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Chapter 10: Problem 56

Saturated vapor from a chemical process condenses at a slow rate on the inner surface of a vertical, thinwalled cylindrical container of length \(L\) and diameter D. The container wall is maintained at a uniform temperature \(T_{s}\) by flowing cold water across its outer surface. Derive an expression for the time, \(t_{f}\), required to fill the container with condensate, assuming that the condensate film is laminar. Express your result in terms of \(D, L,\left(T_{\text {sat }}-T_{s}\right), g\), and appropriate fluid properties. Derive an expression for the time, \(t_{f}\), required to fill the container with condensate, assuming that the condensate film is laminar. Express your result in terms of \(D, L,\left(T_{\text {sat }}-T_{s}\right), g\), and appropriate fluid properties.

Short Answer

Expert verified
The time, \(t_f\), required to fill the container with condensate can be expressed as: \[t_f = \frac{蟺 D^2 蟻_L\nu L^2}{4k_L (T_{\text {sat}} - T_s)(蟻_L - 蟻_v)}\] Here, \(D\) is the container diameter, \(L\) is the container length, \(T_{\text {sat}}\) is the saturation temperature, \(T_s\) is the wall temperature, \(蟻_L\) is the liquid density, \(蟻_v\) is the vapor density, \(k_L\) is the liquid thermal conductivity, and \(谓\) is the liquid kinematic viscosity.

Step by step solution

01

Derive the mass flow rate of condensate

To begin, we need to find the mass flow rate of the condensate. Using Nusselt's film theory for condensation, the mass flow rate per unit length along the vertical wall is given by: \(m' = \frac{k_L \Delta T (蟻_L - 蟻_v)}{谓L}\) Here, \(k_L\) is the thermal conductivity of the liquid, 螖T is the temperature difference across the film, \(蟻_L\) and \(蟻_v\) are the densities of liquid and vapor respectively, 谓 is the kinematic viscosity of the liquid, and L is the length of the wall. Now, 螖T is given by, \(螖T = T_{\text {sat}} - T_s \). Substituting this into the mass flow rate equation, we get: \(m' = \frac{k_L (T_{\text {sat}} - T_s)(蟻_L - 蟻_v)}{谓L}\)
02

Derive the volume flow rate of condensate

Next, we need to find the volume flow rate of the condensate. This can be found by dividing the mass flow rate by the liquid density, \(蟻_L\): \(Q' = \frac{m'}{蟻_L}\) Substitute the expression for \(m'\) from the previous step into the equation: \(Q' = \frac{k_L (T_{\text {sat}} - T_s)(蟻_L - 蟻_v)}{蟻_L\nu L}\)
03

Integrate the volume flow rate with respect to time

Now, we need to find the total volume collected by integrating the volume flow equation with respect to time: \(V_t = \int_0^{t_f} Q'(t) dt\) Since the volume flow rate is assumed to be constant over time, the integration can be simplified: \(V_t = Q't_f\)
04

Equate the total volume collected to the volume of the container

The total volume collected in time \(\(t_f\)\), \(V_t\), should be equal to the volume of the container. The volume of the cylindrical container is given by: \[V_c = \frac{蟺 D^2}{4}L\] Now equate the two volumes to find the expression for time \(t_f\): \(\frac{蟺 D^2}{4}L = Q't_f\) Substitute the expression for \(Q'\) from the previous step: \(\frac{蟺 D^2}{4}L = \frac{k_L (T_{\text {sat}} - T_s)(蟻_L - 蟻_v)}{蟻_L\nu L}t_f\) Lastly, solve for \(t_f\): \[t_f = \frac{蟺 D^2 蟻_L\nu L^2}{4k_L (T_{\text {sat}} - T_s)(蟻_L - 蟻_v)}\] This is the expression for the time, \(t_f\), required to fill the container with condensate in terms of given parameters and fluid properties.

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Key Concepts

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

Nusselt's Film Theory
Understanding Nusselt's film theory is crucial when learning about condensation heat transfer processes. This theory is widely used to analyze the formation of a liquid film during the condensation of a vapor on a surface. According to Nusselt's film theory, a thin film of liquid forms when a saturated vapor comes into contact with a cooler surface. The theory is based on the assumption that the liquid film flows laminarly and the thermal resistance is primarily within the liquid film.

The thermal conductivity of the liquid, the temperature difference across the film, along with other properties such as densities and viscosity, play a significant role in determining the heat transfer rate through the film. This concept is the foundation for deriving certain equations like the mass flow rate of the condensate and the time required for a container to fill with condensate, which we see in the given exercise.

A key formula from the Nusselt's film theory relates the heat transfer coefficient to the mass flow rate and other properties mentioned previously. It is essential for students to understand this theory to not only solve problems but also to grasp how different physical properties and conditions can affect the condensation process.
Mass Flow Rate of Condensate
To analyze the behavior of the condensing vapor, it is imperative to consider the mass flow rate of condensate. This term denotes the amount of mass passing through a certain surface area per unit of time. In the field of heat transfer, specifically in condensation, the mass flow rate of the condensate determines the efficiency of the process.

