Sun’s Radiation Effect on Two Coolers Placed Under a Parasol

A warm sunny day at the beach can be more enjoyable with a steady supply of frosty beverages. This example considers two Styrofoam coolers containing cold beverage cans sitting on a sandy beach. A parasol provides partial shade for one of the coolers over the course of the day. The difference in beverage temperature over time is computed. This tutorial demonstrates the usage of the external radiation source boundary condition, as well as how to model structures exposed to ambient conditions.

Figure 1: A parasol provides shade on the beach. Two Styrofoam coolers contain beverage cans that should remain as cold as possible.

Model Definition

A system of a parasol and two coolers is modeled as shown in Figure 1. The coolers, made of Styrofoam, contain six beverage cans each. The beverage cans are represented by water-filled cylinders with walls modeled as thermally thick thin layers of aluminum. The choice of the thermally thick option is not motivated by the physics, they are in reality thermally thin. With the thermally thick option a slit is defined for the temperature on the walls, so the temperature can differ between the inner and outer faces. This is used to define initial conditions that are discontinuous between the exterior and interior can surfaces. Because aluminum has higher thermal conductivity than the surrounding materials, the Thin Layer – thermally thick condition behaves like a continuity condition as soon as the initial temperature difference vanishes. The initial cans temperature is 1°C. The spacing between the cans and the cooler walls is small, so the model neglects free convection inside the cooler for simplicity.

The parasol primarily provides shade but otherwise has no significant thermal effect on the beverage temperature. For this reason, it is not too important to have a high fidelity model of the parasol. It is only the shadow cast by the parasol that contributes to the beverage temperature profile. The material used for the parasol is acrylic plastic.

The primary source of heat in this model is the solar irradiation, which is included using the External Radiation Source feature. This feature uses the longitude, latitude, time zone, time of year, and time of day to compute the direction of the incident solar radiation over the simulation time. Sun irradiance and temperature values recorded at Caracas, Venezuela (Meteorological data ASHRAE 2017) are chosen for this analysis. Assuming no cloud cover, the solar flux at the surface is about 1000 W/m2. All of the ambient surfaces of the model are included in the solar loading calculation, and shadowing effects are included.

The temperature of the sun is about 5800 K, and it emits primarily short-wavelength infrared and visible light at wavelengths shorter than 2.5 microns. The fraction of this short-wavelength solar radiation that is absorbed by the various materials is quantified by the solar absorptivity. Because the surfaces are at a much lower temperature, they reradiate in the long-wavelength infrared band, at wavelengths above 2.5 microns, and the fraction of reradiated energy is quantified by the surface emissivity. The solar and ambient wavelength dependence of emissivity model is used to account for differing emissivities in different wavelength bands.

There are three ambient temperature conditions in this model. First, the ground at 1 m below the sand surface is assumed to be at a constant temperature of 27°C throughout the day, corresponding to the average water temperature at this location.

The second ambient condition is the surrounding air temperature. There exists a combination of free and forced convection, due to wind, from all exposed surfaces to the ambient air, the temperature of which is assumed to vary sinusoidally through the day. In this application, the Convective Heat Flux boundary condition uses a bulk heat transfer coefficient of 20 W/(m2·K) for all exposed surfaces.

The third boundary condition is the radiative view factor to ambient. The graybody radiative view factors are computed between all exposed faces in the model, and radiative heat transfer is computed between these faces. However, these computed view factors do not sum to unity. There is a significant view factor to surrounding regions that is not modeled; this is the residual view factor. The temperature of the ambient is the same as the ambient air temperature.

Results and Discussion

Figure 2 plots the temperature profile at 4 p.m. Notice the decrease of temperature where the parasol shade stands.

Figure 2: Temperature distribution.

Figure 3 plots the temperature of the beverage inside two of the cans. This shows clearly the advantage of placing the cooler in the shade. At 2 p.m., the parasol shade starts to leave the cooler corresponding to the green curve, which is responsible for the sudden variation in the temperature increase at that moment.

