# Experimental techniques for evaluating the performance of high-blockage
cross-flow turbine arrays

Aidan Hunt and Brian Polagye Pre-print submitted to the 15th European Wave and
Tidal Energy Conference. The version that appeared in the conference
proceedings can be found at https://doi.org/10.36688/ewtec-2023-203.This work
was supported by the United States Advanced Research Projects Agency – Energy
(ARPA-E) under award number DE-AR0001441.A. Hunt and B. Polagye are with the
Department of Mechanical Engineering at the University of Washington, 3900 E
Stevens Way NE, Seattle, WA 98195, U.S.A (e-mail: ahunt94@uw.edu).

###### Abstract

In confined flows, such as river or tidal channels, arrays of turbines can
convert both the kinetic and potential energy of the flow into renewable
power. The power conversion and loading characteristics of an array in a
confined flow is a function of the blockage ratio, defined as the ratio of the
array’s projected area to the channel cross-sectional area. In this work, we
explore experimental methods for studying the effects of the blockage ratio on
turbine performance while holding other variables constant. Two distinct
methods are considered: one in which the array area is held constant and the
channel area is varied, and another in which the array area is varied and the
channel area is held constant. Using both approaches, the performance of a
laboratory cross-flow turbine array in a water tunnel is evaluated at blockage
ratios ranging from 30% to 60%. As the blockage ratio is increased, the
coefficient of performance increases, eventually exceeding the Betz limit and
unity. While similar trends are observed with both experimental approaches, at
high blockage and high tip-speed ratios, the values of the performance and
force coefficients are found to depend on the experimental approach. The
advantages and disadvantages of each approach are discussed. Ultimately, we
recommend investigating blockage effects using a fixed array area and variable
channel area, as this approach does not convolve blockage effects with
interactions between the turbine blades and support structures.

Cross-flow turbine, blockage, array, experiment

##  1 Introduction

The efficiency of a turbine operating in a confined flow, such as a river or
tidal channel, is influenced by how much of the channel the turbine occupies.
The size of the turbine relative to the size of the channel is typically
represented by the blockage ratio, defined as the ratio between the turbine’s
projected area and the channel cross-sectional area:

$\beta=\frac{A_{turbine}}{A_{\mathrm{channel}}}.$ (1)

As the blockage ratio increases, the turbine presents greater resistance to
the oncoming flow, and thus experiences greater thrust. For a constant
volumetric flow rate, this thrust, combined with confinement from the channel
boundaries, yields accelerated flow through the turbine rotor. As a
consequence, a turbine operating in a confined flow produces more power
relative to the same turbine in an unconfined flow [1, 2, 3].

Lateral arrays or “fences” of turbines deployed in river or tidal channels can
harness these blockage effects to enhance power production [4, 5, 3]. However,
since blockage-driven increases in power generation are accompanied by
increases in the forces on the turbine, understanding how both power and loads
scale with the blockage ratio is critical for rotor design [6] and control [7]
Additionally, the blockage ratio in natural channels may vary daily (e.g.,
with the tides), seasonally (e.g., runoff from snowmelt or storms), or as
needed when individual turbines are deactivated for maintenance or to allow
the passage of vessels and marine animals. Therefore, understanding how
changes in blockage will alter turbine hydrodynamics, and thus power
production, is necessary for the management of these systems.

The study of blockage effects is also applicable to turbines that are intended
for use in unconfined environments. Many laboratory settings in which model
turbines are tested, such as wind tunnels or flumes, are inherently confined
flows, and the associated blockage effects will yield augmented performance
[8, 9], even at blockages of 10% and below [10, 11]. Further, in experimental
design, increasing model scale to achieve Reynolds numbers that are more
representative of a full scale turbine generally increases blockage. While
analytical corrections have been developed to predict unconfined turbine
performance using measurements of confined turbine performance [12, 13, 14,
15], most are simplified models based on linear momentum theory, and their
accuracy can vary [16]. Given this, dedicated study of blockage effects,
particularly at the upper end of achievable blockage in practical situations,
is relevant.

Table 1: Summary of experimental studies that consider the effects of varying
the blockage ratio for axial-flow turbines (AFTs) and cross-flow turbines
(CFTs).

Author | Year | Turbine Type | Facility Type |  $\beta$ tested [%] |  $\beta$ varied by changing
---|---|---|---|---|---
[15] | 2009 | AFT | Wind tunnel, water flume | 5, 64 | Test facility
[17] | 2010 | CFT | Water flume | 50, 62.5 | Water depth
[10] | 2011 | CFT | Wind tunnel with removable walls | 2.8†, 10 | Wind tunnel size
[18] | 2011 | AFT | Wind tunnel | 10.2, 20.2, 28.3 | Turbine diameter
[19] | 2011 | CFT | Wind tunnel | 2, 3.5, 8 | Turbine scale
[20] | 2012 | AFT | Wind tunnel | 19, 33.8, 52.8 | Turbine diameter
[21] | 2013 | CFT | Water flume | 24.5 - 49.2† | Water depth
[22] | 2014 | AFT | Water flume | 6.3, 9.9, 14.3, 19.4, 25.4 | Turbine diameter
[23] | 2015 | AFT | Multiple flumes and tow tanks | 1.2, 3.3, 4.8 | Test facility
[24] | 2015 | AFT | Multiple wind tunnels | 8.1, 18.0, 48.1 | Test facility
[11] | 2015 | CFT | Wind tunnel with removable walls | 2.8†, 10 | Wind tunnel size
[25] | 2018 | CFT | Multiple wind tunnels | 3.5, 13.4, 24.7 | Test facility
[16] | 2020 | CFT, AFT | Multiple flumes | AFT: 2, 35. CFT: 3, 36 | Test facility
[26] | 2020 | CFT | Multiple flumes | 14, 36 | Test facility
[9] | 2022 | CFT | Water tunnel | 9.9, 12.4 | Blade span

  * †

Value estimated from provided turbine and channel dimensions.

Table 1 summarizes prior experimental work that has explored how changing the
blockage ratio affects the performance of both axial-flow turbines and cross-
flow turbines. While we focus this review on experimental studies, we
acknowledge that there is a complementary body of numerical work (e.g., [27,
2, 28, 29]). In alignment with theory, an increase in blockage is found to
increase the thrust loading on the turbine [11, 16, 26, 9] and the maximum
efficiency of the turbine [15, 17, 19, 21, 24, 11, 25, 16, 26, 9], as well as
the tip-speed ratio at which this maximum occurs. From a fluid dynamic
standpoint, an increase in blockage is observed to accelerate the flow through
and around the rotor, as well as narrow the turbine wake [10, 22, 26].
Although these key trends are common across these studies, multiple approaches
for varying blockage have been used across these experiments. Some studies
vary $\beta$ by changing the size of the channel; for example, testing the
same turbine in different facilities [15, 24, 25, 16, 26], or altering the
water depth in a flume [17, 21]. Others vary $\beta$ by changing the
dimensions of the turbine itself [18, 19, 20, 22, 9].

