Question

There is currently substantial interest in utilizing eukaryotic algae for the renewable production of several bioenergy carriers, including starches for alcohols, lipids for diesel fuel surrogates, and hydrogen for fuel cells. Algae can convert solar energy into fuels at high photosynthetic efficiencies and can thrive in saltwater systems. Part of an energy balance analysis of such a process is the computation of the penetration of the radiative heat flux in a pool of water. a. Considering the surface of the sun as a black surface at a temperature of 5777 \(\mathrm{K}\), determine the total (i.e., integrated over all wavelengths) solar heat flux incident on the top of the Earth's atmosphere. The radius of the sun is \(6.96 \times 10^{8} \mathrm{~m}\), and a representative value of the sun-to-Earth distance is \(1.496 \times 10^{11} \mathrm{~m}\). Compare your result to the generally accepted annual mean value of \(1366 \mathrm{~W} \cdot \mathrm{m}^{-2}\) and discuss possible reasons for difference. b. Assuming that at a certain location and time, when the sun is in the zenith, the total heat flux arriving on the Earth surface is \(1100 \mathrm{~W} \cdot \mathrm{m}^{-2}\), calculate the balance between incoming and outgoing radiative heat flux at the surface of a pond of stagnant water when the sun is at an angle of \(30^{\circ}\) from the zenith. The total hemispherical emissivity of water is known to vary from \(0.95\) to \(0.963\) in the temperature range from 273 to \(373 \mathrm{~K}\) (Kaviany, 2002 ). c. The absorption coefficient \(\mathbf{\kappa}_{\lambda}\) of clear water is known to depend on wavelength. In the visible range \((0.4-0.7 \mu \mathrm{m})\), it varies from \(0.02\) to \(0.6 \mathrm{~m}^{-1}\) (Modest 2003, p. 416 ). By solving the radiative transfer equation, determine the decrease of intensity of incoming sunlight in a water layer with a depth \(0.1 \mathrm{~m}\) for the case that the sun is at an angle of \(30^{\circ}\) from the zenith.

Short answer

Answer: The solar heat flux incident on the Earth's atmosphere is approximately 1352.56 W m^-2.

Step-by-step solution

Calculate the sun's irradiance using Stefan-Boltzmann Law

The Stefan-Boltzmann Law gives the total radiative heat flux (Q) for a black surface, which can be written as: Q = σT^4 where σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W m^-2 K^-4), and T is the temperature in Kelvin. For the sun, the temperature is T = 5777 K. Plugging this value into the formula, we obtain: Q_sun = (5.67 x 10^-8) * (5777)^4 = 6.32 x 10^7 W m^-2

Calculate the solar heat flux incident on Earth

Using the inverse square law, we can find the solar heat flux incident on Earth's atmosphere: Q_earth = Q_sun * (R_sun / D_se)^2 where R_sun is the radius of the sun (6.96 x 10^8 m), and D_se is the representative distance from the sun to the Earth (1.496 x 10^11 m). Q_earth = (6.32 x 10^7) * [(6.96 x 10^8) / (1.496 x 10^11)]^2 = 1352.56 W m^-2 The result is close to the annual mean value of 1366 W m^-2. The slight difference may be due to factors such as the sun's varying distance from Earth and the atmosphere's absorption and scattering of sunlight. #b. Calculate the balance between incoming and outgoing radiative heat flux at the surface of a pond#

Calculate the incident heat flux on the pond

When the sun is at an angle of 30° from the zenith, the incident heat flux (Q_in) can be calculated as: Q_in = Q_sfc * cos(θ) where Q_sfc is the total heat flux arriving on the Earth surface (1100 W m^-2), and θ is the angle from the zenith (30°). Q_in = 1100 * cos(30°) = 951.93 W m^-2

Calculate the outgoing radiative heat flux from the pond

The outgoing radiative heat flux (Q_out) from the pond can be calculated using Stefan-Boltzmann Law: Q_out = ε_water * σ * T_pond^4 where ε_water is the total hemispherical emissivity of water (0.95 to 0.963) and T_pond is the pond's temperature. Since we are not given the pond's temperature, we will leave the equation as it is.

