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CN114154438A - Three-stage calculation method for cold start of alkali metal heat pipe - Google Patents

Three-stage calculation method for cold start of alkali metal heat pipe Download PDF

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CN114154438A
CN114154438A CN202111486969.0A CN202111486969A CN114154438A CN 114154438 A CN114154438 A CN 114154438A CN 202111486969 A CN202111486969 A CN 202111486969A CN 114154438 A CN114154438 A CN 114154438A
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heat pipe
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王成龙
张泽秦
田智星
郭凯伦
张大林
秋穗正
苏光辉
田文喜
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Xian Jiaotong University
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    • F28D15/02—Heat-exchange apparatus with the intermediate heat-transfer medium in closed tubes passing into or through the conduit walls ; Heat-exchange apparatus employing intermediate heat-transfer medium or bodies in which the medium condenses and evaporates, e.g. heat pipes
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    • F28D15/04—Heat-exchange apparatus with the intermediate heat-transfer medium in closed tubes passing into or through the conduit walls ; Heat-exchange apparatus employing intermediate heat-transfer medium or bodies in which the medium condenses and evaporates, e.g. heat pipes with tubes having a capillary structure
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Abstract

A three-stage calculation method for cold start of an alkali metal heat pipe comprises the following steps of 1, determining geometric parameters and boundary conditions of the alkali metal heat pipe; 2. dividing a calculation control body and setting calculation initial conditions; 3. calculating the temperature change rate of the alkali metal heat pipe wall control body, and regarding the alkali metal heat pipe wall control body as pure heat conduction; 4. calculating the temperature change rate of the alkali metal heat pipe liquid absorption core control body, neglecting the flow of the working medium and regarding the working medium as pure heat conduction; 5. judging the starting stage of the heat pipe according to the temperature of the outermost node of the liquid absorption core of the alkali metal heat pipe; 6. calculating the temperature, speed, density, pressure and gas content of the control body of the vapor area of the alkali metal heat pipe according to different starting stages; 7. solving an equation set by using a Gear algorithm to complete the calculation of the current time node; 8. checking the heat transfer limit of the heat pipe, and updating the heat transfer capacity of the alkali metal heat pipe; 9. repeating steps 3-8 according to the new heat transfer amount until the set total number of time steps is reached. The invention provides suggestions and guidance for the engineering application of the alkali metal heat pipe aiming at the transient calculation of the cold start of the alkali metal heat pipe.

Description

Three-stage calculation method for cold start of alkali metal heat pipe
Technical Field
The invention relates to the technical field of phase change heat exchange equipment, in particular to a cold start three-stage calculation method for an alkali metal high-temperature heat pipe.
Background
The heat pipe is an efficient heat transmission device, has passive working characteristics, cannot influence the whole heat transmission system due to the failure of a single heat pipe, can ensure the inherent safety of the heat transmission system, has large heat transmission capacity, strong heat transfer capacity and high transmission efficiency, can greatly improve the working performance of the system, and is widely applied to the fields of aerospace, chemical engineering, nuclear energy and the like. The heat pipe mainly transfers heat by means of phase change, the working medium absorbs heat at the evaporation section and evaporates into steam, the steam flows to the condensation section through the steam cavity, the heat released at the condensation section is condensed into liquid, and the liquid returns to the evaporation section through capillary action to complete circulation. The working temperature of the high-temperature heat pipe is above 600 ℃, the heat pipe usually adopts alkali metal as a heat pipe working medium, the alkali metal is solid at normal temperature, so a complex cold-state starting process needs to be performed, heat and mass transfer and steam flow coupling calculation in a heat pipe wall, a liquid absorption core and a steam area inside the heat pipe is involved, great difficulty is brought to numerical simulation calculation aiming at the starting stage of the alkali metal heat pipe, and the existing thermal resistance network method mainly aims at the steady-state calculation of the heat pipe, and the accurate simulation of the starting stage of the alkali metal heat pipe is difficult to realize.
Disclosure of Invention
In order to overcome the problems in the prior art, the invention aims to provide a three-stage calculation method for cold start of an alkali metal heat pipe. The invention provides theoretical suggestion and guidance for the development of the alkali metal high-temperature heat pipe and the analysis of the heat and mass transfer mechanism.
