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优化活塞行动改进的发动机性能,优化,活塞,行动,行为,行径,改进,改良,发动机,性能,机能
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Proc. Natl. Acad. Sci. USAVol. 78, No. 4, pp. 1986-1988, April 1981Applied Physical SciencesFinite-time thermodynamics:Engine performance improved by optimized piston motion(Otto cycle/optimized heat engines/optimal control)MICHAEL MOZURKEWICH AND R. S. BERRY Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Illinois 60637 Contributed by R. Stephen Berry, December 29, 1980ABSTRACT The methods of finite-time thermodynamics are used to find the optimal time path of an Otto cycle with friction and heat leakage. Optimality is defined by maximization of the work per cycle; the system is constrained to operate at a fixed frequency,so the maximum power-is obtained. The result is an improvement of about 10% in the effectiveness (second-law efficiency) of a conventional near-sinusoidal engine.Finite-time thermodynamics is an extension ofconventional thermodynamics relevant in principle across the entire span of the subject, from the most abstract level to the most applied. The approach is based on the construction of generalized thermodynamic potentials (1) for processes containing time or rate conditions among the constraints on the system (2) and on the determination of optimal paths that yield the extrema corresponding to those generalized potentials.Heretofore, work on finite-time thermodynamics has concentrated on ratheridealized models (2-7) and on existence theorems (2), all on the abstract side of the subject. This work is intended as a step connecting the abstract thermodynamic concepts that have emerged in finite-time thermodynamics with the practical, engineering side of the subject, the design principles of a real machine. In this report, we treat a model of the internal combustion engine closely related to the ideal Otto cycle but with rate constraints in the form ofthe two major losses found in real engines. We optimize the engine by controlling the time dependence of the volume-that is, the piston motion. As a result, without undertaking a detailed engineering study, we are able to understand how the losses are affected by the time path of the piston and to estimate the improvement in efficiency obtainable by optimizing the piston motion.THE MODELOur model is based on the standard four-stroke Otto cycle. This consists of an intake stroke, a compression stroke, a power stroke, and an exhaust stroke. Here we briefly describe the basic features of this model and the method used to find the optimal piston motion. A detailed presentation will be given elsewhere.We assume that the compression ratio, fuel-to-air ratio, fuel consumption, and period of the cycle all are fixed. These constraints serve two purposes. First, they reduce the optimization problem to finding the piston motion. Also,they guarantee that the performance criteria not considered in this analysis are comparable to those for a real engine. Relaxing any of these constraints can only improve the performance further.We take the losses to be heat leakage and friction. Both of these are rate dependent and thus affect the time response of the system. The heat leak is assumed to be proportional to the instantaneous surface of the cylinder and to the temperature difference between the working fluid and the walls (i.e., Newtonian heat loss). Because this temperature difference is large only on the power stroke, heat loss is included only on this stroke. The friction force is taken to be proportional to the piston velocity, corresponding to well-lubricated metal-on-metal sliding;thus, the frictional losses are directly related, to the square ofthe velocity. These losses are not the same for all strokes. The high pressures in the power stroke make its friction coefficient higher than in the other strokes. The intake stroke has a contribution due to viscous flow through the valve.The function we have optimized is the maximum work per cycle. Because both fuel consumption and cycle time are fixed, this also is equivalent to maximizing both efficiency and the average power.In finding the optimal piston motion, we first separated the power and nonpower strokes. An unspecified but fixed time t was allotted to the power stroke with the remainder of the cycle time given to the nonpower strokes. Both portions of the cycle were optimized with this time constraint and were then combined to find the total work per cycle. The duration t of the power stroke was then varied and the process was repeated until the net work was a maximum.The optimal piston motion for the nonpower strokes takes a simple form. Because of the quadratic velocity dependence of the friction losses, the optimum motion holds the velocity constant during most of each stroke. At the ends of the stroke, the piston accelerates and decelerates at the maximum allowed rate. Because the friction losses are higher on the intake stroke, the optimal solution allots more time to this stroke than to the other two. The piston velocity as a function of time is shown in Fig.1.The power stroke was more difficult to optimize because ofthe presence of the heat leak. The problem was solved by using the variational technique of optimal control theory (8). The formalism yields the equation of motion of the piston as a fourthorder set of nonlinear differential equations. These were solved numerically. The resulting motion is shown in Fig. 1 for the entire cycle.The asymmetric shape of the piston motion on the power stroke arises from the trade-off between friction and heat leak losses. At the beginning of the stroke the gases are hot, capable of yielding high efficiency, and the rate of heat loss is high. It is therefore advantageous to make the velocity high on this part of the stroke. As work is extracted, the gases cool and the rate of heat leakage diminishes relative to frictional losses. Consequently the optimal path moves to lower velocities as the power stroke proceeds.The solutions were obtained first with unlimited acceleration and then with limits on acceleration and deceleration. The latter situation yields a result familiar in other contexts under the name of turnpike solution (9). The system tries to operate as long as possible at its optimal forward and backward velocities, by accelerating and decelerating between these velocities at the maximum rates. In this way, the system spends as much time as possible moving along its best or turnpike path.RESULTSParameters for the computations were taken from ref. 10 or, in the case of the friction coefficient, adjusted to give frictional losses of the magnitude cited in ref. 10. Those parameters are given in Table 1. The results of the calculations of some typical cases are given in Table 2, where they are compared with the conventional Otto cycle engine having the same compression ratio but a standard near-sinusoidal motion. The effectiveness (the ratio of the work done to the reversible work, also called the second-law efficiency) is slightly higher for the optimized engine whose piston-acceleration is limited to 5 x 103 m/sec2 ，the maximum of the conventional engine of the first row. If the piston is allowed to have 4 times the acceleration of the conventional engine, the effectiveness increases 9%; if the acceleration is unconstrained, the improvement in effectiveness goes up to 11%.These values are typical, not the most favorable. If the total losses of the conventional engine are held approximately constant but shifted to correspond to about 80% larger heat loss and about 60% smaller friction loss, the gain in effectiveness goes up, reaching more than 17% above the effectiveness of the corresponding conventional engine.The principal source of the improvement in use of energy in this analysis is in the reduction of heat losses when the working fluid is near its maximum temperature. This is why the improvement is greater for engines with large heat leaks and low friction than for engines with relatively better insulation but higher friction.Finally, it is instructive to examine the path of the piston in time, for the optimized engine and for its conventional counterpart. The position of the piston as a function of time is shown for these two cases in Fig. 2.In closing, let us emphasize the unconventional approach to optimizing a thermodynamic system illustrated by this work. Instead of controlling heat rates, heat capacities, conductances, friction coefficients, reservoir temperatures, or other usual parameters of thermodynamic engines, we have controlled the time path of the engine volume.We thank Dr. Morton Rubin for helpful comments and suggestions. This work was supported in part by a grant from the Exxon Education Foundation.REFERENCE1. Hermann, R. (1973) Geonetry, Physics and Systems (Dekker,New York). Proc. Nati. Acad. Sci. USA 78 (1981)2. Salamon, P., Andresen, B. & Berry, R. S. (1977) Phys. Rev. A 14,2094-2102.3. Curzon, F. L. & Ahlborn, B. (1975) Am. J. Phys. 43, 22-24.4. Andresen, B., Berry, R. S., Nitzan, A. & Salamon, P. (1977)Phys. Rev. A 15, 2086-2093.5. Rubin, M. (1979) Phys. Rev. A 19, 1272-1276, 1277-1289.6. Salamon, P., Nitzan, A., Andresen, B. & Berry, R. S. (1980)Phys. Rev. A 21, 2115-2129.7. Gutkowicz-Krusin, D., Procaccia, I. & Ross, J. (1978) J. Chem. Phys. 69, 3898-3906.8. Hadley, C.F.G. & Kemp, M. C. (1971) Variational Methods in Economics (North-Holland, Amsterdam).9. Sen, A., ed. (1970) Growth Economics (Penguin, Baltimore,MD).10. Taylor, C. F. (1966) The Internal Combustion Engine in Theory and Practice (MIT Press, Cambridge, MA), Vol. 1, pp. 158-164; Vol. 2, pp. 19-20.Proc。 全国。 Acad。 Sci。 美国卷. 78，第 4 页。 1986-1988， 1981 年 4 月应用的物理学有限时间热力学：优化活塞行动改进的发动机性能(奥托循环或优化热引擎或最优控制)MICHAEL MOZURKEWICH 和 R. S. BERRY 化学系和詹姆斯法朗克研究所，芝加哥大学，芝加哥，伊利诺伊州 60637 由 R.斯蒂芬莓果贡献， 1980 年 12 月 29 日摘要： 利用有限时间热力学方法发现奥托循环的优先时间路径及摩擦和热渗漏。 最优性由工作的最大化定义每个周期; 系统被控制在一个固定的内，因此便能获得最大动力。 结果是每一个常规近正弦的发动机改善了大约 10%的效率(第二定律效率)。有限时间热力学是引伸常规热力学相关原则上横跨主题的整个间距，从最抽象的水平到广泛的应用。 方法是根据广义热力学潜力的创立(1)为包含时间或对在限制之中的条件估计在系统之内(2)和在产生对应于那些广义潜力的极值的最佳路径的计算。迄今为止，有限时间热力学的工作集中于较为理想化的模型(2-7)和存在性定理(2)，且全部集中在抽象方面。这项工作是希望作为一个步骤连接在实用的有限时间热力学方面涌现了的抽象热力学概念，工程学方面的课题，一台实用机器的设计的原则。在这个报告中，我们用接近理想的奥多周期来研究内燃机模型，但由于频率限制使得在实际的发动机中是以二主要损失的形式存在。 我们通过“控制”时间改善活塞运动来优化发动机的性能。 结果，没有进行一项详细的工程学研究，我们能够通过受活塞的时间路径的影响和优化活塞行动获得效率的改善的估计来了解是怎么损失的。模 型我们的模型是基于标准的四冲程奥托循环。这包括进气冲程、压缩冲程、作功冲程和排气冲程。 我们在这里简要地描述这个模型和发现优化活塞行动的使用方法及基本特点。 在别处将给一个详细的介绍。我们假设，压缩比、空燃比、燃油消耗率和时间全部是固定的。这些制约因素有两个目的。首先，他们利用减少优化问题来找到活塞运动。 并且，他们保证在这分析没考虑的性能准则与那些是为一个实用的发动机做比较的。 放松这些限制中的任一个可能进一步改善性能。我们采取的损失是热渗漏和摩擦。 这两个是依靠效率来影响系统的时间反应。 热泄漏假设是圆筒的瞬间表面和与在工作流体和墙壁之间的温差比例(即，牛顿热耗)。由于这个温度区别最大是在作功冲程，热渗漏是只包含在这个冲程中。摩擦力与活塞速度成正比，对应于润滑良好的金属表面；因此，摩擦损失也直接与速度正方形有关。 这些损失在所有冲程中是不同样的。高压在作功冲程使它的摩擦系数高于在其他冲程。 进气冲程得益于。我们优选的作用是确定每循环的最大功率。 由于燃料消费和周期是固定的，这也与最大化效率和平均功率是等效的。在寻找优选的活塞行程时，我们首先分离了有能量和无能量的冲程。 非特指，但确定的时间 t 是指作功冲程中无能量冲程剩下的时间。 循环的两个部分优选以一个限制时间和然后结合找到每循环的总工作量。 时间 t 的作功冲程后来变化了，并且这个过程会被重覆，直到净工作量达到最大值。采取一个简单形式来描述无能量冲程的最佳活塞运动。在每个冲程的大多数时间，由于摩擦损失与速度的二次方成比例，最宜的运动依赖于速度常数。 在冲程的末期，活塞以允许的最大效率加速并且减速。 由于摩擦损失在进气冲程较高，与其他两个相比，这个最佳的解决办法是把更多的时间分配到这个冲程。活塞速度与作用时间的关系显示在图 1 中。由于热泄漏的出现，作功冲程更难优选。问题是通过使用最优控制理论的变化技术解决的 (8)。利用实际情况的非线性的微分方程产生活塞的运动方程式。 这些都是实际数值。整个循环运动的结果显示在图 1 上。 图 1 活塞速度与作用时间的关系，从作功冲程开始。最大允许的加速度是 2 x 104 m/sec2。活塞行动的不对称的形状在作功冲程中的摩擦和热泄漏损失之间交替出现。在冲程初气体是热的，能产生高效率，并且散热率高。在作功冲程中得益于活塞速度高。这个冲程被选出，气体冷却率和热泄漏相对于摩擦损失减少。 结果，当作功冲程进行时，最佳路径的移动速度更低。解决的办法在加速度和上首先获得了极大的加速度然后迅速减速。后者情况以“收费公路”解决方案在其他环境下产生一个交叉结果 (9)。在这些速度之间以最高效率进行加速和减速，使系统尽量的在它的最佳的向前和向后速度操作下尽可能延长。 这样，系统花费同样多时间尽可能沿它的最佳路径移动。结 果计算的参量从参考 10 中获取，在给定的摩擦系数下，通过参考 10 中的变量调整摩擦损失的大小。 那些参量在表 1 中给出。一些典型的情况下的计算结果见表 2，但在一个标准近正弦运动下，他们与常规奥托循环的发动机相比有同一压缩比。为了优化发动机使第一列的常规发动机最大值，活塞加速度被限制在 5 x 10 m3/sec2内，使得有效利用率 (有用功与可逆功的比率，也称第二定律效率)稍微提高。 如果发动机的活塞允许有 4 个时间的加速度，有效率将增加 9%；如果加速度是不受强制的，有效率比以前将增加 11%。表 1 发动机参数*发动机参数：压缩比=8在最小容积的活塞位置=1 厘米位移= 7 cm汽缸直径(b) = 7.98 cm汽缸容量(v) = 400 cm 3周期(t) = 33.3 毫秒/3600 转每分钟热力学参量：压缩冲程 作功冲程最初的温度 333K 2795K摩尔气体 0.0144 0.0157恒定热容量 容量 2.5R 3.35R汽缸壁温度(T) = 600 K可逆循环的动能 (WR)= 435.7 J 可逆的能力(WR/I)= 13.1 千瓦损失条件：摩擦系数(a) = 12.9 kg/sec热泄漏系数(K)= 1305 千克/ (度/sec3)每循环的时间损耗和摩擦损失的能量= 50 J *参量数据根据参考书目 10.表 2 结果(所有能量单位用焦耳)在作功冲程上所用的时间；W P在作功冲程完成的工作量；W T，每循环的净工作量；WF，摩擦损失的能量；W Q，工作中的热泄漏损失的能量； Q，热泄漏；T F，作功冲程结束时的温度；，有效利用率。这些改
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