Module 2
Conduction II
Problem
Problem
A 1.6 m diameter sphere is buried in soil with
center at a depth of 5.5 m. Heat is generated in
the sphere at a rate of 580 W. If the conductivity
of the soil is 0.51 W/m0C and the soil surface is
at 60C, Calculate the surface temperature of the
sphere under steady state condition.
UNSTEADY HEAT TRANSFER
• Many heat transfer problems require the understanding of the
complete time history of the temperature variation.
• For example, in metallurgy, the heat treating process can be
controlled to directly affect the characteristics of the processed
materials.
• Annealing (slow cool) can soften metals and improve ductility.
• On the other hand, quenching (rapid cool) can harden the strain
boundary and increase strength. In order to characterize this
transient behavior, the full unsteady equation is needed:
T 1 T
c k T , or
2
2T
t t
k
where = is the thermal diffusivity
c
Biot Number (Bi)
•Defined to describe the relative resistance in a thermal circuit of
the convection compared
hLc Lc / kA Internal conduction resistance within solid
Bi
k 1 / hA External convection resistance at body surface
Lc is a characteristic length of the body
Bi→0: No conduction resistance at all. The body is isothermal.
Small Bi: Conduction resistance is less important. The body may still
be approximated as isothermal
Lumped capacitance analysis can be performed.
Large Bi: Conduction resistance is significant. The body cannot be treated as
isothermal.
Transient heat transfer with no internal
resistance: Lumped Parameter Analysis
Valid for Bi<0.1
Solid
Total Resistance= Rexternal + Rinternal
dT hA T t 0 Ti
GE: T T BC:
dt mc p
Solution: let T T , therefore
d hA
dt mc p
Lumped Parameter Analysis
i Ti T
hA
ln t
i mc p
hA
t
e
m cp
i
T T
m cp - To determine the temperature at a given time, or
t
e hA - To determine the time required for the
Ti T temperature to reach a specified value.
Note: Temperature function only of time and not of space!
Lumped Parameter Analysis
T T hA
T exp( t)
T0 T cV
hA hLc k 1 1
t t Bi 2 t
cV k c Lc Lc Lc
Thermal diffusivity: k (m² s-1)
c
Problem
A 50 cm X 50 cm copper slab 6.25 mm thick has
a uniform temperature of 3000C. Its
temperature is suddenly lowered to 360C.
Calculate the time required for the plate to
reach the temperature of 1080C. Take
density=9000 kg/m3, C = 0/38 kJ/kg0C, k = 370
W/m0C and h = 90 W/m2 0C
Problem
A solid copper sphere of 10 cm diameter
(density=8954 kg/m3, Cp= 383 J/kg k, k=386
W/m k), initially at a uniform temperature ti=
2500C, is suddenly immersed in a well-stirred
fluid which is maintained at a uniform
temperature ta= 500C. The heat transfer
coefficient between the sphere and the fluid is
h=200 W/m2 k. Determine the temperature of
the copper block 5 minutes after immersion.
Problem
An average convective heat transfer coefficient
for flow of 900C air over a flat plate is measured
by observing the temperature time history of a
40 mm thick copper slab (density=9000 kg/m3,
Cp= 0.38 kJ/kg0 C, k=370 W/m0 C) exposed to
900C. In one test run, the initial temperature of
the plate was 2000C and in 4.5 minutes the
temperature decreased by 350C. Find the heat
transfer coefficient for this case. Neglect
internal thermal resistance.
Problem
The heat transfer coefficients for the flow of air at 28OC
over a 12.5 mm diameter sphere are measured by
observing the temperature-time history of a copper ball
of the same dimension. The temperature of copper ball
(c=0.4 KJ/Kg K and density = 8850 kg/m3) was
measured by two thermocouples, one located in the
center and other near the surface. Both the
thermocouples registered the same temperature at a
given instant. In one test the initial temperature of the
ball was 65OC and in 1.15 minute the temperature
decreased by 11OC. Calculate the heat transfer
coefficient for this case. (Ans: 37.17 W/m2 K)
Problem
A steel ball 50 mm in diameter and at 9000C is
placed in still atmosphere of 300C. Calculate the
initial rate of cooling of the ball in OC/min.
Ans: 12OC/min
Problem
A cylindrical ingot 10 cm diameter and 30 cm long
passes through a heat treatment furnace which is 6
m in length. The ingot must reach a temperature of
8000C before it comes out of the furnace. The
furnace gas is at 12500C and ingot initial
temperature is 900C. What is the maximum speed
with which the ingot should move in the furnace to
attain the required temperature? The combined
radiative and convective surface heat transfer
coefficient is 100 W/m2 OC. Take k (steel) = 40
W/mOC and thermal diffusivity of steel = 1.16 x 10-5
m2/s. (Ans: 0.0165 m/s)
Problem
A solid copper cylinder of 7 cm diameter is
initially at a temperature of 25OC and it is
suddenly dropped into ice water. After 3
minutes the temperature of the cylinder is again
measured as 1OC. Determine unit surface
conductance by using lumped heat analysis
method.
