---
title: "Transport Phenomena - MML215"
tags : "SEM4, MME"
---

Weld zone - Martinsite structure


Numerical Oriented

## Importance of Heat Transfer

- Metallurgical Engineering
- Furnace
- Requirement of energy/heat - Fuel/Electricity
- Heat requirement to carry out processes /chemical reactions
- Heat Losses
- Doors, roofs, furnace walls, holes
- Cooling water
- Heat loss with flue gases - gas that emanates from combustion plants
- Selection of suitable insulating material, K - thermal conductivity
- Determination of thermal efficiency - heat input, output, losses
- Design of fuenaces, heat exchangers - heat transfer area, heat capacity of fluids
- Metal Casting
- Solidification of molten metal
- Cooling rates
- Heat Treatment
- Heating rate/ cooling rate
- Homogenization of temperature
- Transformation temperature
- Thermal Stresses
- Quenching medium and cooling rates - water, oil, brine, air
- Reheating of semi-finished products
- Heating rate
- Uniform temperature distibution
- Size and shape of the stock
- Temperature gradients - thermal stresses
- Thermal properties of materials - k
- Step heating
- To avoid development of thermal stresses
- Metal cutting/machining
- Heat generation/lubrication
- Welding
- Solidification of weldments
- Power requirements/heat transfer aspects
- Heat input = $\displaystyle\frac {VI\times60}{S\times1000}KJ/mm$
V - volts(V)
I - current(A)
S - travel speed(mm/min)

Which mechanical property is less sensitive to change in microstructure?


## Conduction Heat Transfer

### Fourier's Law of Heat Conduction

Rate of heat conduction across an isothermal surface is proportional to the temperature gradient normal to the surface

$\dfrac{dQ}{dA} \propto \dfrac{dT}{dx}$

$\dfrac{dQ}{dA} = -K \dfrac{dT}{dx}$

$\frac{dQ}{dA}$ - Heat flux (w/m^2^)
dT - Temperature difference
dx - Distance measure normal to the surface
K - Proportionality constant, Thermal Conductivity (W/m°C)

Negative sign indicates that direction ofheat flow is down the temperature gradient

Fourier's law of heat conducation is applicable to steady state



### Mechanism and effect of temperature on thermal conductivity

- Kinetic Energy of molecules in gases vary with temperature - high temperature molecules have high velocity - continous random motion colliding and exchange energy and momentum
- Movement of a molecule from a high temperature region to a low temperature region results in transport of kinetic energy through collisions with lower energy molecules

Solids
- Lattice vibration
- Transport of free electrons

Vibration energy mode of transport is not as large as electron transport - good eletrical conductors are almost always good heat conductors

### Thermal Conductivity, K (W/mC)

Factors affecting

- Temperature
- K = K~0~(1+αT)
- Purity and alloying
- Alloying additions in metallic materials, decreases K
- Important in design
- Anisotropy
- Varying K in different directions in anisotropic materials

#### Heat flow in a cylindrical shell, tube,or pipe


r~1~ - Internal Radius
r~2~ - Outer Radius
L - Length of cylinder
r - Radius of very thin cylinder concentric with main cylinder with thickness dr at temp T

$Q = \dfrac{2\pi KL(T_1-T_2)}{\text{ln}(r_2/r_1)}$

$R_t = \displaystyle \frac{ln(r_2/r_1)}{2\pi KL}$

#### Radial Heat Flow in a Spherical shell


$Q = \displaystyle-K4r^2\pi\frac{dT}{dr}$

$dT = -\frac{Qdr}{4\pi K r^2}$

$Q = \dfrac{4\pi K (T_1-T_2)r_1r_2}{r_2-r_1}$

$R_{shell} = \displaystyle\frac{r_2-r_1}{4\pi K r_1r_2}$

###### The general Solution procedure consists of the following

1. Known
- State briefly and concisely what is known about the problem.
2. Find
- State briefly what are quantities that must be found.
3. Schematic
- Draw a schematic diagram of the physical system.
4. Assumptions
- List all simplifying assumptions.
5. Properties
- Compile property values needed for calculations.
6. Analysis
- Apply appropriate laws and introduce equations as needed. Substituting numerical values. Check the units before substituting any numerical value. Perform the calculations needed to obtain the desired result.
7. Comments: Discuss your results

### Composite Plain walls with convective boundaries


Driving force $= \frac{1}{h\times a}$

h - Convection heat transfer coefficient of fluid medium in contact with the resistor


