6. Laminar and turbulent boundary layers
|
|
- Angelica Price
- 6 years ago
- Views:
Transcription
1 6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
2 6.1 Some introductory ideas Figure 6.1 A boundary layer of thickness δ Boundary layer thickness on a flat surface δ = fn(u, ρ, µ, x) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
3 6.1 Some introductory ideas The dimensional functional equation for the boundary layer thickness on a flat surface. δ x = fn(re x) ν = µ ρ :kinematic viscosity. Re x : Reynolds number. Re x ρu x µ For a flat surface, where u remains constant. δ x = 4.92 (6.2) Rex = u x ν (6.1) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
4 6.1 Some introductory ideas Figure 6.4 Boundary layer on a long, flat surface with a sharp leading edge. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
5 6.1 Some introductory ideas (u av ) crit : Transitional value of the average velocity. Re critical ρd(u av) crit µ Re xcritical = u x crit ν Transition from laminar to turbulent flow. (6.3) (6.4) Re x,c = John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
6 Thermal boundary layer The wall is at temperature T w k f ( T y y=0) = (T w T )h (6.5) Where k f is the conductivity of the fluid. The following condition defined h within the fluid instead of specifying it as known information on the boundary. ( ) T w T T w T ( ) y y L =0 = hl = Nu L (6.5a) k L f The physical significance of Nu is given by Nu L = hx k f = L δ t (6.6) The nusselt number is inversely proportional to the thickness of the thermal boundary layer δ t. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
7 Thermal boundary layer Figure 6.5 The thermal boundary layer during the flow of cool fluid over a warm plate. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
8 6.2 Laminar incompressible boundary layer on a flat surface Figure 6.7 A steady, incompressible, two-dimensional flow field represented by streamlines, or lines of constant ψ. For an incompressible flow, continuity becomes u x + v y = 0 (6.11) u = u x + v y + w z = 0 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
9 Conservation of momentum Figure 6.9 Forces acting in a two-dimensional incompressible boundary layer. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
10 Conservation of momentum The external forces are : ( τ yx + τ yx y dy ) dx τ yx dx + pdy ( p p ) ( x dx τyx dy = y p ) dxdy x The rate at which A loses x-directed momentum to its surroundings is : ) [ (ρu 2 + ρu2 x dx dy ρu 2 dy + u(ρv) + ρuv ] y dy dx ρuvdx ( ρu 2 = x + ρuv ) dxdy y John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
11 Conservation of momentum We obtain one form of the steady, two-dimensional, incompressible boundary layer momentum equation. u 2 x + uv y A second form of the momentum equation. u u x + v u y = 1 ρ = 1 dp ρ dx + v 2 u y 2 (6.12) dp dx + v 2 u y 2 (6.13) If there is no pressure gradient in the flow, if p and u are constant as they would be for flow past a flat plate, so we obtain, u 2 x + (uv) = u u y x + v u y = v 2 u y 2 (6.15) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
12 The skin friction coefficient The shear stress can be obtained by using Newton s law of viscous shear. τ w = µ u y y=0 = µu Rex x The skin local friction coefficient is defined as : C f τ w ρu /2 2 = (6.33) Rex The overall skin local friction coefficient is based on the average of the shear stress : C f = ReL (6.34) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
13 6.3 The energy equation Using Fourier s law q k T h = = T w T T w T y y=0 (6.35) Pressure variations in the flow are not large enough to affect thermodynamic properties. Density changes result only from temperature changes and will also be small (incompressible behaviour). Temperature varaitions in the flow are not large enough to change k significantly. Viscous stresses do not dissipate enough energy to warm the fluid significantly. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
14 6.3 The energy equation We write conservation of energy in the form. d ( ) ρû dr = ρĥ u n ds dt R S }{{}}{{} rate of internal energy increase in R rate of internal energy and work out of R ( k T ) n ds + q dr (6.36) S }{{} net heat conduction rate out of R For a constant pressure flow field. ρc p T t }{{} energy storage + u T }{{} enthalpy convection = R }{{} rate of heat generation in R k 2 T }{{} + q }{{} heat conduction heat generation (6.37) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
15 6.3 The energy equation In a steady two-dimensional flow field without heat sources, equation 6.37 takes the following forme. With this assumption, 2 T / x 2 2 T / y 2, so the boundary layer form is u T x + v T y = T α 2 y 2 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
