Preliminary Design of a Mini Gas Turbine via 1D Methodology

Abstract

To address the increasing interest towards more environmentally friendly naval transportation and the introduction of IMO2020 restrictions on pollutant emissions onboard ships, the present work details the preliminary design of a mini gas turbine engine, i.e., a gas turbine engine with an output power up to 5 MW, for onboard energy generation. In comparison to conventional propulsion systems, gas turbine units benefit from known compactness, which can be further enhanced by employing single-stage uncooled radial machines, according to similar works in the field. As such, the present paper aims to set up a complete procedure that allows a reliable and fast (i.e., requiring a limited computational effort) preliminary design of one-stage centrifugal compressors and radial turbines operating at a high pressure ratio via the use of classical one-dimensional theory. The aerodynamic design outputs in terms of forces and torques are then used to perform a preliminary mechanical design of the shaft by means of a one-dimensional finite element model with commercial software to estimate the corresponding shaft line stress. Despite some necessary geometrical and modeling simplification of the design problem, which results in the unavailability of detailed information on individual components, the employed procedure nevertheless allows a comprehensive overview of the possibilities in terms of maximum machine performance achievable at an early design stage with the associated limited computational requirements. The design procedure and the geometry achieved for the application are presented along with aerodynamic and structural results.

1. Introduction

Gas turbines (GT) are currently a renowned energy generation system, spanning the fields of inland energy generation [1,2], aircraft propulsion [3,4,5], and, according to the most recent examples in the literature, they seem to be valuable potential alternatives to conventional naval propulsion engines [6,7,8]. A general trend related to such engines towards the introduction of small-sized units can be seen. Inland-distributed generation in the context of the growing amount of renewable energy systems on the grid [1] and key applications such as clean water production via desalinization [9] would benefit from the use of compact combined heat and power (CHP) GT units, from both economic and performance perspectives. A similar need for compact machinery has been observed in the fields of aircraft propulsion [10] and naval onboard energy generation [11,12], where such machines are seen as valid options for compact layouts. Indeed, radial turbomachines benefit from their small overall dimensions, while maintaining interesting performance at low size [4,5,13]. The very first gas turbine engines designed for aviation were based mainly on radial turbomachines [3]. Similarly, gas turbines onboard naval ships mostly benefit from the higher power-to-weight ratio and lower startup times compared to diesel engines [14]. The development of gas turbine systems during the last three decades has shown both the high versatility of small units for distributed generation [15,16] and the high efficiency that is achievable by inland large power plants—up to 60% in combined cycle [17]. To achieve a good compromise between compactness and performance, radial GT units are often characterized by a high pressure ratio (PR) per stage [5,13]. Small units from Capstone realize up to 200 kW by regeneration and pressure ratios on the order of magnitude of four to—more rarely—five. Indeed, increasing this value poses additional aerodynamic challenges, such as transonic flow at compressor inlet near the blade tip [16] or at the turbine outlet [15]. Another remarkable commercial example is represented by Opra Turbines [18], which reach a declared nominal pressure ratio over six, achieved with single-stage radial components in a simple cycle. However, radial turbomachines exploiting PR $>5$ in a single stage are rare in the field and require specific design. As an example, the following table (

Table 1. Design characteristics of a high-PR centrifugal compressor and centripetal turbine. PR indicates the total-to-total and total-to-static pressure ratios as reported by Krain et al. [19] and Sauret [22], respectively.

Parameter	MU	Krain	Jones
Pt,in	[bar]	1.01	5.80
Tt,in	[K]	293.15	1056.50
PR	[ - ]	6.10	5.73
ṁ	[kg/s]	2.55	0.33
n	[rpm]	50,000	106,580
Main	[ - ]	1.30	0.99
${\eta }_{tt}$	[%]	84.00	86.40

) summarizes two specific applications of a transonic centrifugal compressor (documented by Krain et al. [19,20]) and a centripetal high-PR turbine (see Jones [21], Sauret [22] for more detail).

