CN117407634A - Flat plate frosting thickness rapid prediction method based on frosting characteristic curve - Google Patents
Flat plate frosting thickness rapid prediction method based on frosting characteristic curve Download PDFInfo
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Abstract
The invention discloses a flat plate frosting thickness rapid prediction method based on frosting characteristic curves, which relates to the field of frosting prediction and comprises the following steps: step 1: constructing a simplified condition; step 2: establishing a first analytical expression between the thickness of the dimensionless frost layer and the dimensionless frost formation time based on the simplification condition; step 3: constructing a second analytical expression of the dimensionless frosting time and a third analytical expression of the dimensionless frosting layer thickness; step 4: bringing the third analytical expression and the second analytical expression into the first analytical expression to obtain a fourth analytical expression; step 5: the method and the device can reduce the calculated amount of the flat plate frost thickness prediction and improve the efficiency of the flat plate frost thickness prediction.
Description
Technical Field
The invention relates to the field of frosting prediction, in particular to a flat plate frosting thickness rapid prediction method based on frosting characteristic curves.
Background
The novel low-temperature heat exchanger can rapidly cool high-temperature gas flowing at high speed to deep low temperature. When the gas is cooled to a sub-zero temperature, the saturation humidity of the air will be greatly reduced, whereby the water vapour in the air will be caused to sublimate into frost at the low temperature surface of the heat exchange unit. The low-temperature surface frosting action of high-speed airflow flowing through the heat exchange unit belongs to the low-temperature surface frosting problem under the strong convection condition, and is the gas-solid phase transition action of water vapor desublimation. The gas-solid phase transition behavior of the vapor directly sublimating into frost is called dry mode frosting, and liquid water cannot appear in the dry mode frosting process. Because the dry mode frosting on the low-temperature surface can cause blockage of a heat exchanger channel, the control of the dry mode frosting thickness growth condition under frosting conditions such as different incoming flow speeds, incoming flow temperatures, low-temperature surface temperatures, low-temperature element sizes and the like is important to design the heat exchange unit size and the distance of the heat exchanger and determine the cooling strategy of the heat exchanger. For this reason, it is necessary to predict the dry mode frosting behavior of low temperature surfaces under different frosting conditions.
Under the dry mode frosting condition of certain inflow temperature, inflow speed, low-temperature flat plate temperature and low-temperature flat plate length, a numerical calculation prediction method for the change of the frost layer thickness on the low-temperature flat plate with time exists at present. According to the method, an energy equation and a quality equation are constructed according to dimensionalized incoming flow temperature, incoming flow speed, low-temperature flat plate temperature, low-temperature flat plate length and other parameters, and time iterative solution is carried out. However, this numerical calculation prediction method requires iterative solution, which is inconvenient in practical use. In addition, the frosting condition combinations such as different incoming flow temperature, incoming flow speed, low-temperature flat plate temperature, low-temperature flat plate length and the like are required to be calculated independently, so that the calculated amount of the existing prediction method is large, and the prediction efficiency is low.
Disclosure of Invention
The invention aims to reduce the calculated amount of the flat plate frosting thickness prediction and improve the efficiency of predicting the flat plate frosting thickness prediction.
In order to achieve the above object, the present invention provides a method for rapidly predicting a flat plate frost thickness based on a frost characteristic curve, the method comprising:
step 1: constructing a simplified condition;
step 2: establishing a first analytical expression between the thickness of the dimensionless frost layer and the dimensionless frost formation time based on the simplification condition;
step 3: constructing a second analytical expression of the dimensionless frosting time and a third analytical expression of the dimensionless frosting layer thickness;
step 4: bringing the third analytical expression and the second analytical expression into the first analytical expression to obtain a fourth analytical expression;
step 5: and obtaining frosting state information corresponding to the frosting thickness of the flat plate to be predicted, obtaining the equilibrium thickness of the frosting layer and frosting characteristic time based on the frosting state information, and obtaining the corresponding frosting layer thickness by utilizing a fourth analytical expression calculation based on the equilibrium thickness of the frosting layer and the frosting characteristic time.
The method comprises the steps of obtaining a low-temperature flat plate frosting characteristic curve under the condition of strong convection after approximate simplification conforming to physical significance, revealing a unique analytic solution relation between a dimensionless frosting layer thickness and dimensionless frosting time through the curve, rapidly realizing frost layer thickness prediction by utilizing the analytic relation, carrying out iterative calculation in a traditional mode, and obtaining the corresponding frost layer thickness through inputting a corresponding expression at a moment t.
In some embodiments, the simplification conditions are: the ratio of the heat exchange amount of convection to the latent heat release amount of frosting phase change is larger than a first threshold value, the ratio of the incoming flow temperature to the first temperature difference is larger than a second threshold value, the temperature difference between the maximum temperature value and the minimum temperature value of the preset temperature interval is smaller than the preset temperature value, and the first temperature difference is the difference between the dew point temperature and the cold surface temperature.
