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A cell model of effective thermal conductivity for saturated porous media
International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
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A cell model of effective thermal conductivity for saturated porous media Jianting Zhu Department of Civil and Architectural Engineering, University of Wyoming, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 14 February 2019 Received in revised form 11 April 2019 Accepted 27 April 2019 Available online 30 April 2019 Keywords: Effective thermal conductivity Porous media Cell model Thermal conductivity ratio Porosity
a b s t r a c t Effective thermal conductivity of porous materials is of interest in many heat transfer applications. In this study, a new cell model is developed to estimate the effective thermal conductivity of a porous medium consisting of solid particles in a continuous fluid phase. Each solid particle is conceptualized to be enveloped by a spherical fluid cell, which represents the interactions among the solid particles and the fluid phase. The heat conduction equations in both the solid particle and the fluid are separately solved and the effective thermal conductivity is then determined based on the developed temperature profile in the cell. I compare the new model with six experimental data sets as well as four published analytical models for the effective thermal conductivity in the literature. In addition, the developed model is also compared with the results from a detailed pore-scale numerical simulation based on multiple random realizations of porous medium configurations. The developed effective thermal conductivity model compares well with the experimental data and pore scale simulations from random realizations of porous media in the literature. At the two limiting cases of the porosity spectrum, the developed cell model of porous medium effective thermal conductivity approaches that from the Maxwell classical model. When the solid is more conductive than the fluid, over-simplifying the phase interaction underpredicts the effective thermal conductivity. On the other hand, over-simplifying the phase interaction significantly over-predicts the effective thermal conductivity if the fluid phase more conductive. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Thermal conductivity is a critical thermophysical parameter in characterizing the heat transfer process in porous media in many applications in science and engineering, where both fluid and solid phases exist [1–3]. For applications in chemical and petroleum engineering, determining the effective thermal conductivity of a porous medium is of interest to a wide range of applications. Accurate determination of the temperature field such as in a packed bed reactor [4,5] or in a nonisothermal catalyst pellet [6] requires the knowledge of the porous medium effective conductivity. Thermal methods of oil recovery and shale oil operations represent problems for which knowledge of effective thermal conductivities is essential [7,8]. For applications in geoscience and engineering, the thermal conductivity of porous materials is required to determine heat flow in sedimentary basins [9], to simulate heat transport and hydrocarbon formation processes [10], to obtain mechanical properties of sediments [11], and to provide important clues about the nature of the earth materials [12].
E-mail address: [email protected] https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.134 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
In two-phase saturated porous media, the arithmetic and harmonic means were often used as the upper and lower limits of overall effective thermal conductivity [13]. Gonzo [14] proposed two correlation equations for the effective thermal conductivity of porous media with medium and high dense solid phase dispersions and a wide range of phase conductivity ratio values. Wang et al. [15] developed a unifying equation for five fundamental effective thermal conductivity structural models using simple combinatory rules based on structure volume fractions. Gong et al. [16] derived a simple algebraic expression for the thermal conductivity of porous materials that can unify five basic structural models and evaluated the feasibility of the model using the experimental data. An appropriate effective thermal conductivity should be developed so it can be used to approximately represent macroscopic heat conduction behaviors in the porous media [17]. There have been numerous recent studies of effective thermal conductivity models based on various approaches. For example, an effective thermal conductivity was determined by the thermalelectrical analogy theory [18]. Miao et al. [19] developed an analytical solution of axial effective thermal conductivities for saturated dualporosity media, based on the fractal characteristics of pores and
J. Zhu / International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
fractures in the porous media. More recently, Qin et al. [20] proposed an effective thermal conductivity model in porous media with various liquid saturations based on the assumption that the grain size distribution follows the fractal scaling law. The effects of porosity, liquid saturation, and fractal dimensions were evaluated. Deprez et al. [21] proposed a percolation-based effectivemedium approximation (P-EMA) approach to estimate the effective thermal conductivity, permeability and Young’s modulus of a porous medium. The P-EMA approach has been recently extended to determine the effective thermal conductivity of porous materials both in the presence and absence of a liquid phase [22,23]. They compared the P-EMA theory with experimental data of saturationdependent thermal conductivity of porous materials and found good agreement. Ranut [24] presented a review of published theoretical models and empirical correlations and revised mathematical formulation of some of the existing correlations for the prediction of the effective thermal conductivity. Several previous studies started with the physical mechanisms of heat conduction as described by the Laplace’s equation to quantify the temperature distribution around a single particle within a uniform medium and then developed the effective thermal conductivity model for the porous materials. To solve the Laplace’s equation, these studies typically adopted the boundary conditions in the fluid phase at locations far away from the particle surface (i.e., at a radial distance r >> R, where R is the radius of the solid particle) [16,20,25]. While this approach could significantly simplify the solution, it over-simplifies the interaction between solid phase and fluid phase with a net effect of exaggerating the contribution of fluid phase to the effective thermal conductivity. Therefore, a new conceptual model is proposed in this study to relate the effective thermal conductivity to the thermal conductivities of fluid and solid phases of a saturated porous medium. The model is similar to the cell mode to describe fluid flow in a porous medium, in which each solid particle is assumed to be enveloped by a spherical fluid cell, which represents the interactions among the solid particles and fluid. The radius of this hypothetical surface is determined so that the ratio of fluid volume over the total volume is equal to the porosity of the porous medium. To describe fluid flow and the drag characteristics of the porous media, two types of cell model have been developed, namely, the free surface cell model [26] and the zero vorticity cell model [27]. These two models differ only in the boundary conditions at the cell boundary. Happel [26] proposed the cell boundary to be frictionless (zero shear stress) thereby emphasizing the non-interacting nature of cells. Kuwabara [27] suggested the use of the zero vorticity condition at the cell boundary. The cell model has been successfully used to describe flow and resistance of fluids past clusters of particles, bubbles, and drops in Newtonian fluids and in a wide variety of non-Newtonian fluids [28 –31]. The main objective of this study is to determine the effective thermal conductivity of granular porous materials. The model is simple enough for practical applications but should be theoretically justified. The model is built on the similar concept of cell model used to calculate flow and drag in porous media as discussed earlier but is appropriate for heat conduction. In particular, the outer surface of the fluid cell is under the boundary condition with the temperature varying linearly in the direction of heat conduction so that the cell undergoes the applied macroscopic temperature gradient to determine the effective thermal conductivity of the porous medium. The heat conduction equations are separately solved in both the solid particle and the fluid. The effective thermal conductivity model is then developed by using the boundary condition that follows the applied macroscopic temperature gradient on the outer fluid envelope of the cell. The proposed model is compared with published experimental data and with the results from a detailed pore-scale numerical simulation. The
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effects of porous medium porosity and the thermal conductivity ratio of solid phase over fluid phase on the effective thermal conductivity of porous media are discussed. Quantitative analysis is then followed to examine and discuss the differences between the proposed effective thermal conductivity and several published models. 2. Methods In the cell model proposed in this study, one solid particle with the mean diameter of the porous medium particles is enveloped by a spherical fluid cell, which represents the interactions among the solid particles and fluid. The overall heat conduction behavior of the porous medium is assumed to be equivalent to that of this particle-fluid cell. One fundamental requirement of the cell model is that the radius of this hypothetical fluid surface is determined so that the ratio of fluid volume over the total volume is equal to the porosity of the porous medium. The radius of solid particle, R, inside the cell represents the mean radius of the particle assemblage in the porous medium (Fig. 1). The radius of the outer fluid envelope is R1, which is related to the porosity of the porous medium / so that the cell and the porous medium have the same porosity:
ðR1 =RÞ3 ¼ ð1 /Þ1
ð1Þ
The spherical cell is placed in the temperature field with a given temperature gradient of rT ¼ T 22RT1 1 , as shown schematically in Fig. 1. The outer surface of the cell (i.e., r = R1 in Fig. 1) is under the boundary condition with the temperature varying linearly in the direction of z so that the overall temperature gradient is rT. Under steady state conditions, the temperature distribution of a single particle placed within a uniform medium can be described by the Laplace’s equation in a spherical coordinate system as [25]:
1 @ 1 @ @T 1 @2T 2 @T ¼0 r þ sinh þ 2 2 r2 @r @r r2 sinh @h @h r 2 sin h @w
ð2Þ
Assuming asymmetry about the z-axis, the temperature distribution is independent of w and the governing equation reduces to:
Fig. 1. Schematic diagram of the cell model of heat conduction to represent interaction among particles and fluid.
