91Ó°ÊÓ

A steam coil is immersed in a stirred tank. Saturated steam at 7.50 bar condenses within the coil, and the condensate emerges at its saturation temperature. A solvent with a heat capacity of \(2.30 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\cdot} \mathrm{C}\right)\) is fed to the tank at a steady rate of \(12.0 \mathrm{kg} / \mathrm{min}\) and a temperature of \(25^{\circ} \mathrm{C},\) and the heated solvent is discharged at the same flow rate. The tank is initially filled with \(760 \mathrm{kg}\) of solvent at \(25^{\circ} \mathrm{C},\) at which point the flows of both steam and solvent are commenced. The rate at which heat is transferred from the steam coil to the solvent is given by the expression $$\dot{Q}=U A\left(T_{\mathrm{steam}}-T\right)$$ where \(U A\) (the product of a heat transfer coefficient and the coil surface area through which the heat is transferred) equals \(11.5 \mathrm{kJ} /\left(\min \cdot^{\circ} \mathrm{C}\right) .\) The tank is well stirred, so that the temperature of the contents is spatially uniform and equals the outlet temperature. (a) Prove that an energy balance on the tank contents reduces to the equation given below and supply an initial condition. \frac{d T}{d t}=1.50^{\circ} \mathrm{C} / \mathrm{min}-0.0224 T (b) Without integrating the equation, calculate the steady-state value of \(T\) and sketch the expected plot of \(T\) versus \(t,\) labeling the values of \(T_{\mathrm{b}}\) at \(t=0\) and \(t \rightarrow \infty\) (c) Integrate the balance equation to obtain an expression for \(T(t)\) and calculate the solvent temperature after 40 minutes. (d) The tank is shut down for routine maintenance, and a technician notices that a thin mineral scale has formed on the outside of the steam coil. The coil is treated with a mild acid that removes the scale and reinstalled in the tank. The process described above is run again with the same steam conditions, solvent flow rate, and mass of solvent charged to the tank, and the temperature after 40 minutes is \(55^{\circ} \mathrm{C}\) instead of the value calculated in Part (c). One of the system variables listed in the problem statement must have changed as a result of the change in the stirrer. Which variable would you guess it to be, and by what percentage of its initial value did it change?

Step by step solution

01

Energy Balance Equation

Given the expression for the rate at which heat is transferred from the steam coil to the solvent and the other provided data, an energy balance on the tank can be written as: \[\frac{d\left(m C T\right)}{dt}=\dot{m} C \left(T_{\text {stream}}-T\right),\] where m is the mass of the solvent, C is the heat capacity of the solvent, \(T_{\text {steam}}\) is the temperature of the steam, and T is the temperature of the solvent in the tank. This equation can be manipulated to get the form in the exercise statement by substituting values and simplifying.

02

Steady-State Temperature

At steady state, the temperature is constant, so the rate of change of the temperature (dT/dt) is zero. Setting the derivative in the energy balance equation to zero and solving it for the temperature T gives the steady state temperature. This temperature becomes the limit of T as t approaches infinity.

03

Solvent Temperature after Specific Time

Integrating the balance equation results in an expression for \(T(t)\). Substituting the value of t = 40 into this expression yields the solvent temperature after 40 minutes.

04

Analysis After Maintenance

The temperature after the maintenance is different from the value calculated in step 3 due to a change in the system. By comparing the final temperature values before and after the maintenance, an estimation can be made as to which variable might have changed. The variable most likely to have changed is the heat transfer coefficient (UA).

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Key Concepts

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

Heat Transfer Coefficient

The heat transfer coefficient is a crucial entity in chemical engineering, representing the efficiency with which heat can transfer through a boundary between a solid surface and a fluid, or between two fluid layers. In our textbook exercise involving a steam coil immersed in a stirred tank, the heat transfer coefficient, denoted as 'U', is a part of the equation \( \dot{Q} = U A (T_{\mathrm{steam}} - T) \), indicating that the rate of heat transfer \( \dot{Q} \) is directly proportional to both the coefficient itself and the area of contact 'A'. When the heat transfer coefficient is high, this implies that heat can move rather quickly from the steam to the solvent, increasing the solvent's temperature efficiently.

