阅读说明：本技术 一种预测热风干燥过程中球形食品温湿度的模型及其应用 (Model for predicting temperature and humidity of spherical food in hot air drying process and application of model ) 是由 陶阳 朱芮 李丹丹 韩永斌 卢国宁 于 2021-05-10 设计创作，主要内容包括：本发明属于农产品加工技术领域,具体涉及一种预测热风干燥过程中球形食品温湿度的模型及其应用。本发明的模型为在一系列假设条件下构建传热和传质耦合的笛卡尔坐标系下球面几何的一维非稳态方程,并确定初始条件和边界条件。该模型拟合效果较佳,能够较好地预测超声干燥过程中球形食品的温湿度变化,从而在生产过程中进行应用。(The invention belongs to the technical field of agricultural product processing, and particularly relates to a model for predicting the temperature and humidity of spherical food in a hot air drying process and application thereof. The model of the invention is a one-dimensional unsteady state equation of spherical geometry under a Cartesian coordinate system for heat transfer and mass transfer coupling constructed under a series of assumed conditions, and initial conditions and boundary conditions are determined. The model has good fitting effect, and can better predict the temperature and humidity change of the spherical food in the ultrasonic drying process, thereby being applied in the production process.)

1. A model for predicting temperature and humidity of spherical food in a hot air drying process is characterized by comprising the following components:

constructing a one-dimensional unsteady state equation of spherical geometry under a Cartesian coordinate system of heat transfer and mass transfer coupling under a series of assumed conditions, and determining initial conditions and boundary conditions:

the assumption is that:

due to symmetry, there is no humidity and temperature gradient on the center line of the spherical food at first, and the internal heat and mass transfer of the spherical food are carried out by conduction and diffusion, respectively; phase transformation occurs only on the surface of the spherical food; the heat and mass transfer coefficient of the surface of the spherical food is kept unchanged; the path length of moisture diffusion and heat conduction inside the spherical food also changes with volume shrinkage, resulting in changes in boundary conditions;

constructing a heat and mass transfer coupling equation:

ρ_brepresenting the real-time bulk density, which is the ratio of the total mass to the total volume of the spherical food product in kg m^-3；

C_pRepresents the real-time specific heat of a spherical food, and has a unit of J kg^-1 K^-1；

λ represents the heat transfer coefficient in the interior of the spherical food product and has the unit of W m^-1 K^-1；

Q_eRepresenting an internal heat source, unit W m^-3；

ρ_sRepresenting the real-time solid matter density, is the ratio of the dry matter mass to the total volume of the spherical food product in kg m^-3；

D_eRepresents the effective water diffusion coefficient in m² s^-1；

T represents the temperature of the point at which the spheroidal food is to be predicted, in K;

w represents the moisture content of the spot of the ball-shaped food to be predicted, and is one of the results obtained by simulation of the model, and the unit is kg moisture/kg dry matter;

t represents drying time in seconds;

x represents the mass transfer path length, here the length of the point from the center of the sphere where the temperature and humidity of the spherical food are to be predicted, in m;

is the sign of the partial derivative;

initial conditions:

T(x，t＝0)＝T₀(temperature at the beginning of drying)

W(x，t＝0)＝W₀(moisture content at the beginning of drying)

Boundary conditions:

h represents the heat transfer coefficient of air near the surface of the spherical food product and has a unit of W m^-2K^-1；

T_sRepresents the surface temperature of the spherical food, and the unit is K or ℃;

T_airrepresenting the air temperature in K or ℃;

h_fgrepresents the latent heat of water evaporation, and the unit is J/kg;

Q_USrepresents the intensity of ultrasound dissipated as heat from the surface of a spherical food product in Wm^-2；

h_mRepresenting the water mass transfer coefficient, in m s^-1；

C_sRepresents the surface water evaporation concentration of the spherical food, and has a unit of kgm^-3；

C_airRepresents the concentration of water evaporated from the air, and has a unit of kgm^-3；

T represents the temperature of the center point or surface of the spherical food to be predicted, and the unit is K;

w represents the moisture content of the spot to be predicted for the ball, here the dry basis moisture content, and is one of the results obtained from the model simulation, in kg moisture/kg dry matter;

x represents the mass transfer path length where the point at which the temperature and humidity of the spherical food is to be predicted is the length from the center of the sphere in m;

r represents the real-time radius in m during the drying process of the spherical food;

after the above coefficients are determined, the coupled heat and mass transfer model is solved by using a 'pdepe' function in MATLAB.

2. The model for predicting the temperature and the humidity of spherical food in the ultrasonic hot air drying process according to claim 1, wherein the heat and mass transfer coupling equation and partial coefficients in the boundary condition are determined as follows:

formula (1)

a_wThe water activity is determined by isothermal adsorption curve measurement;

t represents the temperature of the center point or surface of the spherical food to be predicted, and the unit is K;

p_VS(T) represents the saturated water vapour pressure at the surface of the spherical food product in Pa;

formula (2)

a_wThe water activity is determined by isothermal adsorption curve measurement;

T_airrepresenting the air temperature in K or ℃;

p_VS(Tair) is the saturated water vapor pressure in air for spherical food products;

RH refers to the relative humidity of air;

p_VSis a temperature dependent function determined by the following equation:

in the above formula, T₁Is temperature in K;

formula (3) C_p＝(0.873+1.256W)×1000

In the above formula, W is the moisture content of the point to be predicted for the ball-shaped food, and is the moisture content on a dry basis, which is one of the results obtained by the model simulation, and the unit is kg moisture/kg dry matter;

formula (4)

Formula (5)

In the above equations (4) and (5), T represents the temperature of the point of the spherical food to be predicted, and the unit is K;

formula (6)

N is the initial dry mass ratio, dry mass/initial mass of the fresh spheroidal food product determined according to the AOAC method; m₀Represents the initial weight of the spherical food product in kg; v_blRepresenting the real-time volume of a spherical food product in m³(ii) a r represents the real-time radius of the spherical food product in m;

formula (7)

M represents the real-time weight of the spheroidal food product; m₀Represents the initial weight of the spherical food product; both units are kg;

formula (8)

M represents the real-time weight of the spherical food in kg; m₀Represents the initial weight of the spherical food product in kg; v_blRepresenting the real-time volume of a spherical food product in m^-3(ii) a r represents the real-time radius of a spherical food product in m.

