Some Thought Experiments
1. Air in a room is Ti = 20oC and the air
outside is To = -10oC. That means:
Everywhere I go in the room the
temperature is Ti and everywhere outside
is To.
What is the wall temperature? Touch the
walltouch the window.
Draw a temp
CHAPTER 5: TRANSIENT
CONDUCTION
Finite Difference Method for
Unsteady Conduction
Unsteady, 2-D Heat Conduction
PDE:
d 2T d 2T
dT
= 2 + 2
dt
dy
dx
We are familiar with the finite difference
mesh (grid) now from our examination of
steady-state 2-D Heat
CHAPTER 4: STEADY STATE,
2-D CONDUCTION
Solution to Multi-Dimensional Problems
Exact (analytical)
Approximation (numerical)
Empirical or graphical techniques
For 2-D, steady with no heat generation,
we get the LaPlace Equation:
d 2T d 2T
+ 2 =0
2
dx
dy
Ca
CHAPTER 3: STEADY STATE,
1-D, CONDUCTION
Heat Balance:
For a slab or wall, we can use
T
x K x + = 0
and we have covered examples concerning
how we can determine the temperature
profile from this
equation.
T1
T2
qx
x
Once the temperature
profile is de
CHAPTER 11: HEAT
EXCHANGERS
This chapter represents the culmination of
the prior chapters that we have dealt with.
It presents theory on one of the most
highly used process unit present in
industry, the heat exchanger. The unit is
used to transfer heat be
WHAT IS HEAT TRANSFER
There are three principle laws upon
which Engineering studies are derived
o Conservation of Mass (Continuity,
Mass Transfer)
o Conservation of Momentum (Fluid
Mechanics, Mass Transfer)
o Conservation of Energy
(Thermodynamics, Heat
Chapter 3. Steady, 1-D Conduction
Contd
Fins Arbitrary Shape
Annular fins and triangular fins are other
popular geometries for extended surfaces.
We need to look at our energy balance
again to see how these geometries are
accounted for.
q" x q" x + dx +q&
CHAPTER 4: STEADY STATE,
2-D CONDUCTION
Empirical Method
This method pre-dates computers,
especially the easy access we now have to
high powered systems. The method lacks
the possible accuracy of numerical
methods, but it is good for first
approximations
CHAPTER 7: EXTERNAL
CONVECTION
More Useful Expressions:
Flow over a Cylinder
Direction of fluid flow perpendicular to the
cylinders central axis
Unlike our study of flow over a plate
where the bulk velocity was constant, i.e.
u, the velocity u increases f
Chapter 3. Steady, 1-D Conduction
Contd
C1 =
(Ti To )
1 1
ri ro
1
(
Ti To )
+ C2
T=
1 1 r
ri ro
using B.C. T = To @ r = ro again, we
finally get,
T To
=
Ti To
1
r
r1o
1
ri
r1o
The heat flow rate is:
T
q = kA(r )
r
2 T
q = k 4r
r
(
)
T 1 Ti To
=
Chapter 3. Steady, 1-D Conduction
Contd
Critical Thickness of Insulation
For small diameter tubes it is sometimes
possible to increase heat loss by adding
insulation on the outer surface!
Ho, T
Ti, ri
To, ro
Conduction thru insulation cylinder:
ro
1
Ti
CHAPTER
11:
HEAT
EXCHANGERS (CONTD)
Methodology
Calculations
for
Heat
Exchanger
Procedures we have discussed:
1) LMTD Method
Require the ability to compute
Tlm
2) NTU-Effectiveness Method
Easier approach to implement
Problem Classes:
Heat Exchanger Desi
CHAPTER 4: STEADY STATE,
2-D CONDUCTION
Finite Difference Method
From last class our layout
i,j+1
i-1,j
i+1,j
i,j
i,j-1
y
Y
x
X
We want to evaluate the LaPlace equation
at node, P(i,j).
For a second-order
derivative, we sum up the two equations
below:
Ti
CHAPTER 6: INTRODUCTION
TO CONVECTION
C p
T
+ C p u T = k 2T + W&
t
But what of the heat transfer through the
air (or other flowing fluids) and how do we
get those convection coefficients that we
have been using so far?