All of these experimental approaches yield similar trends, yet differences in
approach limit a deeper understanding of blockage effects. While differences
in rotor geometry [6, 16] and type of test facility [23] can influence
observed trends, even blockage effects measured in a single test facility with
a single turbine design can be inadvertently convolved with the effects of
other variables. For example, [17] and [21] both change the blockage ratio by
changing water depth in a flume, but in doing so convolve the effects of
blockage with those of the Froude number and proximity to the free surface.
Overall, the variety of approaches employed by prior work motivate a thorough
consideration of experimental methods for studying blockage effects on
turbines and the robustness of these techniques.

In this work, we discuss two approaches by which the blockage ratio of a
turbine array can be varied in a single test facility while holding constant
or minimizing the effects of other dimensionless parameters. Using both
approaches, we evaluate the performance of a cross-flow turbine array at
blockage ratios between 30% and 60%, and compare the results. These blockages
are of importance for associated research employing control co-design to
understand the potential for high-blockage arrays of cross-flow turbines to
significantly reduce cost of energy.

##  2 Background

Consider a row of identical, straight-bladed, vertical-axis cross-flow
turbines operating in a rectangular water channel. Neglecting the area of any
support structures or fixturing, the array blockage ratio for such a system is
given by

$\beta=\frac{A_{\mathrm{turbines}}}{A_{\mathrm{channel}}}=\frac{NHD}{hw}\ \ ,$
(2)

where $N$ is the number of turbines, $D$ is the rotor diameter, $H$ is the
blade span, $h$ is the time-varying channel depth measured at the turbine’s
axis of rotation, and $w$ is the channel width (Fig. 1). To characterize the
array’s performance as a function of $\beta$, there are two distinct
approaches by which $\beta$ can be varied.

### Approach 1: Fixed $A_{\mathrm{turbines}}$, variable $A_{\mathrm{channel}}$

Figure 1: Dimensions and fluid properties for a single straight-bladed cross-
flow turbine in a water channel, as viewed looking downstream (left) and
looking cross-stream (right).

For a fixed array geometry, $\beta$ can be varied by changing the cross-
sectional area of the channel. For both flumes and wind tunnels, it is
possible, though logistically challenging, to reduce $A_{\mathrm{channel}}$ by
changing channel width through the installation of intermediate partitions
[17, 10, 11]. For flumes, $A_{\mathrm{channel}}$ is more easily altered by
varying the water depth ($h$). However, to experimentally isolate the effects
of blockage on array performance while doing so, other non-dimensional flow
parameters must be carefully controlled.

For example, consider an increase in $\beta$ achieved via a decrease in the
water depth. This decrease in $h$ will simultaneously increase the depth-based
Froude number,

$Fr_{h}=\frac{U_{\infty}}{\sqrt{gh}}\ \ ,$ (3)

where $U_{\infty}$ is the freestream velocity and $g$ is the acceleration due
to gravity. The depth-based Froude number represents the balance between
inertial forces in the flow and gravitational forces in the flow. The array’s
proximity to the free surface will also change as $h$ is decreased, which can
be represented by the normalized submergence, $s/h$, where $s$ is the distance
between the free surface and the top of the turbine rotors (Fig. 1). Both
$Fr_{h}$ and $s/h$ have been shown to impact turbine performance [9, 21, 30,
31]. Therefore, if $h$ decreases, $U_{\infty}$ must be decreased to hold
$Fr_{h}$ constant. Similarly, $s$ must be decreased (i.e., the rotors
positioned dimensionally nearer to the free surface) to hold $s/h$ constant at
this new water depth.

However, a decrease in $U_{\infty}$ will also decrease the Reynolds number,
which here is defined with respect to the turbine diameter as

$Re_{D}=\frac{U_{\infty}D}{\nu}\ \ ,$ (4)

where $\nu$ is the kinematic viscosity. The Reynolds number represents the
balance between inertial forces and viscous forces in the flow. The dependence
of turbine performance on the Reynolds number is well documented [32, 33, 34,
9]. Although turbine performance becomes independent of the Reynolds number
above a certain threshold, for cross-flow turbines this is difficult to
achieve at laboratory scale without the use of compressed-air wind tunnels
[32], so in most facilities, the Reynolds number must be held constant to
isolate blockage effects. To compensate for the decrease in $U_{\infty}$
necessitated by holding $Fr_{h}$ constant, the kinematic viscosity $\nu$ can
be increased by changing water temperature ($T$). In this way, $\beta$ may be
varied in water channels, while holding $Fr_{h}$, $s/h$, and $Re_{D}$
constant, without the complications of intermediate partitions to adjust
width.

### Approach 2: Variable $A_{\mathrm{turbines}}$, fixed $A_{\mathrm{channel}}$

Alternatively, $\beta$ can be varied by changing $A_{\mathrm{turbines}}$. For
an array of straight-bladed cross-flow turbines, this can be achieved by
changing $N$, $H$, and/or $D$. However, as for the previous approach,
experimental isolation of blockage effects can be complicated by unintended
effects introduced by changing the turbine geometry or the number of turbines.

For example, if $A_{\mathrm{turbines}}$ is increased by increasing the rotor
diameter, several geometric and flow parameters are simultaneously varied.
First, increasing $D$ decreases the chord-to-radius ratio, $c/R$, which
influences the flow curvature effects (e.g., virtual camber and virtual
incidence) experienced by the blades [35]. Second, increasing $D$ also
increases the diameter-based Reynolds number $Re_{D}$ (4), and necessitates
corresponding increases to $U_{\infty}$ and/or $\nu$ to hold $Re_{D}$ constant
across blockage cases. Finally, for a fixed-width channel, changing $D$
changes both the spacing between adjacent turbines as well as the proximity of
the turbines at the ends of the array to the channel side walls. For proximity
on the order of the blade chord length, this alters the hydrodynamic
interactions between adjacent rotors [36] as well as the lateral boundary
effects on array performance [28].

One could also increase $A_{\mathrm{turbines}}$ by increasing the number of
turbines on a cross-sectional transect. However, as for a change in $D$, for a
fixed-width channel the proximity between turbines and the channel side-walls
decreases, producing the same changes to boundary effects as increasing
diameter. Further, this introduces the potential for new turbine-turbine
interactions, which are similar to turbine-wall interactions, but have an
additional degree of freedom in the rotational phase of adjacent turbines [36,
29]. As a result, if the number of turbines in the array is changed, it can be
difficult to separate the effects of blockage from those of intra-array
interactions and boundary effects.

Conversely, if $A_{\mathrm{turbines}}$ is increased by increasing only $H$
(and each rotors’ position in the water column is adjusted to hold the
submergence depth $s$ constant), then only the rotor aspect ratio, $H/D$, is
simultaneously varied with $\beta$. Given that this method varies fewer
secondary parameters than either $D$ or $N$, changing $A_{\mathrm{turbines}}$
via $H$ alone is a conceptually attractive means of varying array blockage for
fixed $A_{\mathrm{channel}}$. However, this approach would only be effective
at experimentally isolating blockage effects if the effects of changing aspect
ratio on turbine performance are minor by comparison. Prior work by [37] has
shown that the efficiency of a single turbine with blade-end struts at
$\beta=11\%$ is invariant for $H/D=0.95\\!-\\!1.63$, although the range of
invariance likely depends on the type of support structure used (e.g.,
endplates, midspan struts) [38, 39] and may be different for high-blockage
arrays. As $H/D$ is further decreased via a decrease in $H$, it is
hypothesized that hydrodynamic interactions between the blades and the support
structures could become more prominent and alter performance.

