For the general porous insulation materials under the normal condition, the heat transfer flux is composed of four parts, according to the traditional heat transfer theory, that is, heat conduction by the solid matrix itself *λ*
_{
s
}, heat conduction by the filled phases (mainly gas, vapor or other fluent) in the cavities *λ*
_{g}, heat convection between solid phase and filled phases *λ*
_{
c
} and radiation equivalent thermal conduction among the phases *λ*
_{
y
}. Neglecting the coupling effect by all the phases, the effective thermal conductivity of the porous media can be express as:

$$ {\lambda}_{\mathrm{e}}=f\left({\lambda}_{\mathrm{s}},{\lambda}_{\mathrm{g}},{\lambda}_{\mathrm{c}},{\lambda}_{\mathrm{r}}\right) $$

(1)

However, for VIPs, the air pressure in the cavities of the core materials is very small, nearly zero. So the fluent phase is always gas, under the normal temperature. The thermal convection between the solid wall and the filled phases can be ignored.

### The conductivity of solid matrix

The heat conduction for the nonmetal solid matrix can be achieved by the vibration of the lattice morphology. So the certain porous materials, the ETC. is usually regarded as the function of the mean temperature. In fact, the ETC. also has the relationship with the materials density, the micro structure, the characteristics, the porosity, and so on. The calculation formula is so very complex that it is nearly impossible to get the accuracy value. So the value is always provided by vendors or collected from experiment.

### The conductivity of rarefied gas

Free motion of gas molecules makes the heat transfer complex in the open cells of the core porous materials. According to the gas kinetic theory, we can acknowledge that, the smaller the cavity dimension, the harder the gas molecules move in it. For the VIPs core materials, the pore dimension scales are micron or even nanometer level. While the pore dimension is no larger than 100 μm and the air pressure is under 100 Pa, that is, the Reynolds is smaller than 1000 and Knudsen number is bigger than ten, the equivalent conductivity of gas can be ignored, *λ*
_{
c
} ≈ 0.

Based on the rarefied gas heat transfer theory, the ETC. of the rarefied gas embraced in the cavities is given by (Kwon et al. 2009; Kan et al. 2013):

$$ {\lambda}_g=\frac{\lambda_0}{1+2\beta {K}_n} $$

(2)

Where: *λ*
_{0} is the ETC. of static gas under the normal temperature and normal air pressure condition, W/(m · K), here *λ*
_{0} = 0.0230 W/(m · K);*β* is a constant number, which is used to express the grade the gas molecules collide the solid walls, and its value is always between 1.5 and 2, depending on the gas type, core material characteristics and mean temperature, here *β* = 1.5; *K*
_{
n
}, Knudsen number, is the ratio of the gas molecule mean free path *l*
_{
m
} and the mean diameter of the pore *δ*, that is *K*
_{
n
} = *l*
_{
m
}/*δ*; and *l*
_{
m
} is determined by (Kwon et al. 2009):

$$ {l}_m=\frac{K_BT}{\sqrt{2}\pi {d}_g^2{p}_g} $$

(3)

Where *T* is the mean temperature of the porous material in thermodynamic scale, K; *d*
_{
g
} is the diameter of the gas molecule, m, here, for air, *d*
_{
g
} = 3.72 × 10^{−10}
*m*;*K*
_{
B
} is the Boltzmann constant, and *K*
_{
B
} = 1.38 × 10^{−23}
*m* J/K;*P*
_{
g
} is the rarefied air pressure, Pa.

Combined Eqs. (2) and (3), the ECT of rarefied gas embraced in pore can be obtained as the follow:

$$ {\lambda}_g=\frac{\lambda_0}{1+\frac{\sqrt{2}\beta {K}_BT}{\pi {d}_g^2{P}_g\delta }} $$

(4)

### The equivalent thermal conductivity of radiation

Thermal radiation, in the form of electromagnetic wave for energy transfer, can occur without any medium, even in the vacuum case. And the equivalent thermal conductivity of radiation can be calculated by the Eq. (5) (Mendes et al. 2013):

$$ {\lambda}_r=4{l}_c\sigma \left({T}_1+{T}_2\right)\left({T}_1^2+{T}_2^2\right)/3\phi $$

(5)

Where, *l*
_{
c
}is the thickness of the plate core materials, m; *φ* is the attenuation coefficient for porous media, and here *φ* = 445 m^{−1};*σ* is Steve Boltzmann constant, and *σ =* 5.6697 × 10^{−8} W/(K^{4} · m^{2}); *T*
_{
1
}
*,T*
_{
2
} are respectively both sides of the VIPs temperature in thermodynamic scale, K.

### The ETC. of VIPs

To calculate the ETC. of the VIPs, the explicit mathematical model is derived from Eq. (1) to yield the following Eq. (6):

$$ {\lambda}_e=\left(1-\xi \right){\lambda}_s+\xi {\lambda}_g+{\lambda}_r $$

(6)

Where, *ξ* is the porosity of the core materials.

Combined with Eqs. (4), (5) and (6), the ETC. of the VIPs can be expressed as the following:

$$ {\lambda}_{\mathrm{e}}=\left(1-\xi \right){\lambda}_{\mathrm{s}}+\xi \frac{\lambda_0}{1+\frac{\sqrt{2}\beta {K}_BT}{\pi {d}_g^2{P}_g\delta }}+\frac{4{l}_c\sigma \left({T}_1+{T}_2\right)\left({T}_1^2+{T}_2^2\right)}{3\phi } $$

(7)