Putting insulation on a small-diameter cylinder (or sphere) can increase the heat transfer. The diameter of the cylinder (or sphere) must be above a critical radius (r_crit = k_I /h for a cylinder) before insulation will reduce the heat transfer. This is why insulated electric wires can withstand more current than uninsulated ones. At the same wire temperature, insulated wires transfer more heat. Small- versus large-diameter hot-water pipes experience the same phenomenon.

This classroom experiment is designed to demonstrate clearly the effect of sma1l wire radius on insulated- and uninsulated-wire heat transfer rate.

The heat transfer rate q from the inner surface of a tube to the surrounding air is:

q/L / (2π(T_i-T_a)) = 1 / ( 1/hr_o + 1/k_g ln r_o/r_i) (1.1)

Where Fig. 1.1 shows ri and ro to be the inner and outer radii of a tube, respectively, and Ti and Ta the inner and outer air absolute temperatures,respectively. If all quantities are constant except for the expression on the left-hand side of Eq. (1.1) and ro, differentiating Eq. (1.1) with respect to ro results in the critical radius

r_o,crit = k_g /h (1.2)

Equation (1.2) is the condition for (q/L)/(Ti - Ta) to be a maximum since the second derivative of Eq. (1.1) is negative.

1. Small-diameter electric wire
2. Glass tube of i.d. slightly larger than the electric wire, and o.d. approximately 3-5 mm
3. Insulated supports for the wire to be held horizontally with the glass tube occupying half the central portion of the wire
4. Variac to vary the current in the wire.

The experimental setup is shown in Fig. 1.2.

Fig. 1.1. Geometry for a tube.

Fig 1.2 Apparatus

Turn up variac until bare wire glows red or orange. (The portion of the wire in the glass tube will be dark colored indicating a lower temperature than the exposed wire.)

Since i²R (hence heat rate q/A) is essentially the same in both the exposed and glass covered portion of the wire, the thermal resistance from the wire through the glass tube to air is less than from the uninsulated wire to air since it requires less ΔT to transfer essentially the same heat.

The plot below shows eqn 1.1 for k_g/h = 1.0. Note at point A with r_i=0.50 mm adding insulation increases q/ΔT until ro is 3.0 mm or greater, which is a waste of insulation.

This experiment demonstrates why insulated electric wires can carry morecurrent (i2R) than uninsulated wires. Also, hot-water or steam pipes less than around 0.75 in should not be insulated to reduce heat loss. However, cold water pipes should be insulated to prevent condensation and water damage.

An experiment container can be made from a cut cardboard box as shown below:

Here a medium size cardboard box has been trimed. Two 3/8 inch wooden construction dowel pins with small brass eyes screwed into one end have been inserted into 11/32 holes punched into the box sides. These are to be used to support the NiCr wire used in this experiment arranged as shown in fig 1.2. The wire is threaded through a short piece of 3mm hollow glass tubing (about 4 inches) which covers a section of wire in the middle to simulate the critical insulation. The ends of the wire, after passing through the brass eyes are brought down to attach to the Variac. If the Variac doesn't measure voltage and current, then a voltmeter and ammeter need be present in this circuit.