In summary, for fixed $A_{\mathrm{turbines}}$, $\beta$ is best varied by
changing the water depth in the channel, and for fixed $A_{\mathrm{channel}}$,
$\beta$ is best varied by changing the height of the rotors via the blade
span. In this work, both methods are evaluated experimentally and compared.

##  3 Experimental Methods

Table 2: Experimental parameters for each blockage condition tested.

Target Blockage Condition |  | Nominal Flume Parameters | Nominal Non-Dimensional Flow Parameters
---|---|---|---
$\beta$ (%) | $A_{\mathrm{turbines}}\ (\mathrm{m}^{2})$ | $A_{\mathrm{channel}}\ (\mathrm{m}^{2})$ |  $H$ (m) |  $h$ (m) |  $U_{\infty}$ (m/s) |  $s$ (m) |  $T$ [${}^{\circ}\mathrm{C}$] | $s/h$ | $Fr_{h}$ | $Re_{D}$
30.0 | 0.135 | 0.451 | 0.215 | 0.593 | 0.528 | 0.317 | 21.0 | 0.53 | 0.219 | $1.7\\!\times\\!10^{5}$
33.4 | 0.406 | 0.534 | 0.501 | 0.257 | 23.3 | 0.48
36.7 | 0.369 | 0.485 | 0.477 | 0.209 | 25.4 | 0.43
40.1 | 0.338 | 0.445 | 0.457 | 0.169 | 27.3 | 0.38
45.0 | 0.301 | 0.396 | 0.431 | 0.120 | 30.1 | 0.30
50.0 | 0.271 | 0.356 | 0.409 | 0.080 | 32.6 | 0.22
55.0 | 0.246 | 0.324 | 0.390 | 0.048 | 35.0 | 0.15
45.0 | 0.203 | 0.451 | 0.322 | 0.593 | 0.528 | 0.210 | 21.0 | 0.35 | 0.219 | $1.7\\!\times\\!10^{5}$
50.0 | 0.406 | 0.534 | 0.501 | 0.150 | 23.3 | 0.28
55.0 | 0.369 | 0.485 | 0.477 | 0.102 | 25.4 | 0.21
60.0 | 0.338 | 0.445 | 0.457 | 0.062 | 27.3 | 0.14

###  3.1 Test Setup

Figure 2: A rendering of the experimental test-rig, as viewed from upstream.
The array shown is at $\beta=60\%$ with $H=0.322\ \mathrm{m}$, $h=0.445\
\mathrm{m}$, and $s=0.062\ \mathrm{m}$.

Experiments were conducted in the Alice C. Tyler recirculating water flume at
the University of Washington. The flume has a test section that is 0.76 m wide
and 4.88 m long, and can accommodate water depths up to 0.60 m and flow speeds
up to ${\sim}\\!1.1$ m/s. The flume is equipped with both a heater and chiller
for temperature control, and can maintain water temperatures between
$10^{\circ}\mathrm{C}$ and $40^{\circ}\mathrm{C}$ during operation.

The laboratory-scale array consists of two identical straight-bladed cross-
flow turbines. The rotors are each two-bladed and have a diameter of 0.315 m,
defined as the outermost circle swept by the turbine blades. Each blade has a
NACA 0018 profile, a 0.0742 m chord length, and is mounted at a $6^{\circ}$
preset pitch angle as referenced from the quarter chord. The blades are
attached to the central driveshaft of each rotor using thin, hydrodynamic
blade-end struts (NACA 0008 profile, 0.0742 m chord length). The chord-to-
radius ratio is 0.47 and the solidity is 0.15.

The two rotors are integrated into the experimental set-up shown in Fig. 2,
which consists of two identical test-rigs. The top of each turbine’s central
shaft is connected by a flexible shaft coupling (Zero-Max SC040R) to a
servomotor (Yaskawa SGMCS-05BC341) which regulates the rotation rate of the
turbine. The angular position of each turbine is measured via the servomotor
encoder, from which the angular velocity is estimated. The bottom of each
turbine’s central shaft sits in a bearing. The net forces and torques on each
turbine are measured by a pair of 6-axis load cells: an upper load cell (ATI
Mini45-IP65) mounted to the servomotor and fixed to a crossbeam, and a lower
load cell (ATI Mini45-IP68) mounted to the bottom bearing and fixed to the
bottom of the flume via a suction plate. Measurements from the load cells and
servomotor encoders for both turbines are acquired synchronously at 1000 Hz in
MATLAB using a pair of National Instruments PCIe-6353 DAQs.

The freestream velocity is measured using an acoustic Doppler velocimeter
(Nortek Vectrino Profiler) sampling at 16 Hz. The velocimeter sampled a single
cell positioned laterally in the center of the flume, vertically at the array
midplane, and 5 turbine diameters upstream of the array centerline. Velocity
measurements are despiked using the method of [40]. The water depth upstream
of the array is measured at the center of the flume ${\sim}5.8$ turbine
diameters upstream of the array centerline by an ultrasonic free-surface
transducer (Omega LVU 32) sampling at 1 Hz. The water temperature is measured
using a temperature probe (Omega Ultra-Precise RTD) and maintained within $\pm
0.1^{\circ}\mathrm{C}$ of the target value during each experiment.

###  3.2 Test Matrix

Array performance is characterized at blockage ratios ranging from $30\%$ to
$60\%$ using combinations of $A_{\mathrm{turbines}}$ and
$A_{\mathrm{channel}}$. Table 2 summarizes the turbine geometries and flume
conditions used to achieve each $\beta$. Using two different blade spans, two
values of $A_{\mathrm{turbines}}$ are tested. For each value of
$A_{\mathrm{turbines}}$, $\beta$ is varied by changing the water depth, with
corresponding variations in $U_{\infty}$ and $\nu$ to maintain constant
$Fr_{h}$ and $Re_{D}$ across all experiments. To test whether the same
characteristic performance is measured for arrays with identical $\beta$, but
different $A_{\mathrm{turbines}}$ and $A_{\mathrm{channel}}$, both the
$A_{\mathrm{turbines}}=0.135\mathrm{\ m}^{2}$ rotors and the
$A_{\mathrm{turbines}}=0.203\mathrm{\ m}^{2}$ rotors are tested at $45.0\%$,
$50.0\%$, and $55.0\%$ blockage.

