Method for controlling set temperature of electric storage water heater based on three-dimensional fluid heat transfer

Abstract

A method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer includes: constructing a water temperature change function for the electri

Description

TECHNICAL FIELD [0001 ] The present disclosure relates to the technical field of electric water heaters, and specifically, to a method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer. BACKGROUND

[0002] Electric storage water heaters are widely used in households due to their ease of installation and affordable price, which provides great convenience for family bathing. However, inappropriate temperature settings may result in two problems: [00031 (1) Setting the temperature too high can result in excessive electricity waste, especially given the prevalent use of electric water heaters, leading to incalculable energy wastage.

[0004] (2) Conversely, setting the temperature too low can cause discomfort during bathing, potentially impacting the user's health.

[0005] At present, most electric storage water heaters on the market adopt manual adjustment or constant temperature control for temperature setting, which may not effectively achieve a reasonable and efficient set temperature that allows users to enjoy comfortable bathing while maximizing electricity savings. Therefore, in view of the temperature setting issue with electric storage water heaters, it is crucial to determine an appropriate set temperature that minimizes electricity consumption of the electric water heaters while ensuring a sufficient water temperature during the bathing process. SUMMARY

[0006] In order to overcome the shortcomings in the prior art, an objective of the present disclosure is to provide a method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer. The present disclosure provides a control scheme for the set temperature of the electric water heater, that is, calculating the set temperature of the electric water heater, which can not only save electricity but also meet the temperature requirements for user bathing under the condition that the electric water heater is continuously turned on.

[0007] To solve the problems existing in the prior art, the technical solution of the present disclosure is as follows:

[0008] According to a first aspect, the present disclosure provides a method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer, 1 including the following steps: [00091 step (1): constructing a water temperature change function for the electric water heater; [00101 step (2): constructing a three-dimensional fluid heat transfer model; [00111 step (3): constructing an energy balance equation and obtaining a temperature iteration equation of the model; [00121 step (4): establishing a continuity equation; and [00131 step (5): traversing and solving the continuity equation to obtain a minimum set temperature. [00141 According to a second aspect, the present disclosure provides an electronic device, including a processor and a memory. The memory stores machine-executable instructions executed by the processor, and the processor executes the machine-executable instructions to implement the method. [00151 According to a third aspect, the present disclosure provides a machine-readable storage medium. The machine-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions enable the processor to implement the method. [00161 The present disclosure has the following beneficial effects:

[0017] (1) High precision:

[0018] Since the upper and lower portions of the electric water heater have different temperatures, that is, the lower portion is colder and the upper portion is hotter, in the present disclosure, the heat transfer equation and the mass exchange equation are established and solved discretely. This allows for the simulation of water temperature changes during bathing through the vertical flow of water, heat transfer between adjacent nodes, and the heating effect of the heating pipe. As a result, the internal temperature changes of the water heater are accurately replicated, and a suitable set temperature is determined. This approach ensures both comfortable water temperature for users during bathing and electricity savings.

[0019] In the present disclosure, the three-dimensional fluid heat transfer model is reasonably utilized, and it is discretized into a grid simulation. The average temperature is not used to replace the real temperature, resulting in higher accuracy and more practical significance for the model.

[0020] (2) High efficiency:

[0021] In order to address the immaturity of the technology related to set temperature control in the current electric water heater field, the present disclosure utilizes the three-dimensional fluid heat transfer equation in combination with the temperature field's heat transfer model for the inner container. This enables efficient and reliable prediction of the water heater's set temperature. By solving for the appropriate set temperature, both user comfort during bathing and electricity savings 2 can be achieved. [00221 The present disclosure provides a genetic-simulated annealing algorithm constrained based on the temperature difference for determining discrete grid parameters of the inner container of the water heater. This approach effectively addresses the issue of getting stuck in local optima, leading to enhanced accuracy in predicting the set temperature and reduced space complexity. BRIEF DESCRIPTION OF THE DRAWINGS [00231 FIG. lisa flowchart of a method according to the present disclosure; [00241 FIG. 2 is a flowchart of a genetic-simulated annealing algorithm constrained based on a temperature difference; [00251 FIG. 3 is a schematic structural diagram of a water heater; [00261 FIG. 4 is a schematic diagram of a temperature field heat transfer model; [00271 FIG. 5 is a schematic diagram of changes in water temperature during operation of a water heater;

[0028] FIG. 6 is a comparison diagram of the fitness among the present disclosure, the genetic algorithm, the Monte Carlo algorithm and the annealing algorithm;

[0029] FIG. 7 is a diagram of temperature changes at different layers of a water heater predicted according to the present disclosure when bathing at room temperature in winter;

[0030] FIG. 8 is a diagram of temperature changes at different layers of a water heater actually measured when bathing at room temperature in winter;

[0031] FIG. 9 is a diagram of one-day electricity consumption of a water heater at different set temperatures in winter; and

[0032] FIG. 10 is a comparison diagram of temperature prediction errors among the present disclosure, the heat transfer model based on multi-nodes, and the polymerization model based on energy exchange. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0033] A scheme for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer provided in the present disclosure is further described with reference to the accompanying drawings.

[0034] FIG. 1 shows a method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer, including the following steps:

[0035] Step (1): Construct a water temperature change function for the electric water heater.

[0036] Because cold water needs to be filled into the water heater before use, the water must be heated prior to use. It is assumed that the electric storage water heater is heated by two heating 3 pipes arranged vertically, and the water temperature can be approximately regarded as a uniform distribution during the heating process, as shown in FIG. 3. An average heat transfer coefficient of walls of the water heater is assumed to be A a heat dissipation surface area of the water heater to be F, the density and specific heat of water to be and Cp respectively, the water temperature 5 during a heating or cooling process to be and a time variable to be 7 , and a heat balance equation of water during the heating process is constructed as follows: VpCp^ = P-KF(t-tf)

[0037] ^7 (1) 10 20 25

[0038] An initial condition is substituted. is an initial water temperature, and the equation (1) is equivalently transformed into the water temperature heating change function in a non-water outflow period as follows: ^ / +- + ^0-^ - —]Q^-KFt / (VpC ))

[0039] KF KF (2)

[0040] In the water cooling process of the electric water heater, the temperature change equation is similar to the equation (2), and a water temperature cooling function of the electric water heater is as follows:

[0041] t = tf+y0-tf]^-KFT / (VpCpp

[0042] Step (2): Construct a three-dimensional fluid heat transfer model, where step (2) comprises step (2.1) and step (2.2).

