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Chapter 9: Problem 47

A can of engine oil with a length of \(150 \mathrm{~mm}\) and a diameter of $100 \mathrm{~mm}$ is placed vertically in the trunk of a car. On a hot summer day, the temperature in the trunk is \(43^{\circ} \mathrm{C}\). If the surface temperature of the can is \(17^{\circ} \mathrm{C}\), determine heat transfer rate from the can surface. Neglect the heat transfer from the ends of the can.

Short Answer

Expert verified
Answer: The heat transfer rate from the can surface is approximately 24.4 W.

Step by step solution

01

Calculate the Surface Area of the Can

The given can has a cylindrical shape without ends. The surface area of a cylinder without ends is given by the formula: \(A = 2\pi r L\) where \(r\) - the radius of the cylinder \(L\) - the length of the cylinder First, we need to convert the given measurements from millimeters to meters: Length, \(L = 150 \mathrm{~mm} = 0.15 \mathrm{~m}\) Diameter, \(D = 100 \mathrm{~mm} = 0.10 \mathrm{~m}\) Radius, \(r = \frac{D}{2} = \frac{0.10 \mathrm{~m}}{2} = 0.05 \mathrm{~m}\) Now, we can calculate the surface area of the can without ends: \(A = 2\pi (0.05 \mathrm{~m})(0.15 \mathrm{~m}) = 0.0471 \mathrm{~m}^2\)
02

Calculate the Temperature Difference

The temperature difference between the surface of the can and the surrounding air is given by: \(\Delta T = T_\text{air} - T_\text{surface}\) where \(T_\text{air}\) - temperature of the surrounding air in the trunk (K) \(T_\text{surface}\) - surface temperature of the can (K) We first need to convert the given temperatures from Celsius to Kelvin: \(T_\text{air} = 43^{\circ} \mathrm{C} + 273.15 = 316.15 \mathrm{~K}\) \(T_\text{surface} = 17^{\circ} \mathrm{C} + 273.15 = 290.15 \mathrm{~K}\) Now, we can calculate the temperature difference: \(\Delta T = 316.15 \mathrm{~K} - 290.15 \mathrm{~K} = 26 \mathrm{~K}\)
03

Calculate the Heat Transfer Rate

With the surface area, temperature difference, and heat transfer coefficient, we can now calculate the heat transfer rate using the formula: \(q = hA\Delta T\) Using \(h = 20 \mathrm{W/m^2K}\) and substituting the values we calculated before, \(q = (20 \mathrm{W/m^2K})(0.0471 \mathrm{m^2})(26 \mathrm{~K}) = 24.4 \mathrm{~W}\) The heat transfer rate from the can surface is approximately 24.4 W.

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Most popular questions from this chapter

A boiler supplies hot water to equipment through a horizontal pipe. The hot water exits the pipe and enters the equipment at \(98^{\circ} \mathrm{C}\). The outer diameter of the pipe is \(20 \mathrm{~mm}\), and the pipe distance between the boiler and the equipment is \(30 \mathrm{~m}\). The section of the pipe between the boiler and the equipment is exposed to natural convection with air at an ambient temperature of \(20^{\circ} \mathrm{C}\). The hot water flows steadily in the pipe at \(10 \mathrm{~g} / \mathrm{s}\), and its specific heat is \(4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). The temperature at the pipe surface is \(80^{\circ} \mathrm{C}\), and the pipe has an emissivity of \(0.6\) that contributes to the thermal radiation with the surroundings at \(20^{\circ} \mathrm{C}\). According to the service restrictions of the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HG-101), hot water boilers should not be operating at temperatures exceeding \(120^{\circ} \mathrm{C}\) at or near the boiler outlet. Determine whether the water temperature exiting the boiler is in compliance with the ASME Boiler and Pressure Vessel Code.

Consider a double-pane window consisting of two glass sheets separated by a \(1-\mathrm{cm}\)-wide airspace. Someone suggests inserting a thin vinyl sheet between the two glass sheets to form two \(0.5\)-cm-wide compartments in the window in order to reduce natural convection heat transfer through the window. From a heat transfer point of view, would you be in favor of this idea to reduce heat losses through the window?

Exhaust gases from a manufacturing plant are being discharged through a 10 -m-tall exhaust stack with outer diameter of \(1 \mathrm{~m}\). The exhaust gases are discharged at a rate of \(0.125 \mathrm{~kg} / \mathrm{s}\), while temperature drop between inlet and exit of the exhaust stack is $30^{\circ} \mathrm{C}$, and the constant pressure-specific heat of the exhaust gases is \(1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). On a particular calm day, the surrounding quiescent air temperature is \(33^{\circ} \mathrm{C}\). Solar radiation is incident on the exhaust stack outer surface at a rate of $500 \mathrm{~W} / \mathrm{m}^{2}$, and both the emissivity and solar absorptivity of the outer surface are \(0.9\). Determine the exhaust stack outer surface temperature. Assume the film temperature is \(60^{\circ} \mathrm{C}\).

A vertical double-pane window consists of two sheets of glass separated by a \(1.2-\mathrm{cm}\) air gap at atmospheric pressure. The glass surface temperatures across the air gap are measured to be \(278 \mathrm{~K}\) and $288 \mathrm{~K}$. If it is estimated that the heat transfer by convection through the enclosure is \(1.5\) times that by pure conduction and that the rate of heat transfer by radiation through the enclosure is about the same magnitude as the convection, the effective emissivity of the two glass surfaces is (a) \(0.35\) (b) \(0.48\) (c) \(0.59\) (d) \(0.84\) (e) \(0.72\)

A solar collector consists of a horizontal aluminum tube of outer diameter $5 \mathrm{~cm}\( enclosed in a concentric thin glass tube of \)7 \mathrm{~cm}$ diameter. Water is heated as it flows through the aluminum tube, and the annular space between the aluminum and glass tubes is filled with air at $1 \mathrm{~atm}$ pressure. The pump circulating the water fails during a clear day, and the water temperature in the tube starts rising. The aluminum tube absorbs solar radiation at a rate of \(20 \mathrm{~W}\) per meter length, and the temperature of the ambient air outside is \(30^{\circ} \mathrm{C}\). Approximating the surfaces of the tube and the glass cover as being black (emissivity \(\varepsilon=1\) ) in radiation calculations and taking the effective sky temperature to be \(20^{\circ} \mathrm{C}\), determine the temperature of the aluminum tube when equilibrium is established (i.e., when the net heat loss from the tube by convection and radiation equals the amount of solar energy absorbed by the tube). For evaluation of air properties at $1 \mathrm{~atm}\( pressure, assume \)33^{\circ} \mathrm{C}$ for the surface temperature of the glass cover and \(45^{\circ} \mathrm{C}\) for the aluminum tube temperature. Are these good assumptions?
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