Fully-developed flow and heat transfer

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Constant Wall Heat Flux

In this article, we consider the case of fully developed laminar flow and constant properties in a circular tube with a fully developed temperature and concentration profiles. We first consider the case of constant heat rate per unit surface area for steady, laminar, fully developed flow. The energy equation in a circular tube, by neglecting axial heat conduction and viscous dissipation terms, is:

u\frac{\partial T}{\partial x}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]

(1)

For a fully developed flow with constant wall heat flux, eq. (18) in Internal Forced Convection basics can be substituted into the above equation to obtain

u\frac{d{{T}_{m}}}{dx}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]

(2)

The boundary conditions are

\begin{align}& -k\frac{\partial T}{\partial r}={{{{q}''}}_{w}}\begin{matrix}{} & {}  \\\end{matrix}\text{at}\ r={{r}_{o}} \\  & \frac{\partial T}{\partial r}=0\begin{matrix}   {} & {} & {}  \\\end{matrix}\text{at}\ r=0 \\ \end{align}

(3)

Integrating eq. (2) twice and applying the boundary conditions in eq. (3) to get the temperature distribution gives us

T={{T}_{w}}-\frac{2{{u}_{m}}}{\alpha }\frac{d{{T}_{m}}}{dx}\left( \frac{3r_{o}^{2}}{16}-\frac{{{r}^{2}}}{4}+\frac{{{r}^{4}}}{16r_{o}^{2}} \right)

(4)

Using the definition of mean temperature presented in the last section with the above profile for temperature, and assuming constant properties:

{{T}_{m}}=\frac{\int_{A}^{{}}{uTdA}}{\int_{A}^{{}}{udA}}=\frac{2\int_{0}^{{{r}_{o}}}{\pi ruTdr}}{\pi r_{o}^{2}{{u}_{m}}}

Substituting eq. (4) into the above expression yields:

{{T}_{m}}={{T}_{w}}-\frac{11}{96}\left( \frac{2{{u}_{m}}}{\alpha } \right)\frac{d{{T}_{m}}}{dx}r_{o}^{2}

(5)

The heat flux at the wall can be obtained using the above relation for Tm

{{q}_{w}}^{\prime \prime }=h\left( {{T}_{w}}-{{T}_{m}} \right)=h\left( \frac{11}{96} \right)\left( \frac{2{{u}_{m}}}{\alpha } \right)\left( \frac{d{{T}_{m}}}{dx} \right)r_{o}^{2}

(6)

The heat flux at the wall can also be calculated using eq. (4) for the temperature profile and Fourier’s law of heat conduction

{{{q}''}_{w}}={{\left. -k\frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}=\rho {{c}_{p}}{{r}_{o}}\left( \frac{{{u}_{m}}}{2} \right)\left( \frac{d{{T}_{m}}}{dx} \right)

(7)

Combining eqs. (6) and (7) and solving for the heat transfer coefficient, h, yields

\begin{matrix}{} & h=4.364k/D  \\\end{matrix}

or in terms of the Nusselt number,

\begin{matrix}{}\\\end{matrix}Nu=4.364

(8)

Constant Surface Temperature

We begin with the energy equation (equation 29 in Internal Forced Convection Basics) and neglect the effects of axial heat conduction and viscous dissipation. We already showed that for a fully developed flow and temperature profile with constant surface temperature

\frac{\partial T}{\partial x}=\frac{{{T}_{w}}-T}{{{T}_{w}}-{{T}_{m}}}\frac{d{{T}_{m}}}{dx}

The energy equation and boundary conditions, using the fully developed velocity profile (equation 24 in Internal Forced Convection Basics) and the above equation, are

2{{u}_{m}}\left( 1-\frac{{{r}^{2}}}{r_{o}^{2}} \right)\left( \frac{{{T}_{w}}-T}{{{T}_{w}}-{{T}_{m}}} \right)\frac{d{{T}_{m}}}{dx}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]

(9)

\begin{align}
  & r={{r}_{o}},\quad \quad T={{T}_{w}} \\ 
 & r=0,\quad \quad \frac{\partial T}{\partial r}=0\quad \text{or}\quad T=\text{finite} \\ 
\end{align}

(10)

The above equation and boundary conditions have been solved by various techniques in literatures, including separation of variables and infinite series. For additional detailed information the reader should refer to [1][2][3][4].

