1. MODULE-2
HEAT EXCHANGERS

2. ONE DIMENSIONAL STEADY
CONDUCTION
• The simplest example of steady state
conduction in one dimension is the transfer of
heat through a single plane slab.
• Example:x-axis single plane slab
• Many simple problems such as conduction
through the wall of a building ,approximate to
this.

3. T2 < T1 dT
q k
T1 T2 dx
q’’
Conduction through a solid

5. CONDUCTION IN PLANE SLABS:
FOURIER Law applied to conduction through
slab
Q=-kA dT/dx
Q- Rate of heat conduction through the wall,W
k - Thermal conductivity of the wall
material,W/m-K.
A- the area of heat flow taken at right angles
to the direction of flow of heat,m2.
-(dT/-dx) ---- temperature gradient, K/m.
Q(X2-X1)=-KA(T2-T1)=KA(T1-T2)
Q=KA[(T1-T2)/(x2-x1)]=kA/L ∆T

6. Where ∆ T =(T1-T2) =Temperature driving force,k.
L= (x2-x1)= thickness of the wall,m.
Q= ∆ T/R
Where R=L/kA =thermal resistance of the wall k/W.

7. 1.One face of a copper plate 3cm thick is
maintained at 400 0 C,and the other face is
maintained at 100 0 C.How much heat is
transfers through the plate? Thermal
conductivity of copper=370W/m 0 C and the
surface area of the plate is 1m2.
• SOLUTION:
Q=KA( ∆ T)/L=370*1(400-100)/0.03
Q=3.7W

8. CONDUCTION THROUGH SERIES
RESISTANCES
• Conduction through a system of plane slabs of different material
has often to be considered.
• A furnace wall consisting of a layer of firebrick and a layer of
insulating brick is a typical example,such slabs are called composite
walls.
• ∆ T= ∆ T1+ ∆ T2+ ∆ T3---------------(1), ∆ T1=Qa*R1,
• ∆ T2=Qb*R2, ∆ T3=Qc*R3------(2)
• Qa=Qb=Qc=Q---------------(3)
• Adding equations (1 )and (2)
∆ T1+ ∆ T2+ ∆ T3= ∆ T=Q(R1+R2+R3)
Q=(∆ T1+ ∆ T2+ ∆ T3)/(R1+R2+R3) = ∆ T total/R total.
In steady heat flow , all the heat that passes through the first
resistance must pass through the second and in turn through third
(Qa=Qb=Qc)

9. q
Ts1
Ts2
T 2,h 2
T 1 T 2
q
Ts3 R
Ts4
T 1,h 1
KA KB KC
x x=L
1 LA LB LC 1
h 1A kA A kB A kC A h
2
A

10. Where Rtotal=total thermal resistance of the
composite wall
Q= ∆ T/R
Q = ∆ T1/R 1 =∆ T2/R 2 = ∆ T3/R3

11. RADIAL CONDUCTION IN HOLLOW
CYLINDRICAL LAYERS
• Conduction through thick walled pipes is a common
heat transfer problem, and may be treated one-
dimensionally if surface temperatures are uniform.
• To find an expression Q in the radial direction consider
the cross section of a hollow pipe of length(L).
• Q=-kA dT/dr (1)
• A=2πrL (2)
• Substituting (2) in (1) and integrating
• Q={2 πkL/ ln(r0 /ri )}*(Ti-To)= ∆ T/R (note:ln=loge)
R= ln(r0 /ri )/2 πkL= thermal Resistance

12. 1 ln( r2 / r1 ) ln( r3 / r2 ) ln( r4 / r3 ) 1
h1 2 r1 L 2 k L 2 k L 2 kC L h2 2 r4 L
A B

13. Cold fluid
h 2 ,T 2
r1 Ts2
1 d dT
kr
r dr dr
r2
L
Ts1

16. • Q= kA L ∆ T /(r0 /ri )
• A L=2 πL (r0 /ri )/ ln(r0 /ri )
= A0- Ai/ ln(A0/Ai)
rL=ro-ri/ln(ro/ri)= logarithmic mean radius.

17. Multilayer cylindrical system:
1.Athick walled tube of stainless steel [18% Cr,
8%Ni, k=19W . 0 C ]with a 2-cm inner diameter(ID)
and 4-cm outer diameter (OD) is covered with a
3-cm layer of asbestos insulation [k=0.2 W/m . 0
C] If the inside wall temperature of the pipe is
maintained at 600 0 C, and the outside wall
temperature is maintained at 100 0 C, calculate
the heat loss per meter of length.

