CN114997018A - Method for calculating traction force at guide roller with entrained gas - Google Patents

Abstract

The invention discloses a method for calculating the traction force at a guide roller that takes into account entrained gas, which is specifically implemented according to the following steps: Step 1, establishing a theoretical model of the flow field characteristics of the entrained gas; Flow field characteristic analysis; Step 3, calculate the thickness of the gas film; Step 4, establish a traction force model, and conduct a traction force analysis. The method for calculating the traction force at the guide roller with entrained gas is used to find a method for calculating the traction force, realize high-precision transmission, and lay a foundation for realizing high-precision transmission of thin films.

Description

A method for calculating the traction force at the guide roller taking into account the entrained gas

technical field

The invention belongs to the technical field of roll-to-roll manufacturing, and particularly relates to a method for calculating the traction force at a guide roller that takes into account entrained gas.

Background technique

In the roll-to-roll manufacturing process, the transmission route of the film is relatively long, and many rollers are required to control the stable transmission of the film, among which the most distributed is the guide roller. Sufficient traction force can make no speed difference between the film and the guide roller, and ensure the processing quality of film products. When the film is transported at high speed, the surrounding air enters the gap between the film and the guide roller to form entrained gas, which reduces the contact area between the film and the guide roller, and reduces the traction force of the film on the guide roller, resulting in film slippage and Processing defects such as wrinkles are the key issues that restrict the quality of roll-to-roll film products. The stable transmission of the film at the guide roller is still in the stage of qualitative research and testing, and it is urgent to consider the calculation method of the traction force of the entrained gas to ensure the high-precision transmission of the film.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for calculating the traction force at the guide roller that takes into account the entrained gas, so as to quantitatively calculate the traction force, realize high-precision transmission, and lay a foundation for realizing high-quality roll-to-roll manufacturing of flexible electronic products.

In order to achieve the above-mentioned purpose, the technical scheme adopted in the present invention is: a method for calculating the traction force at the guide roller that takes into account the entrained gas, which is specifically implemented according to the following steps:

Step 1, establish a theoretical model of entrained gas flow field characteristics;

Step 2, analyze the flow field characteristics between the film and the circumferential micro-grooved guide roller;

Step 3, calculate the thickness of the gas film;

Step 4, establish a traction force model and conduct traction force analysis.

As a preferred technical solution of the present invention, in the step 1, a theoretical model of the flow field characteristics of the entrained gas is established, and the specific steps are sequentially implemented according to the following steps:

Step 1.1, establish the film deflection equation;

Step 1.2, establish the Reynolds equation for entrained gas;

Step 1.3, perform contact mechanics analysis;

Step 1.4, establish the gas film thickness equation;

Step 1.5, establish boundary conditions.

As a preferred technical solution of the present invention, in the step 2, the analysis of the flow field characteristics between the film and the circumferential micro-grooved guide roller is carried out, specifically according to the following steps:

Step 2.1, perform dimensionless processing of equations; introduce dimensionless parameters to perform dimensionless processing on film deflection equation, Reynolds equation of entrained gas, gas film thickness equation, and boundary conditions.

Introduce dimensionless variables:

In the formula: H is the dimensionless gas film thickness, P is the dimensionless gas pressure, W is the dimensionless film deflection, X is the dimensionless abscissa, Y is the dimensionless ordinate, λ is the dimensionless film width, β is the dimensionless wrap angle, ψ is the shape function of the dimensionless guide roller, ε is the film parameter;

The dimensionless gas film thickness equation is obtained as:

H=W+ψ

Step 2.2, discretize the equation; use the five-point difference format to discretize the equation, and the partial derivatives of the variables in the equation at the nodes are expressed in the central difference format as:

The above five-point difference scheme is used to discretize the dimensionless film deflection equation and the dimensionless Reynolds equation, where i and j represent the x-direction and the y-direction, respectively.

The film deflection equation is discretized as:

in the formula

的第一项离散为：Mode

The first term of the discrete is:

The second discrete is:

The third discrete is:

Substitute the above equations into the discrete Reynolds equation to obtain the dimensionless P _i,j expression at node (i, j):

C _i,j P _i-1,j -A _i,j P _i,j +B _i,j P _i+1,j ++D _i,j P _i,j+1 +E _i,j P _{i, j-1} = F _ij (5)

where:

Rewrite (5) as

利用式H＝W+ψ可根据周围四节点的压力值来计算中心节点的压力值。make

Using the formula H=W+ψ, the pressure value of the central node can be calculated according to the pressure values of the surrounding four nodes.