### Overall heat Transfer coefficient, U (W/m^2^C)

Represents the total thermal resistance in the system
It is denoted by reciprocal of total

For a Crylindrical Shell/pipe with convective boundaries


If planar insulation is increased - the heat loss decreases

If Cylindrical insulation is increased - the heat loss may increase because the area of the heat transfer for convection increases

### Critical Radius of Insulation, r~c~

Addition of insulation does not always bring bring about a **decrease** in heat transfer rate. In fact under certain condition, addition of insulation **increases** heat transfer rate


r~0~ - Outer Radius of the pipe
r~1~ - Radius of insulation
T~0 - Temperature of the outer surface of the pipe
T~∞~ - Temperature of surrounding fluid
K~i~ - Thermal Conductivity of Insulation
L - Length of pipe
h - Convective heat transfer coefficient

$Q = \displaystyle\frac{T_0 - T_\infty}{\frac{ln(r_1/r_0)}{2\pi K_1L}+\frac{1}{2\pi r_iLh}}{}$

Addition of insulation may either decrease or increase heat flow


$r_c = \displaystyle\frac{K_i}{h}$

h - Convective Heat Transfer Coefficient
K~i~ - Thermal Conductivity of insulate

- r~1~ = r~c~ - Minimum value of resistance - max Heat Loss
- r~1~ < r~c~ - Q increases until r~1~ = r~c~ - High K low h
- r~1~ > r~c~ - Q decreases - r~0~ and small r~c~ (good insulating material with low K and high h)

### Effect of variable thermal conductivity

Variation of K with temperature

$Q = K_0A[T_1-T_2][1+\displaystyle\frac\alpha2(T_1+T_2)]$

## 3D Conduction Heat Transfer


To set up an equation for conduction heat transfer when temperature is changing with time and with internal heat generation energy conducted in left face + heat generated = Change in internal energy + energy conducted out of right face

Change in internal Energy = $\rho CA\frac{\partial T}{\partial t}dx$

$Q_x = -KA \frac{\partial T}{\partial x}$



### Thermal Diffusivity, α(m^2^/s)

It is a ratio of thermal conductivity of the material to the columetric heat storage capacity of the material

$α = \dfrac k{\rho C_p}$

### General Heat conduction Equations



- For isotropic material with constant K with internal heat generation unfer 3D non steady state heat transfer condition

- For isotropic material with constant K without internal heat generation under 3D non steady state



## Transint (Unsteady state ) Conduction heat transfer

- Variation of temperature of the body with time
- T = f(x,y,z,t)
- Periodic heat flow - Temperature caries on regular basis
- Non periodic heat flow - Non linear variation of temperature with time
- Heating or cooling of body is dependent on two thermal resistances
- Internal conduction resistance
- Surface Convective resistance

#### Lumped Heat Capacity Analysis

- Negligible internal resistance
- High Conductivity
- Thin wires, thin shells


From Newton's Law of cooling

$Q = \rho C_P V\dfrac{d\theta}{dt} = -hA(T-T_\infty) = -hA\theta$

$\rho C_P V\dfrac{d\theta}{dt} = -hA\theta$

Integrating -
At t = 0, T = T~i~, $\theta = T_i - T_\theta$
At t = t, T = T, $\theta_i = T - T$

$\dfrac{\theta}{\theta_i} = \text{ exp}\left(\dfrac{-hA}{\rho C_P V} t\right)$

$\dfrac{T-T_\infty}{T_i - T_\infty} = e^{\dfrac{-hA}{\rho C_P V}t}$
<!---
-->


Characteristic Length


Instantaneous and Total Heat Flow


Error Frunction Values


Heat Transfer rate
- Instantaneous at a depth
-



# Transfer From extended Surfaces - Fins

$Q = hA(T - T_\infty)$

To increase heat transfer by convection
- Increase h
- Increase A
- Increase $(T-T_\infty)$


Knowledge oftemperature distribution through it

Assumptions
- Thickness of fin is small compared with length and width
- 1D heat conduction
- K fin is constant
- h is constant over entire fin area
- No Internal generation
- Contact resistance is negligible
- Negligible radiation heat transfer

