/*! 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 52 An energy storage system is prop... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 52

An energy storage system is proposed to absorb thermal energy collected during the day with a solar collector and release thermal energy at night to heat a building. The key component of the system is a shelland-tube heat exchanger with the shell side filled with \(n\)-octadecane (see Problem 8.47). (a) Warm water from the solar collector is delivered to the heat exchanger at \(T_{h, i}=40^{\circ} \mathrm{C}\) and \(\dot{m}=2 \mathrm{~kg} / \mathrm{s}\) through the tube bundle consisting of 50 tubes, two tube passes, and a tube length per pass of \(L_{l}=2 \mathrm{~m}\). The thin-walled, metal tubes are of diameter \(D=25 \mathrm{~mm}\). Free convection exists within the molten \(n\)-octadecane, providing an average heat transfer coefficient of \(h_{o}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside of each tube. Determine the volume of \(n\) octadecane that is melted over a 12 -h period. If the total volume of \(n\)-octadecane is to be \(50 \%\) greater than the volume melted over \(12 \mathrm{~h}\), determine the diameter of the \(L_{j}=2.2\)-m-long shell. (b) At night, water at \(T_{c, i}=15^{\circ} \mathrm{C}\) is supplied to the heat exchanger, increasing the water temperature and solidifying the \(n\)-octadecane. Do you expect the heat transfer rate to be the same, greater than, or less than the heat transfer rate in part (a)? Explain your reasoning.

Short Answer

Expert verified
#tag_title#Step 2: Calculate the volume of melted $n$-octadecane#tag_content#After finding the total heat transfer rate, we can calculate the volume of melted \(n\)-octadecane over a 12-hour period by dividing the energy absorbed by the heat of fusion of the \(n\)-octadecane. The energy absorbed can be found by multiplying the heat transfer rate by the duration: \[ E_{absorbed} = Q \times t, \] where \(t = 12 \mathrm{~h}\) is the duration. The volume of melted \(n\)-octadecane can be determined using its heat of fusion, \(h_f\): \[ V_{melted} = \frac{E_{absorbed}}{h_f}, \] where \(h_f\) is the heat of fusion. #tag_title#Step 3: Determine the diameter of the shell#tag_content#To find the diameter of the shell, we need to first determine the total volume of \(n\)-octadecane required. The total volume should be 50% greater than the volume melted over 12 hours: \[ V_{total} = 1.5 \times V_{melted}. \] Since the shell is \(2.2 \mathrm{~m}\)-long, we can calculate its diameter, \(D_{shell}\), using the total volume: \[ D_{shell} = \sqrt{\frac{4V_{total}}{\pi L_j}}, \] where \(L_j = 2.2 \mathrm{~m}\) is the length of the shell. #tag_title#Solution for part (b): Heat transfer rate at night#tag_content#At night, when the cold water flows through the heat exchanger, we expect the heat transfer rate to be less than during the day. This is because the \(n\)-octadecane will start to solidify as it releases its energy, and the heat transfer coefficient for solidified \(n\)-octadecane is lower than that for the molten state. Therefore, the heat transfer rate at night will be less than the heat transfer rate in part (a).

Step by step solution

01

Calculate the total heat transfer rate during the day

First, we need to calculate the total heat transfer rate using the given values for the warm water, tubes, and heat transfer coefficient. The overall heat transfer rate can be expressed as: \[ Q = h_o A_s (T_{h, i} - T_{m}), \] where \(Q\) is the total heat transfer rate, \(h_o\) is the average heat transfer coefficient on the outside of each tube, \(A_s\) is the surface area of the tubes, \(T_{h,i}\) is the inlet temperature of the warm water, and \(T_{m}\) is the mean temperature of the molten \(n\)-octadecane. In this case, you can assume the \(n\)-octadecane to be molten at an approximate constant temperature of \(T_m=30^{\circ}\mathrm{C}\). For \(50\) tubes, each with \(2\) passes and a length per pass of \(2 \mathrm{~m}\), the total tube side length, \(L_t\), can be found using: \[ L_t = 2 \times 2 \times 50 = 200 \mathrm{~m}, \] We can now determine the total surface area of the tubes, \(A_s\): \[ A_s = \pi D L_t, \] where \(D\) is the diameter of each tube. Plugging in the given values, we can calculate the total heat transfer rate, \(Q\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Heat Exchanger
A central component to many industrial and residential heating systems is the heat exchanger. Its purpose is to transfer heat without the need for the materials involved to come into direct contact with each other. In the context of the provided exercise, a shell-and-tube heat exchanger plays a pivotal role, facilitating the transfer of thermal energy from warm water, heated via solar energy, to the phase change material, n-octadecane, which stores this energy efficiently.

