Environmental Engineering work
CE 399-LCT Homework #6 (due Tuesday 11/21 at 10am) Fall 2017
1. The summertime water temperature on the Columbia River has increased about 2 degrees Celsius
over the past century. While the construction of reservoirs is commonly blamed, another possible
cause is that deforestation has removed the shading effect that trees has on small streams,
therefore exposing more water to direct sunlight. Let’s explore this hypothesis in the next few
questions.
a. Using the graph below, estimate the total incident sunlight (daily insolation) per meter
cubed on June 21st. Approximate the curve as a parabola, and then integrate the area
under the curve. The units should be Joules/m2.
b. Assume the extreme condition that a stream is exposed to the sun. What is (i) the total
daily amount of sun energy that is reflected and (ii) the total that is absorbed? For
simplicity, assume that albedo is the noontime value.
c. For a stream that is 0.5m deep, how much will the water temperature increase in a day
due to the incoming radiation? Ignore heat loss, and ignore effects such as groundwater
discharge, snow melt, and along-channel variations in temperature.
d. Assuming that the water temperature is 15oC, what is the daily radiative energy loss, in
Joules? Assume an emissivity of 0.97, and clear conditions.
e. Now, redo part (c) but include the radiation loss. Is this answer realistic? What factors
are being ignored in the heat balance?
f. Some of the radiated energy will be reflected back to the water surface from water
vapor and other gases in the atmosphere, according to this formula:
Back Radiation = , where C = the fractional
cloud cover and Ta is the surface air temperature. Assume a cloud cover of 20% and an
air temperature of 20 degrees Celsius. What is the daily back radiation?
g. Assume also that evaporation is occurring at the stream surface, such that 100 W/m2 is
being lost. An additional 10 W/m2 is lost to conduction. Now, redo part (c ), but now
consider the heat losses and additions from problems d thru f. Is this answer more
realistic?
h. Draw a sketch of the energy balance at the water surface.
Figure 1:
Insolation at 45
degrees latitude,
at different times
of the year.
2. Assume that shading in a stream prevents 80% of the light from reaching the stream surface.
How much will the stream temperature change per day under these conditions? Note that the
vegetation canopy will radiate energy back to the water; assume that you can use the formula
from 1f to estimate this balance, with C = 0.8 (Not strictly speaking correct, but will do the job).
Assume that evaporation/conduction losses are only 20 W/m2 in this scenario. How much will
water temperature change per day? Given your answers to (1) and (2), do you think it possible
that loss of shading could at least explain part of the increase in Columbia River water
temperatures?
3. Consider the simplified global energy balance that might occur after a large comet hits the earth.
Assume that the resulting smoke and dust in the atmosphere absorb 2/3 of the incoming sunlight of
342 W/m2, and that the albedo is reduced to 25%. Convective and evaporative heating of the
atmosphere from the Earth's surface is negligible, as is the energy reflected from the Earth's surface.
The Earth's surface radiates 300 W/m2, all of which is absorbed by the atmosphere. Assuming that
the Earth can be modeled as a blackbody emitter as shown in the figure below, find the following
(equilibrium) quantities:
a. The "nuclear winter" temperature of the surface of the Earth
b. X, the rate at which radiation is emitted from the atmosphere to space
c. Y, the rate of absorption of short-wavelength solar radiation at the Earth's surface
d. Z, the rate at which the atmosphere radiates energy to the Earth's surface