Answer:
The thermal conductivity of the biomaterial is approximately 1.571 watts per meter-Celsius.
Explanation:
Let suppose that thermal conduction is uniform and one-dimensional, the conduction heat transfer (
), measured in watts, in the hollow cylinder is:
![\dot Q = \frac{2\cdot k\cdot L}{\ln \left(\frac{D_{o}}{D_{i}} \right)}\cdot (T_{i}-T_{o})](https://tex.z-dn.net/?f=%5Cdot%20Q%20%3D%20%5Cfrac%7B2%5Ccdot%20k%5Ccdot%20L%7D%7B%5Cln%20%5Cleft%28%5Cfrac%7BD_%7Bo%7D%7D%7BD_%7Bi%7D%7D%20%5Cright%29%7D%5Ccdot%20%28T_%7Bi%7D-T_%7Bo%7D%29)
Where:
- Thermal conductivity, measured in watts per meter-Celsius.
- Length of the cylinder, measured in meters.
- Inner diameter, measured in meters.
- Outer diameter, measured in meters.
- Temperature at inner surface, measured in Celsius.
- Temperature at outer surface, measured in Celsius.
Now we clear the thermal conductivity in the equation:
![k = \frac{\dot Q}{2\cdot L\cdot (T_{i}-T_{o})}\cdot \ln\left(\frac{D_{o}}{D_{i}} \right)](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B%5Cdot%20Q%7D%7B2%5Ccdot%20L%5Ccdot%20%28T_%7Bi%7D-T_%7Bo%7D%29%7D%5Ccdot%20%5Cln%5Cleft%28%5Cfrac%7BD_%7Bo%7D%7D%7BD_%7Bi%7D%7D%20%5Cright%29)
If we know that
,
,
,
,
and
, the thermal conductivity of the biomaterial is:
![k = \left[\frac{40.8\,W}{2\cdot (0.6\,m)\cdot (50\,^{\circ}C-20\,^{\circ}C)}\right]\cdot \ln \left(\frac{0.04\,m}{0.01\,m} \right)](https://tex.z-dn.net/?f=k%20%3D%20%5Cleft%5B%5Cfrac%7B40.8%5C%2CW%7D%7B2%5Ccdot%20%280.6%5C%2Cm%29%5Ccdot%20%2850%5C%2C%5E%7B%5Ccirc%7DC-20%5C%2C%5E%7B%5Ccirc%7DC%29%7D%5Cright%5D%5Ccdot%20%5Cln%20%5Cleft%28%5Cfrac%7B0.04%5C%2Cm%7D%7B0.01%5C%2Cm%7D%20%5Cright%29)
![k \approx 1.571\,\frac{W}{m\cdot ^{\circ}C}](https://tex.z-dn.net/?f=k%20%5Capprox%201.571%5C%2C%5Cfrac%7BW%7D%7Bm%5Ccdot%20%5E%7B%5Ccirc%7DC%7D)
The thermal conductivity of the biomaterial is approximately 1.571 watts per meter-Celsius.