The effect of the environment humidity on the performance of an oscillating water column wave energy converter

Abstract

The sea wave energy has a significant potential to be converted in electrical one. However, there are still many difficulties to its harvesting. The most studied device is the Oscillating Water Column (OWC) wave energy converter, in which the majority of researches consider the air inside the chamber as a dry air. This study aims to investigate the influence of the environment humidity on the behavior and the performance of this device equipped with Wells and Impulse turbines. A numerical model, that uses the first law of thermodynamics to analyze a piston movement in a air chamber with a turbine, is developed to compare results of an ideal gas with those of a real gas. The behavior of the real gas inside the chamber is based on empirical parameters. An typical case of an onshore OWC in atmospheric conditions with high level of humidity of the south of the Brazilian cost is studied. Ranges of wave period and height and turbine characteristic relation for Wells and Impulse turbines are tested. Comparison between dry and real air gases inside the chamber for the same conditions of wave period, amplitude of the free surface oscillation inside the chamber and turbine characteristic relation for Wells and Impulse turbines are analyzed. Vapor precipitation is observed in cases where air pressure inside the chamber is higher than $7.1$ kPa. In general, time-averaged pneumatic powers of real gases are slightly lower ($2\%$ maximum) to those of ideal ones, which indicates that the influence of air humidity on the performance of the device is practically negligible. However, the definitive conclusion depends on the consideration of the coupling between hydrodynamics and aerodynamics in the process.

Appendices

Appendix A: Absolute humidity in inhalation ($Q_t>0$)

Suppose that at instant t there is a volume $V_1$ of the air inside the chamber, whose mass is $m_1=m_{v1}+m_{d1}$, where $m_{v1}$ and $m_{d1}$ are vapor and dry air masses. When the inhalation process takes place, there is an input of mass $\Delta m_0 = \Delta m_{v0} + \Delta m_{d0}$ (vapor and dry air mass increments) entering the chamber, from the atmosphere, between instants t and $t + \Delta t$, see (Fig. 1). Therefore, the absolute humidity r at instant $t+\Delta t$ during the inhalation process is given by

$$\begin{aligned} r = \dfrac{m_v}{m_d} = \dfrac{m_{v1} + \Delta m_{v0}}{m_{d1} + \Delta m_{d0}}. \end{aligned}$$

(34)

On the other hand, the absolute humidity at t is

$$\begin{aligned} r_1 = \dfrac{m_{v1}}{m_{d1}}, \end{aligned}$$

(35)

whereas atmospheric absolute humidity can be calculated by

$$\begin{aligned} r_0 = \dfrac{\Delta m_{v0}}{\Delta m_{d0}}. \end{aligned}$$

(36)

Thus, Eqs. (35) and (36) are used to re-write Eq. (34) as follows

$$\begin{aligned} r = r_1\left[ \dfrac{m_{d1} + \left( \dfrac{r_0}{r_1}\right) \Delta m_d}{m_{d1} + \Delta m_d}\right] . \end{aligned}$$

(37)

The mass of the air inside the chamber at instant t ($m_1$) and the increment of atmospheric air ($\Delta m$) are given by

$$\begin{aligned} \left\{ \begin{array}{ccl} m_1 &{}=&{} m_{d1} + m_{v1} = m_{d1}\left( 1 + r_1\right) = \rho _1\,V_1\\ \\ \Delta m_0 &{}=&{} \Delta m_d + \Delta m_{v} = \Delta m_d \left( 1 + r_0\right) = \rho _0\,Q_t\,\Delta t, \end{array}\right. \end{aligned}$$

(38)

where $\rho _1$ and $\rho _0$ are specific masses of the air inside the chamber at instant t and of the atmosphere, respectively, and $Q_t$ is the turbine flow rate between instants t and $t+\Delta t$. Eqs. (38) can be re-written as follows

$$\begin{aligned} \left\{ \begin{array}{ccl} m_{d1} &{}=&{} \dfrac{\rho \,V_1}{(1+r_1)}\\ \\ \Delta m_d &{}=&{} \dfrac{\rho _0\,Q_t\,\Delta t}{(1 + r_0)}. \end{array}\right. \end{aligned}$$

