One-dimensional calculation method of solidification of lead-bismuth during flow in a tube

Abstract

The application discloses a one-dimensional calculation method for solidification of lead bismuth in a pipe flow process, and comprises the following steps: judging whether the fluid reaches a solidification condition by assuming the existence of a solidification layer and calculating a heat difference in and out of the solidification layer; on the basis of reaching the solidification condition, calculating a solidification mass growth in a unit time step through an energy equation to obtain a solidification parameter of each control body and then calculating a growth amount of the solidification layer; and updating flow channel parameters and adopting an enthalpy-porous medium model to calculate basic parameters of a remaining fluid domain. The method can more accurately study the influence of the solidification problem on system-level parameters while saving calculation resources, thereby providing a future development direction for engineering design and performance optimization.

Description

Technical Field

[0001] This invention belongs to the field of nuclear reactor thermal-hydraulic calculation technology, and specifically designs a one-dimensional calculation method for the solidification of lead-bismuth during the flow of lead-bismuth in a pipe. Background Technology

[0002] Lead-bismuth reactors possess inherent advantages such as good safety, high energy density, and long operational life, making them one of the six preferred reactor types for fourth-generation nuclear energy systems. During reactor shutdown and maintenance, as well as during supercooling of the secondary or emergency cooling systems, or during normal operation, solidification of the lead-bismuth coolant can occur. The resulting changes in flow channel geometry and the interaction between coolant temperatures can cause overall instability in the natural circulation. Furthermore, the volume expansion of solidified lead-bismuth can generate mechanical stress on the internal structure, threatening the safe operation of the reactor. Therefore, it is essential to study the solidification phenomenon of lead-bismuth during its flow within the tubes. Currently, there is limited research on the flow solidification of lead-bismuth, and most existing studies establish three-dimensional computational models. The advantage of three-dimensional models over one-dimensional models is that they can simplify the theoretical model by dividing the data into more comprehensive control volumes; however, this also means a large computational resource requirement. Summary of the Invention

[0003] The purpose of this invention is to provide a one-dimensional calculation method for the solidification of lead-bismuth during flow in a pipe. This method can use computational fluid dynamics and numerical heat transfer techniques to study and calculate the phenomenon of flow-solidification of lead-bismuth in a pipe. Establishing a one-dimensional flow-solidification calculation method is beneficial for saving computational resources while more accurately studying the impact of solidification on system-level parameters, thus providing a future development direction for engineering design and performance optimization.

[0004] A one-dimensional calculation method for the solidification of lead bismuth during flow in a pipe can analyze the temperature and flow rate changes after solidification and the growth of the solidified layer. It has the advantages of convenient operation and accurate calculation.

[0005] Includes the following steps:

[0006] Step 1: Assuming the existence of a solidified layer, calculate the heat difference between the inlet and outlet of the solidified layer to determine whether the fluid has reached the solidification condition. The specific steps are as follows:

[0007] Step 1-1: Compare the wall temperature with the solidus temperature of lead and bismuth. The first condition for lead and bismuth to solidify while flowing in the pipe is that the wall temperature of the pipe containing the lead and bismuth is not higher than the solidus temperature of lead and bismuth, that is:

[0008] T w ≤T s (1)

[0009] in:

[0010] T w —Wall temperature, K;

[0011] T s —The solidus temperature of lead-bismuth, in K;

[0012] If the first-level condition is not met, skip directly to step 3-2; if the first-level condition is met, skip to step 1-2.

[0013] Step 1-2: If the first layer condition is met, proceed to determine the second layer of solidification conditions: First, assume that a solidified layer is attached to the pipe wall. For this solidified layer, the heat difference between the inlet and outlet of the solidified layer per unit time step must satisfy the internal energy change generated by the solidification of fluid lead-bismuth into solid lead-bismuth. Within a time step, the solidification of the fluid will be accompanied by the loss of internal energy; therefore, the heat flowing out of the solidified layer must be greater than the heat entering the solidified layer. For this assumed solidified layer, the heat entering comes from the convective heat transfer of the fluid, and the heat flowing out comes from the heat transfer of the solidified layer to the wall. Therefore, the second layer determination condition is:

[0014] q out -q in >0 (2)

[0015] q out =h wall A s (T sw -T w (3)

[0016] q in =hA f (T f -T s (4)

