Comparing Meshless Local Petrov–Galerkin and Artificial Neural

Renewable and Sustainable Energy Reviews 43 (2015) 521–529

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Renewable and Sustainable Energy Reviews

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Comparing meshless local Petrov–Galerkin and artiﬁcial neural networks methods for modeling heat transfer in cisterns

M. Razavi a, A.R. Dehghani-sanij a, M.R. Khani b,n, M.R. Dehghani c a Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada A1B 3X5 b Department of Environmental Health Engineering, Faculty of Health, Islamic Azad University, Medical Sciences Branch, Tehran, Iran c Department of Architectural Engineering, Science and Arts University, Yazd Branch, Yazd, Iran article info abstract

Article history: Long-term underground cold-water cisterns had been used in old days in the hot and arid regions of Received 27 February 2014 Iran. These cisterns provide cold drinking water during warm seasons for local communities. In this Received in revised form paper, the thermal performance of an underground cold-water cistern during the withdrawal cycles in 15 September 2014 warm seasons is modeled. The cistern is located in the central region of Iran in the city of Yazd. Two Accepted 1 October 2014 approaches are used to model the heat transfer in the mentioned cistern. The first approach is meshless local Petrov–Galerkin (MLPG) method with unity test function and the second approach is artificial Keywords: neural networks (ANN). For the ANN method, the multi layers perceptron (MLP) feed-forward neural Cistern network training by back propagation algorithm is used. Both methods are compared and a good Thermal stratification agreement is observed between the MLPG and ANN results. The results show a stable thermal Numerical stratification in the cistern throughout the withdrawal cycle. The thermal stratification is linear in Meshless local Petrov–Galerkin (MLPG) Artificial neural networks (ANN) lower areas and exponential in upper areas. The exponential trend in the upper area is because of several Energy factors such as: thermal exchange among the upper layers of water and the domed roof, transfer of mass and evaporation due to entry air from the wind towers. & 2014 Published by Elsevier Ltd.

Contents

1. Introduction...... 521 2. Casestudy...... 523 3. MLPG modeling ...... 524 4. ANN modeling ...... 525 5. Results and discussions ...... 526 6. Conclusion and suggestions ...... 528 References...... 528

1. Introduction days, cold nights even in the summer time, low rain and fast evaporation because of the sun and wind, large temperature Climate conditions have always inﬂuenced the life of human difference between shade and sun and the problem of low beings. Cities and villages in hot and arid regions have special available water [1]. People of these regions had invented special architectures to tolerate the climate conditions. Hot and arid architectural passive cooling systems to contradict these weather regions are known by the following speciﬁcations: hot and sunny conditions [2,3] and these systems were widely investigated during the recent decades [4–13]. One of the passive cooling systems is the cold thermal energy

n storage system (CTES). This system employs numerous methods to Corresponding author. E-mail addresses: [email protected] (M. Razavi), store cold thermal energy. For instance, long-term cold thermal [email protected] (A.R. Dehghani-sanij), [email protected] (M.R. Khani). storage systems take advantages of seasonal climate variation in http://dx.doi.org/10.1016/j.rser.2014.10.008 1364-0321/& 2014 Published by Elsevier Ltd. 522 M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529

Nomenclature TH domed roof temperature (1C) w weight matrix of each layer, humidity ratio of air = B length (m) kg kg fi = 1 z vertical axis CPa air speci c heat kJ kg C fi = 2 1 ha heat transfer coef cient W m C = hfg latent heat of vaporization of water kJ kg Greek symbols fi = 2 1 hm mass transfer coef cient W m C H height (m) α thermal diffusivity m2=s = 1 kw thermal conductivity of water W m C ε emissivity = 1 ks thermal conductivity of soil W m C Ω domain 3 = 2 mv evaporation rate kg m s Γ domain boundary r radius axis ϕ interpolation function t time σ Stefan–Boltzmann constant T temperature (1C)

