scholarly journals The Graetz–Nusselt problem extended to continuum flows with finite slip

2015 ◽  
Vol 764 ◽  
Author(s):  
A. Sander Haase ◽  
S. Jonathan Chapman ◽  
Peichun Amy Tsai ◽  
Detlef Lohse ◽  
Rob G. H. Lammertink
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Shear Flow ◽  
Slip Length ◽  
Wall Slip ◽  
Partial Slip ◽  
Scaling Exponent ◽  
Constant Wall Temperature ◽  
Radial Heat ◽  
Transition Points

AbstractGraetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$, i.e. $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$. Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$. For no-shear flow, ${\it\beta}=1/2$. The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$. For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$. A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$.

Convective Heat Transfer of the Flow through a Rotating Circular Straight Pipe

2001 ◽  
Vol 17 (2) ◽  
pp. 79-91
Author(s):  
U. Lei ◽  
Arthur C. Y. Yang
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Reynolds Numbers ◽  
Straight Pipe ◽  
Constant Wall Temperature ◽  
Fully Developed Flow ◽  
Flow Through ◽  
Prandtl Numbers ◽  
Scaling Analyses ◽  
Gradient Condition

ABSTRACTLaminar heat transfer for large ranges of Reynolds numbers, rotational Reynolds numbers, and Prandtl numbers are studied numerically for incompressible fully developed flow in a circular straight pipe, which is rotating constantly about an axis perpendicular to its own axis under the constant wall temperature gradient condition. There exist four types of local Nusselt number distributions associated with the four different flow regimes for different parameters depending on the relative importance of different forces. Correlations of the averaged Nusselt number are also provided. When the Prandtl number is sufficiently large, the temperature distribution in the core is determined essentially by the secondary flow. Scaling analyses are provided for understanding the essential physics of the problem.

On the Convective Heat Transfer in a Tube of Elliptic Cross Section Maintained Under Constant Wall Temperature

1986 ◽  
Vol 108 (1) ◽  
pp. 33-39 ◽  
Author(s):  
M. A. Ebadian ◽  
H. C. Topakoglu ◽  
O. A. Arnas
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Temperature Distribution ◽  
Convective Heat Transfer ◽  
Cross Section ◽  
Convective Heat ◽  
Wall Temperature ◽  
Elliptic Cross Section ◽  
Constant Wall Temperature ◽  
Elliptic Cross

The convective heat transfer problem along the portion of a tube of elliptic cross section maintained under a constant wall temperature where hydrodynamically and thermally fully developed flow conditions prevail is solved in this paper. The successive approximation method is used for the solution utilizing elliptic coordinates. Analytical expressions for temperature distribution and Nusselt number corresponding to the first cycle of approximation are obtained in terms of the ellipticity of the cross section. In the case of a circular section, the first cycle approximation of the Nusselt number is obtained as 3.7288 compared to the exact value of 3.6568. Representative temperature distribution curves are plotted and compared to those corresponding with constant wall heat flux conditions.

Heat Transfer Coefficient in Ducts With Constant Wall Temperature

1983 ◽  
Vol 105 (4) ◽  
pp. 878-883 ◽  
Author(s):  
A. Haji-Sheikh ◽  
M. Mashena ◽  
M. J. Haji-Sheikh
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Heat Transfer Coefficient ◽  
Numerical Calculation ◽  
Transfer Coefficient ◽  
Cross Sections ◽  
Wall Temperature ◽  
Constant Wall Temperature ◽  
Circular Pipes ◽  
The Heat Transfer Coefficient

An analytical method for the numerical calculation of the heat transfer coefficient in arbitrarily shaped ducts with constant wall temperature at the boundary is presented. The flow is considered to be laminar and fully developed, both thermally and hydrodynamically. The method presented herein makes use of Galerkin-type functions for computation of the Nusselt number. This method is applied to circular pipes and ducts with rectangular, isosceles triangular, and right triangular cross sections. A three-term or even a two-term solution yields accurate solutions for circular ducts. The situation is similar for right triangular ducts with two equal sides. However, for narrower ducts, a larger number of terms must be used.

