 Melt-dilution heat balance in a dike

New wall melt enters at the wall melting temperature $T_m$. Besides supplying its latent heat, the flowing magma must warm this added melt. Relative to $T_m$, conservative heat storage is $b(T-T_m)$, so differentiating it retains $(T-T_m)b_t$ and gives the displayed dilution term. Omitting that term requires an additional approximation, such as a small sensible-to-latent heat ratio; it does not follow from equal densities alone.