U-series Disequilibrium Produced by Dynamic Melting

If solid and melt are in equilibrium during partial melting, the conservation law which is concerned with the concentration of a radioactive and radiogenic nuclide in the solid and melt is given by McKenzie (1985) as the following differential equation 

The first term on the right-hand side is the melting term the second term on the right-hand side is the radioactive decay of the nuclide and the last term on the right-hand side represents the radiogenic production by the parent of the nuclide. and Cj are the concentrations of the nuclide 

The concentration of the daughter nuclide in the solid is related to the daughter concentration in the melt Cj by 

Equation (5.31) can be used to model the uranium decay series (U-series) nuclides in the residual melt that is in chemical equilibrium with the solid during dynamic partial melting  

Equation (5.34) is different from Eq. (5.35) and Eq. (5.36) because unlike Th or Ra, has no parent. When melting rate (M ) is constant in the upwelling material and the melt fraction ( ) in equilibrium with the matrix remains constant, both a and F are independent of time (see Eqs. (5.32) and (5.33)) and the system of differential equations (5.34), (5.35) and (5.36) has explicit solutions  

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