## Heat transfer by convection out of the element




## Effectiveness of Fin

Ratio of heat transfer rate with fin to heat transfer rate without fin

$ε_f = \displaystyle\frac{\sqrt{hPKA_C}(T_0-T_\infty)}{hA_C(T_0-T_\infty)}$

$ε_f = \displaystyle\sqrt{\frac{PK}{hA_C}}$

For infinitely long fin

- ε~f~ > 1

### Fin Efficiency, $\eta_f$

It is defined as the ratio of actual heat transferred to the heat transferred if entire fin area is at base temperature

- Measure of thermal performance of fins
- Maximum driving potential for convection is the temperature difference between the base of fin and the surrounding fluid
- Maximum rate of heat transfer would occur if entire fin area is at base themperature - fins have finite temperature gradient along the lenhth of fin

### For straight fin of uniform cross section with no heat loss from the tip

$\eta_f = \displaystyle\frac{\text{tanh }mL}{mL}$

If the width of rectangular fin is much larger than the thickness,perimeter may be aproxiamted as P=2w and

$mL_c = \left[\frac{hP}{kA_C}\right]^{-1/2}L_c$

Multiplying numerator and denominator by L~c~ ^1/2 and introducing corrected profile area A~p~=L~c~ t,


Fin Efiiciency curves


### Fin Design

- Minimize the material and manufacturing cost
- Straight triangular fun
-



# Radiation Heat Transfer

- Heat trasnfer
- All substances with body temperature above absolute temperature continously emit energy in the form of radiation, proportional to the temperature of the body
- heat transfer by radiation occurs in the form of electromagnetic waves
- No material medium is required
- Energy released by a radiating surface is not continous but in the form of discrete packets or quanta of energy called as photons
- At receiving surface, there is reconversion of wave motion into thermal energy
- The general phenomenon of radiation conversion includes the propagation of electomagnetic waves of all wavelengths
- Thermal radiation is limited to 0.1-100µ - Visible, Infrared and a part of UV

Q = Q~r~ + Q~t~ + Q~r~
1 = a + r + t

- Metals absorb radiation within a fraction of µm
- Insulators absorb radiation within a fraction of mm

## Black Body

- Absorbs all the incident radiant energy
- It is a perfect emitter, e = 1
- Emissive power - Thermal radiations emitted by the body per unit surface area per unit time in all directions is called emissive power(E)
- Emissivity - It is the ratio of emissive power of the body to the emissive power of the perfectly black body

$e = \dfrac{E}{E_b}$
$e = f(T,\lambda)$

- Intensity of Radiation (I) - It is the rate at which radiations are emitted in the direction normal to the surface per unit projected area
- Spectral Radiation Intensity $(Ib_\lambda)$ - amount of energy streaming through a unit area projected normal to the given direction per unit time per unt solid angle per unit wavelength
- Total radiation Intensity $Ib$ - Amount of energy emitted over the entire wavelength spectrum from $\lambda = 0$ to $\infty$

Normal Total Emissivity (en)

Ratio of normal component of the total emissive power of surface to the normal component of total emissive power of black body at same temperature and angle of emission

$en = \dfrac{En}{Eb_n}$

Emissivity depends on
- Nature
- Colour
- Texture
- Roughness
- Temperature
- Wavelength
- Angle of emission
- Nature of surface


## Gray Body

$E = \underset{0}{\overset \infty \int}e_\lambda Eb_\lambda d\lambda$

$e_\lambda$ - Monochromatic Emissivity
$Eb_\lambda$ - Black body emissivity

$\displaystyle Eb = \overset{\infty}{\underset{0}{\int}} Eb_\lambda d\lambda = \sigma T^4$

### Stefan Boltzmann Law of Black Body Radiation

The amount of radiant energy emitted per unit time per unit area of black body is proportional to the fourth power of its absolute temperature

Eb = σT^4^
Stefan Boltzmann constant
σ = 5.672x10^-8^ W/m^2^K^4^

$\displaystyle Eb = \overset{\infty}{\underset{0}{\int}} Eb_\lambda d\lambda = \overset{\infty}{\underset{0}{\int}}\frac {C_1}{\lambda^5(e^{C_2/\lambda T}-1)}d\lambda$

- Black body is a diffused emitter

### Wein Displacement Law

$\lambda_{\text{max}}T = C_3$

$C_3 = 0.298 \times 10^{-3} mK$

The displacement of maximum value of radiation to shorter wavelengths and higher temperature is given by Wein's Law

### Kirchhoff's Law

At thermal equilibrium, the emissivity of the body is equal to the absorbivity of the body

Net radiant energy exchange for non black surfaces = (E - aEb)