In some embodiments, the first analytical expression is obtained by:
simulating low-temperature flat plate dry mode frosting, and constructing a mass conservation equation corresponding to the frosting according to the relationship between the mass change rate of the frosting and the phase change rate of the water vapor;
obtaining a characteristic value related to low temperature flat plate dry mode frosting, comprising: dimensionless temperature, dimensionless frost layer thickness, dimensionless frost time, dimensionless humidity, and dimensionless frost layer density;
processing the mass conservation equation based on the dimensionless temperature, the dimensionless frost layer thickness, the dimensionless frost formation time, the dimensionless humidity and the dimensionless frost layer density to obtain a dimensionless mass conservation equation;
if the density and the heat conductivity of the frost layer are unchanged in the frosting process, simplifying a dimensionless mass conservation equation to obtain a first dimensionless mass conservation equation;
and based on the simplification condition, simplifying the first dimensionless mass conservation equation to obtain the first analytical expression.
In some embodiments, based on the simplification conditions, simplifying the first dimensionless mass conservation equation to obtain the first analytical expression includes:
in a preset temperature interval, linearly simplifying the air saturation humidity to obtain a first linear function of the simplified air saturation humidity with respect to temperature;
bringing the expression of the dimensionless frost surface saturated humidity into the first linear function to obtain a second function;
when the ratio of the convection heat exchange amount to the latent heat release amount of frosting phase change is larger than a first threshold value, the second function is arranged to obtain a first relational expression of the non-dimensional frosting surface saturation humidity with respect to the thickness of the non-dimensional frosting layer;
when the ratio of the incoming flow temperature to the first temperature difference is larger than a second threshold value, the first relation is sorted, and a second relation with the non-dimensional frost layer thickness equal to the non-dimensional frost layer saturation humidity in value is obtained;
substituting the second relation into the first dimensionless mass conservation equation to obtain a first analytic expression of the thickness of the dimensionless frost layer with respect to the dimensionless frost formation time.
In some embodiments, the first analytical expression is:
δ＝1-e -τ
wherein delta is the thickness of the dimensionless frost layer, and tau is the dimensionless frost time;
the second analytical expression is:
wherein eta is the characteristic time of frosting, and t is the time of frosting;
the third analytical expression is:
wherein X is f For the thickness of the frost layer, delta b Balancing the thickness of the frost layer;
the fourth analytical expression is:
in some embodiments, the frost layer balances thickness δ b The calculation mode of (a) is as follows:
wherein k is f The thermal conductivity of the frost layer is h is the convective heat transfer coefficient, T a For incoming flow temperature, T d For incoming flow dew point temperature, T w Is the temperature of the low-temperature flat plate;
wherein k is a Is air heat conductivity, L is characteristic length of the low-temperature flat plate along the incoming flow direction,is the average noose number;
wherein Re is L Pr is the Plandt number, u is the incoming air velocity, and v is the incoming air motion coefficient of viscosity.
In some embodiments, the frost characteristic time η is calculated by:
the calculation mode of the initial frost growth rate xi is as follows:
wherein h is m Is the mass transfer coefficient of water vapor, ρ v For incoming flow humidity ρ sa (T w ) Saturated humidity corresponding to low-temperature flat plate temperature, ρ v -ρ sa (T w ) The mass transfer concentration difference of water vapor at the initial frosting moment is h is the convective heat transfer coefficient, c pa Specific heat of air ρ a For air density, le is the lewis number and n is a constant.
In some embodiments, the frosting status information comprises: incoming flow temperature T a Incoming flow dew point temperature T d And low temperature plate temperature T w 。
In some embodiments, the method further comprises: and calculating and predicting the relative error of the thickness of the frost layer, and verifying a prediction method. The accuracy of the prediction method can be calculated by calculating the relative error, so that the feasibility of the method is verified.
In some embodiments, the method further comprises: judging whether the thickness of the frost layer obtained by prediction exceeds a corresponding threshold value, and if so, carrying out early warning. When the predicted frost layer thickness exceeds the threshold value, normal use of the low-temperature heat exchanger can be affected, and the normal use of the low-temperature heat exchanger can be ensured by performing treatment or intervention in advance through early warning.
The one or more technical schemes provided by the invention have at least the following technical effects or advantages:
under the frosting condition meeting the physical meaning represented by approximate simplification, the frosting thickness corresponding to a certain frosting time can be directly calculated based on the relation represented by the frosting characteristic curve, iterative calculation is not needed, and the method can be used for predicting and calculating the frosting thickness of low-temperature flat plate dry mode frosting under the condition of rapid and convenient strong convection, so that the calculated amount of prediction is reduced, and the prediction efficiency is improved.