18. RADIAL CONDUCTION IN HOLLOW
SPHERICAL LAYERS
• Conduction will be in the radial direction if the
temperatures of the inner and outer spherical
surfaces are uniform
Q= -kA dT/dr
A=4πr2
Q=4πkr1r2(T1-T2)/(r2-r1) = ∆ T/R

19. PURPOSE OF INSULATION
• The insulation is defined as a material which retards
the heat flow with reasonable effectiveness.
• The purpose of insulation is two fold
(a) –to prevent the flow of heat from the system to the
surroundings as in the case of steam and the hot water
pipes which are used for air-conditioning in winter
(b) – to prevent the flow of heat from the surroundings
to the system as in the case of brine pipes which are
used for air-conditioning in summer.

20. T2 < T1
q insulator
x
A

21. • The insulations are commonly used for the
following industrial purposes
1. Air-conditioning systems
2. Refrigerators and food preserving stores
3. Preservation of liquid gases
4. Boilers and steam pipes
5. Insulating bricks in furnaces

22. • Factors affecting the thermal conductivity:
The thermal conductivity of insulating materials
is one of the most important physical property
,its low value is required for reduction in heat
flow rate.

23. CRITICAL THICKNESS OF INSULATION
• The addition of small amount of insulation to
small diameter wires or tubes frequently
increases the rate of heat flow through the
tube to the ambient air.
• An experiment showed that the rate of heat
loss increased by the addition of asbestos
sheet.

24. • Critical thickness of insulation for cylinder
q= T1Ta/[{ln(r2/r1)/2πK} + {1/2πr2 ht } ]
1/k * 1/r2 -1/(r22 *ht ) =0
r2 = K/ht
The thickness up to which heat flow increases
and after which heat flow decreases is
termed as critical thickness

25. Asbestos string wound on small diameter glass
tubes or wires generally increases the rate of
heat loss
Rubber covered wires transmit more heat
radially outward than bare wires if the bare
wire has the same emissivity as rubber.
This means that a rubber covered wire can carry
more current than a bare wire for the same
temperature rise in the wire.

26. • If we have relatively good conductors such as
concerete with a K value of 1 kcl/m-hr 0 C .
• The prime purpose of insulation is to provide
protection from electrical hazard, but by using the
proper thickness, the ability of the insulated wire is to
dissipate heat may be greater than that of bare wire.
• r2 – r1 =r1[K /ht r1 –1]
• Critical thickness of insulation for spheres:
• q = T1-Ta/ {(r2 –r1/4πr1r2K ) + (1/ 4πr2 2 ht)}
• r2 =2K/ht

27. MODULE 3
HEAT EXCHANGERS

28. INTRODUCTION
• When a fluid flows past a stationary solid
surface ,a thin film of fluid is postulated as
existing between the flowing fluid and the
stationary surface
• Diagram
• It is also assumed that all the resistance to
transmission of heat between the flowing fluid
and the body containing the fluid is due to the
film at the stationary surface.

29. • The amount of heat transferred Q across this film
is given by the convection equation
Where h: film co-efficient of convective heat
transfer,W/m2K
A: area of heat transfer parallel to the direction of
fluid flow, m2.
T1:solid surface temperature, 0C or K
T2: flowing fluid temperature, 0C or K
∆t: temperature difference ,K

30. • Laminar flow of the fluid is encountered at
Re<2100.Turbulent flow is normally at
Re>4000.Sometimes when Re>2100 the fluid
flow regime is considered to be turbulent
• Reynolds number=
• Prandtl number=
• Nusselt number=
• Peclet number=

31. • Grashof number=
• Where in SI system
• D: pipe diameter,m
• V :fluid velocity,m/s
• :fluid density,kg/m3
• μ :fluid dynamic
viscosity N.s/m2 or kg/m.s
• √:fluid kinematic viscosity, m2/s

32. • K:fluid thermal conductivity,W/mK
• h: convective heat transfer coefficient,W/m2.K
• Cp:fluid specific heat transfer,J/Kg.K
• g:acceleration of gravity m/s2
• ϐ:cubical coefficient of expansion of fluid=
• ∆t:temperature difference between surface and fluid ,K

33. Functional Relation Between
Dimensionless Groups in Convective
Heat Transfer
• For fluids flowing without a change of
phase(i.e without boiling or condensation),it
has been found that Nusselt number (Nu) is a
function of Prandtl number(Pr) and Reynolds
number(Re) or Grashof number(Gr).
• And for forced convection

34. Emperical relationships for Force
Convection
• Laminar Flow in tubes:
• Turbulent Flow in Tubes:For fluids with a
Prandtl number near unity ,Dittus and Boelter
recommend:
• Turbulent Flow among flat plates:

35. Empirical Relationships for natural
convection
• Where a and b are constants Laminar and
turbulent flow regimes have been observed in
natural convection,GrPr<109 Wdepending on
the geometry.
• Horizontal Cylinders:
when 10