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The absorber of a simple flat-plate solar collector with no coverplate consists of a \(2 \mathrm{~mm}\)-thick aluminum plate with \(6 \mathrm{~mm}\)-diameter aluminum water tubes spaced at a pitch of \(10 \mathrm{~cm}\), as shown. On a clear summer day near the ocean, the air temperature is \(20^{\circ} \mathrm{C}\), and a steady wind is blowing. The solar radiation absorbed by the plate is calculated to be 680 \(\mathrm{W} / \mathrm{m}^{2}\), and the convective heat transfer coefficient is estimated to be \(14 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). If water at \(20^{\circ} \mathrm{C}\) enters the collector at \(7 \times 10^{-3} \mathrm{~kg} / \mathrm{s}\) per meter width of collector, and the collector is \(3 \mathrm{~m}\) long, estimate the outlet water temperature. For the aluminum take \(k=200 \mathrm{~W} / \mathrm{m} \mathrm{K}\) and \(\varepsilon=0.20\). (Hint: Evaluate the heat lost by reradiation using Eq. (1.19) with a constant value of \(h_{r}\) corresponding to a guessed average plate temperature. Then apply the steady-flow equation to an elemental length of the collector, and so derive a differential equation governing the water temperature increase along the collector.)

Step by step solution

01

Identify Given Data

First, let's identify all given parameters from the problem statement: - Aluminum plate thickness = 2 mm - Diameter of tubes = 6 mm - Tube pitch = 10 cm - Air temperature = 20掳C - Solar radiation absorbed = 680 W/m虏 - Convective heat transfer coefficient, h_c = 14 W/m虏K - Inlet water temperature = 20掳C - Water mass flow rate per width = 7 脳 10鈦宦� kg/s - Collector length = 3 m - Thermal conductivity of aluminum, k = 200 W/mK - Emissivity, 蔚 = 0.20.

02

Setup the Energy Equation

Since it is a steady-flow process, the energy balance for the control volume of the collector system can be written as:\[ q_{ ext{conv}} + q_{ ext{solar}} - q_{ ext{loss}} = rac{ ext{d}T_w}{ ext{d}x} imes rac{m ext{c}_p}{L}\]where - \( q_{conv} = h_c imes A_s imes (T_p - T_a) \) is the convection heat loss,- \( q_{solar} = 680 ext{ W/m虏} \) absorbed by the plate,- \( q_{loss} = rac{蔚 imes ext{Stefan-Boltzmann Constant} imes A_s imes (T_p^4 - T_s^4)}{1/蔚 + 1} \) accounts for radiative heat loss,- \( T_p \) is the plate temperature, - \( T_a \) is the air temperature, - \( T_w \) is the water temperature,- \( m \) is the mass flow rate, - \( c_p \) is the specific heat of water (approx. 4.18 kJ/kg K at 20掳C),- \( A_s \) is the surface area of the absorber, - and \( L \) is the length of the collector.

03

Estimate Heat Loss by Reradiation

Calculate the heat loss due to reradiation using the formula: \[q_{ ext{loss}} = rac{ ext{蔚} imes ext{蟽} imes A_s imes (T_p^4 - T_s^4)}{1/蔚 + 1},\]where 蟽 is the Stefan-Boltzmann constant (5.67 脳 10鈦烩伕 W/m虏K鈦�), and assuming the sky temperature (T_s) is given typically to be 273 K. Use a guessed average plate temperature (e.g., 323K) for initial calculations. Adjust as necessary.

04

Solve for Water Temperature Differential

Given that the absorbed solar radiation, convective and radiative losses are related to temperature increments:\[q_{ ext{absorb}} - q_{ ext{conv}} - q_{ ext{loss}} = m imes c_p imes rac{dT_w}{dx},\]where \(dT_w/dx\) is the temperature gradient of water along the collector. Integrating this equation along the length (0 to 3 m) of the collector, calculate temperature change.