(39)

Substituting Eqs. (39) into Eq. (37), the absolute humidity inside the chamber (r) during the inhalation process can be determined at instant $t+\Delta t$, based on the previous one ($r_1$).

$$\begin{aligned} r = r_1 \left[ \dfrac{\dfrac{\rho \,V_1}{1+r_1} + \left( \dfrac{r_0}{r_1}\right) \dfrac{\rho _0\,Q_t\,\Delta t}{1+r_0}}{\dfrac{\rho \,V_1}{1+r_1} + \dfrac{\rho _0\,Q_t\,\Delta t}{1+r_0}}\right] . \end{aligned}$$

(40)

Appendix B: Calculation of derivatives of Z, B, $f_0$ and $f_2$

In this appendix, general expressions of derivatives of the compressibility factor (Z), the second coefficient of Virial (B) and temperature correlation functions ($f_0$ e $f_2$) are deduced. These variables are included in the dynamic pressure equation for real gases, Eq. (27).

The first derivative of B with respect to temperature is given by

$$\begin{aligned} B^{\prime } \equiv \dfrac{dB}{dT} = R_0\left( \dfrac{T_c}{p_c}\right) \left[ f_0^{\prime } + \chi _{\text {mol}}^{\prime }\,f_2 + \chi _{\text {mol}}\,f_2^{\prime }\right] , \end{aligned}$$

(41)

where

$$\begin{aligned} \left\{ \begin{array}{cll} f_0^{\prime } &{}=&{} \left[ 0.33 + 0.277\left( \dfrac{T_c}{T}\right) + 0.0363\left( \dfrac{T_c}{T}\right) ^2 + 0.004856\left( \dfrac{T_c}{T}\right) ^7\right] \left( \dfrac{T_c}{T^2}\right) \\ f_2^{\prime } &{}=&{} -\left[ 0.1674 - 0.1832\left( \dfrac{T_c}{T}\right) ^2\right] \left( \dfrac{T_c^6}{T^7}\right) , \end{array} \right. \end{aligned}$$

(42)

calculated from Eqs. (11).

The second derivative of B with respect to temperature is

$$\begin{aligned} B^{\prime \prime } \equiv \dfrac{d^2B}{dT^2} = R_0\left( \dfrac{T_c}{p_c}\right) \left[ f_0^{\prime \prime } + \chi _{\text {mol}}^{\prime \prime }\,f_2 + 2\,\chi _{\text {mol}}^\prime \,f_2^{\prime } + \chi _{\text {mol}}\,f_2^{\prime \prime }\right] , \end{aligned}$$

(43)

with

$$\begin{aligned} \left\{ \begin{array}{cll} f_0^{\prime \prime } &{}=&{} -\left[ 0.66 + 0.831\left( \dfrac{T_c}{T}\right) + 0.1452\left( \dfrac{T_c}{T}\right) ^2 + 0.043704\left( \dfrac{T_c}{T}\right) ^7\right] \left( \dfrac{T_c}{T^3}\right) \\ \\ f_2^{\prime \prime } &{}=&{} \left[ 1.1718 - 1.6488\left( \dfrac{T_c}{T}\right) ^2\right] \left( \dfrac{T_c^6}{T^8}\right) , \end{array} \right. \end{aligned}$$

(44)

where $\chi _{\text {mol}}^{\prime }$ and $\chi _{mol}^{\prime \prime }$ are deduced in "Appendix C". The second mixed derivative of B, with respect time and temperature, is given by

$$\begin{aligned} \dot{B}^{\prime } = R_0\left( \dfrac{T_c}{p_c}\right) \left[ \dot{f}_0^{\prime } + \dot{\chi }_{\text {mol}}^{\prime }\,f_2 + \chi _{\text {mol}}^{\prime }\,\dot{f}_2 + \dot{\chi }_{\text {mol}}^{\prime }\,f_2^{\prime } + \chi _{\text {mol}}\,\dot{f}_2^{\prime }\right] . \end{aligned}$$

(45)