At T = Tb, i.e thermal equilibrium

E - aEb = 0

### Lambert's Consine Law

Diffuse surface radiates energy is such a manner that rate of radiant energy in any particular direction is proportional to cosine of the angle between the direction under consideration and normal to the surface

$Eb_\theta = Eb_n\text{ cos}\theta$

### Plank's law

$\displaystyle Ib_{\lambda n} = \frac {2hc_0^2}{\lambda^5(e^{hc_0/\lambda kT}-1)}$

$\displaystyle Ib_{\lambda n}$= Normal radiation intensity of wavelength.
$h = 6.66356 \times 10^{-34} Js$
$k = 1.3805 \times 10^{-23} J/K$

The monochromatic emissive power of a black body is
$Eb_\lambda = \pi Ib_{\lambda n}$
$Eb_\lambda = \dfrac {C_1}{\lambda^5(e^{C_2/\lambda T}-1)}$

$C_1 = 3.743 \times 10^8 \mu m^4/m^2$
$C_1 = 1.438 \times 10^4 \mu mK$

### Gray surfaces

Emissivity and absorbing ability is independent of wavelength over the spectral region. Real surfaces are not always grey in nature. However most


#### Fractional emissive power over a certain range of wavelength

Total Emissive Power

$Eb = \displaystyle \underset{0}{\overset{\infty}{\int}} Eb_\lambda\,d\lambda$

Emissive power over a range

$Eb_{(\lambda_1 - \lambda_2)} = \displaystyle \underset{\lambda_1}{\overset{\lambda_2}{\int}} Eb_\lambda\,d\lambda = \underset{0}{\overset{\lambda_2}{\int}} Eb_\lambda\,d\lambda - \underset{0}{\overset{\lambda_1}{\int}} Eb_\lambda\,d\lambda$

$F(\lambda_1 - \lambda_2) = \dfrac{1}{\sigma T^4}\left[\underset{0}{\overset{\lambda_2}{\int}} Eb_\lambda\,d\lambda - \underset{0}{\overset{\lambda_1}{\int}} Eb_\lambda\,d\lambda\right]$

$F(\lambda_1 - \lambda_2) = F(0 - \lambda_2) - F(0 - \lambda_1)$

F is defined in terms of '$\lambda T$'

$F(\lambda_1T - \lambda_2T) = F(0 - \lambda_2T) - F(0 - \lambda_1T)$

## Radiant heat exchange between two black surface


A~1~, A~2~ = Areas of black surfaces
dA~1~, dA~2~ = Elementary areas on black surfaces
r = Distance between surfaces
θ~1~, θ~2~= Angles normal to surfaces and line joining them
T~1~, T~2~ = Temperature of two surfaces

## Radiation Shape Factor

View Factor (F)

F~12~ is the ratio of the radiant energy from surface 1 impinging on surface 2 to the total radiation emitted by surface 1

$Q_{12} = A_1F_{12}\sigma T_1^4$
$Q_{21} = A_1F_{21}\sigma T_1^4$

At equilibirum T~1~ = T~2~

$Q_{12} (\text{net}) = A_1F_{12}\sigma T_1^4 - A_2F_{21}\sigma T_2^4$

$A_1F_{12} - A_2F_{21} = 0$

Since $\sigma$ and T~1~ are non zero

$Q_{12} (\text{net}) = A_1F_{12}\sigma (T_1^4 - T_2^4) = A_2F_{21}\sigma (T_1^4 - T_2^4)$

### Radiation Shape factor algebra

The value of radiation shape factor depends upon geometry and orientation of the surface with respect to each other

- Reciprocity Theorem
- A~1~F~12~ - A~2~F~21~ = 0
- If the transmitting surface is subdivided the shape factor for that surface wrt to receiving surface is not equal to sum of individual shape factors
- The shape factor for a transmitting surface to a subdivided receiving surface is sum of the individual shape factors
- For flat or convex surface shape factor with respect to itself (F~11~) is zero
- For concave surface F~11~ is not zero
- In an enclosure with one surface emitting radiation to number of surfaces
F~12~+F~13~+F~14~+-----+F~ij~=1

## Radiant heat exchange between non black bodies

- Involves more complex calculation
- Reflectivity of the body
- e and a are not uniform in all directions and for all wavelengths
- For a gray body, $e_\lambda$ and $a_\lambda$ are constant over entire wavelenth spectrum

Small Gray Bodies
Consider two small gray bodies with e1 e2 as emissivity's. Their size is small compared to distance between them.