In order to achieve the purpose, the invention adopts the following technical scheme to implement:
a three-stage calculation method for cold start of an alkali metal heat pipe comprises the following steps:
step 1: determining the geometric dimension of the heat pipe, a working medium, a pipe wall material, a liquid absorption core structure and boundary conditions of an evaporation section and a condensation section; the evaporation section adopts a second type of boundary condition, the heat flux density is given, the condensation section adopts a third type of boundary condition, and the convection heat transfer coefficient is given; determining the calculated single step time step length and the total step number;
step 2: dividing a heat pipe control body, axially dividing an e-layer control body along an evaporation section, dividing an a-layer control body along a heat insulation section, dividing a c-layer control body along a condensation section, radially dividing w-layer control bodies along the wall of a heat pipe, dividing a p-layer control body along a liquid absorption core of the heat pipe, and dividing 1 layer along a vapor space; initializing parameters of the control body obtained by dividing, and setting the temperature of the control body as a starting initial temperature;
and step 3: calculating the temperature change rate of the pipe wall area of the heat pipe
Figure BDA0003396987730000021
Establishing a two-dimensional heat conduction equation in the pipe wall area of the heat pipe, wherein the control equation is as follows:
Figure BDA0003396987730000022
in the formula: t iswIs the temperature of the wall area of the heat pipe, t is the time, CwIs the volumetric heat capacity, k, of the wall region of the heat pipewIs the heat conductivity coefficient of the wall area of the heat pipe, r is the radial direction of the heat pipe, and z is the axial direction of the heat pipe；
The boundary conditions of control equation (1) are:
an evaporation section:
Figure BDA0003396987730000031
adiabatic section:
Figure BDA0003396987730000032
a condensation section:
hAc(Tw-Tsur)＝Qc (4)
in the formula: a. theeIs the area of the evaporation section of the heat pipe, AcIs the area of the condensation section of the heat pipe, QeFor heating power in the evaporation zone, QcCooling power of the condensing section, h is the convective heat transfer coefficient of the outer surface of the condensing section, and TsurIs ambient temperature;
and 4, step 4: calculating the temperature change rate of the liquid absorption core area of the heat pipe
Figure BDA0003396987730000033
Because the flow velocity in the liquid absorbing core of the heat pipe is very low, the flow of the working medium in the liquid absorbing core is neglected, the area of the liquid absorbing core of the heat pipe is regarded as a mixed solid formed by a static liquid working medium and a solid wire mesh, a two-dimensional heat conduction equation is established in the area of the liquid absorbing core of the heat pipe, and the control equation is as follows:
Figure BDA0003396987730000034
in the formula: t ispIs the temperature of the wick region of the heat pipe, CeffIs the equivalent volumetric heat capacity, k, of the wick region of the heat pipeeffThe equivalent thermal conductivity of the heat pipe wick area;
volumetric heat capacity C of the mixed matrixeffAnd coefficient of thermal conductivity keffThe calculation is made according to the following equation:
Ceff＝εCl+(1-ε)Cs (6)
Figure BDA0003396987730000035
in the formula: clIs the volumetric heat capacity of the liquid working medium, CsIs the volumetric heat capacity, k, of the wick wirelIs the thermal conductivity, k, of the liquid working mediumsThe thermal conductivity of the wick wire mesh is, and epsilon is the porosity of the wick wire mesh;
and 5: judging the starting stage of the heat pipe according to the outermost node temperature of the liquid absorption core of the alkali metal heat pipe: dividing the heat pipe starting into a first starting stage, a second starting stage and a third starting stage, and judging the current starting stage of the heat pipe by comparing the vapor transition temperature of the alkali metal working medium with the temperature of the outermost control body of the liquid absorption core area of the alkali metal heat pipe, namely the interface temperature of the liquid absorption core area and the vapor area, which is obtained by calculation at the last moment, and then collectively calling the gas-liquid interface temperature;
the transition temperature of the alkali metal working fluid is calculated by iterating the formula:
Figure BDA0003396987730000041
in the formula: t istIs the transition temperature of the alkali metal working medium, M is the relative molecular mass of the alkali metal working medium, RuFor ideal gas constants, ρ is the density of the gas, μ is the kinetic viscosity of the gas, D is the diameter of the vapor region of the heat pipe, and for a given heat pipe structure and heat pipe working medium, the transition temperature T of the alkali metal working mediumtIs uniquely determined;
if the temperature of all control bodies on the gas-liquid interface is less than the transition temperature T of the alkali metal working mediumtIf yes, judging to start the first stage; if the gas-liquid interface already has the temperature of the control body more than or equal to the transition temperature T of the alkali metal working mediumtBut still has the gas-liquid interface control body with the temperature less than the transition temperature T of the alkali metal working mediumtIf yes, judging that the second stage is started; if the temperature of all control bodies on the gas-liquid interface is more than or equal to the transition temperature T of the alkali metal working mediumtIf yes, judging to be a third stage of starting;
step 6: calculating the temperature, the speed, the density, the pressure and the gas content of the vapor area control body of the alkali metal heat pipe according to different starting stages:
if the first stage is started, the vapor region is considered to be in a free molecular state, the evaporation and heat transfer of the vapor are ignored, and the adiabatic boundary condition is adopted for the gas-liquid interface:
Figure BDA0003396987730000042
in the formula: t isvIs the temperature of the vapor region of the heat pipe, RgIs the radius of the vapor zone;
then the temperature of the vapor region control body at this point is equal to the temperature of the wick region control body adjacent thereto:
Figure BDA0003396987730000051
in the formula: t isv,iTemperature, T, of ith control body in axial direction for vapor region of heat pipep,iThe temperature of the ith control body along the axial direction of the heat pipe liquid absorption core area is measured;
if the second stage is started, the vapor area control bodies adjacent to the control body with the gas-liquid interface reaching the transition temperature of the alkali metal working medium are considered to be in continuous state flow, the other vapor area control bodies are still in free molecular state, the density, the gas content and the speed parameters of vapor are ignored, and the normalized vapor temperature is obtained in all the continuous flow control bodies through the iteration formula:
Figure BDA0003396987730000052
