Mech 302 Heat Transfer HW3 Solution
1. (Problem 3.126 in the Book) Turbine blades mounted to a rotating disc in a gas
turbine engine are exposed to a gas stream that is at T∞ =1200C and maintains a
convection coefficient of h= 250W /m2 ⋅K over the blade. The blades, which are
fabricated from Iconel, k ≈20W /m⋅K have a length of L = 50mm. The blade profile has a
uniform cross-sectional area of Ac = 6×10−4 and a perimeter of P=110 mm. A propose
blade-cooling scheme, which involves routing air through the supporting disc, is able to
maintain the base of each blade at a temperature of Tb = 300C.
a. If the maximum allowable blade temperature is 1050C and blade tip may be assumed
to be adiabatic, is the proposed cooling scheme satisfactory?
b. For the proposed cooling scheme, what is the rate at which heat is transferred from
each blade to the coolant?
KNOWN: Dimensions and thermal conductivity of a gas turbine blade. Temperature and
convection coefficient of gas stream. Temperature of blade base and maximum allowable blade
temperature.
FIND: (a) Whether blade operating conditions are acceptable, (b) Heat transfer to blade coolant.
SCHEMATIC:
ASSUMPTIONS: (1) One-dimensional, steady-state conduction in blade, (2) Constant k, (3)
Adiabatic blade tip, (4) Negligible radiation.
ANALYSIS: In this question, the most imortant thing to solve the problem, is to use the tables
correctly.
Conditions in the blade are determined by Case B of Table 3.4.
(a) With the maximum temperature existing at x = L, Eq. 3.75 yields
From Table B.1, cosh mL = 5.51. Hence,
And the operating conditions are acceptable.
COMMENTS: Radiation losses from the blade surface and convection from the tip will
contribute to reducing the blade temperatures.
This is the table we use in this problem.
2. (Problem 3.137 in the Book) Circular copper rods of diameter D = 1 mm and length
L= 25 mm are used to enhance heat transfer from a surface that is maintained at
𝑇𝑠, 1 = 100C. One end of the rod is attached to this surface (at x = 0), while the other end
(x = 25 mm) is joined to a second surface, which is maintained at 𝑇𝑠, 2 = 0oC. Air is
flowing between the surfaces (and over the rods) is also at a temperature of T∞ = 0𝐶 ,
and a convection coefficient of h = 100 W/m2 K is maintained.
a. What is the rate of heat transfer by convection from a single copper rod to the air?
b. What is the total rate of heat transfer from a 1 m x 1 m section of the surface at?
100oC, if a bundle of the rods is installed on 4-mm centers?
KNOWN: Dimensions and end temperatures of pin fins.
FIND: (a) Heat transfer by convection from a single fin and (b) Total heat transfer from a
1𝑚2 surface with fins mounted on 4mm centers.
SCHEMATIC:
ASSUMPTIONS: (1) Steady-state, (2) One-dimensional conduction along rod, (3)
Constant properties, (4) No internal heat generation, (5) Negligible radiation.
PROPERTIES: Table A-1, Copper, pure (323K): k ≈ 400 W/m⋅K.
ANALYSIS: (a) By applying conservation of energy to the fin, it follows that
Where the conduction rates may be evaluated from knowledge of the temperature
distribution.
The general solution for the temperature distribution is
The boundary conditions are𝜃(0) ≡ 𝜃𝑜 = 100°𝐶 𝑎𝑛𝑑 𝜃(𝐿) = 0. Hence
𝜃 𝑜 = 𝐶1 + 𝐶2
Now we put the boundary conditions in the general solution which results :
Therefore, 𝐶2 = 𝐶1𝑒2𝑚𝐿
And the temperature distribution has the form
Now that we have derived the temperature distribution with respect to x, the conduction
heat rate can be evaluated by Fourier’s law,
𝑜𝑟, 𝑤𝑖𝑡ℎ 𝑚 = (ℎ𝑃/𝑘𝐴𝑐 )1/ 2 ,
Hence at x = 0,
At x = L
Evaluating the fin parameters:
The conduction heat rates are
And from the conservation relation,
𝑞𝑐𝑜𝑛𝑣 =1.507 W−1.133 W = 0.374 W.
(b) The total heat transfer rate is the heat transfer from N = 250×250 = 62,500 rods and
the
Heat transfer from the remaining (bare) surface (𝐴 = 1𝑚2 − 𝑁𝐴𝑐). Hence,
3. (Problem 3.142 in the Book) Finned passages are frequently formed between
parallel plates to enhance convection heat transfer in compact heat exchanger cores.
An important application is electronic equipment cooling, where one or more air-cooled
stacks are placed between heat-dissipating electrical components. Consider a single
stack of rectangular fins of length L and thickness t, with convection conditions
corresponding to h and T∞ .
a. Obtain expressions for the fin heat transfer rates,𝑞𝑓, 𝑜 𝑎𝑛𝑑 𝑞𝑠, 𝐿, in terms of the base
temperatures, 𝑇𝑜 and 𝑇𝐿
b. In a specific application, a stack that is 200 mm wide and 100 mm deep contains
50 fins, each of length L = 12 mm. The entire stack is made from aluminum, which is
everywhere 1 mm thick. If temperature limitations associated with electrical components
joined to opposite plates dictate maximum allowable plate temperatures of 𝑇𝑜 =
400 𝐾 and TL = 350 K, what are the corresponding maximum power dissipations if h =
150 W/m2 K and 𝑇∞ = 300𝐾 .
KNOWN: Arrangement of fins between parallel plates. Temperature and convection coefficient of air
flow in finned passages. Maximum allowable plate temperatures.
FIND: (a) Expressions relating fin heat transfer rates to end temperatures, (b) Maximum power
dissipation for each plate.
SCHEMATIC:
ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in fins, (3) Constant
properties, (4) Negligible radiation, (5) Uniform h, (6) Negligible variation in 𝑇∞ (7) Negligible contact
resistance.
PROPERTIES: Table A.1, Aluminum (pure), at temperature 375 K, k = 240 W/𝑚2 K.
ANALYSIS: (a) The general solution for the temperature distribution in a fin is
Now we want to derive the temperature distribution, using the boundary conditions:
Now that the temperature distribution is derived with respect to x,the fin heat transfer rate can be
computed:
Maximum power dissipations are therefore