16 6.4 The Prandtl number and the boundary layer thickness To look more closely at the implications of the similarity between the velocity and the thermal boundary layers. h = fn(k, x, ρ, c p, µ, u ) We can find the following number by dimension analysis. Prandtl number Pr ν α Relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers where for laminar flow. Relationship with other dimensionless number. Nu x = fn(re x, Pr) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
17 6.4 The Prandtl number and the boundary layer thickness For simple monatomic gases, Pr = 2 3. For diatomic gases in which vibration is unexcited, Pr = 5 7. As the complexity of gas molecules increase, Pr approaches an upper value of unity. For liquids composed of fairly simple molecules, excluding metals, Pr is of the order of magnitude of 1 of 10. For liquid metals, Pr is of the order of magnitude of 10 2 or less. Boundary layer thickness, δ and δ t, and the Prandtl number When Pr > 1, δ > δ t, and when Pr < 1, δ < δ t. This is because high viscosity leads to a thick velocity boundary layer, and a high thermal diffusivity should give a thick thermal boundary layer. δ ( ν ) = fn δ t α only. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
18 6.5 Heat transfer coefficient for laminar, incompressible flow over a flat surface The following equation expresses the conservation of thermal energy in integrated form. d δt u(t T )dy = q w (6.47) dx 0 ρc p Predicting the temperature distribution in the laminar thermal boundary layer T T = 1 3 y + 1 ( ) 3 y (6.50) T w T 2 δ t 2 δ t John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
19 Predicting the heat flux in the laminar boundary layer Figure 6.14 The exact and approximate Prandtl number influence on the ratio of boundary layer thicknesses. δ t δ = Pr [ 1/3 1 ( δt 2 /14δ 2)] 1 (6.54) 1/ Pr 1/3 So we can write for 0.6 Pr 50. δ t δ = Pr 1/3 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
20 Predicting the heat flux in the laminar boundary layer The following expression gives very accurate results under the assumptions on which it is based : a laminar two-dimensional boundary layer on a flat surface, with T w constant and 0.6 Pr 50 Nu x = 0.332Re 1/2 Pr 1/3 (6.58) High Pr At high Pr, equation 6.58 is still close correct. The exact solution is Nu x 0.339Rex 1/2 Pr 1/3, Pr John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
21 Some other laminar boundary layer heat transfer equations Figure 6.15 A laminar boundary layer in a low-pr liquid. The velocity boundary layer is so thin that u u in the thermal boundary layer. Low Pr In this case, δ t δ, the influence of the viscosity were removed from the problem and for all pratical purposes u = u everywhere. With a dimension analysis, we can find : Nu x = hx k John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
22 Some other laminar boundary layer heat transfer equations We can define a new dimensionless number. Peclet number Pe x Re x Pr = u x (6.61) α Peclet number can be interpreted as the ratio of heat capacity rate of fluid in the b.l. to axial heat conductance of b.l. The exact solution of the boundary layer equations gives, in this case : For Pe x 100, Pr 1 100, Re x Nu x = 0.565Pe 1/2 x (6.62) Nu L = 1.13Pe 1/2 L John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
23 Some other laminar boundary layer heat transfer equations Churchill-Ozoe correlation : For laminar flow over a flat isothermal plate for all Prandtl numbers is the following for Pe x > 100 Nu x = Re 1/2 x Pr 1/3 (1 + (0.0468/Pr) 2/3) 1/4 (6.63) And Nu x = 2Nu x John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
24 Some other laminar boundary layer heat transfer equations Boundary layer with an unheated starting length Figure 6.16 A b.l. with an unheated region at the leading edge. For laminar flow, with x > x 0 Nu x = 0.332Re1/2 x Pr 1/3 (1 (x 0 /x) 3/4) 1/3 (6.64) Uniform wall temp. : h Uniform heat flux : h q T = 1 L L q 1 L L T (x)dx 0 q T = 0 h(x)dx John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
25 The problem of uniform wall heat flux The exact result for Pr 0.6 is Nu x = 0.453Re 1/2 x Pr 1/3 (6.71) Nu L = Re 1/2 L Pr 1/3 Churchill and Ozoe equations for the problem of uniform wall heat flux. For Pe x > Rex 1/2 Pr 1/3 Nu x = (1 + ( /Pr) 2/3) (6.73) 1/4 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