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1 @ 1 @ @T 2 @T r þ sinh ¼0 r 2 @r @r r 2 sinh @h @h
ð3Þ
A general solution to Eq. (3) for the temperature distributions of both solid (Ts) and fluid (Tf) phases can be obtained as [20,25]:
T s ¼ As þ
Bs Ds þ C s rcosh þ 2 cosh r r
ð4Þ
Bf Df T f ¼ Af þ þ C f rcosh þ 2 cosh r r
ð5Þ
where the constants As, Bs, Cs, Ds, Af, Bf, Cf, and Df are to be determined by using the following boundary conditions: (1) at the center of particle, r = 0, Ts – 1, s ks @T @r
@T kf @rf
@T s @h
@T f , @h
¼ and ¼ (2) at the surface of particle, r = R, (3) at the outer spherical surface of fluid envelope where r = R1,
# ( " ) 3 ks =kf 1 1 R1 R1 Q ¼ Q s þ Q f ¼ 4pkf C f R 1 þ2 ln 3 R R ks =kf þ 2 3
ð6Þ
4 pks C s R3 3
ð17Þ
Using the effective medium idea, the effective thermal conductivity can be determined so that the effective medium would conduct the same total volumetric heat flux as the cell:
4 Q ¼ ke V f þ V s rT ¼ ke pR31 rT 3
ð18Þ
From Eq. (17) and Eq. (18), the effective thermal conductivity of the porous medium can be determined as:
ke ¼
T f ðR1 ; hÞ ¼ T 1 þ rT ðR1 þ z1 Þ ¼ T 1 þ rT ðR1 þ R1 coshÞ ¼ T 1 þ rTR1 þ rTR1 cosh
The total volumetric heat flux in the cell is the sum of Qs and Qf:
3kf C f R3
n h 3 R 1 1
3
R
o i k =k 1 1 þ 2 kss =kf þ2 ln RR1 þ ks C s R3 f
R31 rT
ð19Þ
It can be further simplified and re-arranged as:
where rT is the overall temperature gradient. The boundary condition expressed in Eq. (6) is the main point in this study that differs from most previous studies. Using the specified boundary conditions, the temperature distributions can be obtained for the fluid and solid phases, respectively:
ke ¼ kf
T s ðr; hÞ ¼ As þ C s rcosh
ð7Þ
Substituting the relation between R1/R and the porosity / in Eq. (1) into Eq. (20), one has:
ð8Þ
ke 3ks =kf 2 ks =kf 1 ½/ þ ð1 /Þlnð1 /Þ ¼ kf 3 þ / ks =kf 1
ð9Þ
Eq. (21) is the dimensionless effective thermal conductivity of a porous medium based on the conceptual cell model. In the limiting case when / ? 1, it can be shown that:
" T f ðr; hÞ ¼ Af þ C f rcosh 1 þ
kf ks ks þ 2kf
3 # R r
where the four involved constants As, Cs, Af, and Cf are:
As ¼ T 1 þ rTR1 Cs ¼
3rT 3 ks =kf þ 2 þ 1 ks =kf RR1
Af ¼ As ¼ T 1 þ rTR1
ð11Þ
Cf ¼
ks =kf þ 2 rT 3 ks =kf þ 2 þ 1 ks =kf RR1
ð10Þ
ð12Þ
@T s ¼ Cs @z " 3 # @T f ks =kf 1 R ¼ Cf 1 þ 2 r @z ks =kf þ 2
Z
Vs
@T s 4 4pr 2 dr ¼ pks C s R3 3 @z
ð22Þ
which is the same as the classical Maxwell model [32]. The relative difference between the cell model proposed in this study and the Maxwell model [32] can be defined as:
ke =kf
Maxwell
ke =kf cell
ke =kf
ð23Þ
cell
It can be shown that:
2ð1 /Þlnð1 /Þ ks =kf 1 3ks =kf 2/ ks =kf 1 2ð1 /Þlnð1 /Þ ks =kf 1
ð13Þ
ð14Þ
From Eq. (24), the porosity /+ at which the maximum difference of effective thermal conductivity between the two models occurs can be determined from the following equation:
ð15Þ
Similarly, the volumetric heat flux from the fluid is:
Z
@T f 4pr 2 dr @z Vf # ( " ) 3 ks =kf 1 R1 R1 3 1 1 þ2 ¼ 4pkf C f R ln 3 R R ks =kf þ 2
3ks =kf 2/ ks =kf 1 ke ! kf 3 þ / ks =kf 1
ð21Þ
e¼
After determining the temperature gradients, the volumetric heat flux in the solid phase can be determined as follows:
Q s ¼ ks
ð20Þ
e¼
From the developed temperature distributions in both the solid and fluid phases, the temperature gradients in these two phases in the z direction can be obtained, respectively:
i 3 n h 3 o R ks =kf þ 2 RR1 1 þ 6 ks =kf 1 ln RR1 þ 3 ks =kf R1 3 ks =kf þ 2 ks =kf 1 RR1
Q f ¼ kf
ð16Þ
3ks =kf þ ks =kf þ 2 ln 1 /þ 2/þ ks =kf 1 ¼ 0
ð24Þ
ð25Þ
For the limiting case when ks/kf ? 1, /+ can be determined as 0.768. In the following section, the developed cell model is compared with 6 published experimental data sets with discussion of errors along with two classical models. The model is then compared with the effective thermal conductivity results from a pore-scale numerical simulation based on multiple random realizations of porous medium structure. I also quantitatively analyze and discuss the differences between the proposed effective thermal conductivity and the classical Maxwell model, expressed in Eq. (22).
J. Zhu / International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
3. Results and discussion The developed effective thermal conductivity model is first compared with 4 published analytical models based on several different concepts. In the model developed by Ashby [33], the porous materials were cellular solids. The following simple effective thermal conductivity model expressed in Eq. (26) was then developed based on the estimate that one-third of the solid struts lie parallel to each axis.
i ke ks h ¼ ð1 /Þ þ 2ð1 /Þ2=3 þ / kf 3kf
ð26Þ
Revil [34] presented a theoretical model for the thermal conductivity of unconsolidated granular sediments by using a differential effective medium approach based on the similarity between the boundary problems for electrical conductivity and thermal conductivity in porous media.
2 0 13 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m 1m k m m ks ke k k k 4/ s s s s 1m 4/1m þ 0:5 1 @1 þ A5 ¼/ 1 þ kf kf kf kf kf kf ð27Þ where the additional parameter m is the electrical cementation exponent to account for the effect of the topology of the interconnected pore space. For spherical particles, m = 1.5. Krupiczka [35] developed the following effective thermal conductivity based on the correlation with experimental data:
0:280:757log/0:057log ðks =kf Þ ke ks ¼ kf kf
ð28Þ
Note that Eq. (28) diverges when / ? 0 and therefore is not applicable for / < 0.215 as suggested by the model author. The classical Maxwell model [32] relates the effective thermal conductivity to the porosity and thermal conductivity ratio as expressed in Eq. (22): With the exception of Revil [34], all other analytical models used for comparison only need ks/kf and / as the required input. The comparison of the dimensionless effective thermal conductivity of porous medium between the proposed model in this study and 4 published models is shown in Fig. 2. In Fig. 2(a), the compar-
Fig. 2. Comparison of the effective thermal conductivity from this study with that from 4 selected models in the literature when (a) the porosity / = 0.5, and (b) ks/ kf = 100.
