Local Flow Analysis - Flow Over one Tray

The Navier-Stokes (N-S) equations together with the continuity equation, comprise a closed set of equations, the solution of which provides a valid description of laminar and turbulent flows. The turbulent motion has a wide spectrum of eddy sizes, and large and small eddies can coexist in the same volume of fluid. The transport equation, in a general form, can be written as follows:

( ) ( ) ( ^{( )} )

∂

∂ ρ ρ

t ⋅ Φ + div ⋅ ⋅ w r Φ = div Γ

_Φ

grad Φ + S

_{Φ (5.1)}

transient convection diffusion source

where

Φ

is the potential,

Γ

Φ diffusivity, and

S

_Φ the source term, and are given in table 3.

Table 3. Parameters of the general transport equation

Equation

Φ Γ

_Φ

S

_Φ

Engineers are not concerned with all the details of turbulent motion, but rather with its effects on the gross properties of the flow. Consequently, there is no need to solve the instantaneous variables if averaged variables are all that is required. Most turbulence models use the eddy-viscosity concept (

μ

_t

= ⋅ ρ ν

_t) which is a property of the local state of the turbulence. The simplest turbulence model is one which uses a constant value for the eddy viscosity. For dimensional reasons, the effective kinematic viscosity made by the local turbulence is proportional to typical velocity and characteristic length:

ν

_t

= ⋅ ⋅ C w l

_c (5.2)

where C is a constant, w the typical velocity, and

l

_c the characteristic length.

The k-ε two-equation turbulence model has proved the most popular, mainly because it does not require a near-wall correction term. The standard form of the k-ε turbulence model employs the following turbulence transport equation:

kinetic-energy equation:

P

_k is the volumetric production rate of turbulent kinetic energy by shear forces,

G

_b is the volumetric production rate of turbulent kinetic energy by gravitational forces interacting with density gradients (buoyancy production), and

C

1 ,

C

2 and

C

3 are constants [Bradshaw, 1981]. The kinematic turbulent viscosity is defined by:

ν^t C^μ C^d kε

= ⋅ ⋅ ² (5.5)

Two-equation models account for transport effects of velocity and characteristic length, and the distribution of

l

_c is determined by the model. The characteristic length may be recovered from: should be close to zero for stably-stratified flow, and close to 1.0 for unstably-stratified flow.

The volumetric production rate of turbulent kinetic energy by shear forces,

P

_k, is defined as:

and the volumetric production rate of turbulent kinetic energy by gravitational forces interacting with density gradients,

G

b (buoyancy production), is defined as:

G g d

b t i

dz

i

= − ⋅

⋅ ⋅ ν ρ

ρ

Pr

^(5.8)

G

b is negative for stably-stratified (dense below light) layers, so that k is reduced and turbulence damped.

G

_b is positive for unstably-stratified (dense above light) layers, in which therefore, k increases at the expense of gravitational potential energy. This model forms a good compromise between generality and economy of use for many CFD problems.

In order to evaluate the influence of different turbulence models on the flow pattern of geometric arrangements specific for drying rooms, a simplified laboratory model was constructed. This laboratory model consists of one empty tray, having the dimensions length x width x height as 50 x 100 x 9 cm respectively, and placed into a rectangular channel with a cross-section of 100 x 19 cm². The numerical simulation was carried out using the PHOENICS CFD commercial code. Due to the fact that only two velocity components are significant and side effects are negligible, the computational domain was chosen in 2-D, representing a vertical plane, parallel with the main direction of flow and passing through the longitudinal axis of the tray.

The tray, placed in the middle of the domain, was simulated by two vertical walls having 0.5 m distance between them, 9 cm height and 1 cm width, each. The domain was divided into 3,040 cells using a 160 x 19 Cartesian grid. The inlet opening boundary condition is the air velocity, which is supposed to be uniform and equal to 2.87 m/s. The exit boundary condition is the ambient pressure, 101.325 kPa. No-slip boundary condition and appropriate wall functions apply on the solid surfaces. Figure 26 shows the velocity vector distribution for the simulation using k-ε two-equation turbulence model.

Figure 26. Simulation of velocity vectors distribution using two-equation k-ε model.

An experimental arrangement, similar to the one investigated numerically, was set up to carry out air flow measurements in the laboratory. The model consists of a wood tray, having the dimensions: length 0.5 m, width 1.0 m and height 0.1 m, which was inserted in the middle of the test section of a low-speed wind tunnel. The test compartment of the wind tunnel has a cross-section of 1.0 x 0.2 m² and is 1.5 m in length.

The air flow, delivered by a centrifugal blower, passes through a settle room (1.0 x 1.0 m² cross-section, 1.2 m in length) having a series of wire sieves, and enters the test channel

through a convergent nozzle. At the entrance of the test section, the air velocity was measured and found to be uniform and constant having the value 2.87 m/s. The flow field characteristics were measured with a five-hole tube, with a spherical head 8 mm in diameter and a bent shaft, type Schiltknecht f.881, [Ghiaus, 1994]. The measured values of the mean velocity vector field, was plotted using the SURFER graphic commercial code and is presented in figure 27.

Figure 27. Distribution of velocity experimentally measured.

The availability of measurement techniques permitting the description of the spatial structure of the flow allows cross-validation of the results obtained and better suitability of the proposed numerical simulation models. Validation of zero-equation turbulence models showed the closest simulation results with the experimental measurements for a viscosity ratio

ν ν

_{t l}

= 100

.

The two-equation turbulence models did not differ significantly from each another and compared with the experiments. However, the best approximation was given by the basic k-ε model. Figure 28 shows the profiles of the horizontal components of velocity vectors obtained from numerical simulation with different turbulence models compared to actual measured values, in three cross-sections of the test domain.

2

Figure 28 - Continued on next page

2 4 6 8 10 12 14 16 18

-2 0 2 4 6 8 10

Velocity, m/s

Channel height, cm

c) 15 cm after tray

Figure 28. Profiles of velocity horizontal components for constant viscosity and k-ε turbulence models.

We can conclude that the best simulation results for this type of problem can be obtained by using the k-ε turbulence model. It is useful practice, however, to start the numerical simulation with a simple model (constant effective viscosity) and after the configuration problems are solved, the convergence of the solution can be achieved using a more complicated model (k-ε for instance).

In document New Food Engineering (Page 97-101)