CONVECTIVE HEAT TRANSFERmech.sut.ac.ir/People/Courses/18/Chapter2.pdfCONVECTIVE HEAT TRANSFER-CHAPTER 2 By: M. Goharkhah SAHANDUNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

CONVECTIVE HEAT TRANSFER

Mohammad GoharkhahDepartment of Mechanical Engineering, Sahand Unversity of Technology,

Tabriz, Iran

Differential Formulation of the Basic Laws

CHAPTER 2

CONVECTIVE HEAT TRANSFER- CHAPTER 2By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Mathematical Background

Velocity Vector

Velocity Derivative

Cartesian coordinates

cylindrical coordinates

spherical coordinates

Divergence of a Vector

Gradient of Scalar



Mathematical BackgroundTotal Differential and Total Derivative

convective derivative

local derivativeAcceleration in cylindrical coordinates



VISCOUS FLUID STRESSES



FUNDAMENTAL PRINCIPLES

Conservation of Mass

Conservation of Momentum: The Navier-Stokes Equations of Motion

Conservation of Energy



FUNDAMENTAL PRINCIPLES- MASS CONSERVATION

Mass conservation statementfor a control volume

Mcv :mass that is trappedinstantaneously inside thecontrol volume

2 dim. flow 3 dim. flow

material derivative




For negligible temporal and spatial variations in density

This equation is not valid only for incompressible fluids. In fact, it appliesto flows (not fluids) where the density and velocity gradients are such thatthe Dρ/Dt terms are negligible relative to the ρ ∇ · v term.Most of the gas flows encountered in heat exchangers, heated enclosures,and porous media obey the simplified version of the mass conservationprinciple




Cylindrical Coordinate Spherical Coordinate



FUNDAMENTAL PRINCIPLES- MOMENTUM EQUATIONS

Instantaneous force balance on a control volume

n: The direction chosen for analysisvn , Fn : The projections of fluid velocity and

forces in the n direction

Momentum principle or momentum theorem. Control volume formulation of Newton’s second law of motion,

forces are represented by:The normal stress (σx)Tangential stress (τxy ) body force per unit volume (X)



mass conservation=0Navier—Stokes equation

Incompressible and constant viscosity

Y-momentum for a 3dim flow

Vectorial notation

?



MOMENTUM EQUATIONS- Cylindrical Coordinate

MOMENTUM EQUATIONS- Cylindrical Coordinate



MOMENTUM EQUATIONS- Spherical Coordinate

Material derivative and Laplacian operators in spherical coordinates



FUNDAMENTAL PRINCIPLES- Conservation of Energy

First law of thermodynamics

Different forms of energyencountered in flow systems

Energy that can cross a control surfaceMechanical (or electrical) workFlow work: work done by each unit mass of flowing fluid on the control volume or by the control volume, as it flows across the control surfaceHeat: energy transferred by virtue of a temperature gradient

Energy stored in each unit mass as it crosses the control surfaceInternal thermal energy: eKinetic energy 1/2 V2

Potential energy gz, EnthalpyEnergy that can be involved in a change ofenergy storage within a control volumeInternal thermal energy eKinetic energy 1/2 V2

Potential energy gzEnergy generation



?



Work transfer effected by the normal and tangential stresses

Work done per unit time by the normal stresses σx

On the left side of the element:(σx Δy) u

On the right side of the element:[σx + (∂ σx /∂x) x][u + (∂u/∂x) x] y.

The net work transfer rate :[σx(∂u/∂x) + u(∂ σx /∂x)] Δx Δy

Represents the change in kinetic energy of the fluid packet:It is considered negligible relative to the ∂(ρe)/∂t

Assembling Above expressions into the energy conservation statement , and using constitutive relations,

viscous dissipation function

Where does this originate from?




For incompressible and two-dimensional flow

Fourier law of heat conduction

mass conservation=0

How can the energy equation be expressed in terms of enthalpy?