05

Integrate to Find Outlet Temperature

Integrate the energy balance equation across the full length of the collector (3 m):\[rac{螖T_w}{L} = rac{(q_{ ext{absorb}} - q_{ ext{loss}} - q_{ ext{conv}})}{m imes c_p} imes L.\]Substitute known terms into the equation to solve for \( 螖T_w \), and compute the outlet temperature \( T_{ ext{out}} = T_{ ext{in}} + 螖T_w \). Iterate as necessary if the initial guessed average plate temperature affects the convergence of heat loss calculations.

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Key Concepts

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

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. It indicates how quickly heat can pass through a material and is expressed in Watts per meter Kelvin (W/mK). In the context of solar collectors, understanding thermal conductivity is crucial because it affects how effectively heat from the absorbed solar energy transfers through the materials involved.

Aluminum is commonly used in solar collectors due to its high thermal conductivity, which is around 200 W/mK. This means that aluminum can efficiently transfer heat from the solar collector's surface to the water inside the tubes quickly and evenly.

• A high thermal conductivity indicates efficient heat transfer, which is desired for enhancing the solar collector's efficiency.
• Consider the thickness and the thermal conductivity of the materials when designing systems to optimize heat transfer processes.

By maximizing the use of materials with high thermal conductivity, the efficiency of the solar collector is improved as more heat can be transferred into the water over the same period.

Convective Heat Transfer

Convective heat transfer is the process of heat transfer between the surface of the solar collector and the air around it. This process is largely dependent on the flow characteristics of the air and the temperature difference between the air and the collector's surface. A key parameter in this process is the convective heat transfer coefficient, which helps quantify the effectiveness of convection.

In our example, the convective heat transfer coefficient is given as 14 W/m虏K. This value helps us calculate the heat loss from convection, which can significantly influence the solar collector鈥檚 performance. A higher coefficient typically indicates that more heat is being transferred between the collector and the surrounding air, due to faster moving air or larger temperature differences:

• Convective heat loss is calculated using the formula: \[ q_{\text{conv}} = h_c \times A_s \times (T_p - T_a) \]
• \(h_c\) is the convective heat transfer coefficient.
• \(A_s\) is the surface area of the absorber.
• \(T_p\) is the plate temperature.
• \(T_a\) is the air temperature.

Understanding this concept aids in designing solar collectors that minimize unwanted heat losses, therefore optimizing the overall system performance.

Energy Balance

The energy balance in the context of a solar collector refers to accounting for all the heat energy absorbed, lost, and transferred within the system. All these energy elements must add up to maintain a state of equilibrium. This concept forms the basis of calculating the temperature changes within the system.

The energy balance equation combines various factors like absorbed solar radiation, convective heat losses through air contact, and reradiation losses. It considers these major components:

• The energy absorbed from the sun (solar radiation).
• The energy lost through convective heat transfer into the surrounding air.
• The energy balance equation is crucial in designing efficient solar collector systems, helping predict the change in water temperature as it flows through.

The general energy balance formula in our exercise is: \[ q_{\text{conv}} + q_{\text{solar}} - q_{\text{loss}} = \frac{dT_w}{dx} \times \frac{mc_p}{L} \]This equation allows engineers to make informed decisions about the materials and dimensions necessary for optimal heat transfer.

Reradiation Heat Loss

Reradiation heat loss occurs when the solar collector鈥檚 surface emits part of its absorbed heat energy back into the environment. This loss is dependent on the material's emissivity and the difference in temperature between the collector and the surrounding surfaces, like the sky. It's measured using the Stefan-Boltzmann law.

In our scenario, the emissivity (蔚) is 0.20, and reradiation loss can play a significant role in the overall effectiveness of the solar collector. Understanding and calculating this heat loss allows us to troubleshoot and enhance performance. The reradiation loss is given by:

• \[ q_{\text{loss}} = \frac{\varepsilon \times \sigma \times A_s \times (T_p^4 - T_s^4)}{1/\varepsilon + 1} \]
• \(\sigma\) is the Stefan-Boltzmann constant (5.67 脳 10鈦烩伕 W/m虏K鈦�).
• \(T_s\) is the sky temperature, often assumed to be a typical value like 273 K.

By minimizing reradiation heat losses, the collector's efficiency is improved. This is especially important in applications where maintaining high heat retention is critical for maximizing energy harnessing.