Heat transfer

Let's consider a typical case: a finned heat exchanger made of ALUPOR™. This option may be viable, but it is highly unlikely to produce the desired effect for the following reasons:

To achieve maximum efficiency, it is necessary to ensure uniform supply of the coolant into the material volume
Thermal conductivity is significantly lower compared to solid metal
Only the surface layer of pores will be ventilated by the fan
The majority of airflow will follow the path of least resistance – between the fins

Symbols definitions

Constants

R=8,314 462 618 153 24 [J/(mol×K)] - gas constant;

Physical properties

c [J/(kg×K)] - specific heat capacity;
λ [W/(m×K)] - heat conductivity;
μ [Pa×s ] - dynamic viscosity;
ρ [kg/m³] - density;
σ [Sm/m] - electrical conductivity.

Physical values

F_filt [m²] - filtration surface;
j [kg/(m²×s)] - specific mass flux;
Q [J] - heat quantity;
T [K] - absolute temperature;
t [°C] - temperature;
V [m³] - volume;
v [m/s] - velocity;
v_filt [m/s] filtration velocity;
α_S [W/(m²×K)] - surface heat transfer coefficient;
α_V [W/(m³×K)] - volumetric heat transfer coefficient;
τ [s] - time.

ALUPOR™ structure parameters

Π [dimensionless] - porosity;
S_min [dimensionless] - ratio of minimal solid cross-section and filtration area (same as minimal void cross-section of packed bed).
d_p [μm] - mean pore size.
S_sp [m²/m³] - specific surface area.

Subscripts

#_f value related to fluid ;
#_s value related to solid (metal matrix);
#_p value related to pore volume(example: velocity in pores);
#_n value related to necks between pores (example: velocity in necks);
#_Me value related to solid metal;
#_AP value related to ALUPOR™;
#_eff effective property of composite (example:"ALUPOR™-air" system);

Heat conductivity

The heat conductivity of porous metals is significantly lower than solid metals one. It depends on porosity, pore size, and shape. Increasing porosity leads to a decrease in heat conductivity.

The main factors affecting the heat conductivity of porous metals:

Heat conductivity of the metal matrix itself: ALUPOR™ made of pure aluminum will have higher heat conductivity than ALUPOR™ made from aluminum alloys
Presence of closed pores: ALUPOR™ has a structure with open and semi-open porosity. This means that in our case, this factor can be neglected.
Porosity: The greater the volume of pores in the metal, the lower its heat conductivity. This is due to the fact that the air filling the pores has low heat conductivity, which impedes heat transfer through the material.
Minimum cross-section of the metal matrix: This is a bottleneck and actually determines the heat flow through the porous body. It is directly related to porosity. Usually determined from the model of close-packed spheres (fictitious bed). This parameter is what determines the heat conductivity of ALUPOR™.

Heat conductivity of ALUPOR™ is defined as:

λ_{A P} = S_{m i n} \cdot λ_{M e}

Indeed for pure Al we have λ_AP = 0.21 × 238 = 49.98 W/(m×K) , which in accordance with experimental value.

Heat transfer coefficient

For ALUPOR™ one can define two types of heat transfer coefficient: surface α_S and volumetric α_V which are related as:

α_{V} = \frac{α_{S} \cdot S_{у д}}{V}

where
α_V [W/(m³×K)] - volumetric heat transfer coefficient;
α_S [W/(m²×K)] - surface heat transfer coefficient;
S_sp [m²/m³] - specific surface of ALUPOR™;
V [m³] - volume of ALUPOR™ porous section. The process of heat transfer from the pore surface to the cooling medium is characterized by the surface heat transfer coefficient. However, its experimental determination often encounters difficulties due to the lack of data on the dimensions of the internal pore surface of porous bodies. Therefore, when processing experimental data, the volumetric heat transfer coefficient in porous bodies is usually used which can be determined by the formula:

α_{V} = \frac{Δ Q_{τ}}{Δ V (t_{s} - t_{f}) τ}

where
t_s [°C] - temperature of elementary solid volume ΔV;
t_f [°C] - temperature of coolant in that volume;
τ [c] - time, whicch passed while ΔQ_τ [J] have been transferred.
Mean value of volumetric heat transfer coefficient is usually determined experimentally as:

\bar{α_{V}} = \frac{j \cdot F_{f i l t} c_{p} (T_{i n} - T_{o u t})}{V Π Δ \bar{T}}

here
T_in [K] и T_out [K] is temperature values at inlet and outlet correspondingly

Δ \overline{T}

is mean temperature difference between solid and fluid;
Results of experimental measurement are usually treated in terms of dimensionless criteria:

