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Chapter 9: Problem 94

Consider a \(3-\mathrm{m}\)-high rectangular enclosure consisting of two surfaces separated by a \(0.1-\mathrm{m}\) air gap at \(1 \mathrm{~atm}\). If the surface temperatures across the air gap are \(30^{\circ} \mathrm{C}\) and \(-10^{\circ} \mathrm{C}\), determine the ratio of the heat transfer rate for the horizontal orientation (with hotter surface at the bottom) to that for vertical orientation.

Short Answer

Expert verified
Answer: To calculate the ratio of the heat transfer rates for the horizontal orientation (Q_H) to the vertical orientation (Q_V), follow these steps: 1. Calculate the Grashof number for both orientations using the formula: Gr = (g * β * (T_h - T_c) * L^3) / ν^2 2. Calculate the Rayleigh number for both orientations using the formula: Ra = Gr * Pr 3. Determine the Nusselt number for both orientations using the Au-Yang and Chung correlation for horizontal (Nu_H = 0.286 * Ra_H^0.25) and Morgan's correlation for vertical (Nu_V = 0.68 + (0.67 * Ra_V^0.25) / (1 + (0.492/Pr)^(9/16))^(4/9)) 4. Calculate the heat transfer coefficients for both orientations using the formula: h = (Nu * k) / L 5. Calculate the heat transfer rates for both orientations using the formula: Q = h * A * (T_h - T_c) 6. Determine the ratio of the heat transfer rates using the simplified formula: Q_H / Q_V = h_H / h_V The ratio of the heat transfer rates for the horizontal orientation to the vertical orientation can be calculated using the values obtained in these steps.

Step by step solution

01

Calculate the Grashof number for both orientations

First, we need to obtain the Grashof number for both orientations. The Grashof number (Gr) can be calculated using the following formula for natural convection: $$ \mathrm{Gr} = \frac{g \beta (T_h - T_c) L^3}{\nu^2} $$ where \(g\) is the acceleration due to gravity (\(9.81 \mathrm{m/s^2}\)), \(\beta\) is the coefficient of thermal expansion (\(1/T_m\)), \(T_h\) and \(T_c\) are the hot and cold surface temperatures respectively, \(L\) is the characteristic length (height) of the enclosure and \(\nu\) is the kinematic viscosity of air. For horizontal orientation: $$ L = 3 \,\mathrm{m}, \quad T_h = 30^{\circ} \mathrm{C}, \quad T_c = -10^{\circ} \mathrm{C} $$ For vertical orientation: $$ L = 0.1\, \mathrm{m} $$ Calculate the Grashof number for both orientations.
02

Calculate the Rayleigh number for both orientations

Next, we need to calculate the Rayleigh number (Ra) for both orientations. The Rayleigh number can be found using the following formula: $$ \mathrm{Ra} = \mathrm{Gr} \cdot \mathrm{Pr} $$ where Pr is the Prandtl number (\(\sim 0.7\) for air). Calculate the Rayleigh number for both orientations.
03

Determine the Nusselt number for both orientations

Now, we need to determine the Nusselt number for both orientations. For natural convection across an enclosure, we can use the following correlations to obtain the Nusselt number: For horizontal orientation (Au-Yang and Chung correlation): $$ \mathrm{Nu}_H = 0.286 \, \mathrm{Ra}_H^{0.25} $$ For vertical orientation (Morgan's correlation): $$ \mathrm{Nu}_V = 0.68 + \frac{0.67 \, \mathrm{Ra}_V^{1/4}}{\left[ 1 + \left(\frac{0.492}{\mathrm{Pr}}\right)^{9/16}\right]^{4/9}} $$ Calculate the Nusselt numbers for both orientations.
04

Calculate the heat transfer coefficients for both orientations

Next, we need to find the heat transfer coefficients for both orientations. The heat transfer coefficient (h) can be calculated using the following formula: $$ h = \frac{\mathrm{Nu} \cdot k}{L} $$ where \(k\) is the thermal conductivity of air (\(\sim 0.0262 \mathrm{W/(m\cdot K)}\)). Calculate the heat transfer coefficients for both orientations.
05

Calculate the heat transfer rates for both orientations

Now, we can calculate the heat transfer rates (Q) for both orientations. The heat transfer rate can be found using the following formula: $$ Q = h \cdot A \cdot (T_h - T_c) $$ where A is the area of the enclosure. Calculate the heat transfer rates for both orientations.
06

Determine the ratio of the heat transfer rates

Finally, we can find the ratio of the heat transfer rates for the horizontal orientation (Q_H) to the vertical orientation (Q_V) as follows: $$ \frac{Q_H}{Q_V} = \frac{h_H \cdot A_H \cdot (T_h - T_c)}{h_V \cdot A_V \cdot (T_h - T_c)} $$ Since the temperature difference (T_h - T_c) is the same for both orientations, we can simplify the formula to: $$ \frac{Q_H}{Q_V} = \frac{h_H}{h_V} $$ Calculate the ratio of the heat transfer rates.