[0043] Step (2.1): Calculate flows of cold water flowing into and hot water flowing out of a water storage tank.

[0044] The flow F is the volume of water flowing out of the shower head per unit time, which can also be called bathing water consumption per unit time. It is assumed that a water outflow and inflow of the electric storage water heater are the same at each moment to ensure the water storage tank to be always in a full state, thus the flows of cold water flowing into and hot water flowing out of the water storage tank are . It is assumed that a flow of water entering a hot and cold water mixer through a cold water pipe is , a total water inflow is a , a total water outflow is F , thus the following expression is satisfied:

[0045]

[0046] p - P + y A temperature expected for a human body during bathing is assumed to be Thuman, a hot water temperature to be 7\ot(t), a cold water temperature to be Tcoid, and a flow velocity component in a vertical direction to be v . In the cold and hot water mixer, cold water and hot water mixed to obtain the temperature expected for the' human body, and the temperature satisfies the following in the time :

[0047] ( / 3 + Y)tiVpThuman = P^vpT^ + Yt^pT^t^

[0048] An expression of is obtained based on the equations (4) and (5):

[0049] cold+CThottV+T cold)Y T human (6) 5

[0050] An expression of is obtained based on the equation (6):

[0051] _ Inhuman aTCOld , T human Tcold / y\ T hot{t)~Tcold T hot(t)~T cold

[0052] Step (2.2): Establish a three-dimensional fluid heat transfer model, where step (2.2) comprises step (2.2.1) and step (2.2.2).

[0053] Step (2.2.1): Construct a three-dimensional fluid heat transfer equation: 10

[0054] It is assumed that the electric storage water heater starts heating when a water temperature drops to (To — tset), To is a set temperature, and tset is a set value; a temperature stratification phenomenon in an inner container of the water storage tank causes natural convection; during a bathing process, cold water continuously enters the inner container of the water storage tank through a cold water inlet, and hot water flows out from a water outlet at a top portion, resulting in J5. forced convection from bottom to top as a whole; water flow temperature transfer of the electric storage water heater satisfies the three-dimensional fluid heat transfer equation: 3T z 31' 31' <7. ^3'3 32I 31

[0055] &&&&Ty pc (8)

[0056] is a medium thermal diffusivity, is heat generated by a heat source at a point , p is density of water, c is a specific heat capacity of water, and are respectively 20 velocity components of water in directions x, y and z when flowing through a fluid particle.

[0057] Since water flows only along a z-axis, that is, u = 0 and v = 0, for further simplification of the model, only temperature changes in the directions x and z are considered to reduce dimensions, and the equation (8) is further simplified as: 31' 3T 32T 32T F(x,y,z)

[0058] Pc (9) 25

[0059] Step (2.2.2) Establish a working state equation of the water heater.

[0060] It is assumed that heat dissipation of walls of the water heater during bathing is considered as a cold source, a function of the walls is equivalent to the heater, that is, is a result of subtracting heat dissipation power of the walls from heating power of the heater. In this case, the walls are completely insulated, and the following is obtained:

[0061] ST dn x=0,x=842 0,^ dn z=0,z=400 (10) CM

[0062] During bathing, a water temperature at a bottom layer of the water heater is constantly a T\ = t cold water temperature, that is, a room temperature: lz=0 f. [00631 Therefore, the three-dimensional fluid heat transfer model is obtained as follows: ST dT d2T d2T (x,y,z) dt dz dx2 dz2 pc [00641 = 0 5=0,2=400 t\ =tf 12=0 f (11) [00651 In order to describe an entire working process of the water heater clearly, control items are constructed: ______ (0 not bathing [00661 a — ]. , , . (.1 bathing , fO water heater does not heat b = I . , , (12) tl water heater heats [00671 A working state of the water heater is expressed as follows: dT dT d2T d2T^ b* P KF*(T-tr) --= a * (-w--+ a(—- + —-)) +---—

[0068] dt dz dx dz^ Pc VP (13) 57 P2T d2Tx -w--+ a(—+ —-)

[0069] &z indicates convection heat transfer and heat conduction of water P KF*(T^tf) during bathing, Pc indicates heater heating, and P indicates natural heat dissipation of the walls of the water heater.

[0070] Step (3): Construct an energy balance equation and obtain a temperature iteration process of the model.

[0071] The inner container of the water heater tank is divided into M columns in a horizontal direction and rows in a vertical direction, the whole inner container is considered as being composed of ^*4 / nodes, energy transformation of each node is to be solved, and finally a temperature iteration process of each node is obtained. A temperature field heat transfer model is shown in FIG. 4.

[0072] FIG. 5 is a schematic diagram of water temperature changes during operation of the water heater.

[0073] Step (3.1): Calculate an energy change caused by water flow.

[0074] Both the water inlet pipe and the water outlet pipe of the water heater are installed at the bottom of the inner container. The water inlet pipe is shorter, with only a small portion extending out, while the water outlet pipe is longer, reaching near the top of the inner container. Therefore, in the process of water inflow, water gradually moves from the bottom of the inner container to the top. This leads to a transition from high-temperature water to low-temperature water in each node.