The time derivate of $f_2$ is obtained by using Eqs. (11), that is

$$\begin{aligned} \dot{f}_2= & {} -\left[ 0.1674 - 0.1832\left( \dfrac{T_c}{T}\right) ^2\right] \left( \dfrac{T_c^6}{T^7}\right) \dot{T}, \end{aligned}$$

(46)

as well as the mixed derivatives of $f_0$ e $f_2$ are

$$\begin{aligned} \left\{ \begin{array}{cll} \dot{f}_0^{\prime } &{}=&{} - \left[ 0.66 + 0.831\left( \dfrac{T_c}{T}\right) + 0.1452\left( \dfrac{T_c}{T}\right) ^2 + 0.043704\left( \dfrac{T_c}{T}\right) ^7\right] \left( \dfrac{T_c}{T^3}\right) \dot{T}\\ \\ \dot{f}_2^{\prime } &{}=&{} -\left[ 1.6488\left( \dfrac{T_c}{T}\right) ^2 - 1.1718\right] \left( \dfrac{T_c^6}{T^8}\right) \dot{T}. \end{array} \right. \end{aligned}$$

(47)

Finally, the time derivate of Z is given by

$$\begin{aligned} \dot{Z} = \dfrac{1}{R_0}\left[ \dot{B}\left( \dfrac{p}{T}\right) + B\left( \dfrac{\dot{p}}{T} - \dfrac{p}{T^2}\dot{T}\right) \right] . \end{aligned}$$

(48)

Appendix C: The calculation of derivatives of $\chi _{\text {mol}}$

The derivate of $\chi _{\text {mol}}$ with the temperature results in

$$\begin{aligned} \chi _{\text {mol}}^{\prime }= & {} -\dfrac{100}{n\,\varepsilon } \Bigg \{\left( \dfrac{L}{R_v\,T^2}\right) \left[ 1 + \left( \dfrac{p-e_s}{e_s}\right) \right] \left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \nonumber \\&\quad + \left( \dfrac{\rho \,R_v}{\rho \,R_v\,T-p}\right) \left[ \varepsilon + \left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p} \right) \right] \Bigg \}; \end{aligned}$$

(49)

The time derivate of $\chi _{\text {mol}}$ is given by

$$\begin{aligned} \dot{\chi }_{\text {mol}}= & {} \left( \dfrac{100}{n\,\varepsilon }\right) \left[ \dfrac{1}{\,e_s\left( \rho \,R_v\,T-p\right) }\right] \nonumber \\&\quad \Bigg \{\left[ \dot{p}-\dot{e}_s - \left( p-e_s\right) \left[ \left( \dfrac{\dot{n}}{n}\right) +\left( \dfrac{\dot{e}_s}{e_s}\right) \right] \right] \nonumber \\&\quad \times \left( p-\rho \,R_d\,T\right) + \left( p-e_s\right) \Bigg [\dot{p} - R_d\left( \dot{\rho }\,T+\rho \,\dot{T}\right) + \left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \nonumber \\&\quad \times \left[ \dot{p} - R_v\left( \dot{\rho }\,T + \rho \,\dot{T}\right) \right] \Bigg ]\Bigg \} \end{aligned}$$

(50)

The second mixed derivative of $\chi _{\text {mol}}$, with respect time and temperature, is given by