26 6.6 The Reynolds analogy The analogy between heat and momentum transfer can now be generalized. For a flat surface with no pressure gradient : C f is the skin friction coefficient. [ d 1 ( ) u u ( y ) ] δ 1 d = 1 dx u δ 2 C f (x) (6.25) 0 0 u For constant wall temperature case : [ d 1 ( ) ( ) ] u T T y q w δ d = dx u T w T δ t ρc p u (T w T ) (6.74) But the similarity of temperature and flow boundary layers to one another suggests the following approximation, which becomes exact only when Pr = 1. ( T T δ = 1 u ) δ t 1 T w T u 2 C q w f (x) = ρc p u (T w T ) φ 2 (6.75) The result is one instance of the Reynolds-Colburn analogy. h Pr 2/3 = C f ρc p u 2 (6.76) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
27 6.6 The Reynolds analogy For use in Reynold s analogy, C f must be a pure skin friction coefficient. Stanton number : actual heat flux to the fluid St heat flux capacity of the fluid flow St h = Nu x ρc p u Re x Pr (6.77) Stanton mass transfer number : St m Sh ReSc We obtain C f 2 = St = St m This equation is known as the Reynolds analogy. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
28 6.6 The Reynolds analogy For application over a wider range, some corrections are necessary. In particular, the Chilton-Colburn analogies are : For 0.6 < Pr < 60 C f 2 = StPr 2/3 j H For 0.6 < Sc < 3000 C f 2 = St msc 2/3 j m Where j H and j m are the Colburn j factors for heat and mass transfer. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
29 6.7 Turbulent boundary layers Figure 6.17 Fluctuation of u and other quantities in a turbulent pipe flow. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
30 6.7 Turbulent boundary layers We define the actual local velocity : u = u + u. u is the average term and u is instantaneous magnitude of the fluctuation. ū = 1 T T 0 ūdt + 1 T T 0 u dt = ū + ū (6.82) Similary, we have the total shear stress and total fluxes as : ( τ tot = µ ū ) ( y ρu v q tot = k ) T y ρc pv T And the following eddy diffusivities for these processes : ρε M ū y = ρu v τ tot = ρ(ν + ε M ) ū y ε T H y = v T q tot = ρc p (α + ε H ) T y C A ε m y = u C A N A,tot = (D AB + ε m ) C A y John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
31 Turbulence near the wall We define the actual local velocity : u = u + u. u is the average term and u is instantaneous magnitude of the fluctuation. For steady, incompressible, constant property flow with time-averaged variables, the x-momentum, energy and species conservation equations are : ρ ( ū ū ) ū + v x y ( ρc p ū T T + v x y ( ū C A x + v C ) A y = dp dx + ( µ ū ) y y ρu v ) = y = y ( k T y ρc pv T ( C A D AB y v C A ) ) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
32 6.8 Heat transfer in turbulent boundary layers For turbulent flow with Re x up to about 10 7 and above , the local friction factor is correlated by : C f,x = Re 1/5 x For turbulent flow, boundary layer development is dependent on random fluctuation of the fluid, not molecular diffusion, and thus the thermal and species boundary layers do not depend on Pr and Sc. Thus : δ δ t δ c With the Chilton-Colburn analogy, the local Nusselt number in turbulent flow is : for 0.6 < Pr < 60 Nu x = StRe x Pr = Rex 4/5 Pr 1/3 The increase in mixing of the fluid causes the turbulent boundary layer to grow more rapidly than the laminar boundary layer, and have larger friction and convection coefficients. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
33 Mixed boundary layer conditions For a laminar layer flowed by a turbulent layer, integrating the global convection coefficient over the laminar zone (0 < x x c ) and then over the turbulent zone (x c < x L) : h L = 1 { x c L } h lam dx + h turb dx L 0 Re x,c is the critical Reynolds number for transition. Setting Re x,c = , for 0.6 < Pr < 60 and < Re L 10 8 Nu L = (0.037Re 4/5 L 871)Pr 1/3 C f,l = Re 1/5 Re L L For situations where L x c and Re L Re x,c, the average Nusselt number reduce to : Nu L = 0.037Re 4/5 L Pr 1/3 C f,l = 0.074Re 1/5 L In the foregoing equations, the fluid physical properties are evaluated at the film temperature. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34 x c
34 Guidelines of application of convection methods Identify the flow geometry (flat plate, cylinder, etc.) ; Decide whether the local or surface average heat transfer coefficient is required for the problem at hand ; Choose correct reference temperature and evaluated fluid properties at that temperature ; Calculate the Reynolds number to determine if the flow is laminar or turbulent ; Calculate the Prandle number ; Select the appropriate correlation that respects the restrictions on its use ; Double-check your design with a second correlation if application is critical to operation when possible. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34
Fundamental Concepts of Convection : Flow and Thermal Considerations. Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.