The present work aims to set up a procedure that is able to provide a reliable and fast preliminary design of high-pressure-ratio centrifugal compressors and radial turbines for small-scale GTs, assessing the validity of classical, proven theory. This paper follows a previous analysis by the authors [23] concerning the potential application of mini gas turbine power units—ranging from 1 to 10 MW output—as main energy systems onboard ships. The potential of gas turbines as a technology was tested on a large-scale in the recent past on commercial passenger ships, i.e., the notorious case of Millennium Cruises [6], and is the object of promising recent studies on actual courses and modern vessel design [8,24]. Still, there seems to be little work in the field concerning smaller, compact, and modular solutions. Interestingly, the simple Brayton cycle reaches a local optimum efficiency for high temperatures between 1000 and 1200 K, with a PR around 6 [4], which can be realized by a single-stage, uncooled radial GT. Hence, previous works from the authors assessed the thermodynamic cycle and performances related to suitable cycles [11] and identified a proper window of applicability via statistical analysis on modern passenger ship [23]. More specifically, a mini gas turbine unit in combined cycle realizing 5 MW of net electrical power in CHP. The following sections will guide the reader from these findings to the design by means of classical one-dimensional models of the corresponding centrifugal compressor, the power turbine, and the mechanical shaft connecting the components that are properly designed to fulfill the power size and cycle requirements identified in the previous authors’ research. Such methodologies allow for the fast and reliable design and performance prediction of turbomachines [25,26]. A novelty of the present paper consists of the integration of aerodynamic and structural tools for the preliminary design of both the machines and the shaft required to operate them.

2. Thermodynamic Cycle

Thermodynamic cycle analysis has been largely explored in a previous work from the authors, where performances, estimated weight, and dimensions were compared against commercial engines onboard passenger ships [23]. The results identified a single-cycle mini gas turbine ranging from 4 to 5 MW net electric power (MW_el), open to co-generation strategies as a competitive power unit in the field. In a combined cycle, the proposed plant could deliver up to 7 MW_el while fully satisfying the onboard heat demand.

Table 2. Design parameters from gas cycle calculation [23].

Input	MU	Compressor	Turbine
Pt,in	[bar]	1.01	6.26
Tt,in	[K]	293	1223
ṁ	[kg/s]	24.20	24.56
PR	[ - ]	6.50	6.00
${\eta }_{tt}$	[ - ]	0.82	0.90
n	[rpm]	15,000	15,000

summarizes the design objectives derived from cycle design.

Rotational speed $\omega$ and target design efficiencies were chosen considering specific speed ($n_s=\omega ·\sqrt{\dot{m}/{\rho }_{ref}}/\left({\Delta h_{is}}^{0.75}\right)$, where ${\rho }_{ref}$ indicates the compressor rotor total outlet and turbine static exit density, respectively) via classical statistical maps reported in the work of Rodgers [27] and perfected by Van den Braembussche [28] for high-performance centrifugal impellers and similarly, from Glassman [29] and Baskharone [5] for axial and radial turbines. By employing thermodynamic cycle data, a suitable specific speed is selected by searching for the best compromise between machine performances. The calculated values of $n_s$ for the compressor and turbine are 0.35 and 0.68, respectively. The selected rotational speed should not compromise the mechanical integrity of both rotors, as the value reported in Table 2 falls close to and below other high-loaded applications in the field [18,20]. The design choice of a radial turbomachine type grants nearly undoubted advantages for the compressor in terms of compact design [13]. The pressure ratio reported in Table 2 was indeed chosen to allow for a single-stage unit at the cost of additional aerodynamic challenges—mainly, flow recirculation in the compressor due to strong adverse pressure gradients. It is worth noting that the same does not hold for the turbine; for increasing $n_s$, axial gas turbines are usually capable of delivering more power at a higher efficiency [5]. However, at relatively low specific speeds (i.e., $n_s<0.77$) radial gas turbines are still a more efficient solution. Interestingly, the present evaluation—$n_s=0.68$—also suggests the use of a radial turbine and predicts an efficiency range 7% higher than equivalent axial turbines for the same design target.

3. Compressor 1D Design

Centrifugal compressor design was largely based on the data presented in Table 2, which were used as boundary conditions for the implemented procedure. Calculations were carried out by means of an in-house code developed in Matlab, the logic of which is visualized in Figure 1 and summarized in the following:

• From the working condition of the machine, inlet gas thermodynamic state and first guesses of the backsweep angle, impeller tip velocity, and radius are first computed;
• Impeller inlet design is performed following the procedure highlighted by [13];
• Impeller tip design is completed by employing the continuity equation at rotor outlet;
• Loss correlations [30,31] are used in an internal loop to estimate impeller efficiency, up to stable convergence. An external loop updates the number of blades, looking for a good trade-off between efficiency and flow guidance at impeller outlet;
• Upon convergence, diffuser design is performed either via a vaneless duct or wedges. The code returns the global performance of the machine, including the effect of both rotor and stator and the corresponding geometry for the application.

