Figure 80: Plot of the voltage for a section of a domain for a problem with many layers. The dashed rectangle indicates the extent of control volume n. The values and are the reaction coefficients for the layers Kj−1, Kj, Kj+1, Kj+2 Kj+3 shown. First, consider the section of the CV between and where and are xn−½ xn a b the interpolated values of (x) at the layer boundaries. Assuming a piecewise-linear variation in between and the integral can be written as,  xn−1 xn, where where is the value of between and , which is equal to in this case. Similar Ki+2 K Kj equations can be found for the integrals of for the other layers between and xn−½ xn. As a result, adding the integrals for all segment between and gives, xn−½ xn where indicates the segment and is the number of segments between and j xn−½ xn. A similar approach is used to get the integral, where the integral for the segment between and is given by, and where Adding the integrals for all segments between and gives, xn xn+1 The overall integral for the control volume is found by adding Equations and
Figure 80: Plot of the voltage for a section of a domain for a problem with many layers. The dashed rectangle indicates the extent of control volume n. The values Kj−1, Kj, Kj+1, Kj+2 and Kj+3 are the reaction coefficients for the layers shown. First, consider the section of the CV between xn−½ and xn where a and b are the interpolated values of (x) at the layer boundaries. Assuming a piecewise-linear variation in  between xn−1 and xn, the integral can be written as, where where Ki+2 is the value of K between and , which is equal to Kj in this case. Similar equations can be found for the integrals of for the other layers between xn−½ and xn. As a result, adding the integrals for all segment between xn−½ and xn gives, where j indicates the segment and is the number of segments between xn−½ and xn. A similar approach is used to get the integral, where the integral for the segment between and is given by, and where Adding the integrals for all segments between xn and xn+1 gives, The overall integral for the control volume is found by adding Equations and