Fundamental Concepts of Convection : Flow and Thermal Considerations Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.3 6.1 Boundary Layers: Physical Features Velocity Boundary Layer
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More informationFORMULA SHEET. General formulas:
FORMULA SHEET You may use this formula sheet during the Advanced Transport Phenomena course and it should contain all formulas you need during this course. Note that the weeks are numbered from 1.1 to
More informationPrinciples of Convection
Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid
More informationProblem 4.3. Problem 4.4
Problem 4.3 Problem 4.4 Problem 4.5 Problem 4.6 Problem 4.7 This is forced convection flow over a streamlined body. Viscous (velocity) boundary layer approximations can be made if the Reynolds number Re
More informationChapter 6 Fundamental Concepts of Convection
Chapter 6 Fundamental Concepts of Convection 6.1 The Convection Boundary Layers Velocity boundary layer: τ surface shear stress: s = μ u local friction coeff.: C f y y=0 τ s ρu / (6.) (6.1) Thermal boundary
More informationENGR Heat Transfer II
ENGR 7901 - Heat Transfer II Convective Heat Transfer 1 Introduction In this portion of the course we will examine convection heat transfer principles. We are now interested in how to predict the value
More informationSummary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer
1. Nusselt number Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer Average Nusselt number: convective heat transfer Nu L = conductive heat transfer = hl where L is the characteristic
More informationIntroduction to Heat and Mass Transfer. Week 12
Introduction to Heat and Mass Transfer Week 12 Next Topic Convective Heat Transfer» Heat and Mass Transfer Analogy» Evaporative Cooling» Types of Flows Heat and Mass Transfer Analogy Equations governing
More informationCHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW
CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the
More informationTransport processes. 7. Semester Chemical Engineering Civil Engineering
Transport processes 7. Semester Chemical Engineering Civil Engineering 1 Course plan 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume nalysis 4. Differential nalysis of Fluid Flow
More informationConvection. forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds.
Convection The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic,
More informationPHYSICAL MECHANISM OF CONVECTION
Tue 8:54:24 AM Slide Nr. 0 of 33 Slides PHYSICAL MECHANISM OF CONVECTION Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Chapter
More informationConvective Mass Transfer
Convective Mass Transfer Definition of convective mass transfer: The transport of material between a boundary surface and a moving fluid or between two immiscible moving fluids separated by a mobile interface
More informationENGR Heat Transfer II
ENGR 7901 - Heat Transfer II External Flows 1 Introduction In this chapter we will consider several fundamental flows, namely: the flat plate, the cylinder, the sphere, several other body shapes, and banks
More informationMechanical Engineering. Postal Correspondence Course HEAT TRANSFER. GATE, IES & PSUs
Heat Transfer-ME GATE, IES, PSU 1 SAMPLE STUDY MATERIAL Mechanical Engineering ME Postal Correspondence Course HEAT TRANSFER GATE, IES & PSUs Heat Transfer-ME GATE, IES, PSU 2 C O N T E N T 1. INTRODUCTION
More informationGoverning Equations for Turbulent Flow
Governing Equations for Turbulent Flow (i) Boundary Layer on a Flat Plate ρu x Re x = = Reynolds Number µ Re Re x =5(10) 5 Re x =10 6 x =0 u/ U = 0.99 層流區域 過渡區域 紊流區域 Thickness of boundary layer The Origin
More information6.2 Governing Equations for Natural Convection
6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed
More informationOutlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer
Forced Convection Outlines To examine the methods of calculating convection heat transfer (particularly, the ways of predicting the value of convection heat transfer coefficient, h) Convection heat transfer
More informationHEAT TRANSFER BY CONVECTION. Dr. Şaziye Balku 1
HEAT TRANSFER BY CONVECTION Dr. Şaziye Balku 1 CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in the
More informationUNIT II CONVECTION HEAT TRANSFER
UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid
More informationChapter 5 Principles of Convection heat transfer (Text: J. P. Holman, Heat Transfer, 8 th ed., McGraw Hill, NY)
hapter 5 Principles of onvection heat transfer (Tet: J. P. Holman, Heat Transfer, 8 th ed., McGra Hill, NY) onsider a fluid flo over a flat plate ith different temperatures (Fig 5-1) q A ha( T T ) since
