91Ó°ÊÓ

The temperature in a process unit is controlled by passing cooling water at a measured rate through a jacket that encloses the unit. The exact relationship between the unit temperature \(T\left(^{\circ} \mathrm{C}\right)\) and the water flow rate \(\phi(\mathrm{L} / \mathrm{s})\) is extremely complex, and it is desired to derive a simple empirical formula to approximate this relationship over a limited range of flow rates and temperatures. Data are taken for \(T\) versus \(\phi\). Plots of \(T\) versus \(\phi\) on rectangular and semilog coordinates are distinctly curved (ruling out \(T=a \phi+b\) and \(T=a e^{b \phi}\) as possible empirical functions), but a log plot appears as follows: A line drawn through the data goes through the points \(\left(\phi_{1}=25 \mathrm{L} / \mathrm{s}, T_{1}=210^{\circ} \mathrm{C}\right)\) and \(\left(\phi_{2}=40 \mathrm{L} / \mathrm{s},\right.\) \(\left.T_{2}=120^{\circ} \mathrm{C}\right)\). (a) What is the empirical relationship between \(\phi\) and \(T ?\) (b) Using your derived equation, estimate the cooling water flow rates needed to maintain the process unit temperature at \(85^{\circ} \mathrm{C}, 175^{\circ} \mathrm{C},\) and \(290^{\circ} \mathrm{C}\). (c) In which of the three estimates in Part (b) would you have the most confidence and in which would you have the least confidence? Explain your reasoning.

Step by step solution

01

Determine the function form

Since the line drawn through the data points on the log plot is a straight line, it can be concluded that the relationship between \(T\) and \(\phi\) is of the form \(T = a \phi^{b}\), where \(T\) is the unit temperature, \(\phi\) is the water flow rate, and \(a\) and \(b\) are constants to be determined.

02

Solve for the constants a and b

By taking the logarithm of both sides of the equation \(T = a \phi^{b}\), the equation transforms into a linear form and the constants \(a\) and \(b\) can be solved for. This results in \(\log(T) = \log(a) + b \log(\phi)\). Using the given data points, two equations can be formed: \(\log(T_{1}) = \log(a) + b \log(\phi_{1})\) and \(\log(T_{2}) = \log(a) + b \log(\phi_{2})\). These two equations can be solved simultaneously to find the values of \(a\) and \(b\).

03

Apply equation to calculate flow rates

Apply the derived equation with the determined constants to the desired temperatures to find the necessary flow rates to maintain these temperatures. Rearranging the equation to solve for \(\phi\) gives \(\phi = (\frac{T}{a})^{1/b}\). Place the desired temperatures of \(85^{\circ}C, 175^{\circ}C, 290^{\circ}C\) into this equation to calculate the corresponding flow rates.

04

Evaluate the confidence in estimates

The confidence in these estimates would vary based on their proximity to the original data points used to determine the function. Those estimates closer to the data points are more likely to be accurate since the function is an approximation that's most accurate within the range of provided data. Estimates lying within the range of \(\phi_{1}\) to \(\phi_{2}\) would have higher confidence, while those falling outside would have lesser.

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Key Concepts

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

Chemical Engineering

Chemical Engineering is a multidisciplinary branch of engineering that combines the principles of physics, chemistry, and mathematics to process raw materials or chemicals into more useful or valuable forms. In the context of process temperature control, chemical engineers must understand the behavior of systems where heat transfer is essential.

Temperature regulation in reactors or process units is a critical aspect of chemical engineering, as it can significantly affect the rate of reaction and the stability of the products. Cooling water flow, as demonstrated in the exercise, is an integral element of maintaining desired temperatures within industrial processes. Chemical engineers use empirical relationships, such as the one derived in our exercise, to predict how changes in such variables as flow rate might influence the system's temperature.

Process Control

Process control refers to the methods and technologies used to monitor and adjust manufacturing processes. It is crucial for maintaining product quality, optimizing performance, and ensuring safety. In chemical processes, controlling the temperature is often necessary to ensure that the reactions take place within the optimal temperature range.

To achieve this, control systems employ a variety of sensors and actuators. In our exercise, the flow rate of the cooling water is adjusted to control the temperature, reflecting a direct application of process control. By understanding the empirical relationship between temperature and flow rate, engineers can program control systems to adjust variables in real-time, maintaining optimal operating conditions.

Temperature Control

Temperature control remains a cornerstone of ensuring operational efficiency and safety in chemical processes. The ability to precisely control temperature influences reaction rates, product quality, and energy consumption.

Through heat exchange systems like jackets or coils, fluids such as cooling water absorb excess heat, hence regulating the temperature of the process unit. By adjusting the flow rate of the cooling water, the amount of heat removed from the system can be fine-tuned. As seen in our exercise, determining the empirical relationship between these two variables allows for predictive adjustments and is essential for robust temperature control strategies.

Cooling Water Flow Rate

The flow rate of cooling water is a critical variable in process temperature control. It's indicative of how much water passes through a cooling system at any given time and is usually expressed in liters per second (L/s) or gallons per minute (GPM).

In our exercise, a higher flow rate indicates increased heat removal capacity, hence a lower process unit temperature. However, beyond just determining the flow rate, understanding how it impacts temperature across various conditions is essential. The empirical formula derived offers a simplified representation of this complex interaction, enabling engineers to calculate the necessary flow rates for maintaining specific temperatures.

Logarithmic Relationships

Logarithmic relationships are fundamental when the relationship between two variables is multiplicative rather than additive. These relationships often appear in situations dealing with exponential growth or decay, such as cooling and heating processes in chemical engineering.

By applying logarithms, as shown in the step-by-step solution, nonlinear equations can be linearized, facilitating the determination of unknown constants. These constants can then be applied in the empirical formula to estimate other quantities of interest. In the textbook problem, the logarithmic transformation of the temperature and flow rate data allows for a simple, workable relationship between the two variables over the specified range, which is particularly valuable for process control where predictive capability is key.