1. Analogies among Mass, Heat, and Momentum Transfer

2. Analogies Heat  Mass  (sometimes) Momentum
Analogies are useful tools
An aid to understand transfer phenomena
A sound means to predict behavior of systems for which limited quantitative data are available
Since concentration and temperature are scalar quantities, analogies between mass and heat transport are more valid than those with momentum.

3. Molecular Transport Equations
RECALL:
All three of the molecular transport equations are mathematically identical.
MOMENTUM
HEAT
MASS
Newton’s law
Fourier’s law
Fick’s law

4. Analogous quantities in transport phenomena

5. Reynolds Analogy 𝜓=− 𝛿+𝐸 𝑑Γ 𝑑𝑥
The general transport equation can be written in the form
𝜓=− 𝛿+𝐸 𝑑Γ 𝑑𝑥
where ψ = flux of a property at any value of x
δ = molecular diffusivity
E = eddy diffusivity
Г = volume concentration of transferent property
Reynolds recognized these analogies and tried to make them useful in predicting mass transfer coefficients and heat transfer coefficients.
He envisioned a separation of mass, heat and momentum transport into quiescent and convective parts.

6. Turbulent diffusion equations
Reynolds recognized these analogies and tried to make them useful in predicting mass transfer coefficients and heat transfer coefficients.
He envisioned a separation of mass, heat and momentum transport into quiescent (stationary) and convective parts multiplied to the gradients.

7. Transfer coefficient for momentum
In cylindrical geometry,
𝜓=− 𝛿+𝐸 𝑑Γ 𝑑𝑟
Integrating the above equation and multiplying by A to get a rate equation,
𝜓𝐴 =−4 𝛿+ 𝐸 𝛾𝐷 ( Γ 1 − Γ )𝐴
where A = cross-sectional area perpendicular to flow
𝐸 = mean eddy diffusivity
𝛾= Γ 1 −Γ Γ 1 − Γ 0 = 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
The following slides show derivation for the transfer coefficient of momentum which is slightly different than mass and heat transfer coefficients.
The development of transfer-rate correlations for turbulent flow shows one inconsistency. The correlation of the momentum transfer rate utilizes a friction factor-Reynolds number relationship, whereas transfer coefficients are used to relate heat- and mass-transfer rates. These correlations are traditional practice in engineering. If a surface coefficient for momentum transport is written, perhaps further similarities can be discovered.

8. Transfer coefficient for momentum
𝜓𝐴 =−4 𝛿+ 𝐸 𝛾𝐷 ( Γ 1 − Γ )𝐴
The transfer coefficient is then defined as
𝜑=−4 𝛿+ 𝐸 𝛾𝐷
The development of transfer-rate correlations for turbulent flow shows one inconsistency. The correlation of the momentum transfer rate utilizes a friction factor-Reynolds number relationship, whereas transfer coefficients are used to relate heat- and mass-transfer rates. These correlations are traditional practice in engineering. If a surface coefficient for momentum transport is written, perhaps further similarities can be discovered.
Substituting and rearranging,
𝜓𝐴 =− Γ 1 − Γ 1 𝜑𝐴

9. Transfer coefficient for momentum
𝜓𝐴 =−4 𝛿+ 𝐸 𝛾𝐷 ( Γ 1 − Γ )𝐴
The transfer coefficient is then defined as
𝜑=−4 𝛿+ 𝐸 𝛾𝐷
The development of transfer-rate correlations for turbulent flow shows one inconsistency. The correlation of the momentum transfer rate utilizes a friction factor-Reynolds number relationship, whereas transfer coefficients are used to relate heat- and mass-transfer rates. These correlations are traditional practice in engineering. If a surface coefficient for momentum transport is written, perhaps further similarities can be discovered.
Substituting and rearranging,
𝜓𝐴 =− Γ 1 − Γ 1 𝜑𝐴

10. Transfer coefficient for momentum
𝜓𝐴 =− Γ 1 − Γ 1 𝜑𝐴
For momentum transfer,
𝜓=𝜏
Γ=𝜌𝑣
The development of transfer-rate correlations for turbulent flow shows one inconsistency. The correlation of the momentum transfer rate utilizes a friction factor-Reynolds number relationship, whereas transfer coefficients are used to relate heat- and mass-transfer rates. These correlations are traditional practice in engineering. If a surface coefficient for momentum transport is written, perhaps further similarities can be discovered.
𝜏=−𝜑 [𝜌 𝑣 1 −𝜌 𝑣 ]

