Modeling Damage to Limestone Exposed to Atmospheric Pollutants
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sulfate reservoir. Transport of sulfate to the stone interior from the surface reservoir is thought to occur by both diffusive and convective mechanisms and to be driven by repeated cycles of stone surface wetting (by dewfall) and drying (by evaporation) in outdoor exposure. Aqueous sulfate loss along the pore walls is assumed to occur by adsorption, precipitation and / or reaction. The transport and accumulation of sulfate in the limestone interior is modeled as a problem in variably saturated flow through porous media, a complex process influenced by the physical and chemical characteristics of the porous material (limestone), the contaminant (sulfur) and the transport fluid (water). Mathematical description emerges from conservation of mass and definition of the Darcy velocity, the rate of liquid water movement through the porous material [11]. When applied to sulfate transport in limestone, a set of coupled partial differential equations results. The concentration profile of sulfate in the limestone pores as a function of time and position (i.e. depth into the stone) is described by 0
dcS t = V(¢SDVc) - V(cv)-r
(1)
[12] where c is the concentration of sulfate in the pore water (M/L 3 ), 0 is the porosity of the limestone (L 3]L3 ), D is the diffusion coefficient of sulfate in water (L2 /T), v is the Darcy velocity of the water moving through the limestone (L/T) and S is the pore volume saturation (the ratio of pore volume filled with water to the total pore volume). The transport of sulfate between aqueous and solid phases is described by r, the net mass transfer rate of sulfate out of the pore water (M/TL 3). The Darcy velocity is a function of water permeability through limestone and a pressure driving force, v
=
-kkHVH
(2)
where kH is the hydraulic conductivity (a measure of permeability expressed in units of L/T) and H is the pressure driving force, called the piezometric head (L). Note that hydraulic conductivity and head pressure are functions of the pore volume saturation, S [II]. Defining the correct mechanism for sulfate exchange between the pore fluid and pore wall, be it adsorption, precipitation or heterogeneous reaction, greatly influences the interpretation and implications of the model results. In the present study, the sulfate deposition process from fluid to pore wall is modeled as Langmuir adsorption in which all sulfate that adsorbs to the carbonate walls is assumed to be precipitated and immobile. Adsorption isotherm data for sulfate on calcium carbonate was measured in the laboratory and used in the model to determine parameters of the isotherm function [121. Prediction of the sulfate concentration profile requires accounting for the time and spatial distribution of water throughout the pore network. The distribution of water in porous limestone is modeled using the following relationship =
dt
- V. (v)
(3)
where all variables are as previously defined [12]. Numerical solution of the coupled model equations (1) - (3) requires application of appropriate boundary conditions and det
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