[0017] in:

[0018] q out —Heat flowing out of the solidified layer per unit time step, J / s;

[0019] q in —Heat entering the solidification layer per unit time step, J / s;

[0020] h wall —Heat transfer coefficient between the solidified layer and the wall surface, W / (m²) 2 ·K);

[0021] A s —The contact area between the solidified layer and the wall surface, in m² 2 ;

[0022] T sw—Temperature of the solidified layer near the wall, K;

[0023] h — Heat transfer coefficient between the fluid and the solidified layer, W / (m²) 2 ·K);

[0024] A f —The contact area between the fluid and the solidified layer, in meters. 2 ;

[0025] T f — Fluid temperature, K;

[0026] In the initial stage of solidification, it is assumed that:

[0027] A f =A s (5)

[0028] T sw =T s (6)

[0029] If the second-level condition is not met, skip directly to step 3-2; if the second-level condition is met, skip to step 2.

[0030] Step 2: After determining that the fluid lead-bismuth meets the conditions for solidification, the solidification mass increase per unit time step is calculated using the energy equation to obtain the solidification parameters of each control volume, and then the growth of the solidified layer is calculated. The specific steps are as follows:

[0031] Step 2-1: First, calculate the internal energy loss during the solidification process of fluid lead-bismuth per unit time step. The internal energy change during the entire solidification process includes three parts: the internal energy loss ΔU1 when fluid lead-bismuth cools down to the freezing point, the internal energy loss ΔU2 when fluid lead-bismuth transforms into solid lead-bismuth, and the internal energy loss ΔU3 when solid lead-bismuth cools down. Specifically:

[0032]

[0033]

[0034]

[0035]

[0036] in:

[0037] C p,f —Specific heat capacity of liquid lead-bismuth at constant pressure, J / (kg·℃);

[0038] m s —Mass of lead-bismuth solidification, kg;

[0039] L—Latent heat of fusion of lead bismuth, J / kg;

[0040] T m —Temperature at which lead and bismuth solidify, °C;

[0041] Then, the energy equation for the solidified layer is solved to obtain the solidification mass increase at each time step, i.e.:

[0042]

[0043] Step 2-2: After calculating the solidification mass increase per unit time step, assuming that the solidification layer thickness is consistent in each control body, the solidification layer thickness increase in each control body within a unit time step can be calculated based on the density and pipeline parameters.

[0044] Step 3: Update the flow channel parameters and use the enthalpy-porous medium model to calculate the basic parameters of the remaining fluid domain. The specific steps are as follows:

[0045] Step 3-1: Treat the solidified layer that grows within a unit time step calculated in Step 2-2 as the wall surface, and extend the solidified layer temperature to the wall surface temperature to update the pipe flow parameters.

[0046] Step 3-2: For the residual fluid domain, considering the mushy region, the enthalpy-porous medium model is used. That is, for the energy equation, the total latent heat is considered to calculate the total enthalpy based on the sensible enthalpy. For the momentum equation, the source term is added to treat the mushy region as a porous medium. The specific enthalpy equation is as follows:

[0047] H add =h sen +ΔH (12)

[0048]

[0049] ΔH=βL (14)

[0050]

[0051] in:

[0052] H add —Total enthalpy, J / kg;

[0053] h sen —Explicit enthalpy, J / kg;

[0054] ΔH—Latent enthalpy, J / kg;

[0055] h ref —Reference enthalpy, J / kg;

[0056] T ref —Reference temperature, K;

[0057] β—Liquid phase fraction;

[0058] The specific momentum source terms are as follows:

[0059]

[0060] in:

[0061] ε — a constant, to prevent the denominator from being 0;

[0062] A mush —The constant of the pasty region;

[0063] u—fluid velocity, m / s;

[0064] Step 4: After each calculation, check whether the accuracy of the calculation result meets the preset requirements. If it does not meet the requirements, return to step 1 and recalculate. If it meets the requirements, proceed to step 5.

[0065] Step 5: Check whether the current calculation time step has reached the expected number of calculation time steps, and then determine whether to proceed to the next time step. If so, add one unit time step and jump to step 1 to perform the calculation until the calculation result meets the actual requirements.

