Fundamentals of a fluid analysis

Ⅲ. Fluid Analysis

3.1 Fundamentals of a fluid analysis

3.1.1 Governing equations and turbulent model

The Finite Volume Method (FVM) is typically used for Computational Fluid Dynamics (CFD). The FEM divides the shape or space of an object into finite number of elements, while the FVM divides the geometric region into a finite number of volumes. The common point is that the mathematical expressions of object is converted to matrix equations. Therefore, the FVM is used for fluid analysis because it is efficient for calculation of governing equation where convection term exists, and the FEM is used for solving solid problem because the value is calculated at each nodes.

The governing equations considered in the fluid analysis are continuity equation and momentum equation. When considering the temperature, the energy equation is also solved. Each equation is as follows.

^𝜕𝜌

𝜕𝑡+ 𝛻 ∙ (𝜌𝑽⃗⃗ ) = 0 (3-1)

^𝜕

𝜕𝑡(𝜌𝜇) + 𝛻 ∙ (𝜌𝜇𝑽⃗⃗ ) = −^𝜕𝑝

𝜕𝑥+^{𝜕𝜏𝑥𝑥}

𝜕𝑥 +^{𝜕𝜏𝑦𝑥}

𝜕𝑦 +^{𝜕𝜏𝑧𝑥}

𝜕𝑧 + 𝜌𝑓_𝑥 (3-2)

𝜕

𝜕𝑡(𝜌𝑣) + 𝛻 ∙ (𝜌𝑣𝑽⃗⃗ ) = −^𝜕𝑝

𝜕𝑦+^{𝜕𝜏𝑥𝑦}

𝜕𝑥 +^{𝜕𝜏𝑦𝑦}

𝜕𝑦 +^{𝜕𝜏𝑧𝑦}

𝜕𝑧 + 𝜌𝑓_𝑦 (3-3)

𝜕

𝜕𝑡(𝜌𝑤) + 𝛻 ∙ (𝜌𝑤𝑽⃗⃗ ) = −^𝜕𝑝

𝜕𝑧+^{𝜕𝜏𝑥𝑧}

𝜕𝑥 +^{𝜕𝜏𝑦𝑧}

𝜕𝑦 +^{𝜕𝜏𝑧𝑧}

𝜕𝑧 + 𝜌𝑓_𝑧 (3-4)

𝜕(𝜌𝑒)

𝜕𝑡 + ∇ ∙ (ρe𝑽⃗⃗ ) = ρ𝑞̇ +_𝜕𝑥^𝜕 (𝑘^𝜕𝑇_𝜕𝑥) +_𝜕𝑦^𝜕 (𝑘^𝜕𝑇_𝜕𝑦) +_𝜕𝑧^𝜕 (𝑘^𝜕𝑇_𝜕𝑧) − 𝑝 (𝜕𝜇

𝜕𝑥+𝜕𝑣

𝜕𝑦+𝜕𝑤

𝜕𝑧) + λ (𝜕𝜇

𝜕𝑥+𝜕𝑣

𝜕𝑦+𝜕𝑤

𝜕𝑧)

2

+

𝜇 [ 2 (^𝜕𝜇

𝜕𝑥)²+ 2 (^𝜕𝑣

𝜕𝑦)²+ 2 (^𝜕𝑤

𝜕𝑧)²+ (^𝜕𝜇

𝜕𝑦+^𝜕𝑣

𝜕𝑥)²+ (^𝜕𝜇

𝜕𝑧+^𝜕𝑤

𝜕𝑥)²+ (^𝜕𝑣

𝜕𝑧+^𝜕𝑤

𝜕𝑦)²

] (3-5)

First, Eq. (3-1) is a continuity equation and is obtained by the law of conservation of mass. Second,

Eq. (3-2), (3-3), and (3-4) are momentum equation for each direction and are obtained by the Navier- Stokes equation. Third, Eq. (3-5) is an energy equation and is obtained by the Bernoulli equation.^[7] In addition, realizable k − ε turbulence model is used, where k is the turbulent kinetic energy and ε is the turbulent dissipation. Using these two functions, eddy viscosity is determined as shown in Eq. (3- 6).

⁡𝑢_𝑡 = 𝐶_𝑢𝜌^𝑘2

𝜀 (3-6) k and ε are obtained by calculating the transport equation.^[8]

𝜕(𝜌𝑘)

𝜕𝑡 +^{𝜕(𝜌𝑘𝑢𝑖)}

𝜕𝑥_𝑖 = ^𝜕

𝜕𝑥_𝑗[^𝜇𝑡

𝜎_𝑘

𝜕𝑘

𝜕𝑥_𝑗]+2𝜇_𝑡𝐸_𝑖𝑗𝐸_𝑖𝑗− 𝜌𝜖 (3-7)

𝜕(𝜌𝜖)

𝜕𝑡 +^{𝜕(𝜌𝜖𝑢𝑖)}

𝜕𝑥_𝑖 = ^𝜕

𝜕𝑥_𝑗[^𝜇𝑡

𝜎_𝜖

𝜕𝜀

𝜕𝑥_𝑗] + 𝐶_1𝜖^𝜀

𝑘2𝜇_𝑡𝐸_𝑖𝑗𝐸_𝑖𝑗− 𝐶_2𝜀𝜌^𝜖2

𝑘 (3-8)

3.1.2 Flow uniformity

Devices such as SCR chamber used in this study are required to calculate the flow uniformity into the internal catalyst bed. This is because the flow uniformity in the front of the catalyst bed is an important index for the purification efficiency of the exhaust gas.