3. The model for predicting the temperature and humidity of a spherical food product during ultrasonic hot air drying according to claim 1, wherein the effective moisture diffusion coefficient D is_eIs determined by the following method:

formula (9)

W_eqRepresenting the moisture content in kg moisture/kg dry matter in the equilibrium state to be achieved by ultrasonic hot air drying; w₀Represents the initial moisture content in kg moisture/kg dry matter; r is₀Is the initial radius of the spherical food; w_mFor real-time weighingCalculating the water content in kg water/kg dry matter;

when drawing (W)_m-W_eq)/(W₀-W_eq) Deriving D from the slope of the line as logarithmic time_eA value;

the D at different temperatures at different times is deduced from the formula (9)_eValue of D_eAnd real-time temperature into the following formula to obtain D₀、E_aThe value of (c):

formula (10)

D₀The exponential pre-factor representing the Arrhenius equation in m²s^-1；E_aRepresents activation energy in kJ mol^-1；R_gRepresents the gas constant in kJ mol^-1 K^-1(ii) a T represents temperature in K;

substituting the formula (10) into the formula of claim 1, and fitting to obtain a model.

4. The model for predicting the temperature and the humidity of spherical food in the ultrasonic hot air drying process according to claim 1, wherein the heat and mass transfer coefficient determining method is determined by using a metal ball simulation method, and specifically comprises the following steps:

the volume of the metal ball is consistent with that of the spherical food to be measured;

placing the metal ball in an oven and applying ultrasonic waves; assuming a low internal temperature gradient of the metal sphere, lumped analysis can be used to describe the heat transfer process of the metal sphere:

the method for determining the surface heat conduction coefficient h comprises the following steps:

formula (11)

T_CuRepresenting the real-time temperature of the surface of the metal ball, and the unit is K or ℃; t is_airRepresenting the air temperature in K or ℃; t is₀Represents a metal ballThe initial temperature of (a) is in K or ℃; a. the_CuRepresents the surface area of the metal ball in m²；ρ_CuRepresents the density of the metal ball in kg · m^-3，C_p，CuIs the specific heat unit of the metal ball is J.kg^-1k^-1；V_CuRepresents the volume of the metal sphere in m³(ii) a h represents the surface heat transfer coefficient in Wm^-2K^-1(ii) a t represents time in seconds;

carrying out nonlinear fitting on the experimental data and the formula (11) in MATLAB to obtain an equation for calculating h;

h_mthe determination method comprises the following steps:

formula (12)

Where ρ is_airRepresenting the air density in kg m^-3；C_p，airRepresents the specific heat of air, in J kg^-1k^-1(ii) a The h value is calculated by the method; le is the number of Liu Yi Si;

le is calculated as follows:

formula (13)

α_airRepresents the thermal diffusivity of air in m²s^-1；D_airRepresents the diffusivity of water vapor in air, and has the unit of m²·s^-1。

5. The model for predicting the temperature and humidity of spherical food in the ultrasonic hot air drying process according to claim 4,

ultrasonic intensity Q of spherical food surface dissipated in the form of heat_USCalculated by the following method: q was measured due to the lack of constant specific heat data during drying of the spherical food product_USWhen the food is replaced by metal balls with a geometric shape similar to that of spherical food:

formula (14)

Formula (15)

Formula (16) a ═ 4 pi r²

P_UsRepresents the ultrasonic power in W; m is_CuRepresents the weight of the metal ball in kg; c_p，CuIs the specific heat of the metal ball, and has a unit of J.kg^-1k^-1(ii) a A represents the real-time surface area of the spherical food product in m²(ii) a r is the real-time radius of the spherical food product in dry form, in m.

6. The model for predicting the temperature and humidity of spherical food in the ultrasonic hot air drying process according to any one of claims 4 or 5, wherein the metal spheres are selected from copper spheres.

7. The use of the model for predicting the temperature and humidity of spherical food in the ultrasonic hot air drying process according to any one of claims 1 to 6, wherein the model for predicting the temperature and humidity of spherical food in the ultrasonic hot air drying process is used for predicting the average humidity of spherical food in the ultrasonic hot air drying process.

Technical Field

The invention belongs to the technical field of agricultural product processing, and particularly relates to a model for predicting the temperature and humidity of spherical food in a hot air drying process and application thereof.

Background

Ultrasonic waves are a representative non-thermal technique, and the mechanical vibration and cavitation phenomena generated can accelerate the drying process, reduce energy consumption and improve the quality of dehydrated food. There are three different modes of application of ultrasound in drying, including ultrasonic pretreatment, gas-medium ultrasound, and contact ultrasound. In practical applications, the air-borne ultrasonic waves and the contact ultrasonic waves are usually combined with a hot air dryer to dry, so as to enhance the drying process. The energy consumption is reduced and the quality is improved while the time is saved.

Drying is a complex process involving both heat and mass transfer. In order to better understand and control the drying process, it is necessary to know the spatial and temporal variations of the temperature and humidity inside the food material, while the use of mathematical simulations based on physical models is an important tool to explore the underlying mechanisms. However, the current model study of ultrasound-assisted drying is a mass transfer model alone, which ignores the effect of temperature changes on the effective diffusivity of moisture, assuming only a constant mass transfer value, but the actual diffusivity varies with temperature.

In order to enrich basic knowledge of the mechanism of ultrasonic energy assisted hot air drying of spherical food in a gas medium mode and a contact mode and better control the drying process of the spherical food, the invention adopts a heat and mass transfer model which takes the factors of moisture diffusivity change, shrinkage, ultrasonic energy input and the like into consideration to describe the change of the drying temperature and humidity of the spherical food under the irradiation of ultrasonic energy, and the researches are expected to widen the application of the ultrasonic technology in the field of food drying.

Disclosure of Invention

The invention establishes a model for predicting the temperature and the humidity of spherical food in the hot air drying process. Because the spherical structure of the spherical food can be maintained in the whole drying process, technicians can successfully predict the moisture change, the weight change and the temperature change of the inner surface and the outer surface of the spherical food in the drying process through the drying model, so that the ultrasonic hot air drying process can be controlled more accurately, and the quality of the spherical food is ensured.

Firstly, the invention provides a model for coupling the heat transfer and mass transfer processes of spherical food in the ultrasonic hot air drying process, and the temperature and humidity conditions of a certain point of the spherical food in the ultrasonic hot air drying process can be predicted through the model.

Secondly, the present invention provides a method for determining the intensity of the ultrasonic sound QUS dissipated as heat from the surface of a spherical food product, i.e., the actual ultrasonic intensity.

