Discussion: “A Numerical Study of Thermal Dispersion in Porous Media” and “Numerical Determination of Thermal Dispersion Coefficients Using a Periodic Porous Structure”

Abstract

In the two papers 12, Kuwahara et al. investigated thermal dispersion in convective flow in porous media by using a two-dimensional periodic model of porous structure. In their study, a macroscopically uniform flow is assumed to pass through a collection of square rods placed regularly in an infinite space, where a macroscopic temperature gradient is imposed perpendicularly and in parallel to the macroscopic flow direction, respectively. Due to the periodicity of the model, only one structure unit is taken for a calculation domain to resolve an entire domain of porous medium. Exhaustive numerical computations were conducted for thermal dispersion conductivity. The transverse thermal dispersion conductivity results were correlated as the following expression 1: (26a)kdisyykf=0.022 PeD1.71−ε1/4forPeD<10and the longitudinal thermal dispersion conductivity results were correlated as 2(16a)kdisxxkf=0.022 PeD21−εforPeD<10(16b)kdisxxkf=2.7 PeDε1/2forPeD>10where PeD=Pe1−ε1/2. Equation (26a) was also summarized together with their longitudinal thermal dispersion conductivity results, Eq. (16), in their later paper 2. The parameters used in this discussion are defined as the same as those applied by Ref. 12; see also the Nomenclature in this discussion. However, correlations (26a), (16a) and (16b) are questionable. Here is my detailed analysis. According to the definition on the Peclet number given by Refs. 12, the Peclet number Pe should be expressed as (1)Pe=ρCpfumH/kfHowever, the generally applied definition on the Peclet number is 34567(2)Peg=ρCpfumD/kfIf the generally applied definition on the Peclet number, Eq. (D2), is applied to Eqs. (26a), (16a) and (16b), Eqs. (26a), (16a) and (16b) will present the unreasonable results due to the structural unit applied by Kuwahara et al. 12. The structural unit applied by Kuwahara et al. 12 in their numerical experiments is shown in Fig. 1. According to Fig. 1, the total volume of the structural unit is (3)Vt=H2The pore volume of the unit is (4)Vp=H2−D2So, the porosity of the unit can be expressed as (5)ε=Vp/Vt=1−D2/H2From Eq. (D5), we have (6)D/H=1−ε1/2or (7)H=D/1−ε1/2Equation (D7) indicates that H is a function of porosity. Inserting Eq. (D7) into Eq. (D1) yields (8)Pe=ρCpfumD/[kf1−ε1/2]Due to the generally applied definition for the Peclet number Eq. (D2), Eq. (D8) can be written as (9)Pe=Peg/1−ε1/2Equation (D9) establishes the relation between the Peclet number, Eq. (D1), defined by Refs. 12 and the generally applied Peclet number, Eq. (D2) 34567. According to PeD=Pe1−ε1/2 (defined by Refs. 12) and Pe=Peg/1−ε1/2 (i.e. Eq. (D9)), we obtain (10)PeD=PegInserting Eq. (D10) into Eqs. (26a), (16a) and (16b) results in (11)kdisyykf=0.022 Peg1.71−ε1/4forPeD<10(12a)kdisxxkf=0.022 Peg21−εforPeD<10(12b)kdisxxkf=2.7 Pegε1/2forPeD>10where Peg=ρCpfumD/kf is the generally defined definition for the Peclet number given by Eq. (D2). From the above analysis, it is seen that the equations (D11), (D12a) and (D12b) are equivalent to Eqs. (26a) and (16a) and (16b). Equations (D11) and (D12) are transverse and longitudinal thermal dispersion conductivities, respectively, which are expressed in terms of porosity and generally defined Peclet number Peg. However, Eqs. (D11) and (D12) present unreasonable results. For example, the thermal dispersion conductivities given by Eqs. (D11) and (D12a) increase with the increase of porosity. If porosity is 1, Eqs. (D11) and (D12a) present the infinite values for transverse and longitudinal thermal dispersion conductivities, while Eq. (D12b) presents nonzero longitudinal thermal dispersion conductivity. These are inconsistent with physical situation because the thermal dispersion should be reduced with the increase of porosity, and when porosity is 1, there is no dispersion and thermal dispersion conductivity should be zero. In conclusion, it seems to me that the transverse and longitudinal thermal dispersion conductivity correlations, Eq. (26a), Eq. (16a) and (16b) given by Refs. 12, are questionable because they present unreasonable results. This work was supported by the National Natural Science Foundation of China through Grant No. 10272052. D= size of square rod or size of particle H= size of structural unit k= thermal conductivity Pe= Peclet number based on H and the macroscopically uniform velocity defined by Refs. 12Peg= generally defined Peclet number given by Eq. (D2) um= the macroscopically uniform velocity V= volume ε = porosity Subscriptsdis= dispersion f= fluid xx= longitudinal direction yy= transverse direction g= generally applied t= total p= pore