Thermally developing laminar flow

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In this article, we consider problems in which only velocity is fully developed at the point where the heat transfer starts. Furthermore, 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 [1], Lyche and Bird [2] and Blackwell [3] studied the case of fully developed flow with thermal entry effects for non-Newtonian fluids. Sellars et al. [4] 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[5]:
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 Basics of Internal Forced Convection, we will focus on the solution of the energy equation and boundary conditions for a developing temperature profile.

Constant Wall Temperature

The dimensionless energy equation 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]


\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}



{{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. (1), 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}^{+}}}


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)


The substitution of the above equation into eq. (3) yields two ordinary differential equations

X' + λ2X = 0


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



{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) is a simple exponential function of the form {{e}^{-{{\lambda }^{2}}{{x}^{+}}}} while the solution of eq. (6) 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)


where λn are the eigenvalues, Rn are the eigenfunctions corresponding to eq. (6), 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}


{{\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]}


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}}}


\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]



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

The first five terms in eqs. (8) – (11) 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 the following table.

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

n {{\lambda }_{n}}^{2}/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

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

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



\left( {{L}_{T,T}}/D \right)=0.05\operatorname{Re}\Pr

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

The laminar fully developed flow with thermal entry length effects (developing temperature profile) for constant wall heat flux is very similar to that in the case of constant wall temperature, except that the dimensionless temperature and boundary conditions are defined as

\theta =\frac{{{T}_{in}}-T}{{{{{q}''}}_{w}}D/k}


{{{q}''}_{w}}=-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}=\text{constant}


Siegel et al. [6] solved the above problem for laminar fully developed flow using separation of variables and the Strum-Liouville theory, of which the result is presented below:

\theta ={{\theta }^{*}}\left( {{r}^{+}},{{x}^{+}} \right)+4{{x}^{+}}+{{r}^{{{+}^{2}}}}-\frac{{{r}^{{{+}^{2}}}}}{4}-\frac{7}{24}


where θ* is given below

{{\theta }^{*}}\left( {{r}^{+}},{{x}^{+}} \right)=\sum\limits_{n=1}^{\infty }{{{c}_{n}}{{R}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right)}


The eigenvalues, λn, eigenfunctions, Rn, and constants, cn, are presented in the following table.

Table 3 The local friction coefficient for laminar boundary layer flow over a wedge, with U = cxm and an impermeable wall

n λn2/2 Rn(1) cn
1 25.6796 -0.492517 0.403483
2 83.8618 0.395508 -0.175111
3 174.167 -0.345872 0.105594
4 296.536 0.314047 -0.0732804
5 450.947 -0.291252 0.0550357
6 637.387 0.273808 -0.043483
7 855.850 -0.259852 0.035597

The local Nusselt number, based on the above solution, is given below and numerical values are presented in the following table

Table 4 Local Nusselt Number for Thermal Entry Effect with Fully Developed Flow of a Circular Tube with Constant Wall Heat Flux

x+ Nux
0.0025 11.5
0.005 9.0
0.01 7.5
0.02 6.1
0.05 5.0
0.1 4.5
0.2 4.364

N{{u}_{x}}=\frac{\left( 48/11 \right)}{1+\left( 24/11 \right)\sum\limits_{n=0}^{\infty }{{{c}_{n}}\exp \left( -{{\lambda }_{n}}^{2}{{x}^{+}} \right){{R}_{n}}\left( 1 \right)}}


The thermal entrance for constant wall heat flux based on the numerical results presented in the above table is {{x}^{+}}\approx 0.05 or

\left( {{L}_{T,H}} \right)=0.05\left( \operatorname{Re}\Pr  \right)\left( D \right)


where LT,H is the thermal entry length for fully developed flow with constant wall heat flux.

The mean temperature variation can be obtained from the Nusselt number, eq. (17), using the following equation:

{{T}_{w}}-{{T}_{m}}=\frac{{{q}_{w}}^{\prime \prime }}{{{h}_{x}}}=\frac{{{q}_{w}}^{\prime \prime }D}{N{{u}_{x}}k}


The thermal entry length solutions presented above for hydrodynamically fully developed flows are based only on either constant wall temperature or constant heat flux. Although the wall temperature or heat flux may be assumed constant along the tube length in the entrance region for certain internal convection problems, there are cases where the wall temperature or heat flux varies considerably with tube length. In these cases, a decision must be made whether to assume constant wall temperature or constant heat flux, use a mean overall wall temperature or heat flux with respect to length, or attempt to take into account the effect of varying wall temperature or heat flux. If the latter is desirable, the solution can be obtained by superposing the thermal entry length solutions at infinitesimal and finite surface-temperature steps[7].


  1. Whiteman, I. R., and Drake, W. B., 1980, Trans. ASME, Vol. 80, pp. 728-732.
  2. Lyche, B. C., and Bird, R. B., 1956, “Graetz-Nusselt Problem for a Power-Law Non-Newtonian Fluid,” Chem. Eng. Sci., Vol. 6, pp. 35-41.
  3. Blackwell, B. F., 1985, “Numerical Solution of the Graetz Problem for a Bingham Plastic in Laminar Tube Flow with Constant Wall temperature,” ASME J. Heat Transfer, Vol. 107, pp. 466-468.
  4. Sellars, J.R., Tribus, M., and Klein, J.S., 1956, “Heat Transfer to Laminar Flow in a Flat Conduit –The Graetz Problem Extended,” Trans. ASME, Vol. 78, pp. 441-448.
  5. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
  6. Siegel, R., Sparrow, E.M., and Hallman, T.M., 1958, “Steady Laminar Heat Transfer in a Circular Tube with Prescribed Wall Heat Flux,” Appl. Sci. Res., Ser. A, Vol. 7, pp. 386-392.
  7. Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY