Question

A new sustainabily centre building is being planned and is noing to have a photovoltaic surface applied to the entire root area. The photovolaic roof surface is capable of generating electricity from sunlight. The proposed shape of the roof is shown in Figute 1 (a) (b) Figure 1 Proposed shape of the new root: (top) 30 representation showing the roof area in blue: (bottom) plan view showing the principle dimensions as variables There are two figures show the shape of the proposed root. The first illustration is a 30 representation showing the roof area in blue in elevation, with one side of the building in white. The second illustration contains the dimensions of the building. The main area is made up of a rectangle of length b and width c. The entrance hall is an irregular parallelogram consisting of a square of sides a and a triangle of base a and height of d minus a. There are still changes that can be made to the layout of the bulding and these will affect the dimensions of the roof geometry. One considerabion in making this decision is how much power the new roof can potentialy. generate. To assess this, you have been asked to produce a mathernatical model of the maximum power output as a function of the roof size. You will create this modol as you go through the question. a Show that a general formula for the area of the roof (As) can be written as (a) (b) Figure 1 Proposed shape of the new root: (top) 30 representation showing the roof area in blue, (bottom) plan view showing the principle dimensions as variables: There are two figures show the shape of the proposed root The first illustration is a 3D representation showing the roof area in blue in elevation; with one side of the building in white. The second illustration contains the dimensions of the building. The main area is made up of a rectangle of length b and widh c. The entrance hall is an iregular parallelogram consisting of a square of sides a and a triangle of base a and height of d minus a. There are still changes that can be made to the tayout of the building and these will affect the dimensions of the roof geometry. One consideration in making this decision is how much power the new roof can potentially generate. To assess this, you have been asked to produce a mathematical modef of the maximum power output as a function of the roof size You will create this model as you go through the question. a. Show that a general formula for the area of the roof (A-a) can be written as Aᵦₑₓ=1/2a² +bc1/2 ad

using the variables a,b,c and d shown in the figure above Explain each step in obtaining this formula, stating how you have split the roof into different shapes. (Hint for informabon an calculating areas see the addibonal maths respurces Topic 8 localed on the Resources tab of the T192 module website).

Answer 1

To calculate the total area of solar collectors needed to replace a 750 MW power plant with a 2.00% efficiency rate, divide the total power output required (750,000 kW) by the power per square meter reaching the surface (1.30 kW/m²) and divide again by the efficiency rate.

Step-by-step explanation:

Photovoltaic Efficiency and Energy Requirements

When calculating the power per square meter from the sun at Earth's atmosphere, you use the total output of the sun and divide it by the surface area over which this power is spread. For example, the power output of the Sun is given as 4.00 × 1026 W. Since the Earth is located at a certain distance from the Sun, the power per square meter is obtained by dividing this power by the surface area of a sphere with a radius equal to the average distance from the Earth to the Sun (approximately 1 AU).

However, the question we're concerned with involves the installed photovoltaic (PV) systems on Earth's surface. These systems convert sunlight into electricity with a certain efficiency rate. For example, we assume a hypothetical situation where solar collectors replace an electric power plant that generates 750 MW of power. If those collectors have an average conversion efficiency of 2.00% of the maximum power that reaches Earth's surface, assumed to be 1.30 kW/m², we need to calculate the required area of solar collectors.

Let's take the total power needed from the solar collectors to be 750 MW, which is 750,000 kW. We divide this by 1.30 kW/m² to find the total area needed without considering efficiency, then further divide by the efficiency rate (2.00%) to find the actual required area. This results in a substantial area needed for solar panels to meet this power generation requirement. To find the area in km², we convert m² to km² by dividing by 106 (since 1 km² equals 106 m²).