Thermally developing laminar flow

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In the previous section, we considered problems where the velocity and temperature profile were fully developed, so that the heat transfer coefficient was constant with distance along the pipe. In this section, we consider problems in which only velocity is fully developed at the point where the heat transfer starts. Furthermore, as before, we consider two cases of constant wall temperature and wall heat flux, both by assuming uniform temperature at the inlet. Under these conditions, the heat transfer coefficient is not constant but varies along the tube. Whiteman and Drake (1980), Lyche and Bird (1956) and Blackwell (1985) studied the case of fully developed flow with thermal entry effects for non-Newtonian fluids. Sellars et al. (1956) obtained thermal entry length solutions for the case of a Newtonian fluid with constant wall temperature and fully developed flow, which are presented below. The following assumptions are made in order to obtain a closed form solution for heat transfer analysis for fully developed flow and developing temperature profile in a circular tube: 1. Incompressible Newtonian fluid 2. Laminar flow 3. Two-dimensional steady state 4. Axial heat conduction and viscous dissipation are neglected 5. Constant properties This does not mean that one cannot obtain analytical solutions when one or more of the above assumptions is valid, but the solution will be much easier by making the above assumptions. Since the fully developed velocity was already obtained in Section 5.2, we will focus on the solution of the energy equation and boundary conditions for a developing temperature profile.

Constant Wall Temperature

The dimensionless energy eq. (5.33) and boundary conditions using the above assumptions for the case of constant wall temperature are reduced to

\frac{{{u}^{+}}}{2}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\left[ \frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right) \right]

(1)

\begin{align}  & \theta \left( {{r}^{+}},0 \right)=1 \\  & \theta \left( 1,{{x}^{+}} \right)=0 \\  & \theta \left( 0,{{x}^{+}} \right)=\quad \text{finite}\quad \text{or}\quad \frac{\partial \theta }{\partial {{r}^{+}}}\left( 0,{{x}^{+}} \right)=0 \\ \end{align}

(1)

where

{{r}^{+}}=\frac{r}{{{r}_{o}}},\quad \theta =\frac{T-{{T}_{w}}}{{{T}_{in}}-{{T}_{w}}},\quad {{u}^{+}}=\frac{u}{{{u}_{m}}},\quad {{x}^{+}}=\frac{x/{{r}_{0}}}{\operatorname{Re}\Pr }

For a fully developed laminar flow, the parabolic velocity profile previously developed is applicable, i.e.,

u=2{{u}_{m}}\left( 1-\frac{{{r}^{2}}}{r_{o}^{2}} \right)\quad \text{or}\quad {{u}^{+}}=2\left( 1-{{r}^{{{+}^{2}}}} \right)

Substituting the above equation into the energy eq. (5.62), we get

\left( 1-{{r}^{{{+}^{2}}}} \right)\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{{{\partial }^{2}}\theta }{\partial {{r}^{{{+}^{2}}}}}+\frac{1}{{{r}^{+}}}\frac{\partial \theta }{\partial {{r}^{+}}}

(1)

Since the above partial differential equation is linear and homogeneous, one can apply the method of separation of variables. The separation of variables solution is assumed of the form

\theta \left( {{r}^{+}},{{x}^{+}} \right)=R\left( {{r}^{+}} \right)X\left( {{x}^{+}} \right)

(1)

The substitution of the above equation into eq. (5.64) yields two ordinary differential equations

X' + λ2X = 0

(1)

{R}''+\frac{1}{{{r}^{+}}}{R}'+{{\lambda }^{2}}R\left( 1-{{r}^{{{+}^{2}}}} \right)=0

(1)

where

{X}'=\frac{dX}{d{{x}^{+}}},\quad {X}''=\frac{{{d}^{2}}X}{d{{x}^{{{+}^{2}}}}},\quad {R}'=\frac{dR}{d{{r}^{+}}},\quad {R}''=\frac{{{d}^{2}}R}{d{{r}^{{{+}^{2}}}}}

and – λ2 is the separation constant or eigenvalue. The solution for eq. (5.66) is a simple exponential function of the form {{e}^{-{{\lambda }^{2}}{{x}^{+}}}} while the solution of eq. (5.67) is of infinite series referred to by the Sturm-Liouville theory. The solution is of the form

