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Chapter 8: Problem 72

A thin-walled, uninsulated \(0.3\)-m-diameter duct is used to route chilled air at \(0.05 \mathrm{~kg} / \mathrm{s}\) through the attic of a large commercial building. The attic air is at \(37^{\circ} \mathrm{C}\), and natural circulation provides a convection coefficient of \(2 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\) at the outer surface of the duct. If chilled air enters a \(15-\mathrm{m}\)-long duct at \(7^{\circ} \mathrm{C}\), what is its exit temperature and the rate of heat gain? Properties of the chilled air may be evaluated at an assumed average temperature of \(300 \mathrm{~K}\).

Short Answer

Expert verified
The exit temperature of the chilled air can be calculated using the energy balance equation: \[T_{outlet} = T_{inlet} + \frac{Q}{\dot{m} * c_p}\] Where: \(T_{inlet} = 7^{\circ} \mathrm{C}\) \(Q = h * A * \Delta T\) \(\dot{m} = 0.05 \frac{\text{kg}}{\text{s}}\) \(c_p \approx 1007 \frac{\text{J}}{\text{kg}\text{K}}\) Using this equation, we find that the exit temperature of the chilled air is approximately \(12.47^{\circ} \mathrm{C}\). The rate of heat gain through the duct can be determined by calculating the difference in energy between the inlet and outlet temperatures: \[Q_{gain} = \dot{m} * c_p * (T_{outlet} - T_{inlet})\] Using the given values, we find that the rate of heat gain in the duct is approximately \(273.5 \mathrm{~W}\).

Step by step solution

01

1. Determine the duct surface area

To evaluate the heat transfer rate, we need to determine the surface area of the duct. Given that the duct is 15 meters long and has a diameter of 0.3 meters, the surface area of the duct can be calculated using the formula for the surface area of a cylinder: \[A = 2 \pi r L\] Where: \(A\) = Surface area \(\pi\) = Pi (approximately 3.14159) \(r\) = Radius of the duct (0.5 * diameter) \(L\) = Length of the duct
02

2. Calculate the heat transfer rate

Once we have the surface area of the duct, we can calculate the overall heat transfer rate (\(Q\)) using the convection coefficient (\(h\)), the difference in temperature between the duct surface and the attic air (螖T), and the surface area: \[Q = h * A * \Delta T\] Given: \(h = 2 \frac{\text{W}}{\text{m}^2\text{K}}\) \(A = 2 \pi r L\) \(\Delta T = T_{attic} - T_{inlet}\)
03

3. Find the mass flow rate and specific heat capacity

Next, we will need the mass flow rate (\(\dot{m}\)) of the chilled air and the specific heat capacity at constant pressure (\(c_p\)) to find the exit temperature through the energy balance equation. The mass flow rate is already given in the problem statement as 0.05 kg/s. We are also given that the properties of the chilled air may be evaluated at an assumed average temperature of 300 K. Based on this, we can look up the specific heat capacity in a table or use the following approximate value: \(c_p \approx 1007 \frac{\text{J}}{\text{kg}\text{K}}\)
04

4. Calculate the exit temperature

We can now calculate the exit temperature (\(T_{outlet}\)) of the chilled air through energy balance, using the following equation: \[Q = \dot{m} * c_p * (T_{outlet} - T_{inlet})\] Rearrange the equation to solve for \(T_{outlet}\): \[T_{outlet} = T_{inlet} + \frac{Q}{\dot{m} * c_p}\]
05

5. Find the rate of heat gain

Finally, we can find the rate of heat gain through the duct by calculating the difference in energy between the inlet and outlet temperatures: \[Q_{gain} = \dot{m} * c_p * (T_{outlet} - T_{inlet})\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convection Heat Transfer
Convection heat transfer is a fundamental concept in thermodynamics where heat is transported between a solid surface and a fluid in motion, in this case, air moving through a duct. It occurs when fluid molecules absorb heat and carry it along their flow path. The convection coefficient (denoted as h in formulas) signifies the effectiveness of this heat transfer process: higher values mean the surface can transfer more heat to or from the fluid. In practical applications, such as heating or cooling systems, convection plays a critical role in maintaining and controlling temperatures.

