Its dimensionless form is: 2 **2 () sCR s s sf rss Hk CHU tz θ θ θθ τρ ∂∂ =− − + ∂∂ (27) Fig. 9. Comparison of dimensionless energy storage in the ‘washer’, normalized by the ideal maximum energy change in the ‘washer’, due to different methods of solution ( (/2)/ washer i s Bi h d k= =3.0; / 6.0 eq i Dd = ; *2 /[( /2) / ] is tt d α = ; c w = 0.83442) The parameter cluster of /( ) sss kCHU ρ is a dimensionless term. If it is sufficiently large, the axial conduction term in Eq. (27) may not be dropped off. A basic effect of significant axial heat conduction is that it will destroy the thermocline effect—a temperature gradient with hot material being on top of cold. This can lower the thermal storage performance in general. Therefore, to take into account the axial heat conduction effect, a similar correction via the introduction of another factor to the modified heat transfer coefficient is proposed. This results in a new modified heat transfer coefficient of: 11 1/ 1 p s washer c ss hh k Bi w CHU ρ ⎛⎞ ⎜⎟ ⎛⎞ ⎜⎟ = ⎜⎟ + ⎜⎟ ⎝⎠ + ⎜⎟ ⎝⎠ (28) For most thermal storage materials, such as rocks, molten salts, concrete, soil, and sands, the value of /( ) sss kCHU ρ is very small (in the order of 6 110 − × ); while other terms in Eq.(27) are in the order of 1.0. Therefore, the axial heat conduction effect in the thermal storage material in Eq. (27) is negligible. 5.2.7 Application of the model to the storage system with PCM For thermal storage with phase-change involved, the PCM can be enclosed in capsules to form a packed bed as shown in Fig. 2(a), or simply put in a storage tank that has heat transfer tubes inside as shown in Fig. 2(b). The governing equations discussed above are still applicable to the heat transfer at locations where either phase change has not yet occurred or has already been completed. However, at locations undergoing phase change, the energy equations must account for the melting or solidification process (Halawa & Saman, 2011; Wu et al., 2011). The key feature in a melting or solidification process is that the temperature of the material stays constant. Considering the energy balance for the thermal storage material: 2 ()(1) sm f s d hS T T R dt ρεπ Φ −=−Γ − (29) where Γ is the fusion energy of the material, and Φ is the ratio of the liquid mass to the total mass in the control volume of dz. For melting, Φ increases from 0 to 1.0, while for solidification it decreases from 1.0 to 0. Considering the invariant of the temperature of the material during a phase change process, the energy balance equation for HTF is: 2 () ff s mf ff TT hS TT U tz CR ρεπ ∂ ∂ −= + ∂ ∂ (30) Equation (29) and (30) can be reduced to dimensionless equations by introducing the same group of dimensionless parameters: ** 1 () ff mf r tz θ θ θθ τ ∂ ∂ −= + ∂ ∂ (31) * () CR mf r H d dt ψθ θ τ Φ −−= (32) where a new dimensionless parameter ()/ HLs TTC ψ = −Γ is introduced. Since the phase change temperature is known, Eqs. (31) and (32) can be solved separately. 5.3 Numerical methods and solution to governing equations 5.3.1 Solution for the case of no phase change A number of analyses and solutions to the heat transfer governing equations of a working fluid flowing through a packed-bed have been presented in the past (Schumann, 1929; Shitzer & Levy, 1983; McMahan, 2006; Beasley, 1984; Zarty & Juddaimi, 1987). As the pioneering work, Schumann (Schumann, 1929) presented a set of equations governing the energy conservation of fluid flow through porous media. Schumann’s equations have been widely adopted in the analysis of thermocline heat storage utilizing solid filler material inside a tank. His analysis and solutions were for the special case where there is a fixed fluid temperature at the inlet to the storage system. In most solar thermal storage applications this may not be the actual situation. To overcome this limitation, Shitzer and Levy (Shitzer & Levy, 1983) employed Duhamel’s theorem on the basis of Schumann’s solution to consider a transient inlet fluid temperature to the storage system. The analysis of Schumann, and Shitzer and Levy, however, still carry with them some limitations. Their method does not consider a non-uniform initial temperature distribution. For a heat storage system, particularly in a solar thermal power plant, heat charge and discharge are cycled daily. The initial temperature field of a heat charge process is dictated by the most recently completed heat discharge process, and vice versa. Therefore, non-uniform and nonlinear temperature distribution is typical for both charge and discharge processes. To consider a non-uniform initial temperature distribution and varying fluid temperature at the inlet in a heat storage system, numerical methods have been deployed by researchers in the past. To avoid the long mathematical