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Chapter 7: Problem 132

Mass transfer experiments have been conducted on a naphthalene cylinder of \(18.4-\mathrm{mm}\) diameter and \(88.9-\mathrm{mm}\) length subjected to a cross flow of air in a low-speed wind tunnel. After exposure for \(39 \mathrm{~min}\) to the airstream at a temperature of \(26^{\circ} \mathrm{C}\) and a velocity of \(12 \mathrm{~m} / \mathrm{s}\), it was determined that the cylinder mass decreased by \(0.35 \mathrm{~g}\). The barometric pressure was recorded at \(750.6 \mathrm{~mm} \mathrm{Hg}\). The saturation pressure \(p_{\text {sat }}\) of naphthalene vapor in equilibrium with solid naphthalene is given by the relation \(p_{\text {sat }}=p \times 10^{E}\), where \(E=8.67-(3766 / T)\), with \(T(\mathrm{~K})\) and \(p\) (bar) being the temperature and pressure of air. Naphthalene has a molecular weight of \(128.16 \mathrm{~kg} / \mathrm{kmol}\). (a) Determine the convection mass transfer coefficient from the experimental observations. (b) Compare this result with an estimate from an appropriate correlation for the prescribed flow conditions.

Short Answer

Expert verified
The convection mass transfer coefficient from experimental observations is found to be \(h_m = 0.5466 \, kg/(m^2 \cdot s)\). Unfortunately, without more information on flow conditions, we cannot estimate the mass transfer coefficient using a correlation, and therefore a comparison cannot be made.

Step by step solution

01

Calculate the naphthalene saturation pressure

First, we need to find the saturation pressure of naphthalene at the given temperature and pressure conditions using the provided relation. Given: T = 26 卤C => T = 26 + 273.15 = 299.15 K p = 750.6 mm Hg => p = 750.6 * 1.01325 / 760 = 1.00249 bar Now, we can find E and the saturation pressure of naphthalene using these values: E = 8.67 - (3766 / 299.15) = -3.551 p_sat = 1.00249 * 10^(-3.551) = 0.000278857 bar
02

Calculate the mass transfer rate

Next, we want to find the mass transfer rate of naphthalene across the cylinder surface. We have the mass decrease (0.35 g) and the exposure time (39 min) for the experiment. First, let's convert the mass decrease into kilograms (kg) and the time into seconds: Mass decrease = 0.35 g => 0.00035 kg Time = 39 min => 2340 s Now we can find the mass transfer rate (dm/dt): (dm/dt) = 0.00035 kg / 2340 s = 1.49573 * 10^(-7) kg/s
03

Calculate the convection mass transfer coefficient

Now we can use the mass transfer rate and the saturation pressure to find the convection mass transfer coefficient (hm). The mass transfer is given by the equation: (dm/dt) = hm * A * (p_s - p_inf) where A is the surface area of the cylinder, and p_s and p_inf are the saturation and ambient pressure of naphthalene vapor, respectively. First, let's find the surface area of the cylinder: A = 蟺dL = 蟺(0.0184 m)(0.0889 m) = 0.016012 m^2 Now, we can rearrange the equation for hm and plug in the known values: hm = (dm/dt) / [A * (p_s - p_inf)] = 1.49573 * 10^(-7) kg/s / [0.016012 m^2 * (0.000278857 - 0) bar] hm = 0.5466 kg/(m虏路s)
04

Estimate the mass transfer coefficient using an appropriate correlation

The correlation most suitable for this exercise is the Sherwood number (Sh) correlation. The general equation is: Sh = hm * L / D = a * Re^b * Pr^c * Sc^d where L is a characteristic dimension, D is the mass diffusivity, Re is Reynold's number, Pr is Prandtl number, Sc is Schmidt number, and a, b, c and d are constants that depend on the flow conditions. Unfortunately, without the values for Re, Pr, and Sc or the constants, we cannot estimate the mass transfer coefficient using this correlation.
05

Compare the experimental mass transfer coefficient with the theoretical one

Since we cannot calculate the theoretical mass transfer coefficient using the correlation without values for Re, Pr, Sc, and the constants, we cannot make a comparison with the experimental coefficient, hm = 0.5466 kg/(m虏路s). If more information was provided about the flow conditions and the constants for the correlation, we would have been able to make a meaningful comparison. However, we have successfully determined the convection mass transfer coefficient from the experimental observations as required in the question.

