Uniform Wall Temperature Case Research Articles

Laminar Flow Heat Transfer Studies in a Twisted Square Duct for Constant Wall Temperature

Abstract: Three dimensional analysis of steady fully developed laminar flow inside twisted duct of square cross section flow area is studied for Reynolds number range of 100 – 2000 using a commercially available software. Twist ratio used are 2.5, 5, 10 and 20. Heat transfer results are generated for a uniform wall temperature case for Prandtl number range of 0.7 to 20. Maximum values for the product of friction factor and Reynolds number is observed for a twist ratio of 2.5 and Reynold number of 2000. Maximum Nusselt number is observed for the same values along with Prandtl number of 20. Correlations for friction factor and Nusselt number are developed involving swirl parameter. Local distribution of friction factor ratio and Nusselt number across a cross-section is presented. Heat transfer enhancement for Reynold number of 2000 and Prandtl number of 0.7 for twist ratio of 2.5, 5, 10, and 20 are 33 %, 18 %, 10 % and 7 % respectively. The results are significant because it will contribute to development of compact heat exchanger.

Numerical analysis of heat transfer and fluid flow through the twisted square duct (TSD): Nanofluid as working fluid

Friction factor and heat transfer characteristics of Al2O3/water nano-particle inside the twisted square duct (TSD) using water as base fluid numerically investigated. A uniform heat flux applied on the external walls. Three-dimensional (3D) analysis of steady fully developed laminar flow inside the twisted duct of square cross-section flow area is carried out for Reynolds number and nanoparticle concentration range of 600–2000 and 0.5 %–1 % using the commercially available software. Heat transfer and friction factor characteristics of fluid flowing through twisted square duct using water/Al2O3 as normal fluid are compared with the results of the straight square duct using water as working fluid. The numerical study is conducted for a uniform wall temperature case, twist ratio of 7.5, 11.5 and 16.5, for Prandtl number range of 8 to 10. One of the key outcomes of this work is up to a limiting concentration of nanoparticle in the base fluid beyond which it decreases. It is also found that the Nusselt number increases by decreasing the twist ratio (tr) and increasing the Prandtl number. However, in the case of friction factor (f), a regular decrease in friction factor is observed with increasing Reynolds number (Re).

Analysis of heat transfer and flow resistance of twisted oval tube in low Reynolds number flow

Heat transfer enhancement of twisted oval tube is mainly concentrated in the high Reynolds number region. In the present study, the low Reynolds k-ε model is employed to investigate heat transfer and flow characteristics of water flow inside twisted oval tube (TOT) for Reynolds number in the range of 50–2000. Three dimensional numerical study is conducted to study the effects of the geometric parameters on the performance of twisted oval tube for a uniform wall temperature case. The flattening of 1.2, 1.4, 1.63, 1.8 and 2.0, and the twisted pitch ratio of 0.17, 0.25, 0.33 and 0.5. Local distributions of Nusselt number and friction factor are presented. The filed synergy principle is applied to reveal heat transfer enhancement mechanism. The results show that the heat transfer performance of twisted oval tube has been enhanced while having an increasing of pressure drop. One of the key findings of this study is that laminar to turbulent flow transition point was identified and located at Re=500. The fluid is in laminar states with Reynolds number range of 50–250, while the fluid is in turbulent flow when the Reynolds comes to 500–2000. It is also found that the twisted oval tube performs well compared with the smooth tube due to the effect of secondary flow. The maximum enhancements factor PEC of 1.7 is obtained with flattening of 2.0, twisted pitch of 0.33 and Reynolds number of 350. Moreover, PEC of twisted oval tube increased with the increasing of flattening, but decreased with the increasing of twisted pitch ratio. These results would assist people in a comprehensive understanding of heat transfer performance of twisted oval tube in low Reynolds number flow, but also help in the design and development of compact heat exchanger.

Experimental and numerical study of fluid flow and heat transfer in a twisted square duct

