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Transient behaviour of heat transfer with complete evaporation process in Porous Channel with localised heating using non-Darcian flow and LTNE model

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Abstract

This study numerically presents the transient behaviour of complete evaporation process inside porous channel. Based on the modified enthalpy formulation under non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions, solutions have been obtained utilizing the finite volume method for various relevant parameters. The results have been validated with the experiments and they show good agreement between them. The results demonstrate that LTNE model should be employed while modelling the transient complete evaporation process in porous media. The non-Darcy effects have considerable influences on the locations of the initiation and termination of the phase change processes as compared to Darcy flow model. Therefore, the behaviours of the two-phase and the superheated vapour zones, when the steady-state solutions is reached, are substantially changed for non-Darcy flow as compared to Darcy model due this assumption provides an additional mechanism for heat transfer that represented by the heat exchange between the solid and fluid phases. It has been observed that the two-phase zone is considerably extended in the axial direction and this zone has not been occupied a large size towards transverse direction due to the axial velocity is noticeably higher than the transverse velocity, caused by reducing the diffusive energy through the two-phase region. The results indicate that non-Darcian effects leads to considerably extend the size of superheated region towards the outlet of the channel, which is not the case for Darcy model. The non-Darican effects become more pronounced for high Reynolds number and heat flux. The influences of varying in Reynolds number, heat flux, porosity, and solid thermal conductivity are substantial near the heated surface, whereas Darcy number has a minor impact on the solution of complete evaporation process. It can be emphasised from the present predictions that the non-Darcy flow along with LTNE condition is indispensable model for the complete evaporation process, especially under high mass flowrate as well heat flux.

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Notes

  1. These models, i.e., TPMM [4,5,6] are applicable only for low mass flow rate applications.

  2. The solutions have been obtained from the sub-cooled liquid phase to the two-phase mixture, where no superheated vapour zone has been observed.

  3. The saturation temperature varies with the local pressure.

  4. The solid and fluid phases locally coexist at the same temperature.

  5. Two energy equations are solved for each phases and these equations are connected together by the convective heat term.

  6. This model assumes the solid and fluid phases coexist at different temperature.

  7. This term an important factor that has the largest impact on growth of the two-phase region, especially in the case that the temperature of the solid phase exceeds the saturation temperature.

  8. α = l and α = v for liquid and vapour phases, respectively.

  9. Based on the pore diameter, the phase velocities and the properties of the local phase.

  10. The procedure of applying the special remedy is provided in details in the study of Alomar et al. [22].

  11. The exit length has been deliberately selected to overcome the effect of exit boundary condition on the internal solution.

Abbreviations

a s :

Specific surface of the porous medium, 1/m

b :

Body force vector per unit mass, m/s2

\( \overset{\sim }{\mathbf{b}} \) :

Normalized body force vector per unit mass b/g

C p :

Specific heat, J/kgK

d p :

Characteristic pore size of porous matrix, m

D :

Capillary diffusion coefficient, m2/s

ev :

Total evaporated volume fraction

f :

Hindrance function

Fr :

Froude number \( ={u}_{in}/{u}_g={u}_{in}/\sqrt{g\ W}={\mathit{\operatorname{Re}}}_{in}/{\mathit{\operatorname{Re}}}_g \)

g :

Acceleration due to gravitational vector, m/s2

h :

Specific enthalpy, J/kg

h fg :

Latent heat of vaporization =hv, sat − hl, sat, J/kg

h sα :

Convective heat transfer coefficient in the pores, W/m2K

j :

Diffusive mass flux vector, kg/m2s

J :

Capillary pressure function

k rl k rv :

Relative permeabilities for liquid and vapour, respectively

k :

Thermal conductivity, W/mK

K :

Permeability of porous matrix, m2

l :

Length of individual segments, m

L :

Length of the porous channel, m

n :

Exponent of saturation in the expression for relative permeabilities

p :

Effective pressure, Pa

p c :

Capillary pressure, Pa

p e :

Peclet number =uinW/α = RePr

p r :

Prandtl number =μ Cp/k

\( \dot{q}" \) :

Heat flux, W/m2

Q ∗ :

Normalized heat flux \( \dot{q}"W/{\mu h}_{fg} \)

\( {\dot{q}}_{sf}^{\prime \prime \prime } \) :

Heat exchange term between fluid and solid phases, W/m3

R e :

Reynolds number uinW/v

Re g :

Gravitational Reynolds number ugW/vl

Re p :

Reynolds number based on pore diameter and local phase properties

Re K :

Fluid Reynolds number based on the length scale of the permeability

s :

Liquid saturation

T :

Temperature, K

u :

Velocity component vector, m/s

t :

Time, sec

u g :

Gravitational velocity vector \( \sqrt{\mathbf{g}\ W},\mathrm{m}/\mathrm{s} \)

W :

Height of the duct, m

x,y :

horizontal and vertical Coordinates, respectively, m

α :

Thermal diffusivity =k/ρCp,   m2s

β :

Isobaric expansion coefficient, K−1

ΔT :

Temperature difference for relaxation of Γh is in the single-phase regions, K

Δρ :

Difference in densities =ρl − ρv, kg/m3

γ h :

Advection correction coefficient

Γh :

Diffusion coefficient in enthalpy equation, kg/ms

ε :

Porosity

λ :

Relative mobility

μ :

Dynamic viscosity, kg/ms

v :

Kinematics viscosity, m2/s

ρ :

Density, kg/m3

σ :

Surface tension, N/m

\( \overset{\sim }{\sigma } \) :

Normalized surface tension coefficient \( ={\rho}_l\sigma W/{\mu}_l^2 \)

eff :

Effective

f :

Fluid

i, e, h :

Inlet, exit and heated, respectively

in :

Inlet

k :

Kinetic

l :

Liquid

max:

Maximum value

min:

Minimum value

s :

Solid

sat :

Saturation

v :

Vapour

w :

Wall

*:

Dimensionless

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Authors and Affiliations

  1. Engineering Technical College of Mosul, Northern Technical University, Cultural Group Street, Mosul, Iraq

    Omar Rafae Alomar

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Correspondence to Omar Rafae Alomar.

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I hereby declare that there is no conflict of interest regarding the publication of this article entitled, “Transient Behaviour of Heat transfer with Complete Evaporation Process in Porous Channel with Localised Heating using Non-Darcian Flow and LTNE model” in the journal of Heat and Mass Transfer.

Dr.-Ing. Omar Rafae Mahmood Alomar.

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Highlights

• Transient behaviour of complete phase change process has been considered.

• The formulation based on non-Darcian effects along with LTNE conditions has been used.

• Conservation equations have been solved using FVM on non-staggered grid layout.

• The physical model has been considered to be two-dimensional.

• Smoothing of effective diffusion coefficient is required as remedy.

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Alomar, O. Transient behaviour of heat transfer with complete evaporation process in Porous Channel with localised heating using non-Darcian flow and LTNE model. Heat Mass Transfer 57, 1921–1948 (2021). https://doi.org/10.1007/s00231-021-03080-3

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