in the formula: a isccFor cell adjustment factor, hfvW is the latent heat of vaporization of the working medium, the circumference of the vapor region, meThe number of gas-liquid interface control bodies for achieving the transition temperature is not more than e + a and delta liIs the width of the ith control body, TfiTemperature of i-th control body for gas-liquid interface, PfiIs TfiCorresponding saturated vapor pressure, TvIs the temperature of the vapour region, PvIs TvCorresponding saturated vapor pressure, ρvIs TvCorresponding vapor density, AcIs the area of the steam area, and gamma is the heat capacity ratio of the gaseous working medium;
if, for the third stage of actuation, a continuous flow of vapor is deemed to have been fully established in the vapor space, the temperature, velocity, density, pressure and vapor fraction of the vapor zone control volume are obtained by the following equations:
Figure BDA0003396987730000053
Figure BDA0003396987730000054
Figure BDA0003396987730000055
Figure BDA0003396987730000061
Figure BDA0003396987730000062
in the formula: ρ is the vapor density, XqIs vapor gas content, V is vapor velocity, P is vapor pressure, T is vapor temperature, D is vapor zone diameter, upsilon is specific volume of gas-liquid mixed fluid, upsilon isgIs saturated specific vapor volume, upsilonfIs the specific volume of saturated liquid, h is the enthalpy value of gas-liquid mixed fluid, h0Is vapor enthalpy of gas-liquid interface, hfgIs the latent heat of vaporization of the working medium, V0Is the normal velocity of the vapor at the gas-liquid interface,
Figure BDA0003396987730000063
is the mass evaporation rate of the gas-liquid interface, cpIs the constant pressure specific heat capacity of steam, FfAs an interphase friction factor, MfIs a momentum factor, EfIs an energy factor;
specific volume upsilon of gas-liquid mixed fluid, enthalpy value h of gas-liquid mixed fluid and specific volume upsilon of saturated steamgCalculated from the following formula:
υ＝υf+Xq×(υg-υf) (17)
h＝hf+Xq×hfg (18)
Figure BDA0003396987730000064
in the formula: h isfIs the enthalpy of the saturated liquid; mass evaporation rate of gas-liquid interface
Figure BDA0003396987730000065
Calculated from the following formula:
Figure BDA0003396987730000066
in the formula: t isfIs the temperature of the gas-liquid interface, PfIs TfThe corresponding saturated vapor pressure;
normal velocity V of vapor at gas-liquid interface0Calculated from the following formula:
Figure BDA0003396987730000067
in the formula: qintFor input of thermal power to gas-liquid interface, AintIs the area of the gas-liquid interface, pintThe vapor density corresponding to the gas-liquid interface temperature;
interphase friction factor FfMomentum factor MfEnergy factor EfThe following equation is used:
Figure BDA0003396987730000071
Figure BDA0003396987730000072
Figure BDA0003396987730000073
in the formula:
Figure BDA0003396987730000074
Voobtained by the formula (21), D is the diameter of the steam area, and v is the kinematic viscosity of the steam;
and 7: the control equations of each control body are dispersed, and the initial value problem converted into the nonlinear ordinary differential equation set is solved, and the initial value problem has the following form:
Figure BDA0003396987730000075
in the formula:
Figure BDA0003396987730000076
for the solution of the equation at time t,
Figure BDA0003396987730000077
is composed of
Figure BDA0003396987730000078
Is a derivative function of f
Figure BDA0003396987730000079
And
Figure BDA00033969877300000710
the implicit function of the relationship between (a) and (b),
Figure BDA00033969877300000711
is composed of
Figure BDA00033969877300000712
An initial value at time 0; the equation set is respectively solved by adopting a Gear algorithm, and a post-orientation difference grid is adopted according to a time term, and the post-orientation difference grid has a difference equation with the following form:
Figure BDA00033969877300000713
in the formula: σ is a time step and satisfies ti＝t0+ i σ, F is a constructor of F and satisfies
Figure BDA00033969877300000714
With a single step push, the difference equation (23) has a solution of y0,y1,y2,…,yn}; and let G be max F, α be max ti-t0|，β＝max||yi-y0||，
Figure BDA00033969877300000715
Constructing a function vector Zn(t) making Zn(tk)＝ykAnd Z'n(tk)＝F(tk-1,yk-1,ykH) then the remainder Rn(t) is expressed as:
Figure BDA0003396987730000081
giving tolerance errors tol only by ensuring
Figure BDA0003396987730000082
Even if the initial value is questionedThe solution to the problem (22) converges consistently, with the convergence criteria:
Figure BDA0003396987730000083
in actual calculation, the iteration times are specified, if n still does not satisfy the formula (25) after the iteration times are exceeded, the calculation result is judged to be not converged, the calculation time step length sigma is shortened, and calculation is carried out again until the calculation result is converged; if n satisfies the formula (25) within the iteration times, judging that the calculation result is converged, and covering the calculated value as an initial value of a new time step on the current value; if the time step length is shortened to be smaller than the shortening amount, the calculation is still not converged, the output is not converged, and the calculation is stopped; the discrete control equation set of the steam belongs to the boundary value problem of the nonlinear ordinary differential equation set, and the four-order Runge-Kutta method is adopted to solve the boundary value problem, so that the boundary value problem is bound to be converged;
and 8: checking the heat transfer limit of the heat pipe, updating the heat transfer capacity of the alkali metal heat pipe, and considering the sound velocity limit, the carrying limit, the viscosity limit and the capillary limit, calculating according to the following formula:
Figure BDA0003396987730000084
Figure BDA0003396987730000085
Figure BDA0003396987730000086
Figure BDA0003396987730000087
in the formula: qsIs the limit of sound velocity, QxIs the carrying limit, QvIs the viscosity limit, QmIs the capillary limit, ToIs the vapor temperature at the beginning of the evaporation section, poIs ToLower saturated vapor density, μvIs the dynamic viscosity of the steam, σwIs the surface tension of the working medium, rhsIs the hydraulic radius, ρ, of the capillary wicklIs the density of the liquid working medium, dvIs the vapor space diameter, theta is the heat pipe axial inclination angle, LtIs the total length of the heat pipe, LeIs the length of the evaporation section of the heat pipe, FlIs the liquid phase coefficient of friction, FvIs the gas phase coefficient of friction;
if any one of the calculated sound velocity limit and viscosity limit is less than or equal to the heat transfer quantity of the current heat pipe, replacing the heat transfer quantity of the heat pipe with a heat transfer limit value less than or equal to the heat transfer quantity of the current heat pipe; if any one of the calculated carrying limit and the capillary limit is less than or equal to the heat transfer quantity of the current heat pipe, the heat pipe is considered to be failed to start, the start failure is output, and the calculation is stopped; if the calculated sound velocity limit, viscosity limit, carrying limit and capillary limit are all larger than the heat transfer capacity of the current heat pipe, the heat transfer limit is not met;
and step 9: and (5) repeating the steps 3-8 according to the new heat transfer quantity until the set total time step is reached, completing the calculation, and outputting a starting calculation result.
Preferably, the shortened quantity in the shortened computation time step of step 7 is 1 × 10-5s。
Preferably, in step 6, the unit adjustment coefficient a is set for the working alkali metal substance cc1 is taken.
