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Oscillatory Flow in a Mesh-Type Regenerator

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Modeling Transport Phenomena in Porous Media with Applications

Part of the book series: Mechanical Engineering Series ((MES))

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Abstract

A stack of meshes assembled to form a regenerator of a Stirling cycle has been analyzed for determining flow distribution and heat transfer. A non-Darcy, thermal nonequilibrium model is driven by pulsatile flow with hot and cold fluid alternately going past the mesh in opposite directions. The characteristic frequency of pulsation is determined with reference to the time constant of the gas-wire system. The flow is shown to reach dynamic steady state rapidly, when compared to the thermal field. With this result, the flow field is computed by a harmonic analysis technique. The energy equations for the fluid and the solid phases are numerically integrated in time. Calculations continue for a total of 104 cycles. Dense meshes reported in the literature are evaluated against coarse meshes. Results have been presented for two Reynolds numbers, 100 and 10,000, the latter being closer to a practically attainable value in applications. Velocity profiles in dense meshes are seen to be flatter in comparison with coarse meshes. In addition, thermal nonequilibrium effects are quite significant in coarse meshes, pointing to a need for preferring dense meshes in cryocooler applications.

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Abbreviations

\( A_{IF} \) :

Specific area of the porous insert, \( 4(1 - \varepsilon )/d_{w} \)(m−1)

\( A_{F} \) :

Non-dimensional value of \( A_{IF} \) namely \( A_{IF} R \)

\( Bi \) :

Biot number for heat loss to the ambient \( hR/k_{s} \)

\( d_{h} \) :

Hydraulic diameter of the porous insert \( \varepsilon \,d_{w} /(1 - \varepsilon ) \) (m)

\( d_{w} \) :

Mesh wire diameter (m)

\( Da \) :

Darcy number, \( K/R^{2} \)

\( E \) :

Regenerator effectiveness , Eq. (6.44)

\( F_{\varepsilon } \) :

Ergun coefficient

\( f_{j} \) :

jth harmonic of friction factor equal to \( \left| {P_{j} } \right| \)

\( f_{eq} \) :

Total equivalent friction factor , Eq. (6.39)

\( h \) :

Heat transfer coefficient between the regenerator tube and the ambient (W/m2-K)

\( k \) :

Thermal conductivity (W/m-K)

\( K \) :

Permeability (m2)

\( Nu \) :

Fluid-to-solid Nusselt number at the scale of the mesh wire

\( p \) :

Non-dimensional fluid pressure scaled with \( \rho W^{2} \)

\( P_{j} \) :

Pressure gradient for the jth harmonic scaled by \( \rho W^{2} /R \)

\( \Pr \) :

Prandtl number

\( r \) :

Non-dimensional radial coordinate scaled with R

\( R \) :

Tube radius (m)

\( t \) :

Non-dimensional time scaled with \( R/W \)

\( T \) :

Temperature (K)

\( w \) :

Non-dimensional axial velocity scaled with W

\( W \) :

Velocity amplitude of pulsing flow (m/s)

\( \overline{w} \) :

Cross-sectional mean velocity scaled with W

\( z \) :

Non-dimensional axial distance scaled with R

\( \alpha_{f} \) :

Thermal diffusivity of the fluid (m2/s)

\( \beta \) :

Fluid-to-solid specific heat ratio

\( \Gamma \) :

Fluid-to-solid effective thermal conductivity ratio

\( \Delta T \) :

Reference temperature difference \( T_{H} - T_{L}\,\, \text{(K)}\)

\( \lambda \) :

Fluid-to-solid thermal conductivity ratio

ε :

Porosity

\( \varphi_{j} \) :

Velocity amplitude of jth harmonic scaled by W

\( \mu \) :

Dynamic viscosity of the working fluid (Pa-s)

\( \rho \) :

Fluid density (kg/m3)

\( \theta \) :

Non-dimensional temperature \( (T - T_{L} )/(T_{H} - T_{L} ) \)

\( \tau_{c} \) :

Thermal time constant of the porous insert (s)

\( \omega \) :

Pulsing frequency of the flow scaled with \( \omega_{c} \)

\( \omega_{c} \) :

Characteristic frequency \( \pi /\tau_{c} \)(s−1)

\( f \) :

Fluid phase

\( H \) :

Hot half of the oscillation cycle

\( j \) :

jth harmonic

L :

Cold half of the oscillation cycle

\( R \) :

Quantity scaled with R as the length scale

\( w \) :

Quantity scaled with \( d_{w} \) as the length scale

\( s \) :

Solid phase

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

    Malay K. Das

  2. School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA

    Partha P. Mukherjee

  3. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

    K. Muralidhar

Authors
  1. Malay K. Das
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  2. Partha P. Mukherjee
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  3. K. Muralidhar
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Correspondence to Malay K. Das .

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Das, M.K., Mukherjee, P.P., Muralidhar, K. (2018). Oscillatory Flow in a Mesh-Type Regenerator. In: Modeling Transport Phenomena in Porous Media with Applications. Mechanical Engineering Series. Springer, Cham. https://doi.org/10.1007/978-3-319-69866-3_6

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  • DOI: https://doi.org/10.1007/978-3-319-69866-3_6

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