The solution for eqs. (9) and (10) in the form of an infinite series for the temperature and Nusselt number is [2]:

\frac{T-{{T}_{w}}}{{{T}_{in}}-{{T}_{w}}}=\sum\limits_{m=0}^{\infty }{{{C}_{2m}}{{\left( \frac{r}{{{r}_{o}}} \right)}^{2m}}}

(11)

where

\begin{align}
  & {{c}_{o}}=1,\quad \quad {{c}_{2}}=-\frac{1}{4}\lambda _{0}^{2}=-1.828397,\quad \quad {{c}_{2m}}=\frac{\lambda _{0}^{2}}{{{\left( 2m \right)}^{2}}}\left( {{c}_{2m-4}}-{{c}_{2m-2}} \right) \\ 
 & {{\lambda }_{0}}=2.704364 \\ 
\end{align}

The Nusselt number corresponding to the above temperature distribution is

\text{Nu}=\frac{1}{2}{{\lambda }^{2}}=3.657

(12)

The temperature slope for constant wall heat flux at the wall is higher than the temperature slope for constant surface temperature. This effect has resulted in a 16 percent increase in Nusselt number for constant wall flux versus constant wall temperature for a fully developed flow and temperature profile. The two cases of constant wall temperature and constant wall heat flux are special cases of a more general exponential heat flux boundary condition:

{{{q}''}_{w}}=A\exp \left( \frac{1}{2}n{{x}^{+}} \right)

(13)

where A and n are both constants and n can be assumed to be either a positive or negative value, and {{x}^{+}}=\frac{x/{{r}_{o}}}{\operatorname{Re}\Pr }. n = 0 corresponds to constant heat flux at the wall and n = –14.63 corresponds to constant wall temperature.

Shah and London [5] developed the following correlation, which fits the exact solution of eq. (13) within 3 percent for –51.36 < n < 100.

\begin{align}& Nu=4.3573+0.0424n-2.8368\times {{10}^{-4}}{{n}^{2}}+3.6250\times {{10}^{-6}}{{n}^{3}} \\  & -7.6497\times {{10}^{-8}}{{n}^{4}}+9.1222\times {{10}^{-10}}{{n}^{5}}-3.8446\times {{10}^{-12}}{{n}^{6}} \\ \end{align}

(14)

Fully-developed fluid flow and heat transfer in an annulus

The problem of fluid flow and heat transfer in an annulus (see Fig. 1) is also of considerable interest in various applications, including heat exchangers and heat pipes, due to an increase and flexibility in the heating and cooling of surface area. We present the result below for the case in which both the inner wall (radius ri) and outer wall (radius ro) are kept at a constant heat flux for various values of K = ri/ro. K = 0 corresponds to conventional circular tubes, and K =1 corresponds to flow between two parallel planes.

v
Figure 1: Flow and heat transfer in an annulus.


The momentum and energy equations, as well as the boundary conditions for steady, laminar, fully developed flow and temperature profile, by neglecting radial conduction of the wall, axial heat conduction in the fluid, and viscous dissipation, and assuming constant properties, are

\frac{dp}{dx}=\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)

(15)

u\frac{d{{T}_{m}}}{dx}=\alpha \left[ \frac{1}{r}\left( \frac{\partial }{\partial r}\frac{\partial T}{\partial r} \right) \right]

(16)

r={{r}_{i}},\quad \quad u=0,\quad \quad -k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{i}}}}={{q}_{i}}^{\prime \prime }

(17)

r={{r}_{o}},\quad \quad u=0,\quad \quad k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}={{q}_{o}}^{\prime \prime }

(18)

where{{q}_{i}}^{\prime \prime }and{{q}_{o}}^{\prime \prime }are the inner and outer wall heat fluxes, respectively. The velocity profile can be obtained by integrating eq. (5.53) twice and applying no slip boundary conditions at both the inner and outer walls.

\frac{u}{{{u}_{m}}}=\frac{2}{A}\left[ 1-{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}}+B\ln \frac{r}{{{r}_{o}}} \right]

(19)

where

A=1+{{K}^{2}}-B,\quad \quad B=\frac{{{K}^{2}}-1}{\ln K},\quad \quad {{u}_{m}}=\frac{\int_{{{r}_{i}}}^{{{r}_{o}}}{urdr}}{\left( {{r}_{o}}^{2}-{{r}_{i}}^{2} \right)}

