Conservation Equations and the Fundamentals of Heat and Mass

Transfer

1. Transport model, consists of two major parts, governing equations PDE

and the constitutive equation.

2. Constitutive equation , relate diffusion flux to local

material properties

3. Governing equations is the mathematics form of

conservation equation and has universal form.

4. Conservation of equation is a balance equation, relate rate of

accumulation of the quantity to the quantity of it enters or is formed

5. Constitutive equation are empirical and material specific

6. Governing equations are derived based on the concept of

control volume.

7. Control volume can vary in shape and size mean volume and

bounding surface changing with time

8.

9. Fixed Control volume means

10.

11.Rate of accumulation of quantity per CV= net flux per surface + rate of

formation per CV

12.

13.Once these integrations are done , it is function of time

14.Rate of change of concentration

15.A moving control volume which is translating, rotating and deforming.

, it only effect flux (does not influence rate of formation) ,

therefore rate of accumulation=flux in respect to (v-vs)

16. Surface motion leads to additions flux relative to the surface.

17. If the motion is translating the velocity respect to CV, is +ve as

parallel vs is same direction

18.Flux unit

19.

20.For interface problems

21.Inner working = big picture

22.Conservation equations valid at a given point in a continuum obtained

from control volume as we want a given pt where a point volume

is zero

23.CV reduced to a point.

24.

25. , flux relative to surface

26.This derivation ignore interfacial accumulation

27.Total flux as expressed as convective and diffusion contribution.

Convective transport occurs when a constituent of the fluid (mass,

energy, a component in a mixture) is carried along with the fluid. The

amount carried past a plane of unit area perpendicular to the velocity

(the flux) is the product of the velocity and the (concentration or

quantity) /volume

28.

29.

30.EG in membrane both same concentration therefore no gradient la

31.Diffusion due to gradient

32.Conservation of energy :Thermal Effects

i) Energy equation relate to thermal effect, total derivative is taken as

not, but also can be broken down to the partial derivative with

material derivative

ii) Energy equation must be satisfy at the boundaries between pure

material, but mutually insoluble

iii)

iv)

v) At contact point , at interface, meaning there is change

of temperature but same rate, no gradient but got convection, no

net change. Thermal equilibrium but heat flux is not zero, no

diffusion but due to convection or conduction

vi) Convection Boundary Condition

a) Fourier’s law assume conduction heat transfer normal to the

interface

b) Assume phase 1 is solid and phase 2 is liquid

c)

d) Summing of flux (integrate)=h(T-T∞)

e) Phase change

i) Melting and evaporation bulk flow across a phase so

interfacial can’t be used. Diffusion is zero

bulk flow not due to diffusion but convection. Bulk flow

is convection .

ii) , say phase change leads to

bulk flow meaning convection not diffusion

iii) Density different as phase 1 and phase 2 has different

densities

vii)

viii)

Scaling and Approximation Technique

1. Simplification and obtaining approximate solutions

2. Simplification is the concept of scales, become dimensionless. Key to

simplification is by concept of scales

3.

4. Scale of a variable scaled variable =dimension

variable/scale

5. Steady or not steady states depend on time scales.

6. Scaling determine order of the unknown variable magnitude

7. Similarity method, perturbation method (small parameter).

8.

9. Scaled variable

i) Mathematics form in dimensionless, minimizing number of

variables and parameters. From many to two variables , to Bi and

ii) Independent and dependent variables are order of one during

scaling.

iii) Magnitude various terms in an equation revealed by dimensionless

parameter.

iv) Length and time scales are called characteristic length and time.

v) Dimensionless variables also have scales in pure numbers.

vi) How about scales of derivatives.

vii)

viii)

ix) Physical meaning of maximum order of magnitude of

temperature

x)

xi)

xii) Reduction of dimensionality

xiii)

a) Symmetry

i) Symmetry can be exploited

ii) A strategy exclusion of any spatial variable which is not

required by the conservation equation or interfacial

conditions.

iii) In cylindrical form , temperature only depend on any

variable other than r, theta= 0 due to symmetry and z is

ignored due to long path of diffusion

b) Aspect ratio

i) Aspect ratio is ratio of two linear dimension length/width

ii) If insulated

iii) If has short diffusion path it is significant

xiv)