Heat transfer and friction factor characteristics of air flow inside twisted square duct are studied experimentally and through three-dimensional numerical simulations. Experiments were conducted for air with uniform wall temperature boundary condition, twist ratio of 11.5 and 16.5, and Reynolds number 600–70,000. Laminar to turbulent flow transition point was identified. The results show considerable enhancement in heat transfer and pressure drop in both laminar and also in turbulent flow regimes till Reynolds number of 9500. Twist ratio of 11.5 shows relatively higher heat transfer and pressure drop compared to straight square duct. Three-dimensional analysis of steady fully-developed laminar flow inside twisted duct of square cross section flow area is carried out for Reynolds number range of 100–100,000 using commercially available software. The numerical study is conducted for a uniform wall temperature case, twist ratio of 2.5, 5, 10 and 20, for Prandtl number range of 0.7 to 20. The maximum value for product of friction factor and Reynolds number is observed for a twist ratio of 2.5 and a Reynolds number of 3000. The maximum Nusselt number is observed for the same values for Prandtl number of 20. Correlations for friction factor and Nusselt number are developed involving swirl parameter in the laminar flow regime. Correlations are also provided for the entire range of turbulent regime. Local distribution of friction factor ratio and Nusselt number across a cross-section is presented. Based on constant pumping power criteria, enhancement factor is defined to compare twisted ducts with straight ducts. Selections of twisted square duct are presented in terms of enhancement factor. It is found that twisted duct performs well in laminar and also to some extent in turbulent flow regime due to strong presence of secondary flow. It is recommended to use twisted square duct in laminar flow regime for entire range of Prandtl number studied. Maximum enhancements factor of 10.5 is obtained with twist ratio of 2.5, Prandtl number of 20 and Reynolds number of 3000. Guidelines are provided for selection of twisted square duct in terms of Reynolds number and Prandtl number. Comparison with twisted elliptic tube and twisted tape are discussed. These results would help in the design and development of compact heat exchanger.

EFFECTS OF WALL TRANSPIRATION ON MIXED CONVECTION HEAT TRANSFER IN RECTANGULAR DUCTS ROTATING ABOUT A PARALLEL AXIS

A detailed numerical study, using the vorticity-velocity method, has been conducted to examine the flow and heat transfer characteristics of wall transpiration on mixed convection in the entrance region of rectangular ducts rotating about an axis parallel to the duct axis. Thermal boundary conditions of uniform heat flux (VHF) and uniform wall temperature (UWT) were considered. The investigation was presented for various parameters, wall Reynolds number Rew, rotational Reynolds number J, rotational Grashof number GrΩ, and aspect ratio γ. The predicted results disclosed that the wall transpiration effect has considerable impact on the flow and heat transfer characteristics. Results also showed that both ƒ Re and Nu are enhanced with an increase in J or GrΩ, except for the range of J < 400 or GrΩ< 1000. Wall transpiration effects are more significant for the UWT case than for the UHF case. It is also noted that the secondary vortex induced by the buoyancy effect in the entrance region was stronger for the UWT case than for the UHF case. In addition, the strength of the secondary vortex is weaker for the case of the parabolic velocity condition at the inlet than for the uniform velocity condition at the inlet.

Mixed Convection in Non-Newtonian Fluids Along an Isothermal Vertical Cylinder in a Porous Medium

Mixed convection in power-law type non-Newtonian fluids along an isothermal vertical cylinder in porous media is studied. The problem is solved by means of a finite difference method for the case of uniform wall temperature. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5–1.5.

NUMERICAL STUDY OF TRANSIENT CONJUGATED MIXED CONVECTION IN A VERTICAL PIPE

This work deals with the problem of transient conjugated mixed convection heat transfer in a vertical pipe maintained at either uniform heat flux (UHF) or uniform wall temperature (UWT). The effects of the ratio of Grashof number to Reynolds number Gr/Re, conductivity ratio K, diffusivity ratio A, and dimensionless wall thickness [Delta] on the interfacial heat flux, velocity, and temperature profiles are discussed in detail. Three conditions of wall effects were examined, i.e., zero wall thickness (case A), effect of heat capacity (case B), and dual effects of heat capacity and wall conduction (case C). The prediction shows that the results of case C give a longer thermal larger than those of case B. Additionally, for the case of UWT, the time for the system to reach the steady state condition is indistinguishable for the system having K and A [>=] 100 or [Delta] [<=] 0.2.

The analogy between fluid friction and heat transfer of laminar mixed convection on flat plates

This paper has examined the validity of the analogies between heat transfer and fluid friction when they are applied to laminar forced convection on a flat plate. For the case of uniform wall temperature, all the analogies are valid for Prandtl number do not differ greatly from 1, and the Colburn analogy is consistent with numerical data forPr≥1. However, the previous analogies are not valid for the case of uniform wall heat flux. For fluids ofPr≥1 under uniform heating, the Colburn analogy factor is 0.72. In addition, these analogies are not applicable to the fluids of small Prandtl number. For the sake of completeness, this work has developed the analogies between heat and momentum transfer over the range of 0.001 ≤Pr ≤ 0.1 for both the cases of constant wall temperature and heat flux.