\theta \left( {{r}^{+}},{{x}^{+}} \right)=\sum\limits_{n=0}^{\infty }{{{c}_{n}}{{R}_{n}}\left( {{r}^{+}} \right)}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)

(1)

where λn are the eigenvalues, Rn are the eigenfunctions corresponding to eq. (5.67), and cn are constants. The local heat flux, dimensionless mean temperature, local Nusselt number and mean Nusselt number can be obtained from the following equations, using the above temperature distribution

\begin{align}  & {{q}_{w}}^{\prime \prime }=-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}=-k\frac{\left( {{T}_{w}}-{{T}_{in}} \right)}{{{r}_{o}}}{{\left. \frac{\partial \theta }{\partial {{r}^{+}}} \right|}_{{{r}^{+}}=1}} \\  & \text{       }=-\frac{2k}{{{r}_{o}}}\left( {{T}_{w}}-{{T}_{in}} \right)\sum\limits_{n=0}^{\infty }{{{G}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)} \\ \end{align}

(1)

{{\theta }_{m}}=\frac{{{T}_{m}}-{{T}_{w}}}{{{T}_{in}}-{{T}_{w}}}=8\sum\limits_{n=0}^{\infty }{{{G}_{n}}\left[ \frac{\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)}{{{\lambda }_{n}}^{2}} \right]}

(1)

N{{u}_{x}}=\frac{{{h}_{x}}\left( 2{{r}_{o}} \right)}{k}=\frac{-{{q}_{w}}^{\prime \prime }\left( 2{{r}_{o}} \right)}{\left( {{T}_{w}}-{{T}_{in}} \right)k{{\theta }_{m}}}=\frac{-2}{{{\theta }_{m}}}{{\left. \frac{\partial \theta }{\partial {{r}^{+}}} \right|}_{{{r}^{+}}=1}}=\frac{\sum\limits_{n=0}^{\infty }{{{G}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)}}{2\sum\limits_{n=0}^{\infty }{{{G}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)/{{\lambda }_{n}}^{2}}}

(1)

\text{N}{{\text{u}}_{\text{m}}}=\frac{{{{\bar{h}}}_{x}}\left( 2{{r}_{o}} \right)}{k}=\frac{1}{{{x}^{+}}}\int_{0}^{{{x}^{+}}}{N{{u}_{x}}d{{x}^{+}}=}-\frac{1}{2{{x}^{+}}}\ln \left[ 8\sum\limits_{n=0}^{\infty }{\frac{{{G}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)}{{{\lambda }_{n}}^{2}}} \right]

(1)

where

{{G}_{n}}=-\frac{1}{2}{{c}_{n}}{{{R}'}_{n}}\left( 1 \right)

The first five terms in eqs. (5.69) – (5.72) are sufficient to provide accurate solutions to the above infinite series. The eigenvalues, λn and Gn, used to calculate {{q}_{w}}^{\prime \prime } , θm, Nux and Num for the above problem are presented in Table 1.

Table 1 Eigenvalues and Eigenfunctions of a Circular Duct; Thermal Entry Effect with Fully Developed Laminar Flow and Constant Wall Temperature [1]

n λn2/2 Gn
0 3.656 0.749
1 22.31 0.544
2 56.9 0.463
3 107.6 0.414
4 174.25 0.383

Table 2 Nusselt Solution for Thermal Entry Effect of a Circular Tube for Fully Developed Laminar Flow and Constant Wall Temperature

x+ Nux Num θm
0 1
0.001 10.1 15.4 0.940
0.004 8.06 12.2 0.907
0.01 6.00 8.94 0.836
0.04 4.17 5.82 0.628
0.08 3.79 4.89 0.457
0.1 3.71 4.64 0.395
0.2 3.658 4.16 0.190
3.657 3.657 0

Table 2 provides the variations of Nux, Num and θm with distance along the tube. It can be easily observed from Table 2 that the fully developed temperature profile starts at approximately:

{{x}^{+}}=\frac{x/{{r}_{0}}}{\operatorname{Re}\Pr }=0.1

(1)

Therefore, (LT,T / D) = 0.05RePr where LT,T is the thermal entrance length for constant wall temperature The thermal entry length increases as both the Reynolds number and Prandtl number increase. A very long thermal entry length is needed for fluids with a high Prandtl number, such as oil. Therefore, care should be taken to make a fully developed temperature profile assumption for fluids with a high Prandtl number.

Constant Heat Flux at the Wall


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