Thirdly, the invention also provides a method for measuring the surface heat and mass transfer coefficient in the ultrasonic drying process by using the copper ball simulation instead of the spherical food based on the model.

Fourth, the model of the present invention can also predict the average moisture content of a spherical food product during ultrasonic hot air drying.

Based on the above prediction results, data support can be provided for determining the drying time, drying temperature and ultrasonic intensity required by the spherical food in a specific volume, which helps to manage various parameters in the drying process more accurately.

The technical solution of the present invention is further explained below.

In some aspects, the invention discloses a model for predicting the temperature and humidity of spherical food in a hot air drying process, which is characterized by comprising the following steps:

(1) collecting isothermal adsorption curve of spherical food to be dried at certain temperature to determine a_w；

(2) Collecting the total mass of the spherical food at different drying moments, the dry basis water content of the spherical food, and the internal and surface temperatures of the spherical food at different drying moments;

(3) constructing a one-dimensional unsteady equation of spherical geometry under a Cartesian coordinate system of heat transfer and mass transfer coupling under a series of assumed conditions, and determining initial conditions and boundary conditions;

heat and mass transfer coupling equation:

ρ_brepresenting the real-time bulk density, which is the ratio of the total mass to the total volume of the spherical food product in kg m^-3；

C_pRepresentative of a spherical foodReal-time specific heat of the product in Jkg^-1 K^-1；

λ represents the heat transfer coefficient in Wm inside the spherical food^-1 K^-1；

Q_eRepresents an internal heat source in the unit Wm^-3(ii) a According to the actual condition of drying, if no internal heat source exists, Qe is 0.

ρ_sRepresenting the real-time solid matter density, is the ratio of the dry matter mass to the total volume of the spherical food product in kg m^-3；

D_eRepresents the effective water diffusion coefficient in m² s^-1；

T represents the temperature of the point at which the spheroidal food is to be predicted, in K;

w represents the moisture content of the spot of the ball-shaped food to be predicted, and is one of the results obtained by simulation of the model, and the unit is kg moisture/kg dry matter;

t represents drying time in seconds;

x represents the mass transfer path length, here the length of the point from the center of the sphere where the temperature and humidity of the spherical food are to be predicted, in m;

is the sign of the partial derivative;

the assumed conditions include:

due to the symmetry, there is no moisture and temperature gradient on the center line of the spherical food product. The internal heat and mass transfer of the spherical food takes place by conduction and diffusion, respectively; phase transformation occurs only on the surface of the spherical food; (ii) a The heat and mass transfer coefficient of the surface of the spherical food is kept unchanged; volume contraction through p_bAnd ρ_sTo calculate p_bAnd ρ_sRespectively, the ratio of the total mass to the total volume of the spherical food product (volume shrinkage results in bulk density); and the ratio of dry solids mass to total volume. At the same time, the path length (referred to as the sample radius) for moisture diffusion and heat conduction inside the spherical food also changes with volume shrinkage, resulting in a change in boundary conditions. In the heat and mass transfer modelInvolving a change in radius of the spherical food product;

initial conditions:

T(x，t＝0)＝T₀

W(x，t＝0)＝W₀

boundary conditions:

(4) determining coefficients in a heat and mass transfer equation, initial conditions, and boundary conditions;

the coefficients include: c_s，C_atr，ρ_b，C_p，Q_e，λ，ρ_s，D_e，h，h_fg，Q_US，h_m；

(5) Solving the coupling heat and mass transfer model by using a 'pdepe' function in MATLAB and using RMSE and R²And AAD to evaluate the predictive power of the application model;

in some embodiments, the spherical food product surface water vapor concentration C is calculated according to the following relationship_sWater vapor concentration in air C_air：

Wherein, a_wDetermined from the isothermal adsorption curve; pvs (T) in the above formula refers to the value of pvs at temperature T; pvs (Tair) refers to the value of pvs for temperature Tair.

p_VSIs a temperature dependent function determined by the following equation:

in the above formula, p_VS(T) is p at real-time temperature T_VSA numerical value; p is a radical of_VS(T_air) Refers to p at air temperature_VSNumerical values. RH in the formula means relative humidity of air;

formula (3) C_p＝(0.873+1.256W)×1000

In the above formula, W is the moisture content of the spot of the spherical food to be predicted, here the dry basis moisture content, and is one of the results obtained by the model simulation, in kg moisture/kg dry matter;

in the above equations (4) and (5), T represents the real-time temperature of the point where the spherical food is to be predicted, and the unit is K; the initial dry mass ratio of the spheroidal food product, determined according to the AOAC method, is N, i.e. the dry mass/initial mass of the spheroidal food product, p_sThe calculation is as follows:

the dry basis water content W is calculated as follows:

the above equation is converted to:

M＝W×M₀×N+M₀×N

thus, ρ_bRepresented by the following formula:

in the above formula, M₀Represents the initial weight of the spherical food product in kg; v_blRepresenting the real-time volume of a spherical food product in m^-3(ii) a r represents the real-time radius of the spherical food product in m; m represents the real-time weight of the spherical food in kg during the drying process;

effective water diffusion coefficient D_eThe determination method comprises the following steps:

(1)D_ethe values are analytically calculated in a semi-empirical manner:

W_eqrepresenting the moisture content in the equilibrium state to be achieved by ultrasonic hot air drying, namely the moisture content in the dried finished product, and the unit is kg moisture/kg dry matter; w₀Represents the initial moisture content in kg moisture/kg dry matter; r is₀Represents the initial radius of a spherical food product; w_mThe calculated moisture content is weighed in real time, the unit is kg moisture/kg dry matter;

drawing (W)_m-W_eq)/(W₀-W_eq) Deriving D from the slope of the line as logarithmic time_eA value;

the D at different temperatures at different times is deduced from the formula (9)_eValue of D_eAnd the real-time temperature T is substituted into the following formula to obtain D₀、E_aThe value of (c):

D₀the exponential pre-factor representing the Arrhenius equation in m² s^-1；E_aRepresents activation energy in kJ mol^-1；R_gRepresents the gas constant in kJ mol^-1K^-1(ii) a T represents temperature in K or ℃;

substituting the formula (10) into the formula of claim 1, and fitting to obtain a model.

Preferably, the heat and mass transfer coefficients h and h_mAs determined by the metal ball simulation described below.

(1) To integrate the effect of ultrasonic radiation on heat and mass transfer coefficients, copper spheres with a spheroidal food geometry were placed in an oven.

