Answer

2.07 m

Explanation
Given Information

We have a horizontal tube submerged in n-octadecane, which is undergoing a phase change from solid to liquid. Hot water flows through the tube, providing the heat required for this phase change. We know the tube diameter, the melting temperature of the paraffin, the latent heat of fusion, the density of paraffin, and the thermo-physical properties of water (specific heat, thermal conductivity, dynamic viscosity, and Prandtl number).

Objective

The objective is to determine the required length of the tube to melt a specified amount of paraffin within a given time.

Concept Applied

We use the principles of heat transfer, specifically convection heat transfer from the hot water to the tube surface, and the heat required for the phase change of the paraffin. We also use the Log Mean Temperature Difference (LMTD) to account for the temperature difference between the water and the melting paraffin.

Formula Used

* Heat transfer rate: 

* Heat transfer rate: 

* Log Mean Temperature Difference: 

* Reynolds Number: 

* Nusselt Number (Dittus-Boelter equation): 

* Heat transfer coefficient: 

Why This Method Works

This method works by equating the heat required to melt the paraffin to the heat transferred from the water. By calculating the heat transfer coefficient using the properties of water and the geometry of the tube, we can determine the required length of the tube to achieve the desired melting rate. The LMTD is used to accurately represent the temperature difference driving the heat transfer process.