HW# 2 /Tutorial # 2
WRF Chapter 16; WWWR Chapter 17
ID Chapter 3
• Tutorial #2
• WRF#16.2;WWWR#17.13, WRF#16.1; WRF#16.12; WRF#17.39; WRF#16.22.
• To be discussed during the week 27-31 Jan., 2020.
• By either volunteer or class list.
• Homework # 2 (Self practice)
• WWWR #17.15, 17.26
• ID # 3.57
HW# 2 /Tutorial # 2
Hints / Corrections
• Tutorial #2
• WWWR
• #17.39: Line 2: The fins are made of aluminum, they are 0.3cm thick each.
• Homework # 2
• WWWR
• #17.15: You may model the whole heattransfer process as the series/parallelconnection of 5 resistors: R1 internalconvective transfer + R2: Plaster + R3 (PineStuds) and R4 (Fiberglass) in parallel + R5:External convective transfer. The problemdoes not specify which is the “inside” andwhich is the “outside”. Please fix the sidenext to the plaster to be the inside (R1: hencea temperature of 25oC is maintained here) andthe opposite side to be outside (R5: atemperature of -10 degree C is specified here).
• # 17.26: Generate heat uniformly at a rate of 5170 kJ/sm3. the fuel is placed in an environment having a temperature of 370K.
Steady-State Conduction
One-Dimensional Conduction
02 T
Steady-state conduction, no internal generation of energy
0id dTx
dx dx
For one-dimensional, steady-state transfer by conduction
i = 0 rectangular coordinates
i = 1 cylindrical coordinates
i = 2 spherical coordinates
0.0173
1/Rc=1/Ra+1/Rb
Ra
Rb
Equivalent resistance of the parallel resistors Ra and Rb is Rc
, (m)cr cylinder
kr
h
Adapted from Heat and Mass Transfer – A Practical Approach,
Y.A. Cengel, Third Edition, McGraw Hill 2007.
• Thus, insulating the pipe
• may actually increase the
• rate of heat transfer instead
• of decreasing it.
For steady-state conduction in the x direction without internal
generation of energy, the equation which applies is
Where k may be a function of T.
In many cases the thermal conductivity may be a linear function
temperature over a considerable range. The equation of such a
straight-line function may be expressed by
k = ko(1 + ßT)
Where koand ß are constants for a particular material
One-Dimensional Conduction With
Internal Generation of Energy
Plane Wall with Variable Energy
Generation
q = qL [ 1 + ß (T - T L)]. .
The symmetry of the temperature distribution requires a zero
temperature gradient at x = 0.
The case of steady-state conduction in the x direction in a
stationary solid with constant thermal conductivity becomes
Detailed derivation for the transformation
F = C + s q
Detailed Derivation for Equations 17-25
Courtesy by all CN5 Grace Mok, 2003-2004
Detailed Derivation for Equations 17-25
Courtesy by all CN5 Grace Mok, 2003-2004
Heat Transfer from Finned Surfaces
• Temperature gradient dT/dx,
• Surface temperature, T,
• Are expressed such that T is a function of x only.
• Newton’s law of cooling
• Two ways to increase the rate of heat transfer:
– increasing the heat transfer coefficient,
– increase the surface area fins
• Fins are the topic of this section.
Adapted from Heat and Mass Transfer –A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.
conv s sQ hA T T &
Heat transfer from extended
surfaces
For constant cross section and constant thermal conductivity
Where
• Equation (A) is a linear, homogeneous, second-order differential equation with constant coefficients.
• The general solution of Eq. (A) is
• C1 and C2 are constants whose values are to be determined from the boundary conditions at the base and at the tip of the fin.
22
20
dm
dx
qq (A)
2 ; ; c
c
hpT T m A A
kAq
1 2( ) mx mxx C e C eq (B)
Boundary Conditions
Several boundary conditions are typically employed:
• At the fin base
– Specified temperature boundary condition, expressed
as: q(0)= qb= Tb-T∞
• At the fin tip
1. Specified temperature
2. Infinitely Long Fin
3. Adiabatic tip
4. Convection (and
combined convection).
Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition,
McGraw Hill 2007.
How to derive the functional dependence of
for a straight fin with variable cross section area
Ac = A = A(x)?
General Solution for Straight Fin with Three Different Boundary Conditions
In set(a)
Known temperature at x = L
In set(b)
Temperature gradient is zero at x = L
In set(c)
Heat flow to the end of an extended surface by conduction be
equal to that leaving this position by convection.
Detailed Derivation for Equations 17-36 (Case a).
Courtesy by CN3 Yeong Sai Hooi 2002-2003
Detailed Derivation for Equations 17-38 (Case b
for extended surface heat transfer). Courtesy by
CN3 Yeong Sai Hooi, 2002-2003
Detailed Derivation for Equations 17-40 (Case c for extended surface
heat transfer).
Courtesy by all CN4 students, presented by Loo Huiyun, 2002-2003
Detailed Derivation for Equations 17-46 (Case c for extended surface
heat transfer).
Courtesy by all CN4 students, presented by Loo Huiyun, 2002-2003
Infinitely Long Fin (Tfin tip=T) Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.
• For a sufficiently long fin the temperature at the fin
tip approaches the ambient temperature
Boundary condition: q(L→∞)=T(L)-T∞=0
• When x→∞ so does emx→∞
C1=0
• @ x=0: emx=1 C2= qb
• The temperature distribution:
• heat transfer from the entire fin
/( )cx hp kAmx
b
T x Te e
T T
0
c c b
x
dTq kA hpkA T T
dx
Fin Efficiency Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.
• To maximize the heat transfer from a fin the
temperature of the fin should be uniform (maximized)
at the base value of Tb
• In reality, the temperature drops along the fin, and thus
the heat transfer from the fin is less
• To account for the effect we define
a fin efficiency
or
,max
fin
fin
fin
q
q
Actual heat transfer rate from the fin
Ideal heat transfer rate from the fin
if the entire fin were at base temperature
,max ( )fin fin fin fin fin bq q hA T T
Fin Efficiency Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.
• For constant cross section of very long fins:
• For constant cross section with adiabatic tip:
,
,max
1 1fin c b clong fin
fin fin b
q hpkA T T kA
q hA T T L hp mL
,
,max
tanh
tanh
fin c b
adiabatic fin
fin fin b
q hpkA T T mL
q hA T T
mL
mL
Afin = P*L
q
,max
fin
fin
q
q
Fin Effectiveness Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.
• The performance of the fins is judged on the basis of the
enhancement in heat transfer relative to the no-fin case.
• The performance of fins is expressed
in terms of the fin effectiveness efin
defined as
fin fin
fin
no fin b b
q q
q hA T Te
Heat transfer rate
from the surface
of area Ab
Heat transfer rate
from the fin of base
area Ab
finq
no finq
fin
fin
no fin
q
qe
Governing Differential Equation for Circular Fin:
Temperature variation in the R (radial) direction only!
T = T(r)
(RL-Ro)
Problem: Water and air are separated by a mild-steel plane wall. I is
proposed to increase the heat-transfer rate between these fluids by
adding Straight rectangular fins of 1.27mm thickness, and 2.5-cm
length, spaced 1.27 cm apart.
Two and Three - Dimensional
SystemsAnalytical Solution
Analytical solution to any transfer problem must satisfy the
differential equation
•describing the process
•Prescribed boundary conditions
The steady-state temperature distribution in the plate of constant
thermal conductivity must satisfy the differential equation
We obtain an expression in which the variables are separated
(17-57)
Detailed Derivation for Equations 17-57
Courtesy by all CN6 Leow Sheue Ling, 2003-2004
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