Question

A thin-walled capillary tube (diameter=5mm, length=1m) carries steam at 100degreesC The capillary is covered by a later of insulation (thermal conductivity=0.10W/mK) and thickness delta. Assume that the heat transfer coefficient in the outside ir is 25W/m2K and its temp is 25degreesC. Compute and plot the steady state heat loss from the tube as a function of the thickness of insulation used for the range 0<delta<2cm

Answer 1

Solution :

Since the capillary tube is given as thin-walled, it is assumed its wall thickness is negligible. The insulation is covered over the capillary tube. Hence the inner diameter of the insulation is 5mm

Given:

d₁ = 5mm                                    [Inner diameter of the insulation]
r₁ = 2.5mm = 2.5 x 10^-3 m       [Inner radius of the insulation]
= thickness of the insulation
L = 1m                                        [Length of the capillary tube]
K = 0.1 W/m K                           [Thermal conductivity of the insulation material]
h = 25 W/m²K    [Heat transfer coefficient at the outside surface]
T₂ = 25⁰C                                   [Temperature at the outside]
T₁ = 100⁰C                                 [Temperature at the inside ]

We know that,

Heat transfer Q through a cylindrical wall is given by,

Where is is the effective thermal resistance between the two temperatures T₁ and T₂ and is given by they sum of the thermal resistance between inner radius r₁ and outer radius r₂ given by R₁ and the thermal resistance between the outer surface and the surrounding Air given by R₂.

Hence, , and,

        

Hence,

Substituting the values, We get,

From the given data, is the thickness of the insulation. Hence,

Substituting the value of r₂ in the above equation of Q,

The above equation gives the relation between the heat transfer and the thickness of insulation . All other values are known from the given data. Substituting the values of all other variables,

The above equation has two variables - Q and . This equation can be plotted in and Excel sheet for different values of to get corresponding values of Q.

The following plot is represented below in Excel :

It can be noted from the plot that the heat transferred increases in the beginning, reaches a maximum value and then reduces as the insulation thickness increases. This maximum value is called critical thickness.