analysis necessary in analytical solutions, numerical methods used to solve the Schumann equations were discussed in the literature by McMahan (McMahan, 2006, 2007), and Pacheco et al. (Pacheco et al., 2002), and demonstrated in the TRNSYS software developed by Kolb and Hassani (Kolb & Hassani, 2006). Based on the regular finite-difference method, McMahan provided both explicit and implicit discretized equations for the Schumann equations. Whereas the explicit solution method had serious stability issues, the implicit solution method encountered an additional computational overhead, thus requiring a dramatic amount of computation time. The solution for the complete power plant with thermocline storage provided by the TRNSYS model in Kolb’s work (Kolb & Hassani, 2006) cites the short time step requirement for the differential equations of the thermocline as one major source of computer time consumption. To overcome the problems encountered in the explicit and implicit methods, McMahan et al. also proposed an infinite-NTU method (McMahan, 2006, 2007). This model however is limited to the case in which the heat transfer of the fluid compared to the heat storage in fluid is extremely large. The present study has approached the governing equations using a different numerical method (Van Lew et al., 2011). The governing equations have been reduced to dimensionless forms which allow for a universal application of the solution. The dimensionless hyperbolic type equations are solved numerically by the method of characteristics. This numerical method overcomes the numerical difficulties encountered in McMahan’s work—explicit, implicit, and the restriction on infinite-NTU method (McMahan, 2006, 2007). The current model yields a direct solution to the discretized equations (with no iterative computation needed) and completely eliminates any computational overhead. A grid-independent solution is obtained at a small number of nodes. The method of characteristics and the present numerical solution has proven to be a fast, efficient, and accurate algorithm for the Schumann equations. The non-dimensional energy balance equations for the heat transfer fluid and filler material can be solved numerically along the characteristics (Courant & Hilbert, 1962; Polyanin, 2002; Ferziger, 1998). Equation (9) can be reduced along the characteristic ** zt = so that: )( 1 Dt D fr r * f θθ τ θ −= (33) Separating and integrating along the characteristic, the equation becomes: ∫∫ −= * fr r f dt)( 1 d θθ τ θ (34) Similarly, Eq.(12) for the energy balance for the filler material is reposed along the characteristic * z =constant so that: )( H dt d fr r CR * r θθ τ θ −−= (35) The solution for Eq. (35) is very similar to that for Eq. (33) but with the additional factor of CR H . The term CR H is simply a fractional ratio of fluid heat capacitance to filler heat capacitance. Therefore, the equation for the solution of r θ will react with a dampened speed when compared to f θ , as the filler material must have the capacity to store the energy being delivered to it, or vice versa. Finally, separating and integrating along the characteristic for Eq.(35) results in: ∫∫ −−= * fr r CR r dt)( H d θθ τ θ (36) There are now two characteristic equations bound to intersections of time and space. A discretized grid of points, laid over the time-space dimensions will have nodes at these intersecting points. A diagram of these points in a matrix is shown in Fig. 10. In space, there are i = 1, 2, . . . ,M nodes broken up into step sizes of * z Δ to span all of * z . Similarly, in time, there are j = 1, 2, . . . ,N nodes broken up into time-steps of * t Δ to span all of * t . Looking at a grid of the ϑ nodes, a clear picture of the solution can arise. To demonstrate a calculation of the solution we can look at a specific point in time, along * z where there are two points, 1,1 ϑ and 1,2 ϑ . These two points are the starting points of their respective characteristic waves described by Eq. (33) and (36). After the time * t Δ there is a third point 2,2 ϑ which has been reached by both wave equations. Therefore, Eq. (34) can be integrated numerically as: ∫∫ −= 2,2 1,1 2,2 1,1 * fr r f dt)( 1 d ϑ ϑ ϑ ϑ θθ τ θ (37) The numerical integration of the right hand side is performed via the trapezoidal rule and the solution is: * ffrr r ff t 22 1 1,12,21,12,2 1,12,2 Δ θθθθ τ θθ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + =− (38) where 1,1 f θ is the value of f θ at 1,1 ϑ , and 2,2 f θ is the value of f θ at 2,2 ϑ , and similarly so for r θ . The integration for Eq. (36) along * z =constant is: ∫∫ −−= 22 12 22 12 , , , , * fr r CR r dt)]( H [d ϑ ϑ ϑ ϑ θθ τ θ (39) The numerical integration of the right hand side is also performed via the trapezoidal rule and the solution is: * ffrr r CR rr t 22 H 1,22,21,22,2 1,22,2 Δ θ θ θ θ τ θθ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + −=− (40) Equations (38) and (40) can be reposed as a system of algebraic equations for two unknowns, 2,2 f θ and 2,2 r θ , while f θ and r θ at grid points 1,1 ϑ and 1,2 ϑ are known. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ +− −+ r * CR r r * CR f r * r r * f r f r * CR r * CR r * r * 2 tH 1 2 tH 2 t 2 t 1 2 tH 1 2 tH 2 t 2 t 1 1,21,2 1,11,1 2,2 2,2 τ Δ θ τ Δ θ τ Δ θ τ Δ θ θ θ τ Δ τ Δ τ Δ τ Δ (41) Cramer’s rule (Ferziger, 1998) can be applied to obtain the solution efficiently. It is important to note that all coefficients/terms in Eq.(41) are independent of * z , * t , f θ , and r θ , thus they can be evaluated once for all. Therefore, the numerical computation takes a minimum of computing time, and is much more efficient than the method applied in references (McMahan, 2006, 2007). Fig. 10. Diagram of the solution matrix arising from the method of characteristics From the grid matrix in Fig.10 it is seen that the temperatures of the filler and fluid at grids 1,i ϑ are the initial conditions. The temperatures of the fluid and filler at grid 1,1 ϑ are the inlet conditions which vary with time. The inlet temperature for the fluid versus time is given. The filler temperature (as a function of time) at the inlet can be easily obtained using Eq.(12), for which the inlet fluid temperature is known. Now, as the conditions at 1,1 ϑ , 2,1 ϑ , and 1,2 ϑ are known, the temperatures of the rocks and fluid at 2,2 ϑ will be easily calculated from Eq.(41). Extending the above sample calculation to all points in the ϑ grid of time and space will give the entire matrix of solutions in time and space for both the rocks and fluid. While the march of * z Δ steps is limited to 1z * = the march of time * t Δ has no limitation. The above numerical integrations used the trapezoidal rule; the error of such an implementation is not straightforwardly analyzed but the formal accuracy is on the order of )t(O 2* Δ for functions (Ferziger, 1998) such as those solved in this study. 5.3.2 Solutions for the case with phase change For the governing equations of the phase change case, the adopted convention of having the z-direction coordinate always follow the flow direction is preserved, such that for heat charging, z=0 is for the top of a tank, and for heat discharging, z=0 is for the bottom of a tank. The two governing equations (Eq. (31) and Eq.(32)) for the phase change process can be discretized using finite control volume methodology: ** * ** ** ** () () () ( 1) () ** 1 () tt t tt tt fi fi fi fi tt mfi r tz θθθθ θθ τ +Δ +Δ +Δ − +Δ −− −= + ΔΔ (42) ** * ** () * () tt t tt CR i i mfi r H t ψθ θ τ +Δ +Δ Φ −Φ −−= Δ (43) From Eq.(42) the fluid temperature ** () tt fi θ + Δ can be solved, which is then used in Eq. (43) to solve for the fusion ratio ** tt i + Δ Φ . The procedures for finding the solution of phase change problem are as follows: 1. Solve the non-phase-change governing equation analytically using Eq.(12) for the phase change material for the inlet point. 2. Monitor the temperature at each time step as given by Eq.(12), and see if the temperature at a time step is greater than the fusion temperature, if yes, the solution for that and subsequent time steps are to be solved using the phase change equation (Eq. (43) 3. For each time step solved using Eq (43), monitor the fusion ratio, Φ ; when it becomes larger than 1.0 then the solution for that and subsequent time steps are to be solved using the non-phase-change governing equation (Eq.(12)) for the remainder of the required time. 4. March a spatial step forward and repeat all of the above steps. However, now in part (1) of this procedure, Eq.(41) must be used to solve the temperatures of both the fluid and PCM for time steps before phase change starts; and also in part (3) of this procedure Eq.(41) should be used to solve the temperatures of the fluid and PCM for steps after the phase change is over. The repetition of parts (1) to (3) of this procedure is to be continued until all the spatial steps are covered. 