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Electrohydrodynamically Augmented Internal Forced Convection

  • Michal Talmor
  • Jamal Seyed-Yagoobi
Reference work entry

Abstract

Many thermal devices, such as heat exchangers and heat pipes, utilize forced convection of internal flows as their main mechanism for heat transport. This chapter addresses the fundamentals associated with internal flows by providing the mathematical model for the simplest case of forced convection – a laminar flow in a circular tube. As an example for the application of this fundamental theory, the modern topic of electrohydrodynamically driven dielectric liquids and liquid films for heat transfer in internal flows in macro- and microscales is presented.

Nomenclature

Fluid Parameters

\( \overline{v} \)

Fluid velocity vector \( \left[\frac{m}{s}\right] \)

vx , mean

Mean fluid velocity \( \left[\frac{m}{s}\right] \)

P

Pressure [Pa]

T

Temperature [K]

Tmean

Mean temperature [K]

\( \dot{m} \)

Mass flux \( \left[\frac{\mathrm{kg}}{s}\right] \)

τ

Shear stress \( \left[\frac{N}{m^2}\right] \)

Cf

Friction coefficient

RE

Reynolds number

q

Heat flux density \( \left[\frac{W}{m^2}\right] \)

NuD

Nusselt number

Electrical Parameters

\( \overline{E} \)

Electric field vector \( \left[\frac{V}{m}\right] \)

E

Electric field magnitude \( \left[\frac{V}{m}\right] \)

Φ

Electric potential [V]

\( \overline{J} \)

Current density vector \( \left[\frac{C}{m^3s}\right] \)

\( {\overline{f}}_g \)

Force density of gravity \( \left[\frac{N}{m^3}\right] \)

\( {\overline{f}}_{EHD} \)

EHD body force density \( \left[\frac{N}{m^2}\right] \)

ρe

Net charge density, \( {n}_e-{p}_e\ \left[\frac{C}{m^3}\right] \)

ne, pe

-/+ Charge densities \( \left[\frac{C}{m^3}\right] \)

n, p

Ionic species densities [m−3]

ωe

Onsager parameter

F(ωe)

Onsager function

I1

Bessel function, first kind, order one

λ

Heterocharge layer thickness [m]

ζ

Zeta potential [V]

ω

Angular frequency [s−1]

kw

Wave number [m−1]

Constants

h

Heat transfer coefficient \( \left[\frac{W}{m^2K}\right] \)

ρ

Fluid mass density \( \left[\frac{\mathrm{kg}}{m^3}\right] \)

μ

Fluid dynamic viscosity [Pa·s]

cp

Specific heat \( \left[\frac{J}{K}\right] \)

k

Thermal conductivity \( \left[\frac{W}{\mathrm{mK}}\right] \)

α

Thermal diffusivity \( \left[\frac{m^2}{s}\right] \)

ε

Electric permittivity \( \left[\frac{\mathrm{kg}}{m^3}\right] \)

σ

Electric conductivity \( \left[\frac{S}{m}\right] \)

b

Ionic mobility \( \left[\frac{m^2}{\mathrm{Vs}}\right] \)

te

Charge relaxation time [s]

n0

Equilibrium ionic density [m−3]

neq

Equilibrium charge density \( \left[\frac{C}{m^3}\right] \)

kD

Dissociation constant [m−3s−1]

kR

Recombination constant [s−1]

g

Gravitational acceleration \( \left[\frac{m}{s^2}\right] \)

Domain Parameters

t

Time [s]

r, θ, x

Cylindrical coordinates

Ac

Channel cross sectional area [m2]

D

Hydraulic Channel Diameter [m]

r0

Hydraulic channel radius [m]

\( \widehat{n} \)

Normal direction unit vector

Subscripts

Open image in new window

Initial condition or surface

Open image in new window

In the given direction

Open image in new window

Positive or negative

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Multi-Scale Heat Transfer Laboratory, Department of Mechanical EngineeringWorcester Polytechnic InstituteWorcesterUSA

Section editors and affiliations

  • Sumanta Acharya
    • 1
  1. 1.Herff College of Engineering,Department of Mechanical EngineeringThe University of MemphisMemphisUSA

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