Finite Mass Top Boundary

he top boundary may be assumed to have a finite mass, in which case the source concentration starts at an initial value co, increases linearly with time at a rate cr, and then decreases with time as contaminant is transported into the soil and collected by a leachate collection system, if present. In version 7, the gradual conversion of waste into leachate can also be considered by providing a conversion rate half-life.

The concentration at the top boundary is given by:

c(t) = c₀ + c_r t - l ò c(t) dt - 1/H_r ò f(c,t) dt - q_c/H_r ò c(t) dt + R_s/WC (1 - e^-Kt)

where,

  co = initial source concentration at the start time.	

  cr = rate of increase in concentration with time due to 

       the addition of mass to the landfill.

  f(c, t, z=0) = the surface flux (mass per unit area per 

       unit time) passing into the soil at the top boundary.

  qc = the volume of leachate collected per unit area of 

       the landfill per unit time, if there is no 

       leachate collection system qc=0.

  l = first order decay coefficient calculated based on 

      the half-life specified in the Special 

      Features, Radioactive/Biological Decay option, such that  

      l = ln 2 /(half-life for decay).

  Rs = mass of contaminant in the waste available to be 

      transformed into dissolved form over time (per unit volume 

      of waste). The program calculates Rs as follows:

	Rs  = p rw  -  c0 WC

      where,

	p = available (leachable) mass of contaminant in the 	    waste per unit mass of waste (eg. mass of chloride	    in waste/total mass of waste);

	rw = apparent density of the waste (i.e. mass of waste 	    per unit volume of the landfill);

	WC = volumetric water content of the waste.

  k = generation coefficient calculated based on the 

      conversion rate half-life K, such that k = ln 2 / K. A 

      value of k = 0 implies no generation of concentration 

      with time. In the program k = 0 is obtained by specifying 

      K = 0 (this is the default case).

  Hr = reference height of leachate, and represents the volume 

      of leachate (per unit area of landfill) which would 

      contain the total mass of contaminant at a concentration 

      co. And may be defined in one of several ways depending 

      on what other options are being used. Note that generally 

      the program will calculate Hr (i.e. the user will 

      generally not input any value for Hr, but the user does 

      have the power to override the program).

Option (a). If the user specified K is not 0 and WC is not 0 then H_r = WC Hw and represents the actual fluid in the landfill (generally WC corresponds to field capacity, but could vary with time).

Option (b). If the user specified K equals 0 or WC equals 0 then H_r = p r_W H_w / c₀ and this represents the volume of fluid (per unit area of landfill) required to dissolve the leachable mass of contaminant (i.e. p r_W H_w) at the initial concentration c₀.

Option (c). If the user specifies H_r is not 0 then the user specified value overrides the values calculated under option (a) or (b). This is an advanced feature of the program and should not be used without very carefully checking your calculations.

If the rate of increase in concentration c_r is zero and the reference height of leachate H_r is very large, this boundary condition reduces to a constant concentration boundary condition.