3.2 Boundary Layer

Hydrodynamic Entry Length

\[\begin{aligned} \ & \text{Laminar flow:} \qquad & \left(\frac{L}{D}\right)_{e,h} &\approx 0.05\; \text{Re} \\ \ & \text{Turbulent flow:} & \left(\frac{L}{D}\right)_{e,h} &> 10 \end{aligned}\]

Critical Reynolds Number

Critical Reynolds number approximates the location where the flow transitions from laminar to turbulent flow. \[\begin{aligned} \text{Re}_{x,c} \approx 10^5 \qquad & \text{external (flat plate) flow} \\ \text{Re}_{D,c} \approx 10^3 \qquad & \text{internal (duct) flow}\end{aligned}\]

Thermal Entry Length

For laminar flow, \[\left(\frac{L}{D}\right)_{e,t} \approx 0.05\; \text{Re}\; \text{Pr}\]

Comparing thermal boundary layer with hydrodynamic boundary layer, it can be said that if \(\text{Pr}>1\), hydrodynamic boundary layer grows more rapidly. \[\text{Pr} > 1, \quad L_{e,h} < L_{e,t} \qquad \Longrightarrow \quad \delta > \delta_T \ \text{ at any section}\] Inverse is true for \(\text{Pr}<1\).

For most gases the Prandtl number are sufficiently close to unity, and hence the hydrodynamic and thermal boundaries are of similar extent.

  • The relative thickness of the thermal boundary layer \(\delta_{t(x)}\) and velocity boundary layer \(\delta_x\) depend on the magnitude of the Prandtl number for the fluid. For fluids having Prandtl number equal to unity, such as gases, \(\delta_{t(x)} = \delta_x\).

  • The thermal boundary layer is much thicker than the velocity boundary layer for fluids having \(\text{Pr} \ll 1\), such as liquid metals, and is much thinner than velocity boundary layer for fluids having \(\text{Pr} \gg 1\).

Thermally Fully Developed Conditions

Since the existence of convective heat transfer between the surface and the fluid dictates that the fluid temperature must continue to change with \(x\), one might question whether fully developed conditions ever can be reached. The situation is certainly different from the hydrodynamic case, for which \((\partial v/\partial x)=0\) in the fully developed region. In contrast, if there is heat transfer, \((\partial T/\partial x)\) at any radius is not zero. Introducing a dimensionless temperature \((T_w-T)/(T_w-T_m)\), condition for which this ratio becomes independent of \(x\) are known to exist. Although the temperature profile \(T(r)\) continues to change with \(x\), the relative shape of the profile does not change and the flow is said to be fully developed. Instead of \((\partial T/\partial x)=0\), the condition is \[\frac{d}{dx}\left[\frac{T_w(x)-T(r,x)}{T_w(x)-T_m(x)} \right] = 0\] i.e., \((T_w-T)\) changes in the same way as \((T_w-T_m)\).

A fully developed thermally region is possible, if one of two possible surface conditions exist :

  • Uniform temperature (\(T_w\) = constant)

  • Uniform heat flux (\(q_w\) = constant)

For fully developed conditions, \(h/k\) is independent of \(x\).

Two Idealized Tube Surface Conditions

Tube surface conditions: either uniform wall temperature condition (UWT) condition, or a uniform wall heat flux condition (UWH). Refer to Fig.(8).


  • For \(T_w=\text{ constant}\), i.e., \((dT_w/dx)=0\) : Typical example for this case is if the fluid near this surface is undergoing a phase change.

  • For \(q_w=h(T_w-T_m)=\text{ constant}\) : This condition is possible for the case of tube wall getting heated electrically or, if the outer surface were uniformly irradiated.

Film Temperature: In all equations evaluate fluid properties at the film temperature (\(T_f\)) defined as the arithmetic mean of the surface and free-stream temperatures unless otherwise stated. \[T_f = \frac{T_w + T_\infty}{2}\]

Local and Average Heat Transfer Coefficients

The average heat transfer coefficient \(h\) is obtained from local coefficient \(h_x\) by integration as below: \[h = \frac{1}{L}\int_0^L h_x dx\]