Measurement and Analysis of Laminar Heat Transfer Coefficients in Micro and Mini-Scale Ducts and Channels

2014 ◽  
Author(s):  
Y. S. Muzychka ◽  
M. Ghobadi
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Reynolds Numbers ◽  
Transfer Coefficients ◽  
Constant Wall Temperature ◽  
Fully Developed Flow ◽  
Transfer Rates ◽  
Heat Transfer Coefficients ◽  
Advantages And Disadvantages ◽  
Nusselt Numbers

Heat transfer in micro and mini-scale ducts and channels is considered. In particular, issues of thermal performance are considered in systems with constant wall temperature at low to moderate Reynolds numbers or small dimensional scales which lead to conditions characteristic of thermally fully developed flows or within the transition region leading to thermally fully developed flows. An analysis of two approaches to representing experimental data is given. One using the traditional Nusselt number and another using the dimensionless mean wall flux. Both approaches offer a number of advantages and disadvantages. In particular, it is shown that while good data can be obtained which agree with predicted heat transfer rates, the same data can be problematic if one desires a Nusselt number. Other issues such as boundary conditions pertaining to measuring thermally developing and fully developed flow Nusselt numbers are also discussed in detail.

Nusselt Number as Composite Functions of Aspect Ratio and Wall Inclination in Parallelogram Microchannels with Three Types of Thermal Boundary Conditions

2016 ◽  
Vol 33 (4) ◽  
pp. 521-533 ◽  
Author(s):  
T.-M. Liou ◽  
H. Wang ◽  
S.-P. Chan
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Boundary Conditions ◽  
Simple Algorithm ◽  
Working Fluid ◽  
Constant Wall Temperature ◽  
Time Saving ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
New Findings

AbstractIn this study, attention is focused on the numerical simulations of laminar fluid flow and heat transfer in straight smooth-walled parallelogram channels with various aspect ratios (α) and inclined angles (θ). The Reynolds number (Re), characterized by the channel hydraulic diameter and the working fluid of water, is fixed at 100. The examinedαandθrange from 1 to 10 and 45° to 90°, respectively. Their effects on the thermal fluid features are explored under three thermal boundary conditions: constant wall temperature (TBC), constant axial heat transfer rate with constant peripheral temperature (H1BC), and constant wall heat flux (H2BC). The SIMPLE algorithm is employed for velocity–pressure coupling with the algebraic multigrid method, while the second-order upwind scheme is utilized for spatial discretization in pressure term; the momentum and energy equations are solved with a QUICK scheme; Least Squares Cell-Based Gradient Evaluation is applied for predicting scalar values at the cell faces and for computing secondary diffusion terms and velocity derivatives. One of the new findings is that there exists a critical value ofθ= 70° below which the Nusselt number under H2BC increases with increasingαwhereas beyond which the trend reverses, a result distinct from those computed with TBC and H1BC. Moreover, TBC is found to be a time-saving alternative to H1BC. Furthermore, both Nusselt numbers under the three thermal boundary conditions and friction factor timesReare successfully and compactly correlated with α andθto offer useful reference for designing micro-cooling channels.

Heat Transfer Enhancement in Laminar Graetz and Taylor Flows Using Nanofluids

2018 ◽  
Author(s):  
K. Alrbee ◽  
Y. S. Muzychka ◽  
X. Duan
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Aluminum Oxide ◽  
Heat Transfer Enhancement ◽  
Pure Water ◽  
Flow Conditions ◽  
Transfer Enhancement ◽  
Segmented Flow ◽  
Constant Wall Temperature ◽  
Developing Flow

This paper focuses on heat transfer in mini scale tubes under laminar developing flow conditions for a constant wall temperature boundary condition. An experimental study was preformed using Aluminum Oxide nanoparticles (< 50nm) for continuous and segmented fluid streams. A two step method was employed to prepare several samples of aluminum oxide nanofluid with different concentrations 0.25, 0.5 and 1% by volume. Heat transfer enhancement in mini scale tubes (∼1 mm) was assessed using the dimensionless Graetz parameter L*, dimensionless mean wall heat flux q*, and Nusselt number Nu. In this study we investigate the effect of nanofluid concentration on laminar heat transfer enhancement in mini-scale circular tube under continuous and segmented flow using gas as a segmenting medium. The initial results show a maximum of 10–65% enhancement of Nusselt number as compared with pure water under the same conditions as a function of L*. For the upper limit of concentration of 1% Al2O3 nanofluid, the friction factor was found to be less than 5% greater, which means a small sacrifice on pumping power is to be expected. This study provides new insights on the thermal behaviour of nanofluids under laminar developing flow and segmented flow conditions in straight tubes.