Tc > Ts> as heat transfer from solid to surface

xv) the ratio of the temperature

differences indicates the relative thermal resistances. For large Bi,

fluid is isothermal and

xvi) The Biot number (Bi) is a dimensionless quantity used in heat

transfer calculations. It is named after the French physicist Jean-

Baptiste Biot (1774–1862), and gives a simple index of the ratio of

the heat transfer resistances inside of and at the surface of a body.

xvii) Small Biot, the fluid resistance dominates and solid is nearly

isothermal, spatial variation is absence and 3D to zero

dimensional, we can use lumped model instead of distributed

model.

xviii)

xix) Consider steady state transfer from an extended surface or fin to

the surrounding air

xx) Base fin is at and ambient temperature is . Length to width

ratio , y big enough to be not significant assume T(x,z)

xxi)

xxii) , meaning the average temperature of x field of

the x,z fields is equal to average T(z), approximation, summing the

x field and average it so it absorb into

xxiii) average value .

xxiv)

xxv) Simplification Based On Time Scales

1) Temperature or concentration is perturbated at some location

finite time needed so that the effect can be noticed at a distant

from original source of disturbance .

2) Stagnant medium time involved is the characteristics time

3) Characteristics time is essential for diffusion and conduction as

time is needed for diffusion and conduction

4) Characteristic time key factor determining diffusion or

conduction model

5) Fast response use steady state or pseudo steady

6) Slow response model as infinite or semifinite, effect of one or

more boundaries never felt on the time scales

7)

xxvi) For small t , concentration changes from x=0, spread only fraction

the membrane thickness

xxvii) Example 3.4.1

xxviii) The similarity method is a technique , PDE into ODE

xxix) Regular Perturbation

Singular perturbation method

Solution Method for Conduction and Diffusion Problems

1) Finite Fourier Transform (FFT), expanding the solution in term of a set of

known functions. Basis function and then determining the unknown

coefficients in the expansion.

2) FFT to PDE reduces spatial variable until only a two point BV or IV

3) FFT is basically equivalent to separation of variables

4) Point source solutions of DE (Green’s functions) to solve linear problem

5) One spatial dimension be finite, Green’s functions used in unbounded

domain

6)

7)

8)

9)

Fundamental of Fluid Mechanics

1. Linear momentum of a rigid body of mass m and translational velocity

(center-of –mass) velocity is mv

2. , second law of newton , is rate of change of

momentum , where the body is of constant mass, this velocity is a center

of mass velocity –How about the local velocity?

3. F is the net force acting on the rigid body, force exerted on the body by the

surrounding

4. Think in perspective of the control volume not the fluid

5.

6. Any mass crossing the CV carries a certain amount of momentum, by means

of convective transport

7.

8. The evaluation of forces

Fundamental of Fluid Mechanics

1) The linear momentum of constant solid mass m and translational velocity v,

is mv (Physical meaning: Linear momentum of a constant mass solid m

with center of mass velocity v is given by mv)

2) , net force is rate of change of momentum equal to

net force. m is a constant mass

(Physical meaning: F=ma is newton’s second law for a constant mass m,

the rate of change of momentum which is F=mdv/dt is equal to the net

force exerted on the body by surrounding

:the velocity here refer to the center of mass velocity of a body and not

the velocity of the whole body, as accumulation of small bodies

represent the body)

3) Integral and differentiation form is helpful in fluids ((Physical meaning:

integral of the GE means taking the whole lot instead of focusing on a

single small body and taking the whole fluid instead)

Use local velocities instead of center of mass velocity

4) Control volume that has always contain the same mass of fluid (Physical

meaning: since F=ma ,involves is a constant mass , the CV has to

contain the same mass to satisfy the F=ma for a constant mass

condition)

5) If the surface bounding the CV assumed to deform with the flow ,

(Physical meaning: no flux and will always contain same material

which means the bounding surface velocity is equal to local fluid

velocity, this CV is call material volume –same material at all time )

