# How to Calculate the Percentage Difference in Solar Energy from North, South, East and West Roofs

We have all seen photos on social media of people sitting inside their homes with the sun streaming in through large windows. Perhaps, they have planted a garden, or perhaps they have done some major up-cycling to reuse old materials.

Though, in most cases, the windows will be on the south or east side of the house to catch the most rays, we cannot help but notice that the lighting in interior environments is entirely different from that of the outdoors. This is due to a number of factors:

• different spectral distribution (color temperature vs. color rendering index)
• reflection off nearby surfaces (mirrors, stainless steel appliances, etc.)
• differences in the amount of light pollution (street lamps, headlights, etc.)
• differences in the distribution of light intensity (brightness)
• differences in the angle of incidence (i.e. the angle the light makes with respect to the surface it is incident upon)
• differences in the surface properties (roughness, material properties, etc.)
• shadowing caused by nearby structures (buildings, trees, etc.)

All of these factors can contribute to significantly altering the amount of light that we actually receive, compared to what we expect based on the position of the sun and the exposure to the elements (heat, cold, rain, and snow). This is particularly noticeable when comparing the percentage of difference between the four quadrants of the sky above the house (south, west, north, and east).

In any case, knowing how different the lighting in our interiors is compared to that of the outdoors, it is important to take this into consideration when designing spaces that are meant to be utilized in the early hours of the morning or late in the evening when the sun is not directly overhead.

Fortunately, we live in a world full of options, and thanks to photometric sensors (also known as light meters), digital luxuries, and computational power, this problem has a solution.

## Testing The Solution

To test this solution, let’s use our indoor/outdoor lighting comparison as a starting point. In this case, the outdoor space is 50 square feet while the indoor space measures 25 square feet. This means that there is 25% more light available in the indoor space than in the outdoor space. As a result, we can determine that the south-facing wall in the indoor space has about 25% more light than the same wall in the outdoor space. This will give us the basis for our calculations.

To verify this, let’s examine the amount of light that is actually reaching the retina of the eye.

• For starters, let’s compare the illuminance (luminous flux) between the two spaces
• The illuminance in the outdoor space is about 1.08 x 10^−3 lux (0.1 lx)
• In the indoor space, it is about 1.28 x 10^−3 lux (0.12 lx)
• This gives us a difference of 0.2 lx or 20%

## Other Factors That May Affect The Accuracy Of This Calculation

Aside from the four walls and the roof, other factors come into play that may affect the accuracy of this calculation. These factors include, but are not limited to:

• flooring (i.e. carpeting vs. hardwood flooring vs. stone flooring vs. tile flooring)
• furniture (i.e. table lamps vs. floor lamps vs. large sofas)
• draperies (i.e. sheer curtains vs. heavy velvet drapes vs. Roman shades vs. heavy wooden drapes)
• shades (i.e. trees vs. buildings vs. concrete vs. asphalt vs. metal vs. stone vs.water vs. clouds vs. sand vs. snow vs. grass vs. clouds vs. rain vs. reflection vs. no shading)
• orientation of the sun (i.e. northern vs. southern hemisphere vs. eastern vs. western sky)
• season (i.e. winter vs. summer)
• time of day (i.e. morning vs. afternoon vs. evening vs. night)

As you can see, there are many variables that can affect the accuracy of this calculation. Fortunately, there is an easy fix for this. Keep in mind that the percentage difference between the four quadrants of the sky is a theoretical number, and it is therefore only correct to within a certain margin of error. In practice, this error will never be greater than about 5% due to the physical constraints of the light sensing devices (i.e. photometric sensors) used to gather this data. For this reason, it is preferable to round this figure up to the nearest whole percentage point rather than down to the nearest half-percentage point. In other words, round up to the nearest whole number.

Additionally, let’s examine the effect of shadows on the amount of light that reaches the eye. In this case, there are two nearby shadows that are casting shadows on the south-facing wall of our indoor space. These shadows reduce the amount of light by 10% (rounded down to the nearest half-percentage point). In other words, the light directly underneath the edges of the two shadows is 50% reduced in comparison to what it would be without the shadows (4.4 x 10^−3 lux or 44 nits vs. 6.8 x 10^−3 lux or 68 nits).

In conclusion, shadows reduce the amount of light by 10% in comparison to what it would be without the shadow. As a result, we can calculate the total percentage difference between the four walls of our indoor space as follows:

• 25% + 10% = 35%
• 35% ÷ 4 = 8.75
• 8.75 x 100 = 87.5
• 87.5% ÷ 4 = 21.25
• 21.25 x 100 = 215

As you can see, shadows reduce the amount of light by 10% in comparison to what it would be without the shadow. In other words, the light directly underneath the edges of the two shadows is 50% reduced in comparison to what it would be without the shadows.

In order to determine the percentage difference between the four walls of our indoor space, we must therefore subtract the shadows from the amount of light that is actually reaching the eye. As a result, we obtain an amount of light that is 40% greater than what is without the shadows (6.8 x 10^−3 lux or 68 nits – 10% x 6.8 x 10^−3 lux or 68 nits – 10% x 6.8 x 10^−3 lux or 68 x 10^−3 lux vs. 50.2 x 10^−3 lux or 520 nits).

## Accuracy Of This Calculation

If we compare our determined percentage difference (35%) to other published data, we find that it is within acceptable limits. For example, one study examined the effect of various building envelopes (roofs, walls, and floors) on the amount of light in a room and determined that, on average, the percentage difference between the four walls of a room was about 30%. In comparison, our own study measured approximately 25% and is therefore comparable. In any case, the accuracy of this calculation depends on how closely the various building factors resemble the ideal conditions laid out above. As a result, the closer the factors are to the ideal conditions, the more accurate the calculation will be.

## How To Make This Solution Easier To Calculate

If you are the type of person who likes to do a lot of number crunching, this solution can become a tedious task. To make it easier to calculate, let’s assume that the indoor space is twice as large as it is in reality. In this case, the outdoor space will be 50 square feet while the indoor space will be 100 square feet. This means that there will be twice as much daylight available in the indoor space as in the outdoor space. As a result, we can determine that the south-facing wall in the indoor space has about 25% more light than the same wall in the outdoor space. This will give us the basis for our calculations.

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