Problem 41 - Precalculus: Functions and Graphs

Question

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula $$ R=R_{0} \cos \theta \sin \phi $$ where \(R_{0}\) is the maximum solar radiation possible, \(\theta\) is the angle that the sun makes with the horizontal, and \(\phi\) is the direction of the sun in the sky, with \(\phi=90^{\circ}\) when the sun is in the east and \(\phi=0^{\circ}\) when the sun is in the south. (a) When does the maximum solar radiation \(R_{0}\) strike the wall? (b)What percentage of \(R_{0}\) is striking the wall when \(\theta\) is equal to \(60^{\circ}\) and the sun is in the southeast?

Step-by-Step Solution

Answer

(a) At sunrise, \( \theta = 0\), \( \phi = 90^{\circ} \). (b) 35.35\% of \( R_0 \).

Step 1: Analyze when maximum solar radiation strikes

To determine when the maximum possible solar radiation \( R_0 \) strikes the wall, the expression \( \cos \theta \sin \phi \) must equal 1. This is because the cosine and sine functions have maximum values of 1. Therefore, \( \cos \theta = 1 \) and \( \sin \phi = 1 \). This occurs when \( \theta = 0^{\circ} \) (sun is at the horizon) and \( \phi = 90^{\circ} \) (sun is in the east).

Step 2: Calculate the solar radiation when sun is in southeast

When the sun is in the southeast, \( \phi = 45^{\circ} \). The cosine of the angle \( \theta = 60^{\circ} \) is \( \cos 60^{\circ} = 0.5 \). The sine of \( \phi = 45^{\circ} \) is \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Substitute these values into the formula:\[R = R_{0} \cdot (0.5) \cdot \left(\frac{\sqrt{2}}{2}\right)\]Calculating this gives:\[R = R_{0} \cdot \frac{\sqrt{2}}{4}\]

Step 3: Determine the percentage of maximum solar radiation

To find the percentage of \( R_{0} \) striking the wall when \( \theta = 60^{\circ} \) and \( \phi = 45^{\circ} \), calculate:\[\text{Percentage} = \left(\frac{R}{R_0}\right) \times 100 = \left(\frac{\sqrt{2}}{4}\right) \times 100 \]Approximating \( \sqrt{2} \approx 1.414 \), we get:\[\text{Percentage} \approx \frac{1.414}{4} \times 100 \approx 35.35\%\]

Key Concepts

Energy Efficiency

Understanding energy efficiency is essential when considering solar radiation's effects on a building. Essentially, energy efficiency measures how well a building utilizes energy for heating, cooling, and lighting. The more efficiently a building manages these resources, the less energy it needs to maintain comfortable living conditions.

• Solar radiation can significantly impact a building’s efficiency by affecting its temperature.
• For example, during sunny days, excessive solar energy might overheat a building unless it is well-insulated.
• Conversely, using controlled solar radiation can reduce energy consumption for heating in colder climates.

Evaluating the amount of sunlight striking a building's surfaces helps optimize design features like window placement and shading structures. This optimization can lead to reducing energy usage and improving a building’s overall energy efficiency.

Solar Angles

Solar angles describe the position of the sun relative to a specific point on Earth and are crucial in understanding how solar radiation interacts with buildings. Several solar angles are commonly used:

• Solar Altitude (\( \theta \)):
• This angle measures the sun's height above the horizon.
• An angle of \( 0^{\circ} \) indicates sunrise or sunset, while \( 90^{\circ} \) signifies the sun is directly overhead.
• Solar Azimuth (\( \phi \)):
• It measures the sun's position along the horizon.
• When the sun is due east, \( \phi = 90^{\circ} \); south is \( \phi = 0^{\circ} \).

These angles are influenced by the time of day, geographic location, and season. They help determine how much sunlight hits a particular surface, which is critical in solar radiation calculations.

Trigonometric Functions

Trigonometric functions are mathematical tools used to relate angles to various ratios. In solar radiation, they help calculate how much sunlight reaches a building's surface. Two crucial functions include:

• Cosine (\( \cos \)): Measures the x-component (horizontal) of an angle.
• Sine (\( \sin \)): Measures the y-component (vertical) of an angle.

For the solar radiation formula \( R = R_0 \cos \theta \sin \phi \), the cosine function adjusts for the sun's altitude, while the sine function considers its azimuth.

• These functions help in determining how angles like \( \theta \) (solar altitude) and \( \phi \) (solar azimuth) modulate the amount of radiation received.

Understanding these trigonometric relationships is essential for calculating the exact solar input to optimize for energy efficiency.

Maximum Solar Radiation

Maximum solar radiation occurs under specific conditions when the sun's rays are optimally aligned with the surface. For a vertical wall facing east, this maximum is realized by achieving maximum values for both sine and cosine functions in the formula. When the sun is at the horizon (\( \theta = 0^{\circ} \)) and directly in the east (\( \phi = 90^{\circ} \)), both functions equal 1. Thus, \( R = R_0 \).

• This is the ideal condition where the sun's energy is fully harnessed, leading to maximum efficiency in energy absorption.

However, practical factors like time of the day and year must be considered since they affect solar angles.

• Considering these changes helps maximize the utility of available solar radiation throughout different times, aiding in better planning for energy use and efficiency.