Mathematical Characterization of Bénard-Type Geothermal Scenarios Using Discriminated Non-dimensionalization of the Governing Equations
-
M. Cánovas
,
I. Alhama
Abstract
The classical non-dimensionalization process of governing equations is a low-cost method commonly applied for a first approximation to the dimensionless numbers that determine the solution patterns in many problems; however, this procedure fails in complex problems, where it is not even possible to define reference quantities – because they are not established in the statement of the problem – to make the dependent or independent variables dimensionless. The application of discrimination corrects this obstacle and allows suitable dimensionless groups to be defined. These, in turn, have two interesting properties: (i) they are of order of magnitude unity, and (ii) they have a clear meaning in terms of balance of physical quantities that counteract each other in a domain or sub-domain of the problem. In this paper, discriminated non-dimensionalization is applied to geothermal scenarios of Bénard-type convective flow, large horizontal boundary sides under a temperature gradient in porous media, to determine the dimensionless groups that control the steady temperature and stream patterns and, from these, the order of magnitude of the main unknowns of the problem. The results were checked numerically for many cases.
Nomenclature
- cp
specific heat (J/Kg°C)
- Cte
a constant fraction of the unity (dimensionless)
- g
gravitational acceleration (m/s2)
- H
height of the domain (m)
- k
thermal conductivity (W/m°C)
- K
hydraulic conductivity (m/s)
- L
length of the domain (m)
- P
pressure (N/m2),
- q
fluid flow (m/s)
- t
time (s)
- T
temperature (°C)
- u
horizontal velocity (m/s)
- v
vertical velocity (m/s)
- x, y
rectangular coordinates (m)
- α
thermal diffusivity (m2/s)
- β
thermal expansion coefficient (°C-1)
- ψ
Streamfunction (m2/s)
- φ
porosity (dimensionless)
- μ
fluid viscosity (kg m−1s−1)
- ρ
density (kg/m3)
- Subscripts
- 1, 2, …
refer to different arbitrary functions
- b
refers to bottom boundary
- bl
refers to boundary layer
- c
refers to core region of the cell
- DD
refers to discriminate dimensionless number or group
- f
refers to fluid
- hidden
refers to a characteristic or reference quantity not defined in the statement of the problem
- l
refers to left side boundary
- m
average value
- o
Relative to a the reference temperature
- r
refers to right side boundary
- s
refers to solid matrix
- sim
referred to values from numerical simulation
- t
refers to the top boundary
- x, y
refer to spatial direction
- Superscripts
- ′
denotes dimensionless quantity or variable
- *
reference magnitude
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©2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- A Short Remark on WAN Model for Electrospinning and Bubble Electrospinning and Its Development
- ICEEM07: Three-Dimensional Flow Optimization of a Pneumatic Pulsator Nozzle with a Continuous Adjoint
- Homogeneous Sono-Fenton Process: Statistical Modeling and Global Sensitivity Analysis
- Mathematical Characterization of Bénard-Type Geothermal Scenarios Using Discriminated Non-dimensionalization of the Governing Equations
- Sequential Maximum Likelihood Estimation for the Parameter of the Linear Drift Term of the Rayleigh Diffusion Process
- A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System
- Dynamically Induced Magnetic Moment of a Magnetic Dipole System
Articles in the same Issue
- Frontmatter
- A Short Remark on WAN Model for Electrospinning and Bubble Electrospinning and Its Development
- ICEEM07: Three-Dimensional Flow Optimization of a Pneumatic Pulsator Nozzle with a Continuous Adjoint
- Homogeneous Sono-Fenton Process: Statistical Modeling and Global Sensitivity Analysis
- Mathematical Characterization of Bénard-Type Geothermal Scenarios Using Discriminated Non-dimensionalization of the Governing Equations
- Sequential Maximum Likelihood Estimation for the Parameter of the Linear Drift Term of the Rayleigh Diffusion Process
- A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System
- Dynamically Induced Magnetic Moment of a Magnetic Dipole System