More informationMYcsvtu Notes HEAT TRANSFER BY CONVECTION
www.mycsvtunotes.in HEAT TRANSFER BY CONVECTION CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in
More informationChapter 7: Natural Convection
7-1 Introduction 7- The Grashof Number 7-3 Natural Convection over Surfaces 7-4 Natural Convection Inside Enclosures 7-5 Similarity Solution 7-6 Integral Method 7-7 Combined Natural and Forced Convection
More informationCONVECTIVE HEAT TRANSFER
CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW LAMINAR BOUNDARY LAYER FLOW Boundary
More informationENG Heat Transfer II 1. 1 Forced Convection: External Flows Flow Over Flat Surfaces... 4
ENG7901 - Heat Transfer II 1 Contents 1 Forced Convection: External Flows 4 1.1 Flow Over Flat Surfaces............................. 4 1.1.1 Non-Dimensional form of the Equations of Motion.......... 4
More informationCONVECTIVE HEAT TRANSFER
CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 4 HEAT TRANSFER IN CHANNEL FLOW BASIC CONCEPTS BASIC CONCEPTS Laminar
More information10. Buoyancy-driven flow
10. Buoyancy-driven flow For such flows to occur, need: Gravity field Variation of density (note: not the same as variable density!) Simplest case: Viscous flow, incompressible fluid, density-variation
More informationJoule Heating Effect on the Coupling of Conduction with Magnetohydrodynamic Free Convection Flow from a Vertical Flat Plate
Nonlinear Analysis: Modelling and Control, 27, Vol. 12, No. 3, 37 316 Joule Heating Effect on the Coupling of Conduction with Magnetohydrodynamic Free Convection Flow from a Vertical Flat Plate M. A. Alim
More informationHeat processes. Heat exchange
Heat processes Heat exchange Heat energy transported across a surface from higher temperature side to lower temperature side; it is a macroscopic measure of transported energies of molecular motions Temperature
More informationConvection Heat Transfer. Introduction
Convection Heat Transfer Reading Problems 12-1 12-8 12-40, 12-49, 12-68, 12-70, 12-87, 12-98 13-1 13-6 13-39, 13-47, 13-59 14-1 14-4 14-18, 14-24, 14-45, 14-82 Introduction Newton s Law of Cooling Controlling
More informationHeat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay
Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. 18 Forced Convection-1 Welcome. We now begin our study of forced convection
More informationFluid Mechanics II Viscosity and shear stresses
Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small
More informationUnit operations of chemical engineering
1 Unit operations of chemical engineering Fourth year Chemical Engineering Department College of Engineering AL-Qadesyia University Lecturer: 2 3 Syllabus 1) Boundary layer theory 2) Transfer of heat,
More informationChapter 6 Laminar External Flow
Chapter 6 aminar Eternal Flow Contents 1 Thermal Boundary ayer 1 2 Scale analysis 2 2.1 Case 1: δ t > δ (Thermal B.. is larger than the velocity B..) 3 2.2 Case 2: δ t < δ (Thermal B.. is smaller than
More informationMOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow
TRANSPORT PHENOMENA MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow Introduction to Turbulent Flow 1. Comparisons of laminar and turbulent flows 2. Time-smoothed equations of change for incompressible
More informationForced Convection Around Obstacles
Chapter 4 Forced Convection Around Obstacles 4.1. Description of the flow This chapter is devoted to heat transfer on bodies immersed in a stream. We consider a solid characterized by the length scale
More informationForced Convection: Inside Pipe HANNA ILYANI ZULHAIMI
+ Forced Convection: Inside Pipe HANNA ILYANI ZULHAIMI + OUTLINE u Introduction and Dimensionless Numbers u Heat Transfer Coefficient for Laminar Flow inside a Pipe u Heat Transfer Coefficient for Turbulent
More informationMECHANISM BEHIND FREE /NATURAL CONVECTION
CONVECTIVE HEAT TRANSFER By: Prof K. M. Joshi, Assi. Professor, MED, SSAS Institute of Technology, Surat. MECHANISM BEHIND FREE /NATURAL CONVECTION The stagnate layer of fluid in immediate vicinity of
More informationLecture 30 Review of Fluid Flow and Heat Transfer
Objectives In this lecture you will learn the following We shall summarise the principles used in fluid mechanics and heat transfer. It is assumed that the student has already been exposed to courses in
More informationطراحی مبدل های حرارتی مهدي کریمی ترم بهار HEAT TRANSFER CALCULATIONS
طراحی مبدل های حرارتی مهدي کریمی ترم بهار 96-97 HEAT TRANSFER CALCULATIONS ١ TEMPERATURE DIFFERENCE For any transfer the driving force is needed General heat transfer equation : Q = U.A. T What T should
More informationINDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: -- AN No. of Students: 5 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Autumn Semester Examination Sub. Name: Convective Heat and Mass Transfer Instructions:
More informationELEC9712 High Voltage Systems. 1.2 Heat transfer from electrical equipment
ELEC9712 High Voltage Systems 1.2 Heat transfer from electrical equipment The basic equation governing heat transfer in an item of electrical equipment is the following incremental balance equation, with
More informationME 144: Heat Transfer Introduction to Convection. J. M. Meyers
ME 144: Heat Transfer Introduction to Convection Introductory Remarks Convection heat transfer differs from diffusion heat transfer in that a bulk fluid motion is present which augments the overall heat
More informationChapter 3 NATURAL CONVECTION
Fundamentals of Thermal-Fluid Sciences, 3rd Edition Yunus A. Cengel, Robert H. Turner, John M. Cimbala McGraw-Hill, 2008 Chapter 3 NATURAL CONVECTION Mehmet Kanoglu Copyright The McGraw-Hill Companies,
More informationMasters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,
Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =
More informationOutline. Definition and mechanism Theory of diffusion Molecular diffusion in gases Molecular diffusion in liquid Mass transfer
Diffusion 051333 Unit operation in gro-industry III Department of Biotechnology, Faculty of gro-industry Kasetsart University Lecturer: Kittipong Rattanaporn 1 Outline Definition and mechanism Theory of
More informationINSTRUCTOR: PM DR MAZLAN ABDUL WAHID
SMJ 4463: HEAT TRANSFER INSTRUCTOR: PM ABDUL WAHID http://www.fkm.utm.my/~mazlan TEXT: Introduction to Heat Transfer by Incropera, DeWitt, Bergman, Lavine 5 th Edition, John Wiley and Sons Chapter 9 Natural
More informationEmpirical Co - Relations approach for solving problems of convection 10:06:43
Empirical Co - Relations approach for solving problems of convection 10:06:43 10:06:44 Empirical Corelations for Free Convection Use T f or T b for getting various properties like Re = VL c / ν β = thermal
More informationTable of Contents. Foreword... xiii. Preface... xv
Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations...
More informationStudies on flow through and around a porous permeable sphere: II. Heat Transfer
Studies on flow through and around a porous permeable sphere: II. Heat Transfer A. K. Jain and S. Basu 1 Department of Chemical Engineering Indian Institute of Technology Delhi New Delhi 110016, India
More informationConvection Workshop. Academic Resource Center
Convection Workshop Academic Resource Center Presentation Outline Understanding the concepts Correlations External Convection (Chapter 7) Internal Convection (Chapter 8) Free Convection (Chapter 9) Solving
More informationHeat and Mass Transfer Unit-1 Conduction
1. State Fourier s Law of conduction. Heat and Mass Transfer Unit-1 Conduction Part-A The rate of heat conduction is proportional to the area measured normal to the direction of heat flow and to the temperature
More informationChapter 7: External Forced Convection. Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University
Chapter 7: External Forced Convection Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University Objectives When you finish studying this chapter, you should be able to: Distinguish between
More informationExternal Forced Convection. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
External Forced Convection Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Drag and Heat Transfer in External flow Fluid flow over solid bodies is responsible
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationChapter 9 Flow over Immersed Bodies
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes,
More informationLaminar and Turbulent developing flow with/without heat transfer over a flat plate
Laminar and Turbulent developing flow with/without heat transfer over a flat plate Introduction The purpose of the project was to use the FLOLAB software to model the laminar and turbulent flow over a
More informationTransport processes. 7. Semester Chemical Engineering Civil Engineering
Transport processes 7. Semester Chemical Engineering Civil Engineering 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume Analysis 4. Differential Analysis of Fluid Flow 5. Viscous
More informationNonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4,
Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4, 513 524 Effects of Temperature Dependent Thermal Conductivity on Magnetohydrodynamic (MHD) Free Convection Flow along a Vertical Flat Plate
More informationDimensionless Numbers