Backsweep angle was introduced in order to contain the action on the blade surface of Coriolis forces, responsible for the presence of the slip effect [13], which in the present work is accounted for by the classical model from Wiesner [32] for curved blades. The resulting blade bend allows for better machine efficiency at the expense of a slight radius increment. Indeed, the target PR is achieved by consecutively increasing the outlet radius up to the required value of the peripheral velocity [28]. Equations (1) and (2) describe the load factor as a function of the flow coefficient and the backsweep angle for an infinite and finite number of blades, respectively. By fixing the design values of other parameters into Equations (1) and (2), it is possible to determine the backsweep metal angle ${\beta }_{2,M}$ as a function of the number of blades. As can be seen from Figure 2, the required backsweep angle reduces as the number of blades grows due to better flow guidance. On the other hand, a high number of blades leads to increased aerodynamic losses in the channel. Hence, blade number is iterated based on impeller efficiency estimation.

$$\begin{array}{c}\hfill {\psi }_{INF}=2\left[1-{\varphi }_{2}tan\left({\beta }_{2,M}\right)\right]\end{array}$$

(1)

$$\begin{array}{c}\hfill \psi =2\left[1-{\varphi }_{2}tan\left({\beta }_{2,M}\right)-\frac{2\sqrt{cos\left({\beta }_{2,M}\right)}}{Z_{rt}^{0.7}}\right]\end{array}$$

(2)

This number refers to the fractioning of the impeller outlet section and can be achieved by employing splitter blades, as required by the chosen slip model. Finally, impeller maximum tangential speed U and the corresponding diameter D for the specific application were achieved from the Euler work Equation [13].

$$\begin{array}{c}\hfill \frac{{\omega }^2\dot{m}}{\pi \lambda \gamma P_{t,1}{a}_{t,1}}=\frac{{M{a}_{1,s,rel}}^3{\left(sin\left({\beta }_{1,s}\right)\right)}^{2}cos\left({\beta }_{1,s}\right)}{{\left[1+\frac{1}{2}(\gamma -1){M{a}_{1,s,rel}}^2{\left(cos\left({\beta }_{1,s}\right)\right)}^2\right]}^{\frac{1}{\gamma -1}+\frac{3}{2}}}=f(M{a}_{1,s,rel},{\beta }_{1,s})\end{array}$$

(3)

Compressor inlet design was performed via Equation (3) (please refer to the list of symbols for more detail on the quantities involved), obtained by combining the definition of annulus area, continuity, and isentropic relations for compressible flow [13]. This relationship can be applied by following two possible approaches. For a given inlet relative flow angle near the shroud, the problem is already closed, and the equation can be solved to calculate the radius ratio from the factor $\lambda =1-{\left(\frac{r_h}{r_s}\right)}^2$ from the left-hand term. Otherwise, the value of ${\beta }_{1,s}$ maximizing f for a given $Ma$ can be found by employing partial derivatives. Both approaches require a target relative inlet Mach number at the shroud, which has been set from a sensitivity analysis performed on the code. The found values fall in the range of 1 to 1.5, in line with similar examples from the literature for radial compressors at high pressure ratios [19,20,28]. Finally, assuming null tangential velocity of the inlet flow, the code computes impeller inlet velocity components along the radial direction and completes velocity triangles accordingly. Outlet channel height is determined by the continuity equation, the target mass flow rate and the tip diameter calculated previously. To account for the effect of growing boundary layers along the channel, the code requires as input non-dimensional blockage factors to be calibrated at all sections. In the present work, the coefficients for rotor inlet and outlet follow standard practice from the literature [28] for high-performance compressors values range around 5% for impeller inlet and outlet. The program then estimates rotor isentropic total to total efficiency, following the work of Galvas [30] with addictions from Aungier [31] to include the effect of supersonic flows on performance. The employed loss correlations account for incidence, skin friction, diffusion, recirculation, and shocks (the corresponding equations are defined within the original references). Estimated performance affects total enthalpy at the impeller outlet, which is used to update the loss components iteratively and the velocity triangle, until convergence criteria are reached on efficiency variations. Finally, the code proceeds to stator design, either with a bladeless diffuser or wedges depending on user inputs: pressure coefficients and wedged angle from the literature [33,34]. The interposed vaneless space radius ratio is fixed at 1.10 [13] and flow is modeled via the logarithmic spiral approximation [35]. Impeller outlet flow angle is used to determine wedge geometry and the corresponding fluid properties at the outlet, while, in case of a vaneless diffuser, loss coefficients to account for frictional loss are introduced [28]. Diffuser design targets the pressure value set at turbine inlet from the thermodynamic cycle.

