The simplifying physical model for momentum, heat and mass transfer in a vertical refrigerated display case adopted here is a two-dimensional vertical downward cold air jet flowing over the open surface of a vertical open cavity standing in a store, as schematically shown in Fig. 2.1. In order to avoid the boundary conditions in the surrounding to affect our numerical results, the upper and left ambient surroundings are respectively placed at z = 3.30 m and x = 4.61 m, far from the open cavity. At the top and bottom of the cavity there are discharge and return air zones adjacent to the ambient. Initially, the flows in the case and surrounding are assumed to be stationary, isothermal at the same temperature Tamb
and iso-solutal at the same water vapor concentration w1a. Then at time t≥0 a cold air jet is introduced from the discharge grille of the cabinet and injects vertically downwards over the open cavity. At the air discharge the jet flow is assumed to be at a uniform speed Vj and at a lower uniform temperature Tj and uniform relative humidity φj. The width of the jet is bj. Meanwhile, at time t =0 we assume that the temperature of the inner vertical plate of the case is suddenly lowered to a uniform value Tc and maintained at this level thereafter. In the mean time, the upper plate and bottom plate of the cavity is kept adiabatic for t≥0, except at the discharge and return grilles of the cavity. In the present computation we assume Tj =
Tc. The air flow at the steady state long after the jet discharged into the cavity is numerically predicted. In this study the size of the cavity is chosen to be 1.2-m in height and 0.57-m in depth. Note that in the present numerical simulation the computation domain surrounding the cavity is chosen to be large enough to reduce the effects of the store size on the flow in the cavity. The ambient is assumed at a uniform temperature Tamb and uniform relative humidity φamb for t≥0.
2.2 Assumptions and Governing Equations
To simplify the computation, the following assumptions are made:
(1) Flow is two-dimensional, turbulent and incompressible.
(2) Thermal radiation is neglected.
(3) Heat conduction in the wall of the cavity and grilles are neglected in the computation. Besides, heat capacity effects associated with the cavity wall and the grilles are negligible.
(4) The thickness of the cavity wall is very small compared with the cavity height and depth.
(5) Neglect the viscous dissipation effect.
(6) gx = , 0 gz =−g
(7) The pressure gradient and body force terms in the momentum equations are:
/
p z ρg
−∂ ∂ − .
By using the equation of state for ideal gas mixture and assuming a small temperature difference and a low level of water vapor concentration in the flow, the density variation in the air flow can be approximated by Boussinesq approximation as
Where ρ0 is the mixture density evaluated at the referenced temperature T 0 and concentration
w
10, and the referenced conditions are refered to the air surrounding. Moreover, Ma and Mv are the molecular weights of air and water vapor, respectively. the volumetric coefficients of thermal and solutal expansion,1,
The concentration buoyancy force needs to be taken into account here due to the presence of the density variation with the non-uniform species concentration in the flow. Note that the buoyancy force resulting from the concentration difference may assist or oppose the thermal buoyancy force resulting from the temperature variation in the fluid depending on that the concentration expansion coefficient βm is positive or negative. In the present study, we have an adding buoyancy force for the open cavity flow since Ma ＞ Mv (i.e. the buoyancy ratio N ＞ 0). Note that both the thermal and solutal buoyancies tend to induce a counterclockwise flow recirculation in the cavity, which is stronger than the pure thermal buoyancy driven flow. A combined Grashof number GR = Grt + (Pr/Sc)1/2Grm was proposed to character the combined buoyancy effect [23]. But the air jet from the discharge grille can entrain
the air in the cavity to move in a clockwise direction through the viscous shearing effect.
According to Boussinesq’s proposition (1877), the Reynolds stress in a turbulent flow can be approximately related to the gradients of the mean flow and turbulent kinetic energy as
two-equation K− model, the turbulent eddy viscosity for momentum can be ε expressed by K and ε as νt =C Kμ 2/ε . The turbulent Prandtl number and
turbulent Schmidt number are chosen to be Prt =ρνtCp/kt =1 and
/ 1
t t t
Sc =ν D = .
Under these assumptions the basic equations describing the time-average turbulent flow in the open cavity are
Continuity equation-
0 0 0 1 10
The above equations are subjected to the following boundary conditions:
Boundary І-air discharge grille. At the air jet discharge, flow is at uniform speed Vj, temperature Tj , and water vapor concentration w1j
j j C 1 1j b
u = 0, w = - V , T = T = T , w = w at z = H+B =1.74 m; (2.10)
Boundary IІ- inner vertical plate of the cavity. At the inner vertical plate with perforation density dp = 0 we have no-slip condition and an isothermal condition is imposed. Besides, the plate is impermeable. Thus
1
u = w = 0, T = T , C w 0 at x = 0;
x
∂ =
∂ (2.11)
For the inner vertical plate with perforation, flow is at perforation density dp, uniform velocity Vp, temperature Tc and water vapor concentration w1c
p C 1 1
u = V , w = 0, T = T , w =wc at x = 0; (2.12)
Boundary IІІ- upper and bottom plates of the cavity. At the upper and bottom
plates of the cavity we have adiabatic and no-slip conditions. The plates are also
Boundary IV- return grille of the air curtain. At the return grille of the air curtain, the velocity, temperature and concentration gradients are assumed to be very small,
u = 0, w T w1 0 at z = B = 0.54 m;b
z z z
∂
∂ =∂ = =
∂ ∂ ∂ (2.14)
Boundary V- surrounding boundary I- At the upper boundary of the surrounding, the gradient of the air velocity is very small and the air is at ambient temperature and concentration,
Boundary VI- surrounding boundary II- At the outer surrounding horizontally far from the cavity the gradient of the velocity is very small, and the pressure and temperature are assumed at the ambient values,
amb amb 1 1a
Boundary VII- ground boundary- At the ground the flow is subjected to the adiabatic, impermeable and no-slip conditions,
Boundary VIII- other solid boundary- At the other solid boundary the flow is subjected to the adiabatic, impermeable and no-slip conditions,
Region 1- vertical solid surface
These hydrodynamic and thermal boundary conditions are specified in Fig. 2.2.