P _i,j =H _1(i,j) P _i+1,j +H _2(i,j) P _i-1,j +H _3(i,j) P _i,j+1 +H _{4( i,j)} P _i,j-1 -H _5(i,j) F _ij

(7)

After several iterations, a sufficiently accurate approximate solution is finally obtained. The iterative format of the pointwise over-relaxation method is:

P _i,j ^k+1 =(1-ζ)P _i,j ^k +ζ(P _i,j ^k+1 -P _i,j ^k )

(8)

In the formula, ζ is the relaxation factor, which is selected as 1.75 in the present invention, P _i,j ^k is the current calculated pressure value, and P _{i, j} ^k+1 is the new round of calculation pressure value.

Equation (8) is further abbreviated as:

P ^(k+1) =L _ζ P ^k +Q

(9)

where the iteration matrix is:

In the formula, D is the main diagonal matrix, L is the lower triangular matrix, and U is the upper triangular matrix.

The convergence criteria are as follows:

In the formula, m and n are the maximum grid points, k is the current calculated value, and k+1 is the new calculation value;

Step 2.3, solve the domain grid division; first discretize the corner area into grids, and number the grid nodes in the order of the number of columns and rows where they are located. It is divided into m meshes in the x direction of the film transmission, and the node number i is from 1 to m+1; in the y direction of the guide roller, n meshes are divided, and the node number j is from 1 to n+1, a total of m×n grid;

Evenly divide m grids in the x direction, and each grid step is:

Evenly divide n grids in the y direction, and each grid step is:

Step 2.4, set the boundary conditions:

P(X _in ,Y)=p(X _out ,Y)=p(X,-0.5)=p(X,0.5)=0

(1)

W(X _in )=0 W(X _out )=0

(2)

In the formula, the dimensionless gas pressure P at the inlet and outlet is 0, and the dimensionless film deflection W is also 0.

As a preferred technical solution of the present invention, in the step 3, the calculation of the thickness of the gas film is carried out as follows:

Step 3.1, assume that the coordinates of the entrained gas in the inlet and outlet regions are X _in and X _out ;

Step 3.2, assuming initial gas film thickness H ₀ and gas pressure P ₀ ;

Step 3.3, under the control of boundary conditions, use the finite difference method and the successive over-relaxation iterative method to solve the two equations (1) and (7) in step 2.2 until the convergence conditions ε ₁ and ε ₂ are satisfied;

Step 3.4, use the formula H=W+ψ to solve the gas film thickness H, and judge whether the convergence condition ε ₃ is satisfied. The modified initial gas film thickness H ₀ is not satisfied until the convergence condition ε ₃ is satisfied;

Step 3.5, check whether X _in and X _out satisfy the continuity condition formula (3) of step 2.3, if not, correct the inlet and outlet positions, repeat steps 3.2 to 3.4, until the continuity condition is met;

Step 3.6, output the thickness H of the gas film.

As a preferred technical solution of the present invention, in the step 4, a traction force model is established, and traction force analysis is performed, and the specific steps are sequentially implemented according to the following steps:

Step 4.1, perform force analysis on the thin film micro-element, θ is the wrapping angle, R is the radius of the guide roller, T is the film tension, P _a is the atmospheric pressure, h is the thickness of the gas film, P _c is the contact pressure, U _w is the film velocity; in order to derive the traction force expression, the force analysis of the thin film micro-element in the x and y directions is carried out, and the traction force expression is obtained according to the force balance relationship; dP is the entrained gas pressure, dP _c is the contact pressure, dP _a is the atmospheric pressure, dF is the traction force of the guide roller, F _a is the shear force of the entrained gas on the film; the film is balanced in the y direction, and can be obtained:

式(1)化简为：Where: dP _a =P _a Rdθ, dP = PRdθ, let

Equation (1) is simplified to:

dP _c =(T ₀ -PR+P _a R)dθ

(2)

The gas shear force _Fa is so small that it can be ignored. The force balance in the x direction is:

Equation (3) is simplified to:

dF=dT ₀

(4)

Let: dF=μdP _c , then dT ₀ =μdP _c , and substitute formula (2) to get:

Integrate Eq. (5) in the wrap angle region to get:

T+ΔT=e ^μθ (T-PR+P _a R)+PR-P _a R

(6)

Integrating Equation (4) in the wrap angle area, the traction force of the film on the guide roller is obtained as:

F=(T-PR+P _a R)(e ^μθ -1)

(7)

It can be known from formula (7) that the traction force F of the guide roller is related to the film tension T, the entraining gas pressure P, the friction coefficient μ, the wrap angle θ and the radius of the guide roller. The traction force is further analyzed by determining the pressure of the entrained gas.