6. Results from simulations and experimental tests 6.1 Numerical results for the temperature variation in a packed bed The first analysis of the storage system was done on a single tank configuration of a chosen geometry, using a filler and fluid with given thermodynamic properties. The advantage of having the governing equations reduced to their dimensionless form is that by finding the values of two dimensionless parameters ( r τ and CR H ) all the necessary information about the problem is known. The properties of the fluid and filler rocks, as well as the tank dimensions, which determined r τ and CR H for the example problem, are summarized in Table 7. The numerical computation started from a discharge process assuming initial conditions of an ideally charged tank with the fluid and rocks both having the same high temperature throughout the entire tank, i.e. 1 fs θ θ = = . After the heat discharge, the temperature distribution in the tank is taken as the initial condition of the following charge process. The discharge and charge time were each set to 4 hours. The fluid mass flow rate was determined such that an empty (no filler) tank was sure to be filled by the fluid in 4 hours. With the current configuration, after five discharge and charge cycles the results of all subsequent discharge processes were identical—likewise for the charge processes. It is therefore assumed that the solution is then independent of the first-initial condition. The data presented in the following portions of this section are the results from the cyclic discharge and charge processes after 5 cycles. ε r τ CR H H R t 0.25 0.0152 0.3051 14.6 m 7.3 m Fluid (Therminol ® VP-1 ) properties: T H =395 °C; T L =310 °C; f ρ =753.75 kg/m 3 ; C f =2474.5 J/(kg K); k f =0.086W/(m K); m  =128.74 kg/s; f μ =1.8 4 10 Pa s − × ⋅ ; Filler material (granite rocks) properties: s ρ = 2630 kg/m3; C s =775 J/(kg K); k s =2.8 W/(m K); d r = 0.04 m; Table 7. Dimensions and parameters of a thermocline tank (Van Lew et al., 2011) Fig. 11. Dimensionless fluid temperature profile in the tank for every 0.5 hours Shown in Fig. 11 are the temperature profiles in the tank during a discharge process, in which cold fluid enters into the tank from bottom of the tank. The location of * 0z = is at the bottom of a tank for a discharge process. The temperature profile evolves as discharging proceeds, showing the heat wave propagation and the high temperature fluid moving out of the storage tank. The fluid temperature at the exit ( * 1z = ) of the tank gradually decreases after 3 hours of discharge. At the end of the discharge process, the temperature distribution along the tank is shown in Fig. 12. At this time the fluid and rock temperatures, f θ and s θ respectively ( s θ is denoted by r θ when the filler material is rock), in the region with * z below 0.7 are almost zero, which means that the heat in the rocks in this region has been completely extracted by the passing fluid. In the region from * 0.7z = to * 1.0z = the temperature of the fluid and rock gradually becomes higher, which indicates that some heat has remained in the tank. Fig. 12. Dimensionless temperature distribution in the tank after time t * = 4 of discharge (Here r θ is used to denote s θ , as rocks are used as the storage material in the example). Fig. 13. Dimensionless temperature distribution in the tank after time t * = 4 of charge (Here r θ is used to denote s θ , as rocks are used as the storage material in the example) A heat charge process exhibits a similar heat wave propagation scenario. The temperature for the filler and fluid along the flow direction is shown in Fig. 13 after a 4 hour charging process. During a charge process, fluid flows into the tank from the top, where * z is set as zero. It is seen that for the bottom region ( * z from 0.7 to 1.0) the temperatures of the fluid and rocks decrease significantly. A slight temperature difference between heat transfer fluid and rocks also exists in this region. Fig. 14. Dimensionless temperature histories of the exit fluid at z * = 1 for charge and discharge processes The next plots of interest are the variation of f θ at * 1z = as dimensionless time progresses for a charging or discharging process. Figure 14 shows the behavior of f θ at the outlet during both charge and discharge cycles. For the charge cycle, f θ begins to increase when all of the initially cold fluid has been ejected from the thermocline tank. For the present thermocline tank, the fluid that first entered the tank at the start of the cycle has moved completely through the tank at * 1t = , which also indicates that the initially-existing cold fluid of the tank has been ejected from the tank. Similarly, during the discharge cycle, after the initially-existing hot fluid in the tank has been ejected, the cold fluid that first entered the tank from the bottom at the start of the cycle has moved completely through the tank at * 1t = . At * 2.5t = , or t=2.5 hours, the fluid temperature f θ starts to drop. This is because the energy from the rock bed has been significantly depleted and incoming cold fluid no longer can be heated to f θ = 1 by the time it exits the storage tank. The above numerical results agree with the expected scenario as described in section 4. To validate the above numerical method, analytical solutions were obtained using a Laplace Transform method by the current authors (Karaki, et al, 2010), which were only possible for cases with a constant inlet fluid temperature and a simple initial temperature profile. Results compared in Fig. 15 are obtained under the same operational conditions—starting from a [...]