Darcy Flow Analysis and Convection Topology Optimization

Forced Convection for Linear Steady-State Heat Transfer is available via Darcy Flow analysis.

Darcy Flow Analysis is currently only supported for steady-state heat transfer analysis. Forced convection applications include cooling solutions for electric motors, machine tools (casting, forming), heat exchangers, HVAC systems, and cooling for electronic devices including PCBs. Additionally, Topology Optimization is available for steady-state heat transfer with Darcy flow analysis. The topology optimization considers the effect of forced convection for cooling in conjunction with structural steady-state heat transfer analysis. Topology optimization can help optimize cooling channel structures and placement for a wide range of applications.

The flow solution is described by:

$K_{p} p = f_{p}$

Where,

$K_{p}$: Permeability matrix
$p$: Nodal pressure in the structure
$f_{p}$: Pressure load at the inlet

The fluid flow analysis is solved using Darcy’s Law, which describes the flow of a fluid through a porous medium:

$u = - \frac{κ}{μ} \nabla p$

Where,

$u$: Fluid velocity
$κ$: Fluid permeability (this is different from thermal conductivity, represented by $k$ )
$μ$: Fluid dynamic viscosity
$\nabla p$: Pressure differential

The equation can be rewritten as:

$u^{e} = - \frac{κ}{μ} B p^{e}$

Where,

$u^{e}$: Element fluid velocity
$B$: Differential of the shape function
$p^{e}$: Nodal pressure in the element (which is sourced from the flow solution)

The thermal steady-state heat transfer solution is represented by:

$[K_{c} + C (p)] T = f$

Where,

$K {}_{c}$: Conductivity Matrix:
$K_{c} = \sum_{n = 1}^{N_{e}} \int_{Ω^{e}} k B^{T} B d Ω$
$C (p)$: Convection Matrix (which includes flow velocity $u^{e}$ from Darcy's Law):
$C (p) = \sum_{n = 1}^{N_{e}} \int_{Ω^{e}} {\hat{N}}^{T} ρ c_{p} u^{e} B d Ω$
Where,

$f$

Thermal load vector

$T$

Nodal temperature matrix

${\hat{N}}^{T}$

Enhanced shape function

$N_{e}$

Total number of elements

$ρ$

Density

$k$

Thermal conductivity (this is different from fluid permeability, Kappa, represented by $κ$ )

$c_{p}$

Specific heat capacity

$u^{e}$

Element flow velocity from Darcy’s Law

The thermal steady-state solution incorporates forced convection via the Convection Matrix. A topology design space can be defined for a steady-state heat transfer subcase to run the optimization solution which accounts for the forced convection via Darcy flow.