Question

A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has a surface area of \(2 \mathrm{~m}^{2}\) with emissivity and absorptivity of \(0.9\). The surface temperature of the absorber is \(35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at \(500 \mathrm{~W} / \mathrm{m}^{2}\) with a surrounding temperature of \(0^{\circ} \mathrm{C}\). Convection heat transfer coefficient at the absorber surface is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the ambient temperature is \(25^{\circ} \mathrm{C}\). Net heat rate absorbed by the solar collector heats the water from an inlet temperature \(\left(T_{\text {in }}\right)\) to an outlet temperature \(\left(T_{\text {out }}\right)\). If the water flow rate is \(5 \mathrm{~g} / \mathrm{s}\) with a specific heat of \(4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), determine the temperature rise of the water.

Short answer

Answer: The temperature rise of the water in the flat-plate solar collector is approximately 0.022 K.

Step-by-step solution

Calculate the Incident Solar Radiation on the Surface of the Absorber

As given, solar radiation is incident on the absorber at \(500 \mathrm{~W} / \mathrm{m}^{2}\) and the surface area of the absorber plate is \(2 \mathrm{~m}^{2}\). So the total solar radiation energy falling on the surface of the absorber can be calculated as: Solar Radiation Energy = (Radiation Intensity) x (Surface Area) Solar Radiation Energy = \( 500\, \frac{\text{W}}{\text{m}^2} \times 2\, \text{m}^2 =1000\, \text{W}\)

Calculate the Absorbed Solar Radiation

The absorber plate has an absorptivity of \(0.9\). Use this to calculate the absorbed solar radiation energy: Absorbed Energy = (Absorptivity) x (Solar Radiation Energy) Absorbed_Energy = \(0.9 \times 1000\, \text{W} = 900\, \text{W}\)

Calculate the Heat Loss due to Radiation and Convection

To determine the heat loss, we will use the given emissivity, absorber surface temperature, surrounding temperature, and convection heat transfer coefficient. Calculate the heat loss due to radiation energy using the Stefan-Boltzmann Law (\(E = \epsilon \sigma A (T^4 - T_\text{amb}^4)\)): Heat Loss (Radiation) = Emissivity × (Stefan-Boltzmann Constant) × Surface Area × (Absorber Temperature^4 - Surrounding Temperature^4) The Stefan-Boltzmann Constant is approximately \(5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^4\). Convert the temperatures to Kelvin: \(T_\text{absorber} = 35 + 273.15 = 308.15\,\text{K}\) \(T_\text{surrounding} = 0 + 273.15 = 273.15\,\text{K}\) Heat Loss (Radiation) = \(0.9 \times 5.67 \times 10^{-8}\, \frac{\text{W}}{\text{m}^{2} \cdot \mathrm{K}^4} \times 2\, \text{m}^{2} \times (308.15^{4} - 273.15^{4}) = 331.24\, \text{W}\) Now, calculate the heat loss due to convection using: Heat Loss (Convection) = (Convection Heat Transfer Coefficient) × Surface Area × (Absorber Temperature - Ambient Temperature) Convert the temperatures to Celsius: Heat Loss (Convection) = \(5\, \frac{\text{W}}{\text{m}^2\cdot \mathrm{K}} \times 2\, \text{m}^{2} \times (35 - 25) = 100\, \text{W}\)

Calculate the Net Heat Rate Absorbed by the Solar Collector

Combine the absorbed solar radiation energy and the heat loss due to radiation and convection to find the net heat rate absorbed by the solar collector: Net Heat Rate = Absorbed Energy - Heat Loss (Radiation) - Heat Loss (Convection) Net Heat Rate = \(900\, \text{W} - 331.24\, \text{W} - 100\, \text{W} = 468.76\,\text{W}\)

Calculate the Temperature Rise of the Water

Now, use the given water flow rate, specific heat, and net heat rate absorbed to determine the temperature rise of the water: Water Flow Rate = \(5\, \frac{\text{g}}{\text{s}} = 0.005\,\frac{\text{kg}}{\text{s}}\) Specific Heat of Water = \(4.2\, \frac{\text{kJ}}{\text{kg} \cdot \mathrm{K}} = 4200\, \frac{\text{J}}{\text{kg} \cdot \mathrm{K}}\) We will use the formula: Heat Rate = (Mass Flow Rate) × Specific Heat × Temperature Rise Temperature Rise = \(\frac{\text{Heat Rate}}{(\text{Mass Flow Rate}) × (\text{Specific Heat})}\) Temperature Rise = \(\frac{468.76\, \text{W}}{0.005\, \frac{\text{kg}}{\text{s}} \times 4200\, \frac{\text{J}}{\text{kg} \cdot \mathrm{K}}} = 0.022\,\text{K}\) The temperature rise of the water in the flat-plate solar collector is approximately \(0.022\,\text{K}\).

Key concepts

Flat-Plate Solar Collectors

Flat-plate solar collectors are devices designed to capture solar energy and convert it into heat, usually for heating water. Imagine a large flat panel, often black, to absorb maximum sunlight. This panel is called the absorber plate. It is thin and made of materials with high thermal conductivity, allowing heat to be transferred efficiently to the working fluid, like water.

These collectors are widely used because they are simple, durable, and cost-effective. They're ideal for moderate temperature applications such as domestic water heating. The surface of the absorber plate is exposed to sunlight, where it captures solar radiation and transforms it into thermal energy. The absorbed energy is then transferred to the water flowing through tubes attached to the plate's backside. This direct transfer is crucial in maintaining effective performance. Using materials with high absorptivity and emissivity, like those with values close to 0.9, ensures that the maximum amount of solar energy is captured and used effectively.

The design of flat-plate collectors, with their wide surface area, makes them suitable for capturing diffuse solar radiation found in various climatic conditions.

Heat Loss Calculation

To understand how heat is lost in a flat-plate solar collector, we need to consider both convection and radiation losses. Heat loss calculations help in evaluating the efficiency of the solar system.

Start by calculating the heat lost through radiation using Stefan-Boltzmann's Law, which describes how energy is radiated by surfaces. Factors include the material's emissivity, surface area, and temperature difference between the absorber and the surroundings. For instance, with high emissivity and a considerable temperature difference, more heat is lost as radiation.

Next, consider convection heat loss. This occurs when heat transfers between the solid surface of the absorber and the air surrounding it. The convection heat transfer coefficient and the temperature difference between the absorber's surface and the ambient air affect this loss.

By combining these two losses, we determine the total heat loss from the collector. Estimating these accurately ensures the solar collector's design minimizes heat escape, thus maximizing the available energy for warming the water.

Temperature Rise Calculations

The primary goal of a flat-plate solar collector is to warm the water flowing through it, and the temperature rise calculation is crucial in assessing its performance. This is achieved by determining how much the water is heated as it passes through the system.

The temperature rise depends on the net heat rate absorbed by the collector, which includes the total absorbed solar energy less any heat losses. The calculation also requires knowledge of the water flow rate and its specific heat capacity. Essentially, the idea is to quantify how much energy is imparted to the water, raising its temperature from inlet to outlet.

The equation used is based on the principle that the rate of heat gain (absorbed energy) equals the mass flow rate multiplied by the water's specific heat capacity and the temperature increase. Accurate calculations allow us to optimize the system, ensuring that the desired temperature rise is achieved efficiently. This ensures practical applications such as home heating systems meet the expected performance needs.

Convection and Radiation in Solar Systems

In the context of solar systems, understanding convection and radiation is essential for optimizing heat transfer efficiency. Both processes play critical roles in maintaining the balance of heat exchange in solar collectors.

Convection refers to the transfer of heat between the solar panel's surface and the air or fluid flowing over the surface. This process depends on factors like the surface's temperature, ambient conditions, and movement of the fluid. Increasing the convection heat transfer coefficient, such as by using fans to move air, can enhance the system’s performance by reducing the absorber's temperature.

Radiation, however, is about transferring energy in the form of electromagnetic waves. The rate at which radiation heat transfer occurs is highly contingent on the surface’s emissivity and temperature. The difference between the absorber's surface temperature and the surrounding temperature dictates the energy radiated away.

By carefully designing and selecting materials, manufacturers can manage these two heat transfer processes effectively. This balance ensures the solar collector operates efficiently, capturing a significant portion of solar energy for heating purposes.