Scaling heat and mass flow through porous media during pyrolysis

Abstract

The modelling of heat and mass flow through porous media in the presence of pyrolysis is complex because various physical and chemical phenomena need to be represented. In addition to the transport of heat by conduction and convection, and the change of properties with varying pressure and temperature, these processes involve transport of mass by convection, evaporation, condensation and pyrolysis chemical reactions. Examples of such processes include pyrolysis of wood, thermal decomposition of polymer composite and in situ upgrading of heavy oil and oil shale. The behaviours of these systems are difficult to predict as relatively small changes in the material composition can significantly change the thermophysical properties. Scaling reduces the number of parameters in the problem statement and quantifies the relative importance of the various dimensional parameters such as permeability, thermal conduction and reaction constants. This paper uses inspectional analysis to determine the minimum number of dimensionless scaling groups that describe the decomposition of a solid porous material into a gas in one dimension. Experimental design is then used to rank these scaling groups in terms of their importance in describing the outcome of two example processes: the thermal decomposition of heat shields formed from polymer composites and the in situ upgrading of heavy oils and oil shales. A sensitivity analysis is used to divide these groups into three sets (primary, secondary and insignificant), thus identifying the combinations of solid and fluid properties that have the most impact on the performance of the different processes.

Appendix: Deriving the dimensionless groups by inspectional analysis

The general procedure of nondimensionalizing the equations that describe a physical process by inspectional analysis involves the introduction of arbitrary scaling factors. They make a linear transformation from dimensional to dimensionless space. The scaling factors are then grouped into dimensionless scaling groups, and their values are selected to minimize the number of groups.

We define the following linear transformations of every variables from the original dimensional space to a general dimensionless space:

$$\begin{aligned}&x=x^*_1x_D+x^*_2 \quad t=t^*_1t_D+t^*_2 \nonumber \\&\rho _s=\rho ^*_{s1}\rho _{sD}+\rho ^*_{s2} \nonumber \\&T=T^*_1T_D+T^*_2 \quad P=P^*_1P_D+P^*_2\nonumber \\&v_g=v^*_1v_{gD}+v^*_2 \quad q=q^*_1q_D+q^*_2 \end{aligned}$$

(35)

In these transformations, the scale factors are the “*” quantities and the dimensionless variables are those with a subscript “D”. There are 14 scale factors, two for each independent variable ($x$ and $t$) and depend variable ($\rho _s$, $T$, $P$, $v_g$, $q$). The scale factors may be multiplicative (subscript 1) or additive (subscript 2). We substitute (35) into Eqs. (1), (7), (9), (13) and (14) and multiple by selected scale factors to make the equations dimensionless. We obtain:

• Solid decomposition

  

  (36)
  

  (37)
  

  (38)

• Mass conservation

  

  (39)
  

  (40)

• Darcy’s law

  

  (41)
  

  (42)

• Energy equation

  

  (43)

• Fourier’s law

  

  (44)

• Heat flow boundary conditions

  

  (45)

• Mass flow boundary conditions

  

  (46)

• Initial conditions

  

  (47)

The scaling groups that appear in these equations are numbered (e.g., ). Each equation is dimensionless, and the 30 scaling groups are dimensionless too. The next task is to reduce the number of groups.

A large number of scaling groups can be set to zero by chosing the additive factors to be zero or to the initial or final value of the variable. Therefore, we choose:

$$\begin{aligned}&x^*_2=0\quad t^*_2=0 \nonumber \\&\rho ^*_{s2}=\rho _{s,f}\nonumber \\&T^*_2=T_0 \quad P^*_2=0\nonumber \\&v^*_2=0 \quad q^*_2=0 \end{aligned}$$

(48)