The values of $Fr_{h}$ and $Re_{D}$, which are held constant across all
experiments, are constrained by the maximum $U_{\infty}$ at which the highest
blockage arrays can be tested. This velocity, in turn, is constrained by rotor
ventilation (i.e., air entrainment), the onset of which becomes more likely
with decreasing $s/h$ and results in significant performance degradation due
to an increase in form drag on the blades [21, 41]. As turbines at the highest
blockages ($\beta\geq 55.0\%$) necessarily operate close to the free surface,
the maximum allowable $U_{\infty}$ for these cases is set such that any
ventilation occurs well beyond the optimal performance point. While $s/h$
could be held constant across all experiments by adjusting the array
submergence depth, preliminary experiments showed that varying $s/h$ has
minimal effect on array performance until ventilation begins to occur.
Therefore, we instead choose to maximize $s/h$ at each blockage so as to limit
the risk of ventilation as much as possible.

###  3.3 Array Layout and Control

Figure 3: Overhead view of the array layout in the Tyler flume, with key
measured quantities annotated.

An overhead view of the array layout is shown in Fig. 3. The center-to-center
spacing between the turbines is ${\sim}1.2D$, and the array is positioned
laterally such that the blade-to-blade spacing between adjacent turbines is
twice the wall-to-blade spacing (i.e., the walls notionally correspond to
symmetry planes in a larger array). The turbines in the array were operated
under a counter-rotating, phase-locked scheme, wherein both turbines rotate at
the same, constant speed, but in opposite directions, with a constant angular
phase offset, $\Delta\theta$, between them. This control strategy was achieved
by specifying the angular velocities of the rotors, which yields similar time-
average performance to an array with a constant control torque [42]. The
turbines were counter-rotated such that the blades of adjacent rotors pass
nearest each other while moving downstream, which has been shown to augment
performance relative to other rotation schemes [43, 29]. We limit the present
experiments to $\Delta\theta=0^{\circ}$, an operating case in which the
lateral forces and reaction torques for a pair of counter-rotating turbines
are equal and opposite. A closed-loop controller maintained $\Delta\theta$ to
within $1^{\circ}$ of the target value at all rotation rates across all
experiments.

###  3.4 Performance Metrics

Performance metrics are calculated for individual turbines from the measured
quantities shown in Fig. 3. The rotation rate, which is the same for both
turbines, is non-dimensionalized as the ratio of the blade tangential velocity
to the freestream velocity, or the tip-speed ratio

$\lambda=\frac{\omega R}{U_{\infty}}\ \ ,$ (5)

where $\omega$ is the angular velocity of the turbine and $R$ is the turbine
radius. Data are collected at each tip-speed ratio for 60 seconds, and the
time series is cropped to an integer number of turbine rotations before
performance metrics are calculated.

The efficiency (formally, the coefficient of performance) of each turbine is
the mechanical power produced normalized by the kinetic power in the
freestream flow that passes through the turbine’s projected area

$C_{P,i}=\frac{\tau_{i}\omega_{i}}{\frac{1}{2}\rho U_{\infty}^{3}DH}\ \ ,$ (6)

where $\omega_{i}$ and $\tau_{i}$ are the angular velocity and hydrodynamic
torque on turbine $i$, and $\rho$ is the density of the working fluid. The
efficiency of each turbine is a function of the power produced by its blades
and power losses due to parasitic torque on its blade support structures; in
this case, blade-end struts. Because a constant set of blade-end struts is
used for all experiments (i.e., the thickness of the struts is not scaled as
the blade span is changed), the relative impact of these parasitic torques on
turbine efficiency is larger for turbines with shorter blades (i.e., smaller
$A_{\mathrm{turbines}}$ as in Table 2). To account for this, we utilize the
approach of [44, 38] to estimate a blade-level $C_{P}$ for each turbine (i.e.,
the efficiency of the turbine blades in the absence of the support structures)
via superposition as

$\begin{split}C_{P,i,\mathrm{blade}}(\beta,\lambda)&\approx
C_{P,i,\mathrm{turbine}}(\beta,\lambda)\\\
&-C_{P,i,\mathrm{supports}}(\beta,\lambda)\end{split}$ (7)

where $C_{P,i,\mathrm{turbine}}$ is the measured efficiency of the full
turbine $i$, and $C_{P,i,\mathrm{supports}}$ is the measured efficiency of
turbine $i$ with no blades attached.

Structural loads on each turbine are characterized via the thrust and lateral
force coefficients, respectively given as

$C_{F_{X},i}=\frac{F_{X,i}}{\frac{1}{2}\rho U_{\infty}^{2}DH}\ \ ,$ (8)
$C_{F_{Y},i}=\frac{F_{Y,i}}{\frac{1}{2}\rho U_{\infty}^{2}DH}\ \ ,$ (9)

where $F_{X,i}$ and $F_{Y,i}$ are the streamwise force and lateral force,
respectively, on turbine $i$. To estimate blade-level loading for each turbine
in the array, we apply superposition equations for $C_{F_{X},i}$ and
$C_{F_{Y},i}$ analogous to that given for $C_{P,i}$ in (7). However, we note
that, unlike for $C_{P,i}$, the validity of this approach for estimating
blade-level $C_{F_{X},i}$ and $C_{F_{Y},i}$ has not been examined in the
existing literature.

Since the turbines in this array are identical, the array-average performance
metrics are obtained simply as the average of the individual turbine
performance metrics. For example, $C_{P,array}$ is simply the average of
$C_{P,1}$ and $C_{P,2}$ for the full turbine, and the average of
$C_{P,1,\mathrm{blade}}$ and $C_{P,2,\mathrm{blade}}$ at the blade-level. As
noted in Section 3.3, the net lateral force on this array of two identical
counter-rotating turbines is zero since when $\Delta\theta=0^{\circ}$, the
array is symmetric about its centerline. However, by defining the directions
of $F_{Y,1}$ and $F_{Y,2}$ with this counter-rotation in mind as in Fig. 3,
the array-average of $C_{F_{Y}}$ is nonzero and represents the average lateral
force coefficient experienced by an individual rotor.

##  4 Results and Discussion

###  4.1 Non-dimensional Parameters

Figure 4: Time-averaged non-dimensional flow parameters as measured during the
experiments. The marker of each line indicates the $A_{\mathrm{turbines}}$
used to achieve a particular $\beta$. Test points at which ventilation
occurred are marked with ‘x’. The dashed box indicates the step-change in
$Fr_{h}$ and $Re_{D}$ observed for the $\beta=50.0\%$,
$A_{\mathrm{turbines}}=0.135\ \mathrm{m}^{2}$ case without pump frequency
adjustment.