Step 4.2, calculate the traction force of the circumferential microgrooved guide roller. In steady state, the thickness of the gas film between the film and the smooth guide roller is:

Assuming that there is no contact between the film and the guide roll, we get:

P=P _a +T/R (2)

Substituting equation (2) into equation (1), we can get:

where P _a is the atmospheric pressure, U _w is the film speed, Ur is the speed of the guide roller, h is the thickness of the gas film, η is the aerodynamic viscosity, and _R is the radius of the guide roller;

Due to the contact between the film and the guide roller, the effective distance h _eq is used to represent the distance between the film and the micro-grooved guide roller, specifically:

Substituting the formula (4) into the formula F=(T-PR+P _a R)(e ^μθ -1), the traction force of the circumferential micro-groove guide roller is:

From formula (3), it can be known that the traction force F of the guide roller and the film tension T, the sum of the film speed U _w and the speed U _r of the guide roller, the radius R of the guide roller, the friction coefficient μ, the wrap angle θ between the film and the guide roller , The equivalent gas film thickness h _eq between the film and the guide roller is related.

The beneficial effects of the invention are as follows: the invention provides a method for calculating the traction force at the guide roller that takes into account the entrained gas, and based on the N-S equation and the mass conservation equation, the Reynolds equation of the entrained gas between the film and the guide roller is established; Force and moment analysis were performed with shell microelements, and the film deflection equation was derived. The flow field characteristics of the entrained gas between the film and the circumferential micro-grooved guide roller were studied; the traction force model of the circumferential micro-grooved guide roller was established, and the calculation formula of the traction force of the circumferential micro-grooved guide roller was given. The influence and variation of traction force. The invention proposes a method for calculating the traction force at the guide roller which takes into account the entrained gas, so as to ensure the high-precision transmission of the film.

Description of drawings

1 is a five-point difference format diagram of a method of calculating the traction force at a guide roller that takes into account entrained gas in accordance with the present invention;

Fig. 2 is a kind of gas film thickness calculation flow chart of the method for calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

3 is a schematic diagram of film transmission at the guide roller according to a method of calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

4 is a force diagram of a thin film micro-element in the wrap angle area of a method of calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

5 is a graph of the influence of film width and groove width on gas film thickness of a method for calculating the traction force at a guide roller of the present invention that takes into account entrained gas;

6 is a graph of the influence of film transport speed and groove depth on gas film thickness of a method for calculating the traction force at a guide roller of the present invention that takes into account entrained gas;

7 is a graph of the influence of the film tension and the number of grooves on the gas film thickness of a method for calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

8 is a graph of the influence of the radius of the guide roller and the shape of the groove cross-section on the thickness of the gas film in a method for calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

9 is a graph of the influence of the presence or absence of microgrooves on the surface of the guide roller on the traction force of a method for calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

10 is a graph of the influence of the guide roller surface groove width and the number of grooves on the traction force of a method for calculating the traction force at the guide roller that takes into account the entrained gas of the present invention;

Detailed ways

The present invention provides a method for calculating the traction force at the guide roller that takes into account the entrained gas, which is specifically implemented according to the following steps:

Step 1, establish a theoretical model of the entrained gas flow field characteristics, which is specifically implemented according to the following steps:

Step 1.1, establish the deflection equation of the film. From the theory of elasticity, it can be known that the relationship between the force and the stress per unit length is:

The shear force per unit length is:

The moments are as follows:

where σ _x and σ _y are normal stresses, and τ _xy , τ _yx , τ _xz , and τ _yz are shear stresses.

The film deflection equation is obtained by balancing the forces and moments on an infinitesimal cylindrical shell element in the x, y, and z directions.

The balance of forces in the x-direction:

The balance of forces in the y direction:

The balance of forces in the z direction:

where p is the entrained gas pressure, p _a is the atmospheric pressure, and p _c is the contact pressure.

The same can be obtained:

Moment balance equation in the x direction:

Moment balance equation in the y direction:

Moment balance equation in z direction:

Using equations (7) and (8) to solve Q _x , Q _y and substituting them into equation (9), we get:

Using Hooke's law, the relationship between stress and strain is obtained as:

where E is the elastic modulus of the film, v is the Poisson's ratio, ε _x and ε _y are the normal strains in the x and y directions, respectively, and γ _xy is the shear strain.