Ta ambient air temperature (1C)

order to store cold energy during the winter time for using in the air-conditioning system is used during the off-peak periods. The summer time. A comprehensive review on the concept of CTES, chilled water was used throughout the peak time. He showed the application of different types, and their advantages and disadvan- efficiency of the system is almost 90 %. tages had been conducted by Dincer et al. [14] and Saito [15]. Moreover, the thermal stratification in a long-term under- CTES mostly use water as the storage medium. They store cold ground cold-water cistern was studied experimentally by Dehghan water in cold seasons for using during the warm seasons. For this and Dehghani [22,23]. They concluded that the radiation heat purpose, a thermal stratification should develop in the water; exchange between the water surface and ceiling and the con- therefore, the influence of various design parameters on the vective heat and mass transfer from the water surface induced by thermal stratification is the most important factor investigated the airflow have the most impact on the thermal stratification of by numerous researchers. Nelson et al. [16] investigated experi- the cistern. mentally the effects of storage tank dimensions, the physical Hooshmand Aini [24] numerically studied the effect of the properties of walls and mixing effects of fluid flow during charging domed roof and wind towers in the structure of traditional and discharging cycles on the thermal stratification of water in a cisterns of Iran for cooling the water. They studied the water vertical cylinder storage tank. Their results show that by increas- cooling process of cistern in different conditions including differ- ing the length to the diameter ratio of the tank, the degradation of ent wind velocities and modeled the air flow pattern in the cistern the thermal stratification is minimized. However, this effect is using FLUENT commercial software package. Also, Najafi et al. [25] negligible for the aspect ratios more than three. They reported that numerically investigated the fluid flow inside and around the the fluid flow during charging and discharging cycles reduce the cisterns using the FLUENT commercial software package. They stability of the established thermal stratification inside of the investigated the effect of different parameters including the storage tank. Also, they mentioned that using insulation around diameter of the domed roof, wind velocity, inlet and outlet the tank improves the stability of the thermal stratification. geometry of wind tower and its elevation. They obtained the best Borst and Aghamir [17] considered a water storage tank with geometry for having the best performance of cisterns. Khani et al. the approximate volume of 35 m3 that conserved cold water for [26] experimentally studied the performance of a long term cold half of the year. They compared two methods for cooling the water cistern. They considered its effect in the reduction of fuel water. The first method was blowing the cold ambient air in consumption and environmental pollutants. winter over the surface of the water and the second method was Moreover, the history, architecture and functionality of cisterns using a solar collector at night in summer. They showed the first in different regions of Iran were studied by some researchers method is more efficient. [27–39]. The architecture of cisterns in different areas represents The other approach that used in the CTES is cooling the water the architectural identity of the area. Masarrat and Dehghani [40] in storage tanks using the electrical power in the low demand period for using as a cooling system in the time of high-demand for electrical power. Truman et al. [18] were studied one of these water storage tanks using the finite difference method (FDM). Bahadori et al. [19] investigated the thermal characterization of a long-term chilled water cistern in hot and arid regions by the finite difference method assuming a simplified thermal boundary condition at the water surface. Their results show a developed thermal stratification in the cistern. Also, the temperatures of the upper layers of water were a function of the surrounding air temperature and the temperature of the lower layers in the bottom of the cistern remained lower than the ambient temperature. Aghanajafi et al. [20] used the computational fluid dynamics (CFD) to model the thermal behavior of a lake. They showed the temperature and density distributions of water are not stable in the layer of water close to the water surface. Al-Marafie [21] experimentally studied the thermal stratification of a chilled water 3 storage tank with the volume of 7.5 m . For cooling the water, an Fig. 1. A city cistern in Yazd with six wind towers. M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529 523 have classified the cisterns based on criterions such as: application, the water collection rate, the method of water collection, the arrangement of the steps and other spaces, the accessibility method to water, the shape of the reservoir and the dome- coverage, the method of the construction, their position in towns and neighborhoods, size, the method of cooling and ventilating, and the kind of decorations and the material used for construction. They categorized the cisterns based on their applications as follows: city cisterns (Fig. 1), village cisterns (Fig. 2), private

Fig. 4. An en route cistern in Maybod.

