Question

Steam at 80 psia and \(400^{\circ} \mathrm{F}\) is mixed with water at \(60^{\circ} \mathrm{F}\) and 80 psia steadily in an adiabatic device. Steam enters the device at a rate of \(0.05 \mathrm{lbm} / \mathrm{s}\), while the water enters at \(1 \mathrm{lbm} / \mathrm{s}\). Determine the temperature of the mixture leaving this device when the outlet pressure is 80 psia.

Short answer

Answer: The final temperature of the mixture leaving the adiabatic mixing device is approximately \(106.77^{\circ} \mathrm{F}\).

Step-by-step solution

Given information

Steam enters the device at 80 psia, 400°F and 0.05 lbm/s Water enters the device at 80 psia, 60°F, and 1 lbm/s Outlet pressure: 80 psia Our goal is to find the temperature of the mixture leaving this device.

Define the enthalpy of each stream

Define the enthalpy of the incoming steam as \(h_{steam}\), the enthalpy of the incoming water as \(h_{water}\), and the enthalpy of the mixture as \(h_{mixture}\). We need to find these values to use conservation of energy.

Find the properties of steam and water

Use steam tables to find the properties of the incoming steam and water at the given pressures and temperatures. For the steam, at 80 psia and 400°F, we can find: \(h_{steam} = 1203.1 \frac{Btu}{lbm}\) For the water, at 80 psia and 60°F, we can find: \(h_{water} = 28.08 \frac{Btu}{lbm}\)

Apply the conservation of energy

The equation for conservation of energy in this adiabatic mixing process is: $$m_{steam}h_{steam} + m_{water}h_{water} = (m_{steam} + m_{water}) h_{mixture}$$ where, \(m_{steam}\) is the mass flow rate of steam = 0.05 lbm/s \(m_{water}\) is the mass flow rate of water = 1 lbm/s

Solve the conservation of energy equation for \(h_{mixture}\)

Plug the values into the equation above: $$(0.05)(1203.1) + (1)(28.08) = (0.05 + 1) h_{mixture}$$ Solve for \(h_{mixture}\): $$h_{mixture} = \frac{0.05(1203.1) + 1(28.08)}{0.05 + 1} = 62.312 \frac{Btu}{lbm}$$

Find the exit temperature of the mixture

Now, we need to find the temperature corresponding to the calculated \(h_{mixture}\) value. Since the outlet pressure is 80 psia, we will use the steam table at that pressure: 80 psia and enthalpy \(h_{mixture} = 62.312 \frac{Btu}{lbm}\) correspond to a temperature of: \(T_{mixture} = 106.77^{\circ} \mathrm{F}\) (approximately) The temperature of the mixture leaving the device is approximately \(106.77^{\circ} \mathrm{F}\).

Key concepts

Conservation of Energy in Adiabatic Processes

The principle of the conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In thermodynamics, this principle is applied to systems to determine the energy exchange between them and their surroundings. In adiabatic processes, such as the one described in the exercise, there is no heat exchange with the environment - the system is thermally insulated. Hence, the total enthalpy of the incoming steam and water must equal the enthalpy of the mixture leaving the device.

This principle allows us to set up an energy balance equation that accounts for the mass and enthalpy of each stream. By calculating the enthalpy of each component and then applying the conservation of energy, we determine the properties of the mixture at the exit of the adiabatic device. The adiabatic mixing of steam and water without external heat exchange is a practical demonstration of energy conservation in a thermal system.

Enthalpy and Its Role in Thermodynamics

Enthalpy is a measure of the total energy of a thermodynamic system, encompassing both internal energy and the energy required to make room for it by displacing the system's environment. It is a crucial concept in the study of energy transfer during the mixing and heating or cooling of substances.

In our exercise, we refer to the enthalpy values of steam (\(h_{steam}\)) and water (\(h_{water}\)), obtained from steam tables, which are comprehensive references of water and steam properties at various temperatures and pressures. By knowing these values and the mass flow rates of the inputs, we can determine the enthalpy of the resulting mixture (\(h_{mixture}\)). It is important to understand that the enthalpy values are not just abstract numbers but represent the actual energy content within the steam and water, which governs how the two will interact when mixed.

Steam Properties - Understanding Steam Tables

Steam properties are the physical characteristics of steam, such as temperature, pressure, enthalpy, entropy, and specific volume. These properties are essential for the engineering analysis of power plants, heating systems, and any application involving steam.

In the case of an adiabatic mixing process, we use steam tables to find specific properties, such as the enthalpy of steam at a given pressure and temperature (\(400^{\textnormal{o}} \textnormal{F}\) and 80 psia in this problem). Steam tables are indispensable tools that provide these properties in a straightforward manner, removing the need for complex calculations. Material properties like these are the foundational data used to perform the energy balance calculations required to solve our problem.

Mass Flow Rate in Thermal Systems

The mass flow rate is a measure of the amount of mass moving through a cross-section per unit time. This rate is crucial in the analysis of fluid flow systems and is typically measured in units of lbm/s (pounds mass per second) or kg/s (kilograms per second).

In the adiabatic mixing process described in the exercise, we have two different mass flow rates: steam (\(0.05 \textnormal{ lbm/s}\)) and water (\(1 \textnormal{ lbm/s}\)). Accurate measurement and control of these rates are essential for making precise energy balance calculations. The mass flow rates are part of the conservation of energy equation, which allows us to solve for the final temperature of the mixture. Understanding the flow rates helps in predicting system performance and is fundamental in scaling these principles to industrial applications.