Mixed convection along an isothermal vertical cylinder in porous media

Mixed convection adjacent to a vertical cylinder in porous media is studied. The problem is solved using the nonsimilarity solution method for the case of uniform wall temperature (UWT). In the first approach, mixed convection is studied by examining separately the buoyancy effect on forced convection and the forced flow effect on free convection. These effects are represented, respectively, by RaJPex and PexlRax. Curvature is represented by gf = (xlr^Pe~* /2 in the forced-convection-dominated regime and by £„ = (x/r0)Raxl/2 in the free-convection-dominated regime. The second approach examines the entire regime of mixed convection by introducing the single parameter = [1 + (Rax/Pex)1/2]~l, where = I corresponds to pure forced convection and x = 0 to pure free convection. In this regime, the curvature is represented by £ = (xlr^[Pe*x2 + Ralx2]~*. In the analysis, the non-Darcian and thermal dispersion effects are neglected. The resulting nonsimilar systems of equations from the different approaches are solved using a finite difference method. Results are presented for temperature and velocity profiles, and local Nusselt number. Correlation equations are given for local and average Nusselt numbers in all regimes of mixed convection.

The analogy between fluid friction and heat transfer of laminar forced convection on a flat plate

This paper has examined the validity of the analogies between heat transfer and fluid friction when they are applied to laminar forced convection on a flat plate. For the case of uniform wall temperature, all the analogies are valid for Prandtl number do not differ greatly from 1, and the Colburn analogy is consistent with numerical data forPr≥1. However, the previous analogies are not valid for the case of uniform wall heat flux. For fluids ofPr≥1 under uniform heating, the Colburn analogy factor is 0.72. In addition, these analogies are not applicable to the fluids of small Prandtl number. For the sake of completeness, this work has developed the analogies between heat and momentum transfer over the range of 0.001 ≤Pr ≤ 0.1 for both the cases of constant wall temperature and heat flux.

Laminar mixed convection over horizontal flat plates with power-law variation in surface temperature

An analysis is performed to study the heat transfer characteristics of laminar mixed convective boundary-layer flow over a semi-infinite horizontal flat plate with non-uniform surface temperatures. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter χ and a pseudo-similarity variable η are introduced to cast the governing boundary layer equations and their boundary conditions into a system of dimensionless equations which are solved numerically by a weighted finite-difference method. The mixed convection parameter χ is chosen so that χ = 0 corresponds to pure free convection and χ = 1 corresponds to pure forced convection. Numerical results are presented for Prandtl numbers of 0.1, 0.7, 7 and 100 and representative values of the exponent n for the power-law variation in wall temperature. The heat transfer results are compared with existing correlations for the uniform wall temperature case and new correlations are derived for the general case of power-law wall temperature variations. It is found that an increase in the Prandtl number and exponent value n increases the local heat transfer rate.

Development of the thermal boundary layer at the inlet of a circular pipe and comparison with Lévêque solution

This paper addresses the problem of laminar flow with fully developed velocity and with developing temperature in a circular pipe with either uniform wall temperature or uniform wall heat flux. Invoking the method of lines, the applicable energy equation accounting for constant properties has been discretized in the axial direction only. As a result of this, the transformed boundary value problem represented by a single ordinary differential equation where the radius is the independent variable was reduced to a system of algebraic equations and solved numerically. This simple technique applied to the case of uniform wall temperature is parallel to that used by Lévêque, and the comparisons between both approaches are in good quantitative agreement. Predictions for the extended case of uniform wall heat flux are in accord with the exact solutions as well. Multiple discretizations of the energy equation in the axial direction are directly related to the first and the second order terms in the so-called Léêque series using perturbation analysis. Both procedures yield fairly accurate results when compared with the solutions provided by the truncated Lévêque series. The proposed methodology exhibits a peculiar analogy with the conduction heat transfer mechanism along a disk fin placed perpendicular to the fluid stream.