[0066] The present invention has the following beneficial effects:

[0067] 1) It can calculate the impact of solidification of lead and bismuth during flow in different types of pipes under different initial conditions on system-level parameters;

[0068] 2) The calculation method is independent and highly versatile. By changing the physical properties, it can also be applied to the calculation of solidification inside pipes for different materials.

[0069] 3) The calculation method has a short setup period, consumes few computational resources, and produces high-accuracy results, making it suitable for calculating various types of solidification problems; Attached Figure Description

[0070] Figure 1 Schematic diagram of solidification component distribution inside the tube.

[0071] Figure 2 Flowchart of the calculation method of this invention Detailed Implementation

[0072] The following combination Figure 2 The flowchart shown illustrates the invention in further detail, using a typical in-tube lead-bismuth flow solidification calculation process as an example. The distribution of the inner wall, solidified layer, and fluid domain of a single control volume after lead-bismuth solidification in the tube is as follows: Figure 1 As shown.

[0073] The present invention provides a one-dimensional calculation method for the solidification of lead-bismuth during flow within a pipe, comprising the following steps:

[0074] Step 1: At the beginning of the calculation, solidification has not yet occurred, and the solidified layer does not yet exist. In order to analyze the formation and growth of the solidified layer, we assume its existence and calculate the heat difference between the fluid entering and leaving the solidified layer to determine whether the fluid has reached the solidification condition. The specific steps are as follows:

[0075] Step 1-1: Compare the wall temperature with the solidus temperature of lead and bismuth. The first condition for lead and bismuth to solidify during flow inside the pipe is that the wall temperature of the pipe containing the lead and bismuth should not exceed the solidus temperature of lead and bismuth, that is:

[0076] T w ≤T s (1)

[0077] in:

[0078] T w —Wall temperature, K;

[0079] T s —The solidus temperature of lead-bismuth, in K;

[0080] If the first-level condition is not met, skip directly to step 3-2; if the first-level condition is met, skip to step 1-2.

[0081] Step 1-2: If the first layer condition is met, proceed to determine the second layer of solidification conditions. First, assume a solidified layer adheres to the pipe wall. For this solidified layer, the heat difference between the inflow and outflow within a unit time step must satisfy the internal energy change resulting from the solidification of fluid lead-bismuth into solid lead-bismuth. Within a time step, fluid solidification is accompanied by internal energy loss; therefore, the heat flowing out of the solidified layer must be greater than the heat entering it. For this assumed solidified layer, the entering heat comes from convective heat transfer of the fluid, and the outflowing heat comes from heat transfer between the solidified layer and the pipe wall. Therefore, the second layer determination condition is:

[0082] q out -q in >0 (2)

[0083] q out =h wall A s (T sw -T w (3)

[0084] q in =hA f (T f -T s (4)

[0085] in:

[0086] q out—Heat flowing out of the solidified layer per unit time step, J / s;

[0087] q in —Heat entering the solidification layer per unit time step, J / s;

[0088] h wall —Heat transfer coefficient between the solidified layer and the wall surface, W / (m²) 2 ·K);

[0089] A s —The contact area between the solidified layer and the wall surface, in m² 2 ;

[0090] T sw —Temperature of the solidified layer near the wall, K;

[0091] T w —Wall temperature, K;

[0092] h — Heat transfer coefficient between the fluid and the solidified layer, W / (m²) 2 ·K);

[0093] A f —The contact area between the fluid and the solidified layer, in meters. 2 ;

[0094] T f — Fluid temperature, K;

[0095] T s —The solidus temperature of lead-bismuth, in K;

[0096] Before the solidified layer forms, its parameters are unknown. To perform further calculations, appropriate assumptions need to be made regarding these parameters. Therefore, in the initial stage of solidification, the following assumptions are made:

[0097] A f =A s (5)

[0098] T sw =T s (6)

[0099] If the second-level condition is not met, skip directly to step 3-2; if the second-level condition is met, skip to step 2.

