Chapter 7

Consider a person who is trying to keep cool on a hot summer day by turning a fan on and exposing his entire body to airflow. The air temperature is \(85^{\circ} \mathrm{F}\), and the fan is blowing air at a velocity of $6 \mathrm{ft} / \mathrm{s}$. If the person is doing light work and generating sensible heat at a rate of \(300 \mathrm{Btu} / \mathrm{h}\), determine the average temperature of the outer surface (skin or clothing) of the person. The average human body can be treated as a 1 -ft-diameter cylinder with an exposed surface area of \(18 \mathrm{ft}^{2}\). Disregard any heat transfer by radiation. What would your answer be if the air velocity were doubled? Evaluate the air properties at \(100^{\circ} \mathrm{F}\). Answers: 95.1" \(\mathrm{F}, 91.6^{\circ} \mathrm{F}\)

A 12 -ft-long, \(1.5-\mathrm{kW}\) electrical resistance wire is made of \(0.1\)-in-diameter stainless steel $\left(k=8.7 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}-^{\circ} \mathrm{F}\right)$. The resistance wire operates in an environment at \(85^{\circ} \mathrm{F}\). Determine the surface temperature of the wire if it is cooled by a fan blowing air at a velocity of $20 \mathrm{ft} / \mathrm{s}$. For evaluations of the air properties, the film temperature has to be found iteratively. As an initial guess, assume the film temperature to be \(200^{\circ} \mathrm{F}\).

During a plant visit, it was noticed that a \(12-\mathrm{m}\)-long section of a 12 -cm-diameter steam pipe is completely exposed to the ambient air. The temperature measurements indicate that the average temperature of the outer surface of the steam pipe is \(75^{\circ} \mathrm{C}\) when the ambient temperature is \(5^{\circ} \mathrm{C}\). There are also light winds in the area at \(25 \mathrm{~km} / \mathrm{h}\). The emissivity of the outer surface of the pipe is \(0.8\), and the average temperature of the surfaces surrounding the pipe, including the sky, is estimated to be \(0^{\circ} \mathrm{C}\). Determine the amount of heat lost from the steam during a 10 -h-long workday. Steam is supplied by a gas-fired steam generator that has an efficiency of 80 percent, and the plant pays \(\$ 1.05 /\) therm of natural gas. If the pipe is insulated and 90 percent of the heat loss is saved, determine the amount of money this facility will save a year as a result of insulating the steam pipes. Assume the plant operates every day of the year for \(10 \mathrm{~h}\). State your assumptions.

Exposure to high concentration of gaseous ammonia can cause lung damage. To prevent gaseous ammonia from leaking out, ammonia is transported in its liquid state through a pipe \((k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), $D_{i \text {, pipe }}=2.5 \mathrm{~cm}, D_{a \text {, pipe }}=4 \mathrm{~cm}$, and \(\left.L=10 \mathrm{~m}\right)\). Since liquid ammonia has a normal boiling point of \(-33.3^{\circ} \mathrm{C}\), the pipe needs to be properly insulated to prevent the surrounding heat from causing the ammonia to boil. The pipe is situated in a laboratory, where air at \(20^{\circ} \mathrm{C}\) is blowing across it with a velocity of \(7 \mathrm{~m} / \mathrm{s}\). The convection heat transfer coefficient of the liquid ammonia is $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Calculate the minimum insulation thickness for the pipe using a material with $k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\), while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\).

Ice slurry is being transported in a pipe and \(L=5 \mathrm{~m}\) ) with an inner surface temperature of \(0^{\circ} \mathrm{C}\). The ambient condition surrounding the pipe has a temperature of \(20^{\circ} \mathrm{C}\) and a dew point at \(10^{\circ} \mathrm{C}\). A blower is located near the pipe, which provides blowing air across the pipe at a velocity of $2 \mathrm{~m} / \mathrm{s}$. If the outer surface temperature of the pipe drops below the dew point, condensation can occur on the surface. Since this pipe is located near high-voltage devices, water droplets from the condensation can cause an electrical hazard. To prevent accidents, the pipe surface needs to be insulated. Calculate the minimum insulation thickness for the pipe using a material with \(k=0.95 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to prevent the outer surface temperature from dropping below the dew point.