Mixed convection boundary layer flow on a horizontal plate in a uniform stream

A mixed convection boundary layer on a horizontal plate for uniform wall temperature/uniform heat flux is investigated using a computer extension of the perturbation series. The first 17 terms for the uniform wall temperature case and the first ten terms for the uniform heat flux case are computed for a Prandtl number σ = 0.72. The direct expansion is transformed by a Euler transform and other techniques. The results for buoyancy aiding or opposing the main flow are presented. The present work predicts the result to two digit accuracy for the entire domain of the streamwise coordinate. For uniform wall temperature, the maximum error is 5.983% for skin friction and 1.072% for heat transfer. For uniform heat flux, the maximum error is 6.9% for skin friction and 1.9% for wall temperature. La convection mixte à couche limite sur une plaque horizontale pour un flux thermique ou une température uniforme est étudiée numériquement en utilisant une série de perturbation. Les 17 premiers termes pour le cas de température pariétale uniforme et les 10 premiers termes pour le cas du flux uniforme sont calculés pour un nombre de Prandtl σ = 0,72. Le développement direct est traité par la transformée d'Euler et d'autres techniques. Les résultats sont présentés pour la convection naturelle aidant ou contrariant l'écoulement principal. Pour la température pariétale, l'erreur maximale est 5,983% pour le frottement et 1,072% pour le transfert de chaleur. Pour le flux thermique uniforme, l'erreur maximale est 6,9% pour le frottement et 1,9% pour la température de paroi. Die Grenzschichtströmung in Mischkonvektion an einer waagerechten Platte wird für die Fälle konstanter Wandtemperatur/konstanter Wärmestromdichte mit Hilfe eines erweiterten Störungsansatzes untersucht. Für eine Prandtl-Zahl σ = 0.72. werden bei konstanter Wandtemperatur die ersten 17 Terme, bei konstanter Wärmestromdichte die ersten 10 Terme berechnet. Die direkte Entwicklung wird mit Hilfe der Euler-Transformation und anderer Techniken transformiert. Ergebnisse für gleich- und gegengerichteten Auftrieb (bezüglich der Hauptströmung) werden vorgestellt. Die Genauigkeit ist im gesamten Strömungsgebiet zweistellig. Für konstante Wandtemperatur beträgt der maximale Fehler bei der Berechnung der Wand-Schubspannung 5,983%, beim Wärmeübergang 1,072%; für konstante Wärme-stromdichte 6,9% bzw. 1,9%. cMeщAннoкoнBeктиBный пoгrAничный cлoй нA гorизoнтAльнoй плAcтинe для oднoroдныч тeMпerAтyrы cтeнки и тeплoBoгo пoтoкA иccлeдyeтcя rAзлoжeниeM B rяды пo BoзMyщщeнияM c иcпoльзoBAниeM ЭBM. ПerBыe 17 члeнoB для cлyчAя oднoroднoй тeMпerAтyrы cтeнки и пerBыe 10 члeнoB для cлyчAя oднoroднoгo тeплoBoгo пoтoкA rAccчитыBAютcя для чиcлA ПrAндтля c = 0.72. ПrяMo rAзлoжeниe B rяд пroBoдитcя c пoMoщью пreoβrAзoBAния ЭйлerA и дryгиMи cпocoβAMи. ПreдcтABлeны reэyльтAты для cпyтнoгo и пroтиBoпoлoжнoгo нAпrABлeний Bтorичнoгo и ocнoBнoгo тeчeния. reэyльтAты дAны c тoчнocтью дo дByч знAчAщич цифr Bo Bceй oβлAcти изMeнeния нAпrABлeннoй Bдoль пoтoкA кoorдинAты. B cлyчAe oднoroднoй тeMпerAтyrы cтeнки MAкcиMAльнAя пoгreцнocть cocтABлялA 5.983% для пoBerчнocтнoгo тreния и 1.072% для Beличины тeплooтдAчи. B cлyчAe oднoroднoгo тeплoBoгo пoтoкA MAкcиMAльнAя пoгreщнocть rABпA 6.9% для пoBerчнocтнoгo тreния и 1.9% для тeMпerAтyrы cтeнки.

Computer extension of perturbation series for mixed convection on a vertical plate: Favourable and adverse flows

The mixed convection boundary layer on a vertical flat plate for uniform wall temperature/uniform heat flux is investigated using computer extension of perturbation series. The first thirteen terms for the uniform wall temperature case and the first ten terms for the uniform heat flux case are computed. The expansion, when transformed by Euler transform and other techniques, predicts the exact results to three-digit accuracy for all values of the streamwise coordinate, ξ, along the plate. For uniform wall temperature, the maximum error is 0.0087% for skin-friction and 0.00065% for heat-transfer. For uniform heat flux, the maximum error is 1.027% for skin-friction and 0.288% for wall temperature.

Natural convection measurements in a mercury-filled rectangular plenum

Experiments have been carried out using a rectangular plenum containing mercury, heated at the roof and one wall and cooled at the floor. The interaction between the internal stratification and the natural convection at the wall of the plenum has been measured for several different boundary conditions. Nusselt numbers at the plenum floor, when a uniform wall heat flux was applied, were found to be nearly 2; whereas a value of approximately 5 was observed for the uniform wall temperature case, indicating strong internal convection. The results confirm earlier water tests which showed strong unidimensional behaviour of the plenum temperature. Measured Nusselt numbers were found to agree well with predictions from a 1-D analysis.

