# The Mass Conservation Equation Engineering Essay

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The fluid was considered as a mixture of soil gas and radon. The movement of such a mixture is described by a set of conservation equations, including a continuity equation, an energy equation, a momentum equation, and a species equation.

Transport equations, is the general name for conservation of mass (continuity), momentum (Naviar-Stokes equations), thermal energy and concentration of species. These equations within the ventilated room in turbulent model are described in the next section.

For an incompressible and steady state Newtonian flow in a porous medium, the transport equations have the following forms in a Cartesian co-ordinate.

## 5.1.1. The mass conservation equation:

(5.1)

where is the radon added into the mixture in terms of mass, , density and are the velocity components of the velocity vector; , in the x, y , z directions respectively.

## 5.1.2. The momentum conservation equation:

in the x-coordinate;

(5.2a)

in the y-coordinate;

(5.2b)

and in the z-coordinate;

(5.2c)

Where, is the static pressure, is the viscosity of the soil gas, 1.83-10-5 kgm-1s-1 at T=293.

## 5.1.3. The energy conservation equation:

(5.3)

Where, the is the specific heat of the radon and soil gas mixture, and the is the thermal conductivity of the soil matrix.

## 5.1.4. The state equation:

As the flow mixture was treated as an idea gas:

(5.4)

Where, is the specific gas constant, and the density of the mixture:, the temperature of the mixtureand soil: and the thermal expansion coefficient of the mixture: (=1/273 K-1) has the following relation:

(5.5)

## 5.2. Mathematical description of radon transport

As mentioned before two mechanisms, advection and diffusion, leads to radon movement and transport.

## 5.2.1. Soil gas transport:

Radon Indoor air pressure differences induced by ventilation systems, indoor and outdoor temperature and relative humidity can displace soil gases, draw out and accumulate within the buildings. In this work, it is assumed that the disturbance pressure caused by ventilation effects and indoor temperature moves radon soil gas.

In porous media there is an additional sink; Darcy-Forchheimer equation:

Where, is flow velocity vector in m/s, ; the soil permeability in m2, ; air dynamic viscosity , ; the pressure differences (disturbance pressure) between a chamber and the atmosphere (Pa), these disturbance pressures is made by wind, rainfall, temperature and particularly ventilation effects; like in this study, at the boundary conditions. The Forchheimer coefficient is set to be '0' in soil when ksoil<10-9m2, and the equation becomes Darcy's Law and it describes gas flow in porous media:

(5.7)

Disturbance pressure field can be defined as:

(5.8)

Where ; disturbance pressure (Pa), ; air pressure in Pascal, ; air density, ; gravity acceleration in m2/s .

Soil permeability depends on particle diameter within the soil bulk and porosity of soil. Permeability is determined by the following equation:

(5.9)

Where, ; particle diameter in m and ; porosity of soil. Usually is called viscous resistance, i.e: (4.5)

Porosity is also defined as: (5.10)

## 5.2.2.Radon generation and decay rate equation

In this work, it is assumed that all indoor radon comes from the underground dry soil and it is formed in the decay chain of radium. Normally some parts of radon (20- 70%) decays to alpha particle, which is called emanation factor.

The change rate of the number of radon atoms in the partial volume of soil is determined by generation and decay in the decay of radium present within . As radium activity per unit volume of solid is given by , where is the bulk density of dry soil and is the radium activity per mass of soil bulk, the following equation is given:

(5.11)

Where, is radon decay constant in .

To define the rate of change of the number of radon atoms in the air-filled pore space within , , we need to know about emanation fraction of the number of radon atoms produced in the solid material ( soil) which is emanated into the pore air space. Equation (5.11) for pore air filled can be written as:

(5.12)

Using the relation and doing some manipulations the radon production and decay rate equation yield as:

(5.13)

(5.14)

Where is the radon generation rate of radon activity per unit bulk volume and it is assumed time independence and function of radium activity, bulk density and porosity of the material, radon emanation coefficient which are moisture dependence and radon decay constant. The radon concentration, , is in the air filled pore space within . In Eq 5.14, the left hand term is the rate of radon change in air phase, the first right hand term is radon decay rate and the second term is radon production rate.