Figure 3: Beverage temperature over time inside of the two coolers at the left side of the parasol (blue curve) and at the right side (green curve).

Reference

1. F.P. Incropera, D.P. DeWitt, T.L. Bergman, and A.S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, 2006.

Heat_Transfer_Module/Thermal_Radiation/parasol

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Define an analytic function for the time-dependent ambient temperature.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Tavg	27[degC]	300.15 K	Average ambient temperature
dT	3[K]	3 K	Half diurnal temperature variation
dateDay	1	1	Day
dateMonth	1	1	Month
dateYear	2012	2012	Year

Analytic 1 (an1)

In the toolbar, click

and choose .

In the window for , type T_ambient in the text field.

Locate the section. In the text field, type Tavg[1/K]+dT[1/K]*cos(2*pi*(x-14)/24).

Locate the section. In the text field, type h.

In the text field, type K.

Locate the section. In the table, enter the following settings:


Argument	Lower limit	Upper limit
x	0	24*3600

Click

Geometry 1

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 6.

Locate the section. From the list, choose .

In the text field, type -0.5.

Click

This first block corresponds to a large region of sand. The next two blocks are the styrofoam coolers on the sand.

Block 2 (blk2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.3.

In the text field, type 0.22.

In the text field, type 0.18.

Locate the section. From the list, choose .

In the text field, type 0.5.

In the text field, type 0.09.

Click

Block 3 (blk3)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.26.

In the text field, type 0.18.

In the text field, type 0.14.

Click

In the next few steps, you create two six-pack cans by building one cylinder that is duplicated in two 3x2 arrays. Because the cans are located inside the two styrofoam coolers, you need to enable the option to see them.

Click the

button in the toolbar.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.03.

In the text field, type 0.125.

Locate the section. In the text field, type 0.42.

In the text field, type 0.04.

In the text field, type 0.02.

Click

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 3.

In the text field, type 2.

Locate the section. In the text field, type 0.08.

In the text field, type -0.08.

Click

Copy 1 (copy1)

In the toolbar, click

and choose .

Select the objects , , , , , , , and only.

For more convenience, use the button to select the abovementioned objects.

In the window for , locate the section.

In the text field, type -1.5.

Click

Now, create the parasol.

Cone 1 (cone1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.3.

From the list, choose .

In the text field, type 70.

Locate the section. In the text field, type 1.5.

Click

Cone 2 (cone2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 1.4.

Click

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the object only.

Click

Cylinder 2 (cyl2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.05.

In the text field, type 1.7.

Click

Form Union (fin)

In the toolbar, click

Click the

button in the toolbar.

Definitions

The following selection gathers the boundaries of the twelve cans.

Beverage Can Walls

In the toolbar, click

In the window for , type Beverage Can Walls in the text field.

Select Domains 4–7, 9, 10, and 14–19 only.

Locate the section. From the list, choose .

The next selection is for the irradiated surfaces.

Irradiated Surfaces

In the toolbar, click

In the window for , type Irradiated Surfaces in the text field.

Locate the section. From the list, choose .

Select Boundaries 4, 6, 7, 9, 10, 38–40, 56–62, 65–72, 74, 75, and 118 only.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

Materials

Air (mat2)

In the window, under click .

Select Domains 3 and 13 only.

Acrylic plastic (mat3)

In the window, click .

Select Domains 8 and 11 only.

Can Walls

In the window, under click .

In the window for , type Can Walls in the text field.

Locate the section. From the list, choose .

From the list, choose .

Styrofoam

In the toolbar, click

In the window for , type Styrofoam in the text field.

Select Domains 2 and 12 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.05	W/(m·K)	Basic
Density	rho	200	kg/m³	Basic
Heat capacity at constant pressure	Cp	1300	J/(kg·K)	Basic

Sand

In the toolbar, click

In the window for , type Sand in the text field.

Select Domain 1 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.3	W/(m·K)	Basic
Density	rho	1500	kg/m³	Basic
Heat capacity at constant pressure	Cp	800	J/(kg·K)	Basic