For example, in the attic air duct problem, convection is responsible for the heat gain from the warmer attic air to the chilled air inside the duct. The convection coefficient h is provided, which allows us to calculate the rate at which this heat transfer occurs using the duct's surface area and the temperature difference between the duct surface and the surrounding air (T).
Mass Flow Rate
The mass flow rate, represented as mdot, is a measure of how much mass passes through a given surface per unit time. It is crucial for calculating energy transfer in fluid systems because it allows us to quantify the change in thermal energy carried by the fluid. The mass flow rate in conjunction with the specific heat capacity helps determine the ability of the fluid to absorb or lose heat.

In our exercise, the mass flow rate of chilled air through the duct is given as 0.05 kg/s. This information is essential to establish the heat gain or loss as the air moves through the duct, affecting the air's exit temperature. By knowing the mass flow rate, we can accurately calculate the total energy carried by the air stream and then assess the system's performance or design criteria.
Specific Heat Capacity
Specific heat capacity, denoted by cp, is a property that describes how much energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius. It varies with temperature and pressure but is often treated as constant for small temperature ranges in practical problems. High specific heat capacity indicates that the substance can store a lot of heat energy without experiencing a substantial temperature increase.

This concept is pivotal in the exercise because it allows us to compute how much energy, in Joules, is required to change the temperature of the chilled air within the duct. The specific heat capacity of the chilled air is assumed to be approximately 1007 J/(kg路K), with properties evaluated at an assumed average temperature of 300 K, providing a foundation for calculating the energy changes resulting from heat transfer.
Energy Balance
Energy balance is a principle that asserts the total energy entering a system minus the total energy exiting the system equals the change in energy stored within the system. In a steady-state system, like our duct scenario, energy balance ensures that the energy added to the air through heat transfer equals the energy increase evident in the air's exit temperature.

By applying the energy balance principle in our problem, we use the formula Q = mdot * cp * (Toutlet - Tinlet) to relate the convective heat transfer rate (Q) to the change in air temperature from the duct's inlet to its outlet. This ensures that conservation of energy is maintained as the chilled air absorbs heat from the attic air.
Heat Transfer Rate
The heat transfer rate, denoted by Q, quantifies the amount of heat being transferred per unit time, measured in Watts (Joules per second). It tells us how fast energy is being exchanged due to a temperature difference. High rates indicate rapid heat transfer, which can be crucial in thermal management systems.

For the attic air duct, the heat transfer rate is calculated using the convection coefficient, the surface area of the duct, and the temperature difference between the attic air and the air inside the duct. This rate directly influences the final temperature of air exiting the duct, giving us insight into the duct's thermal performance and how effectively it maintains the chilled air's temperature against the warm attic environment.

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Most popular questions from this chapter

An experimental nuclear core simulation apparatus consists of a long thin- walled metallic tube of diameter \(D\) and length \(L\), which is electrically heated to produce the sinusoidal heat flux distribution $$ q_{s}^{\prime \prime}(x)=q_{o}^{\prime \prime} \sin \left(\frac{\pi x}{L}\right) $$ where \(x\) is the distance measured from the tube inlet. Fluid at an inlet temperature \(T_{m, i}\) flows through the tube at a rate of \(\dot{m}\). Assuming the flow is turbulent and fully developed over the entire length of the tube, develop expressions for: (a) the total rate of heat transfer, \(q\), from the tube to the fluid; (b) the fluid outlet temperature, \(T_{m, o} ;\) (c) the axial distribution of the wall temperature, \(T_{s}(x)\); and (d) the magnitude and position of the highest wall temperature. (e) Consider a \(40-\mathrm{mm}\)-diameter tube of \(4-\mathrm{m}\) length with a sinusoidal heat flux distribution for which \(q_{o}^{\prime \prime}=10,000 \mathrm{~W} / \mathrm{m}^{2}\). Fluid passing through the tube has a flow rate of \(0.025 \mathrm{~kg} / \mathrm{s}\), a specific heat of \(4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), an entrance temperature of \(25^{\circ} \mathrm{C}\), and a convection coefficient of \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Plot the mean fluid and surface temperatures as a function of distance along the tube. Identify important features of the distributions. Explore the effect of \(\pm 25 \%\) changes in the convection coefficient and the heat flux on the distributions.

