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Energy optimized Path Unaware Layered Routing Protocol (E-PULRP) for dense 3D Underwater Sensor Network (UWSN) is proposed and analysed in this paper. In the proposed E-PULRP, sensor nodes report events to a stationary sink node using on the fly routing. E-PULRP consists of a layering phase and communication phase. In the layering phase, a layering structure is presented wherein nodes occupy different layers in the form of concentric shells, around a sink node. The layer widths and transmission energy of nodes in each layer are chosen taking into consideration the probability of successful packet transmission and minimization of overall energy expenditure in packet transmission. During the communication phase, we propose a method to select intermediate relay nodes on the fly, for delivering packets from the source node to sink node. We develop a mathematical framework to analyse the energy optimization achieved by E-PULRP. We further obtain expressions for throughput, delay and derive performance bounds for node densities and packet forwarding probabilities, for given traffic conditions. A comparison is made between the results obtained based on simulations and analytical expressions. The energy efficiency is also demonstrated in comparison with existing routing protocol for underwater sensor networks.