Drawings
The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention;
FIG. 1 is a flow chart of a method for rapidly predicting a flat plate frost thickness based on a frost feature curve;
FIG. 2 is a schematic diagram of a fast plate frost thickness prediction flow based on a frost feature curve;
FIG. 3 is a schematic diagram of a dimensionless frosting characteristic curve;
FIG. 4 is a diagram showing the comparison of the growth condition of a dimensionless frost layer and a frost characteristic curve calculated in an iterative manner under a certain state;
FIG. 5 is a graph showing the relative error of the frost formation characteristic curve and the dimensionless frost layer thickness prediction of the iterative calculation result under a certain state.
Detailed Description
In order that the above-recited objects, features and advantages of the present invention will be more clearly understood, a more particular description of the invention will be rendered by reference to the appended drawings and appended detailed description. In addition, the embodiments of the present invention and the features in the embodiments may be combined with each other without collision.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, however, the present invention may be practiced in other ways than within the scope of the description, and the scope of the invention is therefore not limited to the specific embodiments disclosed below.
Example 1
Referring to fig. 1, fig. 1 is a flow chart of a method for rapidly predicting a flat frosting thickness based on frosting characteristic curves, and in an embodiment of the present invention, a method for rapidly predicting a flat frosting thickness based on frosting characteristic curves is provided, and the method includes:
step 1: constructing a simplified condition;
step 2: establishing a first analytical expression between the thickness of the dimensionless frost layer and the dimensionless frost formation time based on the simplification condition;
step 3: constructing a second analytical expression of the dimensionless frosting time and a third analytical expression of the dimensionless frosting layer thickness;
step 4: bringing the third analytical expression and the second analytical expression into the first analytical expression to obtain a fourth analytical expression;
step 5: and obtaining frosting state information corresponding to the frosting thickness of the flat plate to be predicted, obtaining the equilibrium thickness of the frosting layer and frosting characteristic time based on the frosting state information, and obtaining the corresponding frosting layer thickness by utilizing a fourth analytical expression calculation based on the equilibrium thickness of the frosting layer and the frosting characteristic time.
In this embodiment, please refer to fig. 2, fig. 2 is a schematic diagram of a fast plate frost thickness prediction flow based on a frost feature curve, and the method includes the following steps:
step S1: the method for rapidly predicting the frosting thickness of the flat plate is based on a frosting characteristic curve of a flat plate dry mode under the strong convection condition, wherein the dimensionless frosting characteristic curve is shown in figure 3. The basic situation of this frosting characteristic curve is as follows.
Step S1.1: the frosting characteristic curve is based on approximate simplification of three points with real physical meaning: 1) The change of the air saturation humidity with respect to the temperature may be approximately linear in a narrower temperature interval, i.e., a preset temperature interval, such as a temperature interval in which the difference between the highest temperature and the lowest temperature is less than 10K; 2) When the amount of convection heat exchange is far greater than the amount of latent heat released by frosting phase change, the latent heat part in heat transfer can be ignored, namely the ratio of the amount of convection heat exchange to the amount of latent heat released by frosting phase change is greater than a first threshold, the first threshold is usually 10, for example, the amount of convection heat exchange is greater than the amount of latent heat released by frosting phase change by more than an order of magnitude of 10; 3) The incoming flow temperature is much larger than the difference between the incoming flow dew point temperature and the cold face temperature, i.e. the difference between the dew point temperature and the cold face temperature is small, i.e. the ratio of the incoming flow temperature to the first temperature difference is larger than a second threshold value, typically 10, e.g. the incoming flow temperature is more than an order of magnitude larger than the difference between the dew point temperature and the cold face temperature by more than 10 (e.g. the incoming flow temperature is 300K, the difference between the dew point temperature and the cold face temperature is 30K).
Step S1.2: under the condition of approximate simplification in the step S1.1, for low-temperature flat plate dry mode frosting under the strong convection condition, the relationship of the following analytical expression exists between the non-dimensional frost layer thickness delta and the non-dimensional frosting time tau:
δ＝1-e -τ (1)
the analytical expression is irrelevant to frosting conditions such as frosting inflow temperature, frosting inflow speed, low-temperature flat plate temperature, low-temperature flat plate size and the like, and the corresponding relationship of the dimensionless frosting time tau about the dimensionless frost layer thickness delta is unique. The corresponding relation curve of the formula (1) is called a dimensionless frosting characteristic curve, and is called a frosting characteristic curve for short.
The frost formation characteristic curve has the following characteristics that the thickness variation interval of the dimensionless frost layer is 0-1, and the corresponding relation between the dimensionless frost formation time and the thickness of the dimensionless frost layer is shown in table 1. The thickness of the dimensionless frost layer is 0.632 when the dimensionless frost time is 1, 0.865 when the dimensionless frost time is 2, and 0.993 when the dimensionless frost time is 5.