The time-averaged measured values of $\beta$, $Fr_{h}$, $Re_{D}$, and $s/h$ at
each tip-speed ratio are shown in Fig. 4. Across all experiments, the measured
$\beta$ are within $1.5\%$ of the target values in Table 2, and the measured
$Fr_{h}$ and $Re_{D}$ do not deviate more than $5\%$ from the nominal values
in Table 2. However, all of these parameters vary slightly with $\lambda$ due
to turbine-channel interactions. As $\lambda$ increases, the array presents
greater resistance to the flow, causing a reduction of the upstream freestream
velocity and a rise in the upstream free surface, followed by a drop in the
free surface across the turbines as the flow accelerates through the rotors.
This causes $\beta$, $Fr_{h}$ and $Re_{D}$ as measured upstream of the turbine
to decrease slightly with $\lambda$, and $s/h$ as measured upstream of the
turbine to increase slightly with $\lambda$. Most of the variation in these
parameters associated with turbine-channel interaction occurs for $\lambda<2$
(Fig. 4). Rotor ventilation occurred only for the highest tip-speed ratios at
the highest $\beta$ for each $A_{\mathrm{turbines}}$, during which the turbine
blades pierced the free surface during their downstream sweep. The turbine-
channel interaction did not affect the turbulence intensity, which was
${\sim}2\%$ for all test conditions.

During operation, the actual $h$ and $U_{\infty}$ in the flume are a function
of the volume of water in the flume, the pump drive frequency, and the
resistance to flow imposed by the turbines (primarily a function of their
rotation rate). It is theoretically possible to hold $Fr_{h}$ and $Re_{D}$
truly constant across all cases by adjusting the static water depth or pump
drive frequency at each $\beta\\!-\\!\lambda$ set-point to compensate for the
array’s effect on the flow. However, the trial-and-error iteration required to
achieve this would be experimentally intractable. Therefore, for each target
blockage, we choose to set the flume fill and pump frequency to that which
achieves the nominal $h$ and $U_{\infty}$ when no turbines are present, and
report the variation in these non-dimensional parameters during experiments as
in Fig. 4. However, for the case of $\beta=50.0\%$ and
$A_{\mathrm{turbines}}=0.135\ \mathrm{m}^{2}$ a step-change in $U_{\infty}$
was observed near $\lambda=3$, resulting in corresponding sharp decreases in
$Fr_{h}$ and $Re_{D}$ (indicated by the dashed black boxes in Fig. 4). As this
step-change was repeatable and only observed for this configuration, we
attribute this to a unique interaction between the flume and the turbines
under these conditions. Consequently, for this case only, the pump drive
frequency was increased halfway through the test to counteract the velocity
reduction and maintain similar $Fr_{h}$ and $Re_{D}$ to that measured in the
rest of the experiments (Fig. 4).

###  4.2 Array Performance

Figure 5: Time-averaged $C_{P,\mathrm{array}}$, $C_{F_{X},\mathrm{array}}$,
and $C_{F_{Y},\mathrm{rotor}}$ as a function of $\lambda$ for the full-
turbines (left column) and blades only (right column). The marker of each line
indicates the $A_{\mathrm{turbines}}$ used to achieve a particular $\beta$.
The shaded regions indicate the interquartile range of the array- and cycle-
averaged performance at each $\beta$ and $\lambda$ (the vertical span of the
shaded region at each point is similar to the size of the plot markers). Test
points at which ventilation occurred are marked with ‘x’.

Fig. 5a shows the time-averaged array-average efficiencies as a function of
$\beta$ and $\lambda$. In agreement with prior work, $C_{P,\mathrm{array}}$
tends to increase as the array blockage ratio is increased. This trend is
primarily observed for $\lambda>1.5$; at lower $\lambda$,
$C_{P,\mathrm{array}}$ does not vary significantly with blockage.
Additionally, as $\beta$ increases the array produces power over a broader
range of tip-speed ratios, and the $\lambda$ at which maximum
$C_{P,\mathrm{array}}$ occurs increases. Beginning at $\beta=33.4\%$,
$C_{P,\mathrm{array}}$ exceeds the Betz limit [7] and, at $\beta=55.0\%$,
values of $C_{P,\mathrm{array}}$ begin to exceed unity. Such efficiencies are
not violations of energy conservation since the definition of $C_{P}$ given in
(6) considers only the kinetic power that passes through the rotor plane. This
definition neglects the available power associated with the fluid’s potential
energy, which is appreciably drawn down as $\beta$ and thrust increase. Given
the relevance of both the freestream kinetic and potential energy for high-
confinement arrays, a more representative efficiency metric may resemble the
hydraulic efficiency of a hydropower turbine, in which the available power is
a function of volumetric flow rate and net head. However, for comparison with
prior studies, the conventional definition of $C_{P}$ is used here.

Time-averaged $C_{F_{X},\mathrm{array}}$ and $C_{F_{Y},\mathrm{rotor}}$ are
shown in Fig. 5b and Fig. 5c, respectively. As expected from theory and
supported by prior work, the array-average thrust coefficient increases as the
blockage ratio is increased. Similarly, the magnitude of the lateral force
coefficient (which is seldom reported for cross-flow turbines in the
literature) also tends to increase with $\beta$. As for efficiency,
$C_{F_{X},\mathrm{array}}$ and $C_{F_{Y},\mathrm{rotor}}$ do not vary
significantly with blockage at low tip-speed ratios ($\lambda\leq 1$).

As mentioned in Section 4.1, ventilation of the turbine rotors occurred only
at high $\lambda$ for the highest $\beta$ tested for each
$A_{\mathrm{turbines}}$. At these test points, which are well beyond the
maximum efficiency point, foil drag due to ventilation amplifies the decreases
in $C_{P,\mathrm{array}}$ and increases in $C_{F_{X},\mathrm{array}}$ and
$C_{F_{Y},\mathrm{rotor}}$.

Figure 6: Percent difference in performance and force coefficients (relative
to the mean values in Fig. 5) between the $A_{\mathrm{turbines}}=0.135\
\mathrm{m}^{2}$ case and the $A_{\mathrm{turbines}}=0.203\ \mathrm{m}^{2}$
case at 45.0%, 50.0%, and 55.0% blockage. A positive percent difference
implies that the $A_{\mathrm{turbines}}=0.203\ \mathrm{m}^{2}$ case performed
better than the $A_{\mathrm{turbines}}=0.135\ \mathrm{m}^{2}$ case. While the
percent difference in $C_{P,\mathrm{array}}$ does tend to increase with
$\lambda$, the sharp increase near the end of each full turbine curve is a
result of division by small $C_{P,\mathrm{array}}$ values when computing the
percent difference.

###  4.3 Blade-level performance

For $\beta=45.0\%$, $50.0\%$, and $55.0\%$, the results in Fig. 5a-c show that
even when $Fr_{h}$ and $Re_{D}$ are held constant, achieving the same $\beta$
via different values of $A_{\mathrm{turbines}}$ and $A_{\mathrm{channel}}$
yields similar, but not identical, performance. We attribute this to the
relative difference in support structure losses and forces between the two
cases. The $A_{\mathrm{turbines}}=0.135\ \mathrm{m}^{2}$ configuration uses
shorter blades than the $A_{\mathrm{turbines}}=0.203\ \mathrm{m}^{2}$
configuration, but identically sized blade support structures and fixturing.
Therefore, any parasitic torque or drag forces associated with these supports
are normalized by a smaller projected area when power and force coefficients
are calculated. This results in lower power coefficients and higher force
coefficients for the $A_{\mathrm{turbines}}=0.135\ \mathrm{m}^{2}$
configuration than for the $A_{\mathrm{turbines}}=0.203\ \mathrm{m}^{2}$
configuration.

