Math / Geometry

QuestionQ2: It is proposed to illuminate a class room of dimensions $6 \times 8 \times 2.85 \mathrm{~m}$ to an illuminance (E) of 400 lx at the bench level. The specification calls for luminaires having one 1050 mm 40 W fluorescent natural tube with an initial output of 3200 lumens with white metal base an prismatic plastic diffuser (its UF is given in Table -2). Determine the number of luminaires required for this installation when the MF is 0.7 , respectively. The reflection coefficients are: ( $\mathrm{C}=$ $0.70, \mathrm{~W}=0.3, \mathrm{~F}=0.2$ ) Assuming the working table height is 0.85 m \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{6}{|l|}{Room Reflection} & \multicolumn{6}{|l|}{Room Index} \\ \hline c & W & F & 75 & 1.00 & 1.25 & 1.50 & 2.00 & 2.50 & 3.00 & 4.00 & 5.00 \\ \hline \multicolumn{3}{|l|}{\multirow[t]{3}{}{.70 .50 .30 .10}} & 44 & . 50 & .56 & . 60 & 65 & .69 & .72 & .7i5 & .777 \\ \hline & & & . 38 & . 44 & .50 & .54 & . 60 & . 64 & 68 & . 72 & .74 \\ \hline & & & . 33 & 40 & 46 & .50 & . 56 & 61 & . 64 & . 69 & . 72 \\ \hline \multirow[t]{3}{}{. 50} & . 50 & 20 & 39 & 44 & 49 & .5. & . 57 & 60 & 62 & .65 & . 67 \\ \hline & .30 & & 34 & 40 & 44 & 48 & 53 & . 56 & . 59 & 62 & . 64 \\ \hline & . 10 & & .30 & .36 & 41 & 44 & . 50 & .53 & . 56 & . 60 & . 62 \\ \hline \multirow[t]{3}{}{} & . 50 & 20 & 34 & 39 & 42 & 45 & 49 & 51 & . 53 & .55 & . 57 \\ \hline & .30 & & 30 & . 35 & . 39 & 42 & 46 & 49 & .51 & . 53 & . 55 \\ \hline & 10 & & 27 & .32 & . 36 & .39 & 44 & 47 & 49 & 52 & 54 \\ \hline \multirow[t]{2}{}{} & 0.00 & . 0 & 22 & 26 & 29 & . 31 & 34 & . 36 & 38 & 40 & . 41 \\ \hline & & & & & & - & $\rightarrow$ UF & 4 & & & \\ \hline \end{tabular}
Note: The reflection coefficients for a standard room are: $(0.7,0.5,0.2)$

Studdy Solution

STEP 1

1. The classroom dimensions are $6 \, \text{m} \times 8 \, \text{m} \times 2.85 \, \text{m}$ .
2. The desired illuminance at the bench level is $400 \, \text{lx}$ .
3. Each luminaire has a 1050 mm 40 W fluorescent tube with an initial output of 3200 lumens.
4. The maintenance factor (MF) is $0.7$ .
5. The reflection coefficients are: ceiling (C) = 0.70, walls (W) = 0.3, floor (F) = 0.2.
6. The working plane height is 0.85 m.

STEP 2

1. Calculate the room index.
2. Determine the utilization factor (UF) from the table.
3. Calculate the total lumens required.
4. Calculate the number of luminaires needed.

STEP 3

Calculate the room index (RI) using the formula:
$\text{RI} = \frac{\text{Length} \times \text{Width}}{\text{Mounting Height} \times (\text{Length} + \text{Width})}$
The mounting height is the room height minus the working plane height:
$\text{Mounting Height} = 2.85 \, \text{m} - 0.85 \, \text{m} = 2.00 \, \text{m}$
Substitute the values:
$\text{RI} = \frac{6 \times 8}{2 \times (6 + 8)} = \frac{48}{28} \approx 1.71$

STEP 4

Determine the utilization factor (UF) from the table for the given reflection coefficients and room index. Based on the closest room index value of 1.71, we use the value for RI = 1.50, which is approximately $0.50$ .

STEP 5

Calculate the total lumens required using the formula:
$\text{Total Lumens} = \text{Illuminance} \times \text{Area}$
$= 400 \, \text{lx} \times (6 \times 8) \, \text{m}^2 = 400 \times 48 = 19200 \, \text{lumens}$

STEP 6

Calculate the number of luminaires needed using the formula:
$\text{Number of Luminaires} = \frac{\text{Total Lumens}}{\text{Lumens per Luminaire} \times \text{UF} \times \text{MF}}$
$= \frac{19200}{3200 \times 0.50 \times 0.7} = \frac{19200}{1120} \approx 17.14$
Since the number of luminaires must be a whole number, round up to the nearest whole number:
$\text{Number of Luminaires} = 18$
The number of luminaires required is:
$\boxed{18}$

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