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1.42 One method for growing thin silicon sheets for photovoltaic solar panels is to pass two thin strings of high melting temperature material upward through a bath of molten silicon. The silicon solidifies on the strings near the surface of the molten pool, and the solid silicon sheet is pulled slowly upward out of the pool. The silicon is replenished by supplying the molten pool with solid silicon powder. Consider a silicon sheet that is \(W_{\mathrm{si}}=85 \mathrm{~mm}\) wide and \(t_{\mathrm{si}}=150 \mu \mathrm{m}\) thick that is pulled at a velocity of \(V_{\mathrm{si}}=20 \mathrm{~mm} / \mathrm{min}\). The silicon is melted by supplying electric power to the cylindrical growth chamber of height \(H=350 \mathrm{~mm}\) and diameter \(D=300 \mathrm{~mm}\). The exposed surfaces of the growth chamber are at \(T_{s}=\) \(320 \mathrm{~K}\), the corresponding convection coefficient at the exposed surface is \(h=8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the surface is characterized by an emissivity of \(\varepsilon_{s}=0.9\). The solid silicon powder is at \(T_{\mathrm{s}, i}=298 \mathrm{~K}\), and the solid silicon sheet exits the chamber at \(T_{\text {si, } o}=420 \mathrm{~K}\). Both the surroundings and ambient temperatures are \(T_{\infty}=T_{\text {sur }}=298 \mathrm{~K}\). (a) Determine the electric power, \(P_{\text {elec }}\), needed to operate the system at steady state. (b) If the photovoltaic panel absorbs a time-averaged solar flux of \(q_{\text {sol }}^{\prime \prime}=180 \mathrm{~W} / \mathrm{m}^{2}\) and the panel has a conversion efficiency (the ratio of solar power absorbed to electric power produced) of \(\eta=0.20\), how long must the solar panel be operated to produce enough electric energy to offset the electric energy that was consumed in its manufacture?

Step by step solution

01

Calculate mass flow rate of silicon

First, we need to calculate the mass flow rate of the silicon that is being pulled out of the pool. The mass flow rate is given by: \[\dot{m}_{\mathrm{Si}} = W_{\mathrm{si}} \cdot t_{\mathrm{si}} \cdot V_{\mathrm{si}} \cdot \rho_{\mathrm{Si}}\] where \(\rho_{\mathrm{Si}} = 2329 \frac{\mathrm{kg}}{\mathrm{m}^3}\) is the density of silicon. Substituting the given values, we have: \[\dot{m}_{\mathrm{Si}} = (85 \times 10^{-3}\mathrm{m}) \cdot (150 \times 10^{-6}\mathrm{m}) \cdot (20 \times 10^{-3}\frac{\mathrm{m}}{\mathrm{min}}) \cdot (2329 \frac{\mathrm{kg}}{\mathrm{m}^3})\] Now, converting the mass flow rate from \(\frac{\mathrm{kg}}{\mathrm{min}}\) to \(\frac{\mathrm{kg}}{\mathrm{s}}\), we have: \[\dot{m}_{\mathrm{Si}} = \dot{m}_{\mathrm{Si}} \cdot \frac{1\mathrm{min}}{60\mathrm{s}}\]

02

Calculate the rate of energy to heat silicon

The required energy to heat the silicon can be calculated using the equation: \[\dot{Q}_{\mathrm{heat}} = \dot{m}_{\mathrm{Si}} \cdot c_{p,\mathrm{Si}} \cdot (T_{\mathrm{si, o}} - T_{\mathrm{s}, i})\] where \(c_{p,\mathrm{Si}} = 703\) J/kg·K is the specific heat capacity of silicon at constant pressure. Using the values calculated in step 1, we can determine the heating rate.

03

Calculate heat loss through convection and radiation

The rate of heat loss through convection is given by: \[\dot{Q}_{\mathrm{conv}} = h \cdot A_s \cdot (T_s - T_\infty)\] where \(A_s\) is the surface area of the cylindrical growth chamber and can be calculated as: \[A_s = \pi D H\] Calculate \(A_s\) using the given dimensions, and substitute it into the equation for convection heat loss. The rate of heat loss through radiation is given by the Stefan-Boltzmann equation: \[\dot{Q}_{\mathrm{rad}} = \epsilon_s \cdot \sigma \cdot A_s \cdot (T_s^4 - T_\infty^4)\] where \(\sigma = 5.67 \times 10^{-8} \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}^4}\) is the Stefan-Boltzmann constant. Calculate the radiation heat loss using the given values for \(T_s\), \(T_\infty\), and \(\varepsilon_s\).

04

Calculate the required electric power

The required electric power can be determined by summing the rates of energy for heating the silicon and heat loss: \[P_{\mathrm{elec}} = \dot{Q}_{\mathrm{heat}} + \dot{Q}_{\mathrm{conv}} + \dot{Q}_{\mathrm{rad}}\] Calculate \(P_{\mathrm{elec}}\) using the values obtained in the previous steps.

05

Calculate energy generated by the photovoltaic panel

The energy produced by the photovoltaic panel can be calculated as: \[P_{\mathrm{panel}} = q_{\mathrm{sol}}^{\prime\prime} \cdot W_{\mathrm{si}} \cdot L \cdot \eta\] where \(L\) is the length of the solar panel. However, we don't know the length of the solar panel yet. Therefore, we consider energy produced in one second: \[E_{\mathrm{panel}} = P_{\mathrm{panel}} \cdot 1\mathrm{s}\]

06

Calculate time required to offset energy consumed in manufacturing

We can find the time required to offset the energy consumed by dividing the energy consumed during the manufacturing process by the energy generated by the photovoltaic panel in one second: \[t_{\mathrm{offset}} = \frac{P_{\mathrm{elec}} \cdot t_{\mathrm{Si}}}{E_{\mathrm{panel}}}\] Substitute the values for \(P_{\mathrm{elec}}\) and \(E_{\mathrm{panel}}\) into the equation to find \(t_{\mathrm{offset}}\).

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Key Concepts

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

Solar Photovoltaic Panels

Solar photovoltaic panels are devices that convert sunlight into electricity. These panels consist of many solar cells, usually made from silicon, which have the capacity to generate electric currents when exposed to sunlight. Photons from the sun knock electrons loose from their atoms in the silicon, creating an electric flow.
Solar panels are commonly used for residential and commercial energy resources due to their renewable nature and minimal environmental impact. The efficiency of a solar panel is influenced by several factors, including:

• The quality and type of materials used
• The panel's exposure to sunlight
• Environmental conditions such as temperature and shade

Learning about photovoltaic panels is crucial because they are the foundation of solar energy technology, providing paths towards sustainable and clean energy solutions.

Energy Conservation in Manufacturing

Energy conservation in manufacturing focuses on reducing energy consumption without compromising the quality or efficiency of production. In the context of silicon sheet growth for photovoltaic panels, minimizing energy use is critical both for cost effectiveness and environmental sustainability.
Improvements can be made by:

• Optimizing manufacturing processes to reduce waste
• Using high-efficiency equipment that requires less power
• Recapturing waste heat for reuse within the system, thus conserving energy

Energy conservation not only reduces operational costs but also aligns with global efforts to lower carbon emissions, enhancing the sustainability of manufacturing processes.

Heat Transfer Calculations

Heat transfer calculations play a vital role in the manufacturing process, especially for materials like silicon that undergo phase changes. Understanding how heat is transferred within a system helps in designing efficient thermal processes.
Key modes of heat transfer include:

• Conduction: heat transfer through a material by direct contact
• Convection: heat transfer by the movement of fluid (air or liquid)
• Radiation: energy transfer through electromagnetic waves

Accurate heat transfer calculations ensure that the energy supplied is utilized optimally, preventing unnecessary energy loss and improving the overall efficiency of the manufacturing process.

Photovoltaic Efficiency

Photovoltaic efficiency refers to the ability of a solar panel to convert sunlight into usable electricity. It is expressed as a percentage of the sunlight that strikes the panel and is converted into electrical power. The efficiency depends on:

• The purity and type of silicon used
• The design and structure of the photovoltaic cells
• Quality of the installation and maintenance of the panels

Increasing the efficiency of solar panels reduces the need for large installations to generate adequate power, making solar energy more accessible and economically viable. Efforts in research and development focus on increasing efficiency through new materials and technology.

Thermal Radiation

Thermal radiation is a form of heat transfer that occurs when energy is emitted as electromagnetic waves by a body due to its temperature. It doesn’t require a medium, which means it can occur even in a vacuum.
In the growth process of silicon sheets for solar panels, managing thermal radiation is crucial. The growth chamber's design, including its surface properties like emissivity, directly affects how much heat is lost or retained.
By understanding and controlling thermal radiation, manufacturers can reduce energy waste, allowing for a more efficient production process and contributing to the energy conservation goals in manufacturing.

Convection Heat Loss

Convection heat loss occurs when heat is transferred away from a surface by the movement of fluids, such as air or liquid surrounding it. In the context of manufacturing silicon sheets, convection plays a significant role.
Factors influencing convection heat loss include:

• The material's surface area exposed to the fluid
• The temperature difference between the material and the fluid
• The speed and nature of the fluid's movement (laminar or turbulent flow)

By optimizing the growth chamber’s conditions and minimizing unnecessary exposure to surrounding fluids, convection heat losses can be significantly reduced, consequently lowering the energy requirements for production.

Specific Heat Capacity of Silicon

The specific heat capacity of silicon is an essential factor in its manufacturing process, as it defines the amount of energy needed to change its temperature. This property is particularly important when silicon transitions from solid to liquid or vice versa during the photovoltaic panel manufacturing.
Silicon has a specific heat capacity of approximately 703 J/kg·K, which means each kilogram of silicon requires 703 joules for its temperature to rise by one Kelvin.
Understanding this parameter helps in accurate energy calculations, ensuring that enough power is supplied to melt and mold the silicon sheets without excess energy consumption, making the production process energy-efficient and cost-effective.