Question

It is well known that wind makes the cold air feel much colder as a result of the wind-chill effect that is due to the increase in the convection heat transfer coefficient with increasing air velocity. The wind-chill effect is usually expressed in terms of the wind-chill temperature (WCT), which is the apparent temperature felt by exposed skin. For an outdoor air temperature of \(0^{\circ} \mathrm{C}\), for example, the windchill temperature is $-5^{\circ} \mathrm{C}\( with \)20 \mathrm{~km} / \mathrm{h}\( winds and \)-9^{\circ} \mathrm{C}\( with \)60 \mathrm{~km} / \mathrm{h}$ winds. That is, a person exposed to \(0^{\circ} \mathrm{C}\) windy air at \(20 \mathrm{~km} / \mathrm{h}\) will feel as cold as a person exposed to \(-5^{\circ} \mathrm{C}\) calm air (air motion under \(5 \mathrm{~km} / \mathrm{h}\) ). For heat transfer purposes, a standing man can be modeled as a 30 -cm- diameter, 170 -cm-long vertical cylinder with both the top and bottom surfaces insulated and with the side surface at an average temperature of $34^{\circ} \mathrm{C}\(. For a convection heat transfer coefficient of \)15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the rate of heat loss from this man by convection in still air at \(20^{\circ} \mathrm{C}\). What would your answer be if the convection heat transfer coefficient is increased to $30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ as a result of winds? What is the wind-chill temperature in this case? Answers: $336 \mathrm{~W}, 672 \mathrm{~W}, 6^{\circ} \mathrm{C}$

Short answer

Question: Calculate the rate of heat loss from a man in two different scenarios and find the wind-chill temperature in the second scenario if the diameter of the vertical cylinder representing the man is 0.3 m, the height is 1.7 m, the skin temperature is 34°C, and the surrounding air temperature is 20°C. In the first scenario, the convection heat transfer coefficient is 15 W/m²·K, and in the second scenario, the coefficient is increased to 30 W/m²·K due to winds. Answer: To calculate the rate of heat loss in both scenarios, first we need to find the surface area of the vertical cylinder using the formula \(A = \pi \times d \times h\). After finding the surface area, we can use the convection heat transfer formula \(Q = h_c \times A \times (T_s - T_a)\) to find the rate of heat loss in each scenario. Finally, to find the wind-chill temperature in the second scenario, use the formula \(Q = h_c \times A \times (T_s - WCT)\) and rearrange it to solve for the wind-chill temperature.

Step-by-step solution

Calculate the surface area of the cylinder

First, we need to find the surface area of the vertical cylinder (excluding the top and bottom surfaces) to determine the heat loss by convection. We have given the diameter (d=0.3m) and height (h=1.7m) of the cylinder. We can use the formula for the surface area of a cylinder without its top and bottom surfaces: \(Surface \thinspace Area(A) = \pi \times d \times h\)

Calculate the rate of heat loss for the first scenario

Now we can find the rate of heat loss (Q) in the first scenario, where the convection heat transfer coefficient (h_c) is 15 W/m²·K, and the surrounding air temperature (T_a) is 20°C. The average temperature of the man's skin (T_s) is 34°C. We will use the formula for convection heat transfer: \(Q = h_c \times A \times (T_s - T_a)\) Using the given values for h_c, T_s, and T_a and the surface area we calculated earlier, we can find the rate of heat loss.

Calculate the rate of heat loss for the second scenario

For the second scenario, we need to find the rate of heat loss when the convection heat transfer coefficient is increased to 30 W/m²·K due to winds. Just like in step 2, we can use the formula for convection heat transfer and put the given values for h_c, T_s, and T_a and the surface area that we already have: \(Q = h_c \times A \times (T_s - T_a)\)

Calculate the wind-chill temperature

Finally, we need to find the wind-chill temperature (WCT) in the second scenario. To do this, we will use the following formula for wind-chill effect: \(Q = h_c \times A \times (T_s - WCT)\) By rearranging the formula and plugging in the values that we already have, we can calculate the WCT.