cisterns, desert cisterns, mountain cisterns (Fig. 3), citadel cisterns, caravanserai cisterns and en route cisterns (Fig. 4). As mentioned, the cisterns are investigated in different point of views including its capability for water cooling. The heat transfer analysis of cisterns was studied experimentally, analytically and numerically [19–23]. In recent years, neural networks are widely used in heat transfer problems [42–49]. This is a useful method when there is lack of information about the internal processes and boundary conditions of a heat transfer problem. Due to these advantages, neural networks were used for modeling the thermal behavior of cisterns. This method can model the thermal stratiﬁ- cation of the cistern precisely [50–52]. Fig. 2. A village cistern in Zein Abad Village near the route of Yazd to Taft. In this paper, the thermal performance of a long-term underground cold-water cistern are investigated using meshless local Petrov–Galerkin (MLPG) and artiﬁcial neural networks (ANN) methods. In the MLPG method, heat transfer equation in the cistern is solved numerically with the unity test function. Then, the cistern is modeled using multi layers feed-forward neural network training by back propagation algorithm with various calibration methods and structures. Finally, results of both methods are compared with the observations of Dehghan and Dehghani [22,23].

2. Case study

In this paper, an underground cold-water cistern that locates in the Yazd city in the hot and arid central region of Iran is considered. The name of this cistern is Koocheh Biouk Aub-anbar, built in 1835 (Fig. 5). The storage tank of the cistern is a cylinder with the diameter and height of 12 m. The cistern has a domed roof and four wind towers at four corners. The height and cross- sectional area of each wind tower are 10 m and 1 m2, respectively. Cistern’s water is extracted through a tap that locates in the cistern’s wall in 0.9 m above the bottom. A schematic diagram of the cistern is demonstrated in Fig. 6. For measuring the thermal behavior of the cistern, an experimental study was done in winter (Jan. 2002) [23]. The cistern loaded from a nearby well with 15 1C water. It should be mentioned that 2=5 of the cistern was already filled with the water. The water inlet to the cistern was through an aperture on the top of the tank. It caused the water mixed well and reaches a uniform temperature through the cistern. The storage system was left intact until mid April. Then ten days after, the temperature distribution of the tank was measured every 10 days at 8 am and 8 pm. To measure the temperature distribution in vertical direction, 36-temperature sensors, each 0.3 m apart, were placed Fig. 3. A mountain cistern in Fars [41]. inside the tank. Two twenty-channel digital temperature displays 524 M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529 were used to collect data. The cistern was provided drinkable cold- symmetry of the cistern (Fig. 6). Some assumptions are used in this water for about 1000 local residents during six months. In this modeling. Natural convection is assumed negligible inside of the time period, average daily consumption of each person is about cistern. The other assumption is that just the conduction heat 5.5 l [22,23]. In the following sections, the thermal behavior of the transfer exists between the cistern and the surrounding soil. cistern is modeled using MLPG and ANN. Therefore, the governing equation of the system is, ∂2T 1 ∂T ∂2T 1 ∂T þ þ ¼ ð1Þ ∂r2 r ∂r ∂z2 α ∂t 3. MLPG modeling The boundary conditions of the system are as follows: In this section, the MLPG method is used to obtain thermal ∂ T ¼ ð Þ stratification in the cistern and the surrounding soil. The heat ∂ 0 2 r ðB;z;tÞ transfer is modeled in a two dimensional r–z plane because of the ∂ T ¼ ð Þ ∂ 0 3 r ð0;z;tÞ ∂ T ¼ ð Þ ∂ 0 4 z ðr;0;tÞ The last boundary condition that is along the plane of water surface is nonlinear because of the radiation heat transfer between the water surface and the domed roof. Also, convective heat transfer and evaporation exist over the water surface because of the air flow that comes through the wind towers. Considering the mentioned effects, the last boundary condition can be written as follows [53]: ∂ ðm 3 h h T εσT4 Þ ð ; ; Þ εσ 4ð ; ; Þ T ¼ v fg a a H þhaT r H t þ T r H t r r ∂ 0 r B1 z ðr;H;tÞ kw kw kw ð5Þ

