# Dimensional analysis

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## Approach

Dimensional analysis is very important to interpolate the experimental laboratory results (prototype models) to full scale system. Two criteria must be fulfilled to perform such an objective. First, dimensional similarity, in which all dimensions of the prototype to full scale system must be in the same ratio, should be fulfilled. Secondly, dynamic similarity should be met in which relevant dimensionless groups are the same between the prototype model and full scale system.

The convective heat transfer coefficient is a function of the thermal properties of the fluid, the geometric configuration, flow velocities, and driving forces. As a simple example, let us consider forced convection in a circular tube with a length L and a diameter D. The flow is assumed to be incompressible and natural convection is negligible compared with forced convection. The heat transfer coefficient can be expressed as $h = h(k,\mu ,{c_p},\rho ,U,\Delta T,D,L) \qquad \qquad(1)$

where k is the thermal conductivity of the fluid, μ is viscosity, cp is specific heat, ρ is density, U is velocity, and ΔT is the temperature difference between the fluid and tube wall. Equation (1) can also be rewritten as $F(h,k,\mu ,{c_p},\rho ,U,\Delta T,D,L) = 0 \qquad \qquad(2)$

It can be seen from eq. (2) that nine dimensional parameters are required to describe the convection problem. To determine the functions in eq. (1) or (2), it is necessary to perform a large number of experiments, which is not practical. The theory of dimensional analysis shows – as will become evident later – that it is possible to use fewer dimensionless variables to describe the convection problem. According to Buckingham’s Π theorem (Buckingham, 1914), the number of dimensionless variables required to describe the problem equals the number of primary dimensions required to describe the problem minus the number of dimensional variables. Since there are five primary dimensions or units required to describe the problem – mass M (kg), length L (m), time T (sec), temperature T (k), and heat transfer rate Q (W) – four dimensionless variables are needed to describe the problem. These dimensionless variables can be identified using Buckingham’s Π theorem and are formed from products of powers of certain original dimensional variables. Any of such dimensionless groups can be written as $\Pi = {h^a}{k^b}{\mu ^c}c_p^d{\rho ^e}{U^f}{(\Delta T)^g}{D^h}{L^i} \qquad \qquad(3)$

Substituting dimensions (units) of all variables into eq. (3) yields $\begin{array}{l} \Pi = {\left( {\frac{Q}{{{L^2}T}}} \right)^a}{\left( {\frac{Q}{{LT}}} \right)^b}{\left( {\frac{M}{{Lt}}} \right)^c}{\left( {\frac{{Qt}}{{MT}}} \right)^d}{\left( {\frac{M}{{{L^3}}}} \right)^e}{\left( {\frac{L}{t}} \right)^f}{T^g}{L^h}{L^i} \\ = {{\rm{M}}^{c - d + e}}{L^{ - 2a - b - c - 3e + f + h + i}}{t^{ - c + d - f}}{T^{ - a - b - d + g}}{{\rm{Q}}^{a + b + d}} \\ \end{array} \qquad \qquad(4)$

For Π to be dimensionless, the components of each primary dimension must be summed to zero, i.e.: $\begin{array}{l} c - d + e = 0 \\ - 2a - b - c - 3e + f + h + i = 0 \\ - c + d - f = 0 \\ - a - b - d + g = 0 \\ a + b + d = 0 \\ \end{array} \qquad \qquad(5)$

This gives a set of five equations with nine unknowns. According to linear algebra, the number of distinctive solutions of eq. (5) is four (9 − 5 = 4), which coincides with the Π theorem. In order to obtain the four distinctive solutions, we have free choices on four of the nine components. If we select a = d = i = 0 and f = 1, the solutions of eq. (5) become b = 0, c = − 1, e = 1, g = 0, and h = 1 which give us the first nondimensional variable ${\Pi _1} = \frac{{\rho UD}}{\mu } \qquad\qquad(6)$

which is the Reynolds number Π1 = Re.

Similarly, we can set a = 1, c = f = 0 and i = 0 and get the solutions of eq.(5) as b = − 1, d = 0, e = 0, g = 0, and h = 1. The second nondimensional variable becomes ${\Pi _{\rm{2}}} = \frac{{hD}}{k} \qquad\qquad(7)$

which is the Nusselt number (Π2 = Nu).

Following a similar procedure, we can get two other dimensionless variables: ${\Pi _{\rm{3}}} = \frac{{{c_p}\mu }}{k} \qquad \qquad(8)$ ${\Pi _{\rm{4}}} = \frac{L}{D} \qquad \qquad(9)$

which are the Prandtl number ( ${\Pi _3} = \Pr = \nu /\alpha$) and the aspect ratio of the tube.

