Question

(II) Heat conduction to skin. Suppose 150 W of heat flows by conduction from the blood capillaries beneath the skin to the body’s surface area of \({\bf{1}}{\bf{.5}}\;{{\bf{m}}^{\bf{2}}}\). If the temperature difference is 0.50 C°, estimate the average distance of capillaries below the skin surface.

Short answer

The average distance of capillaries below the skin surface is \(1.0 \times {10^{ - 3}}\;{\rm{m}}\).

Step-by-step solution

Understanding of the conduction process

The conduction heat transfer process may be defined as the transfer of heat energy by direct contact between two objects. It can also occur within one object.

Given data

The rate of heat flow is \(\frac{Q}{t} = 150\;{\rm{W}}\).

The surface area of the body is \(A = 1.5\;{{\rm{m}}^{\rm{2}}}\).

The difference in the temperatures is \(\Delta T = 0.50{\rm{^\circ C}}\).

From table 14-4:

The thermal conductivity of the human body is \(k = 0.2\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{s}} \cdot {\rm{m}} \cdot {\rm{^\circ C}}}}} \right.\\} {{\rm{s}} \cdot {\rm{m}} \cdot {\rm{^\circ C}}}}\).

Evaluation of the average distance of capillaries below the skin surface

The expression for the heat conduction rate is given as:

\(\frac{Q}{t} = \frac{{kA\Delta T}}{l}\)

Rewrite the above equation as:

\(l = \frac{{kA\Delta T}}{{\left( {\frac{Q}{t}} \right)}}\)

Substitute the values in the above equation.

\(\begin{array}{c}l = \frac{{\left( {0.2\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{s}} \cdot {\rm{m}} \cdot {\rm{^\circ C}}}}} \right.\\} {{\rm{s}} \cdot {\rm{m}} \cdot {\rm{^\circ C}}}}} \right)\left( {1.5\;{{\rm{m}}^{\rm{2}}}} \right)\left( {0.50{\rm{^\circ C}}} \right)}}{{\left( {150\;{\rm{W}}} \right)}}\\ = 1.0 \times {10^{ - 3}}\;{\rm{m}}\end{array}\)

Thus, the average distance of capillaries below the skin surface is \(1.0 \times {10^{ - 3}}\;{\rm{m}}\).