A 2-kW resistance heater wire with thermal conductivity 15 W/m.°C, diameter 0⋅4 cm, and length 50 cm is used to boil the water by immersing it in water. Assuming the variation of the thermal conductivity of the wire with temperature to be negligible. What is the differential equation that describes the variation of the temperature in the wire during steady operation?

Explanation:

One-Dimensional Heat Flow through a cylinder with Heat Generation:

A hollow Cylinder:

• Consider a hollow cylinder of length L having inner and outer radii r₁ and r₂ respectively in which flow of heat is unidirectional along the radial direction.
• T₁ and T_2­ are temperatures of the inner and outer surfaces of the cylinder respectively.
• In order to determine temperature distribution and heat flow rate, a small element at radius r and thickness 'dr' is considered.
• A heat source present inside the strip is generating 'g' amount of heat per unit volume as shown in Figure

Heat conducted into the element, Q_r = -k(2 × π × r × L) × \(\frac{dT}{dr}\)............................................ (1)

The heat generated in the element, Q_g =  2 × π × r × L × dr × g ............................................ (2)

Heat conducted out of the element, Q_{r + dr} = Q_r + \(\frac{d}{dr}(Q_r)dr\) ............................................ (3)

For steady state condition of heat flow

Heat conducted into the element + Heat generated in the element = Heat conducted out of the element

⇒ Q_r + Q_g = Q_{r + dr}

⇒ Qr + Qg = Qr + \(\frac{d}{dr}(Q_r)dr\) 

⇒ Q_g = \(\frac{d}{dr}(Q_r)dr\) ............................................ (4)

Substituting the values of Q_r and Q_g from equations (1) and (2) in equation (4)

 2 × π × r × L × dr × g = \(\frac{d}{dr}(-2 ~×~\pi~×~r~×~L~×~k~×~\frac{dT}{dr})dr\)

⇒ r × g = \(-k~\times~\frac{d}{dr}(r\frac{dT}{dr})\)

Dividing both sides by k:

⇒ \(\frac{d}{dr}(r\frac{dT}{dr})~+~r\frac{g}{k}~=~0\)

⇒ \(\frac{d^2T}{dr^2}~+~\frac{g}{k}~=~0\)

The above equation can be re-written in the form of:

\(\frac{1}{r}\frac{d}{{dr}}\left( {r\frac{{dT}}{{dr}}} \right) + \frac{g}{k} = 0\)

The above differential equation is the required one that describes the variation of the temperature in the wire during steady operation.