Tðr; H; tÞ¼Ta B1 rr rB2 ð6Þ

In the above analysis, the domed roof temperature (TH)is assumed to be equal to the ambient air temperature (Ta). The outer boundary of the soil is far from the cistern; therefore, this boundary can be assumed an adiabatic boundary condition. Also, the initial temperature of the water and soil are equal to 15 1C and 18 1C, respectively. Fig. 7 shows computational domain and boundary conditions of the cistern and the surrounding soil. The MLPG method is used as a numerical technique for modeling the thermal behavior of the system. By this numerical method, the temperature distributions in the cistern and the surrounding soil are obtained simultaneously. For this purpose, a collection of points is generated in the computational domain (Fig. 8) [54]. The distribution of points can be quite arbitrary. Fig. 5. The view of the underground cold-water cistern. Subsequently, a control volume is generated around each point.

Fig. 6. Schematic representation of the underground cold-water cistern. M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529 525

approximated within Ωi by the following equation [55]:

ð Þ ! 9 i ð ; Þ¼ ∑ ϕðiÞ ð ÞðÞ T r t j Tj t 9 j ¼ 1

! ϕðiÞð Þ; ¼ ð Þ In this equation, j r j 1 1 9 are the usual biquadratic

interpolation functions (i.e. Lagrange polynomials) and Tj; j ¼ 1ð1Þ9 are the nodal values of the unknown temperature at the points. ðiÞ ! Substituting the approximation T ð r ; tÞ into Eq. (8) yields the

discretized equation for Ωi. It should be noted that the discretized equations of the control volumes in the water and the surrounding soil are similar. The only difference is in the respective transport properties. In this research, a domain which consists of 24 24 uniform point distribution in the cistern and the surrounding soil is chosen. Using the same procedure for every control volume yields the system of the discretized equations for all the points within the computational domain. Solving the system of algebraic equations Fig. 7. A schematic representation of the computational domain and boundary gives the unknown temperature at the points during the with- conditions. drawal cycle. The time step of this simulation is assumed 10 days. In each time step, the water level and the properties of the ambient air are assumed to remain constant. Due to small amount of daily water withdrawal in comparison with the capacity of the cistern, the water level does not change considerably in each time step and it shifts to a new position for the next time step. Also, the pervious time step temperature distribution is used as the initial condition for the new time step.

4. ANN modeling

In this section, the thermal behavior of the cistern is modeled using artiﬁcial neural networks (ANN). The proposed ANN method is multi layers perceptron (MLP) feed-forward neural network training by back propagation algorithm. The transfer function that is used in the hidden layer of the network is tangent sigmoid and the transfer function of output layer is linear. The number of neurons in the hidden layer is determined with trial and error process. Initial weights of the network are determined by random numbers with unique distribution. Also, the behavior of variables and their correlation may change and a ﬁx learning rate may not be optimized. Therefore, the learning rate is Fig. 8. A schematic representation of the computational domain, distribution of varying with time and its upper limit is set to 102. Also, the points, and a typical control volume generated around node i for heat transfer in maximum performance function is chosen based on the learning the cistern. rate values. The maximum performance function is the highest acceptable limit of the performance function that is the sum of

A typical rectangular control volume Ωi containing nine points is squared error between modeling and experimental values. This shown in Fig. 8. Eq. (1) is multiplied by the weight function w, and function should be satisfied in each sequence. the resulting expression is integrated over Ωi. After integration by Inputs of the network are as follows: ambient temperature in a parts, the following equation is obtained for a typical control specific date, level of considered points from the bottom of the volume Ωi, cistern, and considered date that is specified for measuring the Z Z Z ambient temperature. Output of this network is the temperature w ∂T ! ! ∂T dΩ ¼ ∇ w ∇ TdΩþ w dΓ ð7Þ in the specified level and at the specified date. For training of the Ω α ∂t Ω Γ ∂n i i i ANN, data of Refs. [22,23] are used. Eventually, MLP networks with various sizes have been used to Γ is the boundary of Ω and n is the direction of outward normal to i i obtain the temperature profile of the cistern. Among various MLP Γ . Choosing unity as the weight function in Eq. (7) yields the i networks, a network with the training function of gradient descent following equation for the control volume [55]: method with variable learning rate and momentum and 20 Z Z ∂ ∂ neurons in the hidden layer presents precise thermal stratification 1 T Ω ¼ T Γ ð Þ α ∂ d ∂ d 8 in the cistern (Fig. 9). In Fig. 10, the differences of experimental Ωi t Γi t temperatures and ANN results in different days and different

