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Chapter 2: Problem 42

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.

Short Answer

Expert verified
The empirical relationship between \(\phi\) and \(T\) is \(T = a \phi^{b}\), where the values of \(a\) and \(b\) are determined by solving the corresponding log equations with the provided data points (\(\phi_{1}=25L/s, T_{1}=210^{\circ}C\) and \(\phi_{2}=40L/s, T_{2}=120^{\circ}C\)). Using this equation, the flow rates necessary to maintain temperatures of \(85^{\circ}C, 175^{\circ}C, 290^{\circ}C\) can be calculated. In terms of confidence, estimates falling within the range of the original data points would be most reliable, while those outside this range would be less dependable, as they extend beyond the range over which the empirical relationship is derived.

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.

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Most popular questions from this chapter

Calculate (a) the weight in \(\mathrm{Ib}_{\mathrm{f}}\) of a \(25.0-\mathrm{lb}_{\mathrm{m}}\) object. (b) the mass in \(\mathrm{kg}\) of an object that weighs \(25 \mathrm{N}\). (c) the weight in dynes of a 10 -ton object (not metric tons).

In modeling the effect of an impurity on crystal growth, the following equation was derived: \(\frac{G-G_{\mathrm{L}}}{G_{0}-G}=\frac{1}{K_{\mathrm{L}} C^{m}}\) where \(C\) is impurity concentration, \(G_{\mathrm{L}}\) is a limiting growth rate, \(G_{0}\) is the growth rate of the crystal with no impurity present, and \(K_{\mathrm{L}}\) and \(m\) are model parameters. In a particular experiment, \(G_{0}=3.00 \times 10^{-3} \mathrm{mm} / \mathrm{min},\) and \(G_{\mathrm{L}}=1.80 \times 10^{-3} \mathrm{mm} / \mathrm{min} .\) Growth rates are measured for several impurity concentrations \(C\) (parts per million, or ppm), with the following results: $$\begin{array}{|c|c|c|c|c|c|}\hline C(\mathrm{ppm}) & 50.0 & 75.0 & 100.0 & 125.0 & 150.0 \\\\\hline G(\mathrm{mm} / \mathrm{min}) \times 10^{3} & 2.50 & 2.20 & 2.04 & 1.95 & 1.90 \\\\\hline\end{array}$$ (For example, when \(\left.C=50.0 \mathrm{ppm}, G=2.50 \times 10^{-3} \mathrm{mm} / \mathrm{min}\right)\). (a) Determine \(K_{\mathrm{L}}\) and \(m,\) giving both numerical values and units. (b) A solution is fed to a crystallizer in which the impurity concentration is 475 ppm. Estimate the expected crystal growth rate in (mm/min). Then state why you would be extremely skeptical about this result.

Sketch the plots described below and calculate the equations for \(y(x)\) from the given information. The plots are all straight lines. Note that the given coordinates refer to abscissa and ordinate values, not \(x\) and \(y\) values. [The solution of Part (a) is given as an example.] (a) A plot of In \(y\) versus \(x\) on rectangular coordinates passes through \((1.0,0.693)\) and \((2.0,0.0)\) (i.e., at the first point \(x=1.0\) and \(\ln y=0.693\) ). (b) A semilog plot of \(y\) (logarithmic axis) versus \(x\) passes through (1,2) and (2,1). (c) A log plot of \(y\) versus \(x\) passes through (1,2) and (2,1). (d) A semilog plot of \(x y\) (logarithmic axis) versus \(y / x\) passes through (1.0,40.2) and (2.0,807.0). (e) A log plot of \(y^{2} / x\) versus \((x-2)\) passes through (1.0,40.2) and (2.0,807.0).

A horizontal drum, a cross-section of which is shown below, is being filled with benzene (density \(\left.=0.879 \mathrm{g} / \mathrm{cm}^{3}\right)\) at a constant rate \(\dot{m}(\mathrm{kg} / \mathrm{min}) .\) The drum has a length \(L\) and radius \(r,\) and the level of benzene in the drum is \(h\). The expression for the volume of benzene in the drum is \(V=L\left[r^{2} \cos ^{-1}\left(\frac{r-h}{r}\right)-(r-h) \sqrt{r^{2}-(r-h)^{2}}\right]\) (a) Show that the equation gives reasonable results for \(h=0, h=r,\) and \(h=2 r\). (b) Estimate the mass of benzene \((\mathrm{kg})\) in the tank if \(L=10 \mathrm{ft}, r=2 \mathrm{ft},\) and \(h=4\) in. (c) Suppose there is a sight glass on the side of the tank that allows observation of the height of liquid in the tank. Use a spreadsheet to prepare a graph that can be posted next to the sight glass so that an operator can estimate the mass that is in the tank without going through calculations like that in Part (b).

The following empirical equation correlates the values of variables in a system in which solid particles are suspended in a flowing gas: $$\frac{k_{g} d_{p} y}{D}=2.00+0.600\left(\frac{\mu}{\rho D}\right)^{1 / 3}\left(\frac{d_{p} u \rho}{\mu}\right)^{1 / 2}$$ Both \((\mu / \rho D)\) and \(\left(d_{p} u \rho / \mu\right)\) are dimensionless groups; \(k_{g}\) is a coefficient that expresses the rate at which a particular species transfers from the gas to the solid particles; and the coefficients 2.00 and 0.600 are dimensionless constants obtained by fitting experimental data covering a wide range of values of the equation variables. The value of \(k_{g}\) is needed to design a catalytic reactor. since this coefficient is difficult to determine directly, values of the other variables are measured or estimated and \(k_{g}\) is calculated from the given correlation. The variable values are as follows: $$\begin{aligned}d_{p} &=5.00 \mathrm{mm} \\\y &=0.100 \quad(\text { dimensionless }) \\\D &=0.100 \mathrm{cm}^{2} / \mathrm{s} \\\\\mu &=1.00 \times 10^{-5} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2} \\\\\rho &=1.00 \times 10^{-3} \mathrm{g} / \mathrm{cm}^{3} \\\u &=10.0 \mathrm{m} / \mathrm{s}\end{aligned}$$ (a) What is the estimated value of \(k_{g} ?\) (Give its value and units.) (b) Why might the true value of \(k_{g}\) in the reactor be significantly different from the value estimated in Part (a)? (Give several possible reasons.) (c) Create a spreadsheet in which up to five sets of values of the given variables ( \(d_{p}\) through \(u\) ) are entered in columns and the corresponding values of \(k_{g}\) are calculated. Test your program using the following variable sets: (i) the values given above; (ii) as above, only double the particle diameter \(d_{p}\) (making it \(10.00 \mathrm{mm}\) ); (iii) as above, only double the diffusivity \(D ;\) (iv) as above, only double the viscosity \(\mu ;(\mathrm{v})\) as above, only double the velocity \(u\). Report all five calculated values of \(k_{g}\).
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