scholarly journals Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels1

2001 ◽  
Vol 124 (2) ◽  
pp. 356-364 ◽  
Author(s):  
Nicolas G. Hadjiconstantinou ◽  
Olga Simek
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Heat Conduction ◽  
Flow Regime ◽  
Wall Temperature ◽  
Slip Flow ◽  
Constant Wall Temperature ◽  
Slip Flow Regime ◽  
Axial Heat ◽  
Axial Heat Conduction

We investigate the constant-wall-temperature convective heat-transfer characteristics of a model gaseous flow in two-dimensional micro and nano-channels under hydrodynamically and thermally fully developed conditions. Our investigation covers both the slip-flow regime 0⩽Kn⩽0.1, and most of the transition regime 0.1<Kn⩽10, where Kn, the Knudsen number, is defined as the ratio between the molecular mean free path and the channel height. We use slip-flow theory in the presence of axial heat conduction to calculate the Nusselt number in the range 0⩽Kn⩽0.2, and a stochastic molecular simulation technique known as the direct simulation Monte Carlo (DSMC) to calculate the Nusselt number in the range 0.02<Kn<2. Inclusion of the effects of axial heat conduction in the continuum model is necessary since small-scale internal flows are typically characterized by finite Peclet numbers. Our results show that the slip-flow prediction is in good agreement with the DSMC results for Kn⩽0.1, but also remains a good approximation beyond its expected range of applicability. We also show that the Nusselt number decreases monotonically with increasing Knudsen number in the fully accommodating case, both in the slip-flow and transition regimes. In the slip-flow regime, axial heat conduction is found to increase the Nusselt number; this effect is largest at Kn=0 and is of the order of 10 percent. Qualitatively similar results are obtained for slip-flow heat transfer in circular tubes.

Slip-Flow Constant-Wall-Temperature Nusselt Number in Circular Tubes in the Presence of Axial Heat Conduction

2001 ◽  
Author(s):  
Olga Simek ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Heat Conduction ◽  
Wall Temperature ◽  
Slip Flow ◽  
Small Scale ◽  
Constant Wall Temperature ◽  
Circular Tubes ◽  
Axial Heat ◽  
Axial Heat Conduction

Abstract We present an investigation of slip-flow constant-wall-temperature convective heat transfer in circular tubes under hydrodynamically and thermally fully developed conditions. Our analysis includes the contribution of axial heat conduction (finite Peclet number) which is important in small scale flows, and has not been included in previous investigations of slip-flow heat transfer. The Nusselt number is found to decrease with increasing Knudsen number for all Peclet numbers in the fully accommodating case, as expected. The effect of axial heat conduction is found to be most important at Kn = 0, and results in an increase in the Nusselt number of the order of 15%; as Kn increases, the effect of axial heat conduction decreases.

Thermally Developing Laminar Liquid Flow and Heat Transfer in Microtubes at Slip Regime

2020 ◽  
Author(s):  
Wenchi Gong ◽  
Jun Shen ◽  
Wei Dai
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Drag Reduction ◽  
Slip Length ◽  
Entrance Length ◽  
Transfer Enhancement ◽  
Slip Boundary ◽  
Slip Regime ◽  
Flow And Heat Transfer ◽  
Thermal Entrance

Abstract Heat transfer enhancement is usually accompanied by an increase in pressure drop. With the implementation of the various drag reduction methods, the researches and applications of DR methods in heat transfer enhancement have attracted more and more attention. The research on drag reduction by introducing superhydrophobic surface shows that the slip regime plays an important role in drag reduction. This study numerically investigated thermally developing laminar liquid flow and heat transfer in microtubes at slip regime, with a hydraulic diameter of 200 μm, a constant heat flux of 105 W/m2, a Re of 100 and a slip length ls ranging from 2 μm to 20 μm. The dimensionless thermal entrance length increases with the increased slip length. The results show that microtube with slip boundary has the larger local Nusselt number and a longer thermal entrance length compared with available experimental data of non-slip boundary. Local and average Nusselt numbers are obtained, and start high and rapidly decrease with the increased dimensionless axial distance. Meanwhile, Nusselt number increases with the increased slip length. The correlations of the dimensionless thermal entrance length and local Nusselt number have mean absolute relative deviation of no more than 0.12% and 1.53% respectively, which can be used to optimize microchannel heat sinks.

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