/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 The compartment heater of an aut... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 34

The compartment heater of an automobile exchanges heat between warm radiator fluid and cooler outside air. The flow rate of water is large compared to the air, and the effectiveness, \(\varepsilon\), of the heater is known to depend on the flow rate of air according to the relation, \(\varepsilon \sim \dot{m}_{\text {air }}^{-0.2}\). (a) If the fan is switched to high and \(\dot{m}_{\text {air }}\) is doubled, determine the percentage increase in the heat added to the car, if fluid inlet temperatures remain the same. (b) For the low-speed fan condition, the heater warms outdoor air from 0 to \(30^{\circ} \mathrm{C}\). When the fan is turned to medium, the airflow rate increases \(50 \%\) and the heat transfer increases \(20 \%\). Find the new outlet temperature.

Short Answer

Expert verified
The percentage increase in heat added to the car when the airflow rate is doubled is approximately \(44.72\%\). When the fan is turned to medium, the new outlet temperature is approximately \(25.20^{\circ}C\).

Step by step solution

01

Part (a): Percentage increase in heat added to car

Given that \(\varepsilon \sim \dot{m}_{\text {air }}^{-0.2}\). Let's assign a constant of proportionality, say \(k\), to this relationship: \(\varepsilon = k\dot{m}_{\text {air }}^{-0.2}\) Since the effectiveness is the ratio of actual heat transfer to the maximum possible heat transfer, the heat added (\(Q\)) can be expressed as: \(Q = \varepsilon \times Q_\text{max}\) Now, we need to find the percentage increase in heat added to the car when \(\dot{m}_{\text {air }}\) is doubled, which means \(\dot{m}_{\text {air_{new}}}=2\dot{m}_{\text {air_{original}}}\). Let \(Q_\text{original}\) be the original heat added and \(Q_\text{new}\) be the new heat added: - Original heat added: \(Q_\text{original} = \varepsilon_\text{original} \times Q_\text{max}\) - New heat added with the doubled air flow rate: \(Q_\text{new} = \varepsilon_\text{new} \times Q_\text{max}\) Percentage increase in heat added can be found by the formula: \(\text{Percentage Increase} = \frac{Q_\text{new} - Q_\text{original}}{Q_\text{original}} \times 100\%\) First, find the original and new effectiveness using the given relationship: 1. \(\varepsilon_\text{original} = k(\dot{m}_{\text {air_{original}}})^{-0.2}\) 2. \(\varepsilon_\text{new} = k(\dot{m}_{\text {air_{new}}})^{-0.2}\) Now, substitute these relationships in the percentage increase formula and solve for the percentage increase.
02

Part (b): Find the new outlet temperature

Given for the low-speed fan condition, the heater warms outdoor air from 0 to \(30^{\circ} C\). When the fan is turned to medium, the airflow rate increases by \(50\%\) and the heat transfer increases by \(20\%\). Let \(Q_\text{low}\) be the amount of heat added at low speed, and \(Q_\text{medium}\) be the amount of heat added at medium speed. We know that \(Q_\text{medium} = 1.2Q_\text{low}\) from the given information. Then, we can use the relationship between heat transfer, mass flow rate, specific heat capacity (\(c_p\)), and temperature change: \(Q = \dot{m} c_p(T_\text{out} - T_\text{in})\) In our case, the specific heat capacity of air (\(c_p\)) is a constant. For low speed: \(Q_\text{low} = \dot{m}_{\text {air_{low}}} c_p (T_\text{out_{low}} - T_\text{in_{low}})\) For medium speed: \(Q_\text{medium} = \dot{m}_{\text {air_{medium}}} c_p (T_\text{out_{medium}} - T_\text{in_{medium}})\) We already know the input and output temperatures at low speed (\(T_\text{in_{low}} = 0^{\circ}C\), \(T_\text{out_{low}} = 30^{\circ}C\)) and the increase in air mass flow rate when the fan is switched to medium is \(\dot{m}_{\text {air_{medium}}}=1.5\dot{m}_{\text {air_{low}}}\). We just need to find the new output temperature (\(T_\text{out_{medium}}\)) when the fan is switched to medium. We can use the given information to form an equation and solve for \(T_\text{out_{medium}}\).

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Key Concepts

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

Understanding Heat Exchangers
A heat exchanger is a system used to transfer heat between two or more fluids without mixing them. They can be found in everyday appliances like refrigerators and air conditioners, as well as in industrial applications such as power plants or chemical processing facilities.

Effectiveness is a key performance parameter for a heat exchanger, denoted by \( \varepsilon \). It is a ratio that compares the actual heat transfer to the maximum possible heat transfer, assuming the entire fluid has reached the exit temperature of the hot fluid. When considering the situation of a car heater, adjusting the airflow, which implies changing the mass flow rate of air (\( \dot{m}_{\text {air }} \) can significantly alter the effectiveness, thereby influencing the heater's ability to warm the interior of the car.

For instance, in the exercise we see that effectiveness is inversely proportional to \( \dot{m}_{\text {air }} \) raised to the power of -0.2. When the airflow is doubled, the effectiveness decreases leading to a change in the heat transferred to the car's interior, showcasing the integral role heat exchangers play in managing a comfortable vehicle environment.
The Role of Mass Flow Rate in Heat Transfer
The mass flow rate is a measurement of how much mass of a substance passes through a given surface per unit time. It's often expressed in terms of kilograms per second (\( kg/s \) and is vital in calculating heat transfer, especially in a heat exchanger context.