Calculate the balance between incoming and outgoing heat flux

The balance between incoming and outgoing heat flux can be calculated as: Balance = Q_in - Q_out Balance = 951.93 - (ε_water * σ * T_pond^4) Since we do not have the pond's temperature and water emissivity, we cannot compute the exact balance. #c. Determine the decrease of intensity of incoming sunlight in a water layer#

Compute the path length of sunlight in water

When the sun is at an angle of 30° from the zenith, the path length of sunlight in water (L) can be calculated as: L = d / cos(θ) where d is the depth of water (0.1 m), and θ is the angle from the zenith (30°). L = 0.1 / cos(30°) = 0.115 m

Calculate the decrease of intensity using Lambert-Beer Law

Lambert-Beer Law states that the decrease in intensity (I_decrease) is proportional to the absorption coefficient (K), path length (L), and the initial intensity (I_0): I_decrease = I_0 * (1 - exp(-K * L)) Since we do not have the initial intensity (I_0), we can leave the equation as it is. I_decrease = I_0 * (1 - exp(-K * 0.115)) As the absorption coefficient (K) varies from 0.02 to 0.6 m^-1 in the visible range (0.4-0.7 μm), the decrease in intensity will depend on the wavelength. In this case, we do not have the wavelength, so we cannot compute the exact decrease in intensity.

Key concepts

Photosynthetic Efficiency in Algae

Bioenergy from algae is an exciting prospect because algae have the ability to convert solar energy into biofuels with high photosynthetic efficiency. Photosynthetic efficiency refers to the percentage of light energy that is converted into chemical energy by photosynthesis in algae. This process is vital for producing bioenergy carriers such as starches, lipids, and hydrogen.

Photosynthetic organisms, like algae, use chlorophyll to capture light energy, usually within the visible spectrum of 400-700 nanometers (nm), which is known as Photosynthetically Active Radiation (PAR). The efficiency of this energy conversion process depends on multiple factors, for example, the wavelength of light, the intensity of the incident light, and the temperature and carbon dioxide concentration of the water where algae grow.

Algae are particularly interesting because of their relatively high photosynthetic efficiency compared to terrestrial plants. While most plants have a photosynthetic efficiency of about 1-2%, some algae can reach efficiencies up to 8%. This is due to their simple cell structure, which allows for more light absorption, and potentially their ability to grow in saline water, which reduces competition for freshwater resources.

Energy Conversion in Algae

Algae convert light energy into chemical energy through photosynthesis slightly differently than terrestrial plants, tailoring their methods to their aquatic environments. This adaptation may involve variations in pigment composition, allowing algae to absorb varying wavelengths of light. The biochemical pathways in some algae also help them fix carbon dioxide more efficiently, further contributing to their higher photosynthetic efficiency.

Solar Heat Flux Computation

Solar heat flux computation is essential in understanding and quantifying the amount of solar energy that reaches the Earth's surface. It involves calculating the energy per unit area received from the sun. For algae bioenergy production, the intensity of solar radiation is a crucial factor since it affects the photosynthetic rate and hence the potential yield of biomass.

The solar heat flux at Earth's surface is not constant and can vary due to geographical location, time of day, season, and atmospheric conditions. The exercise provided evaluates solar heat flux by considering both the Stefan-Boltzmann Law and the inverse square law.

Understanding the Inverse Square Law

In the context of solar heat flux computation, the inverse square law helps to explain why the intensity of solar radiation diminishes with an increase in distance from the sun. It states that the intensity of a radiation field (like sunlight) from a point source (assumed to be the sun) decreases inversely with the square of the distance from the source. This relation is crucial for calculating the correct solar heat flux incident on Earth, which ultimately impacts the cultivation of algae for bioenergy.

Radiative Transfer Equation

The radiative transfer equation (RTE) is a fundamental concept used to describe the propagation of radiation through a medium. In the context of bioenergy from algae, it helps determine the penetration of sunlight through layers of water, which affects the growth and photosynthetic performance of algae.

Clear water absorbs light differently at various wavelengths, and the RTE allows for calculating how much light is absorbed as it travels through water. Solving the RTE requires knowledge of the absorption coefficient for the specific medium, which indicates how much light is absorbed per unit path length at each wavelength.

Application to Algal Ponds

For algae growing in ponds, using the RTE, we can model how sunlight diminishes as it passes through water before reaching the algae. This is essential for algal pond design as it influences the selection of pond depth and algal concentration to optimize light availability for maximal photosynthesis. Furthermore, understanding how light interacts with water can guide the timing and intensity of artificial lighting, potentially augmenting natural sunlight to boost algae growth.