According to the thin shell theory, the relationship between strain and displacement is:

In the circumferential direction, x=Rθ, since the film thickness c is much smaller than the guide roll radius R, the variable in the thickness direction can be omitted. Since N _xy =N _yx , M _xy =M _yx , the stress expressed in displacement is:

为抗弯刚度，本发明中只考虑半径方向的位移，因此u，v可以忽略不计。in

For the bending stiffness, only the displacement in the radial direction is considered in the present invention, so u and v can be ignored.

薄膜是柔性材料，不受弯矩，忽略薄膜的抗弯刚度。接触压力对薄膜挠度影响比较小，忽略接触压力对薄膜挠度的影响。最终薄膜挠度方程化简为：Since the film is not stressed in the width direction, the deflection of the film along the y direction is almost zero, then N _x =T, N _y =0,

The film is a flexible material and is not subject to bending moments, ignoring the bending stiffness of the film. The influence of the contact pressure on the deflection of the film is relatively small, and the influence of the contact pressure on the deflection of the film is ignored. The final film deflection equation simplifies to:

Step 1.2, establish the Reynolds equation for entrained gas, citing the Navier-Stokes equation of viscous incompressible fluid, the N-S equation is simplified to:

Equation (1) can be integrated with z to get:

Using the no-slip boundary condition z=0, u=U ₁ ; when z=h, u=U ₂ , c ₁ , c ₂ can be calculated and substituted into formula (4) to obtain:

Similarly, the velocity in the y direction is:

Define the volume flow in the x direction as:

m _x =ρq _x m _y =ρq _y

(10)

According to fluid mechanics, the fluid mass conservation equation is:

Integrate the variable z and use z=0, w=W ₁ ; z=h, w=W ₂ to get:

Since both the film and guide rollers are in motion,

After scoring:

Substitute m _x , m _y into the above formula, and we can get:

In the formula, U=(U ₁ +U ₂ )/2, and V=(V ₁ +V ₂ )/2.

Since the film and the guide roller move along the x-direction, both V ₁ and V ₂ are 0, and the extrusion term is omitted. The entrained gas is incompressible, that is, ρ remains unchanged, then the Reynolds equation for the entrained gas is as follows:

In the formula, U is the sum of the speed of the guide roller and the film speed, h is the thickness of the air film, and η is the aerodynamic viscosity.

Step 1.3, Contact Mechanical Analysis

Using the Greenwood-Williamson rough contact model, the contact pressure between the film and the guide roll was analyzed.

In the formula, z is the change of roughness, and φ(z) is the distribution of surface roughness peaks, which is assumed to be Gaussian distribution as shown in formula (2). h is the thickness of the air film between the film and the guide roller, p ₀ is the pressure required to make h 0, and the calculation is as follows (3).

为粗糙的密度。对于一般薄膜材料

取值为0.03～0.05。In the formula, E _c is the composite elastic modulus of the film and the surface of the guide roller, which is calculated as formula (4). α is the average surface roughness height, calculated as formula (5). β _t is the average roughness peak radius of the surface of the film and the guide roller, calculated as formula (6).

for rough density. For general thin film materials

The value ranges from 0.03 to 0.05.

where ν _w and ν _r are the Poisson’s ratios of the film and the guide roller, respectively, E _w and _Er are the elastic moduli of the film and the guide roller, respectively, and β _w and β _r are the asperities on the surface of the film and the guide roller, respectively. The radius of curvature, n is the number of surface asperities, and A _a is the contact area.

Step 1.4, establish the gas film thickness equation

When the guide roller is smooth, the deformation of the film in the corner area is equal to the thickness of the gas film. When the surface of the guide roller is grooved, it is necessary to determine the thickness h of the gas film according to the surface structure function δ of the guide roller and the film deflection w. The definition is as follows:

h(x,y)=w(x,y)+δ(x,y) (1)

Using the parabolic model, the surface structure function of the guide roller is defined as follows:

The slot function of the circumferential slot in the present invention is defined as follows:

where _lw is the slot spacing, _gw is the slot width, N is the slot position, and _gd is the slot depth;

Step 1.5, Establish Boundary Conditions

Boundary conditions for the Reynolds equation:

The direction of film movement, the inlet and outlet gas pressures are atmospheric pressure:

In the film width direction, the gas pressure is atmospheric pressure:

p(x _in ,y)=p _a p(x _out ,y)=p _a

Boundary conditions for the film deflection equation:

w(x _in )=0 w(x _out )=0

Step 2, analyze the flow field characteristics between the film and the circumferential micro-grooved guide roller, which is specifically implemented according to the following steps:

Step 2.1, the equation is dimensionless processing. Dimensionless parameters are introduced to carry out dimensionless processing of film deflection equation, Reynolds equation of entrained gas, gas film thickness equation and boundary conditions.

