Matthew J. Traum Mem. ASME Engineer Inc, 4832 NW 76th Rd, Gainesville, FL 32653 e-mail: mtraum@alum.mit.edu Fatemeh Hadi 1 Mem. ASME Mechanical and Manufacturing Engineering Department, Tennessee State University, 3500 John A. Merritt Blvd, Nashville, TN 37209-1561 e-mail: fhadi@tnstate.edu Muhammad K. Akbar Mem. ASME Mechanical and Manufacturing Engineering Department, Tennessee State University, 3500 John A. Merritt Blvd, Nashville, TN 37209-1561 e-mail: makbar@tnstate.edu Extending “Assessment of Tesla Turbine Performance” Model for Sensitivity-Focused Experimental Design The analytical model of Carey is extended and clarified for modeling Tesla turbine per- formance. The extended model retains differentiability, making it useful for rapid evalua- tion of engineering design decisions. Several clarifications are provided including a quantitative limitation on the model’s Reynolds number range; a derivation for output shaft torque and power that shows a match to the axial Euler Turbine Equation; eliminat- ing the possibility of tangential disk velocity exceeding inlet working fluid velocity; and introducing a geometric nozzle height parameter. While nozzle geometry is limited to a slot providing identical flow velocity to each channel, variable nozzle height enables this velocity to be controlled by the turbine designer as the flow need not be choked. To illus- trate the utility of this improvement, a numerical study of turbine performance with respect to variable nozzle height is provided. Since the extended model is differentiable, power sensitivity to design parameters can be quickly evaluated—a feature important when the main design goal is maximizing measurement sensitivity. The derivatives indi- cate two important results. First, the derivative of power with respect to Reynolds number for a turbine in the practical design range remains nearly constant over the whole lami- nar operating range. So, for a given working fluid mass flow rate, Tesla turbine power output is equally sensitive to variation in working fluid physical properties. Second, tur- bine power sensitivity increases as wetted disk area decreases; there is a design trade-off here between maximizing power output and maximizing power sensitivity. [DOI: 10.1115/1.4037967] Introduction This paper extends the work of Carey and colleagues [1–3] to illustrate how Tesla turbine performance equations outlined therein inform practical design of a novel experimental turbine. A unique Tesla turbine was designed to experimentally measure the impact of a single design variable on power output. In turbine design, the goal is typically to maximize power generation, and it is therefore advantageous to select turbine parameters that maxi- mize shaft power output. However, the purpose of this current work is to study how small parameter changes impact shaft power. Thus, the present study seeks to maximize sensitivity of shaft power to a selected design variable; mathematically, the magni- tude of the partial derivate of shaft power with respect to the modulated variable. Thus, conventional Tesla turbine design approaches, which fix variables to maximize power output, have limited utility for the present experimental design. Instead, a dif- ferentiable closed-form analytical model relating geometric and working fluid parameters to Tesla turbine performance is needed. The Tesla turbine was patented in 1913 [4]. Early attempts exist in the hobby literature to develop closed-form analytical solutions relating geometric and working fluid parameters to Tesla turbine performance [5,6]. The extensive Tesla turbine work of Rice and colleagues [7–11] and experiments of others [12–18] provide a rich foundation as do recent review articles [19,20]. A resurgent interest in Tesla turbines for their ability to process two-phase and particulate-laden working fluid characteristic of renewable energy systems (for example, working fluid of low thermodynamic quality from solar [21] or geothermal [22] sources as well as bio- mass combustion products [23]) has also motivated a number of theoretical studies and computational fluid dynamic modeling [24–29]. Other Tesla-type turbine research also exists. Barbarelli et al. [30], for example, posits a tangential flow turbine with a rotating channel containing five deflectors with possibility to achieve higher performance at low rotational velocity compared to conventional turbines. For other low-quality flows, such as low- head flows, use of a propeller pump instead of a conventional tur- bine is suggested [31]. However, the general focus of this whole body of Tesla turbine and related work is measuring and maximiz- ing turbine power output. While other closed-form analytical Tesla turbine models are available [32], we feel the work by Carey [1] represents the most easily modified differentiable closed-form analytical model relat- ing geometric and working fluid parameters to Tesla turbine per- formance. It is therefore extended, utilized, and evaluated in this work, and the resulting new findings are reported. Background Carey’s derivation begins with the Navier–Stokes equations in cylindrical coordinates applied to the rotating disk of a Tesla tur- bine [1,2]. It results in the following form for b W n; Re  m ð Þ, the dimensionless tangential velocity difference between the disk rotor and fluid inside the turbine at any radial location b W n ðÞ¼ e 24n 2 Re  m n Re  m 24 e 24n 2 Re  m þ b W o  Re  m 24 e 24 Re  m   (1) where n ¼ r/r o is a dimensionless radial disk location; b W o is the dimensionless relative disk/fluid velocity at the disk outer radius (n ¼ 1); and Re  m is a modified Reynolds number 1 Corresponding author. Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received September 11, 2017; final manuscript received September 18, 2017; published online October 17, 2017. Editor: Hameed Metghalchi. Journal of Energy Resources Technology MARCH 2018, Vol. 140 / 032005-1 Copyright V C 2018 by ASME