Given the demanding target application, the procedure described was tested and calibrated on a high-PR centrifugal compressor from the literature, documented by Krain et al. [19]. The results are presented in

Table 3. Centrifugal compressor design—procedure calibration [19]. Subscripts 1 and 2 indicate impeller inlet and outlet sections, respectively.

	$r_{1,\mathbf{ml}}$	$b_1$	$r_2$	$b_2$	${\mathit{PR}}_{\mathit{tt},1-2}$	${\mathit{PR}}_{\mathit{ts},1-2}$	${\eta }_{\mathit{is},1-2}$
	[mm]	[mm]	[mm]	[mm]	[ - ]	[ - ]	[%]
Ref.	54.00	48.00	112.00	10.20	6.10	5.70	84.00
RC1D	55.10	45.10	113.20	9.70	6.17	5.97	84.70
% var.	+2.00	+6.30	−1.10	−4.90	+1.20	+0.85	+3.40

and show fair robustness in both geometry and performance prediction, leading to its use for the target of the present work.

Design results are summarized in Figure 3 and

Table 4. Centrifugal compressor design output $Z_{rt}$ specifies the overall number of impeller main and splitter blades. Subscripts: 1—rotor inlet; 2—rotor outlet; 3—diffuser inlet; 4—diffuser outlet.

Parameter	MU	Value	Parameter	MU	Value	Parameter	MU	Value
${\varphi }_2$	[ - ]	0.2	$\psi$	[ - ]	1.41	$n_s$	[ - ]	0.46
${\alpha }_1$	[ ${}^{\circ }$ ]	0.00	$r_2$	[mm]	377	$\Delta x/r_2$	[ - ]	0.52
${\beta }_{1,h}$	[ ${}^{\circ }$ ]	39.53	$b_2/r_2$	[ - ]	0.0817	$Z_{rt}$	[ - ]	13 + 13
${\beta }_{1,s}$	[ ${}^{\circ }$ ]	60.73	$r_{1,s}/r_2$	[ - ]	0.59	$Z_d$	[ - ]	27
${\beta }_{2,M}$	[ ${}^{\circ }$ ]	45.00	$r_{1,h}/r_{1,s}$	[ - ]	0.46	$r_4$	[mm]	570
${\alpha }_2$	[ ${}^{\circ }$ ]	73.00	${\alpha }_4$	[ ${}^{\circ }$ ]	37	$r_3/r_4$	[ - ]	1.40

. Here, channel geometry employs elliptical hub and shroud lines—following literature recommendations [13,28] for the impeller—and a wedge diffuser. This initial design shows that it is indeed possible to achieve the target pressure ratio with a relatively compact diameter. The corresponding velocity triangles at meanline are presented in Figure 3b,c.

A flow coefficient ${\varphi }_2$ and load factor $\psi$, equal to 0.2 and 1.41, respectively, are chosen in the upper limit of typical high-flow radial compressors [36]. Indeed, with a pressure ratio of 6.50, a backsweep angle of 45${}^{\circ }$, and $N{S}_1=0.38$, total-to-total isentropic efficiency is estimated at 83% (specific speed definition from [28]), against a calculated value of 82.6%. The blade number converged to 26 in the present application, similar to existing examples in the literature [19]. Finally, the outlet diffuser radius found reached 570 mm, which determined a reasonable radial compactness of the machine.

4. Turbine 1D Design

Turbine preliminary design principally followed the work of Barsi et al. [15], with improvements to the rotor outlet: statistical tendencies were produced from similar radial turbines in the field and employed to determine nondimensional geometric ratios depending on a global parameter. Geometry was determined by means on an iterative procedure via an in-house Matlab program as described in Figure 4.

Inputs include gas inlet thermodynamic state, flow angle, operating point—set mass flow rate and rotational speed—target expansion ratio, total-total isentropic efficiency, and desired flow angle at outlet mean radius. The procedure also allows for the design of swept blades at the rotor inlet in a similar fashion to compressor impellers. However, the current geometry is expected to benefit from radial blades, aiming both to ease manufacturing and avoid bending stress for a significant mechanical benefit [15,29]. The desired stator inlet flow angle is set from a parallel work in the authors research group [37] concerning radial turbine combustors for power applications ranging between 1 and 10 MW.

