Thermodynamic Behavior and Heat Transfer in Closed Surge Tanks for Hydropower Plants

The main types of heat transfer are conduction, convection, radiation, and phase change (Moran et al. 2012). Of these processes, radiation and phase change are assumed to be negligible for calculation of closed surge tank behavior. Heat transfer is assumed to be dominated by the combination of convection in the air and conduction through the enclosing rock and water, and this process is modeled with the Newton’s empirical law of cooling [Eqs. (5) and (6)].

The RHT method considers lumped heat transfers through rock and water of the surge tank, and is for this study expanded to separate heat transfer to water (subscript $w$) and rock (subscript $r$) separately in order to consider the individual contribution of each (subscript $a$ is used for air).where $h$ = heat transfer coefficient; and $A$ = boundary surface. The modified version is in the following referred to as the MRHT method.

Heat transfer through convectionisassumedto be natural. According to Bejan (1993), the relative magnitude of natural versus forced convection may be quantified as the ratio of the Grashof number [$\mathsf{Gr}=g\mathrm{\Delta }T{L}^3/({\upsilon }^{2}T)$] divided by the Reynolds number ($\mathsf{R}=\mathrm{UL}/\upsilon$) squared, where $L$ is the characteristic length, $U$ is fluid velocity, and $\upsilon$ is kinematic viscosity.From this relationship, it can be shown that the forced convection is negligible compared to the natural convectionfor normal transient in closed surge tanks ($\mathsf{Gr}/{\mathsf{R}}^2\gg 1$).

For heat transfer from air to water, it is assumed that the water holds constant a temperature due to circulation. The heat transfer coefficient for natural convection from air to water may then be calculated from Incropera and Dewitt (2007)where $\mathsf{Nu}$ = Nusselt number; and $\lambda$ = thermal conductivity for air. For heat transfer from air to rock, it is necessary to account for heat transfer resistance and temperature gradient in the rock. The heat transfer coefficient for air to rock may be calculated from Incropera and Dewitt (2007)where $R_r$ = heat transfer resistance defined in Eq. (10); and $l$ = rock layer thickness. Finally, the resulting model for heat transfer in closed surge tanks in the MRHT method becomes

The Nusselt number ($\mathsf{Nu}$) is the only unknown and is determined from laboratory experiments, field measurements, or empirical relationships. Incropera and Dewitt (2007) suggest the following empirical relationship for turbulent air flow ($\mathsf{Gr}>{10}^8$):where $\mathsf{Pr}=c_p\mu /\lambda$ is the Prantl number; $\mu$ is the dynamic viscosity of the fluid; and $k$ is an empirical constant. For large closed surge tanks, the factor $k$ is individual for walls, roof, and floor. Due to the complexity of measuring and calculating each individual surface, the problem may be simplified by assuming lumped factor $k$ for all surfaces.

In order to account for heating and cooling of the rock mass, Eqs. (13) and (14) solve Fourier’s law in order to account for the propagation of heat in the rock

Given an infinite amount of layers, $dT$ becomes zero and the rock temperature reaches a steady state. The necessary amount of layers for reaching a steady state is found by trial and error.

For comparison against the MRHT method, the equation for calculating the heat transfer in the RHT method is given in Eq. (15) as follows:

The expression is converted from imperial to metric units based on the presentation in Graze (1968), which is applied in the benchmark model WHAMO by U.S. Army Corps of Engineers (1998).

The numerical model is developed with the freeware LVTrans 1.7.11, developed by SINTEF research group, which is based on the MOC. This method solves the equations of continuity and motion, and is applied in numerous studies on pipe and tunnel flow (Joung and Karney 2009; De Martino and Fontana 2012; Zhou et al. 2013a, b). For the present study, the software is expanded by including a closed surge tank module, which solves Eqs. (3), (11), and (12) through Newton’s iteration method. Air temperature is calculated with the ideal gas law, as presented in Eq. (2), and rock temperature is calculated with Eqs. (13) and (14). The rock is modeled as 1D layers, with the following thicknesses (cm): 0.02, 0.02, 0.02, 0.02, 0.02, 0.02, 0.03, 0.05, 0.10, 0.20, and 0.40. The module calculates air pressure, air volume, air temperature, rock temperature, water level, and water flow in the closed surge tank. Singular loss, gravity, and pressure forces are included, while inertia and friction loss are neglected due to low water velocity. All simulations are performed with a time-step 0.1 s.

The prototype for the numerical model is the power plant Jukla (40 MW) in western Norway, which utilizes runoff from the glacier Folgefonna. The power plant has two upper reservoirs at different geodetic levels and is constructed with a closed surge tank. The surge tank is an unlined rock cavern constructed by conventional drill and blasting. The layout of the power plant as modeled with LVTrans 1.7.11 is presented in Fig. 1, and key data from the power plant are shown in Table 1.