[0075] Since the cold water flows in at the bottom and the hot water flows out at the top when the water heater is in use, and the water in the inner container continuously flows in the vertical direction, the energy change caused by the water flow is represented as follows: [00761 ^=pc^t(T^-T^ (M) AA (i A [00771 indicates a time interval, M indicates the energy change at a node caused by water flow in a time A?, / indicates a change of the node in the direction x, J indicates a change of the node in the direction z , c indicates the specific heat capacity of water, P indicates the density of water, indicates a calculated inflow, indicates a temperature of a node + at a moment 7 , and indicates a temperature of a node v ' J) at the moment T . [00781 Step (3.2): Calculate heat transfer between adjacent nodes.

[0079] Because of a temperature difference between adjacent nodes, heat transfer occurs in real time, and an energy change caused by heat transfer between the node and each adjacent node per unit time is expressed as follows:

[0080] = Q[Skl+ + s + (15) 1 1 Acol2 Acol2 7 v Aarr2 Aarr2 7J v 7 ,. ..

[0081] indicates an energy change of the node caused by heat transfer in the time At, ” indicates the thermal conductivity of water, indicates a mutual contact area between the nodes in the vertical direction, indicates a mutual contact area between the nodes in the horizontal direction, indicates row spacing between the nodes, and indicates column spacing between the nodes. Calculation is performed according to the following formula: _ (Header1"^ x Header ' ^) (Header^)2 Sarr Header^ 10A / ^co! Headerh,sh

[0082] (16)

[0083] Header g is a vertical height of the inner container of the electric storage water heater, and Header is a horizontal width of the inner container of the electric storage water heater.

[0084] Step (3.3): Calculate a heating effect of the heating pipe. 5

[0085] The heating pipe affects the temperature of each node in the grid. A common electric storage water heater is equipped with two heating pipes: the upper heating pipe and the lower heating pipe. It has been verified that the double-pipe heating can be approximated as uniform 27 0924 heating, meaning that the same amount of heat is distributed to each grid. By considering the two heating pipes as a single heating source with a boundary midpoint, an energy change caused by the heating pipe with heating power of P is expressed as: ^ea = PA^

[0086] 1,3 M*H (17)

[0087] Step (3.4): Calculate a temperature iteration process.

[0088] Before and after the time , the total energy change J of the node is:

[0089] =cPvo(Kj 15

[0090] With reference to the steps (3.1) to (3.3), a sum of all energy changes of the node in the time is equal to the total energy change of the node, and the following is obtained:

[0091] AE . = EE'" + EE™ + AE^ PJ PJ PJ (19)

[0092] The temperature iteration process of each node in a grid at each moment is deduced from the equation (19): 20   

[0093] T^1 =             + _A_           +        +           + 1 J l,J cpvQ cpv0 L Aarr2 Aarr2 J Acol2 Acol2 7J +--—2--Td (20) M*N*c2pv0 'J

[0094] In the foregoing formula, T^j = 7} and <M

[0095] Step (3.5): As shown in FIG. 2, solve division numbers AT and of the inner container of the water storage tank based on a genetic-simulated annealing algorithm constrained based on the temperature difference. [00961 Step (3.5.1): Establish a temperature difference objective function. [00971 It is assumed that ZG [0-^, TV] represents a tOp part an(j zg [O,O.27V] represents a bottom part of the electric storage water heater, node temperatures calculated in the step (3.4) are averaged TT Tt to obtain a top layer temperature up and a bottom layer temperature ldown; TT [00981 the objective function is constructed as follows according to the top layer temperature up rj-iT and the bottom layer temperature ld<™n: / •Dura f = Max( I (?2 TLn [00991 (21) [01001 Dura indicates bathing duration. [01011 Step (3.5.2): Generate an average value of the vertical division numbers and an average value of the horizontal division numbers for the two-dimensional grid of the inner container of the water storage tank.

[0102] Because the sizes of electric storage water heaters vary, it is necessary to design different two-dimensional grid division numbers of the inner container for different sizes of water heaters. Assuming Header 8 t0 be the vertical height (cm) and Header 8 t0 be the horizontal width (cm) n of the inner container of the electric storage water heater, the average value of the vertical A / division numbers and the average value ^7^ of the horizontal division numbers of the two-dimensional grid are respectively: ' aver'7 =\ (Header^11 / 3)} averM = F(Headerl,,nph / 5)1

[0103] 1 1 1 (22)

[0104] Step (3.5.3): Generate an initial population in a normal distribution.

[0105] To maintain a desired level of precision in the division of the grid nodes of the inner container and minimize computational complexity, most of the vertical and horizontal division numbers that yield better results are close to ^7^ and , and satisfy characteristics of the normal distribution, such that the initial population is obtained by using random numbers in the normal distribution.

[0106] First, probability density functions of the normal distribution with and ^7^ as I j I H central axes and yaver and yaver as standard deviations are calculated as follows: 1 _(N-m’erN')z f(N) = . V leaver < 1 JN-aver™')2 f(W = [01071 yJ2^aver ^3)

[0108] Then random numbers u and u between 0 and 1 are separately generated for and according to the principle of equal probability, inverse functions ) and ) of and are calculated, that is, the random numbers following the normal distribution are separately obtained, and the process is repeated 20 times, to finally generate normal random number sets and ga,he,g with a scale of 20.

[0109] The initial population conforming to the normal distribution is obtained after calculating a Cartesian product of the sets ^atherort an(| gather^ . [01101 gather / , x gather^

[0111] Step (3.5.4): Calculate a temperature difference objective function for each solution in the population according to the equation (21), and then obtain an optimal solution ^best of the current population.

[0112] Step (3.5.5): Iteratively obtain the division number of the two-dimensional grid of the inner container of the water storage tank.

[0113] After selection, crossover and mutation, generate a new population, and determine whether the top layer temperature is less affected by the bottom layer temperature in the current optimal f (S) >f (S ) solution, that is, whether the objective function J v ’ J where S indicates a solution in the S = S new population; if yes, update the current optimal solution best , and obtain a new optimal solution through iteration; otherwise, continue to determine whether exp(| / (5) f{Sbest)\)<iandom(Q,\) . Updating and continue to find the optimal S' = 5* S solution; otherwise, update the optimal solution to best random, where random indicates a random solution in the new population, that is, the division number of the two-dimensional grid of the inner container of the water storage tank.