From the boundary conditions, a first estimate of rotor inlet density and a corresponding specific speed are calculated, the latter of which is used to access the tendencies reported in Equations (4) to (12); the calculation sections are numbered from one to four from stator inlet to rotor outlet. All the reported equations are tendencies obtained from available data in the literature and a few commercial models [18,21,22,38,39]. Figure 5 highlights the rarity of this kind of turbine. For the desired application, the closest commercial gas turbine engine is the Opra OP16 [18], realizing 1.8 net MW_el.

$$\begin{array}{c}\hfill \psi =-2.611·n_{s,in}+2.975\end{array}$$

(4)

$$\begin{array}{c}\hfill \frac{D_1}{D_3}=0.704·n_{s,in}+1.068\end{array}$$

(5)

$$\begin{array}{c}\hfill \frac{D_2}{D_3}=-0.622·n_{s,in}+1.312\end{array}$$

(6)

$$\begin{array}{c}\hfill \frac{D_{4,ml}}{D_3}=2.469·n_{s,in}-0.490\end{array}$$

(7)

$$\begin{array}{c}\hfill \frac{b_1}{D_3}=0.079·n_{s,in}+0.027\end{array}$$

(8)

$$\begin{array}{c}\hfill \frac{b_4}{D_3}=1.071·n_{s,in}-0.210\end{array}$$

(9)

$$\begin{array}{c}\hfill Z_{st}=-57.182·n_{s,in}+46.473\end{array}$$

(10)

$$\begin{array}{c}\hfill Z_{rt}=20.897·n_{s,in}+7.018\end{array}$$

(11)

$$\begin{array}{c}\hfill {\Delta }_X=0.135·n_{s,in}+0.289\end{array}$$

(12)

The automatic output of the tendencies is compared and corrected with known geometrical constraints from the literature at every iteration to avoid generating unreasonable shapes. These include hub-to-tip and inlet-to-shroud outlet radius ratios [13], vaneless space, and blade solidity, among others [40]. Rotor diameter is calculated from the load coefficient and target work, then employed to evaluate the dimensional coordinates (x,r) of the meridional channel nodes. At the rotor inlet, the classical Stanitz slip model [41] for radially straight blades is employed to estimate the optimal incidence angle [13,29]. The implemented model follows indications from Dixon [13], as the Coriolis forces at rotor inlet are governed by analogous relations to a compressor impeller outlet. At all sections, meanline analysis is performed to solve the energy equation and turbomachinery kinematics, while a density-based loop on continuity is used to determine static and total quantities according to isentropic flow relationships [13]. Rotor outlet 2D design is performed via the radial equilibrium equation (ISRE). The program employs an analytical solution of this classical equation, achieved by introducing the forced vortex law $C_{\theta }\left(r\right)=k·r$ under the hypothesis of isentropic flow. Meanline tangential velocity is calculated via the desired rotor mean outlet angle from inputs, and the code expands the solution along the radius assuming negligible $C_r$ at turbine outlet. Using the forced vortex law is a design choice. The tangential velocity is set to be null at the blade outlet hub to prevent vortex breakdown, while a positive velocity component in the tangential direction is desirable to energize and stabilize boundary layers at the diffuser inlet. Following the 2D calculation, work exchange is verified by means of the Euler equation. In case of significant differences, this comparison drives small changes in turbine inlet diameter, which starts a new iteration of the procedure.

Turbine outlet diffuser design was carried out employing classical performance maps from the literature for axisymmetric conical diffusers [42]. Two different design choices are presented in

Table 5. Outlet diffuser preliminary design. Parameters according to Dixon [13].

	$P_{t4}$	$P_4$	$C_P$	$N/R_{\mathrm{in}}$	$\mathit{\theta }$	AR	${\mathit{\eta }}_{\mathit{D}}$
	[bar]	[bar]	[ - ]	[ - ]	[ ${}^{\circ }$ ]	[ - ]	[ - ]
Diff1	1.07	0.74	0.83	19	3	2.50	88.60
Diff2	1.10	0.87	0.62	5	5	2.00	83.25

, aiming for best performance and maximum compactness, respectively. The first solution consists of a very demanding diffuser, allowing for a lower turbine outlet pressure, as shown by the higher pressure coefficient, hence maximizing available work. However, pursuing maximum performance necessitates longer outlet channels, which for the current design were deemed as impractical, even when considering an ogive. On the other hand, the second diffuser, proposed in Table 5, represents a compromise between low inlet pressure, performance, and axial compactness, given the context of application [23]. Because of the reliance on maps, this information was used outside of the described procedure to achieve a fine calibration over the applied pressure boundary condition.

Analogously to the compressor, the turbine 1D design procedure was tested on a high-PR radial turbine from the literature [21], as documented in detail by Sauret [22]. The results are presented in

Table 6. Centripetal turbine design—procedure calibration [22]. Numeric convention is analogous to Figure 5.