(Ans: h=1073.21 W/m2K)
Time constant
Problem
Heat flow in semi-infinite solids
• A solid which extends itself infinitely in all
directions of space is known as infinite solid.
• If an infinite solid is split in the middle by a
plane, each half is known as semi infinite
solids.
Transient heat flow in Infinite plate
Problem
Module 3
Forced Convection
Convection
Free or natural convection
(induced by buoyancy May occur with
forces) phase change
Convection (boiling,
condensation)
forced convection (driven
externally)
Heat transfer rate q = h( Ts-T )W
Typical values of h (W/m2K)
h=heat transfer coefficient (W /m2K) Free convection: gases: 2 - 25
(h is not a property. It depends on liquid: 50 - 100
geometry ,nature of flow,
thermodynamics properties etc.)
Forced convection: gases: 25 - 250
liquid: 50 - 20,000
Boiling/Condensation: 2500 -100,000
Forced convection: Non-dimensional groupings
• Nusselt No. Nu = hx / k = (convection heat transfer strength)/
(conduction heat transfer strength)
• Prandtl No. Pr = / = (momentum diffusivity)/ (thermal diffusivity)
• Reynolds No. Re = U x / = (inertia force)/(viscous force)
Viscous force provides the dampening effect for disturbances in the
fluid. If dampening is strong enough laminar flow
Otherwise, instability turbulent flow critical Reynolds number
d d
Laminar Turbulent
FORCED CONVECTION:
external flow (over flat plate)
An internal flow is surrounded by solid boundaries that can restrict the
development of its boundary layer, for example, a pipe flow. An external flow, on
the other hand, are flows over bodies immersed in an unbounded fluid so that the
flow boundary layer can grow freely in one direction. Examples include the flows
over airfoils, ship hulls, turbine blades, etc.
•Fluid particle adjacent to the
les solid surface is at rest
T
•These particles act to retard the
Ts motion of adjoining layers
x q • boundary layer effect
Momentum balance: inertia forces, pressure gradient, viscous forces,
body forces
Energy balance: convective flux, diffusive flux, heat generation, energy
storage
h=f(Fluid, Vel ,Distance,Temp)
How to solve a convection problem ?
• Solve governing equations along with boundary conditions
• Governing equations include
1. conservation of mass
2. conservation of momentum
3. conservation of energy
• In Conduction problems, only (3) is needed to be solved.
Hence, only few parameters are involved
• In Convection, all the governing equations need to be
solved.
large number of parameters can be involved
Boundary layer equations (laminar flow)
• Simpler than general equations because boundary layer is thin
T
U
U
y dT
d
x TW
• Equations for 2D, laminar, steady boundary layer flow
u v
Conservation of mass : 0
x y
u u dU u
Conservation of x - momentum : u v U
x y dx y y
T T T
Conservation of energy : u v
x y y y
dU
• Note: for a flat plate, U is constant, hence 0
dx
Exact solutions: Blasius
d 4.99
Boundary layer thic kness
x Re x
w 0.664
Skin friction coefficien t C f 1
2 U
2
Re x
Re U x , u
x w
y
y 0
UL
L
1 1.328
Average drag coefficien t C D C f dx Re L
L0 Re L
Nux 0.339 Re x Pr
1 1
Local Nusselt number 2 3
N u 0.678 Re L Pr
1 1
Average Nusselt number 2 3
Heat transfer coefficient
• Local heat transfer coefficient:
1 1
Nux k 0.339 k Re x Pr 2 3
hx
x x
• Average heat transfer coefficient:
1 1
Nu k 0.678 k Re L Pr 2 3
h
L L
• Recall: qw h ATw T , heat flow rate from wall
• Film temperature, Tfilm
For heated or cooled surfaces, the thermophysical properties within
temperature of the wall and the free stream; film 2 Tw T
the boundary layer should be selected based on the average
T 1
Turbulent boundary layer
* Re x increases with x. Beyond a critical value of Reynolds number
(Re x Re xc ), the flow becomes transitio nal and eventually turbulent .
U xc
Re xc (For flow over flat plate, xc 5 10 5 )
* Turbulent b.l. equations are similar to laminar ones, but infinitely
more difficult to solve.
