Question

Consider a combined gas-steam power plant. Water for the steam cycle is heated in a well-insulated heat exchanger by the exhaust gases that enter at \(800 \mathrm{K}\) at a rate of \(60 \mathrm{kg} / \mathrm{s}\) and leave at \(400 \mathrm{K} .\) Water enters the heat exchanger at \(200^{\circ} \mathrm{C}\) and \(8 \mathrm{MPa}\) and leaves at \(350^{\circ} \mathrm{C}\) and \(8 \mathrm{MPa}\). If the exhaust gases are treated as air with constant specific heats at room temperature, the mass flow rate of water through the heat exchanger becomes \((a) 11 \mathrm{kg} / \mathrm{s}\) \((b) 24 \mathrm{kg} / \mathrm{s}\) \((c) 46 \mathrm{kg} / \mathrm{s}\) \((d) 53 \mathrm{kg} / \mathrm{s}\) \((e) 60 \mathrm{kg} / \mathrm{s}\)

Short answer

#tag_title#Step 2: Calculate the heat absorbed by the water#tag_content#The heat absorbed by the water is equal to the heat lost by the exhaust gases (assuming no heat loss to the surroundings). Therefore, we can set the heat gained by the water equal to the heat lost by the exhaust gases: $$Q_{water} = Q_{exhaust}$$ #tag_title#Step 3: Determine the mass flow rate of water#tag_content#Now, we need to find the mass flow rate of water. We know the initial and final temperatures of the water passing through the heat exchanger (20°C and 70°C, or 293.15 K and 343.15 K) and the specific heat capacity of water, which is 4.186 kJ/(kg·K). We can use the formula for heat transfer for the water: $$Q_{water} = m_{water} \cdot c_{p,water} \cdot (T_{out,water} - T_{in,water})$$ Since we have found that \(Q_{water} = Q_{exhaust}\), we can find the mass flow rate of water (\(m_{water}\)): $$m_{water} = \frac{Q_{exhaust}}{c_{p,water} \cdot (T_{out,water} - T_{in,water})}$$ Plugging the values into this equation, we get: $$m_{water} = \frac{60 \, kg/s \cdot 1.005 \, kJ/(kg \cdot K) \cdot (800 \, K - 400 \, K)}{4.186 \, kJ/(kg \cdot K) \cdot (343.15 \, K - 293.15 \, K)}$$ #tag_title#Step 4: Calculate the mass flow rate of water#tag_content#Finally, we can solve for the mass flow rate of water: $$m_{water} = \frac{60 \, kg/s \cdot 1.005 \, kJ/(kg \cdot K) \cdot 400 \, K}{4.186 \, kJ/(kg \cdot K) \cdot 50 \, K} = 6.06 \, kg/s$$ Therefore, the mass flow rate of water through the heat exchanger is approximately 6.06 kg/s.

Step-by-step solution

Calculate the heat lost by exhaust gases

First, we need to calculate the heat lost by the exhaust gases as they pass through the heat exchanger. We will use the formula for heat transfer: $$Q = m \cdot c_p \cdot (T_{in} - T_{out})$$ where \(Q\) is the heat transfer, \(m\) is the mass flow rate, \(c_p\) is the specific heat capacity at constant pressure, \(T_{in}\) is the initial temperature, and \(T_{out}\) is the final temperature. We are given the mass flow rate of exhaust gases (\(60 \, kg/s\)) and the initial and final temperatures (\(800 \, K\) and \(400 \, K\)). However, we will also need the specific heat capacity of air at room temperature, which is \(c_p = 1.005 \, kJ/(kg \cdot K)\). Now, we can calculate the heat lost by the exhaust gases: $$Q = 60 \, kg/s \cdot 1.005 \, kJ/(kg \cdot K) \cdot (800 \, K - 400 \, K)$$

Key concepts

Combined Gas-Steam Power Plant

A combined gas-steam power plant is a sophisticated energy conversion system that utilizes both gas turbines and steam turbines to generate electricity. In this type of power plant, the exhaust heat from the gas turbine is harnessed to produce steam, which is then used to power a steam turbine. This process increases the efficiency of the power plant, as it makes use of the waste heat that would otherwise be lost to the environment. The heat exchanger, often a key component in such systems, plays a critical role by transferring the residual heat from the gas turbine exhaust to the water or steam used in the steam cycle.

The beauty of this design lies in its ability to achieve higher thermal efficiency compared to traditional fossil-fuel power plants that rely solely on steam cycles. By employing a heat exchanger, the combined cycle is able to convert more heat into usable work, leading to a reduction in fuel consumption and emissions.

Mass Flow Rate

The term 'mass flow rate' refers to the amount of mass passing through a given surface per unit of time. It is a crucial parameter in many engineering calculations, including the analysis of heat exchangers, as it directly influences the rate of heat transfer within the system. In the case of our combined gas-steam power plant, the mass flow rate of exhaust gases dictates how much heat can be extracted and transferred to the steam cycle.

Considered as one of the primary design aspects for any fluid-related process, the mass flow rate affects the size and performance of the heat exchanger. Higher mass flow rates typically allow for more efficient heat transfer but may require more robust and larger heat exchanger designs to accommodate the increased flow without causing excessive pressure drops.

Specific Heat Capacity

Specific heat capacity, often symbolized by the parameter \(c_p\), is a property that relates the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree (usually in Celsius or Kelvin). This parameter is essential in the calculations related to thermal energy changes in materials, including the working fluids in power plants.

In our context, the specific heat capacity of exhaust gases (treated as air with constant specific heats at room temperature) is used to determine the heat lost to the water in the heat exchanger. The specific heat capacity can vary with temperature, but in many calculations, an average or constant value is used to simplify the analysis. The accuracy of the heat transfer calculation hinges on the correct estimation or selection of the specific heat capacity values for the substances involved.

Heat Transfer Calculation

Heat transfer calculation is a fundamental part of thermodynamics and is particularly essential in designing and analyzing heat exchangers. It involves determining the amount of thermal energy that is transferred from one fluid to another over a period of time.

Using the formula \(Q = m \cdot c_p \cdot (T_{in} - T_{out})\), engineers can calculate the heat transfer occurring within the heat exchanger. Here, \(Q\) depicts the amount of heat transferred, \(m\) represents the mass flow rate of the fluid, \(c_p\) stands for the specific heat capacity, and \(T_{in}\) and \(T_{out}\) are the initial and final temperatures of the fluid, respectively.

Example Calculation

In the given problem, by knowing the mass flow rate of the exhaust gases, specific heat capacity, and the temperature difference, the total energy being transferred from the gas to the water can be calculated. This is crucial for determining the mass flow rate of water that would absorb this energy to reach the desired temperature increase. Understanding and accurately performing these calculations can lead to optimized designs and better efficiency of thermal systems like power plants.