Equation (2) can be rewritten as $F({\Pi _{\rm{1}}},{\Pi _{\rm{2}}},{\Pi _{\rm{3}}},{\Pi _{\rm{4}}}) = 0 \qquad \qquad(10)$

or $Nu = f(Re, Pr, L/D) \qquad\qquad(11)$

Furthermore, if the flow and heat transfer in the tube are fully developed, meaning no change of flow and heat transfer in the axial direction, eq. (10) can be simplified further: $Nu = f(Re, Pr) \qquad \qquad(12)$

It can be seen that the number of nondimensional variables is three, as opposed to the nine dimensional variables in eq. (1). $h = h[k,\mu ,{c_p},\rho ,L,\Delta T,({\rho _\ell } - {\rho _v})g,{h_{\ell v}},\sigma ] \qquad \qquad(13)$

There are 10 dimensional variables in eq. (13) and there are five primary dimensions in the boiling and condensation problem. Therefore, it will be necessary to use (10 − 5) = 5 dimensionless variables to describe the liquid-vapor phase change process, i.e.: $Nu = f(Gr, Ja, Pr, Bo) \qquad \qquad(14)$

where the Grashof number is defined as: $Gr = \frac{{\rho g({\rho _\ell } - {\rho _v}){L^3}}}{{{\mu ^2}}} \qquad \qquad(15)$

The new dimensionless parameters introduced in eq. (14) are the Jakob number, Ja, and the Bond number, Bo, which are defined as $Ja = \frac{{{c_p}\Delta T}}{{{h_{\ell v}}}} \qquad \qquad(16)$ $Bo = \frac{{({\rho _\ell } - {\rho _v})g{L^2}}}{\sigma } \qquad \qquad(17)$

## Dimensionless numbers

The following table provides a summary of the definitions, physical interpretations, and areas of significance of the important dimensionless numbers for transport phenomena in multiphase systems. At reduced-length scale, the effects of gravitational and inertial forces become less important while the surface tension plays a dominant role.

Summary of dimensionless numbers for transport phenomena
 Name Symbol Definition Physical interpretation Area of significance Bond number Bo $({\rho _\ell } - {\rho _v})g{L^2}/\sigma$ Buoyancy force/surface tension Boiling and condensation Brinkman number Br μU2 / (kΔT) Viscous dissipation /enthalpy change High-speed flow Capillary number Ca μU / σ We/Re Two-phase flow Eckert number Ec U2 / (cpΔT) Kinetic energy/enthalpy change High speed flow Fourier number Fo αt / L2 Dimensionless time Transient problems Froude number Fr U2 / (gL) Interfacial force/gravitational force Flow with a free surface Grashof number Gr gβΔTL3 / ν2 Buoyancy force/viscous force Natural convection Jakob number ${Ja_\ell}$ ${c_{p\ell }}\Delta T/{h_{\ell v}}$ sensible heat/latent heat Film condensation and boiling Jav ${c_{pv}}\Delta T/{h_{\ell v}}$ Kapitza number Ka $\mu _\ell ^4g/[({\rho _\ell } - {\rho _v}){\sigma ^3}]$ Surface tension/viscous force Wave on liquid film Knudsen number Kn λ / L Mean free path /characteristic length Noncontinuum flow Lewis number Le α / D Ratio between thermal and mass diffusivities Mass transfer Mach number Ma U / c Velocity/speed of sound Compressible flow Nusselt number Nu hL / k Thermal resistance of conduction/thermal resistance of convection convective heat transfer Peclet number Pe UL / α RePr Forced convection Prandtl number Pr ν / α Rate of diffusion of viscous effect/rate of diffusion of heat convection Rayleigh number Ra gβΔTL3 / (να) GrPr Natural convection Reynolds number Re UL / ν Inertial force/viscous force Forced convection Schmidt number Sc ν / D Rate of diffusion of viscous effect/rate of diffusion of mass Convective mass transfer Sherwood number Sh hmL / D12 Resistance of diffusion /resistance of convection Convective mass transfer Stanton number St h / (ρcpU) Nu/(Re Pr) Forced convection Stanton number (mass transfer) Stm hm / U Sh/(Re Sc) Mass transfer Stefan number Ste ${c_p}\Delta T/{h_{s\ell }}$ Sensible heat/latent heat Melting and solidification Strouhal number Sr Lf / U Time characteristics of fluid flow Oscillating flow /bioengineering Weber number We ρU2L / σ Inertial force/surface tension force Liquid-vapor phase change Womersley number Wr $\sqrt {\omega {R^2}/\nu }$ Radial force /viscous force Bioengineering

The convective heat transfer coefficient can be obtained analytically, numerically, or experimentally, and the results are often expressed in terms of the Nusselt number, in a fashion similar to eqs. (12) and (14).

## References

Buckingham, E., 1914, “On Physically Similar Systems: Illustrations of the Use of Dimensional Equations,” Phys. Rev., Vol. 4, pp. 345-376.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.