(2) Assuming a low internal temperature gradient, lumped analysis can be used to describe the heat transfer process of the copper ball:

determining the surface heat and mass transfer coefficient h of the copper ball:

the h value was calculated by fitting the data non-linearly to the equation in MATLAB. Under different conditions of pure hot air drying, ultrasonic contact, ultrasonic cavitation and the like, the measured h values are different. The h value is determined according to different conditions.

T_CuRepresenting the real-time temperature of the surface of the metal ball, and the unit is K or ℃; t is_airRepresenting the air temperature in the oven in units of K or ℃; t is₀Represents the initial temperature of the metal ball, and the unit is K or ℃; a. the_CuRepresents the surface area of the metal ball in m²；ρ_CuRepresents the density of the metal ball, C_p，CuIs the specific heat unit of the metal ball is J.kg^-1k^-1；V_CuRepresents the volume of the metal sphere in m³(ii) a h represents a surface heat conduction systemNumber in Wm^-2K^-1；

Determining mass transfer coefficient h by using Chilton-Colburn formula_m：

Where ρ is_airRepresents the air density in the oven in kgm^-3；C_p，airRepresents the specific heat of air, in J kg^{- 1}k^-1. Le is the number of lewis, and is calculated as follows:

α_airis the thermal diffusivity of air in m²s^-1，D_airIs the diffusivity of water vapor in air, and has the unit of m²·s^-1。

Actual ultrasonic intensity consumed on spherical food, i.e. ultrasonic intensity Q dissipated as heat on surface of spherical food_USThe method of (3), comprising:

(1) the actual consumption of ultrasonic energy in a spherical food product was measured using a calorimetric method.

(2) Since the spherical food lacks specific heat value, copper balls (25 mm in diameter) of similar geometry to the spherical food were used instead for the measurement.

(3) The copper balls were insulated with a paper box and the ultrasonic power dissipated on the surface of the copper balls was obtained from the following equation, which is considered to be equal to the actual ultrasonic power on the surface of the spherical food product under the applied drying conditions:

in this formula, P_USRepresents the ultrasonic power in W; m is_CuRepresents the weight of the metal ball in kg; c_p，CuIs a specific heat of a metal ballThe bit is J.kg^-1k^-1(ii) a A represents the surface area of the spherical food product in m²(ii) a r is the real-time radius of the spherical food product in dry form, in m. Thus, the ultrasound intensity is calculated as follows:

due to volume contraction, Q_USThe volumetric shrinkage is determined at each sampling time according to a shrinkage mode formula, which varies with decreasing surface of the spherical food item.

Formula (16) a ═ 4 pi r²

In the present invention, the dry basis water content is defined as the ratio of the weight of water to the weight of the dry basis in the following sample. For example, when a sample having a mass of m g is dried sufficiently and then weighs n g, the dry basis moisture content is (m-n)/n × 100%.

It is of course further preferred that the metal balls are selected from copper balls.

The model for predicting the temperature and the humidity of the spherical food in the ultrasonic hot air drying process is applied to predicting the average humidity of the spherical food in the ultrasonic hot air drying process.

Advantageous effects

Firstly, the invention provides a model for coupling the heat transfer and mass transfer processes of spherical food in the ultrasonic hot air drying process, and the temperature and humidity conditions of a certain point of the spherical food in the ultrasonic hot air drying process can be predicted through the model.

Secondly, the present invention provides a method for determining the intensity of the ultrasonic sound QUS dissipated as heat from the surface of a spherical food product, i.e., the actual ultrasonic intensity.

Thirdly, the invention also provides a method for measuring the surface heat and mass transfer coefficient in the ultrasonic drying process by using the metal ball simulation to replace the spherical food based on the model.

Fourth, the model of the present invention can also predict the average moisture content of a spherical food product during ultrasonic hot air drying. The average moisture content is the mass of moisture contained in the whole spheroidal food product at a certain moment/dry mass of the whole spheroidal food product.

The invention also provides a method for predicting the change of the dry basis water content, the center and the surface temperature of the spherical food along with the drying time in the drying process of the spherical food based on the model. The prediction result can provide data support for determining the drying time, the drying temperature and the ultrasonic intensity of the spherical food required by a specific volume, and is helpful for more accurately managing various parameters in the drying process.

Drawings

FIG. 1 shows the shrinkage curves of a spherical food product dried under three different sets of conditions at 65 ℃.

FIG. 2 isothermal adsorption curve of spherical food at 65 ℃.

Fig. 3 is a flow chart of numerical simulation.

FIG. 4 is a graph of the measured average moisture content and the predicted average moisture content for three experiments during hot air drying.

FIG. 5 is a graph of the measured and predicted temperature profiles for three sets of experimental centers (a) and surfaces (b).

FIG. 6 spatial distribution of temperature inside the spherical food product (dotted line indicates outer surface) under three sets of experimental drying.

FIG. 7 spatial distribution of moisture inside the spherical food product (dotted line indicates outer surface) under three sets of experimental drying.

FIGS. 8A, 8B, and 8C are graphs of drying kinetics of spheroidal food products at 50 deg.C, 60 deg.C, and 70 deg.C.

Detailed Description

Embodiments of the present invention will be described in detail below with reference to examples, and it will be understood by those skilled in the art that the following examples are only illustrative of the present invention and should not be construed as limiting the scope of the present invention. The examples, in which specific conditions are not specified, were conducted under conventional conditions or conditions recommended by the manufacturer.

The devices used in the following examples were each commercially available in the following types:

ultrasonic probe (diameter 5mm, 20 kH):

automatic temperature recorder: OMEGA/RDXL12SD, USA

The following examples were conducted with blackberry as the ball food: in the embodiment of the invention, the temperature of the dried blackberry is 65 ℃, the wind speed is 2.0m/s, and the actual output ultrasonic intensity of the device is set to be 180.1W/dm²。

1. Blackberry drying experiment (data temperature and moisture collection)

1.1, collecting an isothermal adsorption curve of blackberries to be dried at 65 ℃;

the purpose of this step is to obtain an isothermal adsorption curve, and therefore pure hot air drying, without ultrasound.