N u_{p} = f (Π, R e_{p}, P r)

where Nusselt number in pores defined as

N u_{p} = \frac{α_{V} d_{p}^{2}}{λ}

and Reynolds number defined by velocity of fluid in pores by following equations:

R e_{p} = \frac{ρ_{f} v_{p} d_{p}}{μ_{f}}

Velocity of fluid in pores and filtration velocity has following relation:

v_{p} = \frac{v_{f i l t}}{Π}

For more details on hydraulics in porous media, please refer to the "Filtration" page.
No corresponding studies of the heat transfer coefficient have been conducted for ALUPOR™. However, since ALUPOR™ is an inverse replica of the sphere packing we can estimate it by known correlations for sintered metals, for instance:

N u_{p} = 0.26 П R e_{p}^{1.2} {\overline{l}}^{- 1}

In above formula we have

\overline{l} = l / l_{0}

as dimensionless thickness of porous layer assuming l₀=1 mm.

The correlation has been experimentally verified with variations in:

Reynolds numbers from 0.04 to 5.0
Porosity from 0.3 to 0.5
Sample thickness from 1.0 to 5.6 mm

The formula allows calculating the volume-averaged heat transfer coefficient in porous metals, which provides only an approximate representation of the local heat transfer coefficients in the porous medium. However, it can be stated that the values of the local heat transfer coefficient on the cooler inlet of porous metal piece exceed the values of the volume-averaged heat transfer coefficient, since even in 1 mm thick samples the temperatures of the gas and porous wall at the outlet are close. The best understanding of local heat transfer coefficients can be obtained from experiments with porous plates of small thickness. The absolute values of this coefficient in 1 mm thick plates are quite high, ranging from 10⁷ — 10⁸ W/(m³·°C).

Electrical Conductivity

It is possible to use resistive heating of ALUPOR™ with electric current. For that end we can use the least electrically and thermally conductive alloys for ALUPOR™ production, such as AMg10 or magnesium casting alloys.
The relationship between thermal conductivity and electrical conductivity in metals and alloys is determined by the Lorentz number.

\frac{λ_{M e}}{σ_{M e}} = L T

Considering porous metal we can use estimation for electrical Conductivity as we did for thermal conductivity:

σ_{A P} = S_{m i n} \cdot σ_{M e}

Lets go ahead and try to estimate Electrical Conductivity of AlMg10. For solid alloy it is 12.8×10⁶ Sm/m, for ALUPOR AlMg10 it is 2.6×10⁶ Sm/m. For comparison we want to give the value of ordinary resistive alloy: Ni80Cr20 - 1.13×10⁶ Sm/m. As we can see, the values are very close.

Heat transfer equations

For numerical simulation of heat transfer in porous media, two main approaches are used:

Equilibrium Method:
The system "Fluid - porous metal" is considered as a single entity. The classical heat transfer equation (Fourier-Kirchhoff) is solved for this system:

ρ_{e f f} \cdot c_{e f f_{}} \cdot (\frac{\partial T}{\partial τ} + ({\vec{υ}}_{filt} \cdot \vec{\nabla}) T) = d i v [λ_{e f f} \cdot g r a d T] + q_{v}

Here, heat capacity, thermal conductivity, and density are averaged effective characteristics for the “Coolant - porous metal” system. The equation is suitable for modeling at low flow velocities and high porosity. In our opinion, its application to AUPOR™ is practical for designing heat exchangers with phase change.
There are ready to go solutions to solve that equation. For instance: ANSYS Fluent, COMSOL Multyphisics etc.

Nonequilibrium method: For the metal matrix and for the coolant, two equations are formulated:

for the fluid — the Fourier-Kirchhoff equation;
for the metal matrix — the Fourier heat conduction equation.

Π \cdot c_{f} \cdot ρ_{f} \cdot (\frac{\partial T_{f}}{\partial τ} + \vec{v_{f i l t}} \cdot \vec{\nabla} T_{f}) = d i v [λ_{f, e f f} \cdot g r a d T_{f}] + q_{f} + α_{V} (T_{s} - T_{f})

(1 - Π) \cdot c_{M e} \cdot ρ_{M e} \cdot \frac{\partial T_{s}}{\partial τ} = d i v [λ_{A P} \cdot g r a d T_{s}] + q_{A P} - α_{V} (T_{s} - T_{f})

two equations are connected by volumetric heat transfer coefficient α_V. We suggest using this system of equations for numerical modeling in the development of heat exchangers products made of AUPOR™, that involves fluid flow. To apply these equations, it will be necessary to program a module for your computational fluid dynamics (CFD) package.

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Heat transfer

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