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Volume Averaging Theory (VAT) based modeling and closure evaluation for fin-and-tube heat exchangers

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Abstract

A fin-and-tube heat exchanger was modeled based on Volume Averaging Theory (VAT) in such a way that the details of the original structure was replaced by their averaged counterparts, so that the VAT based governing equations can be efficiently solved for a wide range of parameters. To complete the VAT based model, proper closure is needed, which is related to a local friction factor and a heat transfer coefficient of a Representative Elementary Volume (REV). The terms in the closure expressions are complex and sometimes relating experimental data to the closure terms is difficult. In this work we use CFD to evaluate the rigorously derived closure terms over one of the selected REVs. The objective is to show how heat exchangers can be modeled as a porous media and how CFD can be used in place of a detailed, often formidable, experimental effort to obtain closure for the model.

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Abbreviations

A :

Area (m2)

A f :

Fin surface area (m2)

A o :

Total surface area (m2)

A w :

Wetted surface (m2)

A wp :

The cross flow projected area (m2)

c p :

Specific heat (J/(kg·K))

D i :

Inner diameter of the tube (m)

D o :

Outer diameter of the tube (m)

D c :

Fin collar outside diameter, D c  = D o  + 2δ f (m)

D h :

Porous media hydraulic diameter (m)

d p :

Diameter of the spherical particles (m)

F 1, F 2 :

Blending function

F p :

Fin pitch (m)

f :

Friction factor

h :

Heat transfer coefficient (W/(m2·K))

k :

Turbulence kinetic energy per unit mass

k f :

Thermal conductivity of the fluid (W/(m·K))

k s :

Thermal conductivity of the solid (W/(m·K))

k T :

Turbulent heat conductivity (W/(m·K))

m :

\( \sqrt {{{\left( {2h} \right)} \mathord{\left/ {\vphantom {{\left( {2h} \right)} {\left( {\lambda_{f} \delta_{f} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\lambda_{f} \delta_{f} } \right)}}} \), parameter

\( \dot{m} \) :

Mass flow rate (kg/s)

\( \left\langle m \right\rangle \) :

Average porosity

N :

The number of tube rows

Nu :

Nusselt number

P k :

Shear production of turbulence

Pr:

Prandtl number

Pr T :

Turbulent Prandtl number

P t :

Transverse tube pitch (m)

P l :

Longitudinal tube pitch (m)

p :

Pressure (Pa)

\( Re_{{D_{c} }} \) :

Reynolds number based on fin collar outside diameter and maximum velocity, \( Re_{{D_{c} }} = {{u_{max} D_{c} } \mathord{\left/ {\vphantom {{u_{max} D_{c} } \nu }} \right. \kern-\nulldelimiterspace} \nu } \)

\( Re_{{D_{h} }} \) :

Reynolds number based on hydraulic diameter and average velocity, \( Re_{{D_{h} }} = {{{{\tilde{\overline{u}}}}D_{h} }}/ \nu \)

R eq :

Equivalent radius for circular fin (m)

r :

Radius of tube, including collar thickness (m)

S :

An invariant measure of the strain rate

S w :

Specific surface of a porous media, \( S_{w} = \partial S_{w} /\Updelta {{\Upomega}} \) (1/m)

S wp :

The cross flow projected area per volume (1/m)

T :

Fluid temperature (K)

T s :

Solid temperature (K)

u :

x direction velocity term (m/s)

w :

z direction velocity term (m/s)

X L :

\( {{\sqrt {\left( {{{P_{t} } \mathord{\left/ {\vphantom {{P_{t} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} + P_{l}^{2} } } \mathord{\left/ {\vphantom {{\sqrt {\left( {{{P_{t} } \mathord{\left/ {\vphantom {{P_{t} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} + P_{l}^{2} } } 2}} \right. \kern-\nulldelimiterspace} 2} \), geometric parameter (m)

X M :

P t /2, geometric parameter (m)

α :

Turbulence model constant or scale attack angle

\( \beta ,\beta^{*} \) :

Turbulence model constant

φ 1 :

Represent any constant in the original k-ω model (\( \sigma_{k1} , \ldots \))

φ 2 :

Represent any constant in the transformed k-ε model (\( \sigma_{k2} , \ldots \))

φ :

Represent the corresponding constant in the SST model (\( \sigma_{k} , \ldots \))

δ f :

Thickness of a fin (m)

μ :

Viscosity (Pa·s)

μ T :

Turbulent eddy viscosity (Pa·s)

ν :

Kinematic viscosity (m2/s)

ν T :

Turbulent kinematic viscosity (m2/s)

ρ :

Density (kg/m3)

σ ɛ :

k-ε turbulence model constant

σ k :

Turbulence model constant for the k equation

σ ω :

k-ω turbulence model constant

τ wL :

Laminar shear stress (N/m2)

τ wT :

Turbulent shear stress (N/m2)

ΔΩ:

The volume of the REV (m3)

ω :

Specific turbulence dissipation rate

~:

A value averaged over the representative volume

–:

An average of turbulent values

^:

Fluctuation of a value

\( \left\langle f \right\rangle_{f} \) :

Means the superficial average of the function f

f :

Fluid phase or fin surface

in:

Air-side inlet

out:

Air-side outlet

T :

Turbulent

s :

Solid phase

1:

A value in the air side

2:

A value in the water side

0:

Evaluated at the wall or surface

b:

Evaluated at the bulk temperature

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Acknowledgments

The support of a DARPA initiated grant within the MACE program, Grant No. W31P4Q-09-1-0005, is gratefully acknowledged. The views, opinions, and/or findings contained in this article are those of the author and should not be interpreted as representing the official views or policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the Department of Defense.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA, USA

    Feng Zhou & Ivan Catton

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  1. Feng Zhou
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Correspondence to Feng Zhou.

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Zhou, F., Catton, I. Volume Averaging Theory (VAT) based modeling and closure evaluation for fin-and-tube heat exchangers. Heat Mass Transfer 48, 1813–1823 (2012). https://doi.org/10.1007/s00231-012-1025-7

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