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ison is shown as a function of the ratio ks/kf when / = 0.5, while Fig. 2(b) shows the relationship with the porosity / when ks/kf = 100. Since the dimensionless effective thermal conductivity is normalized by the fluid phase thermal conductivity, it shows an increasing trend with the increase of the thermal conductivity ratio of the solid phase over fluid phase (i.e., ks/kf) by all the models. However, the extent of the dimensionless effective thermal conductivity increase varies significantly among the models. On the other hand, all the models correctly capture the trend that the effective thermal conductivity decreases with the porosity. Since ks is higher than kf, more fluid presence should decrease the overall effective thermal conductivity. The proposed cell-based model in this study is generally in between two classical models of Maxwell [32] and Krupiczka [35] in the high ks/kf range. It is between the Maxwell model and the recently developed model by Ashby [33], although the values from the Ashby model are significantly higher. Among the three models (i.e., Maxwell, Krupiczka, and Revil) that are close to the present model, I will quantitatively compare the results with some of the available experimental results along with the two classical models of Maxwell [32] and Krupiczka [35] in the following since these two models have the same parameter parsimony requiring only two input parameters (i.e., the thermal conductivity ratio and porosity). The model of Revil [34] involves an additional parameter, m. Fig. 3 presents a comparison between the dimensionless effective thermal conductivity from the proposed model and that from 4 experimental data sets through measurements of the temperature gradient in the literature [36 –38] as a function of porosity /. For the experimental data sets, the porous medium was Quartz sand saturated with brine. The thermal conductivity of brine was kf = 0.59 W m1 °C1., while that of Quartz sand was estimated as ks = 7.2 W m1 °C1. The results of Maxwell [32] and Krupiczka [35] are also shown for quantitative error analysis of different models. The proposed model agrees well with the published experimental data. It can be seen that the Krupiczka model [35] diverges quickly when / becomes small. In the higher / range (/ > 0.215), it is very close to the classical Maxwell model [32], which also underestimates the effective thermal conductivity. Further quantitative analysis of errors for the three presented models illustrates that the root mean square errors (RMSEs) of the dimensionless effective thermal conductivity ke/kf are 1.08, 11.93, and 1.50 for the proposed model, Krupiczka model and Maxwell model, respectively.
Fig. 3. Comparison of the effective thermal conductivity prediction from this study with experimental data from the literature in relation to the porosity /. The model results from Maxwell [32] and Krupiczka [35] are also shown for quantitative analysis. For the experimental data sets, the porous medium was Quartz sands saturated with brine. The thermal conductivity of brine was kf = 0.59 W m1 °C1, while that of Quartz sand was estimated as ks = 7.2 W m1 °C1.
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The proposed effective thermal conductivity model has the smallest errors among the three models. On the other hand, the average errors of ke/kf are 0.367, 1.624, and 1.150 for the proposed cell model, Krupiczka model and Maxwell model, respectively. Overall, the Krupiczka model overpredicts the effective thermal conductivity, while the present model and the Maxwell model under-predict it, evident from their positive average error and negative average error respectively. The present model’s under-prediction of the effective thermal conductivity occurs in the small porosity range, while the classical Maxwell model under-predicts it in the entire experimental porosity range. The severe over-prediction from the Krupiczka model in the small porosity range is not surprising given that this model was developed based on the correlation in the porosity range above 0.215. In Fig. 4, I include an additional comparison with 2 more experimental data sets from Somerton [37] with two different values of the thermal conductivity ratios. In this case, the experimental data of dry oil sands and oil sands saturated with brine were used. For the dry samples, the ratio of ks/kf was 80.9 with ks = 4.45 W m1 °C1 and air kf = 0.055 W m1 °C1. For the brine saturated samples, the ratio of ks/kf was 7.5 with ks = 4.45 W m1 °C1 and brine kf = 0.590 W m1 °C1. It can be seen that the present model predicts the decreasing trend of ke/kf reasonably well with slight under-prediction for the brine saturated samples and over-prediction for the dry samples. Given that there are only two parameters involved in developing the conceptually simple model in this study and the significant scatters in the experimental data of effective thermal conductivity, the developed model captures the overall trend very well. The good agreement illustrates that the model is able to capture the main processes of heat conduction in porous media better than the classical models. Fig. 5 shows a comparison with the pore-scale numerical simulation results from the computational algorithm developed by Askari et al. [39] by which the thermal conduction behavior and the effective thermal conductivity was studied in pore scales. In the study of Askari et al. [39], the granular porous media were re-constructed based on the grain size distribution at different porosities. By solving the steady state heat conduction equation, the temperature distribution, heat transfer rate, and effective thermal conductivity were estimated. For the comparison results in Fig. 5, the thermal conductivity of solid was 7.4 W m1 °C1 and
Fig. 4. Comparison of the effective thermal conductivity prediction from this study with experimental data from Somerton [37] in relation to the porosity / at two different values of the thermal conductivity ratio, ks/kf. For the dry samples, the ratio of ks/kf was 80.9. For the brine saturated samples, the ratio of ks/kf was 7.5.
Fig. 5. Comparison of the effective thermal conductivity prediction from this study with pore scale numerical simulation results by Askari et al. [39] in relation to the porosity /. The thermal conductivity of solid was 7.4 W m1 °C1 and the pores were filled by air with the thermal conductivity of 0.0262 W m1 °C1. The grain diameter was 250 lm with a standard deviation of 30 lm.
the pores were filled by air with the thermal conductivity of 0.0262 W m1 °C1. The mean grain diameter was 250 lm with a standard deviation of 30 lm. The porous media were reconstructed for more than 500 random realizations with porosities ranging from 0.18 to 0.50. Fig. 5 plots the simulated effective thermal conductivities from Askari et al. [39] for various random realizations along with those predicted with the present model and the classical model of Maxwell [32]. It can be seen that while the effective thermal conductivity results from the random realizations show significant scatters, the present model can well capture the overall decreasing trend and is better than the Maxwell model. Similar to the comparison with the experimental observations, the developed simple model slightly under-estimates the effective thermal conductivity at the low porosity range and over-predicts it at the high porosity range. In Fig. 6, I show the dimensionless effective thermal conductivity as a function of the porosity when the solid phase is more conductive than the fluid phase (i.e., ks/kf > 1). In the same figure on the secondary axis, I also plot the percent difference of effective thermal conductivity predictions based on the cell model from this study and that based on the classical Maxwell model [32], calcu-
Fig. 6. The dimensionless effective thermal conductivity and the percent difference of the effective thermal conductivity between the cell model from this study and that from Maxwell [32] when ks/kf > 1.
J. Zhu / International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
lated from Eq. (24). Since the solid particles are more conductive than the fluid, the resulting effective thermal conductivity of the porous medium is always larger than that of the fluid phase, as indicated by the result that ke/kf is always larger than 1. As the porosity increases, which means that the fluid phase becomes more dominant, the effective thermal conductivity of the porous medium also approaches to that of the fluid (i.e., ke/kf approaches 1). This is true for both models. At the two limiting cases of all fluid (i.e., / = 1) and all solid (i.e., / = 0) phases, both the proposed cell model and the Maxwell model correctly predict the fluid phase thermal conductivity and soil phase thermal conductivity, respectively. As a result, the percent difference from these two models approaches zero when / ? 1 and / ? 0, seen from Fig. 6. In the porosity range between these two limiting cases, the percent difference between the two models first increases and then decreases with the porosity. At a certain point of porosity, the percent difference between the two models reaches a peak of either positive or negative maximum difference. It can be seen from Fig. 6 that the Maxwell model underestimates the effective thermal conductivity since it overestimates the effect of fluid phase. The larger the contrast of the thermal conductivities of the two phases, the larger the percent difference between the two models. For example, when ks/ kf = 100, the Maxwell model under-estimates the effective thermal conductivity by more than 30%. Similar to Fig. 6, the dimensionless effective thermal conductivity of the porous medium as a function of the porosity when the solid phase is less conductive than the fluid phase (i.e., ks/kf < 1) is shown in Fig. 7. The percent difference of effective thermal conductivity based on the cell model from this study and that based on the classical Maxwell model [32] is also plotted in Fig. 7. Since the solid particles are less conductive than the fluid now, the resulting effective thermal conductivity of the porous medium is always smaller than that of the fluid phase. As the porosity increases, the effective thermal conductivity of the porous medium also increases to that of the fluid phase (i.e., ke/kf approaches 1). The percent difference between the two models first increases and then decreases with the porosity. In this case, the Maxwell model significantly over-estimates the effective thermal conductivity since it once again exaggerates the effect of fluid phase. The larger the contrast of the thermal conductivities of the two phases, the larger the percent difference between the two models. For example, when ks/ kf = 0.01, the Maxwell model over-estimates the effective thermal conductivity by more than 500%.
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Fig. 8. The porosity (/+) at which the maximum difference of effective thermal conductivity between the proposed cell-based model and that of Maxwell model [32] occurs and the corresponding maximum difference (e (%)) as a function of the thermal conductivity ratio of solid over fluid phases.
