Journal of Thermal Analysis and Calorimetry

, Volume 136, Issue 4, pp 1737–1755 | Cite as

Numerical simulation of a three-layer porous heat exchanger considering lattice Boltzmann method simulation of fluid flow

  • Mojtaba Amirshekari
  • Seyyed Abdolreza Gandjalikhan Nassab
  • Ebrahim Jahanshahi JavaranEmail author


The present study investigates the thermal characteristics of a proposed porous heat exchanger (PHE). This heat exchanger consists of three sections, one high-temperature (HT) section and two heat recovery (HR) sections. Product of combustion as a high-temperature gas mixture enters to HT section in which enthalpy of gas flow is converted to thermal radiation, while in HR sections, the reverse phenomenon occurs. Simulation of fluid flow in porous medium generated by random and regular monodisperse and polydisperse particles is done using combination of the lattice Boltzmann method and smoothed profile method. Because of high-temperature variation in this system, effect of temperature on thermo-physical properties is also considered which has not been studied in previous research studies. Since the gas and solid phases are in non-local thermal equilibrium, separate energy equations are used for these phases. To obtain the radiative term in the solid energy equation, the radiative transfer equation is solved numerically by the discrete ordinates method. The influence of particles array and their sizes on the efficiency of the PHE system is studied. Finally, the effects of various parameters like optical thickness and scattering coefficient on the performance of PHE system are investigated.


Lattice Boltzmann method Smoothed profile method Porous medium Radiation Heat exchanger 

List of symbols


Surface area per unit volume (m2 m−3)


Incoming radiations (W m−2)

\(B_{1,2}^{\prime } = B_{1,2} /\sigma {\text{T}}_{\text{g0}}^{ 4}\)

Non-dimensional incoming radiations

\(Bi = {{hL_{\text{x}} } \mathord{\left/ {\vphantom {{hL_{\text{x}} } {k_{\text{p}} }}} \right. \kern-0pt} {k_{\text{p}} }}\)

Biot number


Sound speed


Specific heat of gas (J kg−1 °C−1)


Obstacle size (m)


Total energy


Discrete particle velocity in LBM


Density distribution function


Fraction function


Shape factor



\(I^{*} = I /\sigma {\text{T}}_{\text{g0}}^{ 4}\)

Non-dimensional intensity


Index of grids in y-direction


Convective heat transfer coefficient (W m−2 °C−1)


Gas thermal conductivity (W m−1 °C−1)


Solid thermal conductivity (W m−1 °C−1)


Length of the porous medium (m)


Height of the porous medium (m)


Molar mass

\(Nu = {{hL_{\text{x}} } \mathord{\left/ {\vphantom {{hL_{\text{x}} } {k_{\text{g}} }}} \right. \kern-0pt} {k_{\text{g}} }}\)

Nusselt number

\(P = {{\tilde{P}} \mathord{\left/ {\vphantom {{\tilde{P}} {\rho u_{{{\text{g}}_{0} }}^{2} }}} \right. \kern-0pt} {\rho u_{{{\text{g}}_{0} }}^{2} }}\)

Non-dimensional pressure


Pressure (Pa)

\(P_{1} = {{hL_{\text{x}} A} \mathord{\left/ {\vphantom {{hL_{\text{x}} A} {\rho_{\text{g}} c_{\text{g}} u_{{{\text{g}}_{0} }} (\Delta x \cdot \Delta y)}}} \right. \kern-0pt} {\rho_{\text{g}} c_{\text{g}} u_{{{\text{g}}_{0} }} (\Delta x \cdot \Delta y)}}\)

Dimensionless group

\(P_{2} = \frac{{_{{{\raise0.7ex\hbox{${K_{\text{p}} }$} \!\mathord{\left/ {\vphantom {{K_{\text{p}} } {L_{\text{x}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${L_{\text{x}} }$}}}} }}{{\sigma T_{{g_{0} }}^{3} }}\)

Dimensionless group

\(P_{3} = {{hL_{\text{x}} A} \mathord{\left/ {\vphantom {{hL_{\text{x}} A} {\sigma T_{{{\text{g}}_{0} }}^{3} (\Delta x \cdot \Delta y)}}} \right. \kern-0pt} {\sigma T_{{{\text{g}}_{0} }}^{3} (\Delta x \cdot \Delta y)}}\)

Dimensionless group

\(P_{4} = {{h_{{{\text{w}}_{\text{p}} }} L_{\text{x}} } \mathord{\left/ {\vphantom {{h_{{{\text{w}}_{\text{p}} }} L_{\text{x}} } {k_{\text{p}} }}} \right. \kern-0pt} {k_{\text{p}} }}\)

Dimensionless group

\(P_{5} = {{h_{{{\text{w}}_{\text{g}} }} L_{\text{x}} } \mathord{\left/ {\vphantom {{h_{{{\text{w}}_{\text{g}} }} L_{\text{x}} } {k_{\text{g}} }}} \right. \kern-0pt} {k_{\text{g}} }}\)

Dimensionless group

\(Pe = {{\rho_{\text{g}} u_{{{\text{g}}_{0} }} c_{\text{g}} L_{\text{x}} } \mathord{\left/ {\vphantom {{\rho_{\text{g}} u_{{{\text{g}}_{0} }} c_{\text{g}} L_{\text{x}} } {k_{\text{g}} }}} \right. \kern-0pt} {k_{\text{g}} }}\)

Peclet number


Radiative heat flux (W m−2)