6) ,

,

7) purpose of integral so is respect to fluid velocity instead of mass velocity

due to summing , integral

8) Momentum in CV

a)

9) Momentum in a moving CV

a)

b) (Physical meaning: the rate of change of momentum + the

convective loss is equal to net force acting on the rigid body, if v=vs,

it is a material volume )

10) Evaluation of forces

i) Body forces

ii) Surface forces are pressure and viscous forces

iii) Surface forces are described using s(n), force per unit area on surface

with a normal surface s(n)

iv)

v) Surface forces is contributed by pressure and viscous stress, if the

fluid is at rest, viscous stress is zero,

vi) P has the same meaning of thermodynamics pressure

vii) , incompressible fluid density is constant, ,

viii)

ix) In static fluid

11) Constitutive equation relates material properties such strain to

viscosity

i) For Newtonian fluid

12) Dynamics pressure why use this , convenience, advantageous to

combine pressure and gravitational term in navier-stokes or Cauchy

momentum is used incompressible fluid with known

boundaries

13)

14) Nondimensionalization and simplification of navier stokes

i) Introducing length, time , velocity , and pressure scales

ii) Characteristic L ,Ls ,Characteristic v ,vs

iii)

iv)

v) Steady flows

a) , derived from viscous term

c)

1) Stream function

a) Solving incompressible fluid, only two non-vanishing

velocity components and two spatial coordinates are

involved.

b) Flow of planar character

c) For 2D described by rectangular coordinates

Dynamics Pressure;

1. Solving incompressible fluid flow with known

boundaries (no free surface i.e. Confined pipe)

advantageous to combine pressure and gravity term of

Navier -Stokes and Cauchy momentum equation using

dynamics pressure.

2. What advantage? solved without referring to the

gravitational term

3. , so that navier-stokes can be expressed as

Unidirectional and Nearly Unidirectional Flow

1) Solutions are organized in term of number of direction and

dimensions

2) Characteristic of flow dimension and direction.

3) Number of direction refer to non-vanishing velocity term

(most crucial)

4) Unidirectional in cylindrical not in rectangular for Couette

5) Only unidirectional flow has exact solution (only one

nonzero velocity component)- Poiseuille Flow (driven by

pressure) in pipe one nonzero is steady flow due to

pressure flow , time dependent

6) Number of dimension refer to spatial coordinates

7) Steady or time dependent also crucial

8) Number of direction most crucial

9) Pressure driven flow in confined flow –Poiseuille flow

10) Entrance and edge effects

11) Viscous flow nearly unidirectional

12) Steady Flow with a Pressure Gradient

i) Fully developed , for an incompressible fluid flowing

in tube or other channel of constant cross section area ,

velocity does not vary with the direction (spatial

independent, no acceleration)

ii) Occur after a distance from inlet , called the entrance

length

iii) Normally unidirectional

iv) Say x direction , the velocity is assumed to be fully

developed

v) , say is unidirectional means

vi)

vii)

viii) If no rotation a fully developed tube will be fully

developed,

ix)

x)

xi)

Scaling and Approximation Technique

1. The PDE of the governing equation is

quite complex and need simplification to

solve it

2. What method to obtain an approximate

method?

3. Experience to find the differential

equation and boundary conditions is like an

art to represent the real model

4. Assumption steady state, 1D but how

this assumption comes from?

5. This assumption comes concept of

scales.

6. Orders of magnitude

a) Order of magnitude; determine

which parameter is significant and

which is not. ~, parameter which is

to a given problem.

b) x~y, difference is less than order

10,<10 (less than order 10)

c) Algebraic sign is ignored as the

order is utmost important say 10,100

and even is -100 it is more important

than 10 in term of variable.

d) ,

good starting point

7. Dimensionless of governing equations to

minimize the number of variables and

parameter

8. Say in a set of parameter and variable ,

we can set different set of quantities to

represent the parameter or variable , like eg

9. Scaling, special dimensionless, making

dependent and independent in order 1 , by

doing this order of parameter or variable

appear and we can do simplification.

10.

11. Length and time scales are important

called characteristic length and time for

steady state or transient or which spatial can

be ignored.

12. Our GE is ODE or PDE our scales must

be also. scale which measure the change .