The porosity at which the maximum percent difference of effective thermal conductivity between the proposed cell model the Maxwell model [32] occurs, which is determined by solving Eq. (25), is shown in Fig. 8 as the solid line along with the actual percent difference between the two models as the dash line. It can be seen that the porosity where the maximum difference occurs increases significantly with the increase of ks/kf, with an asymptotic porosity of 0.768 with ks/kf ? 1. The porosity, /+, and the percent difference, e (%), are both affected by the ratio of solid phase thermal conductivity over fluid phase thermal conductivity. When ks/kf < 1, since the fluid phase is the more significant contributor to the overall effective thermal conductivity, simplifying the interaction between the two phases by applying the boundary condition at r = 1 would significantly exaggerate the contribution from the fluid phase. As a result, it introduces larger errors in the effective thermal conductivity prediction. Under the condition of ks/kf = 0.1, the over-estimation could be as high as 150%. On the other hand, if the solid phase is more conductive than the fluid phase, exaggerating the effect of fluid phase could lead to underestimation of the effective thermal conductivity by over 30%. Note that the effect of particle size distribution is not explicitly addressed in the developed cell model. The basic assumption in the model development is that the cell with a particle surrounded by a fluid envelope with proper boundary conditions is representative of heat conduction characteristics of the porous materials. In addition, the convective heat transfer due to temperature gradient and fluid flow is not considered in the cell model. The combined effects of particle size distribution and convective heat transfer on the temperature distribution in the porous media based on the concept of the cell model deserve future studies.
4. Conclusions
Fig. 7. The dimensionless effective thermal conductivity and the percent difference of the effective thermal conductivity between the cell model from this study and that from Maxwell [32] when ks/kf < 1.
In this study, a new cell model is developed to predict the effective thermal conductivity of a porous medium consisting of solid particles in a continuous fluid phase. The solid particle is enveloped by a spherical fluid cell, which represents the overall interactions among the solid particles and fluid. The radius of this hypothetical fluid cell is uniquely determined so that the ratio of fluid volume over the total volume inside the cell is equal to the porosity of the porous medium. The heat conduction equations are separately solved in both the solid particle and the fluid and the effective thermal conductivity is determined by the
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temperature distribution. I compare the new model with 6 experimental data sets, 4 published models, as well as a pre-scale numerical simulation for the effective thermal conductivity in the literature. The main conclusions are summarized as follows. 1. The developed effective thermal conductivity model compares well with the experimental data in the literature, with the errors being smaller than previously developed models. 2. The developed model is able to capture the variations of the effective thermal conductivity with the porosity with slight under-estimation at the low porosity and over-estimation at the high porosity. 3. At both ends of the porosity spectrum, the developed cell model of effective thermal conductivity approaches that from the classical Maxwell model. 4. At certain porosity between 0 and 1, the difference between the cell model and Maxwell model reaches a maximum. The porosity at which the difference reaches the maximum strongly depends on the contrast of the thermal conductivities of the two phases. 5. When the solid phase is more conductive than the fluid phase, over-simplifying the solid particle interactions under-predicts the effective thermal conductivity. On the other hand, oversimplifying the solid particle interactions significantly overpredicts the effective thermal conductivity if the fluid phase is more conductive. Conflict of interest The author declares no conflict of interest. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.134. References [1] Z.G. Qu, T.S. Wang, W.Q. Tao, T.J. Lu, A theoretical octet-truss lattice unit cell model for effective thermal conductivity of consolidated porous materials saturated with fluid, Heat Mass Transf. 48 (8) (2012) 1385–1395, https://doi. org/10.1007/s00231-012-0985-y. [2] G.H. Tang, C. Bi, Y. Zhao, W.Q. Tao, Thermal transport in nano-porous insulation of aerogel: factors, models and outlook, Energy 90 (2015) 701– 721, https://doi.org/10.1016/j.energy.2015.07.109. [3] D. Niu, L. Guo, H.W. Hu, G.H. Tang, Dropwise condensation heat transfer model considering the liquid-solid interfacial thermal resistance, Int. J. Heat Mass Transf. 112 (2017) 333–342, https://doi.org/10.1016/j. ijheatmasstransfer.2017.04.061. [4] J.J. Carberry, Chemical and Catalytic Reaction Engineering,, Courier Corporation, North Chelmsford, Massachusetts, 2001, p. 642. [5] G.F. Froment, K.B. Bischoff, J. De Wilde, Chemical Reactor Analysis and Design, third ed., Edition, Wiley, New York, 2010, p. 900. [6] L. Lapidus, N.R. Amundson, Chemical Reactor Theory: A Review, Prentice-Hall, Englewood Cliffs, New Jersey, 1977, p. 856. [7] I. Nozad, R.G. Carbonell, S. Whitaker, Heat conduction in multiphase systems— I: theory and experiment for two-phase systems, Chem. Eng. Sci. 40 (5) (1985) 843–855, https://doi.org/10.1016/0009-2509(85)85037-5. [8] R.M. Butler, G.S. McNab, H.Y. Lo, Theoretical studies on the gravity drainage of heavy oil during in-situ steam heating, Can. J. Chem. Eng. 59 (1981) 455–460, https://doi.org/10.1002/cjce.5450590407. [9] R.S. Detrick, R.P. Von Herzen, B. Parsons, D. Sandwell, M. Dougherty, Heat flow observations on the Bermuda rise and thermal models of midplate swells, J. Geophys. Res. 91 (B3) (1986) 3701–3723, https://doi.org/10.1029/ JB091iB03p03701. [10] P.J. Ortoleva, Basin compartments and seals, AAPG Mem. 61 (1994) 3–26. [11] L. Gibiansky, S. Torquato, Rigorous connection between physical properties of porous rocks, J. Geophys. Res. 103 (B10) (1998) 23911–23923, https://doi.org/ 10.1029/98JB02340. [12] M.A. Presley, P.R. Christensen, Thermal conductivity measurements of particulate materials 1, A review, J. Geophys. Res. 102 (1997) 6535–6549, https://doi.org/10.1029/96JE03302.