Convection involves:
External flow
CHAPTER 7: EXTERNAL
CONVECTION
More Useful Expressions:
Flow Across a Bank of Cylinders
Heat transfer involving a bank of
cylindrical tubes in cross flow just like
air flow over the cooling coils of an air
conditioner.
Generally looking for the average h
CHAPTER 4: STEADY STATE,
2-D CONDUCTION
Energy Balance Approach
(Control-Volume Method)
If internal heat generation is known to be
present, it may be better to solve using the
energy balance method (section 4.4.3)
Look at our layout again
i-1,j
i,j+1
i+1,
Chapter 3. Steady, 1-D Conduction
Contd
For a differential element, dx
heat loss = hPdx(T - T , f
)
volume = Adx
q& =
hPdx(T T, f
)
Adx
Consequently,
d 2T hP(T T, f )
=0
2
kA
dx
when, k = constant
Boundary Conditions to solve for
temperature along fin:
A
CHAPTER 11: HEAT
EXCHANGERS (CONTD)
Ex.
Tlm
(
T T ) (T
=
ln(T T ) (T
h,o
c,o
h, o
Tc,i )
h , i Tc , i )
h,i
c,o
The log mean temperature difference
equation given above is that for a cocurrent parallel flow heat exchanger.
Subscripts c and h refer to the
CHAPTER 5: TRANSIENT
CONDUCTION
Determining Temperature using the
Heisler Charts
1) Check the Biot Number (Bi)
hLc
Bi =
k
Is Lumped Parameter Method valid?
o If Bi < 0.1 use LPM instead
2) Determine the characteristic length of
the object, Lc
3) Compute
CHAPTER 7: EXTERNAL
CONVECTION
Goal of this chapter: To derive h for its
used in our conduction studies.
YOU MUST READ CHAPTER 6
Typical form of the expression is:
Nu = A Re Pr
m
n
Prandtl number (Pr): a value based on
the properties of the fluid used an
CHAPTER 5: TRANSIENT
CONDUCTION
This chapter studies a system when the
error is too great to assume steady state
i.e.
dT
0
dt
Lumped Parameter Model
Lumped = no spatial variation (x, y, z) of
temperature.
Temperature is the same throughout. This
is physic
CHAPTER 5: TRANSIENT
CONDUCTION
Lumped Parameter Model
(contd)
Thermal Circuit Form
Total Resistance
Thermal
Capacitance
T T
t
= exp
Ti T
R
*
C
t
where
1
Rt =
hA , and
C = VC p
Thermal Time Constant
1
t = (C pV )
hA
length of time needed till stea
CHAPTER 8: INTERNAL
CONVECTION
We will now direct our studies into
looking at the determination of convection
coefficients for fluid flow within a pipe,
conduit, etc.
The boundary layer is still important the
region which determines the value of the
conve
Chapter 2. Conduction
Dimensions
Heat Flow is a vector quantity it has
magnitude and direction.
q = kAT
where the gradient operator for
temperature is:
T T T
+k
T = grad T = i
+ j
y
z
x
for a Cartesian coordinates system
The components of heat flow are:
Chapter 3. Steady, 1-D Conduction
Contd
The overall heat transfer coefficient in the
case of the above problem would be,
U=
1
Rtotal A
=
1
LA
kA
+ LB
kB
+
LC
kC
The problem given above is an example of
a simple circuit. More complex circuits
can arise whe
CHAPTER 8: INTERNAL
CONVECTION
Correlations for Circular Tubes
Similar to the exterior flow correlations
of Chapter 7, the correlations presented are
based on experimental results (at least for
the turbulent cases) and therefore have a
limited range of ap
CHAPTER 5: TRANSIENT
CONDUCTION
1-D Heat Conduction in a Semi-Infinite
Wall (contd)
We left off with:
=
x
"+2 ' = 0
2 t
Now to solve the ODE with the following
boundary conditions:
B.C. 1
=0 =1
I.C.+B.C. 2 = 0
We have the final equation in its full form:
Review
Lumped Parameter
Limited to objects where the Biot
Number (Bi) exists below 0.1 (error
less than 5%)
hA
General Eqn: T T
s
= exp
t
Ti T
C pV
Thermal Circuit Form:
T T
= exp[ Rt * C * t ]
Ti T
Review
Surface Area, As
Flat Plate of thickn
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