$$\begin{aligned} \dot{\chi }_{\text {mol}}^{\prime }= & {} \left( \dfrac{100}{n\,\varepsilon }\right) \left[ \dfrac{1}{e_s\left( \rho \,R_v\,T-p\right) }\right] \nonumber \\&\quad \Bigg \{-\left[ \dfrac{e_s^{\prime }}{e_s}+\dfrac{\rho \,R_v}{\rho \,R_v\,T-p}\right] \Bigg \{\Bigg [\dot{p}-\dot{e}_s\nonumber \\- & {} (p-e_s)\left[ \left( \dfrac{\dot{n}}{n}\right) + \left( \dfrac{\dot{e}_s}{e_s}\right) \right] \Bigg ] \left( p-\rho \,R_d\,T\right) + (p-e_s)\Bigg [\dot{p}-R_d\left( \dot{\rho }\,T + \rho \,\dot{T}\right) \nonumber \\ \nonumber&\quad +\left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \left[ \dot{p}-R_v\left( \dot{\rho }\,T + \rho \,\dot{T}\right) \right] \Bigg ]\Bigg \}\nonumber \\&\quad +\left[ -\dot{e}_s^{\prime }+e_s^{\prime } \left[ \left( \dfrac{\dot{n}}{n}\right) + \left( \dfrac{\dot{e}_s}{e_s}\right) \right] \right] \left( p-\rho \,R_d\,T\right) -(p-e_s)\left[ \dfrac{\dot{e}_s^{\prime }}{e_s} -\dfrac{\dot{e}_s^{\prime }\,e_s^{\prime }}{e_s^2}\right] \nonumber \\&\quad +(p-e_s)\left[ \left( \dfrac{\dot{n}}{n}\right) + \left( \dfrac{\dot{e}_s}{e_s}\right) \right] \rho \,R_d\nonumber \\- & {} e_s^{\prime }\left[ \dot{p}-R_d\left( \dot{\rho }\,T+\rho \,\dot{T}\right) +\left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \left[ \dot{p}-R_v\left( \dot{\rho }\,T+\rho \dot{T}\right) \right] \right] \nonumber \\- & {} (p-e_s)\Bigg [\dot{\rho }\,R_d + \dfrac{\rho \,R_d\left[ \dot{p}-R_v\left( \dot{\rho }\,T+\rho \,\dot{T}\right) \right] }{\rho \,R_v\,T-p} + \dfrac{R_v\,\dot{\rho }\left( p-\rho \,R_d\,T\right) }{\rho \,R_v\,T-p}\nonumber \\&\quad + \left[ \dfrac{\rho \,R_v\left( p-\rho \,R_d\,T\right) }{\left( \rho \,R_v\,T-p\right) ^2}\right] \left[ \dot{p}-R_v\left( \dot{\rho }\,T+\rho \,\dot{T}\right) \right] \Bigg ]\Bigg \} , \end{aligned}$$

(51)

where derivates of $e_s$ ($\dot{e}_s$, $e_s^{\prime }$, and $\dot{e}_s^{\prime }$) are deduced in “Appendix D”.

Finally, the second derivative of $\chi _{\text {mol}}$, with respect to temperature, is

$$\begin{aligned} \chi _{\text {mol}}^{\prime \prime }= & {} \left( \dfrac{100}{n\varepsilon }\right) \left( \dfrac{L}{T^2}\right) \Bigg \{\left( \dfrac{2}{R_v\,T}\right) \left[ 1+\left( \dfrac{p-e_s}{e_s}\right) \right] \left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \nonumber \\- & {} \left( \dfrac{2\,L}{R_v^2\,T^2}\right) \left( \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) + \left( \dfrac{\rho }{\rho \,R_v\,T - p}\right) \left[ 1+\left( \dfrac{p-e_s}{e_s}\right) \right] \nonumber \\&\quad \times \left( \varepsilon + \dfrac{p-\rho \,R_d\,T}{\rho \,R_v\,T-p}\right) \Bigg \} . \end{aligned}$$

(52)

Appendix D: The calculation of the derivatives of $e_s$

Equations (50) and (51) in “Appendix C” contain derivatives of $e_s$, which can be calculated based on Eq. (3). Therefore, derivatives with temperature, time and mixed (time and temperature) are given by

$$\begin{aligned}&e_s^{\prime } \equiv \dfrac{de_s}{dT} = \dfrac{e_s\,L}{R_v\,T^2}; \end{aligned}$$

(53)

$$\begin{aligned}&\quad \dot{e}_s \equiv \dfrac{de_s}{dt} = e_s\left( \dfrac{L}{R_v}\right) \left( \dfrac{\dot{T}}{T^2}\right) ; \end{aligned}$$

(54)

$$\begin{aligned}&\quad \dot{e}_s^{\prime } = \dfrac{d}{dt}\left( \dfrac{de_s}{dT}\right) = e_s^2\left( \dfrac{L}{R_v}\right) ^2\left( \dfrac{\dot{T}}{T^4}\right) . \end{aligned}$$

(55)

Appendix E: Data of study cases

Table 1 Table with data of Wells and Impulse Turbines