project specific cost of capital and WACC for combined entities

If appraising a project then whether or not it is a combined entity is of no relevance. It is only valid to appraise the project at the WACC is there is no change in the business risk and no material change in the level of gearing. If there is a big change in gearing then we use the APV approach.

The relevant to appraisal of combining entities is when they have different risk, in which case the overall beta of the combined entity is the weighted average of the individual betas.

All of this is explained in my free lectures on here.