[13] C. Clauser, Heat transport processes in the Earth’s crust, Surv. Geophys. 30 (3) (2009) 163–191, https://doi.org/10.1007/s10712-009-9058-2. [14] E.E. Gonzo, Estimating correlations for the effective thermal conductivity of granular materials, Chem. Eng. J. 90 (2002) 299–302, https://doi.org/10.1016/ S1385-8947(02)00121-3. [15] J. Wang, J.K. Carson, M.F. North, D.J. Cleland, A new approach to modelling the effective thermal conductivity of heterogeneous materials, Int. J. Heat Mass Transf. 49 (17–18) (2006) 3075–3083, https://doi.org/10.1016/j. ijheatmasstransfer.2006.02.007. [16] L. Gong, Y. Wang, X. Cheng, R. Zhang, H. Zhang, A novel effective medium theory for modelling the thermal conductivity of porous materials, Int. J. Heat Mass Transf. 68 (2014) 295–298, https://doi.org/10.1016/j. ijheatmasstransfer.2013.09.043. [17] G.K. Batchelor, Transport properties of two-phase materials with random structure, Annu. Rev. Fluid Mech. 6 (1) (1974) 227–255, https://doi.org/ 10.1146/annurev.fl.06.010174.001303. [18] H.-Q. Jin, X.-L. Yao, L.-W. Fan, X. Xu, Z.-T. Yu, Experimental determination and fractal modeling of the effective thermal conductivity of autoclaved aerated concrete: effects of moisture content, Int. J. Heat Mass Transf. 92 (2016) 589– 602, https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.103. [19] T. Miao, S. Cheng, A. Chen, B. Yu, Analysis of axial thermal conductivity of dualporosity fractal porous media with random fractures, Int. J. Heat Mass Transf. 102 (2016) 884–890, https://doi.org/10.1016/j. ijheatmasstransfer.2016.06.048. [20] X. Qin, J. Cai, P. Xu, S. Dai, Q. Gan, A fractal model of effective thermal conductivity for porous media with various liquid saturation, Int. J. Heat Mass Transf. 128 (2019) 1149–1156, https://doi.org/10.1016/j. ijheatmasstransfer.2018.09.072. [21] N. Deprez, D.S. McLachlan, I. Sigalas, The measurement and comparative analysis of the electrical and thermal conductivities, permeability and Young’s modulus of sintered nickel, Solid State Commun. 66 (1988) 869–872, https:// doi.org/10.1016/0038-1098(88)90403-6. [22] B. Ghanbarian, H. Daigle, Thermal conductivity in porous media: percolationbased effective-medium approximation, Water Resour. Res. 52 (1) (2016) 295– 314, https://doi.org/10.1002/2015WR017236. [23] M. Sadeghi, B. Ghanbarian, R. Horton, Derivation of an explicit form of the percolation-based effective-medium approximation for thermal conductivity of partially saturated soils, Water Resour. Res. 54 (2) (2018) 1389–1399, https://doi.org/10.1002/2017WR021714. [24] P. Ranut, On the effective thermal conductivity of aluminum metal foams: review and improvement of the available empirical and analytical models, Appl. Therm. Eng. 101 (2016) 496–524, https://doi.org/10.1016/j. applthermaleng.2015.09.094. [25] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. 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Zhu, Approximate solutions of non-Newtonian flows over a swarm of bubbles, Int. J. Multiph. Flow 30 (10) (2004) 1271–1278, https://doi.org/ 10.1016/j.ijmultiphaseflow.2004.06.007. [31] N. Kishore, R.P. Chhabra, V. Eswaran, Mass transfer from ensembles of Newtonian fluid spheres at moderate Reynolds and Peclet numbers, Chem. Eng. Res. Des. 85 (8) (2007) 1203–1214, https://doi.org/10.1205/cherd06250. [32] J.C. Maxwell, A Treatise on Electricity and Magnetism Vol. 1 (1873) 365. [33] M.F. Ashby, The properties of foams and lattices, Philosophical Transactions of the Royal Society A-Mathematical, Phys. Eng. Sci. 364 (2006) 15–30, https:// doi.org/10.1098/rsta.2005.1678. [34] A. Revil, Thermal conductivity of unconsolidated sediments with geophysical applications, J. Geophys. Res.-Solid Earth 105 (B7) (2000) 16749–16768, https://doi.org/10.1029/2000JB900043. [35] R. Krupiczka, Analysis of thermal conductivity in granular materials, Int. Chem. Eng. 7 (1) (1967) 122–144. [36] M. Zamora, D. 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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A cell model of effective thermal conductivity for saturated porous media Jianting Zhu Department of Civil and Architectural Engineering, University of Wyoming, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 14 February 2019 Received in revised form 11 April 2019 Accepted 27 April 2019 Available online 30 April 2019 Keywords: Effective thermal conductivity Porous media Cell model Thermal conductivity ratio Porosity
a b s t r a c t Effective thermal conductivity of porous materials is of interest in many heat transfer applications. In this study, a new cell model is developed to estimate the effective thermal conductivity of a porous medium consisting of solid particles in a continuous fluid phase. Each solid particle is conceptualized to be enveloped by a spherical fluid cell, which represents the interactions among the solid particles and the fluid phase. The heat conduction equations in both the solid particle and the fluid are separately solved and the effective thermal conductivity is then determined based on the developed temperature profile in the cell. I compare the new model with six experimental data sets as well as four published analytical models for the effective thermal conductivity in the literature. In addition, the developed model is also compared with the results from a detailed pore-scale numerical simulation based on multiple random realizations of porous medium configurations. The developed effective thermal conductivity model compares well with the experimental data and pore scale simulations from random realizations of porous media in the literature. At the two limiting cases of the porosity spectrum, the developed cell model of porous medium effective thermal conductivity approaches that from the Maxwell classical model. When the solid is more conductive than the fluid, over-simplifying the phase interaction underpredicts the effective thermal conductivity. On the other hand, over-simplifying the phase interaction significantly over-predicts the effective thermal conductivity if the fluid phase more conductive. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Thermal conductivity is a critical thermophysical parameter in characterizing the heat transfer process in porous media in many applications in science and engineering, where both fluid and solid phases exist [1–3]. For applications in chemical and petroleum engineering, determining the effective thermal conductivity of a porous medium is of interest to a wide range of applications. Accurate determination of the temperature field such as in a packed bed reactor [4,5] or in a nonisothermal catalyst pellet [6] requires the knowledge of the porous medium effective conductivity. Thermal methods of oil recovery and shale oil operations represent problems for which knowledge of effective thermal conductivities is essential [7,8]. For applications in geoscience and engineering, the thermal conductivity of porous materials is required to determine heat flow in sedimentary basins [9], to simulate heat transport and hydrocarbon formation processes [10], to obtain mechanical properties of sediments [11], and to provide important clues about the nature of the earth materials [12].
E-mail address: [email protected] https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.134 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
In two-phase saturated porous media, the arithmetic and harmonic means were often used as the upper and lower limits of overall effective thermal conductivity [13]. Gonzo [14] proposed two correlation equations for the effective thermal conductivity of porous media with medium and high dense solid phase dispersions and a wide range of phase conductivity ratio values. Wang et al. [15] developed a unifying equation for five fundamental effective thermal conductivity structural models using simple combinatory rules based on structure volume fractions. Gong et al. [16] derived a simple algebraic expression for the thermal conductivity of porous materials that can unify five basic structural models and evaluated the feasibility of the model using the experimental data. An appropriate effective thermal conductivity should be developed so it can be used to approximately represent macroscopic heat conduction behaviors in the porous media [17]. There have been numerous recent studies of effective thermal conductivity models based on various approaches. For example, an effective thermal conductivity was determined by the thermalelectrical analogy theory [18]. Miao et al. [19] developed an analytical solution of axial effective thermal conductivities for saturated dualporosity media, based on the fractal characteristics of pores and
J. Zhu / International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
fractures in the porous media. More recently, Qin et al. [20] proposed an effective thermal conductivity model in porous media with various liquid saturations based on the assumption that the grain size distribution follows the fractal scaling law. The effects of porosity, liquid saturation, and fractal dimensions were evaluated. Deprez et al. [21] proposed a percolation-based effectivemedium approximation (P-EMA) approach to estimate the effective thermal conductivity, permeability and Young’s modulus of a porous medium. The P-EMA approach has been recently extended to determine the effective thermal conductivity of porous materials both in the presence and absence of a liquid phase [22,23]. They compared the P-EMA theory with experimental data of saturationdependent thermal conductivity of porous materials and found good agreement. Ranut [24] presented a review of published theoretical models and empirical correlations and revised mathematical formulation of some of the existing correlations for the prediction of the effective thermal conductivity. Several previous studies started with the physical mechanisms of heat conduction as described by the Laplace’s equation to quantify the temperature distribution around a single particle within a uniform medium and then developed the effective thermal conductivity model for the porous materials. To solve the Laplace’s equation, these studies typically adopted the boundary conditions in the fluid phase at locations far away from the particle surface (i.e., at a radial distance r >> R, where R is the radius of the solid particle) [16,20,25]. While this approach could significantly simplify the solution, it over-simplifies the interaction between solid phase and fluid phase with a net effect of exaggerating the contribution of fluid phase to the effective thermal conductivity. Therefore, a new conceptual model is proposed in this study to relate the effective thermal conductivity to the thermal conductivities of fluid and solid phases of a saturated porous medium. The model is similar to the cell mode to describe fluid flow in a porous medium, in which each solid particle is assumed to be enveloped by a spherical fluid cell, which represents the interactions among the solid particles and fluid. The radius of this hypothetical surface is determined so that the ratio of fluid volume over the total volume is equal to the porosity of the porous medium. To describe fluid flow and the drag characteristics of the porous media, two types of cell model have been developed, namely, the free surface cell model [26] and the zero vorticity cell model [27]. These two models differ only in the boundary conditions at the cell boundary. Happel [26] proposed the cell boundary to be frictionless (zero shear stress) thereby emphasizing the non-interacting nature of cells. Kuwabara [27] suggested the use of the zero vorticity condition at the cell boundary. The cell model has been successfully used to describe flow and resistance of fluids past clusters of particles, bubbles, and drops in Newtonian fluids and in a wide variety of non-Newtonian fluids [28 –31]. The main objective of this study is to determine the effective thermal conductivity of granular porous materials. The model is simple enough for practical applications but should be theoretically justified. The model is built on the similar concept of cell model used to calculate flow and drag in porous media as discussed earlier but is appropriate for heat conduction. In particular, the outer surface of the fluid cell is under the boundary condition with the temperature varying linearly in the direction of heat conduction so that the cell undergoes the applied macroscopic temperature gradient to determine the effective thermal conductivity of the porous medium. The heat conduction equations are separately solved in both the solid particle and the fluid. The effective thermal conductivity model is then developed by using the boundary condition that follows the applied macroscopic temperature gradient on the outer fluid envelope of the cell. The proposed model is compared with published experimental data and with the results from a detailed pore-scale numerical simulation. The
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effects of porous medium porosity and the thermal conductivity ratio of solid phase over fluid phase on the effective thermal conductivity of porous media are discussed. Quantitative analysis is then followed to examine and discuss the differences between the proposed effective thermal conductivity and several published models. 2. Methods In the cell model proposed in this study, one solid particle with the mean diameter of the porous medium particles is enveloped by a spherical fluid cell, which represents the interactions among the solid particles and fluid. The overall heat conduction behavior of the porous medium is assumed to be equivalent to that of this particle-fluid cell. One fundamental requirement of the cell model is that the radius of this hypothetical fluid surface is determined so that the ratio of fluid volume over the total volume is equal to the porosity of the porous medium. The radius of solid particle, R, inside the cell represents the mean radius of the particle assemblage in the porous medium (Fig. 1). The radius of the outer fluid envelope is R1, which is related to the porosity of the porous medium / so that the cell and the porous medium have the same porosity:
ðR1 =RÞ3 ¼ ð1 /Þ1
ð1Þ
The spherical cell is placed in the temperature field with a given temperature gradient of rT ¼ T 22RT1 1 , as shown schematically in Fig. 1. The outer surface of the cell (i.e., r = R1 in Fig. 1) is under the boundary condition with the temperature varying linearly in the direction of z so that the overall temperature gradient is rT. Under steady state conditions, the temperature distribution of a single particle placed within a uniform medium can be described by the Laplace’s equation in a spherical coordinate system as [25]:
1 @ 1 @ @T 1 @2T 2 @T ¼0 r þ sinh þ 2 2 r2 @r @r r2 sinh @h @h r 2 sin h @w
ð2Þ
Assuming asymmetry about the z-axis, the temperature distribution is independent of w and the governing equation reduces to:
Fig. 1. Schematic diagram of the cell model of heat conduction to represent interaction among particles and fluid.