	$r_3$	$b_3$	$r_{4,\mathbf{ml}}$	$b_4$	${\mathit{PR}}_{\mathit{ts},1-4}$	$U_3/C_{\mathbf{sp}}$
	[mm]	[mm]	[mm]	[mm]	[ - ]	[ - ]
Ref.	58.20	6.35	26.00	21.60	5.73	0.70
RT1D	56.30	6.40	24.20	21.20	5.74	0.72
% var.	−3.26	+0.79	−6.92	−1.85	+0.17	+2.85

and show fairly good agreement with the reference literature. As such, the code was used to design the radial turbine suitable for a mini gas turbine in the present work; the results are summarized in Figure 6 and

Table 7. Radial turbine design output: ${\beta }_3$ represents the optimal incidence angle [29] for a radial blade (${\beta }_{3,M}=0$).

Parameter	MU	Value	Parameter	MU	Value	Parameter	MU	Value
${\varphi }_3$	[ - ]	0.34	$\psi$	[ - ]	1.98	$n_s$	[ - ]	0.66
${\alpha }_1$	[ ${}^{\circ }$ ]	22.12	$b_1/r_3$	[ - ]	0.0794	$Z_{rt}$	[ - ]	15
${\alpha }_{4,ml}$	[ ${}^{\circ }$ ]	1.50	${\Delta }_X/r_3$	[ - ]	0.68	$Z_{st}$	[ - ]	25
${\beta }_3$	[ ${}^{\circ }$ ]	−20.00	$r_{4,t}/r_3$	[ - ]	0.63	$r_2/r_3$	[ - ]	1.05
$r_3$	[mm]	424.92	$r_{4,t}/r_{4,h}$	[ - ]	0.40	$r_2/r_1$	[ - ]	0.81

.

Interestingly, the calculated specific speed reflects the prediction of classical maps [5] for uncooled gas turbines, thus further suggesting the use of radial over axial architectures when in search of higher aerodynamic performances in the current application. The predicted rotor inlet relative flow angle is of −20${}^{\circ }$, achievable with a stator outlet angle of 69${}^{\circ }$, in line with literature recommendations [29,40]. Outlet absolute flow angle ranges from 0${}^{\circ }$ at blade hub and does not overcome 10${}^{\circ }$ near the tip. Overall, the turbine should operate with transonic flow between the stator and rotor due to high acceleration through the vanes, while rotor’s maximum outlet relative Mach number is predicted to be 0.8. However, the program does not include corrections due to profile or channel shapes choice, which can further influence flow guidance in transonic conditions, especially near the blade tip [29]. Nevertheless, the achieved meridional channel of Figure 6b is relatively compact and in line with gas turbines in the field with equivalent net power output [43]. Compared to more conventional radial turbomachines, such as micro GTs and turbochargers, the present design is still much larger [15], which poses additional challenges for rotor mechanical integrity. The design choice of an uncooled machine considered in this work allows for manufacturing simplicity at the cost of requiring heavy materials, such as nickel-based alloys, to survive at high temperatures. However, it is almost impossible to capture the 3D complex shape involved to accurately estimate mechanical stress with this design model, while the structural analysis of the radial turbine alone is projected in a future work employing 3D models, and the information presented so far allows for the stress analysis of the shaft.

5. Shaft Preliminary Design

A first design of the shaft was carried out by means of a simple static structural finite element (FE) analysis. The preliminary results presented will need to be verified and updated, accounting for the detailed analysis of the compressor, turbine, and shaft—including the size of the supports required by the dynamic analysis of the shaft line. The geometry essential for a structural analysis with the aid of FE simulation is chosen depending on the bearings adopted. For their choice, only the major load is first considered, which is the turbine torque $M_t$ transmitted towards the compressor and the electrical generator. An initial estimate of the corresponding shaft diameter $D_{t,0}$ can be calculated by considering the power balance of Equation (13) for a circular resisting section under torsional load by means of Equation (14) [44] and it will be verified under more comprehensive loads by new and complex FE model.

$$\begin{array}{c}\hfill M_t=\frac{\dot{m}W}{\omega }\end{array}$$

(13)

$$\begin{array}{c}\hfill D_{t,0}=\sqrt[3]{\frac{16\sqrt{3}}{\pi }·\frac{M_{t}SF}{{\sigma }_Y}}\end{array}$$

(14)

Here, SF indicates the safety factor, normally ranging between two and three for gas turbine rotating components. The evaluation of Equations (13) and (14) yield $M_t=7$ kNm and $D_{t,0}=77$ mm by assuming generic steel as shaft material (${\sigma }_Y=400$ MPa) with a large SF = 3. The subsequent FE analysis allows for an update of this diameter considering all the loads known in the present design phase.