Energy emitted by body 1
= $e_1 A_1 \sigma T_1^4$

Energy incident upon body 2
= $F_{12}e_1 A_1 \sigma T_1^4$

Energy absorbed by body 2 = $a_2F_{12}e_1A_1\sigma T_1^4$

$Q_1 = e_1 e_2 A_1 F_{12}\sigma T_1^4$

Net radiant heat exchange between two bodies
$Q_{12} = e_1e_2A_1F_{12}\sigma (T_1^4 - T_2^4)$

### Small Gray Body in a Large Enclosure

The large enclosure acts like a black body, it absorbs practially all the radiations incident on it and reflect negligible energy to small gray body

Energy emitted by small body 1 and absorbed by large enclosure 2 = $A_1e_1\sigma T_1^4$

Energy emitted by enclosure = $A_2e_2\sigma T_2^4$

Energy incident upon small body = $F_{21}A_2e_2\sigma T_2^4$

$Q_{12}= eA_1\sigma(T^4_1-T^4_2)$

$Q_{\text{net}} = f_{12}F_{12}\sigma A_1(T_1^4 - T_2^4)$


$Q_{\text{n shields}} = \left(\dfrac 1{n+1}\right) Q_{\text{without shield}}$

### Two Infinite parallel planes

Gray surfaces

e~1~ and e~2~ are emissivites of palnes and e~3~ is emissivity of radiation sheild (both sides)paced between them

$Q~12~(without sheild)=A\sigma f~12~ F~12~(T1^4-T2^4)$

With Radiation Shield

$Q_{net} = \dfrac{A\sigma(T_1^4 - T_2^4)}{\left(\dfrac{1}{e_1} + \dfrac{1}{e_3}-1\right)+\left(\dfrac{1}{e_3}+\dfrac{1}{e_2} - 1\right)}$

$Q_{net} = \dfrac{A\sigma (T_1^4 - T_3^4)}{\left(\frac{1}{e_1} + \frac{1}{e_3}-1\right)} = \dfrac{A\sigma (T_3^4 - T_2^4)}{\left(\frac{1}{e_1} + \frac{1}{e_2}-1\right)}$

$T_3^4 = \dfrac{T_1^4\left(\frac{1}{e_3} + \frac{1}{e_2}-1\right)+T_2^4\left(\frac{1}{e_1} + \frac{1}{e_2}-1\right)}{\left(\dfrac{1}{e_1} + \dfrac{1}{e_3}-1\right)+\left(\dfrac{1}{e_3}+\dfrac{1}{e_2} - 1\right)}$

Radiocity(J):
- Total radiant energy leaving a surface per unit time per unit surface area

Irradiations (G):
- $J=E+ \rho G$
- Total radiant energy incident upon a surface per unit time per unit area.Some of it may be reflected to become part of 'J'

$\dfrac 1{A_1F_{12}}$ is space resistance due to distance and geometry of radiating surface

2 Surface resistances for each shiel and one for weach radiating plane

# Convection Heat Transfer

- Mechanism by which heat is transferred between a solid surface and a fluid moving adjacent to it
- Process of energy transfer affected by the circualtion or mixing of fluid medium
- Free or natural convection
- Bulk fluid motion is due to fluid density variation - temperature gradient within the fluid
- Forced convection
- Fluid motion is set by some superimposed velocity field

Analysis of heat transfer by covection requires knowledge of
1. Velocity of fluid flow
2. Temperature field in a fluid flow
3. Type and pattern of fluid flow
4. Velocity distribution or profile
5. Pressure drop experienced during fluid flow
6. Properties of fluid: density, viscosity and their effects

Laminar Flow
- Streamlined flow of fluid
- No transverse displacement of fluid particles
- Transfer of momentum by molecular collision
- Fluid particles retain their position ar any point during flow

Ideal Fluids
- No friction
- Invisicd - zero viscosity
- Incompressible

Real Fluid
- Particles are constantly interacting
- Fluid friction is created
- Shear forces oppose motion of one particle past another
- Viscosity

Turbulent Flow
- Irregular, erratic flow
- Velocity of individual fluid element fluctuates
- Transfer of momentum between large group of molecules
- Pressure and velocity are not constant at any point during flow

Nature of Flow depends on
- Mean flow velocity V
- Density of fluid $\rho$
- Dynamic viscosity of fluid µ
- Characteristic dimensionm diameter d

$R_e = \dfrac{dv\rho}{\mu}$

Through a pipe
- Laminar Flow - R~e~ < 2300
- Turbulent Flow - R~e~ > 4000
Equivalent diameter
$dc = 4\times\dfrac{\text{cross section flow area}}{\text{wetted parameter}}$


## Boundary Layer Concept


Viscous Shear stress
$\tau = µ \dfrac{du}{dy}$

µ - Dynamic Viscosity
$\dfrac{du}{dy}$ - Normal velocity gradient

Characteristics of boundary layer

- Variable thickness in x-direction due to variable velocity gradient along the flow
- Viscous force or shear stress dissipates more and more energy of fluid stream
- Higher velocity - less quantum of boundary layer thickness
- Laminar - Velocity profile - parabolic - Turbulent - velocity profile is flat and boundary layer thickness is large
- Pattern of flow in boundary layer is judged by Reynold's number - critical point depends on surface roughness, plate curvature, intensity of tubulence of free stream flow

At some distances from the entrance of tube


### Thermal Boundary Layer

- Develops when a fluid at a specified temperature flows over a surface that is at a differnt temperature
- The flow region over the surface in which the temperature variation in the direction normal to the surface is significant

$\delta_t$ - distance from the surface at which the temperature difference (T-T~s~) equals 0.99($T_\infty-T_s$)

Momentum Equation for Boundary Layer

$U\dfrac{\delta u}{\delta x} + V\dfrac{\delta u}{\delta y} = \dfrac{\mu}{\rho}\dfrac{\delta^2 u}{\delta^2 y}$

Thickness of boundary layer
$\delta = \dfrac{5x}{\sqrt{R_e}}$

Laminar Boundary layer on flat plate
Drag Coefficient
$C_f = \dfrac{1.328}{\sqrt{R_e}}$

Turbulent flow over flat plate
$C_f = 0.074R_e^{-0.2}$

Drag force experienced
$F_d = \dfrac 12 \rho v^2 C_f/ \text{Area}$

Thermal Boundary Layer
$\delta_t = \dfrac{\delta}{P_r^\frac 13}$

Thickness of velocity boundary layer in turbulent
$\dfrac{\delta}{x} = 0.381Re_x^\frac{-1}5$

Pressure Loss Darcy Weisbach Equation

$\Delta p = f_D\dfrac LD\dfrac{\rho V^2}{2}$

$\Delta p$ - pressure loss in N/m^2
f~D~ - dracy friction factor
L - pipe length in m
D - hydraulic diameter in m
V - fluid diameter in m
$\rho$ - fluid density kg/m^3^

#### Friction factor is evaluated by moody diagram


Fraction force vc Reynolds number

- The friction factor depends not only on Reynold's number but also on roughness of the pipe surface
- Moody diagram gives a plot of friction factor'f' VS Reynolds number 'Re' with relative roughness epsilon/D as a parameter

- Thickness of Boundary Layer,
$\delta = \dfrac{5x}{\sqrt{Re}}$
- For laminar boundry layer on a flat plane
Drag coefficient C~f~ = $\dfrac{1.328}{\sqrt{Re}}$
- For fully turbulant flow over flat plate
Drag coefficient (skin coeff)c~f~ = 0.074Re^-0.2^

Laminar Flow
- Plane Surface - f = 1.292/Re^0.5^
- Circular Pipe - f = 64/R~e~

Turbulent
- Plane Surface - f = 0.074/Re^0.2^
- Circular Pipe - f = 0.316/Re^0.25^

For channels lined with firebrick
f = 0.175/Re^0.12^
For metallic tubes and turbulent flow:
f = 0.03- 0.015

Thickness of velocity boyndary later in tyrbulent flow

$\dfrac \delta x = 0.381 Re_x^{-1}$

The friction fact

Continuity Equation
$V_{\text{rate 1}} = V_{\text{rate 2}}$
$A_1\times v_1 = A_2\times v_2$

$A$ - Area of cross section
$v$ - Rate of flow

#### Dimensionless numbers

- Reynold's number ($R_e$)
$R_e = \frac{D.v.\rho}{\mu} = \frac{D.v}{u}$

$\mu$ = viscocity
$\rho$ = density of the fluid
$D$ = Diameter of the pipe
indicates ratio of inertia force to viscous force.
- Grashoff's Number
$Gr=L^3\rho^2\beta g \Delta T/\mu^2$
$= \text{Buoyant Force} \times \dfrac{\text{Inertia Force}}{\text{Viscous Force}^2}$