To account for disparities in $C_{P,\mathrm{array}}$ between the different
configurations tested at $\beta=45.0\%$, $50.0\%$, and $55.0\%$, we subtract
support structure losses ($C_{P,\mathrm{supports}}$) via (7) to estimate the
$C_{P,\mathrm{array}}$ associated with the blades only, the results of which
are given in Fig. 5d. The agreement in full turbine and blade-only
$C_{P,\mathrm{array}}$ between the two $A_{\mathrm{turbines}}$ configurations
at each blockage is shown in Fig. 6a. For $\beta=45.0\%$, subtracting
$C_{P,\mathrm{supports}}$ improves $C_{P,\mathrm{array}}$ agreement between
the two $A_{\mathrm{turbines}}$ configurations at all $\lambda$. For
$\beta=50.0\%$, subtracting $C_{P,\mathrm{supports}}$ improves agreement in
$C_{P,\mathrm{array}}$ between the two $A_{\mathrm{turbines}}$ configurations
up to $\lambda\approx 3$, but agreement worsens for $\lambda>3$. Similarly,
for $\beta=55.0\%$, subtracting $C_{P,\mathrm{supports}}$ improves agreement
in $C_{P,\mathrm{array}}$ up to $\lambda\approx 3.5$, but agreement worsens
for $\lambda>3.5$.

We attribute the poorer agreement in blade-only $C_{P,\mathrm{array}}$ at high
$\beta$ and $\lambda$ to the difficulty of estimating representative
$C_{P,\mathrm{supports}}$ in the absence of the turbine blades. As described
in Section 3.4, $C_{P,\mathrm{supports}}$ is estimated by testing an array of
bladeless turbines at the same nominal $Fr_{h}$, $s/h$, and $Re_{D}$ as the
array of full turbines. However, the array of bladeless turbines does not
influence the flow field in the same way that the array of full turbines does.
Specifically, as $\beta$ and $\lambda$ are increased, the thrust on the array
increases, resulting in a free surface drop across the rotors and acceleration
of the flow bypassing the array. These changes to the flow field are absent
for an array of bladeless turbines. Consequently, we hypothesize that the
superposition technique in (7) breaks down at higher $\beta$ and $\lambda$,
where turbine-channel interactions are most significant. Despite these
limitations, it is remarkable that (7) improves $C_{P,\mathrm{array}}$
agreement at any $\lambda$ for $\beta=45.0\%$, $50.0\%$, and $55.0\%$ given
that appreciable turbine-channel interactions are observed at these blockage
ratios for lower values of $\lambda$.

Blade-only $C_{F_{X},\mathrm{array}}$ and $C_{F_{Y},\mathrm{rotor}}$ are
similarly estimated via analogous equations to (7), the results of which are
shown in Fig. 5e and Fig. 5f. Unlike for $C_{P,\mathrm{array}}$, subtracting
$C_{F_{X},\mathrm{supports}}$ and $C_{F_{Y},\mathrm{supports}}$ does not
meaningfully change the agreement in $C_{F_{X},\mathrm{array}}$ or
$C_{F_{Y},\mathrm{rotor}}$ between the two $A_{\mathrm{turbines}}$
configurations at $\beta=45.0\%$, $50.0\%$, or $55.0\%$ for any $\lambda$,
implying that the forces on the blades dominate the forces on the support
structures. Additionally, as highlighted in Fig. 6b and Fig. 6c, the
difference in the force coefficients between the two $A_{\mathrm{turbines}}$
configurations does not change significantly at high $\lambda$ (unlike
$C_{P,\mathrm{array}}$). As before, we hypothesize that the lack of force
coefficient agreement between turbines with different aspect ratios is due to
differences between the flow fields experienced by the support structures when
blades are present versus when blades are absent. Further investigation into
techniques for estimating blade-only force coefficients is warranted, but is
outside the scope of the present study.

###  4.4 Evaluation of approaches for varying blockage

As both of the blockage-varying approaches from Section 2 were utilized in the
present experiments, we now consider the advantages and disadvantages of
implementing each approach.

Variation of $A_{\mathrm{channel}}$ via changing the water depth was used to
achieve $\beta=30.0\%-55.0\%$ with $A_{\mathrm{turbines}}=0.135\
\mathrm{m}^{2}$ and $\beta=45.0\%-60.0\%$ with $A_{\mathrm{turbines}}=0.203\
\mathrm{m}^{2}$. The blockages testable with each $A_{\mathrm{turbines}}$ were
constrained by the relative sizes of the flume and the turbines, as well as
the physical limitations of the test facility. In the Tyler flume, the minimum
testable blockage for each $A_{\mathrm{turbines}}$ was set by the maximum
dynamic channel depth of 0.60 m, above which overtopping of the flume walls
occurs. The maximum testable blockage was constrained by the ventilation risk
associated with low $s/h$: the lower the water depth, the higher the $\beta$
that can be achieved, but the closer the turbines are to the surface the
greater the risk of ventilating. However, as described in Section 2, the range
of testable blockages was further constrained to the water depths at which
$Fr_{h}$ and $Re_{D}$ could be matched across all tests via corresponding
adjustments to the freestream velocity and water temperature. Critically,
temperature control is required to avoid convolving blockage effects with
variations in the Reynolds number; as many flumes do not have this capability,
this is a general limitation of the variable-$A_{\mathrm{channel}}$ approach
unless Reynolds-independent performance can be achieved. In the present study,
the relatively wide range of temperatures achievable in the Tyler flume
enabled careful control of non-dimensional flow parameters and an effective
isolation of blockage effects. Even so, the duration of each experiment was
extended by the need to adjust the flume fill and water temperature for each
test.

Given the facility requirements of the variable-$A_{\mathrm{channel}}$
approach, the present experiments also explored how $\beta$ could be varied at
fixed $A_{\mathrm{channel}}$ through variation in $A_{\mathrm{turbines}}$. For
example, as shown in Table 2, array blockage ratios of 40.1% and 60.0% were
achieved simply by testing arrays with $A_{\mathrm{turbines}}=0.135\
\mathrm{m}^{2}$ and $A_{\mathrm{turbines}}=0.203\ \mathrm{m}^{2}$ at the same
nominal water depth (and thus same nominal $A_{\mathrm{channel}}$). Since the
water depth was unchanged, testing different blockages was convenient and fast
since no adjustments to the freestream velocity or temperature were necessary.
In a general case, if no secondary effects are introduced by changing
$A_{\mathrm{turbines}}$, then the range of testable blockages at a given
facility using this approach would be constrained only by 1) the minimum and
maximum blade spans available, and 2) ventilation risk at the highest
blockages.