Compared with the prior art, the invention has the following advantages:
(1) the actual two-dimensional heat transfer of the wall surface of the heat pipe and the liquid absorption core is considered, and the temperature change of the heat pipe from the evaporation section to the heat insulation section to the condensation section can be calculated. (2) The heat and mass transfer of the gas-liquid interface of the heat pipe is calculated by a theoretical formula and is coupled with the heat and mass transfer of the steam area. (3) Five important parameters of the flow speed, pressure, density, gas content and temperature of the steam in the steam space of the heat pipe can be obtained. (4) The limit of the heat transfer limit of the heat pipe to the actual heat transfer quantity of the gas-liquid interface is considered.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic diagram of a modeling area of the present invention.
Fig. 3 is a system control division diagram of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
As shown in FIG. 1, the three-stage calculation method for cold start of an alkali metal heat pipe of the present invention comprises the following steps:
step 1: as shown in fig. 2, since the alkali metal heat pipe is generally manufactured in a symmetrical shape, a half axial cross section of the alkali metal heat pipe is selected as a modeling object, and the geometric size of the heat pipe, the working medium, the pipe wall material, the liquid absorbing core structure, and the boundary conditions of the evaporation section and the condensation section are determined; because the evaporation section of the heat pipe is generally heated by constant heat flux density, and the condensation section is generally cooled by air or coolant through convective heat transfer, the evaporation section adopts a second type of boundary condition, the heat flux density is given, the condensation section adopts a third type of boundary condition, and the convective heat transfer coefficient is given; determining the calculated single step time step length and the total step number;
step 2: as shown in fig. 3, for the modeling object selected in step 1, dividing the heat pipe control bodies, axially dividing the e-layer control bodies along the evaporation section, the a-layer control bodies along the heat insulation section, the c-layer control bodies along the condensation section, radially dividing the w-layer control bodies along the wall of the heat pipe, dividing the p-layer control bodies along the wick of the heat pipe, and dividing 1 layer along the vapor space; obtaining total (e + a + c) x (w + p +1) control bodies, initializing parameters of the control bodies obtained by dividing, and setting the temperature of the control bodies as the initial starting temperature;
and step 3: calculating the temperature change rate of the pipe wall area of the heat pipe
Figure BDA0003396987730000101
Because the heat transfer of the heat pipe wall is a pure heat conduction process, a two-dimensional heat conduction equation is established in the area of the heat pipe wall, and the control equation is as follows:
Figure BDA0003396987730000111
in the formula: t iswIs the temperature of the wall area of the heat pipe, t is the time, CwIs the volumetric heat capacity, k, of the wall region of the heat pipewThe heat conductivity coefficient of the pipe wall area of the heat pipe is shown, r is the radial direction of the heat pipe, and z is the axial direction of the heat pipe;
the boundary conditions of control equation (1) are:
an evaporation section:
Figure BDA0003396987730000112
adiabatic section:
Figure BDA0003396987730000113
a condensation section:
hAc(Tw-Tsur)＝Qc (4)
in the formula: a. theeIs the area of the evaporation section of the heat pipe, AcIs the area of the condensation section of the heat pipe, QeFor heating power in the evaporation zone, QcCooling power of the condensing section, h is the convective heat transfer coefficient of the outer surface of the condensing section, and TsurIs ambient temperature;
and 4, step 4: calculating the temperature change rate of the liquid absorption core area of the heat pipe
Figure BDA0003396987730000114
Because the flow velocity in the liquid absorption core of the heat pipe is very low, and the heat conductivity coefficient of the alkali metal working medium is very high, the diffusion effect is considered to be far greater than the convection effect in the heat transfer process of the liquid absorption core area, the flow of the working medium in the liquid absorption core is neglected, the liquid absorption core area of the heat pipe is considered as a mixed solid formed by a static liquid working medium and a solid wire mesh, the heat transfer process of the liquid absorption core area is considered as a pure heat transfer process, a two-dimensional heat transfer equation is established in the liquid absorption core area of the heat pipe, and the control equation is as follows:
Figure BDA0003396987730000115
in the formula: t ispIs the temperature of the wick region of the heat pipe, CeffIs the equivalent volumetric heat capacity, k, of the wick region of the heat pipeeffThe equivalent thermal conductivity of the heat pipe wick area;
volumetric heat capacity C of the mixed matrixeffAnd coefficient of thermal conductivity keffThe calculation is made according to the following equation:
Ceff＝εCl+(1-ε)Cs (6)
Figure BDA0003396987730000121
in the formula: clIs the volumetric heat capacity of the liquid working medium, CsIs the volumetric heat capacity, k, of the wick wirelIs the thermal conductivity, k, of the liquid working mediumsThe thermal conductivity of the wick wire mesh is, and epsilon is the porosity of the wick wire mesh;
and 5: judging the starting stage of the heat pipe according to the outermost node temperature of the liquid absorption core of the alkali metal heat pipe: dividing the heat pipe starting into a first starting stage, a second starting stage and a third starting stage, and judging the current starting stage of the heat pipe by comparing the vapor transition temperature of the alkali metal working medium with the temperature of the outermost control body of the liquid absorption core area of the alkali metal heat pipe, namely the interface temperature of the liquid absorption core area and the vapor area, which is obtained by calculation at the last moment, and then collectively calling the gas-liquid interface temperature;
the transition temperature of the alkali metal working fluid is calculated by iterating the formula:
Figure BDA0003396987730000122