TABLE 1 correspondence of nondimensional frosting time to nondimensional frosting layer thickness in frosting characteristic curves
Step S2: when the frosting reaches equilibrium, the thickness of the frosting layer is not increased any more, and the thickness of the frosting layer is equal to the equilibrium thickness of the frosting layer. Equilibrium thickness delta of frost layer b The method can be obtained by the following formula:
wherein k is f Is the thermal conductivity of the frost layer, h is the convective heat transfer coefficient and T a To the incoming flow temperature T d For the incoming flow dew point temperature (i.e. the dew point temperature corresponding to the incoming flow humidity), T w Is a low temperature plate temperature.
When the temperature difference between the incoming flow dew point temperature and the low-temperature flat plate temperature is not large, the frost layer density rho f And thermal conductivity k f Can be processed as a constant. h is obtained by：
Wherein k is a Is air heat conductivity, L is characteristic length of the low-temperature flat plate along the flowing direction,To average the number of noose, the average number of noose may be determined by the Reynolds number Re L And Pr, and the average Knoop number in laminar flow is calculated as follows:
in the formula (5), pr of air is 0.7, u is the incoming air speed, and v is the motion viscosity coefficient of the incoming air.
For a certain determined frosting state, all parameters on the right side of the formula (2) are known quantities, and the equilibrium thickness delta of the frosting layer can be obtained b 。
Step S3: the frosting rate at the moment 0 is defined as the initial frosting growth rate xi, and can be obtained by a mass conservation equation at the moment 0, and the mass conservation equation is shown as follows:
wherein h is m Is the mass transfer coefficient ρ of water vapor v For incoming flow humidity ρ sa (T w ) The saturation humidity corresponding to the temperature of the low-temperature flat plate. (ρ) v -ρ sa (T w ) A water vapor mass transfer concentration difference at the time of initial frosting.
h m The method can be obtained according to a Lewis relational expression of heat transfer and mass transfer comparison:
in the formula (7), c pa Specific heat of air ρ a For air density, le is a Lewis number, and for air Le, a value of 1 is generally taken and n is generally taken as 0.33 to 0.4.
For a certain determined frosting state, all parameters on the right side of the formula (6) are known quantities, and then the initial frosting growth rate xi can be obtained.
Step S4: balancing the frost layer to a thickness delta b The characteristic growth rate ζ of the frost layer is divided by the characteristic time of frost formation, denoted as η:
step S5: the dimensionless frost time τ is defined as the frost time t divided by the frost characteristic time η, as shown in the following formula:
step S6: the thickness delta of the dimensionless frost layer is calculated by the formula (1) using the dimensionless frost time tau.
Step S7: the definition of the dimensionless frost thickness delta is the frost thickness X f Divided by the equilibrium thickness delta of the frost layer b The following formula is shown:
the physical meaning represented by the dimensionless frost layer thickness δ is the percentage of frost layer thickness relative to the equilibrium thickness of the frost layer. Then the dimensionless frost layer thickness delta and the frost layer equilibrium thickness delta are known b The thickness X of the frost layer can be calculated by the following formula f 。
X f ＝δ b δ(11)
Step S8: using the formula (9) and the formula (10), the formula (1) can be written as the following expression:
likewise, for a certain determined frosting state, the frost layer has an equilibrium thickness delta b And the frosting characteristic time are determined, so that the frosting layer thickness X can be solved by directly resolving t by using the formula (12) f 。
The effect of the invention is shown in fig. 4 and 5, fig. 4 is a schematic diagram showing the comparison of the growth condition of the non-dimensional frost layer and the frost formation characteristic curve calculated in an iterative manner under a certain state, and it can be seen from fig. 4 that the two non-dimensional frost layer growth curves almost coincide, which indicates that the method for predicting the frost layer thickness by using the frost formation characteristic curve is feasible, and fig. 5 is a graph showing the difference between the frost formation characteristic curve and the frost formation characteristic curve under a certain state (T a ＝300K、T d ＝270K、T w =260K), it can be seen from fig. 5 that the relative error is less than 1%, which indicates that the accuracy of the prediction result using the frosting characteristic curve is high relative to the iterative calculation.
The expression construction process in the present invention is described below:
the invention is based on a one-dimensional simulation method of plate dry mode frosting under strong convection condition (engineering thermophysics journal-2022-01-01, 43 volumes, 001 phase-one-dimensional dry mode frosting simulation research of low-temperature plate normal frost layer under rapid inflow condition), and selects the equilibrium thickness delta of the frost layer b And normalizing parameters such as the initial frost layer growth rate xi, the frost formation characteristic time eta and the like to obtain a dimensionless frost layer thickness and dimensionless frost formation time, and carrying out dimensionless treatment on an energy conservation equation and a mass conservation equation.