In terms of design, the shell-and-tube heat exchanger consists of an outer shell that houses an array of tubes through which the working fluid, in this case, water, is passed. The exercise described a system with 50 tubes and two tube passes, which defines the path fluid travels within the exchanger. The heat is then transferred to the n-octadecane that surrounds these tubes in the shell. The efficiency of this process is quantified by the heat transfer coefficient, which is given as 25 W/m^2.K for the n-octadecane. It's a measure of how well heat is transferred from the tube surfaces to the n-octadecane.

To optimize the function of a heat exchanger, it is vital to ensure the correct surface area to volume ratio to match the system鈥檚 thermal energy requirements. A student looking at the problem will calculate the surface area of the tubes and use this information, along with other values, to determine the necessary volume of the phase change material needed to meet the desired energy storage requirements.
Solar Thermal Energy
Solar thermal energy is a form of energy harnessed from the sun that can be used in a variety of applications, including residential heating and industrial processes. This type of energy is captured using solar collectors, which are designed to absorb solar radiation and convert it to heat. The collected heat can then be utilized immediately or stored for later use 鈥� a concept that is crucial for continuous energy supply despite the periodic nature of sunlight.

In the exercise, warm water is generated by solar collectors and then transferred to a heat exchanger, demonstrating a practical use of solar thermal energy. The energy is not just harvested during the daytime when sunlight is available, but also stored effectively using a phase change material so that it can be released and utilized during the night to heat a building. This emphasizes solar thermal energy's potential for providing a sustainable and renewable source of energy for thermal needs, reducing reliance on fossil fuels and minimizing environmental impact.

While uncovering the mystery behind solar thermal energy collection and usage, students must understand that the efficiency of this process heavily relies on the materials used and the design of the system, including the proper dimensioning of the heat exchanger and the characteristics of the solar collector.
Phase Change Material
Phase change materials (PCMs) have the remarkable ability to store and release large amounts of thermal energy during phase transitions, typically between solid and liquid states. In the scenario provided in the exercise, n-octadecane is used as the PCM and is valuable because of its ability to melt and solidify at temperatures close to the desired comfort levels in building environments. During the phase change process, PCMs can absorb or release a significant amount of 'latent' heat without a substantial change in temperature.

The energy storage system proposed in the exercise takes advantage of this characteristic by using n-octadecane to absorb excess thermal energy during the daytime and releasing it at night to provide heating. The calculation of the volume of n-octadecane that melts over a specified period is foundational to designing a system that meets energy demands. Moreover, this insight into PCMs also reveals why the heat transfer rate might differ between daytime heating and nighttime cooling; the physical properties of n-octadecane shift during phase transitions, impacting the rate of thermal energy exchange.

When students tackle these calculations, they gain a deeper comprehension of the real-world applications of PCMs in thermal energy storage systems. By factoring in the phase change behavior of n-octadecane, they can predict the performance of the storage system under different conditions, such as variations in the flow rates and temperatures of the water used for heating and cooling the PCM.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The human brain is especially sensitive to elevated temperatures. The cool blood in the veins leaving the face and neck and returning to the heart may contribute to thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein and artery running between the chest and the base of the skull for a distance \(L=250 \mathrm{~mm}\), with mass flow rates of \(3 \times 10^{-3} \mathrm{~kg} / \mathrm{s}\) in opposite directions in the two vessels. The vessels are of diameter \(D=5 \mathrm{~mm}\) and are separated by a distance \(w=7 \mathrm{~mm}\). The thermal conductivity of the surrounding tissue is \(k_{\mathrm{r}}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). If the arterial blood enters at \(37^{\circ} \mathrm{C}\) and the venous blood enters at \(27^{\circ} \mathrm{C}\), at what temperature will the arterial blood exit? If the arterial blood becomes overheated, and the body responds by halving the blood flow rate, how much hotter can the entering arterial blood be and still maintain its exit temperature below \(37^{\circ} \mathrm{C}\) ? Hint: If we assume that all the heat leaving the artery enters the vein, then heat transfer between the two vessels can be modeled using a relationship found in Table 4.1. Approximate the blood properties as those of water.