Using the following non-dimensional variables
The basic flow equations can be written in dimensionless form as
Continuity equation-
The above non-dimensional governing equations are subjected to the following
boundary conditions:
Boundary І-air discharge grille. At the air jet discharge, flow is at uniform speed Vj, temperature Tj, and concentration W1j
1 b
U = 0, W = -1, Θ =0, and W = 1 at Z = (H+B ) / H = 1.45; (2.25)
Boundary IІ- inner vertical plate of the cavity. At the inner vertical plate with perforation density dp = 0
U = W = 0, = 0, W1 0 at X=0 X
Θ ∂ =
∂ ; (2.26)
For the condition with perforation density dp
p j 1
U = V /V , W = 0, = 0, WΘ =1 at X=0 (2.27)
Boundary IІІ- upper and bottom plates of the cavity. At the upper and bottom plates of the cavity
Boundary IV- return grilles of the air curtain. At the return grille of the air curtain, the velocity, temperature and concentration gradients are assumed to be very small,
Boundary V- surrounding boundary I- At the upper boundary of the outer surrounding
0, 1, , W = 0 at 1
U W
Z Z Z
∂ = ∂ = Θ = → ∞
∂ ∂ ; (2.30)
Boundary VI- surrounding boundary II- At the outer surrounding horizontally far from the cavity
Boundary VII- ground boundary- at the ground boundary we have adiabatic and no-slip conditions,
Boundary VIII- other solid boundary- at the other solid boundary the flow is subjected to the adiabatic, impermeable and no-slip conditions,
Region 1- vertical solid surface
1
2 2
where ε and νt are respectively the dissipation rate of the turbulent kinetic energy and momentum eddy diffusivity. recommended values [23] for these empirical constants areCμ =0.09,C1ε =1.44,
2 1.92, k 1.0, and 1.3
C ε = σ = σε = .The initial and boundary conditions for the above two equations require some discussion.
Since there is no-slip condition at the solid surface, physically we have 0 and = 0
K = ε at all solid boundaries. At the air curtain discharge, a 5 % turbulence intensity is chosen to represent the disturbance due to a uniform flow through a rectified honeycomb structure. For the air return grille, the small vertical gradients for and K ε are assumed,∂K/∂ = ∂ ∂ = . Small gradients in the z ε/ z 0 vertical and horizontal directions are imposed at the upper surrounding and outer surrounding, which are ∂K/∂ = ∂ ∂ = and z ε/ z 0 ∂K/∂ = ∂ ∂ = , respectively. x ε/ x 0 It is a well-known fact that in the near-wall region using the no-slip condition in the K−ε model adopted here provides unsatisfactory results. An alternative and widely employed way to overcome this deficiency is to introduce the damping
effects through the wall functions. The wall function model can be used only in the near wall region and it uses the empirical laws to approach the velocity profile near wall. In the present prediction, the modified wall functions are employed to replace the direct computation of K and ε from the transport equations given in Equations (2.31) and (2.32) near the solid boundaries.
Strictly speaking, it is supposed to be applied to a point up to the dimensionless distance Y+ ranging from 30 to 130 where Y+=
( )
u yτ /ν . The exact Y+ range in which the wall function is used is automatically determined by the software PHOENICS [25] during the computation. Note that uτ is the resultant friction velocity, uτ = τ ρ0/ . Specifically, the following wall functions are employed to a near-wall layer in local equilibrium, which can be written as follows [25]:( )
where U+ is the non-dimensional centerline velocity, κ is the log-law constant ( 0.41)≈ , and E is a roughness parameter.
Fig. 2.1 Physical model for a vertical refrigerated display case.
(II) Inner isothermal wall
1 1
(V) ,amb V 0, a
T T w w
z
= ∂ = =
∂
11(VI) ,0,amba VTTwwx ∂===∂
Fig. 2.2 Schematic diagram illustrating the geometry and some boundary conditions.
CHAPTER 3
SOLUTION METHOD
The solution method to be used in the present study and the verification of its applicability to predict the transport processes in the open cavity flow considered here are detailed in this chapter. The grid distribution in the computing domain for this numeral scheme is also verified by comparing the predicted results with the open literature.