Your magic does not work on me!

Mistake 1: h_outer = (V_outer + (h_sink * A_inner,external)) / A_outer
Mistake 2:
Code:
    = (V_outer,0 + M_inner,initial/d) / A_outer - (M_inner,initial/d + V_outer,0 - V_outer) / A_inner,external + (V_outer,0 - V_outer)/A_inner,internal         
    = V_outer,0/A_outer + M_inner,initial/(d*A_outer) - M_inner,initial/(d*A_inner,external) - V_outer,0 / A_inner,external + V_outer/A_inner,external + V_outer,0/A_inner,internal + V_outer/A_inner,internal
Correctly:
Code:
h_sink = M_inner / (d*A_inner,external)
       = (M_inner,initial + V_inner * d) / (d*A_inner,external)
Code:
h_diff = h_sink - h_inner
       = (M_inner,initial + V_inner * d) / (d*A_inner,external) - V_inner/A_inner,internal

Code:
h_outer = (V_outer + (h_sink * (A_outer-A_inner,external))) / A_outer
Code:
V_inner = V_outer,0 - V_outer
Code:
h_print = h_outer - h_diff
        = (V_outer + (h_sink * (A_outer-A_inner,external))) / A_outer - (M_inner,initial + V_inner * d) / (d*A_inner,external) + V_inner/A_inner,internal
        = (V_outer + ((M_inner,initial + V_inner * d) / (d*A_inner,external) * (A_outer-A_inner,external))) / A_outer - (M_inner,initial + V_inner * d) / (d*A_inner,external) + V_inner/A_inner,internal
        = (V_outer + ((M_inner,initial + (V_outer,0 - V_outer) * d) / (d*A_inner,external)  * (A_outer-A_inner,external))) / A_outer - (M_inner,initial + (V_outer,0 - V_outer) * d) / (d*A_inner,external) + (V_outer,0 - V_outer)/A_inner,internal
        = (V_outer + ((M_inner,initial/d + V_outer,0 - V_outer) /  (A_inner,external)  * (A_outer-A_inner,external))) / A_outer - (M_inner,initial/d +  (V_outer,0 - V_outer)) / (A_inner,external) + (V_outer,0 -  V_outer)/A_inner,internal
lets call (A_outer-A_inner,external) / A_inner,external = A_correction
        = (V_outer + ((M_inner,initial/d + V_outer,0 - V_outer) * A_correction)) / A_outer - (M_inner,initial/d +  (V_outer,0 - V_outer)) / (A_inner,external) + (V_outer,0 -  V_outer)/A_inner,internal
        = (V_outer * (1-A_correction)/A_outer + V_outer,0/A_outer + M_inner,initial*A_correction/(d*A_outer) - M_inner,initial/(d*A_inner,external) - V_outer,0/A_inner,external + V_outer/A_inner,external + V_outer,0/A_inner,internal - V_outer/A_inner,internal

h_print = V_outer * ((1-A_correction)/A_outer - 1/A_inner,internal + 1/A_inner,external) + <a whole bunch of constants>
So:
Code:
(1-A_correction)/A_outer - 1/A_inner,internal + 1/A_inner,external = 0
(1-((A_outer-A_inner,external) / A_inner,external))/A_outer - 1/A_inner,internal + 1/A_inner,external = 0
1/A_outer - 1/A_inner,external + 1/A_outer - 1/A_inner,internal + 1/A_inner,external = 0
2/A_outer = 1/A_inner,internal
2*A_inner,internal = A_outer
Dont get me wrong. I love that you did the math, never would have done it myself otherway