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.

Ethylene glycol and water, at 60 and \(10^{\circ} \mathrm{C}\), respectively, enter a shell-and-tube heat exchanger for which the total heat transfer area is \(15 \mathrm{~m}^{2}\). With ethylene glycol and water flow rates of 2 and \(5 \mathrm{~kg} / \mathrm{s}\), respectively, the overall heat transfer coefficient is \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Determine the rate of heat transfer and the fluid outlet temperatures. (b) Assuming all other conditions to remain the same, plot the effectiveness and fluid outlet temperatures as a function of the flow rate of ethylene glycol for \(0.5 \leq \dot{m}_{h} \leq 5 \mathrm{~kg} / \mathrm{s}\).

Waste heat from the exhaust gas of an industrial furnace is recovered by mounting a bank of unfinned tubes in the furnace stack. Pressurized water at a flow rate of \(0.025 \mathrm{~kg} / \mathrm{s}\) makes a single pass through each of the tubes, while the exhaust gas, which has an upstream velocity of \(5.0 \mathrm{~m} / \mathrm{s}\), moves in cross flow over the tubes at \(2.25 \mathrm{~kg} / \mathrm{s}\). The tube bank consists of a square array of 100 thin-walled tubes \((10 \times 10)\), each \(25 \mathrm{~mm}\) in diameter and \(4 \mathrm{~m}\) long. The tubes are aligned with a transverse pitch of \(50 \mathrm{~mm}\). The inlet temperatures of the water and the exhaust gas are 300 and \(800 \mathrm{~K}\), respectively. The water flow is fully developed, and the gas properties may be assumed to be those of atmospheric air. (a) What is the overall heat transfer coefficient? (b) What are the fluid outlet temperatures? (c) Operation of the heat exchanger may vary according to the demand for hot water. For the prescribed heat exchanger design and inlet conditions, compute and plot the rate of heat recovery and the fluid outlet temperatures as a function of water flow rate per tube for \(0.02 \leq \dot{m}_{c, 1} \leq 0.20 \mathrm{~kg} / \mathrm{s}\).

As part of a senior project, a student was given the assignment to design a heat exchanger that meets the following specifications: \begin{tabular}{lccc} \hline & \(\dot{m}(\mathrm{~kg} / \mathrm{s})\) & \(T_{m, i}\left({ }^{\circ} \mathrm{C}\right)\) & \(T_{m, \theta}\left({ }^{\circ} \mathrm{C}\right)\) \\ \hline Hot water & 28 & 90 & \(-\) \\ Cold water & 27 & 34 & 60 \\ \hline \end{tabular} Like many real-world situations, the customer hasn't revealed, or doesn't know, additional requirements that would allow you to proceed directly to a final configuration. At the outset, it is helpful to make a first-cut design based upon simplifying assumptions, which can be evaluated to determine what additional requirements and trade-offs should be considered by the customer. (a) Design a heat exchanger to meet the foregoing specifications. List and explain your assumptions. Hint: Begin by finding the required value for \(U A\) and using representative values of \(U\) to determine \(A\). (b) Evaluate your design by identifying what features and configurations could be explored with your customer in order to develop more complete specifications.
See all solutions

Recommended explanations on Physics Textbooks

Dynamics

Read Explanation

Electromagnetism

Read Explanation

Translational Dynamics

Read Explanation

Oscillations

Read Explanation

Kinematics Physics

Read Explanation

Scientific Method Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.