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natita [175]
3 years ago
6

A common method to measure thermal conductivity of a biomaterial is to insert a long metallic probe axially into the center of a

long cylindrical container of the material. The probe is maintained at some uniform temperature Ti, and the outside of the container is maintained at a temperature To. Inside the metallic probe is an electrical heater for which the electrical power is measured. If the diameter of the probe maintained at 50°C is 1 cm, the outer diameter of the container maintained at 20°C is 4 cm, the length of the cylinder container is 60 cm, and the power input is 40.8 W, calculate the thermal conductivity of this material.
Physics
1 answer:
tia_tia [17]3 years ago
8 0

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 (\dot Q), 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})

Where:

k - Thermal conductivity, measured in watts per meter-Celsius.

L - Length of the cylinder, measured in meters.

D_{i} - Inner diameter, measured in meters.

D_{o} - Outer diameter, measured in meters.

T_{i} - Temperature at inner surface, measured in Celsius.

T_{o} - 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)

If we know that \dot Q = 40.8\,W, L = 0.6\,m, T_{i} = 50\,^{\circ}C, T_{o} = 20\,^{\circ}C, D_{i} = 0.01\,m and D_{o} = 0.04\,m, 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)

k \approx 1.571\,\frac{W}{m\cdot ^{\circ}C}

The thermal conductivity of the biomaterial is approximately 1.571 watts per meter-Celsius.

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