Hydrosystems Engineering (EACEE 3250 / 4250)

Spring 2025

Homework #1 (Due Monday, February 24th, 11:59 pm)

Homework Guidelines:

Your solutions to homework assignments will be submitted and graded through Gradescope (see the Gradescope tab on your Courseworks dashboard).

You will have two options for submitting your work in Gradescope, either:  1) upload individual scanned images of your handwritten pages (e.g., using your phone), one or more per question; or 2) upload a single PDF that you create which contains the whole submission (e.g., merge files on your computer or phone with a software of your choice).

Please use the naming convention Lastname_HWxx.pdf when submitting your homework assignment. You may choose to type up your calculations, in which case show all your steps and highlight your solution. Note: During  the upload, Gradescope will ask you to mark which page/s each problem is on (see examplehere). It is important that you follow that step for grading purposes.

It is acceptable to discuss problems with your colleagues, and questions are encouraged during office hours, but all work must be done independently. Make sure to clearly show all work on each problem and that your solutions are  presented in an orderly fashion. It is your responsibility to make your solutions easy to grade.

Topics/Chapters covered:

•   Chapter 1: Important properties of water; Control volume and water balance; Watershed definition and delineation

•   Chapter 2: Atmospheric moisture and temperature; Thermodynamics of air

•   Chapter 3: Basics of radiation, shortwave/longwave radiation, net radiation

Problem #1 (20 pts)

The relatively unique properties of water are among the reasons why it plays such a key role in the Earth's climate. Chief among these properties are the ones that couple energy and water processes. This problem is designed to provide some experience with these properties in the context of simple examples. Provide all answers in standard SI units.

a) Suppose you have a 2 cm x 2 cm x 2 cm ice cube that is frozen at a temperature of 0°C.  How much energy (in Joules) input would be required to melt it?

b) Suppose you are making yourself 1.50 liters of tea but its temperature is 75°C, which is too hot for you to drink. In order to cool its temperature, you immerse five ice cubes of the same dimension as the one in part (a) in your tea. If the energy used to melt the ice came solely from the internal energy (i.e., temperature) of the tea, what would the reduced temperature of your tea be after all five ice cubes have melted (where you may assume tea is made up entirely of water)? For simplicity, in this class you may generally use a value of 1000 kg/m3  for the density of water.

c) The tea will also cool as a result of evaporation. Suppose the evaporation rate from the tea vessel is 5 × 10-4 kg/m2/s and occurs over the tea surface area of 80 cm2. How much more will the temperature of the tea change as a result of evaporative cooling over 10 minutes? Note that you can assume the mass change in tea due to evaporation is negligible here.

Problem #2 (20 pts)

The reservoir behind a small Earth dam in the Midwest can be reasonably represented as a cylinder with a radius of 150 [m]. The water in it stands as it had a uniform depth of 1.0 [m] in May 2014. In May 2015 the water depth stood at 0.7 [m]. During this same period a nearby precipitation gauge measured a cumulative of 0.9 [m] of precipitation. A river feeds this pond. A  river gauge measured an average of 2.0 [m3 hr-1] flow. The local farm uses the reservoir for water supply; during the year the amount used was 0.1 m3 hr-1. What was the evaporative loss from the  lake during the May 2014 to May 2015 period?

Problem #3 (20 pts)

A large room in a museum has a 3-meter-high ceiling and its lateral dimensions are 10 by 15 meters. To preserve the artwork, it is imperative that the specific humidity be kept at 10 g/kg and air temperature maintained between 20ºC and 25ºC. Due to a power outage, the climate control system responsible for maintaining conditions in the room fails.

You are called to help fix the problem and by the time you arrive your measurements indicate that the air relative humidity is 80% with an air temperature of 21ºC and air pressure of 1000 mb. You have a portable dehumidifier that takes in air from the closed room, removes water vapor by condensing the vapor into liquid, and then discharges the dry air back into the closed room. You   run the dehumidifier until the specific humidity is lowered to the required value.

a) What mass of water (in kilograms) will need to be condensed out to reach the required humidity level?

b) What is the amount of energy released (in Joules) as a result of condensing the humidity?

c) Assuming the energy computed in part b) goes into warming the air, what is the expected rise in air temperature due to the dehumidification. Note: The specific heat capacity of air is 1004 J/kg/K. For your calculation, you may assume that the change in air density (or change in air mass) is negligible as a result of the heating/dehumidification.

d) Will the air temperature still be in the necessary range as a result of the dehumidification or will it need to be additionally cooled? Explain.

Problem #4 (20 pts)

Suppose the observed water vapor density from the surface to an altitude of 8 km at an arbitrary location is described by the following exponential decay function:

where the surface vapor density (rv0) is equal to 11 g (H2O) m-3, and H is a length scale describing how quickly the variable decays with height. Suppose H = 4 km at this location.

a) What is the precipitable water (in cm of water) of the 8 km atmospheric column?

b) Increased greenhouse gases are expected to result in an increase in atmospheric temperature. The Intergovernmental Panel on Climate Change (IPCC) reports that the increase in the next 100 years may be between 0.6- 3.4 K. Conceptually, briefly discuss the effect that the increase in temperature will have on maximum water storage in the atmosphere (e.g., is air capable of holding more or less water?). Comment on how this might affect precipitation (frequency and rates).

Problem #5 (20 pts)

In order to plan the day’s irrigation rate, a farmer needs to estimate the maximum potential water loss rate by estimating the absorbed radiative energy at the surface. Over the irrigated field at noon, the ground temperature is generally 41ºC and the air temperature is 33ºC. The air dew point temperature is 16ºC. Assume a surface albedo of 30% and surface emissivity of 0.95.

Based on location and day (Day 186), the cosine of the solar zenith angle at noon is estimated to be 0.974. Assume the atmospheric direct beam shortwave transmissivity at this time is 0.6, the diffuse shortwave scattering coefficient is 0.1 and the sky is cloud-free.

a) Determine the net radiation at the surface. You can assume that εa = 0.809 (rather than use a model to estimate it).

b) Assuming 65% of the net radiation goes into evapotranspiration (i.e., latent heat flux) at the    surface, how much water should the farmer apply (in mm/day) to balance the evapotranspiration flux?