When calculating the mass flow rate using Nusselt's film theory, students must include factors such as thermal conductivity, temperature gradient, liquid and vapor density, and the liquid's viscosity. As seen in the exercise, deriving the mass flow rate forms the initial step to eventually deducing the time required to fill a container with condensate. The equation is not only a functional tool in solving problems but also provides insights into how fast condensation occurs under various conditions. Practically speaking, engineers use this calculation to design systems for processes such as heat exchangers, refrigeration, and power generation systems, where condensation is a key operation.
Laminar Condensate Film
A laminar condensate film is characterized by smooth, orderly fluid motion, which is in contrast to a turbulent flow that is chaotic and mixed. In the context of the exercise, the assumption of a laminar flow is critical for simplifying the analysis and ultimately determining the fill time for the container. The laminar nature of the film means that the fluid particles move in parallel layers, with no disruption between them, which greatly reduces the complexity of the equations involved.

Within a laminar condensate film, heat transfer is primarily through conduction, and the resistance to heat transfer is located within the film itself. This assumption influences the derived expressions for quantities such as the mass and volume flow rates. Recognizing the characteristics of the flow regime is an important step for students learning about heat transfer processes as it impacts the resulting calculations and the interpretation of the physical phenomena. The importance of identifying laminar flow conditions becomes apparent when applying theoretical concepts to real-world scenarios where efficient heat transfer is necessary for system functionality.

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Most popular questions from this chapter

Electrical current passes through a horizontal, 2-mmdiameter conductor of emissivity \(0.5\) when immersed in water under atmospheric pressure. (a) Estimate the power dissipation per unit length of the conductor required to maintain the surface temperature at \(555^{\circ} \mathrm{C}\). (b) For conductor diameters of \(1.5,2.0\), and \(2.5 \mathrm{~mm}\), compute and plot the power dissipation per unit length as a function of surface temperature for \(250 \leq T_{s} \leq 650^{\circ} \mathrm{C}\). On a separate figure, plot the percentage contribution of radiation as a function of \(T_{s}\).

Saturated steam at \(0.1\) bar condenses with a convection coefficient of \(6800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside of a brass tube having inner and outer diameters of \(16.5\) and \(19 \mathrm{~mm}\), respectively. The convection coefficient for water flowing inside the tube is \(5200 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\). Estimate the steam condensation rate per unit length of the tube when the mean water temperature is \(30^{\circ} \mathrm{C}\).

Consider a gas-fired boiler in which five coiled, thinwalled, copper tubes of \(25-\mathrm{mm}\) diameter and \(8-\mathrm{m}\) length are submerged in pressurized water at \(4.37\) bars. The walls of the tubes are scored and may be assumed to be isothermal. Combustion gases enter each of the tubes at a temperature of \(T_{m, i}=700^{\circ} \mathrm{C}\) and a flow rate of \(\dot{m}=0.08 \mathrm{~kg} / \mathrm{s}\), respectively. (a) Determine the tube wall temperature \(T_{s}\) and the gas outlet temperature \(T_{m, o}\) for the prescribed conditions. As a first approximation, the properties of the combustion gases may be taken as those of air at \(700 \mathrm{~K}\). (b) Over time the effects of scoring diminish, leading to behavior similar to that of a polished copper surface. Determine the wall temperature and gas outlet temperature for the aged condition.

A heater for boiling a saturated liquid consists of two concentric stainless steel tubes packed with dense boron nitride powder. Electrical current is passed through the inner tube, creating uniform volumetric heating \(\dot{q}\) \(\left(\mathrm{W} / \mathrm{m}^{3}\right)\). The exposed surface of the outer tube is in contact with the liquid and the boiling heat flux is given as $$ q_{s}^{\prime \prime}=C\left(T_{s}-T_{\text {sat }}\right)^{3} $$ It is feared that under high-power operation the stainless steel tubes would severely oxidize if temperatures exceed \(T_{s s, x}\) or that the boron nitride would deteriorate if its temperature exceeds \(T_{\mathrm{bn}, x^{*}}\) Presuming that the saturation temperature of the liquid \(\left(T_{\text {sat }}\right)\) and the boiling surface temperature \(\left(T_{x}\right)\) are prescribed, derive expressions for the maximum temperatures in the stainless steel (ss) tubes and in the boron nitride (bn). Express your results in terms of geometric parameters \(\left(r_{1}, r_{2}\right.\), \(\left.r_{3}, r_{4}\right)\), thermal conductivities \(\left(k_{\mathrm{ss}}, k_{\mathrm{ba}}\right)\), and the boiling parameters \(\left(C, T_{\text {sial }}, T_{s}\right)\).

A tube of \(2-\mathrm{mm}\) diameter is used to heat saturated water at \(1 \mathrm{~atm}\), which is in cross flow over the tube. Calculate and plot the critical heat flux as a function of water velocity over the range 0 to \(2 \mathrm{~m} / \mathrm{s}\). On your plot, identify the pool boiling region and the transition region between the low- and high-velocity ranges. Hint: Problem \(10.20\) contains relevant information for pool boiling on small-diameter cylinders.
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