in the formula: t istIs the transition temperature of the alkali metal working medium, M is the relative molecular mass of the alkali metal working medium, RuFor ideal gas constants, ρ is the density of the gas, μ is the kinetic viscosity of the gas, and D is the heat pipeDiameter of the vapor region, transition temperature T of the alkali metal working medium for a defined heat pipe structure and heat pipe working mediumtIs uniquely determined;
if the temperature of all control bodies on the gas-liquid interface is less than the transition temperature T of the alkali metal working mediumtIf yes, judging to start the first stage; if the gas-liquid interface already has the temperature of the control body more than or equal to the transition temperature T of the alkali metal working mediumtBut still has the gas-liquid interface control body with the temperature less than the transition temperature T of the alkali metal working mediumtIf yes, judging that the second stage is started; if the temperature of all control bodies on the gas-liquid interface is more than or equal to the transition temperature T of the alkali metal working mediumtIf yes, judging to be a third stage of starting;
step 6: calculating the temperature, the speed, the density, the pressure and the gas content of the vapor area control body of the alkali metal heat pipe according to different starting stages:
if the first stage is started, the steam area is considered to be in a free molecular state, the average free path of steam molecules in the state is far larger than the diameter of the steam area, the heat exchange efficiency among molecules is extremely low, the evaporation and the heat transfer of the steam are ignored, and an adiabatic boundary condition is adopted for a gas-liquid interface:
Figure BDA0003396987730000131
in the formula: t isvIs the temperature of the vapor region of the heat pipe, RgIs the radius of the vapor zone;
then the temperature of the vapor region control body at this point is equal to the temperature of the wick region control body adjacent thereto:
Figure BDA0003396987730000132
in the formula: t isv,iTemperature, T, of ith control body in axial direction for vapor region of heat pipep,iThe temperature of the ith control body along the axial direction of the heat pipe liquid absorption core area is measured;
if the second stage is started, the continuous state flow is considered in the steam area control body adjacent to the control body, the gas-liquid interface of which reaches the transition temperature of the alkali metal working medium, and the other steam area control bodies are still in the free molecular state, at the moment, the pressure gradient on the interface of the continuous state flow and the free molecular state in the steam area is extremely large, and the pressure gradient in the continuous state flow can be ignored and not recorded, so that the temperature drop in the continuous state flow is extremely low, all the continuous state steam can be considered to be at the same normalized temperature, the density, the gas content and the speed parameter of the steam are ignored, and the normalized steam temperature is obtained through the following formula in all the continuous flow control bodies in an iterative manner:
Figure BDA0003396987730000133
in the formula: a isccFor cell adjustment factor, hfvW is the latent heat of vaporization of the working medium, the circumference of the vapor region, meThe number of gas-liquid interface control bodies for achieving the transition temperature is not more than e + a and delta liIs the width of the ith control body, TfiTemperature of i-th control body for gas-liquid interface, PfiIs TfiCorresponding saturated vapor pressure, TvIs the temperature of the vapour region, PvIs TvCorresponding saturated vapor pressure, ρvIs TvCorresponding vapor density, AcIs the area of the steam area, and gamma is the heat capacity ratio of the gaseous working medium;
if the third stage is started, the continuous flow of the vapor in the vapor space is considered to be completely established, the pressure drop in the vapor space along the axial direction cannot be ignored, the NS equation of the compressible vapor needs to be solved, the NS equation of the vapor flow can be simplified into a one-dimensional quasi-steady-state compressible control equation under the assumption that the velocity direction of the vapor on the vapor-liquid interface in the heat pipe is immediately changed from the normal direction to the axial direction after the vapor is generated, and the continuous vapor in the heat pipe flows in a laminar flow manner, and the temperature, the velocity, the density, the pressure and the gas content of the vapor region control body are obtained through the following equations:
Figure BDA0003396987730000141
Figure BDA0003396987730000142
Figure BDA0003396987730000143
Figure BDA0003396987730000144
Figure BDA0003396987730000145
in the formula: ρ is the vapor density, XqIs vapor gas content, V is vapor velocity, P is vapor pressure, T is vapor temperature, D is vapor zone diameter, upsilon is specific volume of gas-liquid mixed fluid, upsilon isgIs saturated specific vapor volume, upsilonfIs the specific volume of saturated liquid, h is the enthalpy value of gas-liquid mixed fluid, h0Is vapor enthalpy of gas-liquid interface, hfgIs the latent heat of vaporization of the working medium, V0Is the normal velocity of the vapor at the gas-liquid interface,
Figure BDA0003396987730000146
is the mass evaporation rate of the gas-liquid interface, cpIs the constant pressure specific heat capacity of steam, FfAs an interphase friction factor, MfIs a momentum factor, EfIs an energy factor;
specific volume upsilon of gas-liquid mixed fluid, enthalpy value h of gas-liquid mixed fluid and specific volume upsilon of saturated steamgCalculated from the following formula:
υ＝υf+Xq×(υg-υf) (17)
h＝hf+Xq×hfg (18)
Figure BDA0003396987730000151
in the formula: h isfIs the enthalpy of the saturated liquid; mass evaporation rate of gas-liquid interface
Figure BDA0003396987730000152
Calculated from the following formula:
Figure BDA0003396987730000153
in the formula: t isfIs the temperature of the gas-liquid interface, PfIs TfThe corresponding saturated vapor pressure;
normal velocity V of vapor at gas-liquid interface0Calculated from the following formula:
Figure BDA0003396987730000154
in the formula: qintFor input of thermal power to gas-liquid interface, AintIs the area of the gas-liquid interface, pintThe vapor density corresponding to the gas-liquid interface temperature;
due to the interphase friction factor FfMomentum factor MfEnergy factor EfReynolds number Re of flow with vapour onlyoIt is therefore possible to find it from the following empirical relationship:
Figure BDA0003396987730000155
Figure BDA0003396987730000156
Figure BDA0003396987730000161
in the formula:
Figure BDA0003396987730000162
Voobtained by the formula (21), D is the diameter of the steam area, and v is the kinematic viscosity of the steam;