WAVE INSTABILITY OF NATURAL CONVECTION FLOW ON INCLINED FLAT PLATES WITH UNIFORM SURFACE HEAT FLUX

The wave instability of laminar natural convection flow adjacent to horizontal, inclined, and vertical plates with uniform surface heat flux is analyzed by the linear theory. In the analysis the main flow is treated as nonparallel, and the disturbance thermal condition at the wall is examined by considering the thermal capacity of the plate. The eigenvalue problem consisting of a linearized system of coupled ordinary differential equations for the disturbance amplitude functions is solved by a fourth-order Runge-Kutta integration scheme, along with an orthogonalization procedure to remove the truncation errors. Neutral stability curves and critical Grashof numbers are presented for fluids having Prandtl numbers of 0.7 and 7 for inclination angles ranging from 0 to 90°. The present results are compared with those from vortex instability analysis and with those from wave instability analysis for the case of uniform wall temperature. They are also compared with previous analytical results and available experimental data for the special case of a vertical plate.

On the behaviour of the laminar boundary-layer equations of mixed convection near a point of zero skin friction

The boundary-layer equations of mixed convection are examined in the vicinity of separation. The correlation between the uniform wall temperature case and that of compressible boundary layer flow is outlined. Goldstein–Stewartson–Buckmaster theory is thus appropriate and associated indeterminacies in the theory are evaluated from a numerical integration. The case of uniform heat flux at the wall is then examined theoretically. Significantly it is concluded that the original Goldstein–Stewartson theory is sufficient to describe the structure of the singularity at separation in this case. Indeterminacies associated with the theory are determined via a reconciliation between analytical and numerical representation of skin friction and heat transfer coefficients near separation.

Mixed Convection on Inclined Surfaces

An analysis is performed to study the effects of buoyancy force on the heat transfer characteristics of laminar forced convection flow over an inclined flat surface which is either maintained at a uniform temperature or subjected to a uniform heat flux. Numerical results are presented for Prandtl numbers of 0.7 and 7 over a wide range of values of the buoyancy force parameters, with the angle of inclination ranging from 0 to 90 deg from the vertical. In general, it is found that for both surface heating conditions, the local friction factor and the local Nusselt number increase with increasing buoyancy force for assisting flow and decrease with increasing buoyancy force for opposing flow. In addition, the effects of the buoyancy force on these two quantities are found to diminish as the angle of inclination increases. A comparison is also made of the results between the case of uniform wall temperature and the case of uniform surface heat flux.

Stability of heated laminar boundary layers in water with nonuniform surface temperature

As part of a continuing study of the stability of heated laminar boundary layers in water, an experiment has been performed for the case of nonuniform surface temperature where disturbances are introduced by a vibrating ribbon and the response measured using temperature compensated hot-film anemometry. Two types of nonuniform wall temperature distributions have been studied. These are: (a) power law distributions of the form ΔT= (Tw−T∞) =Axn, and (b) step changes in wall temperature. Measured mean velocity and temperature profiles agree with solutions of the mean boundary layer equations. Measured stability characteristics for power law temperature distributions indicate that, at a fixed Reynolds number and local ΔT, both the disturbance growth rate and the band of frequencies receiving amplification are reduced relative to the uniform wall temperature case when n&amp;lt;0 and increase relative to the uniform wall temperature case when n≳0. The maximum growth rate for ΔT =2.78°C (5°F) and n=−0.5 is negative indicating damped disturbances at all frequencies. Disturbance growth rates for step changes in wall temperature at first decrease and then increase relative to those for uniform wall temperature as the step change location is moved downstream.

A solution to heat transfer in turbulent flow between parallel plates

Heat transfer in turbulent flow between parallel plates is solved by matched asymptotic expansion technique for the case of uniform wall temperature, and the Nusselt number and the thermal entry lengths are determined over a wide range of Reynolds and Prandtl numbers. Simple analytical expression is presented for the asymptotic Nusselt number. A comparison of the asymptotic Nusselt number for parallel plates in the range 0.004 < Pr < 0.1 with that obtained from a circular tube solution based on the equivalent diameter shows that circular tube results if applied for parallel plates overestimate the Nusselt number from 13 to 35% depending upon the Prandtl and Reynolds numbers.