Then, the groups 3, 9, 12, 15, 22, 23 and 30 are equal to zero. Next, we need to define the multiplicative factors. Setting scaling groups to one usually leaves the final formulation in a compact form that is generally free of constant. Therefore, we choose:

$$\begin{aligned}&x^*_1=L \nonumber \\&\rho ^*_{s1}=\rho _{s,0}-\rho _{s,f} \nonumber \\&T^*_1=\varDelta T = T_1-T_0 \quad P^*_1=P_0\nonumber \\&v^*_1=\frac{K_0P_0}{\mu _{g,0}L} \quad q^*_1=\kappa _s\frac{\varDelta T}{L} \end{aligned}$$

(49)

Thus, the groups 1, 13, 20, 26, 27, 28 and 29 are equal to one. For the multiplicative factor $t^*_1$, various time scales such as the time scale of the chemical reaction or the time scale of heat conduction could be chosen. Here we chose to normalize our time to the time taken for heat to diffuse at initial conditions. This has the advantage that group 18 in Eq. 45 is set to 1 i.e. the rate of change of heat transfer with distance is 1 at initial time.

$$\begin{aligned} t^*_1=\tau =\frac{\rho _{s,0}\gamma _sL^2}{\kappa _s} \Rightarrow {\text {Group }} 18=1 \end{aligned}$$

(50)

Note that the order of the reaction $n$ is an additional parameter. Therefore there remain 15 groups that are not yet defined. These remaining dimensionless groups are no longer arbitrary. They are:

$$\begin{aligned}&D_2 = A\tau \quad D_4 = \frac{E_a}{R\varDelta T} \quad D_5 = \frac{T_0}{\varDelta T} \quad D_6 = \frac{\phi _f}{\phi _0}-1 \nonumber \\&D_7= \frac{K_f}{K_0}-1 \quad D_8 = \frac{K_0P_0\tau }{\phi _0\mu _{g0} L^2} \quad D_{10}=\frac{\rho _{s,0}-\rho _{s,f}}{\rho _{s,0}} \quad D_{11}=\frac{\phi _0P_0}{\rho _{s,0}\gamma _sR\varDelta T} \nonumber \\&D_{14} = \frac{\varDelta T}{\mu _{g,0}}\frac{\partial \mu _g}{\partial T} \quad D_{16} = \frac{\rho _{s,f}}{\rho _{s,0}} \quad D_{17} = \frac{\gamma _g}{\gamma _s} \quad D_{19} = \frac{\varDelta h_{r,0}}{\gamma _s\varDelta T} \nonumber \\&D_{21} =\frac{\phi _0\left( \kappa _g-\kappa _s\right) }{\kappa _s} \quad D_{24} = \frac{\epsilon _s\sigma \varDelta T^3 L}{\kappa _s} \quad D_{25} = \frac{T_i^4}{T_1^{*4}} \end{aligned}$$

(51)

The last task is to minimise the number of groups by identifying dependent groups. We observe that:

$$\begin{aligned} D_{16}&= 1-D_{10}\nonumber \\ D_{25}&= (1+D_5)^4 \end{aligned}$$

(52)

Finally we obtain 13 groups. The system depends only on these groups and the order of reaction. The groups are:

$$\begin{aligned}&D_K = A\tau \quad N_a = \frac{E_a}{R\varDelta T} \quad T^*_0 = \frac{T_0}{\varDelta T} \quad \varDelta m^* = \frac{\rho _{s,0}-\rho _{s,f}}{\rho _{s,0}} \nonumber \\&\delta = \frac{\phi _f}{\phi _0} \quad \xi = \frac{K_f}{K_0} \quad L_e = \frac{\phi _0\mu _{g,0} L^2}{K_0P_0\tau } \quad \rho ^*_g = \frac{\phi _0M_gP_0}{\rho _{s,0}R\varDelta T} \nonumber \\&\varDelta \mu ^*_g = \frac{\varDelta T}{\mu _{g,0}}\frac{\partial \mu _g}{\partial T} \quad \varDelta h^*_r = \frac{\varDelta h_{r,0}}{\gamma _s\varDelta T} \quad \gamma ^*_g = \frac{\gamma _g}{\gamma _s} \quad \varDelta \kappa ^*_g =\frac{\phi _0\left( \kappa _g-\kappa _s\right) }{\kappa _s}\nonumber \\&\epsilon ^* = \frac{\epsilon _s\sigma \varDelta T^3 L}{\kappa _s} \end{aligned}$$