Figure 7 from Stefani et al. [45] plots the maximum allowable peripheral speed DN for different conditions and bearing categories relating to micro-GT plants. This has been applied to the bearing choice of the mini-GT at hand: the initial estimate of $D_{t,0}$ yields a tentative peripheral speed ($D{N}_0=1.15·{10}^6$ mm rpm), which lies in the lowest range of micro-GTs. Although conventional high-precision rolling element bearings (in the red frame of Figure 7) might be an option for application, as far as peripheral speed is considered, the choice of sliding surface bearings (working in full hydrodynamic regime) is more justified, as the machine is more similar in size to conventional axial gas-turbines, which usually adopt such technical solution due to the much longer life and perceptibly higher load-carrying capacity. Therefore, the fluid film hydrodynamic bearings (highlighted by the blue frame in Figure 7) are recommended. Possible cutting-edge alternatives (in the green frame) such as foil and aerostatic or magnetic bearings, will be considered in the next future. Consequently, the shaft geometry must favor the hydrodynamic load-carrying capacity: a hollow shaft with large outer diameter D is well-suited for carrying the high torsional load, having a ratio between the inner and outer diameter (d/D) of less or equal than 0.65. The current choice lies near the upper bound, in agreement with several naval specifications advising a higher diameter for given strength. Figure 8 summarizes the proposed layout. The shaft is made up of a central part with a main diameter D and two lateral ones, representing the shoulders, housing the machines hubs. These are chosen so that the compressor and turbine shaft ends have diameters D_c and D_t, respectively, and by assuming shoulder fillet radii of $0.2·D_{c,t}$ for both machines and related stress concentration factors K equal to 1.5. Therefore, according to Muminović et al. [46], such assumptions yield $D/D_c=D/D_t=1.3$, which corresponds to a main shaft diameter of 100 mm with fillets $r_c=r_t=15$ mm under the conservative assumption $D_t=D_c$. Such dimensions are reasonable, considering the turbine housing shaft end is submitted to $M_t$, while the remaining end experiences a lower load.

Differently, no shaft shoulders are required for hydrodynamic bearings, so the journal diameter is not reduced at the shaft ends where the supports are located. An eventual reduction in the journal diameter related to bearing stability issues emerging from dynamical analysis will be updated in future design phase. The shaft ends serve as journals of the hydrodynamic supports. The axial dimensions are chosen to fit the expected size of the remaining components of the mini-GT unit. The corresponding FE model is depicted in Figure 9, including loads acting on the shaft line. A 2D simulation is carried out by means of commercial FE software (Ansys 2019 R3) and simple beam elements (BEAM188). The model includes proper constraints in the support locations, i.e., zero displacement at the compressor combined (axial/radial) bearing location, zero vertical displacement at the turbine radial bearing position, and zero rotation at the compressor side end. Impeller weights are simulated by the lumped forces $F_c$ and $F_t$, while shaft weight is considered by imposing suitable distributed loads q, $q_c$, and $q_t$, as shown by Figure 9. These are calculated by assuming a high-temperature steel or alloy material having a reference volumetric mass of about $7.8·{10}^3$ kg/m³. The proposed density is a reasonable overall value for nickel superalloys, in agreement with the work of Kutz [47], where values for such materials are reported in the range $7.0\mp 8.5·{10}^3$ kg/m³. After this preliminary design, more detailed analyses will allow us to choose more specifically the proper material for this application. Finally, compressor and turbine torques of $M_c=3.80$ kNm and $M_t=7.00$ kNm. respectively, are determined from the corresponding 1D design and Equation (13).

The pressure rise in the compressor and its sudden decrease in the turbine causes a force imbalance between the operating machines. The relevant axial loads in usual practice are partially compensated by sending pressurized air in the backdisk sealings [44] or by means of more sophisticated technical solutions [48]. For the present application, we propose that a small extraction of the mass flow exiting from the compressor is being used for sealing and force balance. Because the procedures described earlier for the 1D design of the turbine and compressor do not allow consistent simulation of the pressure variations, the axial forces exerted by both compressor and turbine are not known in the present design phase and they are ignored in the preliminary shaft design. After a detailed CFD (3D) simulation of the machine, the shaft structural analysis will be repeated, including these axial forces related to aerodynamic imbalance, which may perceptibly influence the stress distribution in the shaft. The mean operating temperatures of the uncooled compressor and turbine obtained averaging from rotor inlet to outlet are 200 and 700 ${}^{\circ }$C, respectively. The corresponding materials allowable for the application are expected to be titanium and nickel alloys, which allows for the assessment of the pertaining mass of the impellers at about 150 and 300 kg. Because the estimated weights ($F_c=1.47$ kN and $F_t=3.35$ kN) exceed usual radial turbomachinery design applications [45], the proposed layout in Figure 8 introduces bearing supports for both machines. Albeit redundant, such a conservative solution aims to realize a balanced support for the heavy components.