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1 @ 1 @ @T 2 @T r þ sinh ¼0 r 2 @r @r r 2 sinh @h @h
ð3Þ
A general solution to Eq. (3) for the temperature distributions of both solid (Ts) and fluid (Tf) phases can be obtained as [20,25]:
T s ¼ As þ
Bs Ds þ C s rcosh þ 2 cosh r r
ð4Þ
Bf Df T f ¼ Af þ þ C f rcosh þ 2 cosh r r
ð5Þ
where the constants As, Bs, Cs, Ds, Af, Bf, Cf, and Df are to be determined by using the following boundary conditions: (1) at the center of particle, r = 0, Ts – 1, s ks @T @r
@T kf @rf
@T s @h
@T f , @h
¼ and ¼ (2) at the surface of particle, r = R, (3) at the outer spherical surface of fluid envelope where r = R1,
# ( " ) 3 ks =kf 1 1 R1 R1 Q ¼ Q s þ Q f ¼ 4pkf C f R 1 þ2 ln 3 R R ks =kf þ 2 3
ð6Þ
4 pks C s R3 3
ð17Þ
Using the effective medium idea, the effective thermal conductivity can be determined so that the effective medium would conduct the same total volumetric heat flux as the cell:
4 Q ¼ ke V f þ V s rT ¼ ke pR31 rT 3
ð18Þ
From Eq. (17) and Eq. (18), the effective thermal conductivity of the porous medium can be determined as:
ke ¼
T f ðR1 ; hÞ ¼ T 1 þ rT ðR1 þ z1 Þ ¼ T 1 þ rT ðR1 þ R1 coshÞ ¼ T 1 þ rTR1 þ rTR1 cosh
The total volumetric heat flux in the cell is the sum of Qs and Qf:
3kf C f R3
n h 3 R 1 1
3
R
o i k =k 1 1 þ 2 kss =kf þ2 ln RR1 þ ks C s R3 f
R31 rT
ð19Þ
It can be further simplified and re-arranged as:
where rT is the overall temperature gradient. The boundary condition expressed in Eq. (6) is the main point in this study that differs from most previous studies. Using the specified boundary conditions, the temperature distributions can be obtained for the fluid and solid phases, respectively:
ke ¼ kf
T s ðr; hÞ ¼ As þ C s rcosh
ð7Þ
Substituting the relation between R1/R and the porosity / in Eq. (1) into Eq. (20), one has:
ð8Þ
ke 3ks =kf 2 ks =kf 1 ½/ þ ð1 /Þlnð1 /Þ ¼ kf 3 þ / ks =kf 1
ð9Þ
Eq. (21) is the dimensionless effective thermal conductivity of a porous medium based on the conceptual cell model. In the limiting case when / ? 1, it can be shown that:
" T f ðr; hÞ ¼ Af þ C f rcosh 1 þ
kf ks ks þ 2kf
3 # R r
where the four involved constants As, Cs, Af, and Cf are:
As ¼ T 1 þ rTR1 Cs ¼
3rT 3 ks =kf þ 2 þ 1 ks =kf RR1
Af ¼ As ¼ T 1 þ rTR1
ð11Þ
Cf ¼
ks =kf þ 2 rT 3 ks =kf þ 2 þ 1 ks =kf RR1
ð10Þ
ð12Þ
@T s ¼ Cs @z " 3 # @T f ks =kf 1 R ¼ Cf 1 þ 2 r @z ks =kf þ 2
Z
Vs
@T s 4 4pr 2 dr ¼ pks C s R3 3 @z
ð22Þ
which is the same as the classical Maxwell model [32]. The relative difference between the cell model proposed in this study and the Maxwell model [32] can be defined as:
ke =kf
Maxwell
ke =kf cell
ke =kf
ð23Þ
cell
It can be shown that:
2ð1 /Þlnð1 /Þ ks =kf 1 3ks =kf 2/ ks =kf 1 2ð1 /Þlnð1 /Þ ks =kf 1
ð13Þ
ð14Þ
From Eq. (24), the porosity /+ at which the maximum difference of effective thermal conductivity between the two models occurs can be determined from the following equation:
ð15Þ
Similarly, the volumetric heat flux from the fluid is:
Z
@T f 4pr 2 dr @z Vf # ( " ) 3 ks =kf 1 R1 R1 3 1 1 þ2 ¼ 4pkf C f R ln 3 R R ks =kf þ 2
3ks =kf 2/ ks =kf 1 ke ! kf 3 þ / ks =kf 1
ð21Þ
e¼
After determining the temperature gradients, the volumetric heat flux in the solid phase can be determined as follows:
Q s ¼ ks
ð20Þ
e¼
From the developed temperature distributions in both the solid and fluid phases, the temperature gradients in these two phases in the z direction can be obtained, respectively:
i 3 n h 3 o R ks =kf þ 2 RR1 1 þ 6 ks =kf 1 ln RR1 þ 3 ks =kf R1 3 ks =kf þ 2 ks =kf 1 RR1
Q f ¼ kf
ð16Þ
3ks =kf þ ks =kf þ 2 ln 1 /þ 2/þ ks =kf 1 ¼ 0
ð24Þ
ð25Þ
For the limiting case when ks/kf ? 1, /+ can be determined as 0.768. In the following section, the developed cell model is compared with 6 published experimental data sets with discussion of errors along with two classical models. The model is then compared with the effective thermal conductivity results from a pore-scale numerical simulation based on multiple random realizations of porous medium structure. I also quantitatively analyze and discuss the differences between the proposed effective thermal conductivity and the classical Maxwell model, expressed in Eq. (22).
J. Zhu / International Journal of Heat and Mass Transfer 138 (2019) 1054–1060
3. Results and discussion The developed effective thermal conductivity model is first compared with 4 published analytical models based on several different concepts. In the model developed by Ashby [33], the porous materials were cellular solids. The following simple effective thermal conductivity model expressed in Eq. (26) was then developed based on the estimate that one-third of the solid struts lie parallel to each axis.
i ke ks h ¼ ð1 /Þ þ 2ð1 /Þ2=3 þ / kf 3kf
ð26Þ
Revil [34] presented a theoretical model for the thermal conductivity of unconsolidated granular sediments by using a differential effective medium approach based on the similarity between the boundary problems for electrical conductivity and thermal conductivity in porous media.