Unidirectional and Nearly Unidirectional Flow

1. Two fluid dynamics characteristic are dimension and direction , which are

helpful to organizing the solution eg ,

2. Direction is the non-vanishing velocity term

3. Dimension refers to the spatial coordinates

4. Another characteristic is steady or time dependent flow

5. Flow involving incompressible flow, number of direction is most crucial as

6. Most flows are unidirectional meaning ? Only one non

vanishing velocity

7. Viscous flow are normally unidirectional

8. Nearly unidirectional means , inertia term not important and viscous term

important

9. Steady flow with pressure gradient

a)

b)

Laminar Flow High Reynolds Number

1. Reynolds number

2. High Reynolds number means inertial force is more prominent.

3. Involve high velocity and large length and low kinematic viscosities

4. Flow –outer or inviscid

5. Flow- inner or boundary layer regions

6. Boundary layer separation is discussed

7. Viscous force are generally absence in high Reynolds flow

8. High Reynolds number flow

a) Inviscid (outer flow)

1) Reynolds number measure importance or inertial effect

(convective momentum transfer) to viscous effect (diffusion

transfer)

2) High Reynolds number viscous force are absent

3)

4)

9. Dimensionless form of the Navier-Stokes equation

10.

Laminar Boundary Layer Flow (Bejan,2013)

1. Consider heat transfer from solid object to fluid stream in external flow

2. Eg, flat plate of temperature suspended in uniform stream of velocity

and Temperature

3. We want to know a) net force stream on the plate b) heat transfer stream to

plate

4. Stream acts like a drag force on the plate and therefore translate in pressure

drop as the u=0 at y=0, we can carry out force balance analysis

5. Heat transfer solid and fluid also must be answer

6.

7. As observed that fluid layer at y=0 , stuck to solid wall- no slip condition

8. Meaning motionless at y=0, heat conduction by pure conduction

9.

10.At solid wall , no slip u=0, impermeability v=0,T=T0

11.Concept of boundary layer

a)

b) Pradlt’s idea outside the boundary layer, he imagines a free

stream, flow region not affected by obstruction and heating

surface.

c)

d)

e)

f)

g)

h)

i)

j)

12.Scale analysis

k)

l)

13.Scale analysis

1)

2)

14.

The law of heat conduction, also known as Fourier's law, states that the time

rate of heat transfer through a material is proportional to the negative

gradient in the temperature and to the area, at right angles to that gradient,

through which the heat flows. We can state this law in two equivalent forms:

the integral form, in which we look at the amount of energy flowing into or

out of a body as a whole, and the differential form, in which we look at the

flow rates or fluxes of energy locally.

Newton's law of cooling is a discrete analog of Fourier's law, while Ohm's

law is the electrical analogue of Fourier's law.

Differential form

The differential form of Fourier's Law of thermal conduction shows that the

local heat flux density,  , is equal to the product of thermal conductivity,  ,

and the negative local temperature gradient,  . The heat flux density is

the amount of energy that flows through a unit area per unit time.

where (including the SI units)

 is the local heat flux density, W·m−2

 is the material's conductivity, W·m−1·K−1,

 is the temperature gradient, K·m−1.

The thermal conductivity,  , is often treated as a constant, though this is not

always true. While the thermal conductivity of a material generally varies

with temperature, the variation can be small over a significant range of

temperatures for some common materials. In anisotropic materials, the

thermal conductivity typically varies with orientation; in this case   is

represented by a second-order tensor. In non-uniform materials,   varies

with spatial location.

For many simple applications, Fourier's law is used in its one-dimensional

form. In the x-direction,

Integral form

By integrating the differential form over the material's total surface  , we

arrive at the integral form of Fourier's law:

where (including the SI units):

 is the amount of heat transferred per unit time (in W), and

 is an oriented surface area element (in m2)

The above differential equation, when integrated for a homogeneous

material of 1-D geometry between two endpoints at constant temperature,

gives the heat flow rate as:

where

A is the cross-sectional surface area,

 is the temperature difference between the ends,

 is the distance between the ends.

This law forms the basis for the derivation of the heat equation.