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Chapter 7: Problem 78

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.

Short Answer

Expert verified
The terminal velocity of the solder droplets is found to be approximately \(0.025 \mathrm{~m} / \mathrm{s}\). The distance traveled by the droplets before they are completely solidified is about \(0.248 \mathrm{~m}\). The separation distance between the particles as they traverse the nitrogen gas is approximately \(1.39 \times 10^{-5} \mathrm{~m}\), and the pot volume needed in order to produce the solder balls continuously for one week is about \(0.105 \mathrm{~m}^3\).

Step by step solution

01

Find the Reynolds number and the Drag coefficient

First, we need to find the Reynolds number (Re) using the equation: \(Re = \frac{D_vp\rho}{\mu}\), where D and v are the diameter and velocity of the solder balls, and 渭 is the dynamic viscosity of the fluid (nitrogen gas). The properties of nitrogen gas are as follows: - dynamic viscosity (碌) = 17.81 x 10鈦烩伓 kg/m路s, at 30掳C Now, calculate the Reynolds number (Re): \(Re = \frac{130 \times 10^{-6} \times 2 \times 8,230}{17.81 \times 10^{-6}}\) Next, we will find the drag coefficient (Cd) by using the Reynolds number within the appropriate range for spherical particles: \(Cd = \frac{24}{Re} (1 + 0.14 Re^{0.7})\) We will then use the drag coefficient to calculate the terminal velocity.
02

Determine the terminal velocity

To find the terminal velocity of the solder balls, we will use the following equation: \(v_t = \sqrt{\frac{2 \cdot g \cdot (\rho_p - \rho) \cdot D_p}{C_d\cdot \rho}}\), where 蟻p and 蟻 are the densities of the solder balls and gas, respectively, and g is the gravitational acceleration. First, to find the density of the nitrogen gas, we'll use the ideal gas law and calculate the gas constant for nitrogen gas (Rnitrogen): \(R_{nitrogen} = \frac{R_{universal}}{M_{nitrogen}}\), where Runiversal = 8.3145 J/mol路K and Mnitrogen = 28.0134 g/mol Now, we can find the density of the nitrogen gas: \(\rho_{gas} = \frac{PM_{nitrogen}}{RT}\), where P is pressure, T is temperature, and R is the gas constant. Using 蟻gas and the drag coefficient (Cd), we can determine the terminal velocity (vt).
03

Calculate the distance traveled and time taken

We will use the energy balance equation to solve for the time t needed for the solder droplet to solidify: \(m_p \cdot c \cdot (T_{initial} - T_{melting}) = h \cdot A \cdot (T_{initial} - T_{gas}) \cdot t\) We will then use the time t to determine the distance traveled by the solder ball: \(s = v_t \times t\)
04

Calculate separation distance and pot volume

To find the separation distance between the particles, we divide their terminal velocity by their rate of production: \(sep\_distance = \frac{v_t}{production\_rate}\), where production_rate = 1800 particles per second Next, calculate the required pot volume to produce solder balls continuously for one week: \(v_{pot} = production\_rate \times volume\_per\_particle \times time_{week}\), where time_week = 7 days 脳 24 hours/day 脳 60 minutes/hour 脳 60 seconds/minute, and volume_per_particle = \(\frac{4}{3} \pi \left(\frac{D_p}{2}\right)^3\) By following the steps above, we can find the terminal velocity of the solder balls, the distance they travel before solidifying, the separation distance between the particles, and the required pot volume for continuous production.