Introduce dimensionless variables:

where H is the dimensionless gas film thickness, P is the dimensionless gas pressure, W is the dimensionless film deflection, X is the dimensionless abscissa, Y is the dimensionless ordinate, λ is the dimensionless film width, and β is the dimensionless envelope angle, ψ is the shape function of the dimensionless guide roll, and ε is the film parameter.

The dimensionless gas film thickness equation is obtained as:

H=W+ψ

Step 2.2, the equation is discretized. The present invention adopts a five-point difference format discrete equation, as shown in FIG. 1 . The partial derivatives of the variables in the equation at the nodes are expressed in central difference form as:

The above five-point difference scheme is used to discretize the dimensionless film deflection equation and the dimensionless Reynolds equation, where i and j represent the x-direction and the y-direction, respectively.

The film deflection equation is discretized as:

in the formula

的第一项离散为：Mode

The first term of the discrete is:

The second discrete is:

The third discrete is:

Substitute the above equations into the discrete Reynolds equation to obtain the dimensionless P _i,j expression at node (i, j):

C _i,j P _i-1,j -A _i,j P _i,j +B _i,j P _i+1,j ++D _i,j P _i,j+1 +E _i,j P _{i, j-1} = F _ij (5)

where:

Rewrite (5) as

利用式H＝W+ψ可根据周围四节点的压力值来计算中心节点的压力值。make

Using the formula H=W+ψ, the pressure value of the central node can be calculated according to the pressure values of the surrounding four nodes.

P _i,j =H _1(i,j) P _i+1,j +H _2(i,j) P _i-1,j +H _3(i,j) P _i,j+1 +H _{4( i,j)} P _i,j-1 -H _5(i,j) F _ij

(7)

After several iterations, a sufficiently accurate approximate solution is finally obtained. The iterative format of the pointwise over-relaxation method is:

P _i,j ^k+1 =(1-ζ)P _i,j ^k +ζ(P _i,j ^k+1 -P _i,j ^k )

(8)

In the formula, ζ is the relaxation factor, which is selected as 1.75 in the present invention, P _i,j ^k is the current calculated pressure value, and P _{i, j} ^k+1 is the new round of calculation pressure value.

Equation (8) is further abbreviated as:

P ^(k+1) =L _ζ P ^k +Q

(9)

where the iteration matrix is:

In the formula, D is the main diagonal matrix, L is the lower triangular matrix, and U is the upper triangular matrix.

The convergence criteria are as follows:

In the formula, m and n are the maximum grid points, k is the current calculated value, and k+1 is the new calculation value.

Step 2.3, solve the domain grid division, first discretize the corner area into grids, and number the grid nodes in the order of the number of columns and rows where they are located. It is divided into m meshes in the x direction of the film transmission, and the node number i is from 1 to m+1; in the y direction of the guide roller, n meshes are divided, and the node number j is from 1 to n+1, a total of m×n grid.

Evenly divide m grids in the x direction, and each grid step is:

Evenly divide n grids in the y direction, and each grid step is:

Step 2.4, set boundary conditions

P(X _in ,Y)=p(X _out ,Y)=p(X,-0.5)=p(X,0.5)=0

(1)

W(X _in )=0 W(X _out )=0

(2)

In the formula, the dimensionless gas pressure P at the inlet and outlet is 0, and the dimensionless film deflection W is also 0.

Step 3, for the calculation of the thickness of the gas film, the numerical calculation is carried out by using the MATLAB language. The specific calculation process is shown in Figure 2, which is specifically implemented according to the following steps:

Step 3.1, assume that the coordinates of the entrained gas in the inlet and outlet regions are X _in and X _out ;

Step 3.2, assuming initial gas film thickness H ₀ and gas pressure P ₀ ;

Step 3.3, under the control of boundary conditions, use the finite difference method and the successive over-relaxation iterative method to solve the two equations (1) and (7) in step 2.2 until the convergence conditions ε ₁ and ε ₂ are satisfied;

Step 3.4, use the formula H=W+ψ to solve the gas film thickness H, and judge whether the convergence condition ε ₃ is satisfied. The modified initial gas film thickness H ₀ is not satisfied until the convergence condition ε ₃ is satisfied;

Step 3.5, check whether X _in and X _out satisfy the continuity condition of step 2.3, if not, correct the inlet and outlet positions, repeat steps 3.2 to 3.4, until the continuity condition is met;

Step 3.6, output the thickness H of the gas film.

