
| λPCM | = | latent heat of fusion, kJ.kg−1 |
| ΓΦ | = | diffusion coefficient associated with Φ |
| ρL | = | actual density of liquid, kg.m−3 |
| ρS | = | actual density of solid, kg.m−3 |
| μm | = | effective mixture viscosity, kg.m−1.s−1 |
| μl | = | absolute viscosity, kg.m−1.s−1 |
| Φ | = | generalized dependent variable |
| Гeff | = | effective diffusivity, m2s−1 |
| αPCM | = | thermal diffusivity of PCM, m2s−1 |
| β | = | coefficient of thermal expansion, |
| νPCM | = | kinematic viscosity, m2s−1 |
| τ | = | Fourier number, |
| ξ, η | = | axes of nonorthogonal curvilinear coordinate system |
| Subscripts | = | |
| L | = | liquid |
| S | = | solid |
| i | = | inner cylinder |
| o | = | outer cylinder |
| REF | = | reference value |
| w | = | bulk |
| wall | = | bulk |
| Superscripts | = | |
| * | = | nondimensional variables |
Acknowledgments
This work was partially supported by the National Sciences and Engineering Research Council (NSERC) of Canada Discovery Grant RGPIN48158 awarded to M. Hasan of McGill University, Montreal, for which the authors are grateful.