The characteristic curve, namely the fourth analytical expression, is obtained based on a one-dimensional simulation method of the frosting of the plate dry mode under the strong convection condition, and the one-dimensional simulation method of the frosting of the plate dry mode under the strong convection condition comprises the following steps:
one-dimensional simulation of plate dry mode frosting is based on two-point assumptions and simplifications: 1) The formed frost layer is compact, mass transfer inside the frost layer can be ignored, and once the frost layer is formed, the density and the heat conductivity of the frost layer are not changed any more; 2) Because the thickness of the frost layer changes slowly, the interior of the frost layer can be considered to meet the quasi-steady state heat conduction at each moment. Based on the quasi-steady state heat conduction assumption, the heat transfer control equation in the frost layer is:
wherein x is the height of the interior of the frost layer, k f For the frost thermal conductivity at x, T is the temperature at x of the frost.
The energy conservation equation is constructed according to the heat transfer equilibrium relationship at the frost surface as follows:
the left side of equation 14 is the heat conduction quantity in the frost layer, the first term on the right side of equation is the convection heat exchange quantity, and the second term on the right side of equation is the latent heat released when water vapor condenses to form frost. Wherein X is f Is the thickness of the frost layer, h is the convection heat transfer coefficient, T a To the incoming flow temperature T s Is the frost temperature, h m Is the convection mass transfer coefficient ρ of water vapor v For incoming water vapor density (i.e., incoming humidity), γ is the latent heat of sublimation of water vapor ρ sa Is saturated humidity (ρ) sa (T s ) Saturated humidity corresponding to frost temperature).
Constructing a mass conservation equation according to the relation between the mass change rate of the frost layer and the phase change rate of the water vapor:
wherein the left side of the equation 15 is the mass change rate of the frost layer, the right side of the equation is the phase change rate of the water vapor, wherein t is the frosting time and ρ is f The density of the newly generated frost layer at the frost surface at the moment t.
By giving the incoming flow temperature T at time t=0 a Temperature T of frosting face s (time t=0 equals the low temperature plate temperature T w ) Incoming water vapor density ρ v And (3) solving the formula (2) and the formula (3) by numerical iteration to obtain the thickness change condition of the frosted layer of the low-temperature flat plate drying mode frosting related to frosting time.
Selecting a characteristic value for dimensionless treatment, carrying out dimensionless treatment on an energy conservation equation and a mass conservation equation to obtain the dimensionless energy conservation equation and the dimensionless mass conservation equation, wherein the method comprises the following steps of:
using the outflow dew point temperature T d (incoming water vapor density ρ) v Corresponding dew point temperature) and a low temperature plate temperature T w The dimensionless temperature is defined as:
incoming flow dew point temperature T d Surface temperature T of frost layer s And low temperature plate temperature T w The corresponding dimensionless temperature values (ranges) are:
θ d ＝1，0≤θ s ≤1，θ w ＝0 (17)
wherein θ d For incoming flow dew point temperature T d Non-dimensional temperature value, θ s Is the frost layer surface temperature T s Corresponding dimensionless temperature value, theta w At a low temperature plate temperature T w Corresponding dimensionless temperature values;
when the frost formation reaches equilibrium, the thickness of the frost layer is not changed any more, and in the energy conservation equation (15), the thickness of the frost layer is equal to the equilibrium thickness, and the temperature T of the frost surface s Equal to the incoming flow dew point temperature T d And the water vapor phase change release latent heat term (second right term) is 0, then the energy conservation equation (14) can be written as:
wherein,the average heat conductivity when the frost layer reaches the equilibrium thickness is called as the equilibrium heat conductivity of the frost layer for short.
Substituting the dimensionless temperature into the formula (18) and finishing, and writing the temperature into the equilibrium thickness delta of the frost layer b The expression of (2) is as follows:
balancing the frost layer to a thickness delta b As a characteristic thickness of frosting for frosting thickness dimensionless, the normalized dimensionless frosting thickness can be written as:
the initial frost growth rate at time 0 was defined as the characteristic frost growth rate, denoted ζ:
wherein ρ is f0 For the frost layer density at the initial moment of frosting, the frost layer density is simply called initial frost layer density, and the frost layer density is obtained by the temperature T of a low-temperature flat plate w And (5) determining.
Balancing the frost layer to a thickness delta b The characteristic growth rate ζ of the frost layer is divided by the characteristic time of frost formation, denoted as η:
the humidity was treated as follows to obtain dimensionless humidity:
the frost density was treated as follows to obtain a dimensionless frost density:
the thermal conductivity of the frost layer is processed in the following way to obtain the non-dimensional thermal conductivity of the frost layer:
wherein k is f0 The thermal conductivity of the frost layer at the initial moment of frosting is simply called initial frost layer thermal conductivity.
The non-dimensional mass conservation equation can be obtained by substituting the non-dimensional temperature of the formula (16), the non-dimensional frost layer thickness of the formula (20), the non-dimensional frost formation time of the formula (22), the non-dimensional humidity of the formula (23) and the non-dimensional frost layer density of the formula (24) into the mass conservation equation of the formula (15):
substituting the dimensionless temperature of the formula (16), the dimensionless frost layer thickness of the formula (20), the dimensionless humidity of the formula (23) and the dimensionless frost layer thermal conductivity of the formula (25) into an energy conservation equation (14) to obtain a dimensionless energy conservation equation:
gamma is the latent heat of sublimation of water vapor, ρ v Substituting the equilibrium thickness of the frost layer of formula (19) into formula (27) for the incoming water vapor density, omitting the Le number equal to 1, and arranging the energy conservation equation into the dimensionless frost surface temperature theta s The expression form of (a) is as follows:
when the incoming flow dew point temperature T d And low temperature plate temperature T w The physical properties (the density and the heat conductivity of the frost layer) of the frost layer in the frosting process are not greatly different, and the frost layer can be similar to the normal physical properties at the moment, namely, the density and the heat conductivity of the frost layer are not changed in the whole frosting process. The thermal conductivity of the frost layer is constant, the thermal conductivity of the dimensionless frost layer is 1, and the frost layer has equilibrium thickness delta for the thermal conductivity of the frost layer under given frosting conditions b Also a determined value, and can be found by the following equation:
substituting equation (30) into equation (28), the dimensionless energy conservation equation can be further simplified to the form:
the dimensionless mass conservation equation and the dimensionless energy conservation equation can be obtained through the mode.
The method specifically comprises the following steps:
the expression for establishing the low-temperature flat plate dry mode frosting characteristic curve under the strong convection condition is developed based on a flat plate dry mode frosting dimensionless mass conservation equation under the strong convection condition of the ordinary frost layer approximation condition. The non-dimensional mass conservation equation of the plate dry mode frosting under the strong convection condition of the normal frost layer is as follows:
wherein delta is the thickness of the dimensionless frost layer, tau is the dimensionless frost time, omega is the dimensionless humidity of the incoming air,θ s Is the dimensionless temperature of the frost surface, omega sa (θ s ) And the non-dimensional saturated humidity corresponding to the non-dimensional temperature of the surface of the frost layer is represented.
In the dimensionless mass conservation equation, the left side of the equation is the dimensionless frost layer growth rate, the equation is the dimensionless humidity difference (dimensionless excess humidity), and the physical meaning is that the driving force of the dimensionless frost layer growth rate is the dimensionless humidity difference.
In the case of a true physical meaning, a three-term approximate simplification is performed.
The air saturation humidity is a polynomial nonlinear relation with respect to temperature, and no further beneficial information can be obtained by equation (32). But within a narrower temperature interval the change in air saturation humidity with respect to temperature may be approximately linear. Thus, the air saturation humidity is linearly simplified, and the linear function of the simplified air saturation humidity with respect to temperature is as follows:
ρ sa (T)＝ρ sa (T w )+α(T-T w ) (33)
wherein T is temperature, ρ sa (T) the saturation humidity of air at the temperature T, T w Is the slope coefficient of the linear function of the temperature of the low-temperature flat plate and alpha.
Step 2.2: according to the definition of the non-dimensional frost surface saturation humidity, and substituting the definition into the linear relation of the formula (33), the non-dimensional frost surface saturation humidity can be written as the ratio of two temperature differences, and the formula is as follows:
wherein T is d For incoming flow dew point temperature, T s Is the surface temperature ρ of the frost layer v For incoming air humidity ρ s The air saturation humidity corresponding to the surface temperature of the frost layer.
For the temperature difference value on the right term denominator in the formula (34), it can be obtained by the energy conservation equation at the time of frost balance as shown in the following formula:
wherein k is f Is the thermal conductivity of the frost layer, h is the convective heat transfer coefficient and T a For incoming flow temperature, delta b The thickness was balanced for the frost layer.
When the amount of convective heat transfer is much greater than the latent heat release from the frosting phase change, then the latent heat portion of the heat transfer can be ignored. Thus, for the temperature difference value on the right term molecule of formula (34), it can be obtained by the energy conservation equation during frosting, which ignores latent heat:
wherein X is f Is the thickness of the frost layer.
