PoW is a mechanism to keep block chain incorruptable from malicious nodes. Now in a private block chain where all nodes are supposed to be honest, what purpose PoW would serve? Would it not be just a waste of computational power?
You might want to look at this article.
To resume some of the points :
In general, PoW is immoral in it's waste of energy. Other consensus algorithms are available that do are more energy efficient, such as
- PoS (Proof of Stake) used by Ethereum (winner has the most "stake" or "wealth" of some kind)
- PBFT (Practical Byzantine Fault Tolerance) elects a leader. Does not scale due to the high number of messages between nodes.
- Raft (which elects a leader until the leader times-out). Scales well and is efficient. Not BFT (vulnerable to bad actor nodes)
- PoET (Proof of Elapsed Time) (which generates a random wait time and the first timer to expire wins)