Flow Uniformity proposed by Weltens and Velocity RMS (%) are typical methods for calculating the flow uniformity. First, velocity RMS is expressed as follows.^[9]

RMS (%) =¹

μ

̅√^{∫ ρμ(μ−μ̅)2dA}

∫ ρμdA (3-9)

⁡In Eq. (3-9), μ is the velocity at point, 𝜇̅ is the average velocity, ρ is the density of fluid, and A is the area. The value in the root represents the standard deviation of the velocity value and finally RMS value is calculated by dividing the standard deviation by the average value. If the RMS value is 0%, it indicates that the flow is uniform over the entire cross section as shown in Fig. (3-1), and the uniformity of the flow becomes worse as the RMS increases.

Fig. 3-1. Velocity RMS (a) RMS (%)=0 (b) RMS (%)=100

Flow Uniformity is expressed as follows.^[10]

U_F= 1 −^{∑ni=0|Vi−V|Ai}

2AV (3-10) In Eq. (3-10), n is the total number of cells, A is the area, Ai is area at the catalyst cell I, V is the velocity, Vi is unit velocity at the catalyst cell i. Flow Uniformity has value from 0 to 1, which is opposite to RMS. In contrast to RMS, the closer to 1, the flow is uniform.

In this study, the flow uniformity is calculated by Velocity RMS.

3.1.3 Mathematical modeling of catalyst system

In the case of SCR system, the honeycomb type catalyst bed is mainly used in the exhaust gas purification system (Fig. (3-2)). When the computational analysis of the structure with this shape is performed, it is impossible to generate meshes considering the dimension order difference of about 10³~10⁴ due to out of memory.

Fig. 3-2. Geometry of the catalyst bed

The analysis is performed by applying the porous model, which is an equivalent system considering the physical characteristics of the catalyst bed causing pressure drop in the flow direction (Fig. (3-3)).^[11]

Fig. 3-3. Equivalent system of the catalyst bed

To apply the porous model to the computational analysis, two coefficients of Viscous Resistance and Inertial Resistance should be calculated.

First, to consider the pressure drop by the catalyst bed, the pressure increase term is used as shown in Eq. (3-11).

−𝐾_𝑖𝜇_𝑖=^𝑑𝑝

𝑑𝜉_𝑖 (3-11) In Eq. (3-11), 𝜉_𝑖(i = 1,2,3) is the direction of the flow, 𝐾_𝑖 is the transmissivity, 𝜇_𝑖 is the velocity of area in 𝜉_𝑖. 𝐾_𝑖 can be expressed as a function of area velocity magnitude.

𝐾_𝑖 = 𝛼_𝑖|𝑣̅| + 𝛽 (3-12)

The flow inside the catalyst bed flows on the longitudinal direction and not on other direction.

Therefore, 𝜉_𝑖 can be expressed as 𝑥_𝑖, which is the flow direction of the exhause gas. From Eq. (3-11), (3-12), the pressure drop in the porous medium can be expressed as shown in Eq. (3-13).

⁡_𝑑𝑥^𝑑𝑝

𝑖= −(𝛼_𝑖|𝑣̅| + 𝛽)𝑣_𝑖 (3-13) The Eq. (3-13) is expressed as a term that is proportional to the velocity, and a term that is proportional to the square of the velocity. Therefore, by measuring the velocity and the pressure drop in the catalyst bed, the values of the first permeability coefficient (Eq. (3-14)) and second permeability coefficient (Eq.

(3-15)) can be obtained.[12][13][14]

Above Eq. (3-13) can be also expressed as Eq. (3-14). In the governing equation, the porous media are modeled by the addition of a momentum source term to the standard fluid flow equations in Eq (3- 2)~(3-4)

𝑆_𝑖 = − (^𝜇_𝛼̅𝑣_𝑖+ 𝐶₂¹₂𝜌|𝑣̅|𝑣_𝑖) (3-14)

𝛼 = 𝐶₂¹

2𝜌 (3-15)

⁡𝛽 =^𝜇

𝛼̅ (3-16) Inertial Resistance (𝐶₂) can be obtained from Eq. (3-15), and Viscous Resistance (¹

𝛼

̅) can be obtained from Eq. (3-16).

There are two ways to model a catalyst bed as porous medium through above process. The first way is to measure the velocity and pressure drop by experiments,^[15] and the second way is to measure the velocity and pressure drop by CFD.^[16]

In this study, catalyst modeling is performed by CFD and its reliability is verified.