Question

In a 4-stroke single cylinder diesel engine, the inlet valve opens at \(10^\circ\) before TDC and closes at \(40^\circ\) after BDC. The exhaust valve opens at \(25^\circ\) before BDC and closes at \(15^\circ\) after TDC. The percentage of time for which both valves remain closed in one engine cycle is

• A. 32.29
• B. 40.97
• C. 46.53
• D. 75.01

Hint:

Sketch the \(0^\circ\!\to\!720^\circ\) timeline with TDC/BDC at \(0,180,360,540,720^\circ\). Mark open/close angles, take the union of "open" intervals, and subtract from \(720^\circ\) to get the closed duration.

Answer 1

The Correct Option is B

Step 1: Fix angle reference and note the cycle length.
Take \(\theta=0^\circ\) at TDC between exhaust and intake (the valve-overlap TDC). One full 4-stroke cycle spans \(720^\circ\) of crank rotation and ends at the next same TDC (\(720^\circ\)).

Step 2: Map the inlet valve open interval.
Inlet opens \(10^\circ\) BTDC \(\Rightarrow\) at \(-10^\circ\) (i.e., \(710^\circ\) in the current cycle) and closes \(40^\circ\) ABDC of the intake stroke. BDC of intake is at \(180^\circ\); hence close at \(180^\circ+40^\circ=220^\circ\). Therefore, inlet is open on \([710^\circ,720^\circ]\cup[0^\circ,220^\circ]\).

Step 3: Map the exhaust valve open interval.
Exhaust opens \(25^\circ\) BBDC of the power stroke. Power stroke is \(360^\circ \to 540^\circ\); hence open at \(540^\circ-25^\circ=515^\circ\). Exhaust closes \(15^\circ\) ATDC of the next TDC, i.e., at \(720^\circ+15^\circ\). Within our cycle this appears as it being still open on \([0^\circ,15^\circ]\) after \(720^\circ\). Therefore, exhaust is open on \([515^\circ,720^\circ]\cup[0^\circ,15^\circ]\).

Step 4: Find when either valve is open and complement.
Union of open intervals: \[ [0,220]\ \cup\ [515,720]. \] Hence both valves are closed only on \[ [220,515], \] whose length is \(515-220=295^\circ\).

Step 5: Convert to percentage over the \(720^\circ\) cycle.
\[ \text{Percentage closed}=\frac{295}{720}\times 100 =40.972%\ \approx\ 40.97%. \]

Final Answer:
\[ \boxed{40.97%} \]