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An immersed electrical heater is used to raise the temperature of a liquid from \(20^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) in 20.0 min. The combined mass of the liquid and the container is \(250 \mathrm{kg}\), and the mean heat capacity of the system is 4.00 kJ/(kg.'C). The liquid decomposes explosively at 85"C. At 10: 00 a.m. a batch of liquid is poured into the vessel, and the operator turns on the heater and answers a call on his cell phone. Ten minutes later, his supervisor walks by and looks at the computer display of the power input. This what she sees. The supervisor immediately shuts off the heater and charges off to pass on to the operator several brief observations that come to her mind. (a) Calculate the required constant power input \(\dot{Q}(\mathrm{k} \mathrm{W})\), neglecting energy losses from the container. (b) Write and integrate using Simpson's rule (Appendix A.3) an energy balance on the system to estimate the system temperature at the moment the heater is shut off. Use the following data from the recorder chart: $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|} \hline t(\mathrm{s}) & 0 & 30 & 60 & 90 & 120 & 150 & 180 & 210 & 240 & 270 & 300 \\ \hline \dot{Q}(\mathrm{kW}) & 33 & 33 & 34 & 35 & 37 & 39 & 41 & 44 & 47 & 50 & 54 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline t(\mathrm{s}) & 330 & 360 & 390 & 420 & 450 & 480 & 510 & 540 & 570 & 600 \\ \hline \dot{Q}(\mathrm{kW}) & 58 & 62 & 66 & 70 & 75 & 80 & 85 & 90 & 95 & 100 \\\ \hline \end{array}$$ (c) Suppose that if the heat had not been shut off, \(\dot{Q}\) would have continued to increase linearly at a rate of \(10 \mathrm{kW} / \mathrm{min}\). At what time would everyone in the plant realize that something was wrong?

Step by step solution

01

Calculation of the required constant power input

The heat required to increase the temperature of the liquid and container from \(20^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) can be calculated using the formula \[ Q = mc \Delta T \] where \(m\) is the mass, \(c\) is the specific heat capacity and \(\Delta T\) is the change in temperature. Then, to find the power input \(\dot{Q}\), we use \[ \dot{Q} = \frac{Q}{\Delta t} \] where \(\Delta t\) is the change in time.

02

Estimate the system temperature at the moment the heater is shut off

Based on the power input data, we can estimate the heat added to the system over the 10 minute period before the heater was shut off. Simpson's Rule can be applied to numerically integrate the power input as a function of time, giving the total heat added. Then divide the total heat by the product of the mass and the heat capacity to find the increase in temperature. Adding this to the initial temperature will yield the temperature at the time the heater was turned off.

03

Predict the time the liquid would have exploded

Assuming power continues to increase linearly, the rate of increase can be expressed as \(10 \mathrm{kW/min}\). Convert this to \(kW/s\). Continue the numerical integration from the last known power input, stopping when the integrated power input indicates that the temperature would have reached \(85^{\circ}C\). The time corresponding to this temperature is the answer.

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Key Concepts

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

Heat Capacity

The concept of heat capacity is integral in understanding how much heat is required to change the temperature of a substance. At its core, heat capacity is the amount of heat energy required to change the temperature of a given mass of a substance by one degree Celsius. The equation commonly used to calculate this is:

• \( Q = mc \Delta T \)

Here, \( Q \) represents the heat energy in kilojoules, \( m \) is the mass in kilograms, \( c \) is the specific heat capacity in kilojoules per kilogram per degree Celsius, and \( \Delta T \) is the change in temperature in degrees Celsius.
This formula is pivotal when calculating the heat required for heating systems, like the one that was used in your exercise. The mean heat capacity provides an average measurement for the amount of heat a system can absorb before reaching its threshold, keeping the response more predictable across various temperatures.
To understand heat capacity, think of it as the thermal 'holding capacity' of the substance. The higher the heat capacity, the more heat energy the substance can store without a significant change in temperature, which is why it's an essential consideration in thermal management across various fields of study.

Simpson's Rule

Simpson's Rule is a numerical method used to approximate definite integrals, especially useful when dealing with complex data or functions that are not easy to integrate traditionally. In the context of energy balance exercises, such as the one you encountered, Simpson's Rule helps estimate the cumulative impact of variable power input over time.
Simpson's Rule works by approximating the region under a curve as a series of parabolic segments. This method is applied when plotting power versus time, where simple linear or constant functions cannot accurately reflect the real-world alterations in power input over intervals. The formula for Simpson's Rule is:

• \( \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{3n} \left[ f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n) \right] \)

This method involves evaluating the function at multiple points along the interval \( a \) to \( b \), thereby giving a more 'balanced' integration result, which accounts for the variations in data—ideal for estimating the energy added through power input within short, rapid time changes.

Power Input Calculation

Power input calculation is crucial when determining how effectively energy is being delivered into a system over time. In the energy balance exercise, calculating the constant power input \( \dot{Q} \) provides a baseline for understanding the minimum energy required to reach a desired temperature increase in a controlled time period.
The basic equation to determine power input is:

• \( \dot{Q} = \frac{Q}{\Delta t} \)

where \( \dot{Q} \) is the power in kilowatts, \( Q \) is the heat energy in kilojoules, and \( \Delta t \) is the time interval in seconds or minutes. Calculating power input provides critical insights into the performance and requirements of heating systems, especially in scenarios where precision toward temperature control is essential.
In cases where the power input is not constant, careful data analysis—such as monitoring power over specific intervals—becomes necessary. This helps in estimating instantaneous power inputs, which can form a basis for predicting future system behaviors, warning about potential overheating scenarios, or evaluating system efficiency.