To solve for the energy absorbed in each cycle, we can use the relationship between the efficiency of the engine, the work done, and the energy absorbed:

Formula:

$\text{Efficiency} (\eta) = \frac{\text{Work Output (W)}}{\text{Energy Absorbed (Q\textsubscript{in})}}$

Rearranging to find $Q\textsubscript{in}$: $Q\textsubscript{in} = \frac{\text{Work Output}}{\eta}$

The total energy absorbed in each cycle is then the sum of the work done and the heat energy expelled:

$Q\textsubscript{in} = W + Q\textsubscript{out}$

Step-by-Step Solution:

1. Convert Efficiency to Decimal Form: $\eta = 17.8\% = 0.178$

2. Calculate Work Output per Cycle: The power output is given as $P = 4.77 \, \text{kW}$, which is $4770 \, \text{W}$. The engine's work done per cycle can be expressed as: $W = P \cdot t$ Since the time per cycle is not provided, we directly use the relationship involving efficiency: $W = \eta \cdot Q\textsubscript{in}$

3. Solve for $Q\textsubscript{in}$: Rearrange: $Q\textsubscript{in} = \frac{W}{\eta}$ Substituting $Q\textsubscript{out} = 5709 \, \text{J}$: $Q\textsubscript{in} = W + Q\textsubscript{out}$

   Substitute the efficiency:

Given values

power_output_kW = 4.77 # Power in kW efficiency_percent = 17.8 # Efficiency in percentage heat_expelled_J = 5709 # Heat expelled in Joules

Convert to consistent units and values

power_output_W = power_output_kW * 1000 # Convert kW to W efficiency = efficiency_percent / 100 # Convert percentage to decimal

Calculate work output per cycle

Since efficiency = W / Q_in, rearrange Q_in = W / efficiency

Total energy absorbed Q_in = work done + heat expelled

work_done_per_cycle = power_output_W / efficiency # Total energy absorbed

Total energy absorbed per cycle in kJ

energy_absorbed_kJ = (heat_expelled_J + work_done_per_cycle) / 1000 # Convert J to kJ

energy_absorbed_kJ32.5067528089887632.50675280898876successThe energy absorbed in each cycle is approximately 32.51 kJ.

Would you like a further breakdown of these calculations or an explanation of any specific step?

Here are 5 related questions for exploration:

1. How is the efficiency of an engine typically determined experimentally?
2. What happens to the heat energy that is expelled by the engine?
3. How would the energy absorbed change if the efficiency were improved?
4. Can the relationship between power output and cycle time help deduce cycle duration?
5. What are the implications of energy loss in real-world engine applications?

Tip:

Efficiency is a crucial factor in energy systems. Understanding how it's calculated and how it relates to energy flow can help optimize energy usage in engineering systems.