1. HW# 2 /Tutorial # 2 WRF Chapter 16; WWWR Chapter 17 ID Chapter 3
WRF#16.2;WWWR#17.13, WRF#16.1; WRF#17.39.
To be discussed on Jan. 26, 2016.
By either volunteer or class list.

2. HW# 2 /Tutorial # 2 Hints / Corrections
WWWR
#17.39: Line 2: The fins are made of aluminum, they are 0.3cm thick each.

3. Steady-State Conduction

4. One-Dimensional Conduction
Steady-state conduction, no internal generation of energy
For one-dimensional, steady-state transfer by conduction
i = 0 rectangular coordinates
i = 1 cylindrical coordinates
i = 2 spherical coordinates

9. 1/Rc=1/Ra+1/Rb Ra Rb
Equivalent resistance of the parallel resistors Ra and Rb is Rc

16. Thus, insulating the pipe may actually increase the
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.

19. 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

20. One-Dimensional Conduction With Internal Generation of Energy

23. 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

26. Detailed derivation for the transformation
F = C + s q

28. Detailed Derivation for Equations 17-25
Courtesy by all CN5 Grace Mok,

29. Detailed Derivation for Equations 17-25
Courtesy by all CN5 Grace Mok,

30. 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.

31. Heat transfer from extended surfaces

33. 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.
(A)
(B)

35. 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
Specified temperature
Infinitely Long Fin
Adiabatic tip
Convection (and
combined convection).
Adapted from Heat and Mass Transfer –
A Practical Approach, Y.A. Cengel, Third Edition,
McGraw Hill 2007.

39. How to derive the functional dependence of
for a straight fin with variable cross section area
Ac = A = A(x)?

40. General Solution for Straight Fin with Three Different Boundary Conditions

41. 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.

43. Detailed Derivation for Equations 17-36 (Case a).
Courtesy by CN3 Yeong Sai Hooi

44. Detailed Derivation for Equations (Case b for extended surface heat transfer). Courtesy by CN3 Yeong Sai Hooi,

45. Detailed Derivation for Equations 17-40 (Case c for extended surface heat transfer).
Courtesy by all CN4 students, presented by Loo Huiyun,

48. Detailed Derivation for Equations 17-46 (Case c for extended surface heat transfer).
Courtesy by all CN4 students, presented by Loo Huiyun,

49. Boundary condition: q(L→∞)=T(L)-T∞=0 When x→∞ so does emx→∞ C1=0
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

50. 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
Actual heat transfer rate from the fin
Ideal heat transfer rate from the fin
if the entire fin were at base temperature

51. 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:
Afin = P*L

52. q

53. 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
Heat transfer rate from the fin of base area Ab
Heat transfer rate from the surface of area Ab

55. Governing Differential Equation for Circular Fin:
Temperature variation in the R (radial) direction only!
T = T(r)

56. (RL-Ro)

58. Problem: Water and air are separated by a mild-steel plane wall
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.