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ABSTRACT
A numerical study that uses the finite difference method to model the fluid flow and heat transfer of a rectangular natural circulation loop that contains phase change material (PCM) suspensions is presented to investigate how geometric parameters affect the thermal performance. Parametric simulations were performed using different geometrical parameters in the following ranges: the dimensionless length of the heated section = 0.4–1; the relative elevation of the cooled section compared with the heated section = 0.5–2; and the aspect ratio of the loop = 0.25–1. The results determine the important geometric parameters that affect the heat transfer performance of the loop with the PCM suspension. In several of the geometric configurations, the heat transfer performance of the loop is significantly affected by the latent heat contribution associated with the melting/freezing of the PCM particles.
NOMENCLATURE
| AR | = | aspect ratio of the rectangular loop, |
| Bip | = | Biot number of a PCM particle, |
| cp | = | specific heat |
| cv | = | volumetric fraction of PCM particles |
| g | = | acceleration due to gravity |
| h | = | heat transfer coefficient |
| hls | = | latent heat of fusion |
| k | = | thermal conductivity |
| = | thermal conductivity ratio, kb/kf | |
| = | thermal conductivity ratio, kp/kf | |
| = | length of the left and right adiabatic sections of the loop bottom leg | |
| la | = | dimensionless length of the left and right adiabatic sections of the loop bottom leg, |
| = | length of the cooled section | |
| = | dimensionless length of the heated section, | |
| = | length of the heated section | |
| lh | = | dimensionless length of the heated section, |
| = | rectangular loop width | |
| lH | = | dimensionless rectangular loop width, |
| = | total length of the rectangular loop, | |
| = | correction factor for the loop length [Citation18] | |
| lL | = | dimensionless total length of the rectangular loop, 2(lV + lH) or (1 + 2lz)(1 + AR) |
| = | rectangular loop height | |
| lV | = | dimensionless height of the rectangular loop, |
| = | relative elevation of the cooled section compared with the heated section | |
| lz | = | dimensionless mean relative elevation of the cooled section compared with the heated section, |
| Nu | = | Nusselt number, |
| Pr | = | Prandtl number, νb/αb |
| q | = | heat transfer rate |
| q″ | = | local heat flux |
| = | total convection heat transfer rate | |
| RL | = | dimensionless pipe radius, |
| = | modified Rayleigh number, | |
| Re | = | Reynolds number, |
| r+ | = | radial coordinate |
| = | inner radius of the loop pipe | |
| = | radius of a PCM particle | |
| r | = | dimensionless coordinate, |
| rp | = | dimensionless radius of particles, |
| s+ | = | axial coordinate |
| s | = | dimensionless axial coordinate, |
| Sb* | = | modified subcooling factor, |
| Ste* | = | modified Stefan number, |
| T | = | temperature |
| u+ | = | velocity in the axial direction |
| u | = | dimensionless axial velocity, |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ϵh | = | effectiveness of the convection coefficient enhancement, |
| = | effectiveness of the wall temperature reduction, (Tw,h, max, f − Tm)/(Tw,h, max, b − Tm) or | |
| γL | = | correction factor for the loop length, |
| ν | = | kinematic viscosity |
| θ | = | dimensionless temperature, |
| ρ | = | density |
| = | density ratio, ρb/ρf | |
| = | density ratio, ρp/ρf | |
| ξ | = | volumetric phase fraction in a particle |
| Φ | = | latent heat contribution |
| Subscripts | = | |
| b | = | bulk fluid |
| bf | = | bulk to base fluid ratio |
| c | = | cooled section |
| f | = | base fluid (water) |
| h | = | heated section or heat transfer coefficient |
| i | = | inner |
| ℓ | = | liquid phase of PCM |
| lat | = | latent heat |
| m | = | melting point or mean quantity |
| max | = | maximum |
| p | = | particle |
| sen | = | sensible heat |
| w | = | pipe wall |
| Superscripts | = | |
| — | = | surface-averaged quantity |
| * | = | ratio of quantities |
| + | = | dimensional quantity |
NOMENCLATURE
| AR | = | aspect ratio of the rectangular loop, |
| Bip | = | Biot number of a PCM particle, |
| cp | = | specific heat |
| cv | = | volumetric fraction of PCM particles |
| g | = | acceleration due to gravity |
| h | = | heat transfer coefficient |
| hls | = | latent heat of fusion |
| k | = | thermal conductivity |
| = | thermal conductivity ratio, kb/kf | |
| = | thermal conductivity ratio, kp/kf | |
| = | length of the left and right adiabatic sections of the loop bottom leg | |
| la | = | dimensionless length of the left and right adiabatic sections of the loop bottom leg, |
| = | length of the cooled section | |
| = | dimensionless length of the heated section, | |
| = | length of the heated section | |
| lh | = | dimensionless length of the heated section, |
| = | rectangular loop width | |
| lH | = | dimensionless rectangular loop width, |
| = | total length of the rectangular loop, | |
| = | correction factor for the loop length [Citation18] | |
| lL | = | dimensionless total length of the rectangular loop, 2(lV + lH) or (1 + 2lz)(1 + AR) |
| = | rectangular loop height | |
| lV | = | dimensionless height of the rectangular loop, |
| = | relative elevation of the cooled section compared with the heated section | |
| lz | = | dimensionless mean relative elevation of the cooled section compared with the heated section, |
| Nu | = | Nusselt number, |
| Pr | = | Prandtl number, νb/αb |
| q | = | heat transfer rate |
| q″ | = | local heat flux |
| = | total convection heat transfer rate | |
| RL | = | dimensionless pipe radius, |
| = | modified Rayleigh number, | |
| Re | = | Reynolds number, |
| r+ | = | radial coordinate |
| = | inner radius of the loop pipe | |
| = | radius of a PCM particle | |
| r | = | dimensionless coordinate, |
| rp | = | dimensionless radius of particles, |
| s+ | = | axial coordinate |
| s | = | dimensionless axial coordinate, |
| Sb* | = | modified subcooling factor, |
| Ste* | = | modified Stefan number, |
| T | = | temperature |
| u+ | = | velocity in the axial direction |
| u | = | dimensionless axial velocity, |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ϵh | = | effectiveness of the convection coefficient enhancement, |
| = | effectiveness of the wall temperature reduction, (Tw,h, max, f − Tm)/(Tw,h, max, b − Tm) or | |
| γL | = | correction factor for the loop length, |
| ν | = | kinematic viscosity |
| θ | = | dimensionless temperature, |
| ρ | = | density |
| = | density ratio, ρb/ρf | |
| = | density ratio, ρp/ρf | |
| ξ | = | volumetric phase fraction in a particle |
| Φ | = | latent heat contribution |
| Subscripts | = | |
| b | = | bulk fluid |
| bf | = | bulk to base fluid ratio |
| c | = | cooled section |
| f | = | base fluid (water) |
| h | = | heated section or heat transfer coefficient |
| i | = | inner |
| ℓ | = | liquid phase of PCM |
| lat | = | latent heat |
| m | = | melting point or mean quantity |
| max | = | maximum |
| p | = | particle |
| sen | = | sensible heat |
| w | = | pipe wall |
| Superscripts | = | |
| — | = | surface-averaged quantity |
| * | = | ratio of quantities |
| + | = | dimensional quantity |