Fresh blackberries are first dried in a hot air dryer to remove most of the water, ground and transferred to a different glass dryer containing a saturated salt solution (measuring water activity of blackberries in the same container with reagents such as lithium chloride, CH)₃COOK, magnesium chloride, potassium carbonate, potassium iodide, sodium chloride and KCl; all 7 reagents were used). Thymol is added in small amounts to avoid potential microbial contamination. The desiccator was sealed and placed in an air oven at 65 ℃. (since drying is carried out at 65 ℃, isothermal adsorption curve is drawn at 65 ℃ and drying is carried out at which temperature, isothermal adsorption curve is drawn at which temperature)

The weight of the food was monitored periodically until a constant weight was reached (about 40 days). Then, the water content of the blackberry at equilibrium was measured. An adsorption isotherm of the food at 65 ℃ was established using the following Guggenheim-Anderson-De Boer (GAB) model, and the isothermal adsorption curve was used to determine a in the model_wThe value, curve is shown in FIG. 2, where the ordinate We in FIG. 2 is Weq. In fig. 2, the point values are the values obtained from real measurements, and the continuous curve is the line of the fitted equation.

W_eqRepresents the moisture content at equilibrium in kg moisture/kg dry matter; w_mRepresenting the water content of the individual layer in kg water/kg dry matterQuality; c represents a constant in the GAB model; k represents a constant of the GAB model; a is_wRepresents water activity; wherein, W_mBoth the C and K values can be calculated by substituting the formula into MATLAB;

the water activity a of the blackberry can be calculated by the formula_w；

1.2 setting three groups of experiments, collecting the total mass of the blackberries at different drying times in the drying process, calculating the dry basis water content of the blackberries, and simultaneously measuring the internal and surface temperatures of the blackberries at different drying times; (the internal temperature is the temperature of the centre of the blackberry)

The drying process is carried out in a self-assembled hybrid dryer. An ultrasonic probe (20 khz) 5 cm in diameter was inserted into the oven from the top. A sample holder is arranged below the ultrasonic probe. During drying, a sample of a spherical food product, i.e. blackberry (about 3.5 grams), was placed on a sample holder. By adjusting the height of the sample holder, the ultrasonic wave can be switched between the air-dielectric propagation mode and the contact mode. Hot air of a target temperature enters the dryer from the left side and is discharged from the other side. Calorimetry is used to measure the actual applied ultrasonic power, since there is a loss of heat during use. The method comprises the steps of measuring the temperature rise condition of distilled water in a cotton cloth insulated glass beaker with the volume of 100 ml, and calculating the actual power transmitted by the ultrasonic probe through a formula. The ultrasound power was then converted to ultrasound intensity, denoted as W/dm 2. The actual output ultrasonic intensity of the apparatus was measured to be set at 180.1W/dm2, and the air temperature and velocity of the inlet port were 65 ℃ and 2.0m/s, respectively.

The height of the sample holder is adjusted regularly to take into account the shrinkage phenomenon, ensuring that the ultrasonic group spherical food and the ultrasonic probe are in direct contact during the whole drying process. For the cavitation sonication group, the spherical food product was placed 1 cm below the ultrasonic probe. The experimentally derived moisture content of the dry basis and the temperature changes of the center and outer surfaces are used for modeling.

3 sets of blackberry drying experiments (experiments 1-3, shown in table 1) were designed, and the 3 sets of experiments were:

the first group is a hot air group: drying with hot air at 65 ℃ and air speed of 2.0m/s without ultrasonic input;

the second group is a hot air and air medium type ultrasonic group: hot air drying at 65 deg.C and wind speed of 2.0m/s in combination with cavitation ultrasonic drying (blackberry is placed 1 cm below ultrasonic probe), and output ultrasonic intensity is set to 180.1W/dm 2;

the third group is a hot air and contact ultrasonic group: hot air drying at 65 deg.C and air speed of 2.0m/s, and contact ultrasonic drying (the blackberry is always in contact with the ultrasonic probe), with output ultrasonic intensity set to 180.1W/dm 2.

Of the three groups, the three groups had an initial dry basis moisture content of 6.24(g/g dry weight) and a diameter of about 5mm at an air relative humidity of 10%. During the drying process, samples were taken at different drying times, and the dry basis moisture content and the temperatures (K) of the inner and outer surfaces of the spherical food were measured using an automatic temperature recorder (equipped with a probe inserted into the center and surface of a specific location), respectively (see tables 1 and 2). The experimentally measured changes in the average moisture content are shown by the dot values in fig. 4, and the actually measured changes in the internal and external surface temperatures are shown by the solid lines in fig. 5.

The spherical food used in this experiment was perishable, so the samples were stored at-20 ℃ prior to the experiment. Before each experiment, the gel was thawed at room temperature and excess liquid was gently removed from its surface using a tissue paper.

TABLE 1 moisture content of the spherical food product on a dry basis as a function of drying time

TABLE 2 center and surface temperature of the spheroidal products as a function of drying time

During which the weight of the sample is monitored until a constant value is reached; and meanwhile, the total mass of the spherical food at different drying times t is collected, the dry basis water content of the spherical food is calculated, and the internal and surface temperatures of the spherical food at different drying times t are calculated and used for establishing a heat and mass transfer model at the later stage.

1.3 shrinkage determination experiment

The blackberry fruit size, i.e. radius of the blackberry, was measured simultaneously at different stages of drying in three sets of experiments. Despite the water loss, the blackberry for this experiment maintained the spherical geometry during the drying process. Thus, a moisture ratio (current moisture/initial moisture (calculation of moisture as kg moisture/kg dry matter)) and a sample radius ratio (current measured radius/initial radius (r/r)) were established₀) A quantitative relationship representing the change in sample radius ratio as a function of moisture ratio (as shown in figure 1, a shrinkage curve for a spherical food product dried under three different conditions at 65 c) the radius data will be used to determine the path length for heat and mass transfer in modeling. In fig. 1, the point values are the real measured values and the continuous dashed lines are the fitted equations.

The resulting coupling equation is as follows:

and (3) hot air drying group: 0.6068x +0.3472R²＝0.9777

Hot air and air medium type ultrasonic group: 0.4365x +0.5781R²＝0.9459

Hot air and contact ultrasound set: 0.3656x +0.6186R²＝0.9859

In the above formula, y represents r/r of each blackberry group₀X represents W/W₀。

1.4 determination of Heat and Mass transfer coefficients

Since the heat and mass transfer coefficients of the blackberry surface are unknown, and to integrate the effect of ultrasonic radiation on the heat and mass transfer coefficients, copper spheres with a geometry similar to the blackberry (25 mm diameter) were placed in an oven at 65 ℃ under all the above-mentioned drying conditions. The surface temperature of the copper ball was recorded by a K-type thermocouple connected to a data logger (DT-3891G, Shenzhen Guangtai mechanical industries, Ltd., China). (subsequent simulation results show that the coefficients measured by the method are reliable)

The heat transfer process of the copper spheres was described using lumped analysis, assuming low internal temperature gradients, by calculating the h-value by non-linear fitting the data to equation (11) in MATLAB. Wherein the density of the metal balls is rho_CuThat is, the density of copper is 8.96X 103 kg. m^-3The specific heat of the metal ball, i.e., the specific heat C of copper in this example_p，CuIs 390 J.kg^-1k^-1。

Determining the heat and mass transfer coefficient of the surface of the copper ball, namely the heat transfer coefficient h of the simulated surface of the spherical food:

the h value was calculated by fitting the experimental data non-linearly to the equation in MATLAB.