From the last of Maxwell’s relations

For pure substances

coefficient of thermal expansion

How can the energy equation be expressed in terms of temperature




The ‘‘temperature’’ formulation of the first law of thermodynamics

constant fluid conductivity kzero internal heat generation qnegligible viscous dissipation φand negligible compressibility effect βT DP/Dt.

The left-hand side of the energy equation







Extremely viscous flows of the type encountered inlubrication problems or the piping of crude oil




Note the difference between the constant-ρ approximation andincompressible flow.

For incompressible fluids such as water, liquid mercury, andengine oil cp =c.

Do not use cv instead of cp . The derived energy equation represents the first law of

thermodynamics. This law proclaims the conservation of thesum of energy change (the property) and energy interactions(heat transfer and work transfer).

The suggestion that mechanical effects (e.g., work transfer) areabsent from the energy equation when the βT DP/Dt term isabsent is wrong. The presence of cp on the left side of theequation is the sign that each fluid packet expands or contracts(i.e., it does P dV-type work) as it rides on the flow.



Summary

Incompressible and constant viscosity

Energy equation

constant fluid conductivity, zero internal heat generation, negligible viscous dissipation and compressibility effect βT DP/Dt.

Mass conservation

Navier—Stokes equation



Example 1

Steady state:

Axial flow (x-direction only):

Parallel flow:

1

2

3

4

5

6



Example1

7

85

96

At the free surface, y=H, the pressure is uniform equal to :

109

9 11

12

8 13

Boundary conditions

No-slip condition at the inclined surface:

Free surface is parallel to inclined plate:

Negligible shear at the free surface

Specified temperature at the inclined surface:

Specified heat flux at the free surface



Example2



Example2

Continuity equation



Non-dimensional Form of the Governing Equations

Useful information can be obtained without solving the governing equations by rewriting them in dimensionless formIdentify the governing parameters Plan experiments Guide in the presentation of experimental results and theoretical solutions.

Characteristic quantities

To non-dimensionalize the dependent and independent variables, we use characteristicquantities that are constant throughout the flow and temperature fields.

Dimensionless dependent and independent variables:





Dimensionless Form of Continuity

Dimensionless Form of the Navier-Stokes Equations of Motion

Dimensionless Form of the Energy Equation

(1) Incompressible, constant conductivity

(2) Ideal gas, constant conductivity and viscosity




From the nondimensional form of the equations, the temperature solution for convection problem is obtained as:

The Re is associated with viscous flow while the Pr is a heat transfer parameter which is a fluid property. The Gr represents buoyancy effect and the Ec number is associated with viscous dissipation and is important in high speed flow and very viscous fluids.

In dimensional formulation the following six quantities and five properties affect the solution. In dimensionless formulation these factors are consolidated into four dimensionlessparameters: Re, Pr, Gr and Ec.

The number of parameters can be reduced in two special cases: 1. If fluid motion is dominated by forced convection (negligible free convection), the

Grashof number can be eliminated. 2. If viscous dissipation is negligible, the Eckert number can be dropped. Thus under

these common conditions the solution is simplified to:




It is concluded from dimensionless temperature that geometrically similarbodies have the same dimensionless velocity and temperature solutions if thesimilarity parameters are the same for all bodies.

By identifying the important dimensionless parameters governing a givenproblem, experimental investigations can be planned accordingly. Instead ofvarying the relevant physical quantities, one can vary the similarity parameters.This will vastly reduce the number of experiments needed. The same is true ifnumerical results are to be generated.

Presentation of results such as heat transfer coefficient, pressure drop, anddrag, whether experimental or numerical, is most efficiently done whenexpressed in terms of dimensionless parameters.



Non-dimensional Form of the heat transfer coefficient

Non-dimensional Form of the heat transfer coefficient

This suggests how experiments should be planned and provides an appropriate form for correlation equations for the Nusselt number.

Negligible buoyancy and viscous dissipation

Free convection with negligible dissipation

Special cases:



Another way to obtain h as a function of dimensionless variables is to usethe Pi Theorem.

