Question

For a cylindrical annulus whose inner and outer surfaces are maintained at 30 ºC and 40 ºC, respectively, a heat flux sensor measures the magnitude of the heat flux at 3cmr= as 40 W/m2. Assume that the temperature distribution is steady and depends only on the radial coordinate. Under the hypothesis that the thermal conductivity is independent of temperature, what is k if a) the inner and outer radii

Answer 1

`k=0.12\ln(r_2/r_1)`
`\frac {W}{ m^(\circ) C}`

where
`r_1` and
`r_2` be the inner radius, outer radius of the annalus.

Step-by-step explanation:

Let
`r_1`,
`r_2` and
`L` be the inner radius, outer radius and length of the given annulus.

Temperatures at the inner surface,
`T_1=30^(\circ)C\\` and at the outer surface,
`T_2=40^(\circ)C`.

Let q be the rate of heat transfer at the steady-state.

Given that, the heat flux at r=3cm=0.03m is

`40 W/m^2`.

`\Rightarrow (q)/((2\pi*0.03* L))=40`

`\Rightarrow q=2.4\pi L \;W`

This heat transfer is same for any radial position in the annalus.

Here, heat transfer is taking placfenly in radial direction, so this is case of one dimentional conduction, hence Fourier's law of conduction is applicable.

Now, according to Fourier's law:

`q=-kA(dT)/(dr)\;\cdots(i)`

where,

K=Thermal conductivity of the material.

T= temperature at any radial distance r.

A=Area through which heat transfer is taking place.

Here,
`A=2\pi rL\;\cdots(ii)`

Variation of temperature w.r.t the radius of the annalus is

`\frac {T-T_1}{T_2-T_1}=(\ln(r/r_1))/(\ln(r_2/r_1))`

`\Rightarrow (dT)/(dr)=(T_2-T_1)/(\ln(r_2/r_1))* (1)/(r)\;\cdots(iii)`

Putting the values from the equations (ii) and (iii) in the equation (i), we have

`q=(2\pi kL(T_1-T_2))/(\LN(R_2/2_1))`

`\Rightarrow k= (q\ln(r_2/r_1))/(2\pi L(T_2-T_1))`

`\Rightarrow k=((2.4\pi L)\ln(r_2/r_1))/(2\pi L(10))` [as
`q=2.4\pi L`, and
`T_2-T_1=10 ^(\circ)C`]

`\Rightarrow k=0.12\ln(r_2/r_1)`
`\frac {W}{ m^(\circ) C}`

This is the required expression of k. By putting the value of inner and outer radii, the thermal conductivity of the material can be determined.