An incandescent lightbulb is an inexpensive but highly inefficient device that converts electrical energy into light. It converts about 10 percent of the electrical energy it consumes into light while converting the remaining 90 percent into heat. (A fluorescent lightbulb will give the same amount of light while consuming only one-fourth of the electrical energy, and it will last 10 times longer than an incandescent lightbulb.) The glass bulb of the lamp heats up very quickly as a result of absorbing all that heat and dissipating it to the surroundings by convection and radiation. Consider a 10 -cm-diameter, 100 -W lightbulb cooled by a fan that blows air at \(30^{\circ} \mathrm{C}\) to the bulb at a velocity of $2 \mathrm{~m} / \mathrm{s}\(. The surrounding surfaces are also at \)30^{\circ} \mathrm{C}$, and the emissivity of the glass is \(0.9\). Assuming 10 percent of the energy passes through the glass bulb as light with negligible absorption and the rest of the energy is absorbed and dissipated by the bulb itself, determine the equilibrium temperature of the glass bulb. Assume a surface temperature of \(100^{\circ} \mathrm{C}\) for evaluation of \(\mu_{x}\). Is this a good assumption?

A 0.4-m-diameter spherical tank of negligible thickness contains iced water at \(0^{\circ} \mathrm{C}\). Air at \(25^{\circ} \mathrm{C}\) flows over the tank with a velocity of \(3 \mathrm{~m} / \mathrm{s}\). Determine the rate of heat transfer to the tank and the rate at which ice melts. The heat of fusion of water at \(0^{\circ} \mathrm{C}\) is \(333.7 \mathrm{~kJ} / \mathrm{kg}\).

stream is to be measured by a thermocouple with a spherical junction. Due to the nature of this experiment, the response time of the thermocouple to register 99 percent of the initial temperature difference must be within $5 \mathrm{~s}\(. The properties of the thermocouple junction are \)k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\rho=8500 \mathrm{~kg} / \mathrm{m}^{3}\(, and \)c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. The hot air has a freestream velocity and temperature of $3 \mathrm{~m} / \mathrm{s}\( and \)140^{\circ} \mathrm{C}$, respectively. If the initial temperature of the thermocouple junction is \(20^{\circ} \mathrm{C}\), determine the thermocouple junction diameter that would satisfy the required response time of \(5 \mathrm{~s}\). Hint: Use lumped system analysis to determine the time required for the thermocouple to register 99 percent of the initial temperature difference (verify the application of this method to this problem).

A thermocouple with a spherical junction diameter of \(1 \mathrm{~mm}\) is used for measuring the temperature of a hydrogen gas stream. The properties of the thermocouple junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8500 \mathrm{~kg} / \mathrm{m}^{3}\(, and \)c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. The hydrogen gas, behaving as an ideal gas at \(1 \mathrm{~atm}\), has a free-stream temperature of \(200^{\circ} \mathrm{C}\). If the initial temperature of the thermocouple junction is $10^{\circ} \mathrm{C}$, evaluate the time for the thermocouple to register 99 percent of the initial temperature difference at different free-stream velocities of the hydrogen gas. Using appropriate software, perform the evaluation by varying the free-stream velocity from 1 to \(100 \mathrm{~m} / \mathrm{s}\). Then, plot the thermocouple response time and the convection heat transfer coefficient as a function of free-stream velocity. Hint: Use the lumped system analysis to determine the time required for the thermocouple to register 99 percent of the initial temperature difference (verify the application of this method to this problem).

A glass \((k=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) spherical tank is filled with chemicals undergoing exothermic reaction. The reaction keeps the inner surface temperature of the tank at \(80^{\circ} \mathrm{C}\). The tank has an inner radius of \(0.5 \mathrm{~m}\), and its wall thickness is $10 \mathrm{~mm}$. Situated in surroundings with an ambient temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of $70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, the tank's outer surface is being cooled by air flowing across it at \(5 \mathrm{~m} / \mathrm{s}\). In order to prevent thermal burn on individuals working around the container, it is necessary to keep the tank's outer surface temperature below $50^{\circ} \mathrm{C}$. Determine whether or not the tank's outer surface temperature is safe from thermal burn hazards.