1 06.10.2017, 09:49 Dimensionless Numbers A. Salih Dept. of Aerospace Engineering IIST, Thiruvananthapuram The nondimensionalization of the governing equations of fluid flow is important for both theoretical
More informationDAY 19: Boundary Layer
DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence
More informationSo far, we have considered conduction, which is the mechanism of heat
cen58933_ch06.qxd 9/4/2002 12:05 PM Page 333 FUNDAMENTALS OF CONVECTION CHAPTER 6 So far, we have considered conduction, which is the mechanism of heat transfer through a solid or a quiescent fluid. We
More informationTwo Phase Thermal Boundary Layer Flow
Two Phase Thermal Boundary Layer Flow P. K. Tripathy Deptt. Of Mathematics & Science, U. C. P. Engg. School, Berhampur (Orissa), INDIA S. K. Mishra Deptt. Of Mathematics, R. C. M. Science College, Khallokote
More informationLaminar Flow. Chapter ZERO PRESSURE GRADIENT
Chapter 2 Laminar Flow 2.1 ZERO PRESSRE GRADIENT Problem 2.1.1 Consider a uniform flow of velocity over a flat plate of length L of a fluid of kinematic viscosity ν. Assume that the fluid is incompressible
More information7.6 Example von Kármán s Laminar Boundary Layer Problem
CEE 3310 External Flows (Boundary Layers & Drag, Nov. 11, 2016 157 7.5 Review Non-Circular Pipes Laminar: f = 64/Re DH ± 40% Turbulent: f(re DH, ɛ/d H ) Moody chart for f ± 15% Bernoulli-Based Flow Metering
More informationFluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 17 Laminar and Turbulent flows
Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 17 Laminar and Turbulent flows Welcome back to the video course on fluid mechanics. In
More informationBasic Fluid Mechanics
Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible
More informationFLUID MECHANICS PROF. DR. METİN GÜNER COMPILER
FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 5. FLOW IN PIPES 5.1.3. Pressure and Shear Stress
More informationInternal Flow: Heat Transfer in Pipes
Internal Flow: Heat Transfer in Pipes V.Vuorinen Aalto University School of Engineering Heat and Mass Transfer Course, Autumn 2016 November 15 th 2016, Otaniemi ville.vuorinen@aalto.fi First about the
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 19 Turbulent Flows Fausto Arpino f.arpino@unicas.it Introduction All the flows encountered in the engineering practice become unstable
More informationBOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW
Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email:
More informationCHME 302 CHEMICAL ENGINEERING LABOATORY-I EXPERIMENT 302-V FREE AND FORCED CONVECTION
CHME 302 CHEMICAL ENGINEERING LABOATORY-I EXPERIMENT 302-V FREE AND FORCED CONVECTION OBJECTIVE The objective of the experiment is to compare the heat transfer characteristics of free and forced convection.
More informationTankExampleNov2016. Table of contents. Layout
Table of contents Task... 2 Calculation of heat loss of storage tanks... 3 Properties ambient air Properties of air... 7 Heat transfer outside, roof Heat transfer in flow past a plane wall... 8 Properties
More informationTurbulence Solutions
School of Mechanical, Aerospace & Civil Engineering 3rd Year/MSc Fluids Turbulence Solutions Question 1. Decomposing into mean and fluctuating parts, we write M = M + m and Ũ i = U i + u i a. The transport
More informationNumerical Investigation of Thermal Performance in Cross Flow Around Square Array of Circular Cylinders
Numerical Investigation of Thermal Performance in Cross Flow Around Square Array of Circular Cylinders A. Jugal M. Panchal, B. A M Lakdawala 2 A. M. Tech student, Mechanical Engineering Department, Institute
More informationChemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017
Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Objective: Text: To introduce the basic concepts of fluid mechanics and heat transfer necessary for solution of engineering
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More informationConvection. U y. U u(y) T s. T y
Convection Heat transfer in the presence of a fluid motion on a solid surface Various mechanisms at play in the fluid: - advection physical transport of the fluid - diffusion conduction in the fluid -
More informationFluid Dynamics and Balance Equations for Reacting Flows
Fluid Dynamics and Balance Equations for Reacting Flows Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Balance Equations Basics: equations of continuum mechanics balance equations for mass and
More information1 Introduction to Governing Equations 2 1a Methodology... 2
Contents 1 Introduction to Governing Equations 2 1a Methodology............................ 2 2 Equation of State 2 2a Mean and Turbulent Parts...................... 3 2b Reynolds Averaging.........................