- Prandtl Number Pr

$Pr = \dfrac{\mu C_p}{K} = \dfrac v\alpha$

$P_r= \frac{\text{kinematic viscosity}}{\text{Thermal diffusivity}}$

Kinematic Viscosity - Momentum transport by molecular friction and thermal diffusivity represent heat energy transport by conduction

Thus Pr provides measure of realtive effectiveness of Momentum and Energy transport by diffusion

- Nusselt No
- $Nu = \dfrac{hL}{K}$
K - Thermal Conductivity of Liquid Medium
L - Dimension
h - Heat Transfer Coefficient
- Indicates ratio of temperature gradient at surface to an overall reference temperature gradient
- Stanton No
$St= \frac{Nu}{Re.P_r}$
or
$St= \frac{Nu}{P_e}$
where $P_e$ = Peclet number

- Indicates ratio of convection heat transfer coefficient to heat transfer coefficient to heat flow per unit temperature rise due to velocity of fluid

### Forced Convection

#### Flow over plates


- Laminar Flow
- $Nu_x= 0.332 Re_x^{0.5} Pr^{0.33}$
- $Nu = 0.664 Re^{0.5}Pr^{0.33}$
- Proportion of fluid at T~m~
- Restrictions - $Re \geqslant 40000$ & $Pr > 0.6$
- Turbulent Flow
- $Ne = 0.036 Re^{0.8}Pr^{0.33}$
- Proportion of Fluid at T~m~
- Restrictions - Re > $5\times 10^5$

### Liquid Metal Convection Heat Transfer

- High heat transfer rates
- High thermal conductivity of liquid metal as compared to fluids
- Prandtl number for liquid metals is verlow, of the orfer of 0.01 so thermal boundary layer thickness should be substantially larger than hydrodynamic
- $\frac{\delta}{\delta t}$ is small & velocityprofile has a blunt shape


- Seider and Tate Correlation
$Nu = 1.86(Re.Pr)^{\frac 13}(\frac{D}{L})^{\frac{1}{3}}(\frac{\mu}{\mu_w})^{0.14}$

Fluid properties are ev aluated at bulk mean temperature
Pe=Peclet No= Re.Pr =$\frac{D.V.\rho.Cp}{k}$



### Physical state of heat exchanging process
- Condenser
- Hot fluid at constant temperature gives heat to cold fluid
- hot fluid loses its latent heat
- Evaporator
- Cold fluid (boiling water)evaporates at constant temperature and temperature of hot gas decreases

#### Performance variation in heat exchanger
- Mass flow rate
- Specific heat
- inlet and outlet temperature
- Surface Area
- K
- Degree of scaling


# Mass Transfer

- Mass in transit as a resualt of a species concentration difference in a mixture
- Modes of Mass Transfer
- Diffusion Mass Transfer (Molecular)

## Fick's Law of Diffusion

Analogus to the heat transfer by diffusion, i.e. conduction, the rate equation for mass diffusion is given by Fick's Law, for transfer of species A in a binary mixture of A+B

$\dot J_A = -P\cdot D_{AB}\nabla m_a$

$J_A^* = -c\cdot D_{AB}\nabla x_a$

$\dot J_A$ - Mass flux of species A (kg s^-1^ m^-2^) - Amount of A transferred per unit time per unit are perpendicular to direction mass transfer

D~AB~ - MAss Diffusivity (Binary diffusion coefficient)

# Heat Exchangers

- used to transfer heat between two fluids, which may be in direct contact or indirect contact seprated by a wall. Ex boilers,radiators,evaporators,ect.
- Types
- Direct
- Indirect or no direct
- Arrangement of flow path
- Parallel Flow
- Poorest from thermodynamical point of view
- Large heat transfer area is required
- ▲ T value decreases over length
- Constant performance over wide range of flow rates
- Colder fluid can never reach exit temperatures of hot fluid
- 

- counter Flow
- Thermodynamically better arrangement
- Requires small heat transfer area
- ▲T is approxiamtely constant over length of tube
- Problem to make compact heat exchanger
- Cold fluid may reach the exit temperature and exceed it
-
- Cross flow
- Intermediate performance between Parallel and Counter parallel Heat Exchanger
- Permits use of extended surfaces
- it is compact
- if one if the fluid is gas it is used