Previousarrow_backNextarrow_forward

Problem Statement

Consider a system containing two pressure vessels at differing temperatures as in the experiments presented by Pamuk and Özdemir (2012). The vessels are connected via a pipe that contains a filter consisting of close-packed steel spheres as shown in Figure 1. Predict the velocity and temperature of the fluid inside the filter. The pipe is 0.304 m in length and 0.0514 m in diameter. The fluid inside the system is liquid water at approximately 30 degrees Celsius. Each of the steel spheres are 1 mm in diameter.

Figure 1: Schematic of the pressure vessel system for which a custom MOOSE-based application will be designed to solve Pamuk and Özdemir (2012).

For this tutorial, the outlined portion of the pipe of length , shown in Figure 1, is of particular interest. This region shall serve as the problem domain .

Governing Equations

To solve this problem, the following physics must be considered:

  • Conservation of Mass:

(1)

  • Conservation of Energy:

(2)

  • Darcy's Law:

(3)

The variables shown in Eq. (1), Eq. (2), and Eq. (3) denote the properties listed in Table 1.

Table 1: Disambiguation of problem variables.

SymbolProperty
Velocity (Volumetric Flux) Vector
Heat Capacity
Temperature
Time
Porosity
Thermal Conductivity
Permeability Tensor
Dynamic Viscosity
Pressure
Density
Gravity Vector

If a zero-gravity condition is assumed, i.e., if , and if the divergence-free condition of Eq. (1) is imposed onto Eq. (3), then it follows that the two unknowns, and , must satisfy the following system of two PDEs:

(4)

The system overall heat capacity, density, and thermal conductivity are weighted by the contributions of the individual materials and each depend on the porosity of the packed steel sphere medium. These three relationships are defined by the following:

(5)(6)(7)

Here, denotes specific heat and the subscripts and refer to the fluid material (water) and solid material (steel), respectively.

Material Properties

The material properties of the fluid () and the solid () are given in Table 2. The permeability of the packed steel sphere medium that is present throughout the pipe is assumed to be isotropic. Pamuk and Özdemir (2012) provides the following relationship:

(8)

where is the diameter () of the spheres and denotes the scalar permeability (), which is a linear function of .

Table 2: Material property values.

PropertyValueUnits
Viscosity of water,
Steel sphere diameter,
Density of water, 996
Density of steel, 8000
Thermal conductivity of water, 0.600
Thermal conductivity of steel, 18.00
Specific heat capacity of water, 4180
Specific heat capacity of steel, 466
Previousarrow_backNextarrow_forward

References

  1. Mehmet Turgay Pamuk and Mustafa Özdemir. Friction factor, permeability and inertial coefficient of oscillating flow through porous media of packed balls. Experimental thermal and fluid science, 38:134–139, 2012.[BibTeX]