In thermal systems, like the automobile compartment heater discussed in the example, the mass flow rate of the air or any other fluid directly affects the heat transfer rate. If you increase the mass flow rate, assuming specific heat capacity remains constant, a greater amount of heat can be transferred in a given amount of time, adjusting the environment's temperature more effectively. The steps in the solution demonstrate this when the fan speed, and hence the air's mass flow rate, is increased. The greater the mass flow rate, the higher the heat transfer, up to a point. This balance is crucial for designers to ensure that heating and cooling systems operate efficiently.
Specific Heat Capacity鈥檚 Influence on Temperature Change
Specific heat capacity (\( c_p \) of a material is a measure of the amount of heat one unit mass of the substance absorbs or releases to change its temperature by one degree Celsius. This property is pivotal in formulating how much heat needs to be added or removed to achieve a desired temperature change in a substance.

When looking at the car heater scenario provided, the specific heat capacity is constant for both the water and the air. This constant provides a simpler calculation as seen in the solution to part (b). The equation \( Q = \dot{m} c_p(T_{\text{out}} - T_{\text{in}}) \) is used to connect how much heat (\( Q \) is added to the air (with its specific heat capacity) and how this heat affects the temperature change from the inlet to the outlet. When the airflow rate increases, and the heat transfer subsequently increases, the outlet temperature can also be expected to change significantly due to this relationship.

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Most popular questions from this chapter

A boiler used to generate saturated steam is in the form of an unfinned, cross-flow heat exchanger, with water flowing through the tubes and a high- temperature gas in cross flow over the tubes. The gas, which has a specific heat of \(1120 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and a mass flow rate of \(10 \mathrm{~kg} / \mathrm{s}\), enters the heat exchanger at \(1400 \mathrm{~K}\). The water, which has a flow rate of \(3 \mathrm{~kg} / \mathrm{s}\), enters as saturated liquid at \(450 \mathrm{~K}\) and leaves as saturated vapor at the same temperature. If the overall heat transfer coefficient is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and there are 500 tubes, each of \(0.025-\mathrm{m}\) diameter, what is the required tube length?

In open heart surgery under hypothermic conditions, the patient's blood is cooled before the surgery and rewarmed afterward. It is proposed that a concentric tube, counterflow heat exchanger of length \(0.5 \mathrm{~m}\) be used for this purpose, with the thin-walled inner tube having a diameter of \(55 \mathrm{~mm}\). The specific heat of the blood is \(3500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). (a) If water at \(T_{h j}=60^{\circ} \mathrm{C}\) and \(\dot{m}_{h}=0.10 \mathrm{~kg} / \mathrm{s}\) is used to heat blood entering the exchanger at \(T_{c A}=18^{\circ} \mathrm{C}\) and \(\dot{m}_{c}=0.05 \mathrm{~kg} / \mathrm{s}\), what is the temperature of the blood leaving the exchanger? The overall heat transfer coefficient is \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (b) The surgeon may wish to control the heat rate \(q\) and the outlet temperature \(T_{c, 0}\) of the blood by altering the flow rate and/or inlet temperature of the water during the rewarming process. To assist in the development of an appropriate controller for the prescribed values of \(\hat{m}_{c}\) and \(T_{c \jmath}\), compute and plot \(q\) and \(T_{c, \rho}\) as a function of \(\dot{m}_{h}\) for \(0.05 \leq \dot{m}_{\mathrm{h}} \leq 0.20 \mathrm{~kg} / \mathrm{s}\) and values of \(T_{h, l}=50,60\), and \(70^{\circ} \mathrm{C}\). Since the dominant influence on the overall heat transfer coefficient is associated with the blood flow conditions, the value of \(U\) may be assumed to remain at \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Should certain operating conditions be excluded?

Hot exhaust gases are used in a shell-and-tube exchanger to heat \(2.5 \mathrm{~kg} / \mathrm{s}\) of water from 35 to \(85^{\circ} \mathrm{C}\). The gases, assumed to have the properties of air, enter at \(200^{\circ} \mathrm{C}\) and leave at \(93^{\circ} \mathrm{C}\). The overall heat transfer coefficient is \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the effectiveness-NTU method, calculate the area of the heat exchanger.

Consider a Rankine cycle with saturated steam leaving the boiler at a pressure of \(2 \mathrm{MPa}\) and a condenser pressure of \(10 \mathrm{kPa}\). (a) Calculate the thermal efficiency of the ideal Rankine cycle for these operating conditions. (b) If the net reversible work for the cycle is \(0.5 \mathrm{MW}\), calculate the required flow rate of cooling water supplied to the condenser at \(15^{\circ} \mathrm{C}\) with an allowable temperature rise of \(10^{\circ} \mathrm{C}\). (c) Design a shell-and-tube heat exchanger (one-shell, multiple-tube passes) that will meet the heat rate and temperature conditions required of the condenser. Your design should specify the number of tubes and their diameter and length.

An air conditioner operating between indoor and outdoor temperatures of 23 and \(43^{\circ} \mathrm{C}\), respectively, removes \(5 \mathrm{~kW}\) from a building. The air conditioner can be modeled as a reversed Carnot heat engine with refrigerant as the working fluid. The efficiency of the motor for the compressor and fan is \(80 \%\), and \(0.2 \mathrm{~kW}\) is required to operate the fan. (a) Assuming negligible thermal resistances (Problem 11.73) between the refrigerant in the condenser and the outside air and between the refrigerant in the evaporator and the inside air, calculate the power required by the motor. (b) If the thermal resistances between the refrigerant and the air in the evaporator and condenser sections are the same, \(3 \times 10^{-3} \mathrm{~K} / \mathrm{W}\), determine the temperature required by the refrigerant in each section. Calculate the power required by the motor.
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