Example Beer Lambert Law Calculations
Problem Statement #1: Estimate the transmissivity of solar radiation of several wavelengths through the upper one foot layer of ocean water, using the following values for the absorption coefficient of water (from Stewart's data1):
violet light (shortest wavelength visible light): α = 0.01 cm-1 = 0.305 ft-1
red light (longest wavelength visible light): α = 1 cm-1 = 30.5 ft-1
short wavelength infrared radiation: α = 1 cm-1 = 30.5 ft-1
long wavelength infrared radiation: α = 10,000 cm-1 = 305,000 ft-1
Solution: For violet light (α = 0.01 cm-1): T = I/Io = 10-αl = 10-(0.305*1) = 0.495
For red light and the shortest wavelength infrared radiation (α = 1 cm-1): T = I/Io = 10-αl = 10-(30.5*1) = 3.2 x 10-35 (nearly zero!!)
The rest of the infrared spectrum will transmit even less light through 1 foot of water, because α is larger,
For the longest wavelength infrared radiation (α = 10,000 cm-1): T = I/Io = 10-αl = 10-(305,000*1) = (even closer to zero!)
Discussion: This shows that over most of the solar spectrum, nearly all of the solar radiation is absorbed by the upper one foot of ocean water. It also confirms that a "water wall," as used in some passive solar heating systems should work well for absorbing and storing solar energy that strikes it. This of course has been confirmed in practice.
Problem Statement #2: In the area near Honolulu, Hawaii, estimate the amount of solar energy absorbed per day by one square mile of the upper layer of the ocean.
Solution: From the article, "Estimating Solar Radiation Rate to the Surface of a Solar Panel in the U.S." The rate of solar radiation striking a horizontal surface in Honolulu, Hawaii is 3.9 KWh/day/m2. Converting units, or from the reference in the article, "Solar Insolation Data for Passive Solar Heating at Your Location in the U.S.," the same figure is 1240 Btu/day/ft2.
Converting units: 1 mi2 = 2.79 X107 ft2 = 2.59 x 106 m2
Neglecting the effect of reflection, virtually all of the incident solar radiation will be absorbed by the upper layer of ocean water as calculated above, thus:
Energy absorbed = (3.9)(2.59 x 106) kWhr/day = 1.01 x 107 kWhr/day
Also: Energy absorbed = (1240)(2.79 x 107) Btu/day = 3.46 x 1010 Btu/day