The local heat transfer coefficients and Nusselt numbers are

\begin{align}
  & {{h}_{i}}=\frac{{{q}_{i}}^{\prime \prime }}{{{T}_{i}}-{{T}_{m}}},\text{  }{{h}_{o}}=\frac{{{q}_{o}}^{\prime \prime }}{{{T}_{o}}-{{T}_{m}}} \\ 
 & N{{u}_{i}}=\frac{{{h}_{i}}{{D}_{h}}}{k},\text{  }N{{u}_{o}}=\frac{{{h}_{o}}{{D}_{h}}}{k} \\ 
\end{align}

where the hydraulic diameter, Dh = 2(ro – ri). The energy eq. (5.54) can also be solved using the velocity profile given by eq. (5.57) in a similar manner as presented for a circular tube, except that it can involve lengthy algebra. Since the energy eq. (5.54) is linear and homogeneous, the principle of superposition was used to obtain the solution of the above problem as a sum of two problems. One problem is the outer wall heated uniformly and the inner wall insulated. The second problem is the inner wall heated uniformly and the outer wall insulated. The principle of superposition can be applied to linear homogeneous differential equations as long as the summation of the governing equations and boundary conditions for each subset problem will add to the original problem. Kaka? and Yucel (1974) performed a numerical solution for fully developed velocity and temperature profile. Table 5.1 shows the inner and outer wall Nusselt number for various K values obtained by Kaka? and Yucel (1974). The inner and outer wall Nusselt numbers, based on the average wall heat flux ratio, can be calculated from the results in Table 5.1 using the following equations:

N{{u}_{i}}=\frac{N{{u}_{ii}}}{1-\left( {{{{q}''}}_{o}}/{{{{q}''}}_{i}} \right){{\theta }_{i}}^{*}}

(20)

N{{u}_{o}}=\frac{N{{u}_{oo}}}{1-\left( {{{{q}''}}_{i}}/{{{{q}''}}_{o}} \right){{\theta }_{o}}^{*}}

(21)

where Nuii is defined as the inner wall Nusselt number when the inner tube is heated and outer wall is insulated. Similarly, Nuoo is the outer tube Nusselt number when the outer wall is heated and inner tube is insulated. \theta _{i}^{*}and\theta _{o}^{*}are defined as influence coefficients and are a function of K only for laminar flow.

Table 1 Nusselt number for fully developed flow and temperature profile with constant wall heat flux in an annulus, K = ri /ro [6]

K Nuii Nuoo θi* θo*
0 4.364 0
0.10 11.900 4.834 1.3835 0.0562
0.25 7.735 4.904 0.7932 0.1250
0.50 6.181 5.036 0.5288 0.2160
1.00 5.384 5.384 0.3460 0.3460

Fully-developed flow and heat transfer in tubes of noncircular cross section

The fluid and heat transfer solution for tubes of noncircular cross section with a fully developed flow and temperature profile can be obtained by solving the energy equation for the particular geometry. The results for some of the more conventional geometries obtained by Shah and London (1974)[7] are presented in Table 2. There are three different boundary conditions in this table; “H1” which refers to the circumferentially constant wall temperature and the axial constant wall heat flux boundary condition, “H2” is for both axially and circumferentially constant heat flux at the wall, and “T” represents a constant wall temperature boundary condition. For the case of a symmetrically heated straight duct having no corners and a constant peripheral curvature, e.g. parallel planes and circular pipe, H1 and H2 boundary conditions are the same.

Table 2 Solutions for Nusselt Number and Friction Coefficient for fully developed laminar flow and temperature profile for various geometries [7]
Solutions for Nusselt Number and Friction Coefficient for fully developed laminar flow and temperature profile for various geometries.
  1. Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ.
  2. 2.0 2.1 Kakaç, S., Shah, R., Aung, W., 1987, Handbook of Single-Phase Convective Heat Transfer, John Wiley, New York.
  3. Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY
  4. Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ.
  5. Shah, R. K.; London, A. L., 1978, “Laminar Flow Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Irvine, T. F. and Harnett, J.P., Eds., Academic Press, San Diego, CA.
  6. Kakaç, S., and Yucel, O., 1974, Laminar Flow Heat Transfer in an Annulus with Simultaneous Development of Velocity and Temperature Fields, Technical and Scientific Council of Turkey, TUBITAK, ISITEK No. 19, Ankara, Turkey.
  7. 7.0 7.1 Shah, R. K.; London, A. L., 1974, “Thermal Boundary Conditions and Some Solutions for Laminar Duct Flow Forced Convection,” ASME J. Heat Transfer Vol. 96, pp. 159-165.