To obtain the discretized equation of the control volume Ωi heights are shown. As it is clear, the difference is less than 0.5 1C that contains nine points, the unknown temperature ﬁeld is in most of the points. 526 M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529

Fig. 11. Variation of the temperature of the bottom and top layers of the water and the average ambient temperature during the discharge period.

Fig. 9. Scatter plot for modeling values respect to experimental values, training function is gradient descent with variable learning rate and momentum (20 neurons in hidden layer).

Fig. 12. The 24-h variation of the ambient temperature.

Fig. 10. The difference between experimental data and ANN results. temperature at the bottom and top layers of the cistern during the entire withdrawal cycle are shown in Fig. 11. It is clear that the 5. Results and discussions water temperature at the bottom layer where the tap is located varies from 11.9 to 13.1 1C. Also, the maximum and the minimum The thermal stratification of the cistern is obtained using the temperatures of the top layer during this time are equal to 18 1C MLPG method with unity test function. Subsequently, the thermal and 15 1C, respectively. Moreover, the 24-h variation of the behavior of the cistern is modeled using artificial neural networks ambient temperature for selected days during the withdrawal (ANN). The results are compared with the temperature distribu- cycle is depicted in Fig. 12. It is clear that the ambient temperature tions that were obtained through the experimental research. fluctuates between 30 1C and 42 1C in this time period. For solving the algebraic equations and obtaining the temperature Fig. 13 shows a comparison between the MLPG, ANN and the distribution, the following parameters and transport properties have experimental results for the temperature distribution at different been used. The soil density, thermal conductivity and its thermal levels inside the cistern for selected days during the withdrawal diffusivityinhotandaridregionofYazdhavebeenassumedtobe cycle. It is observed that the difference of obtained temperature equal to 103 kg=m 3,0:25 W=m2 1C and 2:58 10 7 m2=s, profiles from MLPG, ANN, and experimental methods are very low. respectively [19]. The water properties are kw ¼ 0:56 W=m 1C, The results show that a stable thermal stratification is developed 7 2 αw ¼ 1:37 10 m =sandhfg ¼ 2480 kJ=kg. The outside ambient and preserved in the cistern throughout the withdrawal cycle. Two temperature data are obtained from the Yazd Weather Bureau. The different regions can be distinguished in the temperature profile. A heat transfer coefficient, ha, for May and October is equal to 3.8 and main portion of the profile, beginning from the bottom of cistern up 1.25 W=m2 1C,respectively[19]. Linear interpolation is used for the to approximately 2=3 of the water height, is linear; while, in the months in between. The rate of evaporation from the water surface is upper region, i.e., close to the water surface, the temperature varies mv1 ¼ hm ðws waÞ by defining the mass transfer coefficient as exponentially. The deviation of the upper portion of temperature ha hm ¼ [19]. The humidity ratio of the ambient air and saturated distribution from linear profile is due to radiation heat exchange with Cpa air are wa and ws,respectively. the domed roof and evaporative heat and mass transfer at the water By substituting the above parameters and solving the system of surface. It is also observed that during the withdrawal cycle when the algebraic equations, the unknown temperature at the points outside temperature reaches as high as 42 1C, cold-water with during the withdrawal cycle are obtained. The variations of water temperature varying from 12 to 13 1C is available from the cistern. M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529 527

Fig. 13. Comparison between the MLPG, ANN and experimental methods for modeling the temperature distribution of the cistern (a) June, (b) July, (c) August.

Fig. 14. Mean daily temperatures of water and the soil surrounding the cistern [19]. 528 M. Razavi et al. / Renewable and Sustainable Energy Reviews 43 (2015) 521–529

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