* We will mainly use correlatio ns based on experiment al data :
C f 0.059 Rex 0.2 (Re x 5 10 5 )
C D 0.072 Re L
1
Re L
0.072 Re0xc.8 1.328 Re0xc.5
Nux 0.029 Re0x.8 Pr
1
3
N u 0.036 Re0L.8 Pr 3 Pr 3 0.036 Re0xc.8 0.664 Re0xc.5
1 1
Nu k
* Calculate heat trans fer coefficien t in usual way : h etc.
x
Problem
Air at 200C, at a pressure of 1 bar is flowing over
a flat plate at a velocity of 3 m/s. If the plate is
maintained at 600C, calculate the heat transfer
per unit width of the plate. Assuming the length
of the plate along the flow of air as 2 m.
Problem
Air at 200C at atmospheric pressure flows over a flat
plate at a velocity of 3 m/s. If the plate is 1m wide
and 800C, calculate the following at x=300 mm.
i) Hydrodynamic boundary layer thickness
ii) Thermal boundary layer thickness
iii) Local friction coefficient
iv) Average friction coefficient
v) Local heat transfer coefficient
vi) Average heat transfer coefficient
vii) Heat transfer
Home work Problem
Air at 200C at atmospheric pressure flows over a
flat plate at a velocity of 3.5 m/s. If the plate is
0.5 m wide and at 600C, calculate the following
at x=0.4 m
i) Boundary layer thickness
ii) Local friction coefficient
iii) Average friction coefficient
iv) Shearing stress due to friction
Forced convection flow across cylinders and
spheres
Problem
Air at 150C, 30 km/h flows over a cylinder of 400
mm diameter and 1500 mm height with surface
temperature of 450C. Calculate the heat loss.
Problem
Air at 300C, 0.2 m/s flows across a 120W electric
bulb at 1300C. Find heat transfer and power lost
due to convection if bulb diameter is 70 mm.
Problem
Air at 400C flows over a tube with a velocity of
30 m/s. The tube surface temperature is 1200C,
Calculate the heat transfer coefficient for the
following cases
i) Tube could be square with a side of 6 cm.
ii) Tube is circular cylinder of diameter 6 cm
Fully Developed Flow
• Laminar Flow in a Circular Tube:
The local Nusselt number is a constant throughout the fully developed
region, but its value depends on the surface thermal condition.
– Uniform Surface Heat Flux (qs ) :
NuD hD 4.36
k
– Uniform Surface Temperature (Ts ):
NuD hD 3.66
k
• Turbulent Flow in a Circular Tube:
– For a smooth surface and fully turbulent conditions Re D 10,000 , the
Dittus – Boelter equation may be used as a first approximation:
NuD 0.023Re4D/ 5 Pr n n 0.3 Ts Tm
n 0.4 Ts Tm
– The effects of wall roughness and transitional flow conditions Re D 3000
may be considered by using the Gnielinski correlation:
NuD
f / 8 Re D 1000 Pr
1 12.7 f / 8 Pr 2 / 3 1
1/ 2
Problem
Water flows inside a tube of 20 mm diameter
and 3 m long at a velocity of 0.03 m/s. The
water gets heated from 400C to 1200C while
passing through the tube. The tube wall is
maintained at constant temperature of 1600C.
Find heat transfer.
Problem
Water at 500C enters 50 mm diameter and 4 m
long tube with a velocity of 0.8 m/s. The tube
wall is maintained at a constant temperature of
900C. Determine the heat transfer coefficient
and the total amount of heat transferred if exit
water temperature is 700C.
Problem
Water at 300C, 20 m/s flows through a straight
tube of 60 mm diameter. The tube surface is
maintained at 700C and outer temperature of
water is 500C. Find the heat transfer coefficient
from the tube surface to the water, heat
transferred and the tube length.
Problem
In a condenser, water flows through 200 thin
walled circular tubes having inner diameter 20
mm and length 6 m. The mass flow rate of water
is 160 kg/s. The water enters at 300C and leaves
at 500C. Calculate the average heat transfer
coefficient.
Problem
Air at 2 bar pressure and 600C is heated as it
flow through a tube of diameter 25 mm at a
velocity of 15 m/s. If the wall temperature is
maintained at 1000C, find the heat transfer per
unit length of the tube. How much would be the
bulk temperature increase over one meter
length of the tube.
Problem
Air at 300C, 6 m/s flows in a rectangular section
of size 300 x 800 mm. Calculate the heat leakage
per meter length per unit temperature
difference.
Natural convection
Module 4
Homework Problem
A vertical plate of 0.75 m height is at 1700C and
is exposed to air at a temperature of 1050C and
one atmosphere. Calculate
i) Mean heat transfer coefficient (Ans:4.24
W/m2K)
ii) Rate of heat transfer per unit width of the
plate (Ans:Q = 206.8 W)
Empirical Correlations : Horizontal Plate
•Define the characteristic length, L as W
L
2
•Upper surface of heated plate, or Lower surface of cooled plate :
Nu L 0.54 Ra1L/ 4 104 RaL 107
Nu L 0.15 Ra1L/ 3 107 RaL 1011
•Lower surface of heated plate, or Upper surface of cooled plate :
Nu L 0.27 1/ 4
Ra L 10 5
Ra L 10 10
Ts T
Note: Use fluid properties at the film temperature
Tf
2
Problem
A horizontal plate of 800 mm long, 70 mm wide
is at a temperature of 1400C and is immersed in
a large tank full of water at 600C. Determine the
total heat loss from the plate.
