Question

Hot exhaust gases of a stationary diesel engine are to be used to generate steam in an evaporator. Exhaust gases \(\left(c_{p}=1051 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enter the heat exchanger at \(550^{\circ} \mathrm{C}\) at a rate of \(0.25 \mathrm{~kg} / \mathrm{s}\) while water enters as saturated liquid and evaporates at \(200^{\circ} \mathrm{C}\left(h_{f g}=1941 \mathrm{~kJ} / \mathrm{kg}\right)\). The heat transfer surface area of the heat exchanger based on water side is \(0.5 \mathrm{~m}^{2}\) and overall heat transfer coefficient is \(1780 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water.

Short answer

Question: Calculate the rate of heat transfer, the exit temperature of the exhaust gases, and the rate of evaporation of water in an evaporator heat exchanger given the following information: overall heat transfer coefficient (U), heat transfer surface area (A), initial temperature of the exhaust gas (T_in), mass flow rate of the exhaust gas (m), and specific heat capacity of the exhaust gas (c_p).

Step-by-step solution

Use the heat transfer formula to find the rate of heat transfer

The heat transfer equation is given by: \(q = UA(T_{in} - T_{out})\), where \(q\) is the rate of heat transfer, \(U\) is the overall heat transfer coefficient, \(A\) is the heat transfer surface area, \(T_{in}\) is the initial temperature and \(T_{out}\) is the final temperature. We have the values for \(U\), \(A\), and \(T_{in}\). We need to find \(q\) and \(T_{out}\). First, let's find \(q\). We can rewrite the heat transfer equation as: \(q = mA(T_{in} - T_{out})=m_water \times h_{fg}\), where \(m\) is the mass flow rate of the exhaust gas, \(A\) is the specific heat capacity of exhaust gas, \(T_{in}\) is the initial temperature of exhaust gas, \(T_{out}\) is the final temperature of exhaust gas, and \(m_water\) and \(h_{fg}\) are (respectively) the mass flow rate of water and the specific enthalpy of water. Rearrange the equation to solve for \(q\): \(q = \frac{m_water \times h_{fg}}{A}\).

Find the change in temperature of exhaust gases

Solve the equation we found in step 1 for \(m_water\): \(m_water = \frac{q}{h_{fg}}\). Now we can use the heat transfer equation to find the change in temperature of the exhaust gases: \(\Delta T = T_{in} - T_{out} = \frac{q}{mc_p}\).

Calculate the exit temperature of exhaust gases

To find the final temperature of the exhaust gases (\(T_{out}\)), we can subtract the change in temperature (\(\Delta T\)) found in step 2 from the initial temperature (\(T_{in}\)): \(T_{out} = T_{in} - \Delta T\).

Calculate the rate of evaporation of water

Now, we can calculate the rate of evaporation of water by using the equation from step 1 and the heat transfer equation we used in step 2: \(Rate\ of\ evaporation = \frac{m_water}{\Delta T} = \frac{q}{h_{fg}\Delta T}\).

Key concepts

Evaporator

An evaporator is a key component in a heat transfer system, where its main function is to facilitate the phase change of a liquid into vapor. In many industrial applications, like the generation of steam from water, evaporators play a crucial role. This process begins with supplying heat to a liquid, which increases its temperature until it reaches its boiling point. At this stage, additional heat is required to convert the liquid into vapor, without further increasing its temperature.
This process is vital in many applications and is used extensively in refrigerating plants and power stations.

• In our exercise, water acts as the liquid entering the evaporator in a saturated liquid state at 200°C.
• The evaporator in this scenario uses heat from exhaust gases to evaporate the water.

Understanding the function of an evaporator helps in grasping how heat energy can be utilized effectively in transforming liquid into vapor, serving various industrial needs.

Heat Exchanger

A heat exchanger is a system used to transfer heat between two or more fluids. In simple terms, heat exchangers are devices that allow heat from a fluid (a liquid or a gas) to pass to a second fluid without the two fluids having to mix or come in contact.
They are used widely in space heating, refrigeration, air conditioning, power stations, chemical plants, petrochemical plants, petroleum refineries, natural gas processing, and sewage treatment.

• The role of a heat exchanger in our given scenario is to facilitate the transfer of heat from hot exhaust gases to the water in the evaporator.
• The heat exchanger allows these processes to occur efficiently and without the direct interaction between the exhaust gas and the water.

This transfer process is crucial because it ensures that the exhaust gases can efficiently provide heat to evaporate the water, demonstrating the functional importance of the heat exchanger in real-world applications.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient (\( U \)) is a measure of how easily heat is transferred across materials in a heat exchanger setup. It encompasses all the resistances encountered by heat transfer, including the conduction and convection resistances. The higher the overall heat transfer coefficient, the more effective the material is at transferring heat.
In calculation terms, the overall heat transfer coefficient is utilized in determining the rate of heat transfer (\( q \)) using the formula:\[ q = UA(T_{in} - T_{out}) \]where \( U \) is the overall heat transfer coefficient, \( A \) is the heat transfer surface area, and \( T_{in} \) and \( T_{out} \) are the input and output temperatures.

• In the exercise, \( U \) is given as 1780 W/m²·K, which is crucial for calculating how much heat is being transferred from the exhaust gases to the water.

This coefficient is used to appreciate how efficient the heat exchanger is in transferring energy, and plays a vital role in designing energy-efficient thermal systems.

Latent Heat of Vaporization

Latent heat of vaporization (\( h_{fg} \)) is the amount of heat required to convert a unit mass of a liquid into vapor without a change in temperature. This concept is fundamental in thermodynamics and is critical in processes that involve phase changes from liquid to vapor.
The value of latent heat of vaporization varies depending on the substance being vaporized and the pressure and temperature conditions. It is calculated in kilojoules per kilogram (kJ/kg).

• In our problem, the latent heat of vaporization for water at 200°C is given as 1941 kJ/kg. This value is used to determine how much heat in total is necessary for each kilogram of water to be evaporated by the heat source.

The latent heat of vaporization is essential in understanding how much energy is required in heating processes, especially in designing industrial applications like evaporators where efficient energy use is paramount.