Basics of internal forced convection

It is important to clarify some basic definitions, terminologies and criteria that are often used in internal convective heat and mass transfer. These include:
1. Mean velocity, temperature, and concentration
2. Fully developed flow, temperature, and concentration profiles
3. Hydrodynamic, thermal, and concentration entrance lengths
Figure 5.1(a) shows the development of a velocity profile inside a duct or tube with uniform inlet velocity for laminar flow of an incompressible Newtonian fluid. The velocity profile at some distance away from the tube’s inlet no longer changes along the flow direction, where it is referred to as the fully developed flow condition. The fully developed condition is often met at some distance away from the inlet. However, there are also applications in which fully developed flow is never reached. Momentum, thermal, and concentration boundary layers form on the inside surface of the tube. The thickness of the layers increases in a similar manner as boundary layer flow over a flat plate (which was presented in detail in Chapter 4). Figure 5.1(a) shows how the momentum boundary layer builds up in a pipe along the flow direction. At some distance away from the inlet, the boundary layer fills the flow area. The flow downstream from this point is referred to as fully developed flow since the velocity slope does not change after this point. The distance downstream from the inlet to where the flow becomes fully developed is called the hydrodynamic entrance length. If the flow is laminar (Re < 2300 for flow inside circular tubes), the fully developed velocity is a parabolic shape. It should be noted that the fluid velocity outside the boundary layer increases with x, which is required to satisfy the conservation of mass (or continuity) equation.

The center line velocity finally reaches a value two times the inlet velocity, uin, for fully developed, steady, incompressible, laminar flow inside tubes. It should be noted that the hydrodynamic entrance length for fully developed flow does not start from the point where the friction coefficient,

 $c_{f}=\frac{\tau _{w}}{\rho u_{in}^{2}/2}$ (1)

does not change along the flow. The friction coefficient variation for laminar flow inside a circular tube with uniform inlet velocity is shown in Fig. 5.1(b). The friction coefficient is highest at the entrance and then decreases smoothly to a constant value, corresponding to fully developed flow. Two factors cause the friction coefficient to be higher in the entrance region of tubes than in the fully developed region. The first factor is the larger velocity gradient at the entrance on the wall. The gradient decreases along the pipe and becomes constant before the velocity becomes fully developed. The second factor is the velocity outside the boundary layer, which must increase to satisfy the conservation of mass or continuity equation. Accelerating velocity in the core produces an additional drag force when its effect is considered in the friction coefficient. The turbulent velocity profile and friction coefficient variation for a circular pipe are shown in Fig. 5.2. Even for a very high inlet velocity, the boundary layer will be laminar over a part of the entrance. This transition from laminar to turbulent is clearly shown by the sudden increase in momentum boundary layer thickness as shown in Fig. 5.2(a). The friction coefficient variation for turbulent flow in a pipe entrance is shown in Fig. 5.2(b). The hydrodynamic entry length required for fully developed flow should be obtained by a complete solution of the flow and thermal field in the entrance region. A rule of thumb to judge whether or not the flow is fully developed for circular pipes is

 $\frac{L_{H}}{D}\ge 0.05\operatorname{Re}\text{ for laminar flow}$ (1)
 $\frac{L_{H}}{D}\ge 0.625\operatorname{Re}^{0.25}\text{ for turbulent flow}$ (1)

where LH is the hydrodynamic length and the Reynolds number is defined by$\operatorname{Re}=\frac{u_{m}D}{\nu }$

A similar behavior is expected for thermal cases with the thermal boundary layer growth at the entrance of a tube as shown in Figure 5.3, which corresponds to a case where there may be an unheated length in which the velocity is fully developed before heating starts. One expects that the thermal boundary layer increases in the thermal entry region before the heat transfer coefficient becomes constant. It should be noted that the requirement for a fully developed thermal region is that the dimensionless temperature, θ,

$\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}$

does not change with distance along the flow direction. The mean and centerline temperatures are Tm and Tc, respectively. Similar requirements exist for the fully developed concentration profile where θ is replaced with $\varphi$,

$\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}$

where the mean and centerline concentrations (or mass fractions) are cm and cc, respectively. In the subsequent sections, we use the following definitions to mathematically define the fully developed flow, temperature, and concentration profiles: Fully developed flow profile

 $\frac{u}{u_{c}}\quad \ \text{or}\quad \ \frac{u}{u_{m}}=f\left( \frac{r}{r_{o}} \right)$ (1)

Fully developed temperature profile

 $\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}=g\left( \frac{r}{r_{o}} \right)$ (1)

Fully developed concentration profile

 $\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}=h\left( \frac{r}{r_{o}} \right)$ (1)

We can now define the local heat and mass transfer coefficients (h and hm) based on the mean temperature or concentration.

 ${q}''_{w}=h\left( T_{w}-T_{m} \right)=-\left. k\frac{\partial T}{\partial r} \right|_{r=r_{o}}$ (1)
 $\dot{{m}''}_{w}=h_{m}\left( \omega _{w}-\omega _{m} \right)=\left. -\rho D\frac{\partial \omega }{\partial r} \right|_{r=r_{o}}$ (1)

where D is mass diffusivity. Since we define the fully developed temperature profile as when the non-dimensional temperature profile (TwT) / (TwTm)

is invariant in the flow direction (x-direction), we can write the following equation:

 $\left. \frac{\partial }{\partial r}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right) \right|_{r=r_{o}}=\text{constant}=\frac{\left. -\frac{\partial T}{\partial r} \right|_{r=r_{o}}}{T_{w}-T_{m}}=\frac{h}{k}=\text{constant}$ (1)

The above conclusion, that the local heat transfer coefficient is constant along the flow direction for a fully developed temperature profile, is only valid for constant wall heat flux or constant wall temperature conditions. The requirement for the dimensionless temperature to be invariant for a fully developed temperature profile can also be presented as

 $\frac{\partial }{\partial x}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)=0$ (1)

Differentiating the above equation yields

 $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}-\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{w}}{dx}+\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ (1)

In external flow, the heat and mass transfer coefficients are usually defined by a driving differential (Tw - T∞) or (ωw – ω∞) where T∞ and ω∞ are the temperature and mass fraction of the fluid in the free stream (far away from the wall). In most cases, T∞ and ω∞ are known and constant for external flows. However, in internal flow configurations, there is not usually a well-defined temperature or concentration (mass fraction), except at the inlet and/or the boundaries. In internal flow, the temperature and concentrations may change both in the axial direction and perpendicular to the flow direction. Therefore, there are several choices available for the driving differential for temperature and concentration in internal flow. The most common choice for defining the driving temperature or concentration is based on mean temperature or concentration (mass fraction or mass density). The mixed mean fluid temperature or concentration is defined at a given local axial location based on the convective thermal energy or mass balance, i.e.,

 $T_{m}=\frac{1}{Au_{m}\rho _{m}c_{p,m}}\int_{A}^{{}}{uT\rho c_{p}}dA$ (1)
 $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{u\rho _{A}}dA$ (1)

where ρA,m

is the mean mass density for a given component A, and


ρm

is the mean density for the fluid where the mean velocity is defined as

 $u_{m}=\frac{1}{A\rho _{m}}\int_{A}^{{}}{u\rho dA}$ (1)

Assuming constant properties for mean velocity, temperature and mass density in the above equation, we obtain

 $u_{m}=\frac{1}{A}\int_{A}^{{}}{udA}$ (1)
 $T_{m}=\frac{1}{Au_{m}}\int_{A}^{{}}{uTdA}$ (1)
 $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{\rho _{A}udA}$ (1)

We will now focus our attention on two special conventional boundary conditions; constant wall heat flux and constant surface temperature. First, consider the constant heat flux or heat rate at the wall, which occurs in many applications such as electronic cooling, electric resistance heating, and radiant heating. From eq. (5.9), since h and q''w

are constant, we can conclude Tw – Tm = constant. Differentiating leads to