By assuming a working temperature of the shaft made of alloy steel equal to 450 ${}^{\circ }$C (average between compressor and turbine values), its material properties may be the following [49]: Young modulus $E=180$ GPa and Poisson ratio $\nu =0.28$. Results of the FE structural analysis are presented in Figure 9, where the reference frame (x, y) is adopted to ease readability alongside the normalized coordinate $\xi =x/L$, $L=1577$ mm being the total shaft length. A maximum displacement of 0.77 mm is measured at roughly $\xi =0.60$ along the negative y axis. The shear force is negligible all over the shaft length and is then ignored. Therefore, the main stress source is torsion due to torque, which is presented in Figure 10a. Maximum shear stress is found to be constant, equal to ${\tau }_{max}=95$ MPa in proximity of the turbine housing shaft part ($0.7<\xi <0.77$), which is the section with the minimum diameter $D_t$.

As shown by the diagram in Figure 10b, the maximum absolute value of the bending stress is ${\sigma }_{max}=27.3$ MPa at $\xi =0.77$—the same as the maximum shear stress location. Therefore, this location can be identified as having the most relevant Von Mises stress ${\sigma }_{VM}=167$ MPa. Moreover, if considering stress concentration (K = 1.5) and assuming the bending stress in the shoulder section $(\xi =0.7$) as $K{\sigma }_{max}$ (a conservative choice, reasonably applicable as the shoulder is close to the location $\xi =0.77$), the Von Mises stress reaches a ${\sigma }_{VMK}=170$ MPa. Because the reduction factor of the yield stress for a structural steel is roughly 0.7 at 450 ${}^{\circ }$C [50], if the shaft has the strength of an average alloy steel at room temperature (${\sigma }_y=600$ MPa), an approximate choice of its yield stress may be ${\sigma }_y=400$ MPa. Hence, the safety factor (SF) of the proposed design is 2.3, which lies in the permissible range of $2<SF<3$. To summarize, the present structural design can be carried out by considering the torque as a primary load if the axial force due to pressure imbalance into the operating machines is neglected. The results should be confirmed by simulating the pressure evolution in the compressor and turbine components by means of 3D CFD models to assess the shaft axial load reasonably.

6. Conclusions

The current context of naval propulsion and the increasing interest in more sustainable naval transport opens up the possibility of employing gas turbines as onboard engines and, more specifically, the use of smaller units in a modular layout. In the pursuit of compactness for onboard application, radial gas turbines seem to be a promising, yet challenging solution. Because the design of these kind of machines can become particularly complex as far as high pressure ratios are concerned, to reach the desired power, the present paper showed a preliminary procedure through in-house Matlab codes, that is able to achieve a reliable 1D machine design and allows a designer to assess a large number of configurations in a short time. Indeed, the developed codes proved to be able to facilitate the aerodynamic design of high-pressure-ratio compressors and turbines, as well as the static structural analysis of the shaft required to connect the components via a simple FE model of the shaft loads. Despite the lack of detailed information on the machines involved, the developed platform was capable of providing an overview of the required machine, by producing a preliminary geometry suitable to the application needs and assessing aerodynamic performance, structural integrity of the shaft, and a following tentative choice of bearings. This provides useful information during the preliminary design procedure, allowing for the discrimination of different options when considering new types of machines. Another achievement of the present work lies in the tendencies found for the main turbine design parameters, obtained from the available data in literature and few commercial models, to overcome an apparent lack of documentation related to high-pressure-ratio turbomachines. The procedure presented here allows for the definition of a preliminary architecture of a compressor and turbine, and it could be further extended in the future by integrating combustion chamber design. The natural subsequent step will be 3D design through commercial software, which will allow a more detailed perspective on the increase in the correlation’s reliability. Indeed, the results of CFD simulations could be used to find correlations properly suited for the present methods, i.e., equations based on a larger number of independent variables, which are exactly those taken into consideration for the 1D design procedure.