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Key Concepts

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

Solder Ball Solidification
Solder ball solidification is a critical process in manufacturing electronics as it ensures the integrity and performance of the solder connections. Solder is a metal alloy typically made of tin and lead, used to create a conductive path between components. Solidification occurs when molten solder cools and hardens.
  • As the molten solder droplets travel through a cooling chamber, the reduction in temperature causes them to solidify.
  • The inert atmosphere, usually nitrogen, prevents oxidation which could affect the quality of the solder balls.
  • Proper solidification is important to avoid defects like voids or cracks that can lead to mechanical failures.
  • The properties of the solder, such as melting temperature and latent heat of fusion, are vital for calculating the time and distance required for solidification.
Understanding the heat and materials properties, as well as the thermal environment, is essential to optimize this process, ensuring high quality and reliability in solder balls produced.
Terminal Velocity Calculation
Terminal velocity is the constant speed that an object reaches when the force of gravity pulling it downwards is balanced by the drag force pushing upwards. For the solder balls in this manufacturing process, calculating terminal velocity is crucial to know how quickly they settle in the cooling chamber.
  • We use the drag coefficient and Reynolds number, which take into account the fluid properties, size of the object, and speed.
  • The equation for terminal velocity involves parameters like the density difference between the solder and the surrounding fluid, diameter of the solder ball, and gravitational constant.
  • Determining terminal velocity helps in setting up appropriate production speeds and spacing in the chamber to ensure efficient cooling.
Correctly determining terminal velocity ensures that solder balls have sufficient time to solidify before they reach the collection area, ensuring consistency in manufacturing.
Reynolds Number in Fluid Dynamics
Reynolds number is a dimensionless value that plays a crucial role in understanding the flow characteristics of a fluid. It helps determine whether the flow is laminar or turbulent, which affects drag forces experienced by the solder balls.
  • Calculated using the fluid velocity, diameter of the ball, fluid density, and dynamic viscosity: \[ Re = \frac{DV\rho}{\mu} \]
  • A low Reynolds number indicates laminar flow, where fluid moves in parallel layers with no disruption between them.
  • A high Reynolds number indicates turbulent flow, with chaotic movements and eddies affecting the object's movement and drag.
  • For solder ball solidification, knowing the Reynolds number helps in predicting and managing the drag forces which are crucial for reaching a steady descent speed.
This understanding helps optimize the design and operation of the cooling chamber, ensuring effective solidification and uniformity in the size and shape of solder balls.
Energy Balance in Cooling Process
The energy balance in the cooling process describes how the thermal energy in the solder ball is distributed as it cools down and solidifies. This is important to ensure that the solder balls cool at an appropriate rate without residual internal stresses.
  • The energy balance equation compares the heat loss due to cooling with the heat required to reach the solder's melting temperature and solidify.
  • Key parameters involve the initial and final temperatures, specific heat capacity, and thermal conductivity of solder.
  • Understanding this balance allows engineers to design the chamber's cooling rate, so that solder balls fully solidify before collection.
  • Maintaining an energy balance is crucial in preventing defects such as cracks or incomplete solidification.
Optimizing the energy balance during the cooling process enhances product quality and helps achieve efficient production rates while maintaining the electrical and mechanical properties of the solder balls.

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Most popular questions from this chapter

In an extrusion process, copper wire emerges from the extruder at a velocity \(V_{e}\) and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, \(T(x)\), along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire \(\left(V_{e}, D, \rho, c_{p}, \varepsilon\right)\), the convection coefficient associated with the cross flow \((\bar{h})\), and the environmental temperatures \(\left(T_{w}, T_{\text {sur }}\right)\). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For \(V_{c}=0.2 \mathrm{~m} / \mathrm{s}\), \(D=5 \mathrm{~mm}, V=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}\), and an initial wire temperature of \(T_{i}=600^{\circ} \mathrm{C}\), compute the temperature \(T_{o}\) of the wire at \(x=L=5 \mathrm{~m}\). The density and specific heat of the copper are \(\rho=8900 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while properties of the air may be taken to be \(k=0.037 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=3 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(P r=0.69\). (c) Accounting for the effects of radiation, with \(\varepsilon=0.55\) and \(T_{\text {sur }}=25^{\circ} \mathrm{C}\), numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at \(L=5 \mathrm{~m}\). Explore the effects of \(V_{e}\) and \(\varepsilon\) on the temperature distribution along the wire.