[0114] Step (3.5.6): If a number of iterations reaches the maximum, stop the iteration and output the optimal solution, and substitute the optimal solution (N, M) into the temperature iteration process of the model; otherwise, proceed to the step (3.5.3).

[0115] Step (4): Establish a continuity equation.

[0116] Since a lowest temperature at the top of the water heater is to be higher than a lowest bathing temperature Twarm, an objective function and a constraint condition are obtained based on the water temperature change function for the electric water heater, the three-dimensional fluid heat transfer model and the temperature iteration process (20):

[0117] VpC— = P-KF(t-tf) dr VpC ± = -KF(t-tf) dr dT dT --1- w— dt dz d^T d2T = £(--T +--T dx2 dz2 dT dT --Fw-- dt dz q2t d2T --- L I 1 ___________ dx2 dz2 q pc 101181 ( Minm ) rj-t rrt Ut=to + 9OO 'warm t <t0,T <T0 -5 t<t0,T>T0 -5 t0<t <t0 + 900, To >T >T0-5 t0<t<t0 + 900, T <To - 5 (25) V pC — = P-KF(t-tf)

[0119] dT indicates a water temperature change when a power supply is C\J VpCp — = ^KF(^tf) turned on, indicates a water temperature change during natural heat OdT ST &T d2T --1- w— —1^(—— -l--—) dissipation, indicates a water temperature change when the water CM temperature is not lower than the set temperature by 5°C during bathing, dT dT d2T d2T q + W — £( ,+ 2 ) t ®z Pc indicates a water temperature change when the water temperature is lower than the set temperature by 5°C during bathing, Tt=to+9OO >Twarm indicates the constraint condition, and Twarm indicates a minimum temperature of hot water required for bathing in different seasons. 15

[0120] Step (5): Traverse and solve the continuity equation to obtain a minimum set temperature and electricity consumption, where step (5) comprises 5-1, 5-2, 5-3 and 5-4. T

[0121] 5-1: Initialize randomly generated 0.

[0122] 5-2: Assume that a minimum value of the top layer temperature of the water heater at each moment during bathing is the same as a bathing water temperature, and calculate the top layer 20 temperatures of the water heater at different moments during bathing according to the objective function (25).

[0123] 5-3: Screen out a minimum value from the top layer temperatures at different moments, J1 — determine whether the minimum value is higher than Twarm, if yes, update 0 0 , where a indicates a temperature step, make an iteration count=count+l, and return to the step 5-2; otherwise, T T । d determine whether a current iteration count is 1; if yes, update 0-0 , make count=count+l, T and return to the step 5-2; otherwise, output 0 obtained in a last iteration as an optimal set temperature T .

[0124] 5-4: Substitute the optimal set temperature T into the temperature iteration process (20), further obtain a heating time thot of the electric water heater of the day, and multiply the power of the electric water heater with thot to obtain electricity consumption.

[0125] Example

[0126] [1] Implementation conditions

[0127] Combining with the scheme of the present disclosure, two representative seasons (summer and winter) are selected for experimentation. Specifically, 60 L and 100 L electric water heaters of a certain brand are used as experimental conditions, with heating power of 800 W, 1200 W, and 1500 W. Assuming that each user takes a bath at 8 p.m., the indoor temperature in summer is set as the lowest temperature of the day, that is, 24°C, and the indoor temperature in winter is set as the lowest temperature of the day, that is, 4°C. Additionally, the water temperature at the beginning of bathing is set at the set temperature minus 5°C, with an average bathing time of 15 minutes, a time step of 0.1 s, Twarm in summer of 37°C, Twarm in winter of 45°C, an initial water heater temperature of 20°C, a heat transfer coefficient of 0.879, a heat dissipation area of 1.08 m2, a water thermal conductivity a of 0.59, and the outflow and inflow of 0.00008 m . Under these conditions, calculations are performed to determine the lowest set temperature and the minimum electricity consumption required to enable bathing at any time in both summer and winter.

[0128] The following examples are described using a 60 L and 1500 W water heater operating in winter at room temperature, and other specifications are not described in detail herein.

[0129] [2] Implementation steps

[0130] According to the step (1), the first step is to establish the heat balance equation for a nonoutflow heating and cooling process based on the factors that affect the change in water temperature. Relevant parameters such as an initial water temperature of 20°C, an ambient temperature of 4°C, a heat transfer coefficient of 0.879, and a heat dissipation area of 1.08 are then substituted into the water temperature heating change function during the non-outflow period in the step (1), to obtain an equation that describes the change in water temperature over time during the actual heating process of the water heater.

[0131] ^(0=1600.078-1580.078^^3767^^ 1 070^(-3.767^0^) [01321 is the water temperature cooling change function of the water heater. [01331 According to the step (2), the addition of cold water to the electric water heater is considered as a process of uniform diffusion of the heat source. To maintain a constant total water volume in the electric water heater, the total water inflow must be equal to the total water outflow, that is, u By substituting the corresponding parameters, w = 0.00008 anj / / ^0.00008 are obtained. Given the constant changes of Thot,Tcoid during the heating and cooling process, the expression / = 0.00 0 0 8 37 Tcold that describes the flow of cold water flowing in and hot water Thot~Tcold flowing out of the water storage tank can be derived. Then, due to the natural convection and forced convection, a three-dimensional heat transfer model can be established, and the working state equation of the water heater can be obtained by substituting the water inflow and outflow into the three-dimensional heat transfer model: 5T ar d2T b 0.94932*(A - A)

[0134] 5 / fir2 fiz2 2800 60000 (27)

[0135] According to the step (3) and its related expressions, the two-dimensional grid structure of the inner container of the water heater can be divided. For each node, the energy balance equation of water flow up and down can be derived based on time dispersion: mij

[0136]

[0137] The energy balance equation for heat transfer between the nodes is:

[0138]

[0139] aa;.. ^+1 (29)

[0140]

[0141]

[0142]

[0143] The energy balance equation of heating effect of the heating pipes is: A,, 252000 A .. .. MN (30) The total energy change in the specified time step is: 252000 ..., aa;, =------cc+1^7;n y a a * m v y y' Based on the internal energy balance equation of the water heater, the following can be obtained: „ . . A1 AA.. = AA .. + AA .. + AA...