Step 4, establish a traction force model, and conduct traction force analysis, which is implemented according to the following steps:

Step 4.1, perform force analysis on the thin film micro-element. Figure 3 shows the motion model of the film and the guide roller, θ is the wrap angle, R is the radius of the guide roller, T is the film tension, P _a is the atmospheric pressure, h is the film thickness, P _c is the contact pressure, U _w is the film speed. In order to derive the expression of the traction force, the thin-film micro-element shown in Figure 4 is selected, and the force analysis of the thin-film micro-element in the x and y directions is carried out, and the expression of the traction force is obtained according to the force balance relationship. dP is the entrained gas pressure, dP _c is the contact pressure, dP _a is the atmospheric pressure, dF is the traction force of the guide roller, and _Fa is the shear force of the entrained gas on the film. The film is force balanced in the y direction, we can get:

式(1)化简为：Where: dP _a =P _a Rdθ, dP = PRdθ, let

Equation (1) is simplified to:

dP _c =(T ₀ -PR+P _a R)dθ

(2)

The gas shear force _Fa is so small that it can be ignored. The force balance in the x direction is:

Equation (3) is simplified to:

dF=dT ₀

(4)

Let: dF=μdP _c , then dT ₀ =μdP _c , and substitute formula (2) to get:

(5)

Integrate Eq. (5) in the wrap angle region to get:

T+ΔT=e ^μθ (T-PR+P _a R)+PR-P _a R

(6)

Integrating Equation (4) in the wrap angle area, the traction force of the film on the guide roller is obtained as:

F=(T-PR+P _a R)(e ^μθ -1)

(7)

It can be seen from formula (7) that the traction force F of the guide roller is related to the film tension T, the entrained gas pressure P, the friction coefficient μ, the wrap angle θ and the radius of the guide roller. Traction characteristics were further analyzed by determining the pressure of the entrained gas.

Step 4.2, calculate the traction force of the circumferential microgrooved guide roller. In a steady state, the thickness of the gas film between the film and the smooth guide roller is:

Assuming that there is no contact between the film and the guide roll, we get:

P=P _a +T/R (2)

Substituting equation (2) into equation (1), we can get:

In the formula, _Pa is the atmospheric pressure, the film speed U _w , the speed of the guide roller Ur and h are the thickness of the film, η is the aerodynamic viscosity, and _R is the radius of the guide roller.

In the actual situation, there is contact between the film and the guide roller. This patent uses the effective distance h _eq to represent the distance between the film and the micro-groove guide roller, then:

Substituting the formula (4) into the formula F=(T-PR+P _a R)(e ^μθ -1), the traction force of the circumferential micro-groove guide roller is:

From formula (3), it can be known that the traction force F of the guide roller and the film tension T, the sum of the film speed U _w and the speed U _r of the guide roller, the radius R of the guide roller, the friction coefficient μ, the wrap angle θ between the film and the guide roller , The equivalent gas film thickness h _eq between the film and the guide roller is related.

In a method for calculating the traction force at the guide roller of the present invention that takes into account the entrained gas: the function of step 1 is to study the flow field characteristics of the entrained gas and analyze the influence of the circumferential micro-groove parameters on the thickness of the entrained gas film.

Using the G-W contact theory, the contact mechanics between the film and the guide roller was analyzed. The principle is: through analysis, we can effectively calculate the thickness of the gas film, and the thickness of the gas film and the equivalent roughness divide the film and the guide roller into three lubrication states.

The advantages of using fluid lubrication theory and elastic thin shell theory are that the Reynolds equation and the film deflection equation of entrained gas can be effectively deduced respectively. Defining the parameters of the groove with circumferential rectangular section, and then based on the above theory, can provide a theoretical basis for analyzing the characteristics of the flow field.

It can be seen from Figure 5 that the width of the groove increases and the thickness of the gas film between the film and the guide roller decreases. When the groove depth is constant, the groove width increases, the more entrained gas is discharged, and the smaller the gas film thickness. When the width of the film increases, the thickness of the air film between the film and the guide roll becomes larger. It can be seen from Figure 6 that the thickness of the gas film between the film and the guide roller decreases as the groove depth increases. As the film speed increases, the thickness of the gas film between the film and the guide roller increases, because the higher the film speed, the larger the amount of entrained gas. It can be seen from Figure 7 that the more the number of grooves, the smaller the thickness of the gas film between the film and the guide roller, and the entrained gas is discharged from the more grooves. As the film tension increases, the thickness of the gas film between the film and the guide roll decreases. The film tension increases, the pressure of the film on the guide roller increases, and more entrained gas is squeezed out, so the film thickness decreases. It can be seen from Figure 8 that the thickness of the gas film between the film and the guide roller increases as the radius of the guide roller increases. It can be seen from Fig. 9 that the grooves on the surface of the guide roller have a significant effect on improving the traction force of the guide roller. It can be seen from Fig. 10 that as the groove width increases, the traction force of the guide roller increases. The groove width gradually decreased from 100 μm to 250 μm, and gradually increased from 250 μm to 400 μm. The greater the number of grooves per unit length, the greater the pulling force of the guide rollers.