However, as shown in Fig. 5, turbines tested at similar $\beta$, $Fr_{h}$, and
$Re_{D}$, but different $A_{\mathrm{turbines}}$ and $A_{\mathrm{channel}}$,
exhibit small, but appreciable differences in power and force coefficients.
The superposition-based support structure subtraction techniques for
reconciling these differences begin to break down for $C_{P,\mathrm{array}}$
at high $\beta$ and $\lambda$, and do not have any effect on disagreements in
$C_{F_{X},\mathrm{array}}$ or $C_{F_{Y},\mathrm{rotor}}$ across
$A_{\mathrm{turbines}}$. While, for these experiments, the disparities in
$C_{P,\mathrm{array}}$, $C_{F_{X},\mathrm{array}}$ and
$C_{F_{Y},\mathrm{rotor}}$ between the different $A_{\mathrm{turbines}}$ are
small relative to the overall blockage effects, the scale of these disparities
likely depends on the specific blade spans and support structures used [38,
37, 39]. Consequently, interpreting trends in blockage effects obtained via a
variable-$A_{\mathrm{turbines}}$ approach requires better models for support
structure effects than simple experimental superposition. The uncertainty
associated with the variable-$A_{\mathrm{turbines}}$ approach can only be
quantified by changing $A_{\mathrm{turbines}}$ at constant $\beta$, as was
performed in this study at $\beta=45.0\%$, $50.0\%$, and $55.0\%$. To do so
while holding $Fr_{h}$ and $Re_{D}$ constant requires either a facility with
temperature control or a facility capable of velocities of ${\sim}10$ m/s to
achieve Reynolds independence. Consequently, the facility requirements for
fully interpreting results obtained via the variable-$A_{\mathrm{turbines}}$
method negate the principal advantage of this method. Therefore, the
variable-$A_{\mathrm{channel}}$ fixed-$A_{\mathrm{turbines}}$ approach is most
robust.

##  5 Conclusion

In this work, we explored two experimental methods for characterizing the
effects of blockage ratio on the performance of an array of two, straight-
bladed cross-flow turbines operating in a water channel. For fixed
$A_{\mathrm{turbines}}$ and variable $A_{\mathrm{channel}}$, the blockage
ratio is most easily varied by changing the water depth, with corresponding
changes in the freestream velocity and temperature to hold the Reynolds number
and Froude numbers constant. For fixed $A_{\mathrm{channel}}$ and variable
$A_{\mathrm{turbines}}$, the blockage ratio is most appropriately varied by
changing the blade span, as several secondary effects are introduced if the
turbine diameter or number of turbines are changed. A laboratory-scale array
operating at blockages between 30% and 60% is tested using both approaches.
While similar trends in efficiency and force are observed regardless of
approach, the values of the array-average performance and force coefficients
vary with the method used to achieve a particular $\beta$. For future
experimental studies focusing on blockage effects, we recommend that the
blockage ratio is varied by changing $A_{\mathrm{channel}}$ with fixed
$A_{\mathrm{turbines}}$ while holding the Reynolds and Froude numbers
constant. Although this method requires the use of a flume with temperature
control or the ability to achieve Reynolds-invariant turbine performance, we
find that it was the most robust approach since changes in blade-support
structure interactions associated with changes in $A_{\mathrm{turbines}}$ can
be difficult to quantify.

We recommend that future studies investigate more robust methods for
estimating the parasitic losses and drag forces of turbine blade support
structures such that blade-only efficiency and force coefficients can be
better estimated at high blockage and tip-speed ratio, allowing more accurate
comparisons of blockage effects to be drawn across turbines. The development
of such a method would improve the reliability of the
fixed-$A_{\mathrm{channel}}$ variable-$A_{\mathrm{turbines}}$ experimental
approach, and facilitate studies of blockage effects at a wider range of test
facilities. Additionally, as ventilation was a constraint on experimental
design for both experimental methods, future work should evaluate how the
onset and effects of ventilation are influenced by the blockage ratio, the
Froude number, and the normalized submergence depth.

## Acknowledgement

The authors would like to thank Gregory Talpey and Gemma Calandra for their
assistance in commissioning the high-blockage test-rig, as well as help with
data collection. The authors would also like to thank Abigale Snortland for
several insightful discussions regarding support structure torque and force
subtraction techniques.

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# Mass-Radius Relationships for Exoplanets II:
Grüneisen Equation of State for Ammonia

Damian C. Swift<EMAIL_ADDRESS>Condensed Matter and Materials Division,
Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore,
California 94550, USA

(September 18, 2011; modified October 13, 2011 – LLNL-JRNL-505357)

###### Abstract

We describe a mechanical equation of state for NH3, based on shock wave
measurements for liquid ammonia. The shock measurements, for an initial
temperature of 203 K, extended to 1.54 g/cm3 and 38.6 GPa. The shock and
particle speeds were fitted well with a straight line, so extrapolations to
higher compressions are numerically stable, but the accuracy is undetermined
outside the range of the data. The isentrope through the same initial state
was estimated, along with its sensitivity to the Grüneisen parameter. Mass-
radius relations were calculated for self-gravitating bodies of pure ammonia,
and for differentiated ammonia-rock bodies. The relations were insensitive to
variations in the Grüneisen parameter, indicating that they should be accurate
for studies of planetary structure.

ammonia, shock, equation of state, planetary structure

##  1 Introduction

Ammonia, NH3, is a common molecule in ice giant planets, which appear to be
found widely throughout the galaxy (Schneider, 2011). The equation of state
(EOS) of ammonia is therefore important for our understanding of planetary
structures and their evolution, potentially to pressures of order 1 TPa for
the base of the ice-rock interface in icy exoplanets. An accurate EOS for
ammonia is also needed for studies of hypervelocity impacts, such as meteoroid
collisions with ice giants. Furthermore, ammonia is a simple prototype for
bonds occurring in chemical explosives, for which densities from up to around
twice that of zero-pressure solids are of interest for shock initiation and
detonation.

Although shock compression experiments have been performed on ammonia to
pressures of several tens of gigapascals (Marsh, 1980), the only equation of
state readily available is SESAME 5520 (Holian, 1984), based on National
Bureau of Standards gas phase data (Haar & Gallagher, 1978), and is tabulated
to a maximum density of 0.765 g/cm3, which is barely greater than the zero-
pressure density for liquid ammonia. Quasistatic compression experiments have
been performed in which the density and sound speed were measured along
isotherms (Abramson, 2008; Li et al, 2009), but the highest pressures reported
have reached only a few gigapascals.

##  2 Empirical Grüneisen equation of state

Shock experiments have been reported previously in which the shock and
particle speeds $u_{s}$ and $u_{p}$ were measured for a range of shock
pressures, for liquid ammonia at an initial temperature of 203 K (Marsh, 1980)
(Fig. 1). The uncertainties in $u_{s}$ and $u_{p}$ were approximately 1%.
These data can be fitted by a straight line fit

$u_{s}=c_{0}+s_{1}u_{p}$ (1)

with

$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle 2.00\pm
0.13\quad(6.7\%)$ $\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle
1.511\pm 0.039\quad(2.6\%),$

where the standard errors shown are from the residual fitting error,
neglecting the uncertainty in measurement. The experimental measurements
exhibited a slightly curved trend, but the number of points was not great
enough to justify a higher-order fit. Solving the Rankine-Hugoniot equations
for a steady shock (Zel’dovich & Raizer, 1966) using the fitted parameters
rather than the individual shock measurements, the highest shock pressure was
38.6 GPa, giving a mass density of 1.54 g/cm3. The observed sound speed in
liquid ammonia at 203 K is 1.9535 km/s (Bowen & Thompson, 1968), which is
consistent with the extrapolated Hugoniot data. (Fig. 1)

Figure 1: Principal shock Hugoniot of liquid ammonia (initial temperature 203
K): experimental measurements and least-squares fit. The point at zero
particle speed is the observed sound speed, which was not included in the fit.
The curve labelled WCS is the Woolfolk-Cowperthwaite-Shaw universal liquid
equation of state, whose sole fitting parameter is the sound speed at zero
pressure.