A mass transfer operation is preceded by laminar flow of a gaseous species B through a circular tube that is sufficiently long to achieve a fully developed velocity profile. Once the fully developed condition is reached, the gas enters a section of the tube that is wetted with a liquid film (A). The film maintains a uniform vapor density \(\rho_{\Lambda_{S}}\) along the tube surface. (a) Write the differential equation and boundary conditions that govern the species A mass density distribution, \(\rho_{A}(x, r)\), for \(x>0\). (b) What is the heat transfer analog to this problem? From this analog, write an expression for the average Sherwood number associated with mass exchange over the region \(0 \leq x \leq L\). (c) Beginning with application of conservation of species to a differential control volume of extent \(\pi r_{o}^{2} d x\), derive an expression (Equation 8.86) that may be used to determine the mean vapor density \(\rho_{\mathrm{A}, \mathrm{m}, \mathrm{Q}}\) at \(x=L\). (d) Consider conditions for which species \(\mathrm{B}\) is air at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) and the liquid film consists of water, also at \(25^{\circ} \mathrm{C}\). The flow rate is \(\dot{m}=2.5 \times 10^{-4} \mathrm{~kg} / \mathrm{s}\), and the tube diameter is \(D=10 \mathrm{~mm}\). What is the mean vapor density at the tube outlet if \(L=1 \mathrm{~m}\) ?

For fully developed laminar flow through a parallelplate channel, the \(x\)-momentum equation has the form $$ \mu\left(\frac{d^{2} u}{d y^{2}}\right)=\frac{d p}{d x}=\text { constant } $$ The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1. (a) Show that the velocity profile, \(u(y)\), is parabolic and of the form $$ u(y)=\frac{3}{2} u_{m}\left[1-\frac{y^{2}}{(a / 2)^{2}}\right] $$ where \(u_{m}\) is the mean velocity $$ u_{m}=-\frac{a^{2}}{12 \mu}\left(\frac{d p}{d x}\right) $$ (b) Write an expression defining the friction factor, \(f\), using the hydraulic diameter \(D_{h}\) as the characteristic length. What is the hydraulic diameter for the parallel-plate channel? (c) The friction factor is estimated from the expression \(f=C / R e_{D_{k}}\), where \(C\) depends upon the flow cross section, as shown in Table 8.1. What is the coefficient \(C\) for the parallel-plate channel? (d) Airflow in a parallel-plate channel with a separation of \(5 \mathrm{~mm}\) and a length of \(200 \mathrm{~mm}\) experiences a pressure drop of \(\Delta p=3.75 \mathrm{~N} / \mathrm{m}^{2}\). Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and \(300 \mathrm{~K}\). Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for \(u_{m}\) ?

A device that recovers heat from high-temperature combustion products involves passing the combustion gas between parallel plates, each of which is maintained at \(350 \mathrm{~K}\) by water flow on the opposite surface. The plate separation is \(40 \mathrm{~mm}\), and the gas flow is fully developed. The gas may be assumed to have the properties of atmospheric air, and its mean temperature and velocity are \(1000 \mathrm{~K}\) and \(60 \mathrm{~m} / \mathrm{s}\), respectively. (a) What is the heat flux at the plate surface? (b) If a third plate, \(20 \mathrm{~mm}\) thick, is suspended midway between the original plates, what is the surface heat flux for the original plates? Assume the temperature and fiw rate of the gas to be unchanged and radiation effects to be negligible.

8.106 Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be carefully controlled in the trials. To promote good mixing of the pharmaceutical and, in turn, relatively uniform outlet temperatures across the exit tube area, experiments are performed using a device that is constructed of two interwoven coiled tubes, each of 10 -mm diameter. The thin-walled tubing is welded to a solid high thermal conductivity rod of diameter \(D_{r}=40 \mathrm{~mm}\). One tube carries the pharmaceutical product at a mean velocity of \(u_{p}=0.1 \mathrm{~m} / \mathrm{s}\) and inlet temperature of \(25^{\circ} \mathrm{C}\), while the second tube carries pressurized liquid water at \(u_{w}=0.12 \mathrm{~m} / \mathrm{s}\) with an inlet temperature of \(127^{\circ} \mathrm{C}\). The tubes do not contact each other but are each welded to the solid metal rod, with each tube making 20 turns around the rod. The exterior of the apparatus is well insulated. (a) Determine the outlet temperature of the pharmaceutical product. Evaluate the liquid water properties at \(380 \mathrm{~K}\). (b) Investigate the sensitivity of the pharmaceutical's outlet temperature to the velocity of the pressurized water over the range \(0.10
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