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Chapter 7: Problem 105

Consider the plasma spraying process of Problems \(5.25\) and \(7.82\). For a nozzle exit diameter of \(D=\) \(10 \mathrm{~mm}\) and a substrate radius of \(r=25 \mathrm{~mm}\), estimate the rate of heat transfer by convection \(q_{\text {cany }}\) from the argon plasma to the substrate, if the substrate temperature is maintained at \(300 \mathrm{~K}\). Energy transfer to the substrate is also associated with the release of latent heat \(q_{\text {lat }}\), which occurs during solidification of the impacted molten droplets. If the mass rate of droplet impingement is \(\dot{m}_{p}=0.02 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2}\), estimate the rate of latent heat release.

Short Answer

Expert verified
To estimate the rate of heat transfer by convection \(q_{\text{cany}}\), we first calculate the area available for heat transfer as \(A = 1.96 \times 10^{-3} \mathrm{m}^2\). Then, we use the convective heat transfer formula \(q_{\text{cany}} = h_{\text{conv}} \times A \times \Delta T\), assuming \(h_{\text{conv}}\) and \(\Delta T\) are known. For the rate of latent heat release \(q_{\text{lat}}\), we use the given mass rate of droplet impingement \(\dot{m}_{p} = 0.02 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^2\) and the formula \(q_{\text{lat}} = \dot{m}_{p} \times L \), but we cannot estimate the rate of latent heat release without information about the material or the actual latent heat value \(L\).

Step by step solution

01

Calculate the rate of heat transfer by convection

To estimate the rate of heat transfer by convection, we can first determine the total area available for heat transfer, using the given substrate radius: Area \(A = \pi r^2 \) Using the given substrate radius of \(r = 25 \mathrm{~mm}\), we can calculate the area: \(A = \pi (25 \times 10^{-3} \mathrm{m})^2 = 1.96 \times 10^{-3} \mathrm{m}^2\) Now, we need to find the convective heat transfer rate \(q_{\text{cany}}\). We will assume that the convective heat transfer coefficient \(h_{\text{conv}}\) and the temperature difference between the plasma and the substrate \(\Delta T\) are known. Then, we can use the convective heat transfer formula: \(q_{\text{cany}} = h_{\text{conv}} \times A \times \Delta T\) In this problem, we don't have enough information to calculate \(h_{\text{conv}}\) and \(\Delta T\). So, we will assume that they are known and use the formula to estimate the convective heat transfer rate.
02

Estimate the rate of latent heat release

To find the rate of latent heat release, we will use the mass rate of droplet impingement \(\dot{m}_{p}\), which is given as: \(\dot{m}_{p} = 0.02 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^2\). The rate of latent heat release \(q_{\text{lat}}\) can be estimated as: \(q_{\text{lat}} = \dot{m}_{p} \times L \) where \(L\) is the latent heat of the molten droplets. However, without information about the material being sprayed or the actual latent heat value, we cannot provide an estimate for the rate of latent heat release.

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Key Concepts

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

Plasma Spraying
Plasma spraying is an advanced coating process that uses a high-temperature plasma jet to melt and accelerate material particles onto a substrate. This process involves passing a working gas, often argon, through an electric arc, generating exceptionally high temperatures. The heat from the plasma then melts the coating material.
  • The molten droplets are sprayed onto a surface at high velocity, forming a coating layer.
  • This technique is widely used for protecting surfaces from corrosion, heat, and wear.
  • It is flexible enough to be applied to metals, ceramics, and even certain plastics.
The effectiveness of plasma spraying lies in its ability to create thick, dense, and strongly bonded coatings. Such characteristics make it invaluable in various industries, including aerospace, automotive, and biomedical fields, where enhanced surface properties are crucial.
Convective Heat Transfer
Convective heat transfer is the mechanism of heat being carried away by the movement of fluid or gas particles. In the context of plasma spraying, convective heat transfer occurs when heat is transferred from the hot plasma to the cooler substrate surface.
  • This transfer depends heavily on the convective heat transfer coefficient (
  • The coefficient describes the efficiency of heat transfer due to convection and varies with fluid properties and flow conditions.
  • The temperature difference (<

    螖罢

    ) between the hot plasma and the substrate is also a critical factor.
To calculate the rate of convective heat transfer (

q_{ ext{cany}}

), we use the formula: \[ q_{ ext{cany}} = h_{ ext{conv}} \times A \times 螖罢\] This equation shows that larger areas and greater temperature differences lead to higher heat transfer rates, assuming a known
Latent Heat Release
Latent heat release is a crucial part of the plasma spraying process, specifically when dealing with molten particles. This form of heat transfer doesn't necessarily change the temperature but rather involves the phase transition of materials.
  • In spraying, droplets of molten material impact the substrate and solidify, releasing their latent heat.
  • Latent heat is the energy absorbed or released during a phase change from liquid to solid, without a change in temperature.
The rate of latent heat release (

q_{ ext{lat}}

) can be calculated using the formula:\[ q_{ ext{lat}} = \dot{m}_{p} \times L\] where \dot{m}_{p} is the mass rate of droplet impingement, and

L

is the latent heat. Knowing the properties of the sprayed material is necessary to determine the actual latent heat value. Understanding this concept ensures that coatings are not only structurally sound but also efficient in thermal management.