Heat flux in x-direction


Heat flux in y-direction

\(Q_{\text{rad}} = {{q_{\text{rad}} } \mathord{\left/ {\vphantom {{q_{\text{rad}} } {\sigma T_{{{\text{g}}_{0} }}^{4} }}} \right. \kern-0pt} {\sigma T_{{{\text{g}}_{0} }}^{4} }}\)

Dimensionless radiative heat flux

\(r = {{L_{\text{x}} } \mathord{\left/ {\vphantom {{L_{\text{x}} } {L_{\text{y}} }}} \right. \kern-0pt} {L_{\text{y}} }}\)

Aspect ratio


Particle radius


Particle position vector

\(Re_{{{\text{L}}_{\text{x}} }} = {{u_{{{\text{g}}_{0} }} L_{\text{x}} } \mathord{\left/ {\vphantom {{u_{{{\text{g}}_{0} }} L_{\text{x}} } \upsilon }} \right. \kern-0pt} \upsilon }\)

Reynolds number

\(Re_{{{\text{d}}_{\text{p}} }} = {{u_{{{\text{g}}_{0} }} d_{\text{p}} } \mathord{\left/ {\vphantom {{u_{{{\text{g}}_{0} }} d_{\text{p}} } \upsilon }} \right. \kern-0pt} \upsilon }\)

Reynolds number


Direction vector in RTE


Temperature (°C)

\(T_{\infty }\)

Ambient temperature (°C)

\(T_{{{\text{g}}_{0\prime } }}\)

Gas temperature at duct’s inlet (°C)


Velocity along x-direction (m s−1)

\(u_{{{\text{g}}_{0} }}\)

Gas velocity at duct’s inlet (m s−1)


Velocity along y-direction (m s−1)

\(\bar{U} = {u \mathord{\left/ {\vphantom {u {u_{{{\text{g}}_{0} }} }}} \right. \kern-0pt} {u_{{{\text{g}}_{0} }} }}\)

Non-dimensional x velocity

\(\bar{V} = {v \mathord{\left/ {\vphantom {v {u_{{{\text{g}}_{0} }} }}} \right. \kern-0pt} {u_{{{\text{g}}_{0} }} }}\)

Non-dimensional y velocity


Coordinate along the flow direction (m)


Mole fraction

X = \({{L_{\text{x}} } \mathord{\left/ {\vphantom {{L_{\text{x}} } {L_{\text{y}} }}} \right. \kern-0pt} {L_{\text{y}} }}\)

Non-dimensional length


Coordinate perpendicular to the flow direction (m)


Mass fraction

Greek symbols


Particle velocity direction

\(\beta = \sigma_{\text{a}} + \sigma_{\text{s}}\)

Extinction coefficient

\(\mathop \nabla \limits^{*} = L_{\text{x}} \nabla\)

Non-dimensional gradient operator

\(\Delta x\)

Grid spacing along x-axis (m)

\(\Delta y\)

Grid spacing along y-axis (m)

\(\Delta_{{\upeta_{\text{x}} }} = {{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} {L_{\text{x}} }}} \right. \kern-0pt} {L_{\text{x}} }}\)

Non-dimensional grid spacing along x-axis

\(\Delta_{{\upeta_{\text{y}} }} = {{\Delta y} \mathord{\left/ {\vphantom {{\Delta y} {L_{\text{x}} }}} \right. \kern-0pt} {L_{\text{x}} }}\)

Non-dimensional grid spacing along y-axis

\(\delta t\)

Time step

\(\delta x\)

Lattice spacing


Energy square





\(\eta_{\text{x}} = {x \mathord{\left/ {\vphantom {x {L_{\text{x}} }}} \right. \kern-0pt} {L_{\text{x}} }}\)

Non-dimensional x coordinate

\(\eta_{\text{y}} = {y \mathord{\left/ {\vphantom {y {L_{\text{x}} }}} \right. \kern-0pt} {L_{\text{x}} }}\)

Non-dimensional y coordinate


Kinematical viscosity (m2 s−1)



\(\varphi (x,t)\)

Concentration function


Scattering phase function


Gas density (kg m−3)


Wall reflection coefficient


Stephan-Boltzmann coefficient (W m−2 K−4)


Absorption coefficient (m−1)


Scattering coefficient (m−1)

\(\theta_{{{\text{g}},{\text{p}}}} = {{T_{{{\text{g}},{\text{p}}}} } \mathord{\left/ {\vphantom {{T_{{{\text{g}},{\text{p}}}} } {T_{{{\text{g}}_{ 0} }} }}} \right. \kern-0pt} {T_{{{\text{g}}_{ 0} }} }}\)

Non-dimensional temperature


Non-dimensional relaxation time

\(\tau_{0} = \beta L_{\text{x}}\)

Optical thickness

\(\tau_{1} = \sigma_{\text{a}} L_{\text{x}}\)

Non-dimensional parameter

\(\tau_{2} = \sigma_{\text{s}} L_{\text{x}}\)

Non-dimensional parameter


Weighting constant


Scattering albedo


Dynamic viscosity


Particle interfacial thickness

Sub- and superscripts


Black body




Exit of the porous matrix




Inlet of the porous matrix








Incoming velocity direction


Outgoing radiation direction

\({\text{m}}^{\prime }\)

Incoming radiation direction


Outgoing velocity direction


Downstream direction


Upstream direction


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • Mojtaba Amirshekari
    • 1
  • Seyyed Abdolreza Gandjalikhan Nassab
    • 1
  • Ebrahim Jahanshahi Javaran
    • 2
    Email author
  1. 1.Department of Mechanical EngineeringShahid Bahonar University of KermanKermanIran
  2. 2.Department of Energy, Institute of Science and High Technology and Environmental SciencesGraduate University of Advanced TechnologyKermanIran

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