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#17.39.

• To be discussed on Jan. 26,

2021.

• By either volunteer or class

list.

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.

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

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.