2 0 13 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m 1m k m m ks ke k k k 4/ s s s s 1m 4/1m þ 0:5 1 @1 þ A5 ¼/ 1 þ kf kf kf kf kf kf ð27Þ where the additional parameter m is the electrical cementation exponent to account for the effect of the topology of the interconnected pore space. For spherical particles, m = 1.5. Krupiczka [35] developed the following effective thermal conductivity based on the correlation with experimental data:
0:280:757log/0:057log ðks =kf Þ ke ks ¼ kf kf
ð28Þ
Note that Eq. (28) diverges when / ? 0 and therefore is not applicable for / < 0.215 as suggested by the model author. The classical Maxwell model [32] relates the effective thermal conductivity to the porosity and thermal conductivity ratio as expressed in Eq. (22): With the exception of Revil [34], all other analytical models used for comparison only need ks/kf and / as the required input. The comparison of the dimensionless effective thermal conductivity of porous medium between the proposed model in this study and 4 published models is shown in Fig. 2. In Fig. 2(a), the compar-
Fig. 2. Comparison of the effective thermal conductivity from this study with that from 4 selected models in the literature when (a) the porosity / = 0.5, and (b) ks/ kf = 100.
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ison is shown as a function of the ratio ks/kf when / = 0.5, while Fig. 2(b) shows the relationship with the porosity / when ks/kf = 100. Since the dimensionless effective thermal conductivity is normalized by the fluid phase thermal conductivity, it shows an increasing trend with the increase of the thermal conductivity ratio of the solid phase over fluid phase (i.e., ks/kf) by all the models. However, the extent of the dimensionless effective thermal conductivity increase varies significantly among the models. On the other hand, all the models correctly capture the trend that the effective thermal conductivity decreases with the porosity. Since ks is higher than kf, more fluid presence should decrease the overall effective thermal conductivity. The proposed cell-based model in this study is generally in between two classical models of Maxwell [32] and Krupiczka [35] in the high ks/kf range. It is between the Maxwell model and the recently developed model by Ashby [33], although the values from the Ashby model are significantly higher. Among the three models (i.e., Maxwell, Krupiczka, and Revil) that are close to the present model, I will quantitatively compare the results with some of the available experimental results along with the two classical models of Maxwell [32] and Krupiczka [35] in the following since these two models have the same parameter parsimony requiring only two input parameters (i.e., the thermal conductivity ratio and porosity). The model of Revil [34] involves an additional parameter, m. Fig. 3 presents a comparison between the dimensionless effective thermal conductivity from the proposed model and that from 4 experimental data sets through measurements of the temperature gradient in the literature [36 –38] as a function of porosity /. For the experimental data sets, the porous medium was Quartz sand saturated with brine. The thermal conductivity of brine was kf = 0.59 W m1 °C1., while that of Quartz sand was estimated as ks = 7.2 W m1 °C1. The results of Maxwell [32] and Krupiczka [35] are also shown for quantitative error analysis of different models. The proposed model agrees well with the published experimental data. It can be seen that the Krupiczka model [35] diverges quickly when / becomes small. In the higher / range (/ > 0.215), it is very close to the classical Maxwell model [32], which also underestimates the effective thermal conductivity. Further quantitative analysis of errors for the three presented models illustrates that the root mean square errors (RMSEs) of the dimensionless effective thermal conductivity ke/kf are 1.08, 11.93, and 1.50 for the proposed model, Krupiczka model and Maxwell model, respectively.
Fig. 3. Comparison of the effective thermal conductivity prediction from this study with experimental data from the literature in relation to the porosity /. The model results from Maxwell [32] and Krupiczka [35] are also shown for quantitative analysis. For the experimental data sets, the porous medium was Quartz sands saturated with brine. The thermal conductivity of brine was kf = 0.59 W m1 °C1, while that of Quartz sand was estimated as ks = 7.2 W m1 °C1.
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The proposed effective thermal conductivity model has the smallest errors among the three models. On the other hand, the average errors of ke/kf are 0.367, 1.624, and 1.150 for the proposed cell model, Krupiczka model and Maxwell model, respectively. Overall, the Krupiczka model overpredicts the effective thermal conductivity, while the present model and the Maxwell model under-predict it, evident from their positive average error and negative average error respectively. The present model’s under-prediction of the effective thermal conductivity occurs in the small porosity range, while the classical Maxwell model under-predicts it in the entire experimental porosity range. The severe over-prediction from the Krupiczka model in the small porosity range is not surprising given that this model was developed based on the correlation in the porosity range above 0.215. In Fig. 4, I include an additional comparison with 2 more experimental data sets from Somerton [37] with two different values of the thermal conductivity ratios. In this case, the experimental data of dry oil sands and oil sands saturated with brine were used. For the dry samples, the ratio of ks/kf was 80.9 with ks = 4.45 W m1 °C1 and air kf = 0.055 W m1 °C1. For the brine saturated samples, the ratio of ks/kf was 7.5 with ks = 4.45 W m1 °C1 and brine kf = 0.590 W m1 °C1. It can be seen that the present model predicts the decreasing trend of ke/kf reasonably well with slight under-prediction for the brine saturated samples and over-prediction for the dry samples. Given that there are only two parameters involved in developing the conceptually simple model in this study and the significant scatters in the experimental data of effective thermal conductivity, the developed model captures the overall trend very well. The good agreement illustrates that the model is able to capture the main processes of heat conduction in porous media better than the classical models. Fig. 5 shows a comparison with the pore-scale numerical simulation results from the computational algorithm developed by Askari et al. [39] by which the thermal conduction behavior and the effective thermal conductivity was studied in pore scales. In the study of Askari et al. [39], the granular porous media were re-constructed based on the grain size distribution at different porosities. By solving the steady state heat conduction equation, the temperature distribution, heat transfer rate, and effective thermal conductivity were estimated. For the comparison results in Fig. 5, the thermal conductivity of solid was 7.4 W m1 °C1 and
Fig. 4. Comparison of the effective thermal conductivity prediction from this study with experimental data from Somerton [37] in relation to the porosity / at two different values of the thermal conductivity ratio, ks/kf. For the dry samples, the ratio of ks/kf was 80.9. For the brine saturated samples, the ratio of ks/kf was 7.5.
Fig. 5. Comparison of the effective thermal conductivity prediction from this study with pore scale numerical simulation results by Askari et al. [39] in relation to the porosity /. The thermal conductivity of solid was 7.4 W m1 °C1 and the pores were filled by air with the thermal conductivity of 0.0262 W m1 °C1. The grain diameter was 250 lm with a standard deviation of 30 lm.
the pores were filled by air with the thermal conductivity of 0.0262 W m1 °C1. The mean grain diameter was 250 lm with a standard deviation of 30 lm. The porous media were reconstructed for more than 500 random realizations with porosities ranging from 0.18 to 0.50. Fig. 5 plots the simulated effective thermal conductivities from Askari et al. [39] for various random realizations along with those predicted with the present model and the classical model of Maxwell [32]. It can be seen that while the effective thermal conductivity results from the random realizations show significant scatters, the present model can well capture the overall decreasing trend and is better than the Maxwell model. Similar to the comparison with the experimental observations, the developed simple model slightly under-estimates the effective thermal conductivity at the low porosity range and over-predicts it at the high porosity range. In Fig. 6, I show the dimensionless effective thermal conductivity as a function of the porosity when the solid phase is more conductive than the fluid phase (i.e., ks/kf > 1). In the same figure on the secondary axis, I also plot the percent difference of effective thermal conductivity predictions based on the cell model from this study and that based on the classical Maxwell model [32], calcu-
Fig. 6. The dimensionless effective thermal conductivity and the percent difference of the effective thermal conductivity between the cell model from this study and that from Maxwell [32] when ks/kf > 1.