Non-dimensional Form of the heat transfer coefficient- The Pi Theorem


h depends on several parameters. The firs step is to find them.

?





The buoyancy force:









Now we can apply the Pi Theorem



















Other Dimensionless Numbers



Physical Interpretation of the Dimensionless Numbers-Reynolds Number

Physical Interpretation of the Dimensionless Numbers-Reynolds Number

The rate momentum passes through this area



Physical Interpretation of the Dimensionless Numbers-Nusselt Number

Physical Interpretation of the Dimensionless Numbers-Nusselt Number



Physical Interpretation of the Dimensionless Numbers-Grashof Number







Physical Interpretation of the Dimensionless Numbers-Prandtl Number

Physical Interpretation of the Dimensionless Numbers-Prandtl Number





Scale Analysis

Scale analysis or scaling, is a procedure by which estimates of usefulresults are obtained without solving the governing equations.

The object of scale analysis is to use the basic principles of convective heattransfer to produce order-of-magnitude estimates for the quantities ofinterest. This means that if one of the quantities of interest is the thicknessof the boundary layer in forced convection, the object of scale analysis is todetermine whether the boundary layer thickness is measured in millimetersor meters

Scale analysis goes beyond dimensional analysis (whose objective is todetermine the dimension of boundary layer thickness, namely, length).

Scaling is accomplished by assigning order of magnitude values todependent and independent variables in an equation.



Scale Analysis- Example1

A problem from the field of conduction heat transfer

we see a plate plunged at t = 0 into a highlyconducting fluid, such that the surfaces ofthe plate instantaneously assume the fluidtemperature T∞ = T0 + ΔT. we areinterested in estimating the time needed bythe thermal front to penetrate the plate,that is, the time until the center plane ofthe plate ‘‘feels’’ the heating imposed onthe outer surfaces.




The penetration time compares well with any order-of-magnitude interpretation of the exact solution to this classical problem

We focus on a half-plate of thickness D/2 and the energy equation for pure conduction in one direction:




Melting Time of Ice SheetAn ice sheet of thickness L is at the freezing temperature Tf . One side is suddenly maintained at temperature To which is above the freezing temperature. The other side is insulated.

The entire sheet melts when xi =L. The largesttemperature difference is T0 -Tf .

exact solution

Conservation of energy at the melting front gives



Scale Analysis- Rules

• Rule 1. Always define the spatial extent of the region in which you performthe scale analysis. In the first example, the size of the region of interest isD/2. In other problems, such as boundary layer flow, the size of the region ofinterest is unknown; the scale analysis begins by selecting the region andby labeling the unknown thickness of this region δ. Any scale analysis of aflow or a flow region that is not uniquely defined is nonsense.

• Rule 2. One equation constitutes an equivalence between the scales of twodominant terms appearing in the equation. In the examples, the left-handside of the equation could only be of the same order of magnitude as theright-hand side. The two terms appearing in the equation are the dominantterms. In general, the energy equation can contain many more terms, notall of them important. The reasoning for selecting the dominant scalesfrom many scales is condensed in rules 3–5.



Scale Analysis, Rules

• Rule 3. If in the sum of two terms, the order of magnitude of one term is greater than the order of magnitude of the other term, then the order of magnitude of the sum is dictated by the dominant term:

c = a + b O(a) > O(b) ⇒ O(c) = O(a)

• Rule 4. If in the sum of two terms, the two terms are of the same order of magnitude, then the sum is also of the same order of magnitude:

O(a) = O(b) ⇒ O(c) ∼ O(a) ∼ O(b)

• Rule 5. In any product, the order of magnitude of the product is equal to the product of the orders of magnitude of the two factors

p = ab ⇒ O(p) = O(a)O(b), r =a/b ⇒ O(r) =O(a)/O(b)

Documents

CONVECTIVE HEAT TRANSFERmech.sut.ac.ir/People/Courses/18/Chapter2.pdfCONVECTIVE HEAT TRANSFER-CHAPTER 2 By: M. Goharkhah SAHANDUNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING