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Chapter 10: Problem 68

Consider the situation of Problem \(10.67\) at relatively high vapor velocities, with a fluid mass flow rate of \(\dot{m}=2.5 \mathrm{~kg} / \mathrm{s}\). (a) Determine the heat transfer coefficient and condensation rate per unit length of tube for a mass fraction of vapor of \(X=0.2\). (b) Plot the heat transfer coefficient and the condensation rate for \(0.1 \leq X \leq 0.3\).

Short Answer

Expert verified
The heat transfer coefficient (h) can be calculated using the Nusselt Number (Nu), hydraulic diameter (D), and thermal conductivity (k) of the fluid: \[h = \frac{k \times Nu}{D}\] The condensation rate per unit length of the tube (Q') is given by: \[Q' = \frac{h \times \Delta T}{L}\] For a mass fraction of vapor of \(X = 0.2\), first calculate the mass flow rate of vapor and liquid using the total mass flow rate (\(\dot{m}\)). Then, determine the heat transfer coefficient (h), and the condensation rate per unit length of the tube (Q') using the expressions derived. To plot the heat transfer coefficient and condensation rate for \(0.1 \leq X \leq 0.3\), use software like Excel, Matlab, or any other plotting tool and create a graph that shows the relationship between heat transfer coefficient and condensation rate at different mass fractions of vapor. Analyze the trend and the relationship between the heat transfer coefficient and condensation rate at different mass fractions of vapor to gain insight into the tube's performance under the given conditions.

Step by step solution

01

Calculate the Total Mass Flow Rate of Vapor and Liquid

First, we need to calculate the mass flow rate of vapor and liquid using the total mass flow rate and mass fraction of vapor. The mass flow rate of vapor (\(\dot{m}_v\)) can be determined using the given mass fraction of vapor (X) and the total mass flow rate of the fluid (\(\dot{m}\)): \[\dot{m}_v = X \times \dot{m}\] The mass flow rate of liquid (\(\dot{m}_l\)) can be calculated using the total mass flow rate (\(\dot{m}\)) and the mass flow rate of vapor (\(\dot{m}_v\)): \[\dot{m}_l = \dot{m} - \dot{m}_v\]
02

Determine the Heat Transfer Coefficient

To determine the heat transfer coefficient (h), we need to use the mass flow rates of vapor and liquid along with any given information about the properties of these fluids (such as specific heat, latent heat of condensation, or the boiling point). For simplicity, let's assume that we are dealing with a simple case where these properties can be incorporated into a constant, called the Nusselt Number (Nu). The heat transfer coefficient can be determined using the Nusselt Number, the hydraulic diameter (D), and the thermal conductivity (k) of the fluid: \[h = \frac{k \times Nu}{D}\]
03

Determine the Condensation Rate Per Unit Length of the Tube

Now that we have the heat transfer coefficient, we can determine the condensation rate per unit length of the tube (Q') by using the following relationship: \[Q' = \frac{h \times \Delta T}{L}\] where 螖T is the temperature difference between the vapor and the tube wall, and L is the length of the tube. If required, adjust this formula to incorporate any given information about the fluid properties, such as specific heat, latent heat of condensation, or boiling point temperatures.
04

Calculate the Heat Transfer Coefficient and Condensation Rate for the Given Mass Fraction of Vapor

After obtaining the expressions for the heat transfer coefficient (h) and the condensation rate (Q'), calculate their values using the mass flow rates of vapor and liquid, the mass fraction of vapor, and any given properties of the fluid.
05

Plot the Heat Transfer Coefficient and Condensation Rate for the Given Range of Mass Fractions of Vapor

Now that we have a better understanding of the relationships between the mass flow rates, mass fractions, heat transfer coefficients, and condensation rates, we can use these expressions to generate a plot of the heat transfer coefficient and the condensation rate for mass fractions of vapor ranging from 0.1 to 0.3. We can use software like Excel, Matlab, or any other plotting tool to generate this plot. Create two sets of data points for the heat transfer coefficient and condensation rate for the given range of mass fractions of vapor (0.1 to 0.3) using their respective expressions. Then, plot these data points on the same graph to visualize the relationship between the heat transfer coefficient and the condensation rate at different mass fractions of vapor. Make sure to properly label the axes and provide a legend to differentiate between the two data sets. Once the plot is created, analyze the trend and the relationship between heat transfer coefficient and condensation rate at different mass fractions of vapor. This will provide insight into how the tube performs at various mass fractions of vapor under the given conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Fraction of Vapor
In fluid dynamics, understanding the mass fraction of vapor is crucial when dealing with mixtures that contain both liquid and vapor phases. The mass fraction of vapor ( \(X\),) is essentially a ratio that represents the proportion of the vapor mass compared to the total mass of the fluid mixture. It is calculated as follows:
  • Determine the total mass flow rate of the fluid mixture ( \(\dot{m}\),).
  • Calculate the mass flow rate of vapor ( \(\dot{m}_v = X \times \dot{m}\),).
  • The remaining mass flow rate corresponds to the liquid phase ( \( \dot{m}_l = \dot{m} \) \( - \dot{m}_v \),).
The mass fraction helps in analyzing the characteristics of the fluid mixture, such as its behavior during phase change processes like condensation. For instance, knowing the mass fraction of vapor can help in determining how much of the vapor will condense into liquid under specific conditions. This is fundamental for calculating the overall heat transfer occurring within a system that deals with mixed-phase fluids.
Condensation Rate
Condensation is a process where vapor turns into liquid. The condensation rate is crucial in many engineering applications, as it relates to the rate at which vapor is turning into liquid per unit length of a surface. It can impact the efficiency of heat exchangers and other thermal systems. The condensation rate per unit length of a tube ( \(Q'\),) can be calculated using the heat transfer coefficient and the temperature difference:
  • Heat transfer coefficient ( \(h\),) plays a key role in this calculation.
  • The formula to find condensation rate is given as: \[Q' = \frac{h \times \Delta T}{L}\], where \(\Delta T\) is the temperature difference, and \(L\) is the tube length.
This equation signifies that the condensation rate depends on how well heat is transferred through the system, which is often expressed through \(h\). A higher heat transfer efficiency leads to a higher condensation rate. The adjustment of \(\Delta T\) and the other parameters can influence how efficiently vapor is condensed into liquid, which is essential for optimizing the performance of cooling systems.
Nusselt Number
The Nusselt Number ( \(Nu\),) is a dimensionless number that characterizes the type of heat transfer occurring in a fluid. It bridges the convective heat transfer to conductive heat transfer across a boundary layer. The formula used to calculate the heat transfer coefficient with the Nusselt number is:
  • Heat transfer coefficient ( \(h\),) is derived using the equation \[h = \frac{k \times Nu}{D}\], where \(k\) is the thermal conductivity, and \(D\) is the hydraulic diameter of the tube.
  • Simply, \(Nu\) helps compare the convective heat transfer to conductive within the boundary layer.
A large Nusselt number typically indicates efficient convective heat transfer relative to conduction; this is critical for designing systems that require high-efficiency heat exchange. In practice, knowing the Nusselt number helps engineers optimize the design of heat exchangers and cooling systems to improve performance and reduce energy consumption.

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Most popular questions from this chapter

The condenser of a steam power plant consists of a square (in-line) array of 625 tubes, each of \(25-\mathrm{mm}\) diameter. Consider conditions for which saturated steam at \(0.105\) bars condenses on the outer surface of each tube, while a tube wall temperature of \(17^{\circ} \mathrm{C}\) is maintained by the flow of cooling water through the tubes. What is the rate of heat transfer to the water per unit length of the tube array? What is the corresponding condensation rate?

A technique for cooling a multichip module involves submerging the module in a saturated fluorocarbon liquid. Vapor generated due to boiling at the module surface is condensed on the outer surface of copper tubing suspended in the vapor space above the liquid. The thin-walled tubing is of diameter \(D=10 \mathrm{~mm}\) and is coiled in a horizontal plane. It is cooled by water that enters at \(285 \mathrm{~K}\) and leaves at \(315 \mathrm{~K}\). All the heat dissipated by the chips within the module is transferred from a \(100-\mathrm{mm} \times 100-\mathrm{mm}\) boiling surface, at which the flux is \(10^{5} \mathrm{~W} / \mathrm{m}^{2}\), to the fluorocarbon liquid, which is at \(T_{\text {sait }}=57^{\circ} \mathrm{C}\). Liquid properties are \(k_{l}=0.0537\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, c_{p, l}=1100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, h_{f g}^{\prime} \approx h_{f g}=84,400 \mathrm{~J} / \mathrm{kg}\), \(\rho_{l}=1619.2 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{v}=13.4 \mathrm{~kg} / \mathrm{m}^{3}, \sigma=8.1 \times 10^{-3}\) \(\mathrm{N} / \mathrm{m}, \mu_{l}=440 \times 10^{-6} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), and \(P r_{l}=9\). (a) For the prescribed heat dissipation, what is the required condensation rate \((\mathrm{kg} / \mathrm{s})\) and water flow rate \((\mathrm{kg} / \mathrm{s})\) ? (b) Assuming fully developed flow throughout the tube, determine the tube surface temperature at the coil inlet and outlet. (c) Assuming a uniform tube surface temperature of \(T_{s}=53.0^{\circ} \mathrm{C}\), determine the required length of the coil.

Consider a horizontal, \(D=1\)-mm-diameter platinum wire suspended in saturated water at atmospheric pressure. The wire is heated by an electrical current. Determine the heat flux from the wire at the instant when the surface of the wire reaches its melting point. Determine the corresponding centerline temperature of the wire. Due to oxidation at very high temperature, the wire emissivity is \(\varepsilon=0.80\) when it burns out. The water vapor properties at the film temperature of \(1209 \mathrm{~K}\) are \(\rho_{v}=0.189 \mathrm{~kg} / \mathrm{m}^{3}, c_{p, v}=2404 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), \(v_{v}=231 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, k_{v}=0.113 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

The role of surface tension in bubble formation can be demonstrated by considering a spherical bubble of pure saturated vapor in mechanical and thermal equilibrium with its superheated liquid. (a) Beginning with an appropriate free-body diagram of the bubble, perform a force balance to obtain an expression of the bubble radius, $$ r_{b}=\frac{2 \sigma}{p_{\mathrm{sta}}-p_{l}} $$ where \(p_{\text {sat }}\) is the pressure of the saturated vapor and \(p_{l}\) is the pressure of the superheated liquid outside the bubble. (b) On a \(p-v\) diagram, represent the bubble and liquid states. Discuss what changes in these conditions will cause the bubble to grow or collapse. (c) Calculate the bubble size under equilibrium conditions for which the vapor is saturated at \(101^{\circ} \mathrm{C}\) and the liquid pressure corresponds to a saturation temperature of \(100^{\circ} \mathrm{C}\).

A 10-mm-diameter copper sphere, initially at a uniform temperature of \(50^{\circ} \mathrm{C}\), is placed in a large container filled with saturated steam at \(l\) atm. Using the lumped capacitance method, estimate the time required for the sphere to reach an equilibrium condition. How much condensate \((\mathrm{kg})\) was formed during this period?
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