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Most popular questions from this chapter

A computer code is being developed to analyze a temperature sensor of \(12.5-\mathrm{mm}\) diameter experiencing cross flow of water with a free stream temperature of \(80^{\circ} \mathrm{C}\) and variable velocity. Derive an expression for the convection heat transfer coefficient as a function of the sensor surface temperature \(T_{s}\) for the range \(20

The cylindrical chamber of a pebble bed nuclear reactor is of length \(L=10 \mathrm{~m}\), and diameter \(D=3 \mathrm{~m}\). The chamber is filled with spherical uranium oxide pellets of core diameter \(D_{p}=50 \mathrm{~mm}\). Each pellet generates thermal energy in its core at a rate of \(\dot{E}_{g}\) and is coated with a layer of non-heat-generating graphite, which is of uniform thickness \(\delta=5 \mathrm{~mm}\), to form a pebble. The uranium oxide and graphite each have a thermal conductivity of \(2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The packed bed has a porosity of \(\varepsilon=0.4\). Pressurized helium at 40 bars is used to absorb the thermal energy from the pebbles. The helium enters the packed bed at \(T_{i}=450^{\circ} \mathrm{C}\) with a velocity of \(3.2 \mathrm{~m} / \mathrm{s}\). The properties of the helium may be assumed to be \(c_{p}=5193 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), \(k=0.3355 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2.1676 \mathrm{~kg} / \mathrm{m}^{3}, \mu=4.214 \times\) \(10^{-5} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, \operatorname{Pr}=0.654\). (a) For a desired overall thermal energy transfer rate of \(q=125 \mathrm{MW}\), determine the mean outlet temperature of the helium leaving the bed, \(T_{o}\), and the amount of thermal energy generated by each pellet, \(\dot{E}_{g^{*}}\) (b) The amount of energy generated by the fuel decreases if a maximum operating temperature of approximately \(2100^{\circ} \mathrm{C}\) is exceeded. Determine the maximum internal temperature of the hottest pellet in the packed bed. For Reynolds numbers in the range \(4000 \leq R e_{D} \leq 10,000\), Equation \(7.81\) may be replaced by \(\varepsilon \bar{j}_{H}=2.876 R e_{D}^{-1}+0.3023 R e_{D}^{-0.35}\).

Worldwide, over a billion solder balls must be manufactured daily for assembling electronics packages. The uniform droplet spray method uses a piezoelectric device to vibrate a shaft in a pot of molten solder that, in turn, ejects small droplets of solder through a precision-machined nozzle. As they traverse a collection chamber, the droplets cool and solidify. The collection chamber is flooded with an inert gas such as nitrogen to prevent oxidation of the solder ball surfaces. (a) Molten solder droplets of diameter \(130 \mu \mathrm{m}\) are ejected at a velocity of \(2 \mathrm{~m} / \mathrm{s}\) at an initial temperature of \(225^{\circ} \mathrm{C}\) into gaseous nitrogen that is at \(30^{\circ} \mathrm{C}\) and slightly above atmospheric pressure. Determine the terminal velocity of the particles and the distance the particles have traveled when they become completely solidified. The solder properties are \(\rho=8230 \mathrm{~kg} / \mathrm{m}^{3}\), \(c=240 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=38 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{s f}=42 \mathrm{~kJ} / \mathrm{kg}\). The solder's melting temperature is \(183^{\circ} \mathrm{C}\). (b) The piezoelectric device oscillates at \(1.8 \mathrm{kHz}\), producing 1800 particles per second. Determine the separation distance between the particles as they traverse the nitrogen gas and the pot volume needed in order to produce the solder balls continuously for one week.

Consider an air-conditioning system composed of a bank of tubes arranged normal to air flowing in a duct at a mass rate of \(\dot{m}_{a}(\mathrm{~kg} / \mathrm{s})\). A coolant flowing through the tubes is able to maintain the surface temperature of the tubes at a constant value of \(T_{s}

Dry air at \(35^{\circ} \mathrm{C}\) and a velocity of \(15 \mathrm{~m} / \mathrm{s}\) flows over a long cylinder of \(20-\mathrm{mm}\) diameter. The cylinder is covered with a thin porous coating saturated with water, and an embedded electrical heater supplies power to maintain the coating surface temperature at \(20^{\circ} \mathrm{C}\). (a) What is the evaporation rate of water from the cylinder per unit length \((\mathrm{kg} / \mathrm{h}+\mathrm{m})\) ? What electrical power per unit length of the cylinder \((\mathrm{W} / \mathrm{m})\) is required to maintain steady-state conditions? (b) After a long period of operation, all the water is evaporated from the coating and its surface is dry. For the same free stream conditions and heater power of part (a), estimate the temperature of the surface.
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