Applying Boundary Conditions


A boundary condition is simply a statement of the relationship between two variables.  Ours boundary conditions will come from knowing the concentrations of caffeine next to the surface of the coffee ground and in the bulk coffee.  By applying them to the equation we just found, we will be able to find an equation that describes the concentration of caffeine at any point within l from the surface of the coffee ground.  We are able to state:

            at X = 0;   C = Csat

            at X = l;    C = Cbulk

In the previous page, we found the equation C = X2 / (2 * DAB) + P *  / DAB + Q.  Using our first boundary condition, let us plug in 0 for X and Csat for C.  Thus we get:

            Csat = 0 /  (2 * DAB) + P * 0 / DAB + Q        simplified:

            Csat = Q                                                         we know know the value of Q.  Thus:

            C = X2 / (2 * DAB) + P * X / DAB + Csat

Now, using the second boundary condition, let us plug in l for X and Cbulk for C.  This produces:

            Cbulk = l2 /  (2 * DAB) + P * l / DAB + Csat        simplified:

            P = DAB * (Cbulk - Csat) / l  - l / 2                       This is the value of P.  Therefore:

            C = X2 / (2 * DAB) + [DAB * (Cbulk - Csat) / l  - l / 2] * X / DAB + Csat        Which can be written:

              C = X2 / (2 * DAB) + [(Cbulk - Csat) / l  - l / (2 * DAB)] * X  + Csat                 Plugging in for the variables:

            C = 2.28 * 104 * X2 - 1.41132 * 109 * X + 1.77

This plugging in any number for X between 0 and l you will get the concentration at that distance from the surface of the coffee ground.

cooking home

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