and 7: the control equations of each control body are dispersed, and the initial value problem converted into the nonlinear ordinary differential equation set is solved, and the initial value problem has the following form:
Figure BDA0003396987730000163
in the formula:
Figure BDA0003396987730000164
for the solution of the equation at time t,
Figure BDA0003396987730000165
is composed of
Figure BDA0003396987730000166
Is a derivative function of f
Figure BDA0003396987730000167
And
Figure BDA0003396987730000168
the implicit function of the relationship between (a) and (b),
Figure BDA0003396987730000169
is composed of
Figure BDA00033969877300001610
An initial value at time 0; the system of equations is a system of ill-conditioned equations, and therefore solved separately using Gear algorithms, taking a backward difference format for the time term, which has a difference equation of the form:
Figure BDA00033969877300001611
in the formula: σ is a time step and satisfies ti＝t0+ i σ, F is a constructor of F and satisfies
Figure BDA00033969877300001612
With a single step push, the difference equation (23) has a solution of y0,y1,y2,…,yn}; and let G be max F, α be max ti-t0|，β＝max||yi-y0||，
Figure BDA00033969877300001613
Constructing a function vector Zn(t) making Zn(tk)＝ykAnd Z'n(tk)＝F(tk-1,yk-1,ykH) then the remainder Rn(t) can be expressed as:
Figure BDA00033969877300001614
giving tolerance errors tol only by ensuring
Figure BDA00033969877300001615
The solution of the initial value problem (22) can be converged uniformly, and the convergence condition is as follows:
Figure BDA00033969877300001616
in actual calculation, the iteration times are specified, if n still does not satisfy the formula (25) after the iteration times are exceeded, the calculation result is judged to be not converged, the calculation time step length sigma is shortened, and calculation is carried out again until the calculation result is converged; if n satisfies the formula (25) within the iteration times, judging that the calculation result is converged, and covering the calculated value as an initial value of a new time step on the current value; if the time step length is shortened to be smaller than the shortening amount, the calculation is still not converged, the output is not converged, and the calculation is stopped; the discrete control equation set of the steam belongs to the boundary value problem of the nonlinear ordinary differential equation set, and the four-order Runge-Kutta method is adopted to solve the boundary value problem, so that the boundary value problem is bound to be converged;
and 8: checking the heat transfer limit of the heat pipe, updating the heat transfer capacity of the alkali metal heat pipe, and considering the sound velocity limit, the carrying limit, the viscosity limit and the capillary limit, calculating according to the following formula:
Figure BDA0003396987730000171
Figure BDA0003396987730000172
Figure BDA0003396987730000173
Figure BDA0003396987730000174
in the formula: qsIs the limit of sound velocity, QxIs the carrying limit, QvIs the viscosity limit, QmIs the capillary limit, ToIs the vapor temperature at the beginning of the evaporation section, poIs ToLower saturated vapor density, μvIs the dynamic viscosity of the steam, σwIs the surface tension of the working medium, rhsIs the hydraulic radius, ρ, of the capillary wicklIs the density of the liquid working medium, dvIs the vapor space diameter, theta is the heat pipe axial inclination angle, LtIs the total length of the heat pipe, LeIs the length of the evaporation section of the heat pipe, FlIs the liquid phase coefficient of friction, FvIs the gas phase coefficient of friction;
if any one of the calculated sound velocity limit and viscosity limit is less than or equal to the heat transfer quantity of the current heat pipe, replacing the heat transfer quantity of the heat pipe with a heat transfer limit value less than or equal to the heat transfer quantity of the current heat pipe; if any one of the calculated carrying limit and the capillary limit is less than or equal to the heat transfer quantity of the current heat pipe, the heat pipe is considered to be failed to start, the start failure is output, and the calculation is stopped; if the calculated sound velocity limit, viscosity limit, carrying limit and capillary limit are all larger than the heat transfer capacity of the current heat pipe, the heat transfer limit is not met;
and step 9: and (5) repeating the steps 3-8 according to the new heat transfer quantity until the set total time step is reached, completing the calculation, and outputting a starting calculation result.

Claims (3)

1. A three-stage calculation method for cold start of an alkali metal heat pipe is characterized by comprising the following steps: the method comprises the following steps:
step 1: determining the geometric dimension of the heat pipe, a working medium, a pipe wall material, a liquid absorption core structure and boundary conditions of an evaporation section and a condensation section; the evaporation section adopts a second type of boundary condition, the heat flow density is given, the condensation section adopts a third type of boundary condition, and the convection heat transfer coefficient is given; determining the calculated single step time step length and the total step number;
step 2: dividing a heat pipe control body, axially dividing an e-layer control body along an evaporation section, dividing an a-layer control body along a heat insulation section, dividing a c-layer control body along a condensation section, radially dividing w-layer control bodies along the wall of a heat pipe, dividing a p-layer control body along a liquid absorption core of the heat pipe, and dividing 1 layer along a vapor space; initializing parameters of the control body obtained by dividing, and setting the temperature of the control body as a starting initial temperature;
and step 3: calculating the temperature change rate of the pipe wall area of the heat pipe
Figure FDA0003396987720000011
Establishing a two-dimensional heat conduction equation in the pipe wall area of the heat pipe, wherein the control equation is as follows:
Figure FDA0003396987720000012
in the formula: t iswIs the temperature of the wall area of the heat pipe, t is the time, CwIs the volumetric heat capacity, k, of the wall region of the heat pipewHeat conducting system for heat pipe wall areaThe number r is the radial direction of the heat pipe, and the number z is the axial direction of the heat pipe;
the boundary conditions of control equation (1) are:
an evaporation section:
Figure FDA0003396987720000013
adiabatic section:
Figure FDA0003396987720000014
a condensation section:
hAc(Tw-Tsur)＝Qc (4)
