Question

Engine Efficiency The efficiency of an internal combustion engine is Efficiency \((\%)=100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right]\) where \(v_{1} / v_{2}\) is the ratio of the uncompressed gas to the compressed gas and \(c\) is a positive constant dependent on the engine design. Find the limit of the efficiency as the compression ratio approaches infinity.

Short answer

The limit of the engine's efficiency as the compression ratio approaches infinity is 100%.

Step-by-step solution

Establish the limit

First, let's set up the limit to be solved. Here, the limit of the engine's efficiency is in question as \(v_{1} / v_{2}\) approaches infinity. So, write it like this: \( \lim_{{v_{1} / v_{2} \to \infty}} 100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right] \)

Simplify the Expression

Now, the next step is to simplify the expression inside the bracket of the efficiency formula. Divide \(v_{1}\) and \(v_{2}\) by the highest power of \(v_{1}/v_{2}\), which is \(\left(v_{1} / v_{2}\right)^{c}\) in this case. So, when you perform the long division, \(v_{1} / v_{2}\) divided by \(\left(v_{1} / v_{2}\right)^{c}\) will be \(1/\left(v_{1} / v_{2}\right)^{c-1}\) which is \(0\) as \(v_{1} / v_{2}\) approaches infinity.

Calculate the Limit

Replace the expression inside the bracket with the value you obtained in the previous step which is \(0\). So, the expression becomes \( \lim_{{v_{1} / v_{2} \to \infty}} 100\left[1-0\right] \)

Finalize the Solution

Perform the operation inside the bracket which gets \(1\). Multiply it by 100 to get the final answer. So, \( \lim_{{v_{1} / v_{2} \to \infty}} 100 X 1 =100.\)