(53)

The dimensionless groups satisfy the scaling requirements for the one-dimensional problem. We can demonstrate that they are independent by using the method of elementary row operations descibed in [27]. We obtain the following form of the dimensionless equation:

• Solid decomposition

  $$\begin{aligned}&\frac{\partial \rho _{sD}}{\partial t_D} = D_K\rho _{sD}^n\exp \left( -\frac{N_a}{T_D+T^*_0}\right) \end{aligned}$$

  (54)
  $$\begin{aligned}&\phi _D = 1+\left( 1-\delta \right) \left( 1-\rho _{sD}\right) \end{aligned}$$

  (55)
  $$\begin{aligned}&K_D = 1+\left( 1-\xi \right) \left( 1-\rho _{sD}\right) \end{aligned}$$

  (56)

• Mass conservation

  $$\begin{aligned}&\frac{\partial }{\partial t_D}\left( \phi _D\rho _{gD}\right) = -\frac{1}{L_e}\frac{\partial }{\partial x_D}\left( \rho _{gD} v_{gD}\right) -\varDelta m^*\frac{\partial \rho _{sD}}{\partial t_D}\end{aligned}$$

  (57)
  $$\begin{aligned}&\rho _{gD}=\rho ^*_g\left( \frac{P_D}{T_D+T^*_0}\right) \end{aligned}$$

  (58)

• Darcy’s law

  $$\begin{aligned} v_{gD}&= \frac{K_D}{\mu _{gD}}\frac{\partial P_D}{\partial x_D}\end{aligned}$$

  (59)
  $$\begin{aligned} \mu _{gD}&= 1+\varDelta \mu ^*_g\left( T_D-T^*_0\right) \end{aligned}$$

  (60)

• Energy equation

  $$\begin{aligned} \left( 1-\varDelta m^* \left( 1-\rho _{sD}\right) +\phi _D\rho _{gD}\gamma ^*_g\right) \quad \frac{\partial T_D}{\partial t_D}&= -\frac{1}{L_e}\rho _{gD}\gamma ^*_g v_{gD}\frac{\partial T_D}{\partial x_D}-\frac{\partial q_D}{\partial x_D}\nonumber \\&\quad -\varDelta m^*\left( \varDelta h^*_r +\left( 1-\gamma ^*_g\right) T_D\right) \frac{\partial \rho _{sD}}{\partial t_D} \end{aligned}$$

  (61)

• Fourier’s law

  $$\begin{aligned} q_D=-\left( 1+\phi _D\varDelta \kappa _g\right) \frac{\partial T_D}{\partial x_D} \end{aligned}$$

  (62)

• Heat flow boundary conditions

  $$\begin{aligned}&{\text {at}} \quad x_D=0 \quad \forall t_D \nonumber \\&q_D=Q^*_i - \epsilon ^*\left( T_D+T^*_0\right) ^4 \quad {\text {or}} \quad T_D = 1 \nonumber \\&{\text {at}} \quad x_D=1 \quad \forall t_D \nonumber \\&q_D= - \epsilon ^*\left( T_D+T^*_0\right) ^4 \quad {\text {or}} \quad T_D = 0 \end{aligned}$$

  (63)

• Mass flow boundary conditions

  $$\begin{aligned}&{\text {at}} \quad x_D= 0 \quad \forall t_D \nonumber \\&P_D=0 \quad {\text {or}} \quad v_{gD} = 0 \nonumber \\&{\text {at}} \quad x_D=L \quad \forall t_D \nonumber \\&P_D=0 \quad {\text {or}} \quad v_{gD} =0 \end{aligned}$$

  (64)

• Initial conditions

  $$\begin{aligned}&\rho _{sD}=1 \nonumber \\&P_D=1 \quad {\text {at}} \quad t_D=0 \quad \forall x_D\nonumber \\&T_D=0 \end{aligned}$$

  (65)