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ABSTRACT
In this paper, a numerical study focusing on the thermal behavior of carbon dioxide within the supercritical thermodynamic region is performed. For that, the well-known Graetz problem, which considers the fluid flow between two parallel plates with a uniformly distributed heat flux on both walls and a specified pressure difference across the channel, is studied. The numerical solution is obtained by discretizing the energy equation within a two-dimensional rectangular domain using a finite-volume formulation and enforcing spatially variable thermophysical properties throughout the channel. The boundary conditions are selected such that the pressure and temperature of the working fluid are slightly above the critical point (Tc = 304.13 K and Pc = 7.34 MPa for carbon dioxide). The figure of merit used, the Nusselt number, is shown to be highly sensitive to changes in key thermodynamic properties, such as the specific heat. However, the results revealed that the specific heat could not be simply determined as a function of the local temperature and pressure. Instead, the specific heat must be corrected with an additional term, which depends on thermodynamic properties and the flow field – the modified specific heat is referred to as c*. The results obtained show that the agreement between the traditionally defined specific heat and c* is satisfactory up to pressure drops across the channel of roughly 10 kPa. For larger values of pressure difference, the Nusselt number is severely underestimated if the specific heat is not properly corrected.
Nomenclature
| A | = | area [m2] |
| b | = | distance between plates [m] |
| c | = | specific heat [J/(kg K)] |
| D | = | diameter [m] |
| f | = | generic term for physical or thermodynamic property |
| k | = | thermal conductivity [W/(m K)] |
| h | = | specific enthalpy [J/kg] |
| p | = | pressure [Pa] |
| q | = | heat flow [J/kg] |
| q′ | = | heat flow per unit length [W/m] |
| q″ | = | heat flux [W/m2] |
| n | = | normal coordinate [m] |
| Nu | = | Nusselt number [–] |
| P | = | pressure [MPa] |
| T | = | temperature [K] |
| u | = | velocity [m/s] |
| x | = | Cartesian coordinate [m] |
| x* | = | dimensionless coordinate [–] |
| y | = | Cartesian coordinate [m] |
| Subscripts | = | |
| crit | = | critical |
| a | = | analytical |
| b | = | bulk |
| c | = | conventional |
| h | = | hydraulic |
| i | = | inlet |
| m | = | mean |
| p | = | indicates constant pressure |
| s | = | surface, suggested |
| T | = | indicates constant temperature |
| ρ | = | indicates constant density |
| Greek Symbols | = | |
| β | = | coefficient of thermal expansion [1/K] |
| δ | = | inexact differential [–] |
| ρ | = | density [kg/m³] |
| μ | = | dynamic viscosity [Pa s] |
Nomenclature
| A | = | area [m2] |
| b | = | distance between plates [m] |
| c | = | specific heat [J/(kg K)] |
| D | = | diameter [m] |
| f | = | generic term for physical or thermodynamic property |
| k | = | thermal conductivity [W/(m K)] |
| h | = | specific enthalpy [J/kg] |
| p | = | pressure [Pa] |
| q | = | heat flow [J/kg] |
| q′ | = | heat flow per unit length [W/m] |
| q″ | = | heat flux [W/m2] |
| n | = | normal coordinate [m] |
| Nu | = | Nusselt number [–] |
| P | = | pressure [MPa] |
| T | = | temperature [K] |
| u | = | velocity [m/s] |
| x | = | Cartesian coordinate [m] |
| x* | = | dimensionless coordinate [–] |
| y | = | Cartesian coordinate [m] |
| Subscripts | = | |
| crit | = | critical |
| a | = | analytical |
| b | = | bulk |
| c | = | conventional |
| h | = | hydraulic |
| i | = | inlet |
| m | = | mean |
| p | = | indicates constant pressure |
| s | = | surface, suggested |
| T | = | indicates constant temperature |
| ρ | = | indicates constant density |
| Greek Symbols | = | |
| β | = | coefficient of thermal expansion [1/K] |
| δ | = | inexact differential [–] |
| ρ | = | density [kg/m³] |
| μ | = | dynamic viscosity [Pa s] |
Acknowledgments
The authors are sincerely thankful to our sponsor, ANEEL/PETROBRAS, which allowed the development of this study under project #0050.0080593.12.9.