(صورة توضيحية فقط)
The thermal efficiency ([latex]\eta_{th}[/latex]) of an ideal Otto cycle is a function of the compression ratio ([latex]r[/latex]) and the specific heat ratio ([latex]\gamma[/latex]) of the working fluid. The formula is [latex]\eta_{th} = 1 – \frac{1}{r^{\gamma-1}}[/latex]. This equation shows that efficiency increases with the compression ratio, providing a fundamental principle for engine design and performance optimization.
The derivation of the Otto cycle thermal efficiency formula begins with the general definition of thermal efficiency for any heat engine: [latex]\eta_{th} = \frac{W_{net}}{Q_{in}} = 1 – \frac{Q_{out}}{Q_{in}}[/latex], where [latex]W_{net}[/latex] is the net work output, [latex]Q_{in}[/latex] is the heat added, and [latex]Q_{out}[/latex] is the heat rejected. For the Otto cycle, heat is added at constant volume (process 2-3) and rejected at constant volume (process 4-1). Therefore, [latex]Q_{in} = m c_v (T_3 – T_2)[/latex] and [latex]Q_{out} = m c_v (T_4 – T_1)[/latex], where [latex]m[/latex] is the mass of the working fluid, [latex]c_v[/latex] is the specific heat at constant volume, and [latex]T[/latex] represents the temperature at the numbered states of the cycle.
Substituting these into the efficiency equation gives [latex]\eta_{th} = 1 – \frac{T_4 – T_1}{T_3 – T_2}[/latex]. To simplify this in terms of volumes, we use the relationships for the isentropic processes (1-2 and 3-4). For an isentropic process, [latex]TV^{\gamma-1} = \text{constant}[/latex]. Thus, [latex]\frac{T_2}{T_1} = (\frac{V_1}{V_2})^{\gamma-1} = r^{\gamma-1}[/latex] and [latex]\frac{T_3}{T_4} = (\frac{V_4}{V_3})^{\gamma-1} = r^{\gamma-1}[/latex]. This implies [latex]\frac{T_2}{T_1} = \frac{T_3}{T_4}[/latex] or [latex]\frac{T_4}{T_1} = \frac{T_3}{T_2}[/latex]. Rearranging the efficiency equation to [latex]\eta_{th} = 1 – \frac{T_1(T_4/T_1 – 1)}{T_2(T_3/T_2 – 1)}[/latex] and substituting the temperature ratio equality, the terms in the parentheses cancel out. This leaves [latex]\eta_{th} = 1 – \frac{T_1}{T_2}[/latex]. Finally, using the isentropic relation [latex]\frac{T_1}{T_2} = (\frac{V_2}{V_1})^{\gamma-1} = \frac{1}{r^{\gamma-1}}[/latex], we arrive at the final formula: [latex]\eta_{th} = 1 – \frac{1}{r^{\gamma-1}}[/latex].
الكفاءة الحرارية لدورة أوتو
(إذا كان التاريخ غير معروف أو غير ذي صلة، على سبيل المثال "ميكانيكا الموائع"، يتم توفير تقدير تقريبي لظهوره الملحوظ)
الصور والتنزيلات بالحجم الكامل متاحة فقط، ومجانية تمامًا، للأعضاء المسجلين.
> تسجيل الدخول <