After the formula (35) and the formula (36) are finished, the formula (34) is substituted, and the relational expression of the non-dimensional frost surface saturation humidity with respect to the non-dimensional frost layer thickness can be obtained:
since the frost temperature is between the cold temperature and the dew point temperature (T w ≤T s ≤T d ) When the incoming flow temperature is far greater than the difference between the dew point temperature and the cold face temperature, namely:
T a ＞＞(T d -T w ) (38)
when the temperature difference ratio of formula (37) may be approximately equal to 1, namely:
(T a -T s )/(T a -T d )＝1 (39)
substituting the formula (39) into the formula (37) to obtain a relational expression that the saturated humidity value of the dimensionless frost surface is equal to the thickness of the dimensionless frost layer:
ω sa (θ s )＝δ (40)
substituting equation (40) into the dimensionless mass conservation equation of equation (32), i.e., replacing the dimensionless frost surface saturation humidity in the dimensionless mass conservation equation with the dimensionless frost layer thickness, the following relationship can be obtained:
after the integral solution of the formula (41), an analytical expression of the thickness of the non-dimensional frost layer with respect to the non-dimensional frost formation time is obtained:
δ＝1-e -τ (42)
therefore, the non-dimensional frost layer thickness can be directly resolved by the resolution formula (42) without iteratively solving given initial conditions.
In the formula (42), only two variable parameters of the dependent variable non-dimensional frosting time tau and the dependent variable non-dimensional frost layer thickness delta are provided, and the relation between the non-dimensional frosting time and the non-dimensional frost layer thickness is fixed. The curve of the non-dimensional frosting time with respect to the thickness of the non-dimensional frosting layer is unique, and therefore, the curve is called a non-dimensional frosting characteristic curve, which is abbreviated as frosting characteristic curve.
Expanding the dimensionless frost layer thickness and the dimensionless frost formation time in the formula (42), and reducing to an expression of the frost layer thickness with respect to the frost formation time:
wherein delta b The frost characteristic time is calculated by the equilibrium thickness delta of the frost layer b Divided by the initial frost growth rate ζ.
Under the frosting condition meeting the physical meaning represented by approximate simplification, the frost thickness X at the time t can be directly calculated by the formula (43) f Without the need for iterative computations. The frosting characteristic time is endowed with the meaning of a time constant, and the frosting characteristic time characterizes the frosting speed. The characteristic time and the characteristic thickness respectively represent the frosting speed and the maximum thickness of the frosting layer, and are the most important and most visual two quantities in dynamic frosting.
The rapid prediction of the flat frosting thickness can be realized by the fourth analytical expression of the expression 43.
In some embodiments, the method further comprises: and calculating and predicting the relative error of the thickness of the frost layer, and verifying a prediction method. The accuracy of the prediction method of the present invention can be calculated by calculating the relative error, so as to verify the feasibility of the method, as shown in fig. 4 and 5.
In some embodiments, the method further comprises: judging whether the thickness of the frost layer obtained by prediction exceeds a corresponding threshold value, and if so, carrying out early warning. Such as predicting a frosting condition: incoming flow temperature T a Incoming flow dew point temperature T d And low temperature plate temperature T w And the thickness d1 of the frost layer corresponding to a certain time t1 in the future is judged to be whether d1 exceeds the threshold value Y of the thickness of the frost layer of the low-temperature heat exchanger, when the predicted thickness of the frost layer exceeds the threshold value, the normal use of the low-temperature heat exchanger is influenced, the normal use of the low-temperature heat exchanger is ensured by early warning and can be treated or intervened in advance, and if the predicted thickness exceeds the threshold value, the temperature of incoming flow is regulated or the temperature of a low-temperature flat plate is regulated in advance.
In some embodiments, the method can also be applied to heat exchanger design, for example, working condition data and environment data of the heat exchanger can be obtained in advance, frosting state data of the heat exchanger can be obtained through the data, the frosting thickness corresponding to the heat exchanger when the heat exchanger works can be predicted through the frosting state data of the heat exchanger, and the heat exchange unit size and the distance of the designed heat exchanger are designed through the maximum frosting thickness.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the following claims be interpreted as including the preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (10)
1. A method for rapidly predicting a flat frosting thickness based on frosting characteristic curves, the method comprising:
step 1: constructing a simplified condition;
step 2: establishing a first analytical expression between the thickness of the dimensionless frost layer and the dimensionless frost formation time based on the simplification condition;
step 3: constructing a second analytical expression of the dimensionless frosting time and a third analytical expression of the dimensionless frosting layer thickness;
step 4: bringing the third analytical expression and the second analytical expression into the first analytical expression to obtain a fourth analytical expression;
step 5: and obtaining frosting state information corresponding to the frosting thickness of the flat plate to be predicted, obtaining the equilibrium thickness of the frosting layer and frosting characteristic time based on the frosting state information, and obtaining the corresponding frosting layer thickness by utilizing a fourth analytical expression calculation based on the equilibrium thickness of the frosting layer and the frosting characteristic time.
2. The method for rapidly predicting the frosting thickness of a flat plate based on the frosting characteristic curve according to claim 1, wherein the simplification condition is as follows: the ratio of the heat exchange amount of convection to the latent heat release amount of frosting phase change is larger than a first threshold value, the ratio of the incoming flow temperature to the first temperature difference is larger than a second threshold value, the temperature difference between the maximum temperature value and the minimum temperature value of the preset temperature interval is smaller than the preset temperature value, and the first temperature difference is the difference between the dew point temperature and the cold surface temperature.