[0100] Step 2: After determining that the fluid lead-bismuth meets the conditions for solidification, the solidification mass increase per unit time step is calculated using the energy equation. In reality, the solidified layer thickness varies within each control volume, but the thickness distribution is simplified in the one-dimensional system program, assuming no difference in solidified layer thickness within a single control volume. Therefore, the solidification layer growth of each control volume can be obtained from the solidification mass increase per unit time step and the density at the corresponding temperature. The specific steps are as follows:

[0101] Step 2-1: First, calculate the internal energy loss during the solidification process of fluid lead-bismuth per unit time step. The internal energy change during the entire solidification process includes three parts: the internal energy loss ΔU1 when fluid lead-bismuth cools down to the freezing point, the internal energy loss ΔU2 when fluid lead-bismuth transforms into solid lead-bismuth, and the internal energy loss ΔU3 when solid lead-bismuth cools down. Specifically:

[0102]

[0103]

[0104]

[0105]

[0106] in:

[0107] C p,f —Specific heat capacity of liquid lead-bismuth at constant pressure, J / (kg·℃);

[0108] m s —Mass of lead-bismuth solidification, kg;

[0109] L—Latent heat of fusion of lead bismuth, J / kg;

[0110] T m —Temperature at which lead and bismuth solidify, °C;

[0111] Then, the energy equation for the solidified layer is solved to obtain the solidification mass increase at each time step, i.e.:

[0112]

[0113] Step 2-2: After calculating the solidification mass increase per unit time step, assuming that the solidification layer thickness in each control body is consistent, the solidification layer thickness increase in each control body within a unit time step can be calculated based on the density and pipeline parameters.

[0114] Step 3: After calculating the new solidified layer thickness, the newly grown solidified layer is regarded as a new wall. In the next calculation, the parameters of this solidified layer are used to replace the original wall parameters, while the flow channel parameters are updated. The enthalpy-porous medium model is then used to calculate the basic parameters of the remaining fluid domain. The specific steps are as follows:

[0115] Step 3-1: Treat the solidified layer that grows within a unit time step calculated in Step 2-2 as the wall surface, and extend the solidified layer temperature to the wall surface temperature to update the pipe flow parameters.

[0116] Step 3-2: For the residual fluid domain, considering the mushy region, the enthalpy-porous medium model is used. That is, for the energy equation, the total latent heat is considered to calculate the total enthalpy based on the sensible enthalpy. For the momentum equation, the source term is added to treat the mushy region as a porous medium. The specific enthalpy equation is as follows:

[0117] H add =h+ΔH (12)

[0118]

[0119] ΔH=βL (14)

[0120]

[0121] in:

[0122] H add —Total enthalpy, J / kg;

[0123] h sen —Explicit enthalpy, J / kg;

[0124] ΔH—Latent enthalpy, J / kg;

[0125] h ref —Reference enthalpy, J / kg;

[0126] T ref —Reference temperature, K;

[0127] β—Liquid phase fraction

[0128] The specific momentum source terms are as follows:

[0129]

[0130] in:

[0131] ε — a constant, to prevent the denominator from being 0;

[0132] A mush —The constant of the pasty region;

[0133] u—fluid velocity, m / s;

[0134] Step 4: After each calculation, check whether the accuracy of the calculation result meets the preset requirements. If it does not meet the requirements, return to step 1 and recalculate. If it meets the requirements, proceed to step 5.

[0135] Step 5: Check whether the current calculation time step has reached the expected number of calculation time steps, and then determine whether to proceed to the next time step. If so, add one unit time step and jump to step 1 to perform the calculation until the calculation result meets the actual requirements.