Improving the heat transfer coefficient can involve several methods. Increasing the surface area through which the heat is to be transferred, enhancing the material thermal conductivity, and reducing any physical barriers, like mineral scale on the steam coil, can increase 'U'. Moreover, in the exercise, when the thin mineral scale is removed from the steam coil, the improvement of heat transfer efficiency can be deduced by comparing the temperature measurements before and after maintenance. An increased final solvent temperature would suggest an enhanced heat transfer coefficient, which affects the rate at which the solvent reaches its steady-state temperature.

On the other hand, a decrease in the solvent’s temperature could imply that the heat transfer coefficient has been reduced, which is what might happen if the surface area ‘A’ were to decrease or if an insulating layer formed on the coil, impeding the flow of heat.

Steady-State Temperature

The concept of steady-state temperature refers to the point at which the temperature within a system no longer changes with time. It represents an equilibrium state where the rate of energy input into the system equals the rate of energy loss. In our example, the steady-state temperature is the temperature the solvent reaches after a sufficiently long period of operation of the steam coil. Mathematically, at steady state, the derivative of temperature with respect to time (dT/dt) is zero.

Understanding the behavior of steady-state temperature is essential for designing and optimizing chemical processes. Engineers can predict the conditions under which a system will stabilize, which is particularly relevant when dealing with continuous flow processes like the stirred tank scenario. In the exercise, the students are tasked with finding the steady-state value of the solvent’s temperature by setting the rate of change to zero in the balance equation. This task aids in visualizing the thermal equilibrium state of the solvent.

The steady-state temperature also serves as a referential point for identifying any deviations or inefficiencies within a process, such as those caused by unexpected insulating materials or changes to the heat transfer coefficient, as seen in the tank maintenance analysis.

Integration of Balance Equations

The integration of balance equations is a fundamental process in chemical engineering that permits the quantitative description of changes in a system over time. These balance equations are derived from conservation laws, such as the conservation of energy, which is the bedrock of our textbook problem.

In the context of the stirred tank example, the students are tasked to integrate an energy balance differential equation to obtain an expression for the solvent temperature as a function of time, denoted as \( T(t) \). Integration transforms the rate of temperature change into an equation that can predict the solvent temperature at any point in time during the process, given the initial conditions.

The full integration of such equations often requires mathematical techniques and assumptions about the system's behavior, such as its initial condition. In the exercise, an initial temperature is provided, aiding in setting the necessary boundary condition for the integration. This practice strengthens the understanding of transient phenomena in chemical processes, enabling engineers to forecast process outcomes and make informed decisions regarding operation and control.

Steam Condensation

The process of steam condensation is pivotal in the exercise at hand. It involves the phase change of steam from its gaseous state back into liquid water, known as condensate. This phase transition releases a significant amount of latent heat, which is effectively transferred to the solvent in the tank.

Condensation is influenced by various factors including pressure and temperature. At a given pressure, steam condenses at its saturation temperature, which is the temperature where steam exists in equilibrium with water at the same pressure. In our steam coil and tank scenario, the saturated steam condenses inside the coil, releasing heat to the solvent, emphasized in the equation \( \dot{Q}=U A(T_{\mathrm{steam}}-T) \). The efficiency of condensation directly affects the heat transfer rate, where more effective condensation means a higher rate of heat input into the solvent.

Furthermore, the condition of the steam coil surface impacts the rate of condensation. For instance, if scale or fouling occurs on the coil surface, this can impede heat transfer, slowing the condensation process and therefore affecting the amount of heat transferred to the solvent. This is why the removal of such scaling becomes an essential maintenance task, as highlighted in the problem when examining the system variables after routine maintenance.