๐Ÿ” Search & Filter All 51 Engineering Calculators โ†’

๐Ÿงช Coupled & Numerical Systems Reference

Standard formulations, transport analogies, boundary equations, and mathematical solvers.

๐Ÿ“š Recommended Textbooks

Essential texts for engineering classrooms and industrial consulting portfolios:

TP

Transport Phenomena

Bird, Stewart, & Lightfoot (BSL)

The standard textbook for coupled transport, establishing momentum, energy, and mass balances in parallel. Crucial for understanding shell balances and diffusion profiles.

Advanced Academic / Research
HT

Fundamentals of Heat and Mass Transfer

Incropera, DeWitt, Bergman, & Lavine

Chapters 6 & 14 provide exceptional, intuitive introductions to transport boundary layer analogies, Fick's laws, and Chilton-Colburn relations.

Beginner / Intermediate Academic

๐Ÿ“„ Key Publications & Papers

Subject Reference Paper Used in Calculator
Transport Analogies Chilton, T. H., & Colburn, A. P. (1934). Mass transfer (absorption) coefficients - prediction from data on heat transfer and fluid friction. Industrial & Engineering Chemistry. Heat & Mass Analogy
Boundary Layers Lewis, W. K. (1922). The evaporation of a liquid into a gas. ASME Transactions. Introduces the Lewis relation for humidification. Heat & Mass Analogy
Transient Conduction Eigenvalues Incropera et al. Chapter 5: One-term approximation transcendental eigenvalue equations for walls, cylinders, and spheres. 1D Transient Conduction

๐Ÿ“Š Governing Equations & Analogies

Chilton-Colburn Analogy (j-factors)

Relates momentum, heat, and mass transfer in turbulent boundary layers:

$$j_H = j_D = \frac{f}{2}$$ $$j_H = St \cdot Pr^{2/3} = \frac{h}{\rho C_p U_\infty} \cdot Pr^{2/3}$$ $$j_D = St_m \cdot Sc^{2/3} = \frac{h_m}{U_\infty} \cdot Sc^{2/3}$$

Where $St$ is the thermal Stanton number, $St_m$ is the mass Stanton number, $f$ is friction factor, $Pr$ is Prandtl number, and $Sc$ is Schmidt number.

Nusselt vs. Sherwood Analogy

Boundary equations match identically between thermal and concentration domains:

$$Nu_L = C \cdot Re_L^m \cdot Pr^n \iff Sh_L = C \cdot Re_L^m \cdot Sc^n$$ $$\frac{h}{h_m} = \rho C_p \cdot Le^{1-n} \approx \rho C_p \cdot Le^{2/3}$$

Where $Sh_L = h_m L / D_{AB}$ is the Sherwood number, and $Le = Sc/Pr = \alpha/D_{AB}$ is the Lewis number.

Transient Lumped Capacitance (Biot Number)

Determines if internal conduction resistance within a solid is negligible relative to surface convection:

$$Bi = \frac{h L_c}{k_s} < 0.1$$

Where $L_c = V/A_s$ is the characteristic solid length. If $Bi < 0.1$, the solid temperature is spatially uniform, and transient heating/cooling decays exponentially: $$\frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left( - \frac{h A_s}{\rho V C_p} t \right)$$

๐Ÿง  Pedagogical & Numerical Notes

1. Ratios of Transport Coefficients

Dimensionless numbers define relative transport rates:

  • Prandtl Number ($Pr = \nu/\alpha$): Ratio of momentum diffusivity to thermal diffusivity.
  • Schmidt Number ($Sc = \nu/D_{AB}$): Ratio of momentum diffusivity to mass diffusivity.
  • Lewis Number ($Le = \alpha/D_{AB}$): Ratio of thermal diffusivity to mass diffusivity.

2. Numerical Solvers in Our Engines

Because many engineering relations are implicit (non-linear), our Fortran solvers implement robust numerical root-finding methods:

  • Newton-Raphson Method: Used for the Colebrook-White equation. Requires the derivative of the residual function: $$x_{new} = x_{old} - \frac{F(x_{old})}{F'(x_{old})}$$ Converges quadratically when starting near a root.
  • Bisection Method: Used for moist air saturation boundaries and wet-bulb temperatures. Highly stable, trapping roots in a continuously halving interval.

๐ŸŒ Academic & Online Resources

  • MIT OpenCourseWare Transport: Core lecture materials on momentum, heat, and mass transport equations.
  • COMSOL Multiphysics: Learning center articles explaining coupled fluid-thermal numerical models.