Claims

1. A one-dimensional calculation method for the solidification of lead-bismuth during flow in a pipe, characterized in that: A one-dimensional calculation method for lead-bismuth flow solidification is proposed, which can analyze the temperature and flow rate changes after solidification of lead-bismuth during the flow process, as well as the growth of the solidified layer. It has the advantages of convenient operation and accurate calculation. Includes the following steps: Step 1: Assuming the existence of a solidified layer, calculate the heat difference between the inlet and outlet of the solidified layer to determine whether the fluid has reached the solidification condition. The specific steps are as follows: Step 1-1: Compare the wall temperature with the solidus temperature of lead and bismuth. The first condition for lead and bismuth to solidify while flowing in the pipe is that the wall temperature of the pipe containing the lead and bismuth is not higher than the solidus temperature of lead and bismuth, that is: T w ≤T s (1) in: T w —Wall temperature, K; T s —The solidus temperature of lead-bismuth, in K; If the first-level condition is not met, skip directly to step 3-2; if the first-level condition is met, skip to step 1-2. Step 1-2: If the first layer condition is met, proceed to determine the second layer of solidification conditions: First, assume that a solidified layer is attached to the pipe wall. For this solidified layer, the heat difference between the inlet and outlet of the solidified layer per unit time step must satisfy the internal energy change generated by the solidification of fluid lead-bismuth into solid lead-bismuth. Within a time step, the solidification of the fluid will be accompanied by the loss of internal energy; therefore, the heat flowing out of the solidified layer must be greater than the heat entering the solidified layer. For this assumed solidified layer, the heat entering comes from the convective heat transfer of the fluid, and the heat flowing out comes from the heat transfer of the solidified layer to the wall. Therefore, the second layer determination condition is: q out -q in >0 (2) q out =h wall A s (T sw -T w ) (3) q in =hA f (T f -T s ) (4) in: q out —Heat flowing out of the solidified layer per unit time step, J / s; q in —The amount of heat entering the solidification layer per unit time step, J / s; h wall —Heat transfer coefficient between the solidified layer and the wall surface, W / (m²) 2 ·K); A s —The contact area between the solidified layer and the wall surface, in m² 2 ; T sw —Temperature of the solidified layer near the wall, K; h — Heat transfer coefficient between the fluid and the solidified layer, W / (m²) 2 ·K); A f —The contact area between the fluid and the solidified layer, in meters. 2 ; T f — Fluid temperature, K; In the initial stage of solidification, it is assumed that: A f =A s (5) T sw =T s (6) If the second-level condition is not met, skip directly to step 3-2; if the second-level condition is met, skip to step 2. Step 2: After determining that the fluid lead-bismuth meets the conditions for solidification, the solidification mass increase per unit time step is calculated using the energy equation to obtain the solidification parameters of each control volume, and then the growth of the solidified layer is calculated. The specific steps are as follows: Step 2-1: First, calculate the internal energy loss during the solidification process of fluid lead-bismuth per unit time step. The internal energy change during the entire solidification process includes three parts: the internal energy loss ΔU1 when fluid lead-bismuth cools down to the freezing point, the internal energy loss ΔU2 when fluid lead-bismuth transforms into solid lead-bismuth, and the internal energy loss ΔU3 when solid lead-bismuth cools down. Specifically: in: C p,f —Specific heat capacity of liquid lead-bismuth at constant pressure, J / (kg·℃); m s —Mass of lead-bismuth solidification, kg; L—Latent heat of fusion of lead bismuth, J / kg; T m —Temperature at which lead and bismuth solidify, °C; Then, the energy equation for the solidified layer is solved to obtain the solidification mass increase at each time step, i.e.: Step 2-2: After calculating the solidification mass increase per unit time step, assuming that the solidification layer thickness is consistent in each control body, the solidification layer thickness increase in each control body within a unit time step can be calculated based on the density and pipeline parameters. Step 3: Update the flow channel parameters and use the enthalpy-porous medium model to calculate the basic parameters of the remaining fluid domain. The specific steps are as follows: Step 3-1: Treat the solidified layer that grows within a unit time step calculated in Step 2-2 as the wall surface, and extend the solidified layer temperature to the wall surface temperature to update the pipe flow parameters. Step 3-2: For the residual fluid domain, considering the mushy region, the enthalpy-porous medium model is used. That is, for the energy equation, the total latent heat is considered to calculate the total enthalpy based on the sensible enthalpy. For the momentum equation, the source term is added to treat the mushy region as a porous medium. The specific enthalpy equation is as follows: H add =h sen +ΔH (12) ΔH=βL (14) in: H add —Total enthalpy, J / kg; h sen —Explicit enthalpy, J / kg; ΔH—Latent enthalpy, J / kg; h ref —Reference enthalpy, J / kg; T ref —Reference temperature, K; β—Liquid phase fraction; The specific momentum source terms are as follows: in: ε — a constant, to prevent the denominator from being 0; A mush —The constant of the pasty region; u—fluid velocity, m / s; Step 4: After each calculation, check whether the accuracy of the calculation result meets the preset requirements. If it does not meet the requirements, return to step 1 and recalculate. If it meets the requirements, proceed to step 5. Step 5: Check whether the current calculation time step has reached the expected number of calculation time steps, and then determine whether to proceed to the next time step. If so, add one unit time step and jump to step 1 to perform the calculation until the calculation result meets the actual requirements.

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