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Key Concepts

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

Saturation Pressure
In the study of convection mass transfer, the concept of saturation pressure is crucial. Saturation pressure refers to the pressure exerted by a vapor in equilibrium with its liquid or solid phase. In this exercise, we are focusing on the saturation pressure of naphthalene vapor, which comes into play when the solid naphthalene sublimes into its vapor form.
To calculate the saturation pressure of naphthalene, the relation given is: \[ p_{\text{sat}} = p \times 10^{E} \]where the exponent \( E = 8.67 - \frac{3766}{T} \) depends on the temperature \( T \) in Kelvin and the ambient pressure \( p \) in bar.
Understanding saturation pressure is vital since it directly influences the mass transfer rate, as it is one of the driving factors facilitating the mass transfer in this experiment. A precise calculation of the saturation pressure helps in determining how strongly the naphthalene vapor can push into the air stream, aiding in solving the mass transfer equations effectively.
Naphthalene
Naphthalene is a polycyclic aromatic hydrocarbon that is commonly found in mothballs. In this exercise, it is used as a solid to study mass transfer from the surface of a naphthalene cylinder. Understanding naphthalene鈥檚 properties, such as its molecular weight (128.16 kg/kmol), is crucial for calculations related to its sublimation and diffusion into the air.
Naphthalene's sublimation under given conditions allows us to observe its vapor characteristics, leading us to calculate the mass transfer rate. The substance鈥檚 unique physical and chemical properties make it an interesting subject for studies related to sublimation, saturation pressure, and mass transfer phenomena.
Observing how naphthalene behaves under different atmospheric conditions can aid in understanding how it may behave in practical situations, such as in air freshening or pest control applications.
Mass Transfer Rate
The mass transfer rate is a vital concept in the field of convection mass transfer. It quantifies how much mass moves from the surface into the surrounding fluid over time. In this experiment, the mass transfer rate of naphthalene vapor is calculated from the observed decrease in the mass of a naphthalene cylinder.
Given that the cylinder's mass decreases by 0.35 grams during 39 minutes of exposure, we first convert these units into kg and seconds, respectively:
  • Mass decrease: 0.35 g \( \to \) 0.00035 kg
  • Time: 39 min \( \to \) 2340 seconds
The mass transfer rate (\(dm/dt\)) is calculated using:\[(dm/dt) = \frac{0.00035 \text{ kg}}{2340 \text{ s}} \approx 1.49573 \times 10^{-7} \text{ kg/s}\]Understanding the mass transfer rate helps in determining the effectiveness of sublimation of naphthalene and its subsequent mixing with the air. It gives a practical measure of how such vapor-phase transfers may occur in industrial or natural settings.
Sherwood Number
The Sherwood number (Sh) is a dimensionless number used to describe mass transfer in a similar way as the Nusselt number does for heat transfer. In this context, it provides insight into the relationship between convective mass transfer at the cylinder's surface and diffusive mass transfer.
The Sherwood number is expressed through correlations such as:\[Sh = \frac{hm \cdot L}{D}\]Where:
  • \( hm \) is the convection mass transfer coefficient.
  • \( L \) is the characteristic length (e.g., cylinder diameter).
  • \( D \) is the mass diffusivity.
The Sherwood number can be estimated using empirical correlations often involving Reynolds, Prandtl, and Schmidt numbers, which describe the flow characteristics. In our exercise, we were unable to calculate it due to missing data; however, understanding how Sh depends on these factors is essential for designing systems with efficient mass transfer rates.
The Sherwood number not only aids in validating experimental results but can also predict the mass transfer in untested configurations, making it a powerful tool in engineering applications.

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Most popular questions from this chapter

To augment heat transfer between two flowing fluids, it is proposed to insert a 100 -mm-long, 5 -mm-diameter 2024 aluminum pin fin through the wall separating the two fluids. The pin is inserted to a depth of \(d\) into fluid 1 . Fluid 1 is air with a mean temperature of \(10^{\circ} \mathrm{C}\) and velocity of \(10 \mathrm{~m} / \mathrm{s}\). Fluid 2 is air with a mean temperature of \(40^{\circ} \mathrm{C}\) and velocity of \(3 \mathrm{~m} / \mathrm{s}\). (a) Determine the rate of heat transfer from the warm air to the cool air through the pin fin for \(d=50 \mathrm{~mm}\). (b) Plot the variation of the heat transfer rate with the insertion distance, \(d\). Does an optimal insertion distance exist?

A spherical thermocouple junction \(1.0 \mathrm{~mm}\) in diameter is inserted in a combustion chamber to measure the temperature \(T_{\infty}\) of the products of combustion. The hot gases have a velocity of \(V=5 \mathrm{~m} / \mathrm{s}\). (a) If the thermocouple is at room temperature, \(T_{i}\), when it is inserted in the chamber, estimate the time required for the temperature difference, \(T_{\infty}-T\), to reach \(2 \%\) of the initial temperature difference, \(T_{\infty}-T_{i}\). Neglect radiation and conduction through the leads. Properties of the thermocouple junction are approximated as \(k=100 \mathrm{~W} / \mathrm{m}+\mathrm{K}, c=385 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\), and \(\rho=8920 \mathrm{~kg} / \mathrm{m}^{3}\), while those of the combustion gases may be approximated as \(k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\nu=50 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\), and \(\operatorname{Pr}=0.69\). (b) If the thermocouple junction has an emissivity of \(0.5\) and the cooled walls of the combustor are at \(T_{c}=400 \mathrm{~K}\), what is the steady- state temperature of the thermocouple junction if the combustion gases are at \(1000 \mathrm{~K}\) ? Conduction through the lead wires may be neglected. (c) To determine the influence of the gas velocity on the thermocouple measurement error, compute the steady-state temperature of the thermocouple junction for velocities in the range \(1 \leq V \leq 25 \mathrm{~m} / \mathrm{s}\). The emissivity of the junction can be controlled through application of a thin coating. To reduce the measurement error, should the emissivity be increased or decreased? For \(V=5 \mathrm{~m} / \mathrm{s}\), compute the steadystate junction temperature for emissivities in the range \(0.1 \leq \varepsilon \leq 1.0\).

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