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 :
Proof of work is not required on a permissioned or private blockchain.
For the more efficient operation, for example Proof-of-Authority can be used, like in Ethereum Kovan testnet.
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)