3. The method for rapidly predicting the frosting thickness of a flat plate based on the frosting characteristic curve according to claim 2, wherein the first analytical expression is obtained by the following steps:
simulating low-temperature flat plate dry mode frosting, and constructing a mass conservation equation corresponding to the frosting according to the relationship between the mass change rate of the frosting and the phase change rate of the water vapor;
obtaining a characteristic value related to low temperature flat plate dry mode frosting, comprising: dimensionless temperature, dimensionless frost layer thickness, dimensionless frost time, dimensionless humidity, and dimensionless frost layer density;
processing the mass conservation equation based on the dimensionless temperature, the dimensionless frost layer thickness, the dimensionless frost formation time, the dimensionless humidity and the dimensionless frost layer density to obtain a dimensionless mass conservation equation;
if the density and the heat conductivity of the frost layer are unchanged in the frosting process, simplifying a dimensionless mass conservation equation to obtain a first dimensionless mass conservation equation;
and based on the simplification condition, simplifying the first dimensionless mass conservation equation to obtain the first analytical expression.
4. The method for rapidly predicting a flat frosting thickness based on frosting characteristic curves according to claim 3, wherein the simplifying the first quantitative mass conservation equation based on the simplifying conditions to obtain the first analytical expression comprises:
in a preset temperature interval, linearly simplifying the air saturation humidity to obtain a first linear function of the simplified air saturation humidity with respect to temperature;
bringing the expression of the dimensionless frost surface saturated humidity into the first linear function to obtain a second function;
when the ratio of the convection heat exchange amount to the latent heat release amount of frosting phase change is larger than a first threshold value, the second function is arranged to obtain a first relational expression of the non-dimensional frosting surface saturation humidity with respect to the thickness of the non-dimensional frosting layer;
when the ratio of the incoming flow temperature to the first temperature difference is larger than a second threshold value, the first relation is sorted, and a second relation with the non-dimensional frost layer thickness equal to the non-dimensional frost layer saturation humidity in value is obtained;
substituting the second relation into the first dimensionless mass conservation equation to obtain a first analytic expression of the thickness of the dimensionless frost layer with respect to the dimensionless frost formation time.
5. The method for rapidly predicting the frosting thickness of a flat plate based on the frosting characteristic curve according to claim 1, wherein the first analytical expression is:
δ＝1-e -τ
wherein delta is the thickness of the dimensionless frost layer, and tau is the dimensionless frost time;
the second analytical expression is:
wherein eta is the characteristic time of frosting, and t is the time of frosting;
the third analytical expression is:
wherein X is f For the thickness of the frost layer, delta b Balancing the thickness of the frost layer;
the fourth analytical expression is:
6. the method for rapidly predicting the frosting thickness of a flat plate based on frosting characteristic curve according to claim 5, wherein the equilibrium thickness delta of the frosting layer is as follows b The calculation mode of (a) is as follows:
wherein k is f The thermal conductivity of the frost layer is h is the convective heat transfer coefficient, T a For incoming flow temperature, T d For incoming flow dew point temperature, T w Is the temperature of the low-temperature flat plate;
wherein k is a Is air heat conductivity, L is characteristic length of the low-temperature flat plate along the incoming flow direction,is the average noose number;
wherein Re is L Pr is the Plandt number, u is the incoming air velocity, and v is the incoming air motion coefficient of viscosity.
7. The method for rapidly predicting the frosting thickness of a flat plate based on the frosting characteristic curve according to claim 5, wherein the calculation mode of the frosting characteristic time eta is as follows:
the calculation mode of the initial frost growth rate xi is as follows:
wherein h is m Is the mass transfer coefficient of water vapor, ρ v For incoming flow humidity ρ sa (T w ) Saturated humidity corresponding to low-temperature flat plate temperature, ρ v -ρ sa (T w ) The mass transfer concentration difference of water vapor at the initial frosting moment is h is the convective heat transfer coefficient, c pa Specific heat of air ρ a For air density, le is the lewis number and n is a constant.
8. The method for rapidly predicting the frosting thickness of a flat plate based on the frosting characteristic curve according to claim 1, wherein the frosting state information comprises: incoming flow temperature T a Incoming flow dew point temperature T d And low temperature plate temperature T w 。
9. The method for rapidly predicting the frosting thickness of a flat plate based on frosting characteristic curves according to claim 1, further comprising: and calculating and predicting the relative error of the thickness of the frost layer, and verifying a prediction method.
10. The method for rapidly predicting the frosting thickness of a flat plate based on frosting characteristic curves according to claim 1, further comprising: judging whether the thickness of the frost layer obtained by prediction exceeds a corresponding threshold value, and if so, carrying out early warning.
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