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

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Abstract

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.

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Abbreviations

\(A\) :

Surface area per unit volume (m2 m−3)

\(B_{1,2}\) :

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

\(c_{\text{s}}\) :

Sound speed

\(c_{\text{g}}\) :

Specific heat of gas (J kg−1 °C−1)

\(d_{\text{p}}\) :

Obstacle size (m)

e:

Total energy

\(e_{\upalpha}\) :

Discrete particle velocity in LBM

\(f\) :

Density distribution function

F :

Fraction function

F :

Shape factor

I :

Intensity

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

Non-dimensional intensity

\(j\) :

Index of grids in y-direction

\(h\) :

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

\(K_{\text{g}}\) :

Gas thermal conductivity (W m−1 °C−1)

\(K_{\text{p}}\) :

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

\(L_{\text{x}}\) :

Length of the porous medium (m)

\(L_{\text{y}}\) :

Height of the porous medium (m)

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

\(\tilde{P}\) :

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

\(q_{\text{rad}}\) :

Radiative heat flux (W m−2)

\(q_{\text{x}}\) :

Heat flux in x-direction

\(q_{\text{y}}\) :

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

R :

Particle radius

R i :

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

\(\hat{s}_{\text{i}}\) :

Direction vector in RTE

\(T\) :

Temperature (°C)

\(T_{\infty }\) :

Ambient temperature (°C)

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

Gas temperature at duct’s inlet (°C)

\(u_{\text{g}}\) :

Velocity along x-direction (m s−1)

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

Gas velocity at duct’s inlet (m s−1)

\(v_{\text{g}}\) :

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

\(x\) :

Coordinate along the flow direction (m)

\(x_{\text{i}}\) :

Mole fraction

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

Non-dimensional length

\(y\) :

Coordinate perpendicular to the flow direction (m)

\(y_{\text{i}}\) :

Mass fraction

\(\alpha\) :

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

\(\varepsilon\) :

Energy square

\(\varepsilon\) :

Emissivity

\(\varepsilon\) :

Effectiveness

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

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

\(\nu\) :

Kinematical viscosity (m2 s−1)

\(\varphi\) :

Porosity

\(\varphi (x,t)\) :

Concentration function

\(\phi\) :

Scattering phase function

\(\rho_{\text{g}}\) :

Gas density (kg m−3)

\(\rho_{\text{w}}\) :

Wall reflection coefficient

\(\sigma\) :

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

\(\sigma_{\text{a}}\) :

Absorption coefficient (m−1)

\(\sigma_{\text{s}}\) :

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

\(\tau\) :

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

\(w\) :

Weighting constant

\(\omega\) :

Scattering albedo

\(\mu\) :

Dynamic viscosity

\(\zeta_{{}}\) :

Particle interfacial thickness

\({\text{b}}\) :

Black body

\({\text{B}}\) :

Bottom

\({\text{e}}\) :

Exit of the porous matrix

\({\text{g}}\) :

Gas

i:

Inlet of the porous matrix

p:

Solid

T:

Top

eq:

Equilibrium

in:

Incoming velocity direction

m:

Outgoing radiation direction

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

Incoming radiation direction

\({\text{out}}\) :

Outgoing velocity direction

\(+\) :

Downstream direction

\(-\) :

Upstream direction

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

    Mojtaba Amirshekari & Seyyed Abdolreza Gandjalikhan Nassab

  2. Department of Energy, Institute of Science and High Technology and Environmental Sciences, Graduate University of Advanced Technology, Kerman, Iran

    Ebrahim Jahanshahi Javaran

Authors
  1. Mojtaba Amirshekari
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  2. Seyyed Abdolreza Gandjalikhan Nassab
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  3. Ebrahim Jahanshahi Javaran
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Correspondence to Ebrahim Jahanshahi Javaran.

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Amirshekari, M., Gandjalikhan Nassab, S.A. & Jahanshahi Javaran, E. Numerical simulation of a three-layer porous heat exchanger considering lattice Boltzmann method simulation of fluid flow. J Therm Anal Calorim 136, 1737–1755 (2019). https://doi.org/10.1007/s10973-018-7794-6

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