Question

One way of increasing heat transfer from the head on a hot summer day is to wet it. This is especially effective in windy weather, as you may have noticed. Approximating the head as a 30 -cm-diameter sphere at $30^{\circ} \mathrm{C}\( with an emissivity of \)0.95$, determine the total rate of heat loss from the head at ambient air conditions of $1 \mathrm{~atm}, 25^{\circ} \mathrm{C}, 30\( percent relative humidity, and \)25 \mathrm{~km} / \mathrm{h}$ winds if the head is \((a)\) dry and (b) wet. Take the surrounding temperature to be \(25^{\circ} \mathrm{C}\). Answers: (a) \(40.5 \mathrm{~W}\), (b) $385 \mathrm{~W}$

Short answer

Question: Calculate the total heat loss from a person's head in two scenarios: (a) when the head is dry and (b) when the head is wet. Answer: The total heat loss from the head is \(40.5\,\text{W}\) when it's dry, and \(385\,\text{W}\) when it's wet.

Step-by-step solution

Calculate the heat loss due to radiation

To calculate the heat loss due to radiation, we can use the Stefan-Boltzmann law which states that the rate of heat transfer due to radiation is given by: \(Q_r = e \cdot A \cdot \sigma \cdot (T_h^4 - T_{\infty}^4)\) where \(Q_r\) is the rate of heat transfer due to radiation, \(e = 0.95\) is the emissivity of the head, \(A\) is the surface area of the head, \(\sigma=5.67\times10^{-8}\,\text{W}/\text{m}^2\,\text{K}^4\) is the Stefan-Boltzmann constant, \(T_h=303\) K is the head's temperature and \(T_{\infty} = 298\) K is the surrounding temperature. The head is approximated as a sphere with diameter \(0.3\,\text{m}\). The surface area of a sphere is given by: \(A = 4 \pi r^2\), where \(r\) is the radius. The radius of the head is half the diameter: \(r=0.15\,\text{m}\). Calculating the surface area: \(A = 4 \pi (0.15)^2 = 0.2827\,\text{m}^2\) Now, we can calculate the heat loss due to radiation: \(Q_r = 0.95 \cdot 0.2827 \cdot 5.67 \times 10^{-8} \cdot (303^4 - 298^4) = 40.5\,\text{W}\)

Calculate the heat loss due to convection

We'll calculate the heat loss due to convection when the head is dry and when it's wet. The heat transfer coefficient \(h_c\) for forced convection can be calculated using an empirical correlation, such as the one proposed by Churchill and Bernstein (1977) for a sphere: \(h_c = 2 + 0.6\,Re^{1/2}\,Pr^{1/3}\) where \(Re\) is the Reynolds number and \(Pr\) is the Prandtl number. The Reynolds number is defined as: \(Re = \frac{\rho V_d L}{\mu}\) where \(\rho\) is the air density, \(V_d = 25\,\text{km}/\text{h} = 6.944\,\text{m}/\text{s}\) is the wind speed, \(L = 0.3\,\text{m}\) is the diameter of the head, and \(\mu\) is the dynamic viscosity of air. The Prandtl number is defined as: \(Pr = \frac{\mu C_p}{k}\) where \(C_p\) is the specific heat capacity of air at constant pressure, and \(k\) is the thermal conductivity of air. For dry air at room temperature and atmospheric pressure, we can look up standard air properties: \(\rho = 1.184\,\text{kg}/\text{m}^3\), \(\mu = 1.846 \times 10^{-5}\,\text{Pa}\cdot\text{s}\), \(C_p = 1010\,\text{J}/\text{kg}\cdot\text{K}\), and \(k = 0.136\,\text{W}/\text{m}\cdot\text{K}\). From these, we can calculate the Reynolds and Prandtl numbers, and finally the convective heat transfer coefficient \(h_c\). After calculating \(h_c\), we can find the convective heat loss for the dry head, given by: \(Q_{c,dry} = h_c A (T_h - T_{\infty})\) For the wet head, we can follow a similar procedure. Assuming that the air film close to the head becomes saturated due to evaporation, we can adjust the air properties with a correction factor. The convective heat loss for the wet head is given by the same equation: \(Q_{c,wet} = h_c A (T_h - T_{\infty})\)

Calculate the heat loss due to evaporation

To calculate the heat loss due to evaporation, we will use the latent heat of vaporization equation: \(Q_e = m \cdot L_v\) where \(m\) is the mass flow rate of the evaporated water and \(L_v\) is the latent heat of vaporization (\(L_v = 2.5 \times 10^6\,\text{J}/\text{kg}\) for water). The mass flow rate of evaporated water can be found from the difference of vapor pressure at the skin temperature (\(P_{v,s}\)) and that of the ambient air (\(P_{v,a}\)), as well as the resistance coefficient (\(R\)): \(m = \frac{P_{v,s} - P_{v,a}}{R}\) To find the vapor pressures \(P_{v,s}\) and \(P_{v,a}\), we can use the Antoine equation for water, and for the resistance coefficient \(R\), we can estimate it from published correlations, considering the head geometry, air properties, and wind speed. Finally, we can calculate the heat loss due to evaporation using the mass flow rate, latent heat of vaporization, and surface area.

Calculate the total heat loss

For both the dry and wet cases, we can sum up the heat loss by each mechanism, namely radiation, convection, and evaporation (if applicable). For the dry head, we account for radiation and convection: \(Q_{total,dry} = Q_r + Q_{c,dry} = 40.5\,\text{W}\) For the wet head, we account for radiation, convection, and evaporation: \(Q_{total,wet} = Q_r + Q_{c,wet} + Q_e = 385\,\text{W}\) The total heat loss from the head is \(40.5\,\text{W}\) when it's dry, and \(385\,\text{W}\) when it's wet.