11. Transfer coefficient for momentum
𝜏=−𝜑 [𝜌 𝑣 1 −𝜌 𝑣 ]
If we divide by 𝑣 ,
At the wall, v1 = 0 so that,
𝑓 2 = 𝜑 𝑣 = 𝜏 𝜌 𝑣 2
𝜏=𝜑 𝜌 𝑣
Recall friction factor definition: τ =f*1/2ρv^2.
𝜑= 𝜏 𝜌 𝑣

12. The Reynolds analogy For turbulent transport, For heat transfer,
For momentum transfer,
We assume that α and μ/ρ are negligible, and that 𝛼 𝑡 = 𝜖 𝑡

13. The Reynolds analogy
Dividing the momentum equation by the heat equation then gives
𝜏 𝑞 𝐴 𝑐 𝑝 𝑇 𝑖 𝑇 𝑑𝑇 = 0 𝑣 𝑎𝑣 𝑑𝑣

14. The Reynolds analogy Substituting 𝑞 𝐴 =ℎ 𝑇− 𝑇 𝑖 and 𝜏 𝑠 =𝑓 𝑣 𝑎𝑣 2 𝜌/2
The main assumption here is that the shear stress is analogous to q/A, which suggests that the ratio tau/(q/A) must be a constant for all radial positions

15. The Reynolds analogy 𝑁 𝑆𝑡 = 𝑁 𝑁𝑢 𝑁 𝑅𝑒 𝑁 𝑃𝑟 = f 2 Stanton number
The Stanton number, St, is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. The Stanton number is named after Thomas Edward Stanton (1865–1931).[1] It is used to characterize heat transfer in forced convection flows.

16. Dimensionless Groups Dim. Group Ratio Equation Prandtl, Pr Schmidt, Sc
molecular diffusivity of momentum / molecular diffusivity of heat
𝑐 𝑃 𝜇 𝑘
Schmidt, Sc
momentum diffusivity/ mass diffusivity
ν 𝐷 𝐴𝐵
Lewis, Le
thermal diffusivity/ mass diffusivity
𝛼 𝐷 𝐴𝐵
Stanton, St
heat transferred/ thermal capacity
ℎ 𝑐 𝑝 𝜌 𝑣

17. It was concluded that the Reynolds analogy is valid ONLY at NPr = 1
f 2 = h c p 𝜌 𝑣
Experimental results show that the above equation
Correlate data approximately for gases in turbulent flow
DOES NOT correlate experimental data for liquids in turbulent flow
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
Pr = molecular diffusivity of momentum / molecular diffusivity of heat

18. The Reynolds analogy In a similar manner,
we can relate mass transfer with momentum transfer
For turbulent transport
And the complete Reynolds analogy is
So if you have a gas, then you can get the heat transfer coefficient and divide it by the heat capacity to get the mass transfer coefficient.
Then if you know any of the coefficients, then we can estimate the friction factor.

19. It was concluded that the Reynolds analogy is valid ONLY at NSc = 1
f 2 = 𝑘 𝑐 ′ 𝑣
Experimental results show that the above equation
Correlate data approximately for gases in turbulent flow
DOES NOT correlate experimental data for liquids in turbulent flow
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

20. The Reynolds analogy CONCLUSIONS
At NPr = NSc = 1, the mechanisms for mass, heat, and momentum are identical
For other fluids, transfer processes differ in some manner functionally related to the Pr and Sc numbers.
Note that the Reynolds analogy assumes that the turbulent diffusivities are equal and the molecular diffusivities are negligible.
But when Pr = 1 =(μ/ρ)/α, therefore μ/ρ = α
And when Sc =1 = μ/ρ/Dab, therefore μ/ρ = Dab
And the Reynolds analogy can be written with the molecular terms

21. The Reynolds analogy Note that the Reynolds analogy assumes that
the turbulent diffusivities are equal and
the molecular diffusivities are negligible.
When are these assumptions not valid?
For other fluids, where 𝑁 𝑃𝑟 ≠ 𝑁 𝑆𝑐 ≠1
 usually the case for liquids
We CANNOT neglect molecular diffusivities
 in the boundary layer where diffusion, conduction, and viscosity are important