To determine which formula accurately represents the compression ratio of an engine, let's analyze each given formula step-by-step.
### Formula 1
[tex]\[ r = \frac{V_d + V_c}{V_c} \][/tex]
This formula suggests that the compression ratio [tex]\( r \)[/tex] is the ratio of the total volume (displaced volume [tex]\( V_d \)[/tex] plus clearance volume [tex]\( V_c \)[/tex]) to the clearance volume [tex]\( V_c \)[/tex].
By substituting some example values, say [tex]\( V_d = 10 \)[/tex] and [tex]\( V_c = 1 \)[/tex], we can calculate:
[tex]\[ r = \frac{10 + 1}{1} = \frac{11}{1} = 11 \][/tex]
This reflects the correct understanding of a compression ratio, where the total volume at the bottom of the stroke is compared to the clearance volume at the top of the stroke.
### Formula 2
[tex]\[ r = \frac{V_c - V_d}{V_d} \][/tex]
This formula suggests that the compression ratio [tex]\( r \)[/tex] is the ratio of the difference between the clearance volume [tex]\( V_c \)[/tex] and the displaced volume [tex]\( V_d \)[/tex] to the displaced volume [tex]\( V_d \)[/tex].
Using the same values, [tex]\( V_d = 10 \)[/tex] and [tex]\( V_c = 1 \)[/tex]:
[tex]\[ r = \frac{1 - 10}{10} = \frac{-9}{10} = -0.9 \][/tex]
This value is negative and does not make physical sense in the context of compression ratios, which should be positive values.
### Formula 3
[tex]\[ r = V_d + V_c \][/tex]
This formula suggests that the compression ratio [tex]\( r \)[/tex] is the sum of the displaced volume [tex]\( V_d \)[/tex] and the clearance volume [tex]\( V_c \)[/tex].
Using the example values [tex]\( V_d = 10 \)[/tex] and [tex]\( V_c = 1 \)[/tex]:
[tex]\[ r = 10 + 1 = 11 \][/tex]
While this formula gives a result, it misses the correct concept of comparing total volume to the clearance volume. It simply adds the two volumes without forming a ratio.
### Conclusion
The formula that accurately represents the compression ratio of an engine is:
[tex]\[ r = \frac{V_d + V_c}{V_c} \][/tex]
This formula correctly describes the compression ratio in terms of the volume relationship between the displaced volume and the clearance volume.
