SOLUTION: Automotive trades: Total piston displacement is given by the following formula: P.D. = piD^2LN (pi is over 4) where: P.D. = piston displacement in cubic in. D = diame

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Question 115561This question is from textbook
: Automotive trades:
Total piston displacement is given by the following formula:
P.D. = piD^2LN (pi is over 4)

where:
P.D. = piston displacement in cubic in.
D = diameter of bore of cylinder
L= length of stroke, in inches
N = Number of cylinders
A. P.D. = 400 cu in. L = 4.5 in. N = 8
B. P.D. = 392.7 cu. in. L = 4 in. N = 6 This question is from textbook

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
Given the formula for total piston displacement as:
.

.
Let's analyze this formula just a little to see what it is telling us.
.
The part of the formula that is is just the old geometry formula for finding
the area of a circle when you are given the diameter of the circle. In this case you are given
the diameter of the bore, and you are calculating the cross-sectional area of the cylinder.
.
Next you multiply that area by the length of the stroke. What you are finding by doing this
is the volume displacement for a single cylinder. What you have done this far is just the
same as the old geometry problem of finding the volume of right cylinder by multiplying
the area of its circular base times the height of the cylinder.
.
Finally you multiply this by volume for a single cylinder in the car by the number of cylinders
in the engine to find the total volume displacement or Piston Displacement.
.
You give two problems in which you specify the total Piston Displacement, the length of the stroke,
and the number of cylinders. I guess that you want to find the diameter of the cylinder bore.
.
For the first problem the total PD is 400 cubic inches ...(about 6.6 liter - a gas guzzler).
The length of the stroke is 4.5 inches, and there are 8 cylinders. Substituting all these
values into the equation you get:
.

.
The L and the N multiply out to give 4.5 * 8 = 36. Substitute this into the equation:
.

.
Divide both sides of this equation by 36 to get rid of the factor 36 on the right side.
On the left side 400 divided by 36 becomes 11.11111111. And on the right side the 36 is
divided by 36 which equals 1. So the reduced equation is:
.

.
Get rid of the denominator of 4 on the right side by multiplying both sides of the equation
by 4 to get:
.

.
Divide both sides of the equation by which approximately equals 3.141592654.
The left side of the equation becomes:
.
and this divides out to give
.
On the right side the becomes just a factor of 1. So the equation you are left
with is:
.

.
Solve for D by taking the square root of both sides to get:
.

.
So the answer to this problem is that the bore has a diameter of 3.76126389 inches. You
can round off this answer to the accuracy you need.
.
The second problem is the same ... only the numbers have changed ... This time the equation with
the appropriate numbers inserted is:
.

.
And the stroke times the number of cylinders is 24 ... making the equation:
.

.
Divide both sides by 24:
.

.
Multiply both sides by 4:
.

.
Divide by
.

.
Solve for D by taking the square root of both sides:
.

.
So this time the diameter of the cylinder bore is 4.564359983 inches ... (probably not
a very economical 6 cylinder engine).
.
Hope this is what you were looking for and it helps you to understand the problem.