T_CuRepresenting the real-time temperature of the surface of the copper ball, and the unit is K or ℃; t is_airRepresenting the air temperature in K or ℃; t is₀Represents the initial temperature of the copper ball, and the unit is K or ℃; acu represents the surface area of the copper ball in m²(ii) a Density of copper rho_CuHas a density of 8.96X 103 kg.m^-3Copper C_p，CuSpecific heat of 390 J.kg^-1k^-1；V_CuRepresents the volume of the copper ball in m³(ii) a t is the time to dry, in units of s.

Mass transfer coefficient of matter, namely water mass transfer coefficient h_m:

Where ρ is_airRepresents the air density in kgm^-3The value is 1.029kg m^-3；C_p，airRepresents the specific heat of air, in Jkg^-1k^-1The value is 1.0 multiplied by 10³Jkg^-1k^-1(ii) a The h value is obtained by the calculation; le is the number of Liu Yi Si;

furthermore, h and h are added during the entire drying process_mAre considered constant and assume the same boundary of the surface exposed to the hot air during the modeling process.

Where Le is calculated as follows:

wherein alpha is_airRepresents the thermal diffusivity of air in m²s^-1。D_airRepresents the diffusivity of water vapor in air, and has the unit of m²·s^-1。

α_airAnd D_airRespectively, are 2.5X 10^-5m²s^-1And 2.18X 10^-5m²·s^-1。

h and h_mThe values in the three different cases are different and need to be determined separately for each case: hot air drying, hot air and air medium type ultrasonic treatment, hot air and contact ultrasonic treatment

1.5 determination of effective moisture diffusion coefficient

Wherein D is_eDetermined by the following equation:

W_eqrepresents the moisture content at equilibrium in kg moisture/kg dry matter; w₀Represents the initial moisture content in kg moisture/kg dry matter; d₀Represents arrhenibExponential pre-factor of the Uss equation in m² s^-1；r₀Represents the initial radius of the blackberry in m; t is the drying time in seconds. W_mIs the actually measured humidity at a certain drying time point, i.e. kg moisture/kg dry matter obtained by weighing.

Before using equation (9), a series of assumptions are required, including a constant De value, uniform humidity and temperature distribution, and no external mass transfer resistance.

To meet these assumptions, the blackberry was cut into pellets of only 5mm in diameter. The pellets were then dried at different temperatures (50, 60 and 70 ℃) in the three different cases described above, and the drying kinetics at 50, 60 and 70 ℃ are shown in FIGS. 8A, 8B and 8C, respectively, where the diamonds: drying with hot air; square: carrying out hot air and air medium type ultrasonic treatment; triangle: hot air and contact ultrasonic treatment. The wind speed was adjusted to the highest level (5 m/s) in order to minimize the external mass transfer resistance of the pellet surface. Weighing was continued until the drying process reached equilibrium. When drawing (W)_m-W_eq)/(W₀-W_eq) The De value is derived from the slope of the line as a logarithmic curve over time.

Then, the effect of temperature on De was studied using an arrhenius type temperature correlation equation:

D₀the exponential pre-factor representing the Arrhenius equation in m² s^-1；E_aRepresents activation energy in kJ mol^-1；R_gRepresents the gas constant in kJ mol^-1K^-1(ii) a T represents the real-time temperature in K or ℃.

Then substituting the De values at different temperatures and the corresponding temperatures into the formula (10) for fitting to obtain D₀The value of Ea, Rg.

1.6 determination of the actual ultrasound intensity consumed on the blackberry under the Hot air and air-mediated ultrasonic treatment and the Hot air and contact ultrasonic treatment, i.e.the ultrasound intensity dissipated as heat at the surface of the spherical food

The actual ultrasonic energy consumed on the blackberry was measured using a calorimetry method. Since blackberries lack a constant specific heat value during drying, copper balls (25 mm diameter) of similar geometry were used for the measurement and insulated with a paper box.

The temperature rise on the ball surface was monitored during hot air and air-jet ultrasound and hot air and contact ultrasound treatment with an output intensity of 180.1W/dm2 using the ultrasonic probe mentioned in section 1.1. Then, the ultrasonic power consumed on the surface of the copper sphere is obtained by the following equation:

P_USrepresents the ultrasonic power dissipated on the surface of the copper sphere in units of W; m is_CuRepresents the weight of the copper ball, and the unit is kg; c_p，CuIs the specific heat of copper;

here, the actual power of the hot air and air-mediated ultrasound and the hot air and contact ultrasound was 0.245 and 0.299W, respectively. The ultrasound intensity was calculated as follows:

formula (16) a ═ 4 pi r²

Q_USRepresenting the intensity of the ultrasonic waves dissipated as heat at the blackberry surface in W m^-2(ii) a A represents the surface area of the spherical food product in m²(ii) a Where r is the real-time radius of the blackberry as the food is dried.

Qus changes with decreasing surface of spherical food due to volume shrinkage, which is determined at each sampling time according to the shrinkage formula.

2. Heat and mass transfer model establishment in blackberry drying process

The blackberry drying is a process in which heat transfer and mass transfer are simultaneously carried out, water is diffused from the inside of the blackberry to the outside, and heat is conducted from the outside to the inside.

And constructing a heat and mass transfer model based on the law of conservation of energy, the Fourier law and the Fick second law.

Before performing the mathematical simulation, the following assumptions were made:

due to the symmetry, initially there was no humidity and temperature gradient on the centre line of the blackberry. The internal heat and mass transfer of the blackberry is carried out by conduction and diffusion, respectively;

the phase change only occurs on the blackberry surface;

the initial moisture content, temperature and composition of the blackberries used in each set of experiments were consistent;

the heat and mass transfer coefficient of the blackberry surface is kept unchanged;

wherein the volume shrinks by p_bAnd ρ_sTo calculate. Rho_bIs the ratio of real-time total mass to real-time total volume of the blackberry, rho_sRefers to the ratio of the mass of the blackberry dry solids to the total volume in real time. The path length (referred to as the sample radius) for moisture diffusion and heat conduction within the blackberry also changes with volume shrinkage, leading to changes in boundary conditions.