Claims (5)

1. a method for calculating the traction force at the guide roller place that takes into account entrained gas, is characterized in that, is specifically implemented according to the following steps: Step 1, establish a theoretical model of entrained gas flow field characteristics; Step 2, analyze the flow field characteristics between the film and the circumferential micro-grooved guide roller; Step 3, calculate the thickness of the gas film; Step 4, establish a traction force model and conduct traction force analysis. 2. a kind of traction force calculation method at the guide roller place considering entrained gas according to claim 1, is characterized in that, in described step 1, set up entrained gas flow field characteristic theoretical model to be implemented sequentially according to the following steps: Step 1.1, establish the film deflection equation; Step 1.2, establish the Reynolds equation for entrained gas; Step 1.3, perform contact mechanics analysis; Step 1.4, establish the gas film thickness equation; Step 1.5, establish boundary conditions. 3. a kind of traction force calculation method at the guide roller place considering entrained gas according to claim 2, it is characterized in that, in described step 2, carry out the flow field characteristic analysis between film and circumferential micro-groove guide roller, Specifically, follow the steps below to implement: Step 2.1, perform dimensionless processing of equations; introduce dimensionless parameters to perform dimensionless processing on film deflection equation, Reynolds equation of entrained gas, gas film thickness equation, and boundary conditions. Introduce dimensionless variables:

In the formula: H is the dimensionless gas film thickness, P is the dimensionless gas pressure, W is the dimensionless film deflection, X is the dimensionless abscissa, Y is the dimensionless ordinate, λ is the dimensionless film width, β is the dimensionless wrap angle, ψ is the shape function of the dimensionless guide roller, ε is the film parameter; The dimensionless gas film thickness equation is obtained as: H=W+ψ Step 2.2, discretize the equation; use the five-point difference format to discretize the equation, and the partial derivatives of the variables in the equation at the nodes are expressed in the central difference format as:

The above five-point difference scheme is used to discretize the dimensionless film deflection equation and the dimensionless Reynolds equation, where i and j represent the x-direction and the y-direction, respectively. The film deflection equation is discretized as:

in the formula

Mode

The first term of the discrete is:

The second discrete is:

The third discrete is:

Substitute the above equations into the discrete Reynolds equation to obtain the dimensionless P _i,j expression at node (i, j): C _i,j P _i-1,j -A _i,j P _i,j +B _i,j P _i+1,j ++D _i,j P _i,j+1 +E _i,j P _{i, j-1} = F _ij (5) where:

Rewrite (5) as

make

Using the formula H=W+ψ, the pressure value of the central node can be calculated according to the pressure values of the surrounding four nodes. P _i,j =H _1(i,j) P _i+1,j +H _2(i,j) P _i-1,j +H _3(i,j) P _i,j+1 +H _{4( i,j)} P _i,j-1 -H _5(i,j) F _ij (7) After several iterations, a sufficiently accurate approximate solution is finally obtained. The iterative format of the pointwise over-relaxation method is: P _i,j ^k+1 =(1-ζ)P _i,j ^k +ζ(P _i,j ^k+1 -P _i,j ^k ) (8) In the formula, ζ is the relaxation factor, which is selected as 1.75 in the present invention, P _i,j ^k is the current calculated pressure value, and P _{i, j} ^k+1 is the new round of calculation pressure value. Equation (8) is further abbreviated as: P ^(k+1) =L _ζ P ^k +Q (9) where the iteration matrix is:

In the formula, D is the main diagonal matrix, L is the lower triangular matrix, and U is the upper triangular matrix. The convergence criteria are as follows:

In the formula, m and n are the maximum grid points, k is the current calculated value, and k+1 is the new calculation value; Step 2.3, solve the domain grid division; to solve the domain grid division, first discretize the corner area into a grid, and number the grid nodes in the order of the number of columns and rows where they are located. Divided into m meshes in the x direction of film transport, node number i is from 1 to m+1; divided into n meshes in the y direction of the guide roller axis, node number j is from 1 to n+1, a total of m*n grid; Evenly divide m grids in the x direction, and each grid step is:

