Question

Carbon dioxide gas is heated as it flows steadily through a 2.5-cm-diameter pipe. At the inlet, the pressure is 2 bar, the temperature is \(300 \mathrm{~K}\), and the velocity is \(100 \mathrm{~m} / \mathrm{s}\). At the exit, the pressure and velocity are \(0.9413\) bar and \(400 \mathrm{~m} / \mathrm{s}\), respectively. The gas can be treated as an ideal gas with constant specific heat \(c_{p}=0.94 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). Neglecting potential energy effects, determine the rate of heat transfer to the carbon dioxide, in \(\mathrm{kW}\).

Short answer

Rate of heat transfer is approximately 38342.49 kW.

Step-by-step solution

- Write Down Known Values

List all the given data from the problem: Inlet conditions: Pressure: \(P_1 = 2 \text{ bar} = 200 \text{ kPa}\) Temperature: \(T_1 = 300 \text{ K}\) Velocity: \(V_1 = 100 \text{ m/s}\) Exit conditions: Pressure: \(P_2 = 0.9413 \text{ bar} = 94.13 \text{ kPa}\) Velocity: \(V_2 = 400 \text{ m/s}\) Specific heat: \(c_p = 0.94 \text{ kJ/kg}\cdot\text{K}\) Diameter of the pipe: \(d = 2.5 \text{ cm} = 0.025 \text{ m}\)

- Calculate Mass Flow Rate

First, calculate the cross-sectional area of the pipe: \[ A = \frac{\pi d^2}{4} \] \[ A = \frac{\pi (0.025)^2}{4} = 4.91 \times 10^{-4} \text{ m}^2 \] Use the ideal gas equation to determine the densities at the inlet and exit: \[ \rho_1 = \frac{P_1}{R T_1} \,,\; \rho_2 = \frac{P_2}{R T_2} \] Given that CO2 has a specific gas constant \(R = 0.1889 \text{ kJ/kg} \cdot \text{K}\): \[ \rho_1 = \frac{200 \text{ kPa}}{0.1889 \cdot 300 \text{ K}} = 3.526 \text{ kg/m}^3 \] Recalling the mass flow rate equation \( \dot{m} = \rho_1 A V_1 \): \[ \dot{m} = 3.526 \cdot 4.91 \times 10^{-4} \cdot 100 = 0.1733 \text{ kg/s} \]

- Determine Temperatures at Inlet and Exit

Given the specific heat \(c_p\) and the temperatures \(T_1\) and \(T_2\), consider the energy balance equation for a control volume: Neglecting potential energy changes and assuming steady flow: \[ \dot{Q} = \dot{m} c_p (T_2 - T_1) + \frac{\dot{m}}{2} (V_2^2 - V_1^2) \] Rearrange for determining \(T_2\): Given: At exit: \( P_2 = 94.13 \text{ kPa} \,,\; V_2 = 400 \text{ m/s} \)

- Calculate Exit Temperature

To find exit temperature: Use energy equation for gases based on ideal gas properties: \[ T_2 = \left( \frac{P_2}{P_1} \right) \left( \frac{V_2}{V_1} \right) T_1 \] \[ T_2 = \left( \frac{94.13}{200} \right) \cdot \frac{400\text{ m/s}}{100\text{ m/s}} \cdot 300 \text{ K} = 564.78 \text{ K} \]

- Calculate Heat Transfer Rate

Using \( \dot{Q} = \dot{m} c_p (T_2 - T_1) + \frac{\dot{m}}{2} (V_2^2 - V_1^2) \): Substitute values: \[\dot{Q} = 0.1733 \cdot 0.94 \cdot (564.89 - 300) + \frac{0.1733}{2} \cdot ((400^2) - (100^2)) \] \[ = 0.1733 \cdot 0.94 \cdot (264.89) + \frac{0.1733}{2} \cdot (160000) \] \[ = (43.08) + (13866.4) = 745.9167 + 13866.4 \approx 38342.49 \text{ kW}\]

Key concepts

steady flow process

In thermodynamics, a steady flow process is a situation where fluid flows through a control volume (like a pipe or other device) steadily over time. The key characteristic of this process is that all properties of the fluid (mass flow rate, pressure, temperature, etc.) do not change with time. This is crucial for many engineering applications because it allows the use of steady-state equations to simplify calculations and designs.

During a steady flow process, the mass that enters the control volume equals the mass that exits. In our exercise, this means the amount of carbon dioxide entering and leaving the pipe per unit of time is constant. We assume there are no gross imbalances of mass within the control volume.

A steady flow energy equation can be applied, which balances the heat transfer, work done, and changes in energy of the fluid as it moves through the system. The steady flow process assumption streamlines our computation as we dealt with velocities, temperatures, and pressures at the inlet and exit without considering time-dependent changes.

ideal gas law

The ideal gas law is an equation of state for an ideal gas. It connects pressure, volume, temperature, and the number of moles of the gas. The equation is given by:

\[ PV = nRT \]
Where:

• P is the pressure of the gas.
• V is the volume of the gas.
• n is the number of moles of the gas.
• R is the universal gas constant (8.314 J/mol·K).
• T is the temperature of the gas.

In the steady flow process described in our exercise, the ideal gas law helps us compute the density of carbon dioxide both at the inlet and the outlet. Knowing that for an ideal gas, \[ \rho = \frac{P}{RT}\] we used it to find the density by substituting the given pressures and temperatures.

Using these densities, we calculated the mass flow rate through the pipe, which is essential for determining the heat transfer rate.

specific heat capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). For an ideal gas, it is important to recognize the distinction between specific heat at constant volume (\[c_v\]) and specific heat at constant pressure (\[c_p\]). In our problem, specific heat at constant pressure (\[c_p\]) is given as 0.94 kJ/kg·K.

The use of \[c_p\] in our heat transfer calculation is crucial as it incorporates the work done by the gas when it expands (or contracts) at constant pressure. The heat transfer rate \[\dot{Q}\] in a steady flow process can be affected by changes in both temperature and kinetic energy. As formulated in the energy balance equation:

\[\dot{Q} = \dot{m} c_p (T_2 - T_1) + \frac{\dot{m}}{2} (V_2^2 - V_1^2)\] where:

• \[\dot{m}\] is the mass flow rate,
• \[T_1\] and \[T_2\] are the inlet and exit temperatures, respectively,
• \[V_1\] and \[V_2\] are the velocities at the inlet and exit, respectively.

This equation helps us to determine how much heat is added into or removed from the system, which directly influences the thermal efficiency of the process. In the given exercise, specific heat capacity provided the necessary factor to calculate the change in internal energy and, thereby, the total heat transfer rate.