传递过程原理-附件-英文版
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Thus, Fick's First Law states: J = -D [ ∂C(x,t) / ∂x ] where J is the flux, D is the diffusion constant for the material that is diffusing in the specific solvent, and ∂C(x,t)/∂x is the concentration gradient. The diffusion constant of a material is also referred to as 'diffusion coefficient' or simply 'diffusivity.' It is expressed in units of length2/time, such as µm2/hour. The negative sign of the right side of the equation indicates that the impurities are flowing in the direction of lower concentration.
We define the curl of V as:
Flat brackets mean “take determinant”
i j Vz V y Vx Vz V y Vx j V curlV i z x k x y x y z y Vx V y
传递过程原理-附录
刘相东
绕圆柱流模拟
射流模拟
雷诺试验
绕圆柱流演示
流线演示
Shorthand Notation in Vector Equations
f x
= partial differential operator Means: differentiate contents of brackets with respect to x and treating all other variables as constants Scalar product By definition the scalar product of two vectors A (=A= (Ax,Ay,Az)) and B (= B = (Bx,By,Bz)) is given by: where is the angle between the two vectors A and B. Vector product By definition the vector product of two vectors A and B is given by:
f f f f f f f gradf i j k x , y , z x y z
The scalar product of del with a vector V is known as the divergence of V, and is written as: V y Vx Vz .V divV x y z
j Ay By
k Az Bz
where i, j, k are unit vectors in the x, y, z dir., respectively
We can define a vector operator called “del” , written as: It is more complicated than normal partial i j k , , x y z x y z differentials as it is a vector ! We can define the product of del with a scalar f as:
Fick's First Law
Whenever an impurity concentration gradient, ∂C/∂x, exists in a finite volume of a matrix substance (the silicon substrate in this context), the impurity material will have the natural tendency to move in order to distribute itself more evenly within the matrix and decrease the gradient. Given enough time, this flow of impurities will eventually result in homogeneity within the matrix, causing the net flow of impurities to stop. The mathematics of this transport mechanism was formalized in 1855 by Fick, who postulated that the flux of material across a given plane is proportional to the concentration gradient across the plane.
The flux J1 of impurities entering a section of a bar with a concentration gradient is different from the flux J2 of impurities leaving the same section. From the law of conservation of matter, the difference between J1 and J2 must result in a change in the concentration of impurities within the section (assuming that no impurities are formed or consumed in the section). This is Fick's Second Law, which states that the change in impurity concentration over time is equal to the change in local diffusion flux, or ∂C(x,t)/∂t = - ∂J/∂x or, from Fick's First Law, ∂C(x,t)/∂t = ∂(D∂C(x,t)/∂x)/∂x. If the diffusion coefficient is independent of position, such as when the impurity concentration is low, then Fick's Second Law may be further simplified into the following equation: ∂C(x,t)/∂t = D ∂2C(x,t)/∂x2.
k z Vz
Another important expression is “del squared”, also known as the “Laplacian”:
2f .f div grad f f f f , , . , , x y z x y z f f f x x y y z z 2f 2f 2f 2 2 x y z 2
Fick's Laws
Diffusion the movement of a chemical species from an area of high concentration to an area of lower concenቤተ መጻሕፍቲ ባይዱration, is one of the two major processes by which chemical species or dopants are introduced into a semiconductor (the other one being ion implantation) . The controlled diffusion of dopants into silicon to alter the type and level of conductivity of semiconductor materials is the foundation of forming a p-n junction and formation of devices during wafer fabrication. The mathematics that govern the mass transport phenomena of diffusion are based on Fick's laws.
Ay Bz Az B y i A B A B e A B sin Az Bx Ax Bz Ax Ax B y Ay Bx Bx Unit vector perpendicular
to A and B
A.B Ax Bx Ay B y Az Bz A B cos
Fick's Second Law
Fick's First Law does not consider the fact that the gradient and local concentration of the impurities in a material decreases with an increase in time, an aspect that's important to diffusion processes.
We define the curl of V as:
Flat brackets mean “take determinant”
i j Vz V y Vx Vz V y Vx j V curlV i z x k x y x y z y Vx V y
传递过程原理-附录
刘相东
绕圆柱流模拟
射流模拟
雷诺试验
绕圆柱流演示
流线演示
Shorthand Notation in Vector Equations
f x
= partial differential operator Means: differentiate contents of brackets with respect to x and treating all other variables as constants Scalar product By definition the scalar product of two vectors A (=A= (Ax,Ay,Az)) and B (= B = (Bx,By,Bz)) is given by: where is the angle between the two vectors A and B. Vector product By definition the vector product of two vectors A and B is given by:
f f f f f f f gradf i j k x , y , z x y z
The scalar product of del with a vector V is known as the divergence of V, and is written as: V y Vx Vz .V divV x y z
j Ay By
k Az Bz
where i, j, k are unit vectors in the x, y, z dir., respectively
We can define a vector operator called “del” , written as: It is more complicated than normal partial i j k , , x y z x y z differentials as it is a vector ! We can define the product of del with a scalar f as:
Fick's First Law
Whenever an impurity concentration gradient, ∂C/∂x, exists in a finite volume of a matrix substance (the silicon substrate in this context), the impurity material will have the natural tendency to move in order to distribute itself more evenly within the matrix and decrease the gradient. Given enough time, this flow of impurities will eventually result in homogeneity within the matrix, causing the net flow of impurities to stop. The mathematics of this transport mechanism was formalized in 1855 by Fick, who postulated that the flux of material across a given plane is proportional to the concentration gradient across the plane.
The flux J1 of impurities entering a section of a bar with a concentration gradient is different from the flux J2 of impurities leaving the same section. From the law of conservation of matter, the difference between J1 and J2 must result in a change in the concentration of impurities within the section (assuming that no impurities are formed or consumed in the section). This is Fick's Second Law, which states that the change in impurity concentration over time is equal to the change in local diffusion flux, or ∂C(x,t)/∂t = - ∂J/∂x or, from Fick's First Law, ∂C(x,t)/∂t = ∂(D∂C(x,t)/∂x)/∂x. If the diffusion coefficient is independent of position, such as when the impurity concentration is low, then Fick's Second Law may be further simplified into the following equation: ∂C(x,t)/∂t = D ∂2C(x,t)/∂x2.
k z Vz
Another important expression is “del squared”, also known as the “Laplacian”:
2f .f div grad f f f f , , . , , x y z x y z f f f x x y y z z 2f 2f 2f 2 2 x y z 2
Fick's Laws
Diffusion the movement of a chemical species from an area of high concentration to an area of lower concenቤተ መጻሕፍቲ ባይዱration, is one of the two major processes by which chemical species or dopants are introduced into a semiconductor (the other one being ion implantation) . The controlled diffusion of dopants into silicon to alter the type and level of conductivity of semiconductor materials is the foundation of forming a p-n junction and formation of devices during wafer fabrication. The mathematics that govern the mass transport phenomena of diffusion are based on Fick's laws.
Ay Bz Az B y i A B A B e A B sin Az Bx Ax Bz Ax Ax B y Ay Bx Bx Unit vector perpendicular
to A and B
A.B Ax Bx Ay B y Az Bz A B cos
Fick's Second Law
Fick's First Law does not consider the fact that the gradient and local concentration of the impurities in a material decreases with an increase in time, an aspect that's important to diffusion processes.