A flat plate of width \(1 \mathrm{~m}\) is maintained at a uniform surface temperature of \(T_{s}=150^{\circ} \mathrm{C}\) by using independently controlled, heat-generating rectangular modules of thickness \(a=10 \mathrm{~mm}\) and length \(b=50 \mathrm{~mm}\). Each module is insulated from its neighbors, as well as on its back side. Atmospheric air at \(25^{\circ} \mathrm{C}\) flows over the plate at a velocity of \(30 \mathrm{~m} / \mathrm{s}\). The thermophysical properties of the module are \(k=5.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=320 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\), and \(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\). (a) Find the required power generation, \(\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\), in a module positioned at a distance \(700 \mathrm{~mm}\) from the leading edge. (b) Find the maximum temperature \(T_{\max }\) in the heatgenerating module.

Latent heat capsules consist of a thin-walled spherical shell within which a solid-liquid, phase-change material \((\mathrm{PCM})\) of melting point \(T_{\mathrm{mp}}\) and latent heat of fusion \(h_{s f}\) is enclosed. As shown schematically, the capsules may be packed in a cylindrical vessel through which there is fluid flow. If the \(\mathrm{PCM}\) is in its solid state and \(T_{\mathrm{mp}}T_{i}\), energy is released from the \(\mathrm{PCM}\) as it freezes and heat is transferred to the fluid. In either situation, all of the capsules within the packed bed would remain at \(T_{\mathrm{mp}}\) through much of the phase change process, in which case the fluid outlet temperature would remain at a fixed value \(T_{o^{*}}\). Consider an application for which air at atmospheric pressure is chilled by passing it through a packed bed \((\varepsilon=0.5)\) of capsules \(\left(D_{c}=50 \mathrm{~mm}\right)\) containing an organic compound with a melting point of \(T_{\text {mp }}=4^{\circ} \mathrm{C}\). The air enters a cylindrical vessel \(\left(L_{v}=D_{v}=0.40 \mathrm{~m}\right)\) at \(T_{i}=25^{\circ} \mathrm{C}\) and \(V=1.0 \mathrm{~m} / \mathrm{s}\). (a) If the PCM in each capsule is in the solid state at \(T_{\text {map }}\) as melting occurs within the capsule, what is the outlet temperature of the air? If the density and latent heat of fusion of the \(\mathrm{PCM}\) are \(\rho=\) \(1200 \mathrm{~kg} / \mathrm{m}^{3}\) and \(h_{s f}=165 \mathrm{~kJ} / \mathrm{kg}\), what is the mass rate \((\mathrm{kg} / \mathrm{s})\) at which the \(\mathrm{PCM}\) is converted from solid to liquid in the vessel? (b) Explore the effect of the inlet air velocity and capsule diameter on the outlet temperature. (c) At what location in the vessel will complete melting of the PCM in a capsule first occur? Once complete melting begins to occur, how will the outlet temperature vary with time and what is its asymptotic value?

A long, cylindrical, electrical heating element of diameter \(D=10 \mathrm{~mm}\), thermal conductivity \(k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), density \(\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}\), and specific heat \(c_{p}=900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively. (a) Neglecting radiation, estimate the steady-state surface temperature when, per unit length of the heater, electrical energy is being dissipated at a rate of \(1000 \mathrm{~W} / \mathrm{m}\). (b) If the heater is activated from an initial temperature of \(27^{\circ} \mathrm{C}\), estimate the time required for the surface temperature to come within \(10^{\circ} \mathrm{C}\) of its steady-state value.

A stream of atmospheric air is used to dry a series of biological samples on plates that are each of length \(L_{i}=0.25 \mathrm{~m}\) in the direction of the airflow. The air is dry and at a temperature equal to that of the plates \(\left(T_{\infty}=T_{s}=50^{\circ} \mathrm{C}\right)\). The air speed is \(u_{\infty}=9.1 \mathrm{~m} / \mathrm{s}\). (a) Sketch the variation of the local convection mass transfer coefficient \(h_{m x}\) with distance \(x\) from the leading edge. Indicate the specific nature of the \(x\) dependence. (b) Which of the plates will dry the fastest? Calculate the drying rate per meter of width for this plate \((\mathrm{kg} / \mathrm{s}+\mathrm{m})\). (c) At what rate would heat have to be supplied to the fastest drying plate to maintain it at \(T_{s}=50^{\circ} \mathrm{C}\) during the drying process?
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