[0144] y m,J C,J h,J (32)

[0145] Through the above energy balance equation, change curves of the top layer temperature, the middle layer temperature, and the bottom layer temperature in the temperature field divided into two-dimensional grid nodes for the inner container of the water storage tank related to N can be i r XT obtained. Then, the optimal values of LV ’ v are solved according to the genetic-simulated annealing algorithm with constrained based on the temperature. First, a random initial water temperature value To-6OC obtajne(|, anc| then aver =20 anj 1 03.H be calculated. The initial population 1(5,9),(11,30),(19,20),(20,21),...} js ranc[omiy generated according to inverse function values in the normal distribution. [01461 Next, the current optimal solution is selected based on the energy balance / •900 f = Max{I (r -Vdom equation and the constraint condition Jo . Each iteration generates a new generation of temperature node arrangement population through selection, mutation, and crossover operations. If the new optimal solution of the temperature node arrangement is better than the current optimal solution, the optimal solution is updated to the new value; otherwise, if exp(| f(S) - f(Sbes1) |) >random^, 1) is satisfied, the optimal solution is replaced by a random temperature node arrangement; otherwise, the next iteration starts. After 30 iterations, the optimal solution (^^)-(20,20) for temperature node in the grid directions is calculated, and (NyM') substitutec| into the energy balance equation to obtain the final temperature iteration process of the model: 63(7"‘-V,) = 63+^, / )+0.01092^(7^ -7,^)+1.68(7^-T’)

[0148] According to the obtained model temperature iterative formula, the time-dependent curves of the top layer temperature, middle top temperature, and bottom top temperature can be calculated for an initial set temperature.

[0149] According to the step (4), utilizing the water temperature change function and the previously derived temperature iteration formula, the continuity equation can be formulated on the basis of the three-dimensional heat transfer model and the water temperature change curve of the water heater:

[0150] Min( / ) \T >45 1 ?=?o+9OO 7( / ) = 1600.078-1580.078^3 t< zt0,T<T0-5 7( / ) = -1580.078^3 767*10^ t < ' t T >T — 5 dT ST „^cco / d2T d2T — + w— = 0.14558(—- + —-) dt dz dx2 dz2 7 <t<t0+9QQJ0 >T>T0-5 dT dT d2T d2T __ + w__ = o.14558(—^- + —^) + 0.34095 dt dz dx2 dz" < / < / 0+900,7 <70-5 [01511 L (34) [01521 According to the step (5), based on the first four formulas of the continuity equation, the change curve of the outflow water temperature of the water heater during bathing at an initial set T temperature 0 can be obtained. Then, starting from the reasonably estimated temperature of 55°C, traversing is conducted in increments of the set temperature step ±0 । C t0 find a solution target T of the continuity equation, until an initial set temperature 0 that satisfies the fifth constraint T_ >45 condition of the continuity equation (that is, the outflow water temperature of the y _ pQ water heater is always higher than 45°C during bathing) is found. Ultimately, 0 - satisfies the specified constraint, that is, the optimal set temperature is 61°C. According to the heating power P = 150017 of water heater, natural heat dissipation, and heating time of 3.2753 hours caused by cold water injection, the electricity consumption of the day can be calculated as 4.913 kwh.

[0153] [3] Measuring instruments and methods

[0154] The actual temperature measurement for each layer of the inner container of the water heater is carried out using PT1000 patch temperature sensors. These sensors are suitable for temperature measurement in the water heater, offering high precision and a wide temperature range of [-50°C, 200°C]

[0155] The temperature measurement method involves removing the thermal insulation foam covering the inner container of the water heater and attaching patch temperature probes to the upper and lower walls of the inner container. Once the probes are secured, the peeled position is refilled with foam. The walls of the inner container have good heat transfer properties, which allows to consider the real-time temperature measured on the walls as the corresponding water temperature at that position on the inner container. To collect temperature data, the PT 1000 patch temperature sensors are connected to an Arduino device, to record the temperature at each moment.

[0156] A B23 111-400 electricity meter is used to measure electricity consumption. This type of meter offers high measurement accuracy, low power consumption, and easy configuration.

[0157] The electricity consumption is measured by connecting the B23 111-400 electricity meter to the household circuit, turning off all other electrical appliances connected to the circuit, and 18 recording the readings on the electric meter once the water heater completes its operation cycle. [01581 [4] Implementation results [01591 FIG. 8 shows the water temperature change curve of each layer in the inner container of the water heater, as actually measured using the PT 1000 patch temperature sensors and the temperature measurement method during the 15-minute bath at 8 p.m. with an indoor temperature of 22°C. FIG. 7 is a water temperature change curve predicted using the model temperature iterative formula provided in the step (3) of the present disclosure. It can be concluded that the actual temperature of each layer of the water heater is generally consistent with the predicted value during the bathing process, from the beginning to the end of 15 minutes, with an initial temperature set at 61 °C. This is possible due to the discretization and gridding of the inner container of the water heater, which effectively divides the inner container into the top layer, middle layer, and bottom layer. In this way, temperature of each layer can be accurately calculated as it changes over time when the three-dimensional fluid heat transfer equation is used to predict the temperature. These findings highlight reliable accuracy and practical significance of the present disclosure.