in the formula: a. theeIs the area of the evaporation section of the heat pipe, AcIs the area of the condensation section of the heat pipe, QeFor heating power in the evaporation zone, QcCooling power of the condensing section, h is the convective heat transfer coefficient of the outer surface of the condensing section, and TsurIs ambient temperature;
and 4, step 4: calculating the temperature change rate of the liquid absorption core area of the heat pipe
Figure FDA0003396987720000021
Because the flow velocity in the liquid absorbing core of the heat pipe is very low, the flow of the working medium in the liquid absorbing core is neglected, the area of the liquid absorbing core of the heat pipe is regarded as a mixed solid formed by a static liquid working medium and a solid wire mesh, a two-dimensional heat conduction equation is established in the area of the liquid absorbing core of the heat pipe, and the control equation is as follows:
Figure FDA0003396987720000022
in the formula: t ispIs the temperature of the wick region of the heat pipe, CeffIs the equivalent volumetric heat capacity, k, of the wick region of the heat pipeeffThe equivalent thermal conductivity of the heat pipe wick area;
volumetric heat capacity C of the mixed matrixeffAnd heat conduction systemNumber keffThe calculation is made according to the following equation:
Ceff＝εCl+(1-ε)Cs (6)
Figure FDA0003396987720000023
in the formula: clIs the volumetric heat capacity of the liquid working medium, CsIs the volumetric heat capacity, k, of the wick wirelIs the thermal conductivity, k, of the liquid working mediumsThe thermal conductivity of the wick wire mesh is, and epsilon is the porosity of the wick wire mesh;
and 5: judging the starting stage of the heat pipe according to the outermost node temperature of the liquid absorption core of the alkali metal heat pipe: dividing the heat pipe starting into a first starting stage, a second starting stage and a third starting stage, comparing the vapor transition temperature of the alkali metal working medium with the temperature of the outermost control body of the liquid absorption core area of the alkali metal heat pipe, namely the interface temperature of the liquid absorption core area and the vapor area, which is obtained by calculation at the last moment, and then collectively calling the gas-liquid interface temperature to judge the current starting stage of the heat pipe;
the transition temperature of the alkali metal working fluid is calculated by iterating the formula:
Figure FDA0003396987720000031
in the formula: t istIs the transition temperature of the alkali metal working medium, M is the relative molecular mass of the alkali metal working medium, RuFor an ideal gas constant, ρ is the density of the gas, μ is the kinetic viscosity of the gas, D is the diameter of the vapor region of the heat pipe, and for a given heat pipe structure and heat pipe working medium, the transition temperature T of the alkali metal working mediumtIs uniquely determined;
if the temperature of all control bodies on the gas-liquid interface is less than the transition temperature T of the alkali metal working mediumtIf yes, judging to start the first stage; if the gas-liquid interface has the temperature of the control body which is greater than or equal to the transition temperature T of the alkali metal working mediumtBut still have gas-liquid interactionThe temperature of the interface control body is less than the transition temperature T of the alkali metal working mediumtIf yes, judging that the second stage is started; if the temperature of all control bodies on the gas-liquid interface is more than or equal to the transition temperature T of the alkali metal working mediumtIf yes, judging to be a third stage of starting;
step 6: calculating the temperature, the speed, the density, the pressure and the gas content of the vapor area control body of the alkali metal heat pipe according to different starting stages:
if the first stage is started, the vapor region is considered to be in a free molecular state, the evaporation and heat transfer of the vapor are ignored, and the adiabatic boundary condition is adopted for the gas-liquid interface:
Figure FDA0003396987720000032
in the formula: t isvIs the temperature of the vapor region of the heat pipe, RgIs the radius of the vapor zone;
then the temperature of the vapor region control body at this point is equal to the temperature of the wick region control body adjacent thereto:
Figure FDA0003396987720000033
in the formula: t isv,iTemperature, T, of ith control body in axial direction for vapor region of heat pipep,iThe temperature of the ith control body along the axial direction of the heat pipe liquid absorption core area is measured;
if the second stage is started, the steam area control bodies adjacent to the control body with the gas-liquid interface reaching the transition temperature of the alkali metal working medium are considered to be in continuous state flow, the other steam area control bodies are still in free molecular state, the density, the gas content and the speed parameters of the steam are ignored, and the normalized steam temperature is obtained in all the continuous flow control bodies through the iterative formula:
Figure FDA0003396987720000041
in the formula: a isccFor cell adjustment factor, hfvW is the latent heat of vaporization of the working medium, the circumference of the vapor region, meThe number of gas-liquid interface control bodies for achieving the transition temperature is not more than e + a and delta liIs the width of the ith control body, TfiTemperature of i-th control body for gas-liquid interface, PfiIs TfiCorresponding saturated vapor pressure, TvIs the temperature of the vapor region, PvIs TvCorresponding saturated vapor pressure, ρvIs TvCorresponding vapor density, AcIs the area of the steam area, and gamma is the heat capacity ratio of the gaseous working medium;
if, for the third stage of actuation, a continuous flow of vapor is deemed to have been fully established in the vapor space, the temperature, velocity, density, pressure and vapor fraction of the vapor zone control volume are obtained by the following equations:
Figure FDA0003396987720000042
Figure FDA0003396987720000043
Figure FDA0003396987720000044
Figure FDA0003396987720000045
Figure FDA0003396987720000046
in the formula: ρ is the vapor density, XqIs vapor void fraction, V is vapor velocity, P is vapor pressure, and T is vapor temperatureD is the diameter of the steam area, upsilon is the specific volume of the gas-liquid mixed fluid, upsilongIs saturated specific vapor volume, upsilonfIs the specific volume of saturated liquid, h is the enthalpy value of gas-liquid mixed fluid, h0Is vapor enthalpy of gas-liquid interface, hfgIs the latent heat of vaporization of the working medium, V0Is the normal velocity of the vapor at the gas-liquid interface,