Statkraft AS is the owner and operator of the power plant, which has been in operation since 1974. The length of the headrace between Dravladalsvatn reservoir and the junction point is 5,804 m, and the length from Langavatn reservoir to the junction point is 3,320 m. The length from the connection point to the turbines is 724 m.

There are numerous brook inlets in the tunnel system, which are added in the numerical model as a lumped volume.

The data set was collected in May 1979 by Statkraft AS. By switching from the lower upstream reservoir to the higher reservoir, the air pressure in the closed surge tank was doubled during 40 min. The duration of the filling process is mainly governed by filling several connection tunnels and brook inlets.

The water level and air pressure measurements from the closed surge tank are presented in the result section. The temperature in air and rock was not measured. The water temperature in the Jukla waterway was 275 K at the time of measurement.

The turbines were closed during the entire event. Initially, the water flow in the system was zero, the intake gate to Langavatn reservoir was closed, and the pressure in the waterway was governed by the water level in Dravladalsvatn reservoir. The power plant operation is separated into the following main events: (1) the intake gates for Dravladalsvatn reservoir close, (2) the intake gates for Langavatn reservoir are opened slowly in order to fill the dry connection tunnel, (3) the dry tunnel is filled after 40 min, and (4) the system reaches the steady state after 5 h.

Water level measurements are conducted with a mercury U-pipe system as show in Fig. 2. The U-pipe is connected to two steel pipes with a diameter of 6 mm that lead into the closed surge tank. The elevation of the two pipes in the surge tank is known, and the water level is calculated from the pressure difference.

The rock cavern geometry is known and air pressure is calculated from the pressure balance between upper reservoir, water level in the surge tank, and air pressure.

The uncertainties of the significant parameters are $\mathrm{\Delta }{h}_1=\pm 0.005\mathrm{m}$, $\mathrm{\Delta }{h}_2=\pm 0.001\mathrm{m}$, $\mathrm{\Delta }{\rho }_{Hg}=\pm 10\mathrm{kg}/{\mathrm{m}}^3$, and $\mathrm{\Delta }{p}_{Z_o}=1\mathrm{kPa}$. The rest of the parameters have a negligible effect on the uncertainty. The parameter $h_1$ has a larger uncertainty compared to $h_2$ due to water evaporation in the U-pipe. The total uncertainty of the water level and air pressure is $\mathrm{\Delta }{Z}_w=\pm 0.08\mathrm{m}$ and $\mathrm{\Delta }{p}_{\text{air}}=\pm 0.8\mathrm{kPa}$.

Fig. 3 presents the observed and simulated water level in the closed surge tank. For the MRHT method, the empirical factor $k$ is found to be 0.05 (-). For comparison, results obtained with the RHT method as expressed in Eq. (15) are provided. For the polytrophic relationship, simulations are performed with $n=1.4$ (adiabatic), $n=1.2$ (intermittent), and $n=1.0$ (isothermal). The total amount of heat transfer in the MRHT and isothermal simulations is approximately 1.6 GJ. The total amount of heat transferred is 1.4 GJ in the RHT simulation, 0.7 GJ in the intermittent polytrophic simulation, and 0 GJ in the adiabatic simulation.

The gradient of water level rise is high during the filling process and decreases rapidly after the filling process is complete. After completion, the water level increases slowly as heat energy in the air disperses into the enclosing rock and water.

Fig. 4 presents simulated and observed air pressure in the closed surge tank during the slow transient event. The pressure builds up during the filling process, peaks immediately after the filling is complete, and thereafter declines as heat energy dissipates into the surrounding media. Fig. 4 show that the difference in calculated pressure between the different thermodynamic models is limited, as pressure is governed by the upstream water level. In comparison, the water level simulations show larger differences as it is dependent on both pressure and temperature in the air pocket.

As can be seen, the observed data reveal a peak in the pressure immediately after completing the filling process. The simulations also show a peak in the pressure at this point, but at smaller magnitude. The water level does not indicate such a peak, and the cause of the pressure peak therefore needs to be related to another physical phenomena. This is discussed further in the next section.

The simulated air temperatures are compared in Fig. 5. Additionally, rock temperatures at selected depths are presented as calculated with the MRHT method. The air temperatures show a peak at the end of the filling, and thereafter remain constant for the polytrophic simulations. For the RHT and MRHT methods, the air temperature eventually cools down and moves toward equilibrium as heat is transferred to water and rock.

For the polytrophic simulations, the air temperature is highest at the time of the pressure peak (end of the filling process). For the RHT and MRHT simulations, the temperature peaks slightly before, at the time when heat transfer out of the system is equal to heat generation due to work done by the air. The results from the MRHT and adiabatic simulations are approximately equal during the first 20 min. The rock temperature is seen to be affected until 400 mm deep, at which depth the temperature is stable at 275 K during the entire event.