Flow through Porous Beds

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Original Contributor: Kathryn L. McCarthy, UC Davis

Abstract

In food processing, there are a number of applications that incorporate the movement of fluid through a porous media. Porous media, for both food and nonfood applications, is the term applied to material that has interconnected non-solid space (or pores) within a solid matrix. Examples include the flow of solvent through solid particles in extraction processes (e.g., coffee, tea, and sugar), wort separation in a lauter tun (e.g., beer production) and whey separation from cheese curd. The characterization and quantification of flow in these systems will ensure higher food quality through the ability to better tailor processing conditions and product formulation. This paper provides an approach to modeling these systems based on Darcy’s law.


Darcy’s law (1856) describes laminar flow through porous media by a linear relationship between volumetric flow rate through a cross sectional area and the hydraulic gradient. Darcy's law states that the specific discharge rate, , is proportional to the hydraulic gradient; the hydraulic conductivity, , is the proportionality constant. For the one-dimensional flow of fluid in the vertical positive downward direction, taken as the -direction, the expression is

Mathematically, this expression (Eqn 1) is similar to Fourier’s law of heat transfer and Fick’s law of diffusive mass transfer. Figure 1 illustrates the flow geometry to be modeled.

Image:PorousBed.png

Figure 1. Schematic of flow through a porous bed.




The total hydraulic head, , is composed of two components, hydraulic head due to pressure () and hydraulic head due to gravity ()

In other words, the hydraulic head, , is the sum of the pressure head and the gravitational head. In this model no other forces are accounted for. The negative sign in front of is due to the positive downward coordinate system (Philip, 1957).

As an example of numerical values, Darcy’s law (Eqn 1) is an appropriate model for flow through porous matrix that does not compress during steady state flow (e.g., a bed of glass beads). The slope of (the water velocity through the column) versus the pressure difference per unit length (given as hydraulic head gradient) yields the value of the hydraulic conductivity, . Table 2 gives these values and the corresponding coefficient of determination (R2). As the bead size increases, the hydraulic conductivity increases.

Table 1. Hydraulic conductivity values, , as a function of bead size for a column diameter of 4.75 cm at flow rates from 100 – 500 mL/min.
Bead Size Column height (cm) Porosity K (cm/s) R2
0.5 mm 8.5 0.39 0.389 0.982
1 mm 9.0 0.37 0.824 0.989
2 mm 8.7 0.38 2.530 0.937

In addition to Darcy’s law, the equation of continuity describes fluid flow through the porous bed for the transient draining process. Written in terms of moisture content on a volumetric basis, (volume fluid/volume solids), the continuity expression in the -direction is


This expression equates the rate of decrease of volumetric water content to the gradient in flux. Substituting Darcy’s law in terms of both hydraulic head components (Eqns 1-2) for the flux term, q, yields


Although Darcy’s law was developed to describe flow in saturated media, the concept has been extended to flow in unsaturated media as well (Hillel, 1980). When material is saturated, all pores are filled with fluid and conducting, which results in maximal conductivity. As material desaturates, pores become air filled and the conductive portion of the material decreases. As a result, an important difference between unsaturated and saturated flow is the form of the hydraulic conductivity. Darcy’s law can be extended to unsaturated flow with the provision that the hydraulic conductivity is a function of moisture content. In addition, the specific water capacity, c, is introduced. This variable expresses the gradient of moisture with respect to hydraulic head due to pressure

As a result, the convection flow expression (Eqn 4) becomes


The focus, here, is on porous matrix that does not compress during either steady state flow or draining process. For this type of system, the following boundary and initial conditions apply:


 & \theta =\theta _{0}\text{   at  t}=\text{0} \\ 
& \theta =\theta _{1} \text{at}     z=0 \\ 
& \theta =\theta _{0}   \text{at}  z=L \text{  } \\ 

\end{align}~~~~~~~~~~(7)">


where is the saturated moisture content, and is surface moisture content of the bed. In words, the drainage process starts at an initial time with a fully saturated bed. Drainage of fluid is gravity-driven and fluid leaves the bottom of the bed. For this model, the system geometry is one-dimensional or of dimensions that the downward flow can be approximated as one-dimensional. During the entire process, the bottom of the bed (at ) remains fully saturated.

A limiting case allows a continued analogy to heat transfer and diffusive mass transfer. This limiting case is the physical situation in which the draining path is relatively short and the average moisture content is relatively high. The ratio / is termed the hydraulic diffusivity, . For simplicity, the hydraulic conductivity, , and the hydraulic diffusivity, , are taken as a constant and the resulting partial differential equation becomes



Though mathematically identical to Fick’s second law of diffusion, Eqn 8 describes convective flow, not diffusive flow.

If resistance to flow is present at the bottom of the bed, the appropriate boundary conditions would include a mass transfer coefficient, . The boundary conditions given in Eqn 7 become


  {} & \theta =\theta _{\text{0}}\text{   at  t}=\text{0}  \\
  {} & \theta =\theta _{1}\text{   at  }z=0  \\
  {} & D\frac{\partial \theta }{\partial z}=-h_{m}\left( \theta -\theta _{1} \right)\text{ at  }z=L\text{ }  \\

\end{array}~~~~~~~~~~(9)">



For Eqn 8, with either boundary conditions (Eqn 7 or Eqn 9), the analytical solution is obtained by a straight-forward approach using separation of variables to yield a series solution with exponential and trigometric functions (Crank, 1979).

For illustration, Fig. 2 gives the solution to Eqn 8 for a hydraulic diffusivity of over a time interval of 30 s. The bed was initially saturated at a 0.67 volumetric moisture content (0.4 void volume for water/0.6 volume of solids). The solution at long times is a linear profile between zero volumetric moisture content at the surface and the saturated condition at the bottom of the bed.

Image:SolEqn8.png

Figure 2. Solution to Eqn 8 for a hydraulic diffusivity of , over a time interval of 30 s.


Figure 3 gives the fraction of the total fluid leaving the column over time under the same conditions.

Typically for flow through porous beds, the hydraulic diffusivity and the hydraulic conductivity are not constants. The general form of the partial differential equation (PDE) given in Eqn 6 is

Equivalently,

A second limiting case can be considered; this case is the situation in which the hydraulic conductivity as a function of position is constant. With the derivative of with respect to position a constant (Eqn 10), an analytical solution is obtained by dividing the problem into the sum of two solutions: the steady state solution that satisfies the nonhomogeneous term in the PDE and the unsteady state solution that is solved for the homogeneous PDE (Haberman, 2004).

As the modeling of and progresses to increasing complexity, numerical solutions using either finite difference or finite element methods provide spatial and temporal solutions. This approach is needed for variable hydraulic diffusivity and hydraulic conductivity functions.

Image:FracTotFluid.png

Figure 3. Fraction of the total fluid leaving the column over 30 seconds.


References

CRANK, J. 1979. The Mathematics of Diffusion, 2nd edition. Clarendon Press, Oxford.

HABERMAN, R. 2004. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th Ed. Pearson Education, Inc., Upper Saddle River, NJ.

HILLEL, D. 1980. Fundamentals of Soil Physics. Academic Press, Inc., Orlando, FL.

PHILIP, J.R. 1957. The theory of infiltration: 1. The infiltration equation and its solution. Soil Science 83:345-357.

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