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Most popular questions from this chapter

Consider a 1.2-m-high and 2 -m-wide glass window with a thickness of $6 \mathrm{~mm}\(, thermal conductivity \)k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and emissivity \)\varepsilon=0.9$. The room and the walls that face the window are maintained at \(25^{\circ} \mathrm{C}\), and the average temperature of the inner surface of the window is measured to be $5^{\circ} \mathrm{C}\(. If the temperature of the outdoors is \)-5^{\circ} \mathrm{C}$, determine \((a)\) the convection heat transfer coefficient on the inner surface of the window, \((b)\) the rate of total heat transfer through the window, and (c) the combined natural convection and radiation heat transfer coefficient on the outer surface of the window. Is it reasonable to neglect the thermal resistance of the glass in this case?

In a plant that manufactures canned aerosol paints, the cans are temperature- tested in water baths at \(60^{\circ} \mathrm{C}\) before they are shipped to ensure that they withstand temperatures up to \(55^{\circ} \mathrm{C}\) during transportation and shelving (as shown in Fig. P9-51). The cans, moving on a conveyor, enter the open hot water bath, which is \(0.5 \mathrm{~m}\) deep, $1 \mathrm{~m}\( wide, and \)3.5 \mathrm{~m}$ long, and they move slowly in the hot water toward the other end. Some of the cans fail the test and explode in the water bath. The water container is made of sheet metal, and the entire container is at about the same temperature as the hot water. The emissivity of the outer surface of the container is \(0.7\). If the temperature of the surrounding air and surfaces is \(20^{\circ} \mathrm{C}\), determine the rate of heat loss from the four side surfaces of the container (disregard the top surface, which is open). The water is heated electrically by resistance heaters, and the cost of electricity is \(\$ 0.085 / \mathrm{kWh}\). If the plant operates $24 \mathrm{~h}\( a day 365 days a year and thus \)8760 \mathrm{~h}$ a year, determine the annual cost of the heat losses from the container for this facility.

A 0.6-m \(\times 0.6-\mathrm{m}\) horizontal ASTM A240 410S stainless steel plate has its upper surface subjected to convection with cold, quiescent air. The minimum temperature suitable for the stainless steel plate is $-30^{\circ} \mathrm{C}$ (ASME Code for Process Piping, ASME B31.3-2014, Table \(\mathrm{A}-1 \mathrm{M}\) ). If heat is added to the plate at a rate of $70 \mathrm{~W}$, determine the lowest temperature that the air can reach without causing the surface temperature of the plate to cool below the minimum suitable temperature. Evaluate the properties of air at $-50^{\circ} \mathrm{C}$. Is this an appropriate temperature at which to evaluate the air properties?

Determine the \(U\)-factor for the center-of-glass section of a double-pane window with a \(13-\mathrm{mm}\) airspace for winter design conditions. The glazings are made of clear glass having an emissivity of \(0.84\). Take the average airspace temperature at design conditions to be $10^{\circ} \mathrm{C}$ and the temperature difference across the airspace to be \(15^{\circ} \mathrm{C}\).

The side surfaces of a 3 -m-high cubic industrial furnace burning natural gas are not insulated, and the temperature at the outer surface of this section is measured to be \(110^{\circ} \mathrm{C}\). The temperature of the furnace room, including its surfaces, is \(30^{\circ} \mathrm{C}\), and the emissivity of the outer surface of the furnace is \(0.7\). It is proposed that this section of the furnace wall be insulated with glass wool insulation $(k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ wrapped by a reflective sheet \((\varepsilon=0.2)\) in order to reduce the heat loss by 90 percent. Assuming the outer surface temperature of the metal section still remains at about \(110^{\circ} \mathrm{C}\), determine the thickness of the insulation that needs to be used. The furnace operates continuously throughout the year and has an efficiency of 78 percent. The price of the natural gas is \(\$ 1.10 /\) therm ( 1 therm \(=105,500 \mathrm{~kJ}\) of energy content). If the installation of the insulation will cost \(\$ 550\) for materials and labor, determine how long it will take for the insulation to pay for itself from the energy it saves.
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