Critical Thickness of Insulation | Thermal Engineering

Contrary to the common belief that addition of insulating material on a surface always brings about a decrease in the heat transfer rate, there are instances when the addition of insulation to the outside surfaces of cylindrical or spherical walls (geometries which have non-constant cross-sectional areas) does not reduce the heat loss.

In fact, under certain circumstances, it actually increases the heat flow upto a certain thickness of insulation. To establish this fact, consider a thin-walled metallic cylinder of length I, radius r_i and transporting a fluid at temperature t_i which is higher than the ambient temperature t₀. Surrounding this cylinder is an annular section of insulating sheathing of thickness (r – r_i) and thermal conductivity k.

With stipulations of:

(i) Steady state conditions

(ii) One-dimensional heat flow only in the radial direction

(iii) Negligible thermal resistance due to cylinder wall

(iv) Negligible radiation exchange between outer surface of insulation and surroundings the heat transmission can be expressed as-

where h_i and h₀ are the film coefficients at the inner and outer surface respectively. The denominator represents the sum of thermal resistance to heat flow. The values of k, r_i, h_i and h₀ are constant; therefore the total thermal resistance will depend upon thickness of insulation which specifies the outer radius r of the arrangement.

An examination of equation 3.32 would reveal that with increase in r (i.e., thickness of insulation), the thermal resistance of insulating increases but the resistance due to convection coefficient at the outer surface drops. The thermal resistance due to inside film coefficient remains unaffected with change in radius r. Obviously, addition of insulation material can either increase or decrease the rate of heat transmission depending upon a change in the total resistance with outer radius r.

To determine whether the foregoing result maximises or minimises the total resistance, the second derivative needs to be calculated-

Then r = k/h_o represents the condition for minimum resistance and consequently maximum heat flow rate. The insulation radius at which resistance to heat flow is minimum is called the critical radius. The critical radius, designated by r_c is dependent only on the thermal quantities k and h_o. Thus-

The fact that heat flow rate attains a maximum at r opposing effects; increasing r increases the thermal resistance of the insulation layer but decreases the thermal resistance of the surface area. At r = r_c the total resistance reaches a minimum. Apparently, a pipe carrying a high temperature fluid will lose more heat (compared to a bare pipe) if the conductivity and thickness of insulation are improperly chosen. Dependence of heat loss on the thickness of insulation has been shown in Fig. 3.31.

Two cases of practical interest are:

(i) r_i < r_c: the addition of insulation to a bare pipe leads to increasing heat transfer until the outer radius of insulation becomes equal to the critical radius. This may be attributed to the fact that in the range r_i < r_c, the progressive decrease in the convection resistance with addition of insulation predominates over the correspondence increase in conduction resistance. The net result is drop in total resistance and consequently the heat loss increases.

Any further increase in insulation thickness causes the heat loss to decrease from this peak value. However until a certain amount of insulation (r* represented by point b) is added, the heat loss is still greater than that for the bare pipe. Evidently an insulation thickness greater than (r* – r_i) must be added to reduce the heat loss below the uninsulated rate.

The phenomenon of increase in heat transmission with addition of insulation is most likely to occur when insulating materials of poor quality are applied to pipes and wires of small radius. Such a situation is used to advantage in the insulation of electrical wires and cables.

The electrical wires are given a coating of insulation with the prime objective to provide protection from electrical hazards. However an increase in the rate of heat dissipation can be made feasible and the conductors maintained within safe temperature limits by a proper choice of the insulation thickness. That permits some increase in the current carrying capacity of the cable.

(ii) r_i > r_c: The effect of wall thickness dominates and the overall thermal resistance increases. Sheathing of insulation then acts as lagging and that obstructs the flow of heat.

Heat insulation is the main objective in steam and refrigeration pipes. For insulation to be properly effective in restricting heat transmission, the outer radius r_o must be greater than or equal to the critical radius. If r_o < r_c no useful purpose will be served with the chosen material for insulation.

Following a similar approach the critical radius of insulation for a sphere can be worked out as:

Example 1:

“Addition of insulating material does not always bring about a decrease in the heat transfer rate for geometries with non-constant cross-section area.” Comment upon the validity of this statement.

A pipe of outside diameter 20 mm is to be insulated with asbestos which has a mean thermal conductivity of 0.1 W/m-deg. The local coefficient of convective heat to the surroundings is 5 W/m²-deg. Comment upon the utility of asbestos as the insulating material.

What should be the minimum value of thermal conductivity of insulating material to reduce heat transfer?

Solution:

The critical radius of insulation for optimum heat transfer from pipe is given by:

For insulation to be properly effective in restricting heat transmission, the pipe radius r₀ must be greater than or equal to the critical radius r_c. Here r₀ < r_c and as such there is no point in using asbestos as the insulating material. Addition of asbestos insulation will increase the heat transfer rate and that is not desirable. An insulating material with smaller thermal conductivity need to be employed.

For insulation to be effective, the pipe radius should be greater than the critical radius, i.e.,

Apparently, the maximum conductivity of insulation permitted is 0.05 W/m-deg.