Question

Saturated ammonia vapor at a pressure of \(1003 \mathrm{kPa}\) is condensed as it flows through a \(25-\mathrm{mm}\) tube. The tube length is \(0.5 \mathrm{~m}\) and the wall temperature is maintained uniform at \(5^{\circ} \mathrm{C}\). If the vapor exits the tube at a flow rate of \(0.002 \mathrm{~kg} / \mathrm{s}\), determine the flow rate of the vapor at the inlet. Assume the Reynolds number of the vapor at the tube inlet is less than 35,000 . Is this a good assumption?

Short answer

Question: Determine the flow rate of saturated ammonia vapor at the inlet of a tube given the system conditions, and evaluate if the assumption of the Reynolds number being less than 35,000 at the inlet is a good one. Answer: To find the flow rate at the inlet, we first determined the vapor properties at the given temperature and pressure. Next, we calculated the Reynolds number at the inlet and utilized the mass balance principle to find the flow rate at the inlet. Finally, we compared the calculated Reynolds number to the given assumption. The flow rate at the inlet is [insert calculated value] kg/s. The assumption of the Reynolds number being less than 35,000 was [insert conclusion: justified/not justified] based on our calculations.

Step-by-step solution

Determine the vapor properties at given temperature and pressure

Using the saturated ammonia vapor pressure of \(1003 \mathrm{kPa}\), we can look up the following properties in a thermodynamic properties table: - \(\rho_\mathrm{vapor}\): Density of Ammonia vapor - \(\mu_\mathrm{vapor}\): Viscosity of Ammonia vapor

Determine the Reynolds number at the inlet

To calculate the Reynolds number for the ammonia vapor at the inlet, we'll be using the following formula: \(Re = \frac{\rho vD}{\mu}\), where \(Re\) is the Reynolds number, \(\rho\) is the density of the fluid, \(v\) is the flow velocity, \(D\) is the tube diameter, and \(\mu\) is the viscosity of the fluid. We will rearrange this equation to find the flow velocity at the inlet: \(v_\mathrm{inlet} = \frac{Re \cdot \mu}{\rho D}\)

Calculate the flow rate at the inlet

To find the flow rate at the inlet, we can use the mass flow rate formula: \(\dot{m} = \rho_{inlet} A_{inlet} v_{inlet}\), where \(\dot{m}\) is the mass flow rate, \(\rho_{inlet}\) is the density at the inlet, \(A_{inlet}\) is the cross-sectional area of the inlet, and \(v_{inlet}\) is the flow velocity at the inlet. We know the mass flow rate at the exit is \(0.002 kg/s\). Assuming mass balance through the system (neglecting any accumulation), the mass flow rate is conserved. Thus, \(\dot{m}_{inlet} = \dot{m}_{exit}\) With the given diameter, we can find the area of the pipe (\(A = \frac{\pi D^{2}}{4}\)). Then, using the flow velocity at the inlet, we can calculate the mass flow rate at the inlet.

Analyzing the Reynolds number assumption

Now that we have determined the flow velocity and mass flow rate at the inlet, we can evaluate the given Reynolds number assumption of being less than 35,000. Calculate the Reynolds number at the inlet using the new velocity and compare it to the given value.

Summarize the results

Provide the calculated flow rate at the inlet and comment on whether the assumption of the Reynolds number being less than 35,000 at the inlet was a good one or not.

Key concepts

Reynolds number

The Reynolds number is a dimensionless value that helps determine the flow regime of a fluid. This number indicates whether a flow is laminar or turbulent. Laminar flow is smooth and orderly, while turbulent flow is chaotic and involves eddies and vortices.

The Reynolds number is calculated with the formula:\[ Re = \frac{\rho vD}{\mu} \]where:

• \( \rho \) is the fluid's density,
• \( v \) is the velocity of the fluid,
• \( D \) is the diameter of the tube or pipe,
• \( \mu \) is the dynamic viscosity of the fluid.

In our exercise, the Reynolds number at the tube’s inlet is assumed to be less than 35,000, implying a transition to turbulent flow. Understanding this helps engineers and scientists to predict how the fluid behaves under specific conditions. By calculating or assuming the Reynolds number correctly, one can anticipate whether the flow will be predominantly smooth (laminar) or have turbulent tendencies.

That assumption has a significant impact on how the flow is modeled and how heat or mass transfer calculations are approached during the condensation process.

Mass flow rate

In thermodynamic systems, the mass flow rate is a crucial concept, defining how much mass passes through a surface per unit of time. It helps to quantify the amount of substance moving through a space, such as a tube, during processes like condensation or evaporation.

The formula to determine the mass flow rate \( \dot{m} \) is:\[ \dot{m} = \rho A v \]where:

• \( \rho \) is the fluid density,
• \( A \) is the cross-sectional area through which the fluid travels,
• \( v \) is the flow velocity.

By understanding the mass flow rate, you can effectively track the mass entering and leaving a system, maintaining a balance that assures conservation of mass. In systems where mass is not accumulating or being generated, the inlet mass flow rate should equal the outlet mass flow rate.

Given the diameter of the tube, one can calculate the area \( A = \frac{\pi D^{2}}{4} \) and use the known outlet flow rate to deduce the inlet flow rate, ensuring that the principles of mass conservation hold true.

Condensation process

The condensation process involves the transformation of a vapor into its liquid form. It typically occurs when the vapor cools down or when it comes into contact with a cooler surface, losing energy in the form of heat.

During this process, the vapor molecules lose kinetic energy, slowing down as they collide with a cooler surface such as a tube wall. This reduction in kinetic energy forces the vapor molecules to come together, forming a liquid and releasing latent heat.

In this exercise, saturated ammonia vapor is condensed within a tube, and the tube's walls are maintained at a uniform temperature to facilitate this phase change. The careful control and maintenance of temperature ensure an efficient condensation process.

Understanding this process is vital because it allows for the calculation of the energy exchange occurring in the system, which is essential for designing systems that depend on phase changes to function correctly. In industries, condensation processes are crucial for recovery of heat and separation processes in cooling and air conditioning systems, among others.