More informationNumber of pages in the question paper : 05 Number of questions in the question paper : 48 Modeling Transport Phenomena of Micro-particles Note: Follow the notations used in the lectures. Symbols have their
More informationMODULE CODE: ENGG08021 INTRODUCTION TO THERMOFLUIDS. Date: 15 January 2016 Time: 10:00 12:00
School of Engineering & Computing Session 2015-16 Paisley Campus Trimester 1 MODULE CODE: ENGG08021 INTRODUCTION TO THERMOFLUIDS Date: 15 January 2016 Time: 10:00 12:00 Attempt FOUR QUESTIONS IN TOTAL
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationME 331 Homework Assignment #6
ME 33 Homework Assignment #6 Problem Statement: ater at 30 o C flows through a long.85 cm diameter tube at a mass flow rate of 0.020 kg/s. Find: The mean velocity (u m ), maximum velocity (u MAX ), and
More informationBoundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity
Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity Tobias Knopp D 23. November 28 Reynolds averaged Navier-Stokes equations Consider the RANS equations with
More informationOUTCOME 2 - TUTORIAL 1
Unit 4: Heat Transfer and Combustion Unit code: K/60/44 QCF level: 5 Credit value: 5 OUTCOME - TUTORIAL Heat transfer coefficients Dimensional analysis: dimensionless groups; Reynolds, Nusselt, Prandtl,
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationThermal and Fluids in Architectural Engineering
hermal and Fluids in Architectural Engineering 12. Convection heat transfer Jun-Seo Par, Dr. Eng., Prof. Dept. of Architectural Engineering Hanyang Univ. Where do we learn in this chaper 1. Introduction
More informationHeat Transfer Convection
Heat ransfer Convection Previous lectures conduction: heat transfer without fluid motion oday (textbook nearly 00 pages) Convection: heat transfer with fluid motion Research methods different Natural Convection
More informationFall 2014 Qualifying Exam Thermodynamics Closed Book
Fall 2014 Qualifying Exam Thermodynamics Closed Book Saturated ammonia vapor at 200 O F flows through a 0.250 in diameter tube. The ammonia passes through a small orifice causing the pressure to drop very
More informationPhone: , For Educational Use. SOFTbank E-Book Center, Tehran. Fundamentals of Heat Transfer. René Reyes Mazzoco
8 Fundamentals of Heat Transfer René Reyes Mazzoco Universidad de las Américas Puebla, Cholula, Mexico 1 HEAT TRANSFER MECHANISMS 1.1 Conduction Conduction heat transfer is explained through the molecular
More informationFluid Mechanics Theory I
Fluid Mechanics Theory I Last Class: 1. Introduction 2. MicroTAS or Lab on a Chip 3. Microfluidics Length Scale 4. Fundamentals 5. Different Aspects of Microfluidcs Today s Contents: 1. Introduction to
More informationOE4625 Dredge Pumps and Slurry Transport. Vaclav Matousek October 13, 2004
OE465 Vaclav Matousek October 13, 004 1 Dredge Vermelding Pumps onderdeel and Slurry organisatie Transport OE465 Vaclav Matousek October 13, 004 Dredge Vermelding Pumps onderdeel and Slurry organisatie
More informationEffects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium
Transport in Porous Media (2006) 64: 1 14 Springer 2006 DOI 10.1007/s11242-005-1126-6 Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium
More informationFluid Mechanics. Chapter 9 Surface Resistance. Dr. Amer Khalil Ababneh
Fluid Mechanics Chapter 9 Surface Resistance Dr. Amer Khalil Ababneh Wind tunnel used for testing flow over models. Introduction Resistances exerted by surfaces are a result of viscous stresses which create
More informationLectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 6
Lectures on Nuclear Power Safety Lecture No 6 Title: Introduction to Thermal-Hydraulic Analysis of Nuclear Reactor Cores Department of Energy Technology KTH Spring 2005 Slide No 1 Outline of the Lecture
More informationTheoretical and Experimental Studies on Transient Heat Transfer for Forced Convection Flow of Helium Gas over a Horizontal Cylinder
326 Theoretical and Experimental Studies on Transient Heat Transfer for Forced Convection Flow of Helium Gas over a Horizontal Cylinder Qiusheng LIU, Katsuya FUKUDA and Zheng ZHANG Forced convection transient
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationChapter 3 Lecture 8. Drag polar 3. Topics. Chapter-3
Chapter 3 ecture 8 Drag polar 3 Topics 3.2.7 Boundary layer separation, adverse pressure gradient and favourable pressure gradient 3.2.8 Boundary layer transition 3.2.9 Turbulent boundary layer over a
More information