... of heat transfer fluid and thermal storage material, as well as the packing porosity in a thermocline tank The design analysis using the general charts provided in the present study will include the following steps: 1 Select a minimum required volume for a thermocline tank using Eqs (1) and (3) 2 Choose a radius, R, and the corresponding height, H, from the minimum volume decided in step (1) Using... die-casting dies and moulds are made of hard materials to increase their lifetime The recent developments in cutting tools for turning and milling and the processes of high speed machining allow to machine harder materials than before, but EDM still remains the only available process for machining many hard materials One of the important areas of EDM application is in mould and dies making industries... selected as the dielectric fluid in the present study 424 Developments in Heat Transfer The product of machining, the tiny debrtis should be removed from the machining zone in order to avoid sparks between the electrode and the debrtis This is done by effective flushing system of the dielectric fluid to remove the debrits away from the machining zone Suction flushing of dielectric fluid is preferred... minimal effect on the material removal rate (MRR) EDM is gaining popularity due to the following reasons: Machining of Complex Shapes: Complex cavities can often be machined without difficulties by EDM It can be used to machine material with small or odd shaped holes, a large number of holes, holes having shallow entrance angles, intricate cavities, or intricate contour Adaptability to Micro-Machining:... pp 883 – 893 Incropera, F.P., and DeWitt, D.P., 2002, Introduction to Heat Transfer, fourth ed John Wiley and Sons, Inc Jeffreson, C.P., 1972, Prediction of breakthrough curves in packed beds: 1 applicability of single parameter models, American Institute of Chemical Engineers, 18(2), pp 409416 Karaki, W., Van Lew, J.T., Li, P.W., Chan, C.L., Stephens, J., 2010, Heat transfer in thermocline storage... radius which is dependent on the current intensity and pulse on-time duration rsp = 2040 × (I)0.43 × (ton )0.44 (6) 422 Developments in Heat Transfer where rsp is heat input radius in µm, I is current intensity in A and ton is the spark time in µs In the work of Bulent et al., 2006 and Ozgedik & Cogun, 2006 they used Eq 7 to estimate the energy released due to a single spark Esp = I ×V × ton (7) where... Journal of Solar Energy Engineering, MAY 2 011, Vol 133/021003 Wu, S.M., Fang, G.Y., Liu, X 2 011, Dynamic discharging characteristics simulation on solar heat storage system with spherical capsules using paraffin as heat storage material, Renewable Energy, 36 (2 011) 119 0 -119 5 Wyman, C., Castle, J., Kreith, F., 1980, A review of collector and energy storage technology for intermediate temperature applications,... assuming that the heat from the plasma channel is transferred to the workpiece or electrode by conduction only About 90% of the total energy liberated is conducted to the discharge gap and it was distributed equally between the electrode and workpiece DiBitonto et al, 1989 approximated the heat source by a point instead of a disk for conducting heat to the interior The point heat source can result in. .. is applied to the workpiece; it eliminates the mechanical stresses, chatter and vibration problem during machining (Ho and Newman, 2003) This allows the effective machining of very thin, delicate and fragile workpiece without distortion The large cutting force of the materials removal processes on the convectional machines is absent in EDM The electrode is the main part of the EDM process, which is connected... machine a particular cavity Usually the electrode is connected to the negative terminal and the workpiece is connected to the positive terminal of the power supply which is called direct polarity However, depending on the combination of the electrode and the work materials, reverse polarity is also sometimes used The machine used in the present study is shown in Fig 3 Fig 3 EDM die sinking machine . phase-change involved, the PCM can be enclosed in capsules to form a packed bed as shown in Fig. 2(a), or simply put in a storage tank that has heat transfer tubes inside as shown in Fig. 2(b) thermocline tank (Van Lew et al., 2 011) Fig. 11. Dimensionless fluid temperature profile in the tank for every 0.5 hours Shown in Fig. 11 are the temperature profiles in the tank during a. well as the packing porosity in a thermocline tank. The design analysis using the general charts provided in the present study will include the following steps: 1. Select a minimum required