Various universal EOS have been proposed for different classes of material. It
is interesting to compare with the ‘universal liquid EOS’ of Woolfolk et al
(1973), whose only material-specific parameter is the sound speed at zero
pressure. This EOS does not reproduce the shock data for ammonia, which is
softer and more linear than the universal EOS (Fig. 1).

The fit to the shock Hugoniot can be used to predict the mechanical equation
of state, using the Hugoniot as a reference curve (McQueen et al, 1970)

$\displaystyle p(\rho,e)$ $\displaystyle=$ $\displaystyle
p_{r}(\rho)+\Gamma(\rho)\left[e-e_{r}(\rho)\right]$ (2) $\displaystyle
p_{r}(\rho)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{c_{0}^{2}\rho_{r}\rho(\rho-\rho_{r})}{\left[\rho+s_{1}(\rho-\rho_{r})\right]^{2}}$
(3) $\displaystyle e_{r}(\rho)$ $\displaystyle=$ $\displaystyle
e_{0}+\frac{1}{2}p_{r}(\rho)\left(\frac{1}{\rho_{r}}-\frac{1}{\rho}\right),$
(4)

where $\rho_{r}$ is the initial density on the reference curve, and
$p_{r}(\rho)$ was derived for zero initial pressure, as here. Other
experiments are required to determine $\Gamma(\rho)$, such as sound speed
measurements on the Hugoniot, a shock Hugoniot from a different initial state,
or ramp compression. However, $\Gamma$ can be estimated from the slope of the
shock Hugoniot as $2s_{1}-1$, which is accurate for cubic crystals (Skidmore,
1965). Thus $\rho_{r}=0.725$ g/cm3 and $\Gamma\simeq 2.022$.

Given the mechanical EOS, the isentrope through any state can be calculated by
integrating the $-p\,dv$ work numerically (Swift, 2008). Isentropes calculated
from Grünseisen EOS fitted to shock data typically behave unphysically at high
compression, where the assumption that the Grüneisen parameter is a function
of density only breaks down. For the ammonia fit, the breakdown occurred at a
mass density of 2.145 g/cm3. The isentrope was well-behaved to several
terapascals, though its accuracy was undetermined. The isentrope should be
reasonably accurate at least up to the peak compression in the shock data,
which equates to 22.1 GPa on the isentrope. To investigate the sensitivity to
the Grüneisen parameter, isentropes were calculated for the nominal value
above, and for values 10% lower and higher. With this variation in $\Gamma$,
the pressure varied by 10% at 20 GPa, rising to 25% at 500 GPa. (Fig. 2)

Figure 2: Principal isentrope of liquid ammonia (initial temperature 203 K)
deduced from mechanical equation of state fitted to principal shock Hugoniot,
showing sensitivity to assumed Grünseisen parameter.

##  3 Mass-radius relationships

Mass-radius relationships were calculated using the deduced EOS, for a self-
gravitating body comprising pure ammonia and also for differentiated bodies
consisting of a rocky core and an ammonia mantle, using the numerical methods
described previously (Swift et al, 2011). Separate mass-radius curves were
constructed for the nominal and perturbed values of the Grüneisen parameter.
In all cases, the temperature at the surface was taken to be 203 K, to match
the initial state in the shock experiments. The rocky core was modeled using
an EOS for basalt, SESAME 7530 (Barnes & Lyon, 1988), as was done previously
(Swift et al, 2011).

The variations in $\Gamma$ made a negligible difference to the mass-radius
relations. At high masses, the mass-radius relation for pure ammonia
asymptoted to a power-law behavior $R=\alpha M^{\beta}$ with $\alpha=1.4395\pm
0.0005$ and $\beta=0.32889\pm 0.00004$. For an incompressible material,

$M=\frac{4}{3}\pi r^{3}\rho_{0},$ (5)

giving $\beta=1/3$. The difference in the fitted value is small but
significant; $\alpha$ is considerably less than the incompressible value of
$\left(\frac{4}{3}\pi\rho_{0}\right)^{-1/3}$. The mass-radius relation was
also deduced using the SESAME EOS, which matched that from the Grüneisen EOS
up to 0.1 M${}_{\mbox{E}}$, above which point the extrapolation beyond the
bounds of the table gave unphysical behavior. (Figs 3 to 5.)

Figure 3: Mass-radius relation deduced from equation of state for liquid
ammonia (surface temperature 203 K), also showing relation for incompressible
material and least-squares fit to the relation, which is dominated by high
pressure behavior. Figure 4: Variation of central pressure with mass, for
liquid ammonia (surface temperature 203 K). Figure 5: Mass-radius relations
for differentiated ammonia-rock bodies. The percentages are the mass fraction
of ammonia in the body.

The planetary radius for pure ammonia did not exhibit a maximum within the
range of masses investigated. The range of compressions explored by the shock
experiments was equivalent to the central pressure in bodies of pure ammonia
up to around 2/3 M${}_{\mbox{E}}$(1.5 R${}_{\mbox{E}}$). However, the mass-
radius relation is accurate for significantly larger bodies, because the mass
and volume are dominated by matter at much lower pressures until the average
density exceeds 1.5 g/cm3 or so: approximately 4 M${}_{\mbox{E}}$, 2.5
R${}_{\mbox{E}}$, and a core pressure of 100 GPa. The relation may be accurate
for even larger bodies, but it has not been validated by EOS experiments.

##  4 Conclusions

The relation between shock and particle speeds in liquid ammonia appears
linear to within the scatter in the data up to pressures of at least 39 GPa. A
Grüneisen mechanical equation of state was constructed using the principal
Hugoniot of initial state zero pressure and 203 K as a reference, and
estimating the Grüneisen parameter $\Gamma$ from the slope of the Hugoniot.
Isentropes were calculated through the same state, the sensitivity to $\Gamma$
rising with pressure.

Mass-radius relations were calculated for self-gravitating bodies consisting
of ammonia, and differentiated ammonia-rock mixtures. The mass-radius
relations were insensitive to variations in $\Gamma$, indicating that the
relations should be reliable for comparison to planetary measurements, to
central pressures substantially above those reached in the shock experiments.

## Acknowledgements

This work was performed under the auspices of the U.S. Department of Energy
under contract DE-AC52-07NA27344.

## References

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