## 5.2.3. Radon diffusion and convection transport equation

Radon migrates through soil and building materials pores by diffusion and advection mechanisms. However, the main entry mechanism is the convective flow from the pores in soil through cracks, and the flow increases with increasing negative differences in pressure across the floor and walls. Radon concentration in soil gas depends on the radium content and the physical characteristics of the soil, such as its grain-size distribution, moisture, porosity and especially the permeability of the soil. Two basic mechanisms of radon transport within building materials are demonstrated in Figure 5.1. Because of concentration gradient diffusion mechanism in a particular medium before decaying is done by the random molecular motion.

## Figure 5.1. Radon diffusion and advection mechanisms

## 5.2.3.1. Diffusion mechanism

In order to enter the indoor air, radon gas must first transported through the larger air-filled pores within the building material, so that a fraction of these reaches the building-air interface before decaying and then by the air flow enters indoor. Like any fluid substance there is a tendency to migrate in a direction opposite to that of the increasing concentration gradient within the material. Diffusion mechanism is stated by Fick's law and is written as:

(5.15)

Where , , is the diffusive flux density vector of radon activity in air filled pore space, ( is the radon diffusion coefficient in air phase. and is time-mean concentration of radon in air phase.

The bulk diffusion flux density in the water phase, , can also be written as:

(5.16)

Where ( is the diffusion coefficient for radon activity in water phase, comparing and it can be seen that is much smaller than . It is clear to say that moisture in air can affect radon atoms diffusing transport.

## 5.2.3.2.. Advection mechanism

Transport of radon activity caused by the bulk movement of soil gas is usually called advective transport. The bulk advective flux density of radon activity in the air filled pore space,

, is obtained by multiplying equation 5.7 by the pore air radon activity concentration:

(5.17)

Where is the advective flow density in unit of ; (m2) is the intrinsic soil permeability; (Pa ) is the pressure field; and μ (Pa s) is the dynamic viscosity of the radon.

Radon advection transport in the water phase as in the case of diffusive transport can take place. Because of considering dry soil in this thesis, this part of radon transport is discarded.

The case of radon transport in soil has a sufficiently low Reynolds number (Re=0.01), laminar fluxes may be induced due to a pressure gradient. As mentioned before, this gradient could be created mainly by changes in environmental conditions by means of ventilation systems and heating and air-conditioned systems in dwellings.

Also combination of these two time dependent equation is written as:

(5.18)

## 5.3. Radon transport equation

Radon mass transfer or continuity equation, also called as the total radon transport equation, is an integration of radon production, radon decay rate equation, Eq. 5.14, and radon diffusion transport equation, advection transport equation, Eq. 5.18, written as the following conservation equation:

(5.19)

In mass balance equation , therefore Eq 4.15, yields the following equation (Loureiro CO, 1987):

(5.20)

## 5.4. Analytical solutions of radon transport equation

In one dimensional problem under certain conditions, such as in the area infinitely distant from the pressure disturbance, the radon transport equation, Eq. 5.20, has analytical solutions. Using z as spatial variable and , in this case, this equation is reduced to:

(5.21)

This condition for boundary condition is known as the infinite distant boundary. The results of differential equation solution can be used compared by the solution with the numerical results at this boundary. The model's validity can be partially confirmed by comparing analytical and numerical solutions.

To solve this equation, assuming that, and we have the follows:

(5.21)

Therefore the solution of this differential equation is:

(5.22)

And

(5.23)

To solve this differential equation and considering boundary condition (at results:

(5.23)

We can find, , and .

This solution shows that the radon concentration along the boundary only depends on diffusion coefficient and radon decay constant.

## 5.5. Diffusion length

One of the variables in radon transport is radon diffusion length which depends on soil and materials structure.

For one dimensional problem, using z as spatial variable, radon diffusion length can be derived from Eq. 5.23:

(5.24)

Where is the diffusion length (m):

(5.25)

Where is the effective diffusion coefficient and in air phase is introdeuced as:

(5.26)

The diffusion length is defined as the characteristic distance transferred by the radon atoms during on half life.

## 5.6. Radon exhalation rate

Another variable of indoor radon transport is radon exhalation rate from soil or building material. The radon concentration in the chamber (room) with volume V is described by the equation (Man, o.a., 1997; Petropoulos, o.a., 2001) ,

(5.27)

Where is the radon concentration in the chamber at time t , the initial radon content at h, the total radon decay rate and ventilation rate in , the radon exhalation rate , the exhalation area and the volume of the house or any chamber.

At the steady state situation Eq. 5.27 is reduced to:

(5.28)

This relation clearly indicates that indoor radon concentration in given chamber only depends on air change rate. In this thesis we use this analytic relation to confirm numeric results at 4 distinct air change rates.