Question

Water vapor \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) is heated during a steady-flow process at 1 atm from 298 to \(3000 \mathrm{K}\) at a rate of \(0.2 \mathrm{kg} / \mathrm{min}\). Determine the rate of heat supply needed during this process, assuming \((a)\) some \(\mathrm{H}_{2} \mathrm{O}\) dissociates into \(\mathrm{H}_{2}, \mathrm{O}_{2},\) and \(\mathrm{OH}\) and \((b)\) no dissociation takes place.

Short answer

Question: Determine the rate of heat supply required for heating water vapor in a steady-flow process from an initial temperature of 298 K to a final temperature of 3000 K at a rate of 0.2 kg/min, in two cases (a) when some H2O dissociates into H2, O2, and OH, and (b) when no dissociation takes place. Use the first law of thermodynamics to find the heat supply rate for both cases. Answer: To find the rate of heat supply for both cases, we applied the first law of thermodynamics in the form of Q_dot = m_dot × (h2 - h1), where Q_dot is the rate of heat supply, m_dot is the mass flow rate, and h1 and h2 are the initial and final enthalpies of water vapor. After calculating the mass flow rate and finding the initial and final enthalpies from the property tables, we can plug in the known values for both cases to find the respective rate of heat supply Q_dot(a) and Q_dot(b).

Step-by-step solution

Determine the initial and final enthalpies

First, we need to find the initial and final enthalpies of water vapor. For this, we can refer to the property tables (available in thermodynamics books), given the initial and final temperatures of 298 K and 3000 K respectively.

Calculate the rate of water vapor flow

Next, we find out the mass flow rate. We have been given that the rate of water vapor flow is 0.2 kg/min, so we convert this value into kg/s as follows: mass flow rate (m_dot) = 0.2 kg/min × (1 min/60 s) = 1/300 kg/s

Apply the first law of thermodynamics

As previously mentioned, the first law of thermodynamics for this process can be simplified to: Q_dot = m_dot × (h2 - h1) Where Q_dot is the rate of heat supply, m_dot is the mass flow rate, and h1 and h2 are the initial and final enthalpies of water vapor.

Calculate the rate of heat supply (a)

Plug in the known values for mass flow rate and enthalpies from Step 1 and Step 2: Q_dot(a) = m_dot × (h2 - h1) Case (b): No dissociation takes place

Determine the initial and final enthalpies (b)

For this case, we need to find the initial and final enthalpies of water vapor without dissociation. For this, we can again refer to the property tables (available in thermodynamics books), given the initial and final temperatures of 298 K and 3000 K respectively.

Calculate the rate of water vapor flow (b)

For this case, the mass flow rate remains the same as in Case (a), which is m_dot = 1/300 kg/s.

Apply the first law of thermodynamics (b)

Just like in Case (a), the first law of thermodynamics can be simplified to: Q_dot = m_dot × (h2 - h1) Where Q_dot is the rate of heat supply, m_dot is the mass flow rate, and h1 and h2 are the initial and final enthalpies of water vapor without dissociation.

Calculate the rate of heat supply (b)

Plug in the known values for mass flow rate and enthalpies from Step 1 and Step 2: Q_dot(b) = m_dot × (h2 - h1) Now, we have the rate of heat supply for both cases (a) and (b).

Key concepts

First Law of Thermodynamics

The first law of thermodynamics, also known as the principle of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a steady-flow process, such as the heating of water vapor in the textbook exercise, the first law dictates that the heat added to the system will increase the internal energy of the water vapor, causing a change in enthalpy.

Mathematically, the first law can be represented as \( Q_\text{dot} = m_\text{dot} \times (h_2 - h_1) \), where \( Q_\text{dot} \) is the rate of heat supply, \( m_\text{dot} \) is the mass flow rate, and \( h_1 \) and \( h_2 \) are the initial and final enthalpies. Recognizing the importance of this law allows us to calculate the amount of heat required to bring about a desired temperature change in a substance like water vapor.

Enthalpy of Water Vapor

Enthalpy is a thermodynamic property that reflects the total heat content of a system, incorporating both internal energy and the energy associated with pressure and volume. Specifically for water vapor, enthalpy varies with temperature and pressure conditions. To solve problems involving the heating of water vapor, one must refer to thermodynamic tables or equations that relate enthalpy to temperature at a constant pressure.

Enthalpy is particularly important when considering phase changes, like evaporation and condensation, but also when only temperature changes occur, as with the heating of water vapor in the exercise. Knowing the initial and final enthalpies of water vapor at 298 K and 3000 K respectively allows us to determine the energy changes involved in this steady-flow heating process.

Heat Supply Rate

The heat supply rate, denoted as \( Q_\text{dot} \), is a measure of how much heat energy is supplied to a substance per unit of time. It is a crucial concept for understanding how energy transfers impact the temperature and phase of substances within a thermodynamic process.

In practical terms, knowing the heat supply rate is fundamental when designing heating and cooling systems, such as boilers or refrigerators. By calculating \( Q_\text{dot} \) using the mass flow rate and the change in enthalpy, as done in the textbook solution, we can predict the performance of such systems under various conditions and ensure they operate efficiently.

Chemical Dissociation in Thermodynamics

Chemical dissociation occurs when molecules such as \( \mathrm{H}_2\mathrm{O} \) break down into simpler molecules or atoms, often due to high temperatures or pressures. This dissolution affects the thermodynamic properties of the substance, including its enthalpy. In the textbook exercise, under the assumption of dissociation, the water vapor breaks into \( \mathrm{H}_2, \mathrm{O}_2, \) and \( \mathrm{OH} \) at the elevated temperature of 3000 K.

Understanding dissociation is crucial in high-temperature thermodynamics because it can significantly alter the amount of heat required to achieve a particular process. Therefore, calculating the enthalpies for scenarios with and without dissociation gives us insight into how chemical changes can influence thermodynamic behaviors of substances.