Analogies among Mass, Heat, and Momentum Transfer

Analogies

Heat Mass (sometimes) Momentum

Analogies are useful tools

1. An aid to understand transfer phenomena2. A sound means to predict behavior of systems for

which limited quantitative data are available

Molecular Transport Equations

RECALL:driving forcerate of transport = resistance

( )xyx

d vdy

( c T)y pq d

A dy

* AAy AB

dcJ Ddy

MOMENTUM HEAT MASSNewton’s law Fourier’s law Fick’s law

Analogous quantities in transport phenomena

Reynolds Analogy

The general transport equation can be written in the form

𝜓=− (𝛿+𝐸 ) 𝑑 Γ𝑑𝑥where ψ = flux of a property at any value of x

δ = molecular diffusivityE = eddy diffusivityГ = volume concentration of transferent property

Turbulent diffusion equations

Transfer coefficient for momentum

Integrating the above equation and multiplying by A to get a rate equation,

where A = cross-sectional area perpendicular to flow = mean eddy diffusivity = ratio of the difference in concentration of transferent property between the wall and the mean value and the mean value of the fluid to the maximum difference between the wall and the center

D = diameter

In cylindrical geometry,

Transfer coefficient for momentum

(𝜓 𝐴 )=−4(𝛿+𝐸 )𝛾 𝐷 (Γ1−Γ ) 𝐴

The transfer coefficient is then defined as

𝜑=−4(𝛿+𝐸 )𝛾 𝐷

(𝜓 𝐴 )=−Γ 1− Γ1𝜑 𝐴

Substituting and rearranging,

Transfer coefficient for momentum

(𝜓 𝐴 )=−4(𝛿+𝐸 )𝛾 𝐷 (Γ1−Γ ) 𝐴

The transfer coefficient is then defined as

𝜑=−4(𝛿+𝐸 )𝛾 𝐷

(𝜓 𝐴 )=−Γ 1− Γ1𝜑 𝐴

Substituting and rearranging,

Transfer coefficient for momentum

(𝜓 𝐴 )=−Γ 1− Γ1𝜑 𝐴

For momentum transfer,

Γ=𝜌𝑣𝜓=𝜏

𝜏=−𝜑 [𝜌𝑣1−𝜌 𝑣]

Transfer coefficient for momentum

𝜏=−𝜑 [𝜌𝑣1−𝜌 𝑣]

At the wall, v1 = 0 so that,

𝜏=𝜑𝜌𝑣

𝜑=𝜏𝜌𝑣

If we divide by ,

𝑓2 =

𝜑𝑣 =

𝜏𝜌𝑣2

The Reynolds analogy

For turbulent transport,For heat transfer,

For momentum transfer,

We assume that α and μ/ρ are negligible, and that

The Reynolds analogy

Dividing the momentum equation by the heat equation then gives

( 𝜏𝑞𝐴 )𝑐𝑝∫𝑇 𝑖

𝑇

𝑑𝑇=∫0

𝑣𝑎𝑣

𝑑𝑣

The Reynolds analogy

Substituting and

The Reynolds analogy

Stanton number

𝑁 𝑆𝑡=𝑁 𝑁𝑢

𝑁𝑅𝑒𝑁 𝑃𝑟= f2

Dimensionless Groups

Dim. Group Ratio Equation

Prandtl, Pr molecular diffusivity of momentum / molecular diffusivity of heat

Schmidt, Sc momentum diffusivity/ mass diffusivity

Lewis, Le thermal diffusivity/ mass diffusivity

Stanton, St heat transferred/ thermal capacity

The Reynolds analogy

Experimental results show that the above equation1. Correlate data approximately for gases in turbulent flow2. DOES NOT correlate experimental data for liquids in turbulent flow3. DOES NOT correlate experimental data for any fluids in laminar flow

* 0.6 < NPr for gases < 2.5

It was concluded that the Reynolds analogy is valid ONLY at NPr = 1

f2=

hc p𝜌 𝑣

The Reynolds analogy

In a similar manner, we can relate mass transfer with momentum transfer

And the complete Reynolds analogy is

For turbulent transport

The Reynolds analogy

Experimental results show that the above equation1. Correlate data approximately for gases in turbulent flow2. DOES NOT correlate experimental data for liquids in turbulent flow3. DOES NOT correlate experimental data for any fluids in laminar flow

* NSc for gases ~ 1.0

It was concluded that the Reynolds analogy is valid ONLY at NSc = 1

f2=𝑘𝑐′

𝑣

The Reynolds analogy

CONCLUSIONS1. At NPr = NSc = 1, the mechanisms for mass, heat, and momentum are

identical2. For other fluids, transfer processes differ in some manner

functionally related to the Pr and Sc numbers.

The Reynolds analogy

Note that the Reynolds analogy assumes that 1. the turbulent diffusivities are equal and2. the molecular diffusivities are negligible.

1. For other fluids, where usually the case for liquids2. We CANNOT neglect molecular diffusivities in the boundary layer where diffusion, conduction, and viscosity are important

When are these assumptions not valid?