Problem
A thin 80 cm long and 8 cm wide horizontal
plate is maintained at a temperature of 1300C in
a large tank full of water at 700C. Estimate the
rate of heat input into the plate necessary to
maintain the temperature of 1300C.
Empirical Correlations : Long Horizontal Cylinder
•Very common geometry (pipes, wires)
•For isothermal cylinder surface, use general form equation for computing
Nusselt #
hD
NuD CRaDn
k
Problem
A horizontal wire of 3 mm diameter is
maintained at 1000C and is exposed to air at
200C. Calculate
i) Heat transfer coefficient
ii) Maximum current. Take resistance of wire as
7 ohm/m.
Problem
A steam pipe 10 cm outside diameter runs
horizontally in a room at 230C. Take the outside
surface temperature of pipe as 1650C.
Determine the heat loss per meter length of the
pipe.
Problem
A sphere of diameter 20 mm is at 3000C and is
immersed in air at 250C. Calculate the
convective heat loss.
BOILING HEAT TRANSFER
• Evaporation occurs at the liquid–vapor interface when
the vapor pressure is less than the saturation pressure of
the liquid at a given temperature.
• Boiling occurs at the solid–liquid interface when a liquid
is brought into contact with a surface maintained at a
temperature sufficiently above the saturation
temperature of the liquid.
107
Boiling heat flux from a solid surface to the fluid
excess temperature
Classification of boiling
• Boiling is called pool boiling in the
absence of bulk fluid flow.
• Any motion of the fluid is due to
natural convection currents and the
motion of the bubbles under the
influence of buoyancy.
• Boiling is called flow boiling in the
presence of bulk fluid flow.
• In flow boiling, the fluid is forced to
move in a heated pipe or over a
surface by external means such as a
pump.
108
Subcooled Boiling
• When the
temperature of the
main body of the
liquid is below the
saturation
temperature.
Saturated Boiling
• When the
temperature of the
liquid is equal to the
saturation
temperature.
109
POOL BOILING
In pool boiling, the fluid is not forced to flow by
a mover such as a pump.
Any motion of the fluid is due to natural
convection currents and the motion of the
bubbles under the influence of buoyancy.
Boiling Regimes and the
Boiling Curve
Boiling takes different forms, depending on
the DTexcess = Ts Tsat
110
111
Module 5
Heat Exchangers
Introduction to Heat Exchangers
What Are Heat Exchangers?
Heat exchangers are designed to transfer heat
from a hot flowing stream to a cold flowing
stream.
Why Use Heat Exchangers?
Heat exchangers and heat recovery is often used
to improve process efficiency.
What are heat exchangers for?
Heat exchangers are practical devices used to
transfer energy from one fluid to another
To get fluid streams to the right temperature for the
next process
– reactions often require feeds at high temp.
To condense vapours
To evaporate liquids
To recover heat to use elsewhere
To reject low-grade heat
To drive a power cycle
Recuperators/Regenerators
Recuperative:
Has separate flow paths for each fluid
which flow simultaneously through the
exchanger transferring heat between
the streams
Regenerative
Has a single flow path which the hot
and cold fluids alternately pass
through.
The schematic of a shell-and-tube heat exchanger (one-shell pass
and one-tube pass).
Shell-Side Flow
Plate-Fin Exchanger
Made up of flat plates (parting sheets) and corrugated sheets
which form fins
Brazed by heating in vacuum furnace
Heat Exchanger Analysis
Log mean temperature difference (LMTD)
method
.
Want a relation Q UADTm
Where DTmis some mean DT between hot and cold fluid
Problem
• In a counter flow double pipe heat exchanger,
oil is cooled from 850C to 550C by water
entering at 250C. The mass flow rate of oil is
9,800 kg/h and specific heat of oil is 2000 J/kg
K. The mass flow rate of water is 8000 kg/h
and specific heat of water is 4180 J/kg K.
Determine the heat exchanger area and heat
transfer rate for an overall heat transfer co-
efficient of 280 W/m2K.
Problem
The flow rates of hot and cold water streams
running through a parallel flow heat exchanger
are 0.2 kg/s and 0.5 kg/s respectively. The inlet
temperatures on the hot and cold sides are 750C
and 200C respectively. The exit temperature of
hot water is 450C. If the individual heat transfer
coefficients on both sides are 650 W/m2 0C,
calculate the area of the heat exchanger.
(Ans: Area = 2.66 m2)