Figure FDA0003396987720000056
is the mass evaporation rate of the gas-liquid interface, cpIs the constant pressure specific heat capacity of steam, FfAs an interphase friction factor, MfIs a momentum factor, EfIs an energy factor;
specific volume upsilon of gas-liquid mixed fluid, enthalpy value h of gas-liquid mixed fluid and specific volume upsilon of saturated steamgCalculated from the following formula:
υ＝υf+Xq×(υg-υf) (17)
h＝hf+Xq×hfg (18)
Figure FDA0003396987720000051
in the formula: h isfIs the enthalpy of the saturated liquid; mass evaporation rate of gas-liquid interface
Figure FDA0003396987720000052
Calculated from the following formula:
Figure FDA0003396987720000053
in the formula: t isfIs the temperature of the gas-liquid interface, PfIs TfThe corresponding saturated vapor pressure;
normal velocity V of vapor at gas-liquid interface0Calculated from the following formula:
Figure FDA0003396987720000054
in the formula: qintFor input of thermal power to gas-liquid interface, AintIs the area of the gas-liquid interface, pintThe vapor density corresponding to the gas-liquid interface temperature;
interphase friction factor FfMomentum factor MfEnergy factor EfThe following equation is used:
Figure FDA0003396987720000055
Figure FDA0003396987720000061
Figure FDA0003396987720000062
in the formula:
Figure FDA0003396987720000063
Voobtained from the formula (21), D is the vapor zone diameter, and ν is the kinematic viscosity of the vapor;
and 7: the control equations of each control body are dispersed, and the initial value problem converted into the nonlinear ordinary differential equation set is solved, and the initial value problem has the following form:
Figure FDA0003396987720000064
in the formula:
Figure FDA0003396987720000065
for the solution of the equation at time t,
Figure FDA0003396987720000066
is composed of
Figure FDA0003396987720000067
Is a derivative function of f
Figure FDA0003396987720000068
And
Figure FDA0003396987720000069
the implicit function of the relationship between (a) and (b),
Figure FDA00033969877200000610
is composed of
Figure FDA00033969877200000611
An initial value at time 0; the equation set is solved by adopting a Gear algorithm respectively, and a backward difference format is adopted for a time term, wherein the equation set has a difference equation with the following form:
Figure FDA00033969877200000612
in the formula: σ is a time step and satisfies ti＝t0+ i σ, F is a constructor of F and satisfies
Figure FDA00033969877200000613
With a single step push, the difference equation (23) has a solution of y0,y1,y2,…,yn}; and let G be max F, α be max ti-t0|，β＝max||yi-y0||，
Figure FDA00033969877200000614
Constructing a function vector Zn(t) making Zn(tk)＝ykAnd Z'n(tk)＝F(tk-1,yk-1,ykH) then the remainder Rn(t) is expressed as:
Figure FDA00033969877200000615
giving tolerance errors tol only by ensuring
Figure FDA00033969877200000616
Even if the solution of the initial value problem (22) converges consistently, the convergence condition is:
Figure FDA0003396987720000071
in actual calculation, the iteration times are specified, if n still does not satisfy the formula (25) after the iteration times are exceeded, the calculation result is judged not to be converged, the calculation time step length sigma is shortened, and calculation is carried out again until the calculation result is converged; if n satisfies the formula (25) within the iteration times, judging that the calculation result is converged, and covering the calculated value as an initial value of a new time step on the current value; if the time step length is shortened to be smaller than the shortening amount, the calculation is still not converged, the output is not converged, and the calculation is stopped; the discrete control equation set of the steam belongs to the boundary value problem of the nonlinear ordinary differential equation set, and the four-order Runge-Kutta method is adopted to solve the boundary value problem, so that the boundary value problem is bound to be converged;
and 8: checking the heat transfer limit of the heat pipe, updating the heat transfer capacity of the alkali metal heat pipe, and considering the sound velocity limit, the carrying limit, the viscosity limit and the capillary limit, calculating according to the following formula:
Figure FDA0003396987720000072
Figure FDA0003396987720000073
Figure FDA0003396987720000074
Figure FDA0003396987720000075
in the formula: qsIs the limit of sound velocity, QxIs the carrying limit, QvIs the viscosity limit, QmIs the capillary limit, ToIs the vapor temperature at the beginning of the evaporation section, poIs ToLower saturated vapor density, μvIs the dynamic viscosity of the steam, σwIs the surface tension of the working medium, rhsIs the hydraulic radius, ρ, of the capillary wicklIs the density of the liquid working medium, dvIs the vapor space diameter, theta is the heat pipe axial inclination angle, LtIs the total length of the heat pipe, LeIs the length of the evaporation section of the heat pipe, FlIs the liquid phase coefficient of friction, FvIs the gas phase coefficient of friction;
if any one of the calculated sound velocity limit and viscosity limit is less than or equal to the heat transfer quantity of the current heat pipe, replacing the heat transfer quantity of the heat pipe with a heat transfer limit value less than or equal to the heat transfer quantity of the current heat pipe; if any one of the calculated carrying limit and the capillary limit is less than or equal to the heat transfer quantity of the current heat pipe, the heat pipe is considered to be failed to start, the starting failure is output, and the calculation is stopped; if the calculated sound velocity limit, viscosity limit, carrying limit and capillary limit are all larger than the heat transfer capacity of the current heat pipe, the heat transfer limit is not met;
and step 9: and (5) repeating the steps 3-8 according to the new heat transfer quantity until the set total time step is reached, completing the calculation, and outputting a starting calculation result.
2. The cold start three-stage calculation method for the alkali metal heat pipe according to claim 1, wherein: step 7, taking 1 × 10 shortened samples in the shortened calculation time step-5s。
3. The cold start of an alkali metal heat pipe as claimed in claim 1The three-stage calculation method is characterized in that: in step 6, for the alkali metal working medium, the unit adjustment coefficient acc1 is taken.
CN202111486969.0A 2021-12-07 2021-12-07 Three-stage calculation method for cold start of alkali metal heat pipe Pending CN114154438A (en)

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