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lated from Eq. (24). Since the solid particles are more conductive than the fluid, the resulting effective thermal conductivity of the porous medium is always larger than that of the fluid phase, as indicated by the result that ke/kf is always larger than 1. As the porosity increases, which means that the fluid phase becomes more dominant, the effective thermal conductivity of the porous medium also approaches to that of the fluid (i.e., ke/kf approaches 1). This is true for both models. At the two limiting cases of all fluid (i.e., / = 1) and all solid (i.e., / = 0) phases, both the proposed cell model and the Maxwell model correctly predict the fluid phase thermal conductivity and soil phase thermal conductivity, respectively. As a result, the percent difference from these two models approaches zero when / ? 1 and / ? 0, seen from Fig. 6. In the porosity range between these two limiting cases, the percent difference between the two models first increases and then decreases with the porosity. At a certain point of porosity, the percent difference between the two models reaches a peak of either positive or negative maximum difference. It can be seen from Fig. 6 that the Maxwell model underestimates the effective thermal conductivity since it overestimates the effect of fluid phase. The larger the contrast of the thermal conductivities of the two phases, the larger the percent difference between the two models. For example, when ks/ kf = 100, the Maxwell model under-estimates the effective thermal conductivity by more than 30%. Similar to Fig. 6, the dimensionless effective thermal conductivity of the porous medium as a function of the porosity when the solid phase is less conductive than the fluid phase (i.e., ks/kf < 1) is shown in Fig. 7. The percent difference of effective thermal conductivity based on the cell model from this study and that based on the classical Maxwell model [32] is also plotted in Fig. 7. Since the solid particles are less conductive than the fluid now, the resulting effective thermal conductivity of the porous medium is always smaller than that of the fluid phase. As the porosity increases, the effective thermal conductivity of the porous medium also increases to that of the fluid phase (i.e., ke/kf approaches 1). The percent difference between the two models first increases and then decreases with the porosity. In this case, the Maxwell model significantly over-estimates the effective thermal conductivity since it once again exaggerates the effect of fluid phase. The larger the contrast of the thermal conductivities of the two phases, the larger the percent difference between the two models. For example, when ks/ kf = 0.01, the Maxwell model over-estimates the effective thermal conductivity by more than 500%.
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Fig. 8. The porosity (/+) at which the maximum difference of effective thermal conductivity between the proposed cell-based model and that of Maxwell model [32] occurs and the corresponding maximum difference (e (%)) as a function of the thermal conductivity ratio of solid over fluid phases.
The porosity at which the maximum percent difference of effective thermal conductivity between the proposed cell model the Maxwell model [32] occurs, which is determined by solving Eq. (25), is shown in Fig. 8 as the solid line along with the actual percent difference between the two models as the dash line. It can be seen that the porosity where the maximum difference occurs increases significantly with the increase of ks/kf, with an asymptotic porosity of 0.768 with ks/kf ? 1. The porosity, /+, and the percent difference, e (%), are both affected by the ratio of solid phase thermal conductivity over fluid phase thermal conductivity. When ks/kf < 1, since the fluid phase is the more significant contributor to the overall effective thermal conductivity, simplifying the interaction between the two phases by applying the boundary condition at r = 1 would significantly exaggerate the contribution from the fluid phase. As a result, it introduces larger errors in the effective thermal conductivity prediction. Under the condition of ks/kf = 0.1, the over-estimation could be as high as 150%. On the other hand, if the solid phase is more conductive than the fluid phase, exaggerating the effect of fluid phase could lead to underestimation of the effective thermal conductivity by over 30%. Note that the effect of particle size distribution is not explicitly addressed in the developed cell model. The basic assumption in the model development is that the cell with a particle surrounded by a fluid envelope with proper boundary conditions is representative of heat conduction characteristics of the porous materials. In addition, the convective heat transfer due to temperature gradient and fluid flow is not considered in the cell model. The combined effects of particle size distribution and convective heat transfer on the temperature distribution in the porous media based on the concept of the cell model deserve future studies.
4. Conclusions
Fig. 7. The dimensionless effective thermal conductivity and the percent difference of the effective thermal conductivity between the cell model from this study and that from Maxwell [32] when ks/kf < 1.
In this study, a new cell model is developed to predict the effective thermal conductivity of a porous medium consisting of solid particles in a continuous fluid phase. The solid particle is enveloped by a spherical fluid cell, which represents the overall interactions among the solid particles and fluid. The radius of this hypothetical fluid cell is uniquely determined so that the ratio of fluid volume over the total volume inside the cell is equal to the porosity of the porous medium. The heat conduction equations are separately solved in both the solid particle and the fluid and the effective thermal conductivity is determined by the
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temperature distribution. I compare the new model with 6 experimental data sets, 4 published models, as well as a pre-scale numerical simulation for the effective thermal conductivity in the literature. The main conclusions are summarized as follows. 1. The developed effective thermal conductivity model compares well with the experimental data in the literature, with the errors being smaller than previously developed models. 2. The developed model is able to capture the variations of the effective thermal conductivity with the porosity with slight under-estimation at the low porosity and over-estimation at the high porosity. 3. At both ends of the porosity spectrum, the developed cell model of effective thermal conductivity approaches that from the classical Maxwell model. 4. At certain porosity between 0 and 1, the difference between the cell model and Maxwell model reaches a maximum. The porosity at which the difference reaches the maximum strongly depends on the contrast of the thermal conductivities of the two phases. 5. When the solid phase is more conductive than the fluid phase, over-simplifying the solid particle interactions under-predicts the effective thermal conductivity. On the other hand, oversimplifying the solid particle interactions significantly overpredicts the effective thermal conductivity if the fluid phase is more conductive. Conflict of interest The author declares no conflict of interest. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.134. References [1] Z.G. Qu, T.S. Wang, W.Q. Tao, T.J. Lu, A theoretical octet-truss lattice unit cell model for effective thermal conductivity of consolidated porous materials saturated with fluid, Heat Mass Transf. 48 (8) (2012) 1385–1395, https://doi. org/10.1007/s00231-012-0985-y. [2] G.H. Tang, C. Bi, Y. Zhao, W.Q. Tao, Thermal transport in nano-porous insulation of aerogel: factors, models and outlook, Energy 90 (2015) 701– 721, https://doi.org/10.1016/j.energy.2015.07.109. [3] D. Niu, L. Guo, H.W. Hu, G.H. Tang, Dropwise condensation heat transfer model considering the liquid-solid interfacial thermal resistance, Int. J. Heat Mass Transf. 112 (2017) 333–342, https://doi.org/10.1016/j. ijheatmasstransfer.2017.04.061. [4] J.J. Carberry, Chemical and Catalytic Reaction Engineering,, Courier Corporation, North Chelmsford, Massachusetts, 2001, p. 642. [5] G.F. Froment, K.B. Bischoff, J. De Wilde, Chemical Reactor Analysis and Design, third ed., Edition, Wiley, New York, 2010, p. 900. [6] L. Lapidus, N.R. Amundson, Chemical Reactor Theory: A Review, Prentice-Hall, Englewood Cliffs, New Jersey, 1977, p. 856. [7] I. Nozad, R.G. Carbonell, S. Whitaker, Heat conduction in multiphase systems— I: theory and experiment for two-phase systems, Chem. Eng. Sci. 40 (5) (1985) 843–855, https://doi.org/10.1016/0009-2509(85)85037-5. [8] R.M. Butler, G.S. McNab, H.Y. Lo, Theoretical studies on the gravity drainage of heavy oil during in-situ steam heating, Can. J. Chem. Eng. 59 (1981) 455–460, https://doi.org/10.1002/cjce.5450590407. [9] R.S. Detrick, R.P. Von Herzen, B. Parsons, D. Sandwell, M. Dougherty, Heat flow observations on the Bermuda rise and thermal models of midplate swells, J. Geophys. Res. 91 (B3) (1986) 3701–3723, https://doi.org/10.1029/ JB091iB03p03701. [10] P.J. Ortoleva, Basin compartments and seals, AAPG Mem. 61 (1994) 3–26. [11] L. Gibiansky, S. Torquato, Rigorous connection between physical properties of porous rocks, J. Geophys. Res. 103 (B10) (1998) 23911–23923, https://doi.org/ 10.1029/98JB02340. [12] M.A. Presley, P.R. Christensen, Thermal conductivity measurements of particulate materials 1, A review, J. Geophys. Res. 102 (1997) 6535–6549, https://doi.org/10.1029/96JE03302.
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