Accordingly, the following heat and mass transfer coupling model was constructed:

ρ_brepresents real-time bulk density (bulkdensity), which refers to the ratio of total mass of blackberry to total volume, where total volume is real-time total volume in kg m^-3；

C_pRepresents the real-time specific heat (specific heat) of a spherical food product, determined by the formula relating to the water content, described hereinafter, in units of Jkg^-1 K^-1；

λ represents the heat transfer coefficient inside the spherical food, and is determined by the following formula related to moistureConstant, unit Wm^-1 K^-1；

Q_eRepresents an internal heat source in the unit Wm^-3In the present embodiment, no internal heat source is added, so this value is considered to be 0;

ρ_srepresenting the real-time solid matter density, refers to the ratio of the mass of dry matter of a spherical food product to the total volume, where the total volume is the real-time total volume in kg m^-3；

D_eRepresenting the effective water diffusion coefficient, determined by the temperature-dependent equation described below, in m² s^-1；

T represents temperature in K (Kelvin);

w represents the moisture content, one of the results obtained from the model simulation, in kg moisture/kg dry matter;

t represents drying time in units of s;

x represents the length of a mass transfer path, and the length of a point which is to predict the temperature and the humidity of the spherical food from the center of the sphere is represented by m;

is the sign of the partial derivative;

in the case of ultrasonic enhanced convection drying, an internal heat source or heat sink Q_eIs considered to be 0 because the ultrasonic energy is input from the external environment.

The initial conditions are written as:

T(x，t＝0)＝T₀

W(x，t＝0)＝W₀

the boundary conditions are written as:

λ represents the heat transfer coefficient in the interior of the spherical food product, unit W m^-1K^-1；

h represents the surface heat transfer coefficient of the spherical food and has the unit of W m^-2K^-1；

T_sRepresents the surface temperature of the spherical food, and the unit is K or ℃;

T_airrepresenting the temperature of air in the oven, and the unit is K or ℃;

h_fgrepresents the latent heat of water evaporation, and the unit is J/kg;

D_erepresents the effective water diffusion coefficient in m² s^-1；

ρ_sRepresents the density of the solid matter in kgm^-3；

Q_USRepresents the intensity of ultrasound dissipated as heat from the surface of a spherical food product in Wm^-2；

h_mRepresenting the water mass transfer coefficient, in m s^-1；

C_sRepresents the surface water evaporation concentration of spherical food, and has unit of kg m^-3；

C_airRepresents the concentration of water evaporated in the air, and the unit is kg m^-3；

T represents temperature in K or ℃;

w represents the moisture content, here dry basis moisture content, which is one of the results obtained from the model simulation, in grams moisture/gram dry matter;

x represents the length of a mass transfer path, and the length of a point which is to predict the temperature and the humidity of the spherical food from the center of the sphere is represented by m;

the formula V shows that the conduction heat transfer speed of the blackberry surface is equal to the convection heat transfer of hot air to the blackberry surface, the water evaporation heat transfer and the ultrasonic radiation heat transfer.

Formula VI formula indicates that the moisture migrating to the blackberry surface by diffusion is equal to the moisture that is transferred from the surface to the air by convection.

Wherein the water vapor concentration C on the surface of the spherical food_sAnd the concentration C of steam in the air_airThe ideal gas law is used to express (Eq. (1) and (2)):

C_srepresents the surface water evaporation concentration of the spherical food, and has a unit of kgm^-3；

C_airRepresents the concentration of water evaporated in the air, and the unit is kg m^-3；

RH stands for air relative humidity;

p_VSrepresents the saturated water vapor pressure on the blackberry surface or in the air, and the unit is Pa;

T_airrepresents the air temperature in units of K or ℃;

here, a on the surface of the sample_wThe values are determined from the adsorption isotherm relationship, while the water vapor pressure is calculated using the following equation:

other thermophysical parameters, e.g. specific heat C_pLatent heat of water evaporation h_fgAnd thermal conductivity λ, determined as follows:

formula (3) C_p＝(0.873+1.256W)×1000

Density of solid matter rho_s：

Dry basis water content W:

M₀represents the initial weight of the spherical food product in kg; n is the initial dry mass ratio of the fresh spheroidal food product as determined according to the AOAC method, in this example N is 0.138; v_blRepresents the volume of the spherical food product in m^-3(ii) a r represents the radius of the whole blackberry fruit or blackberry sphere in m; m represents the real-time weight of the spherical food in kg;

equation (7) is converted to:

M＝W×M₀×N+M₀×N

thus, ρ_bRepresented by the following formula:

after the parameters are determined, the equations and the parameters are substituted into MATLAB, and the model is solved by using a 'pdepe' function in the equation and the parameters.

After the model is determined, the humidity prediction of the three blackberries in this embodiment using model prediction is shown by the solid line in fig. 4, and the temperature is shown by the dashed line in fig. 5. Prediction of temperature and humidity: relevant parameters are input into MATLAB software, and the software can predict the average moisture content and corresponding temperature change of the blackberry at each drying moment according to the established model. And using the predicted value and the measured valueR²And calculating RMSE and AAD so as to judge the accuracy of the model.

In the present model, the number of radial grid points is optimized to minimize the RMSE value. Using two other indices R simultaneously²And AAD to evaluate the predictive capabilities of the application model. R²RMSE and AAD are expressed as:

D_i，prepresents predictive data on temperature or moisture content in units of (Korkg moisture/kg dry matter); d_i，eExperimental data on temperature or moisture content are shown in units of (Korkg moisture/kg dry matter);

3. numerical simulation process

After the above parameters are determined according to the model, numerical simulation can be performed, and the flow is as follows. The flow chart is shown in fig. 3. By simulating the above mentioned data in MATLAB, simulated temperature and moisture values will be obtained, as will AAD, R for true and simulated moisture and temperature²RMSE values. These three values need to be within certain limits to make the modeling capability of the model closer (AAD less than 20%, R)²Above 0.97, the smaller the RMSE the better).

The parameters obtained by the correlation simulation in this example are as follows:

in this embodiment, since there is no internal heat source, Qe is 0.

TABLE 3 Arrhenius type temperature correlation equation for effective diffusion rate of blackberry moisture under hot air drying with and without sonication

Based on table 3, the arrhenius relationship can describe the change in De (as a function of temperature) and then be used in the heat and mass transfer model.

