Chapter 2 - In depth

# 2.2.2 Dimensionless numbers

In fluid flows (gas or liquid), dimensionless numbers are used to characterize the flow regime (e.g. laminar vs turbulent, viscous vs inertial, conductive vs convective heat transfer). These numbers typically indicate a ratio of the magnitudes of forces acting on a fluid. For specific flow regimes, relations exist to determine heat transfer coefficients, typically via the Nusselt number.

## Re (Reynolds)

The Reynolds number is the ratio of inertial to viscous forces within a fluid. At very low Reynolds number (Re<<1), a flow is completely dominated by its viscosity, often called creeping or Stokes flow. Such low Reynolds numbers are not encountered in gas flows.

At larger Reynolds numbers (Re>1), inertial effects due to velocity and acceleration of the fluid play a role. Up to some threshold, the flow is laminar. Above a transition regime (typically 2000<Re<3000 for pipe flow, but strongly dependent on geometry), the flow will transition into turbulence. The transition into turbulence strongly increases the convective heat transfer coefficient from a solid surface, due to the increased mixing in the fluid.

The Reynolds number is defined as

$$Re = \frac{u L}{\nu}$$

Where u is fluid velocity (m/s), ν is the kinetic viscosity (m2/s) and L is a characteristic length scale, such as pipe diameter or gap size (m).

## Pr (Prandtl)

In fluid flow along the wall of a solid body, we may consider two types of “boundary layers”. The thermal boundary layer is the depth (or, distance to the wall) to which the fluid is affected by the heat flow to or from the wall. It can be seen as the thermal penetration depth. It depends on the thermal diffusivity of the fluid.  Likewise, the velocity boundary layer is the depth to which the effect of the wall can be seen in the velocity profile of the fluid flow. The velocity boundary layer is a function by the fluid’s viscosity.  The ratio of the two boundary layers is thus called the Prandtl number, and is purely a function of the fluid properties, and not the specific flow conditions. The ratio is indicative of the relative roles of conduction (away from the wall and into the bulk of the fluid) and convection (in the flow direction) in a flowing fluid.

In low Pr fluids, heat is quickly conducted away from the wall, into the bulk of the fluid stream, where fluid velocity is fairly uniform. In high Pr fluids the velocity boundary layer is thicker, and heat is not conducted into the free stream velocity area of the fluid. Heat is transported in the velocity boundary layer, not in the bulk of the fluid stream. For most gases, Pr is around unity. This means that the both conductive and convective heat diffusion play a role in the heat transfer in a flowing gas.

The Prandtl number is defined as:

$$Pr = \frac{\nu}{\alpha}$$

Where ν is the kinetic viscosity (m2/s) and α is the thermal diffusivity (m2/s).

## Nu (Nusselt)

The Nusselt number is defined as the ratio of convective to conductive heat transfer from a boundary. Typically, this number is used in reverse with respect to the other numbers listed here. That is: for many flow regimes, empirical relations exist to determine the Nusselt number of the flow. Those relations are often given in terms of the other dimensionless numbers listed here. Then, the definition of the Nusselt number is used to arrive at a value for the convective heat transfer coefficient.

The Nusselt number is defined as:

$$Nu = \frac{h L}{k}$$

Where h is the convective heat transfer coefficient (W/m2K), L is a characteristic length (m), and k is the thermal conductivity of the fluid (W/mK).

## Gr (Grashof)

The Grashof number is the ratio of buoyancy to viscous forces in natural convection. Larger Gr means a stronger natural convection. There is a transition to turbulent flow in the range 108<Gr<109.

The Grashof number is defined as:

$$Gr = \frac{g \beta (T_{w} – T_{\infty}) y^3}{\nu^2}$$

Where g is gravitational acceleration (9.81 m/s2 on earth) β the fluid’s thermal expansion coefficient (1/K), Tw and T  are the wall and fluid bulk temperatures (K), y is the vertical length scale (m) and ν is the kinetic viscosity (m2/s).

## Ra (Rayleigh)

The Rayleigh number is important in natural convection flows. It characterizes the flow regime for natural convection, similar to the Reynolds number. Very low Ra indicates natural convection is insignificant (conductive heat transfer is dominant). Following that is a laminar convective flow regime, followed by a turbulent convective flow regime at Ra>1013 (very roughly).  It plays an important role in empirical relations for the value Nu, in which the relevant relations for different regimes are based on the value of Ra.

The Rayleigh number is defined as the product of Gr and Pr:

$$Ra = Gr Pr = \frac{g \beta (T_{w} – T_{\infty}) y^3}{\alpha \nu}$$

Where Gr and Pr are the Grashof and Prandtl numbers. The other variables are explained in the descriptions of Gr and Pr.

## Gz (Graetz)

The Graetz number relates to internal forced convection (e.g. pipe flow) and often appears in empirical Nusselt relations regarding these flows. It also can be used to determine if a flow is thermally fully developed. That is, the thermal boundary layer is developed and no longer changes downsteam. Thermally developed flows have Gz>1000.

The Graetz number is defined as:

$$Gz = \frac{D}{L} Re Pr$$

Where D is pipe diameter (m) and L is length (m). Re and Pr are the Reynolds and Prandtl numbers.