[0160] FIG. 9 shows the electricity consumption of the water heater measured using the B23 111-400 electricity meter when the indoor temperature was 22°C, the set temperature ranged between f61°C 69°C1 L ’ J, considering natural heat dissipation of hot water and a 15-minute bath at 8 p.m. The data collected by the meter reveals that the electricity consumption increases as the set temperature increases, and the water heater's daily electricity consumption reaches its minimum at the set temperature. If the set temperature is below 61 °C, the resulting hot water temperature during bathing may be lower than the desired temperature for the human body. Conversely, if the set temperature exceeds 61 °C, it will lead to unnecessary electricity wastage. The present disclosure can accurately predict set temperature due to the consideration of energy exchange resulting from the vertical flow of water and various heat transfers. By utilizing the three-dimensional fluid heat transfer equation, these factors are comprehensively incorporated, resulting in a highly realistic simulation of real-world conditions.

[0161] Based on the above setting conditions and implementation methods, the water heaters with varying capacities and heating powers are subjected to repeated simulation under summer and winter conditions. Starting with the predicted temperature calculated in the present disclosure, a step length of 0.1°C is employed, and a temperature sensor is utilized to measure the real-time water temperature on the top layer of the water heater. The real initial set temperature of the water heater that consistently maintains the top layer temperature of the water heater higher than the temperature expected for the human body within 15-minute bath is then determined. The electricity consumption in this case is measured using an electricity meter, thereby yielding the experimental data presented in Table 1. The data in Table 1 illustrates that under different seasonal temperatures and water heater storage and power capacities, the measured and calculated values of heating energy consumption exhibit close agreement, with a maximum relative error of no more than 2%. Additionally, the actual optimal set temperature aligns well with the calculated value, with a maximum relative error of no more than 1.6%.

[0162] The comparison presented in FIG. 6 demonstrates that in contrast to other algorithms such as the Monte Carlo algorithm and traditional genetic algorithm, the method of the present disclosure yields a more stable model. It also effectively separates the top layer temperature from the bottom layer temperature, enhancing the accuracy and effectiveness of the solution. Moreover, the method's selection of the number of temperature nodes reduces space complexity while maximizing model accuracy. This is attributed to the incorporation of the annealing algorithm in the present disclosure. This algorithm allows for the acceptance of the temperature node arrangement scheme with low fitness, to overcome local optima during the crossover process. Through a sufficient number of iterations, the algorithm has a high probability of converging towards the global optimal solution. As a result, it successfully separates the temperature layers within the inner container of the water heater.

[0163] FIG. 10 shows a percentage error comparison between actual values of the top layer temperature measured by the temperature sensor when the 60 L and 1500 W water heater is used for bathing at the set temperature in winter, and top layer temperatures calculated using the temperature iterative formula of the present disclosure and two other heat transfer models. In comparison with the heat transfer model based on multiple nodes and the aggregation model based on energy exchange, the present disclosure shows superior performance in terms of the percentage error between the predicted value and the actual value of the top layer temperature of the electric water heater during bathing. The error consistently remains lower than 2%, effectively demonstrating the effectiveness of the present disclosure. This is due to the fact that the present disclosure takes into account the energy exchange between nodes in the electric water heater and derives the corresponding temperature iterative formula, but also considers the effects of natural convection and forced convection. By establishing a three-dimensional fluid heat transfer equation, the present disclosure accurately predicts and calculates the temperature change curve of each layer. This further demonstrates the considerable reliability of the present disclosure.

[0164] Table 1 Comparison between measured and calculated values of set temperature and heating electricity consumption

[0165] (L) T 1 warm (°C) P(W) Optimal set temperature (°C) Electricity consumption of the day (kwh) Measured value Calculated value Error Measured value Calculated value Error 60 37 800 44.2 44.5 0.67% 1.221 1.253 0.16% 1200 45.0 45.3 1.55% 1.305 1.312 0.53% 1500 45.2 45.8 1.32% 1.317 1.334 1.29% 45 800 60.8 60.9 0.16% 4.819 4.874 1.14% 1200 60.9 61.4 0.82% 4.896 4.901 0.10% 1500 62.0 61.0 0.16% 4.906 4.913 0.17% 100 37 800 44.9 45.0 0.22% 1.893 1.902 0.47% 1200 45.3 45.8 1.10% 1.913 1.921 0.49% 1500 45.5 45.9 0.87% 1.989 2.029 1.80% 45 800 61.4 61.3 0.16% 5.116 5.109 0.13% 1200 62.0 61.8 0.33% 5.210 5.203 0.20% 1500 62.1 62.2 0.16% 5.245 5.214 0.57%

