Assuming that:
The heat generation is uniform throughout the heating element, Nichrome wire.
The cross-sections are insulated and heat transfer is taking place only in the radial direction.
All the heat generated is conducted, there is energy storage.
Now, from the properties of Nichrome:
The melting temperature of Nichrome,
Thermal conductivity,
Given that:
The power generated by the heating element,
Diameter,
So, radius,
Length,
The volume of the Nichrome wire,
Heat generation rate per unit volume,
With the assumptions made above, this is the case of heat transfer in one direction.
Let be the temperature at the radius .
Now, the heat generated within the cylinder of radius is conducted in a radially outward direction. i.e
[where K is the thermal conductivity oof Nichrome]
, where is constant.
Let the surface temperature is , i.e at .
Putting this boundary condition to get , we have
This is the temperature profile within the Nichrome wire, which is maximum at .
So, the maximum temperature,
(1) The water is boiling, to the temperature of the water is, .
The total heat generated within the heating element is convected to the water from the surface. i.e
where, is the convective heat transfer constant (Given).
So, the surface temperature is .
From equation (i), the maximum temperature is at the center of the wire which is
(2) In this case, the temperature of the superheated water vapor, (Given)
The heat transfer coefficient between the superheated water vapor and the heating surface is, .
Similarly,
So, the surface temperature is .
From equation (i), the maximum temperature is at the center of the wire which is
(3) The melting temperature of Chromium is .
So, the 1st case when the heating element is surrounded by water is safe as the maximum temperature within the element is below the melting temperature.
But, it the 2nd case, the heating element will melt out as the maximum temperature is much higher than the melting temperature of the element.