TABLE 4 convection heat and mass transfer coefficient at 65 deg.C on the surface of spherical food

Table 4 shows the convective heat and mass transfer coefficients for the hot air and air-mediated ultrasonic group and the hot air and contact ultrasonic group with the surrounding hot air. The h and hm values under the treatment of hot air and contact ultrasound (contact ultrasound for short) are the highest, and then the treatment of hot air and air-medium type ultrasound (air-medium type ultrasound for short) and the non-ultrasonic hot air drying treatment are carried out, which shows that the contact ultrasound effectively promotes the heat-moisture exchange between the blackberry surface and the air compared with the air-medium type ultrasound, thereby strengthening the drying dynamics. Furthermore, h and hm were considered constant throughout the drying process of the same set of experiments, assuming that the surface boundary exposure to hot air was the same during modeling.

TABLE 5 prediction accuracy of coupled heat and mass transfer model

Using experimental and predictive data of surface temperature, R of the heat transfer model was calculated²RMSE and AAD values, the results show that predictive models are available.

The coupling heat and mass transfer model is successfully solved by utilizing the shrinkage curve, the adsorption isotherm, the water diffusivity related to the internal temperature of the blackberry and the heat and mass transfer coefficient of the blackberry surface, and the figure 4 and the figure 5 are compared graphically. Meanwhile, the accuracy index is summarized in table 5. It can be seen that although there are some differences in certain areas, the applied model is able to predict the changes in temperature and moisture content during the drying process with acceptable accuracy, as can be seen from figure 4. It is noted that the number of grids made in the radial direction of the heat and mass transfer path significantly affects the prediction results of the model used. Here, when solving the partial differential equation using the pdepe function, the radius of the spherical food is divided into 121 pieces.

As can be seen from fig. 6, the higher heat transfer coefficient under sonication allowed the temperature of the sonicated samples to equilibrate earlier as the drying time increased and the moisture flux on the surface of the spherical food decreased. All these results are in good agreement with the experimental curves of surface and core temperatures under different treatments shown in fig. 5, demonstrating that the coupled heat and mass transfer model with applied thermodynamic parameters and assumptions is a good representation of the actual temperature evolution. FIG. 7 shows the water distribution of blackberries at different drying times during the drying process, unlike the temperature distribution, at the initial stage of drying, the water distribution inside the blackberry is not uniform, and the water gradient decreases with the increase of the drying time. This phenomenon is common because the outer surface around the hot air always dries out first, and then the concentration gradient drives the moisture from the inside to the surface. Meanwhile, the attenuation of the moisture concentration gradient along with the drying process means that the driving force of mass transfer is weakened and the drying rate is reduced due to moisture loss. The visualization of the simulation results also reflects the effect of contact and air-borne sonication on the movement of water within the blackberry. The interior of the blackberry that had been subjected to contact and air-mediated sonication dried earlier than the interior of the blackberry that had not been subjected to sonication. After drying for 120 minutes, the moisture concentration gradient within the contact sonicated blackberry almost disappeared. According to einstein's equation, the high temperature rise rate results in rapid decay of the moisture concentration gradient under contact sonication as the molecular diffusivity increases with temperature. Furthermore, the difference in moisture content between the centre and the surface of the blackberry during drying is plotted in figure 7. It is clearly observed that the contact sonication reduces the differences most rapidly, followed by air-borne sonication and pure hot air drying.

In addition, the total energy required to reduce the moisture content of the spheroidal food to 1.0g/g DM was 2.67 kW.h for air drying alone, 1.89 kW.h for air-mediated ultrasound, and 1.38 kW.h for contact ultrasound. Therefore, the ultrasonic wave accelerated heat and mass transfer can reduce energy consumption, is beneficial to the drying process, and shows the potential of being applied to industrial drying as a green technology. Furthermore, the energy consumption of the contact sonication was 27.0% lower than that of the gas-mediated sonication.

The running code of the model is as follows:

MATLAB enforcement code

function pdex7

m＝2；

x＝linspace(0,0.01,121)

t＝[0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500]；

sol＝pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t)；

u1＝sol(:,:,1)；

u2＝sol(:,:,2)；

disp(u1)；

disp(u2)；

a＝u2

b＝u1

c＝mean(a,2)

d＝b(:,121)

e＝b(:,1)

f＝a(1,:)

figure

surf(x,t,u1)

title('u1(x,t)')

xlabel('Distance x')

ylabel('Time t')

figure

surf(x,t,u2)

title('u2(x,t)')

xlabel('Distance x')

ylabel('Time t')

function[c,f,s]＝pdex4pde(x,t,u,DuDx)

a＝(u(2)*0.003*0.138+0.003*0.138)/0.000004189；

b＝(0.837+1.256*u(2))*1000；

c＝[a*b；1]；

f＝[0.149*60+0.493*60*u(2)/(1+u(2))；1.000*2.608*10^(-6)*exp(-1159/u(1))].*

DuDx；

s＝[0；0]；

function u0＝pdex4ic(x)

u0＝[298；6.25]；

function[pl,ql,pr,qr]＝pdex4bc(xl,ul,xr,ur,t)

pl＝[0；0]；

ql＝[1；1]；

m＝-14.69*ur(2)*ur(2)+8.203*ur(2)-0.3625；％-37.6*ur(2)*ur(2)+13.12*ur(2)-0.3625；％％0.5742*exp(0.6402*ur(2))-6.547*exp(-32.79*ur(2))；％0.4417*exp(0.4417-54.47*exp(-43.92*ur(2)))；

n＝exp(-0.0058/ur(1)+1.391-0.04864*ur(1)+0.00004176*ur(1)*ur(1)-1.445*10^(-8)*ur(1)*ur(1)*ur(1)+6.545*log(ur(1)))；

w＝((647.3-ur(1))/(647.3-273.15))^0.3298；

pr＝[636.6*(ur(1)-338)-2501.05*2.1667*0.678*w*(m*n/ur(1)-7.187)；

-0.678*0.0021667*(m*n/ur(1)-7.187)]；

qr＝[0.149*60+0.493*60*ur(2)/(ur(2)+1)；(0.003*0.138/0.000004189)*1.000*2.608*10^(-6)*exp(-1159/ur(1))]；

s＝[0；0]；

The above applications illustrate that the prediction method of the present invention provides data support for determining the drying time, drying temperature, and ultrasonic intensity required for a spherical food product in a specific volume, and helps to manage various parameters in the drying process more precisely.

It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention.