energies Article Validated Analytical Modeling of Diesel Engines Intake Manifold with a Flexible Crankshaft Salah A. M. Elmoselhy 1 , Waleed F. Faris 2, * and Hesham A. Rakha 3   Citation: Elmoselhy, S.A.M.; Faris, W.F.; Rakha, H.A. Validated Analytical Modeling of Diesel Engines Intake Manifold with a Flexible Crankshaft. Energies 2021, 14, 1287. https://doi.org/10.3390/ en14051287 Academic Editors: Constantine Rakopoulos and Ali Elkamel Received: 17 November 2020 Accepted: 25 January 2021 Published: 26 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 CFisUC, Department of Physics, University of Coimbra, P-3004 516 Coimbra, Portugal; salah.elmoselhy@msm.nl 2 Department of Mechanical Engineering, International Islamic University Malaysia, Gombak, Kuala Lumpur 53100, Malaysia 3 Virginia Tech Transportation Institute, Virginia Polytechnic Institute and State University, 3500 Transportation Research Plaza, Blacksburg, VA 24061, USA; hrakha@vt.edu * Correspondence: wfaris@vt.edu Abstract: The flexibility of a crankshaft exhibits significant nonlinearities in the analysis of diesel engines performance, particularly at rotational speeds of around 2000 rpm. Given the explainable mathematical trends of the analytical model and the lack of available analytical modeling of the diesel engines intake manifold with a flexible crankshaft, the present study develops and validates such a model. In the present paper, the mass flow rate of air that goes from intake manifold into all the cylinders of the engine with a flexible crankshaft has been analytically modeled. The analytical models of the mass flow rate of air and gas speed dynamics have been validated using case studies and the ORNL and EPA Freeway standard drive cycles showing a relative error of 7.5% and 11%, respectively. Such values of relative error are on average less than those of widely recognized models in this field, such as the GT-Power and the CMEM, respectively. A simplified version for control applications of the developed models has been developed based on a sensitivity analysis. It has been found that the flexibility of a crankshaft decreases the mass flow rate of air that goes into cylinders, resulting in an unfavorable higher rate of exhaust emissions like CO. It has also been found that the pressure of the gas inside the cylinder during the intake stroke has four elements: a driving element (intake manifold pressure) and draining elements (vacuum pressure and flow losses and inertial effect of rotating mass). The element of the least effect amongst these four elements is the vacuum pressure that results from the piston’s inertia and acceleration. The element of the largest effect is the pressure drop that takes place in the cylinder because of the air/gas flow losses. These developed models are explainable and widely valid so that they can help in better analyzing the performance of diesel engines. Keywords: nonlinear dynamics; fluid mechanics; diesel engine performance; exhaust emissions; modeling and simulation Article Highlights 1. Validated Analytical model of the mass flow rate of air goes into cylinders with a flexible crankshaft; 2. Flexibility of the crankshaft decreases the mass flow rate of air that goes into cylinders, resulting in unfavorable higher rate of exhaust emission; 3. Cylinder pressure during intake stroke has four elements: a driving element (intake manifold pressure) and draining elements (vacuum pressure, flow losses and inertial effect of rotating mass). 1. Introduction The continual demand for diesel powertrain development is due to the distinguished operating features the diesel powertrains exhibit [1]. As a cost-effective tool for developing Energies 2021, 14, 1287. https://doi.org/10.3390/en14051287 https://www.mdpi.com/journal/energies