Evenly divide n grids in the y direction, and each grid step is:

Step 2.4, set the boundary conditions: P(X _in ,Y)=p(X _out ,Y)=p(X,-0.5)=p(X,0.5)=0 (1) W(X _in )=0 W(X _out )=0 (2)

In the formula, the dimensionless gas pressure P at the inlet and outlet is 0, and the dimensionless film deflection W is also 0. 4. a kind of traction force calculation method at the guide roller place considering entrained gas according to claim 3, is characterized in that, in described step 3, carry out gas film thickness calculation and carry out specifically as follows: Step 3.1, assume that the coordinates of the entrained gas in the inlet and outlet regions are X _in and X _out ; Step 3.2, assuming initial gas film thickness H ₀ and gas pressure P ₀ ; Step 3.3, under the control of boundary conditions, use the finite difference method and the successive over-relaxation iterative method to solve the two equations (1) and (7) in step 2.2 until the convergence conditions ε ₁ and ε ₂ are satisfied; Step 3.4, use the formula H=W+ψ to solve the gas film thickness H, and judge whether the convergence condition ε ₃ is satisfied; the modified initial gas film thickness H ₀ is not satisfied, until the convergence condition ε ₃ is satisfied; Step 3.5, check whether X _in and X _out satisfy the continuity condition formula (3) of step 2.3, if not, correct the inlet and outlet positions, repeat steps 3.2 to 3.4, until the continuity condition is met; Step 3.6, output the thickness H of the gas film. 5. a kind of traction force calculation method at the guide roller place that takes into account entrained gas according to claim 4, it is characterized in that, in described step 4, establish traction force model, carry out traction force analysis, specifically according to the following steps to implement in turn: Step 4.1, perform force analysis on the thin film micro-element, θ is the wrapping angle, R is the radius of the guide roller, T is the film tension, P _a is the atmospheric pressure, h is the thickness of the gas film, P _c is the contact pressure, U _w is the film velocity; in order to derive the traction force expression, the force analysis of the thin film micro-element in the x and y directions is carried out, and the traction force expression is obtained according to the force balance relationship; dP is the entrained gas pressure, dP _c is the contact pressure, dP _a is the atmospheric pressure, dF is the traction force of the guide roller, F _a is the shear force of the entrained gas on the film; the film is balanced in the y direction, and can be obtained:

Where: dP _a =P _a Rdθ, dP = PRdθ, let

Equation (1) is simplified to: dP _c =(T ₀ -PR+P _a R)dθ (2) The gas shear force _Fa is so small that it can be ignored. The force balance in the x direction is:

Equation (3) is simplified to: dF=dT ₀ (4) Let: dF=μdP _c , then dT ₀ =μdP _c , and substitute formula (2) to get:

Integrate Eq. (5) in the wrap angle region to get: T+ΔT=e ^μθ (T-PR+P _a R)+PR-P _a R (6) Integrating Equation (4) in the wrap angle area, the traction force of the film on the guide roller is obtained as: F=(T-PR+P _a R)(e ^μθ -1) (7) It can be seen from formula (7) that the traction force F of the guide roller is related to the film tension T, the entrained gas pressure P, the friction coefficient μ, the wrap angle θ and the radius of the guide roller. The traction characteristics are further analyzed by determining the pressure of the entrained gas; Step 4.2, calculate the traction force of the circumferential microgrooved guide roller. In a steady state, the thickness of the gas film between the film and the smooth guide roller is:

Assuming that there is no contact between the film and the guide roll, we get: P=P _a +T/R (2) Substituting equation (2) into equation (1), we can get:

where _Pa is the atmospheric pressure, the film speed U _w , the speed of the guide roller Ur and h are the film thickness, η is the aerodynamic viscosity, and _R is the radius of the guide roller; Due to the contact between the film and the guide roller, the effective distance h _eq is used to represent the distance between the film and the micro-grooved guide roller, specifically:

Substituting the formula (4) into the formula F=(T-PR+P _a R)(e ^μθ -1), the traction force of the circumferential micro-groove guide roller is:

From formula (3), it can be known that the traction force F of the guide roller and the film tension T, the sum of the film speed U _w and the speed U _r of the guide roller, the radius R of the guide roller, the friction coefficient μ, the wrap angle θ between the film and the guide roller , The equivalent gas film thickness h _eq between the film and the guide roller is related.

Images