Claims

1. A method for controlling a set temperature of an electric storage water heater based on three-dimensional fluid heat transfer, comprising the following steps:5 step (1): constructing a water temperature change function for the electric water heater;step (2): constructing a three-dimensional fluid heat transfer model, wherein step (2)comprises:step (2.1): calculating flows of cold water flowing into and hot water flowing out of a water storage tank; and10 step (2.2): establishing the three-dimensional fluid heat transfer model, wherein step (2.2)comprises:step (2.2.1): constructing a three-dimensional fluid heat transfer equation, whereinit is assumed that the electric storage water heater starts heating when a water temperature drops to (To — tSet), To is a set temperature, and tset is a set value; a 15 temperature stratification phenomenon in an inner container of the water storage tankcauses natural convection; during a bathing process, cold water continuously enters the\|' inner container of the water storage tank through a cold water inlet, and hot water flows out* from a water outlet at a top portion, resulting in forced convection from bottom to top as awhole; water flow temperature transfer of the electric storage water heater satisfies the simplified three-dimensional fluid heat transfer equation:ol ol .11 oP Ipx.y.z)__(_ _______ ___ £ / __|__\ _|__~ ~ ' dt dz dx2 dz2 2 pcwherein is a medium thermal diffusivity, ^x,y,z) is heat generated by a heat source at a point , P is density of water, c is a specific heat capacity of water, and w is a velocity component of water in direction z when flowing through a fluid particle; and25 step (2.2.2): establishing a working state equation of the water heater, whereinit is assumed that heat dissipation of walls of the water heater during bathing is considered as a cold source, a function of the walls is equivalent to the heater, wherein, T* (*, y,z) js a resuq of subtracting heat dissipation power of the walls from heating power of the heater; in this case, the walls are completely insulated, and the following is obtained:30dn x=0,x=842z=0,z=4000(10)during bathing, a water temperature at a bottom layer of the water heater is constantly a coldt\ =twater temperature, that is, a room temperature: lz=0 ’ \the three-dimensional fluid heat transfer model is obtained as follows:51' 51' ^52T 521\ ,— + w— = ^(—- +—-) + 5 dt dz dx" dz2(x,y,z) pcx=0,x=842z=0,z=400(11)in order to describe an entire working process of the water heater clearly, control items are5 constructed:(0 a — j inot bathing bathingwater heater does not heat water heater heatsa working state of the water heater is expressed as follows:ST dtdT 5T_ d T5z + a W + dz2 } +Vp(13)whereinoT dz82T 52T.—F --F )Sr dz indicates convectionheat transfer and heat conduction ofwater during bathing, Pc indicates heater heating, andVpindicates natural heatdissipation of the walls of the water heater;step (3): constructing an energy balance equation and obtaining a temperature iteration processof the model, wherein15 the inner container of the water heater is divided into M columns in a horizontal direction androws in a vertical direction, the whole inner container is considered as being composed of N*M nodes, energy transformation of each node is to be solved, and finally a temperature iteration process of each node is obtained;step (4): establishing a continuity equation, wherein20 since a lowest temperature at the top portion of the water heater is to be higher than a lowest bathing temperature Twarm-. an objective function and a constraint condition are obtained based on the water temperature change function for the electric water heater, the three-dimensional fluid heat transfer model and the temperature iteration process:foO T = cit P—KF(t—tf) t < / T< T -5 VpCpT = dr —KF(t — tf) t< ■T -5 dT dT --1- w— dt dz ^d^T d2Tx — o 2 + o 2 ) dx dz <t<h + 9OO,7o >T>T0-5 dT ,dfT_ dt dz t,,d2T d2T q dx2 dz2 pc 7 <t<t. + 900,7 <T0 -5 (25)Min^’ ">Tt=to+9OO 1warmVpCp— = P^KF(t^tf)wherein dr indicates a water temperature change when a power supplyVpCp^-=-KF{t-tf)is turned on, dT indicates a water temperature change during natural heatdT dT drT d2T--h W— — H--—)5 dissipation, dz dx dz indicates a water temperature change when the watertemperature is not lower than the set temperature by 5°C during bathing, — dT dT ,F2T d2T, qot oz ox cz pc in(iicates a water temperature change when the water temperatureis lower than the set temperature by 5°C during bathing, Tt=to+9OO >Twarm indicates the constraint X] condition, and Twarm indicates a minimum temperature of hot water required for bathing in 10 different seasons; andstep (5): traversing and solving the continuity equation to obtain a minimum set temperature and electricity consumption, wherein step (5) comprises:Tstep (5.1): initializing randomly generated 0;step (5.2): assuming a minimum value of top layer temperatures of the water heater at 15 different moments during bathing is the same as a bathing water temperature, and calculating the top layer temperatures of the water heater at different moments during bathing according to the objective function (25);step (5.3): screening out the minimum value from the top layer temperatures at different moments, determining whether the minimum value is higher than Twarm, if yes, updating p = T a20 0 0 , wherein a indicates a temperature step, making an iteration count=count+1, andreturning to the step (5.2); otherwise, determining whether a current iteration count is 1, if yes, updating ^o~ h+a making count=count+l and returning to the step (5.2); otherwise,outputting 0 obtained in a last iteration as an optimal set temperature T , andstep (5.4): substituting the optimal set temperature T into the temperature iteration process, further obtaining a heating time thot of the electric water heater of the day, and multiplying the power of the electric water heater with thot to obtain electricity 5 consumption.27 09 242. The method according to claim 1, wherein the step (1) specifically comprises: assuming that the electric storage water heater is heated by two heating pipes arranged vertically, considering the water temperature to be approximately uniformly distributed during the 10 heating process, assuming an average heat transfer coefficient of the walls of the water heater to beK, a heat dissipation surface area of the water heater to be the density and specific heat of water to be P and Cp respectively, the water temperature during a heating or cooling process to be -, and a time variable to be T , and establishing a heat balance equation of water during the heating process as follows: dtV pC — = P KF(t-tf) d^ (1) substituting an initial condition , wherein is an initial water temperature, and equivalently transforming the equation (1) into a water temperature heating change function in a non-water outflow period as follows: p pt = tf^--+ [T --1 expC^ATr / (V pC„\)KF f KF p (2)20 obtaining a water temperature cooling change function for the electric water heater in a water temperature cooling process of the electric water heater as follows:t = tf + [?o -tf]Q^(-KFTI {VpCp))3. The method according to claim 2, wherein the step (2.1) specifically comprises: assuming 25 that a water outflow and inflow of the electric storage water heater are the same at each moment to ensure the water storage tank to be always in a full state, thus the flows of cold water flowing into and hot water flowing out of the water storage tank are ; and assuming that a flow of water entering a hot and cold water mixer through a cold water pipe is a total water inflow is a , a total water outflow is , thus the following expression is satisfied:27 09^24a = 7 + / y= + / 0-)assuming a temperature expected for a human body during bathing to be Thuman, a hot water temperature to be ThotCO, a cold water temperature to be Tcoid, a flow velocity component in a vertical direction to be v, wherein in the cold and hot water mixer, cold water and hot water are 5 mixed to obtain the temperature expected for the human body, and the temperature satisfies the following in the time ?1:(P +Y)hvpThuman PhvpTcoid + YhvpTho^t) (5)obtaining an expression of based on the equations (4) and (5):g = aT hotV)+T cold)^ (qT human10 obtaining an expression of based on the equation (6):_ ^human~a^ cold human-^ cold s^x T cold Thot(t)~T cold4. The method according to claim 3, wherein the step (3) specifically comprises:step (3.1): calculating an energy change caused by water flow, whereinsince the cold water flows in at the bottom and the hot water flows out at the top when the water heater is in use, and the water in the inner container continuously flows in the vertical direction, the energy change caused by the water flow is represented as follows:= (14)wherein A / indicates a time interval, iJ indicates the energy change at a node caused 20 by water flow in the time A / , z indicates a change of the node in the direction x, indicates a change of the node in the direction z, c indicates the specific heat capacity of water, indicates the density of water, indicates a calculated inflow, indicates a temperature of a node (z + J) at a moment T, and indicates a temperature of a node - at the moment T ;step (3.2): calculating heat transfer between adjacent nodes, wherein25 because of a temperature difference between adjacent nodes, heat transfer occurs in real time, and an energy change caused by heat transfer between the node and each adjacent node per unit time is expressed as follows:Ei,j 51 kl( 6col2 + acol2 ) + k2\ Aarr2 + Aarr2 ( ' / ^End (iwherein hJ indicates the energy change of the node ' caused by heat transfer in thestime , n indicates the thermal conductivity of water, H indicates a mutual contact area between5the nodes in the vertical direction, n indicates a mutual contact area between the nodes in thehorizontal direction, ^arr indicates row spacing between the nodes, indicates column spacing between the nodes, and calculation is performed according to the following formula:_ (Header^ x Header,ensth)H “ N_ (Header1™®11)2k2 "MA Header1'"®'^arr =---------10 MA , HeaderhighAcol =---------8A' (16)wherein Header g js a vertical height of the inner container of the electric storage water heater, and Header1™®1 js a horizontal width of the inner container of the electric storage water heater;step (3.3): calculating a heating effect of the heating pipe:^j^heaan energy change 7 7 caused by the heating pipe with heating power of P is expressed as:kEhea = PA(l’J M*N (17)step (3.4): calculating a temperature iteration process:wherein before and after the time , the total energy change ’’7 of the node is;(18)with reference to the steps (3.1) to (3.3), a sum of all energy changes of the node in the timeis equal to the total energy change of the node, and the following is obtained:AEtj = + +the temperature iteration process of each node in a grid at each moment is deduced from theequation (19):q Ji-Xj TiJ , Ti+i,j TifK , rij , TU+1 Ti,h--1--ph (............................................................;—।--—) । ^k?\----—।--r;— cpv0 cpv0 Aarrz Aarrz Acolz AcolzPM m-----------o--r i tM*N*czpv0(20)wherein T^j = Tt and j<M anjstep (3.5): solving division numbers and of the inner container of the water storage tank based on a genetic-simulated annealing algorithm constrained based on the temperature difference.27 09:

245. The method according to claim 4, wherein the step (3.5) specifically comprises:5 step (3.5.1): establishing a temperature difference objective function, whereinit is assumed that -e -represents a top part and represenfS a bottompart of the electric storage water heater, node temperatures calculated in the step (3.4) are averaged7^ 7"to obtain a top layer temperature up and a bottom layer temperature ;TTthe objective function is constructed as follows according to the top layer temperature up and10 the bottom layer temperature ldown: pDurawherein Dura indicates bathing duration;step (3.5.2): generating an average value of the vertical division numbers and an average value of the horizontal division numbers for the two-dimensional grid of the inner container of the water storage tank, whereinAT Mthe average value aver of the vertical division number 1N and the average value aver of the horizontal division number M for the two-dimensional grid of the inner container are calculated as follows:averN = (Headerh'sh / 3)”javerM = [(Headerlmgt11 / 5)^step (3.5.3): generating an initial population in a normal distribution, whereinNmost of the vertical and horizontal division numbers that yield better results are close toand , and satisfy characteristics of the normal distribution, such that the initial population is obtained by using random numbers in the normal distribution;N M25 first, probability density functions of the normal distribution with aver and aver as centralaxes and and as standard deviations are calculated as follows:fW =,nJN^aver^Y g 2averN(N-aver)' 2 averM102025I y / ZTraver (23)then random numbers nN and uM between 0 and 1 are separately generated for and according to the principle of equal probability, inverse functions and ) of(N) and J are calculated, that is, the random numbers following the normal distribution are separately obtained, and the process is repeated 20 times, to finally generate normal random number sets Sathero« and Sather«T with a scale of 20;the initial population conforming to the normal distribution is obtained after calculating a Cartesian product of the sets £ather°" and gather^:gather^ x gather^step (3.5.4): calculating a temperature difference objective function for each solution in thepopulation according to the equation (21), and then obtaining an optimal solution best of the current population;step (3.5.5): iteratively obtaining the division number of the two-dimensional grid of the inner container of the water storage tank:after selection, crossover and mutation, generating a new population, and determining whether the top layer temperature is less affected by the bottom layer temperature in the current optimal solution, that is, whether the objective function > / (¾^), wherein S indicates a solution inthe new population; if yes, updating the current optimal solution best , and obtaining a new optimal solution through iteration; otherwise, continuing to determine whether expf / f.S'j-H.S^J |) < / ^^ veg^ sypping updating and continuing to find the optimalsolution; otherwise,s =s         supdating the optimal solution to random, wherein ra”dom indicates a random solution in the new population; andstep (3.5.6): if a number of iterations reaches the maximum, stopping the iteration andoutputting the optimal solution, and substituting the optimal solution ' iteration process of the model; otherwise, proceeding to the step (3.5.3).into the temperature6. An electronic device, comprising a processor and a memory, wherein the memory stores machine-executable instructions executed by the processor, and the processor executes the machineexecutable instructions to implement the method according to any one of claims 1 to 5.5 7. A machine-readable storage medium, storing machine-executable instructions, whereinwhen the machine-executable instructions are called and executed by the processor, the machineexecutable instructions enable the processor to implement the method according to any one of claims 1 to 5.10CM

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