COVENANT UNIVERSITY
NIGERIA
TUTORIAL KIT
OMEGA SEMESTER
PROGRAMME: MECHANICAL
ENGINEERING
COURSE: MCE 524
DISCLAIMER
The contents of this document are intended for practice and leaning purposes at the
undergraduate level. The materials are from different sources including the internet
and the contributors do not in any way claim authorship or ownership of them. The
materials are also not to be used for any commercial purpose.
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MCE 524: Heat Transfer
Contributor:
Dr.S.O Oyedepo
Q1
Calculate the rate of heat loss from a furnace wall per unit area. The wall is constructed from
an inner layer of 0.5-cm-thick steel (k = 40 W/m K) and an outer layer of 10-cm zirconium
brick (k = 2.5 W/m K) as shown in Fig. Q1. The inner- surface temperature is 900 K and the
outside surface temperature is 460 K. What is the temperature at the interface?
Fig. Q1: Schematic diagram of furnace wall
Q2
The outside surface of a cylindrical cryogenic container is at – 10°C. The outside radius is 8 cm.
There is a heat flow of 65.5 W/m, which is dissipated to the surroundings both by radiation and
convection. The convection coefficient has a value of 4.35 W/m2K. The radiation factor F = 1.
Determine the surrounding temperature.
Q3
In a solar flat plate heater some of the heat is absorbed by a fluid while the remaining heat
is lost over the surface by convection the bottom being well insulated. The fraction absorbed is
known as the efficiency of the collector. If the flux incident has a value of 800 W/m2 and if the
collection temperature is 60°C while the outside air is at 32°C with a convection coefficient
of 15 W/m2K, determine the collection efficiency. Also find the collection efficiency if
collection temperature is 45°C.
Q4
Choose the correct statement in each question.
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(i)
A pipe carrying steam at about 300°C traverses a room, the air being still at 30°C. The major
fraction of the heat loss will be by (a) conduction to the still air (b) convection to the air (c)
radiation to the surroundings (d) conduction and convection put together.
(ii)A
satellite in space exchanges heat with its surroundings by (a) conduction (b) convection (c)
radiation (d) conduction as well as convection.
(iii)
For the same temperature drop in the temperature ranges of 300–400°C the heat flow rate
will be highest by (a) conduction process (b) convection process (c) radiation process (d) other
factors should be known before any conclusion.
(iv)
In the cold season a person would prefer to be near a fire because (a) the conduction from
the fire will be better (b) the convection will be better if he is near the fire (c) direct unimpeded
radiation will provide quick warmth (d) combined conduction and convection will be better.
(a)
A finned tube hot water radiator with a fan blowing air over it is kept in rooms during winter.
The major portion of the heat transfer from the radiator to air is due to: (a) radiation
(b)
convection (c) conduction (d) combined conduction and radiation.
(v) For
a specified heat input and a given volume which material will have the smallest temperature
rise (Use data book if necessary) (a) steel (b) aluminium (c) water (d) copper.
Q5
A thin metal sheet receives heat on one side from a fluid at 80°C with a convection
coefficient of 100 W/m2K while on the other side it radiates to another metal sheet parallel to it.
The second sheet loses heat on its other side by convection to a fluid at 20°C with a convection
coefficient of 15 W/m2K. Determine the steady state temperature of the sheets. The two sheets
exchange heat only by radiation and may be considered to be black and fairly large in size.
Q6
A steel tube having k = 46 W/m · ◦C has an inside diameter of 3.0 cm and a tube wall thickness
of 2 mm. A fluid flows on the inside of the tube producing a convection coefficient of 1500
W/m2 · ◦C on the inside surface, while a second fluid flows across the outside of the tube
producing a convection coefficient of 197 W/m2 · ◦C on the outside tube surface. The inside fluid
temperature is 223◦C while the outside fluid temperature is 57◦C. Calculate the heat lost by the
tube per meter of length.
Q7
A furnace wall is of three layers, first layer of insulation brick of 12 cm thickness of conductivity
0.6 W/mK. The face is exposed to gases at 870°C with a convection coefficient of 110 W/m 2K.
This layer is backed by a 10 cm layer of firebrick of conductivity 0.8 W/mK. There is a contact
resistance between the layers of 2.6 × 10–4 m2 °C/W. The third layer is the plate backing of 10
mm thickness of conductivity 49 W/mK. The contact resistance between the second and third
layers is 1.5 × 10–4 m2 °C/W. The plate is exposed to air at 30°C with a convection coefficient
of 15 W/m2K. Determine the heat flow, the surface temperatures and the overall heat transfer
coefficient.
Q8
Steam having a quality of 98% at a pressure of 1.37 X 105 N/m2 is flowing at a velocity of 1 m/s
through a steel pipe of 2.7-cm OD and 2.1-cm ID. The heat transfer coefficient at the inner
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surface, where condensation occurs, is 567 W/m2 K. A dirt film at the inner surface adds a unit
thermal resistance of 0.18 m2 K/W. Estimate the rate of heat loss per meter length of pipe if (a)
the pipe is bare, (b) the pipe is covered with a 5-cm layer of 85% magnesia insulation. For both
cases assume that the convection heat transfer coefficient at the outer surface is 11 W/m 2 K and
that the environmental temperature is 21°C. Also estimate the quality of the steam after a 3-m
length of pipe in both cases.
Q9
A fin in the form of a ring of 0.25 mm thickness and 15 mm OD and 15 mm long is used on an
electric device to dissipate heat. Consider the outer surface alone to be effective and exposed to
air at 25°C with a convection coefficient of 40 W/m2K. The conductivity of the material is 340
W/mK. If the heat output is 0.25 W and if the device is also of the same OD, determine the
device temperature with and without the fin.
Q10
A solar collector plate is exposed to a flux of 900 W/m2. Heat is collected by water pipes fixed at
12 cm pitch with a water temperature of 48°C. The plate is 2 mm thick and has a conductivity of
204 W/mK. If the losses over the plate is accounted by a convection coefficient of 15 W/m2K to
air at 30°C, determine the maximum temperature in the plate and also the rate of heat collection
by the water per pitch width and 1 m length.
Q11
For the boundary conditions for the plate shown in Fig. Q11 determine using analytical method
the temperature at the midpoint p, under steady two dimensional conduction. (use up to 5 terms
in the series summation).
Fig. Q11. Problem model
Q12
A rectangle 0.5 m × 1 m has both the 1 m sides and one 0.5 m side at 200°C. The other side is
having a temperature distribution given by T = 200 + 400 sin (π x/0.5) where x is in m and T in
°C. Locate the y values at x = 0.5 m at which the temperatures will be 300, 400, 500°C. Also
locate the values of x for y = 1 m at which these temperatures occur.
Q13
The temperature distribution and boundary condition in part of a solid is shown in Fig. Q13.
Determine the Temperatures at nodes marked A, B and C. Determine the heat convected over
surface exposed to convection. k = 1.5 W/mK.
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Fig. Q13.
Q14
A part of a solid with temperatures at the nodes and the boundaries are shown in Fig. Q14.
Determine the temperature at node A and also the heat flow over the convecting surface. The top
surface is exposed to convection at 300°C with h = 10 W/m2K.
Fig. Q14
Q15
Nitrogen at a pressure of 0.1 atm flows over a flat plate with a free stream velocity of 8 m/s. The
temperature of the gas is – 20°C. The plate temperature is 20°C. Determine the length for the
flow to turn turbulent. Assume 5 × 105 as critical Reynolds number. Also determine the thickness
of thermal and velocity boundary layers and the average convection coefficient for a plate length
of 0.3 m. Properties are to be found at film temperature.
Q16
A thin conducting plate separates two parallel air streams. The hot stream is at 200°C and 1 atm
pressure. The free stream velocity is 15 m/s. The cold stream is at 20°C and 2 atm pressure and
the free stream velocity is 5 m/s. Determine the heat flux at the mid - point of the plate of 1 m
length.
Q17
Air at 1 atm with a temperature of 500°C flows over a plate 0.2 m long and 0.1 m wide. The
Reynolds number is 40,000. (Flow is along the 0.2 m side). Determine the rate of heat transfer
from the plate at 100°C to air 50°C. If the velocity of flow is doubled and the pressure is
increased to 5 atm, determine the percentage change. The properties of air are read from tables
and interpolated for film temperature of 75°C.
Q18
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A radioactive sample is to be stored in a protective box with 4-cm-thick walls and interior
dimensions of 4 cm X 4 cm X 12 cm. The radiation emitted by the sample is completely
absorbed at the inner surface of the box, which is made of concrete. If the outside temperature of
the box is 25°C but the inside temperature is not to exceed 50°C, determine the maximum
permissible radia- tion rate from the sample, in watts.
Q19
A surface with A = 2 cm2 emits radiation as a blackbody at T= 1000 K. (a) Calculate the
radiation emitted into a solid angle subtended by 0 ≤ ∅ ≤ 2𝜋 and 0 ≤ 𝜃 ≤ 𝜋⁄6 (b) What
fraction is the energy emitted into the above solid angle of that emitted into the entire
hemispherical space?
Q20
Greenhouse effect is nothing but trapping of radiation by letting in radiation of short wavelength and shutting out radiation of long wavelength. A green house has a roof area of 100 m2
perpendicular to the solar inclination. The material has a transmissivity of 0.9 up to a wavelength of 4 µm and zero beyond. The solar flux has a value of 800 W/m2. The total wall area is
600 m2. It the inside is to be maintained at 22°C while the outside is at – 5°C, determine the
maxi- mum value of overall heat transfer coefficient for heat flow through the walls. The
temperature of solar radiation may be taken as 5000 K.
MODEL ANSWERS
A1
Assumptions
Assume that steady state exists, neglect effects at the corners and edges of the wall, and assume
that the surface temperatures are uniform.
The rate of heat loss per unit area can be calculated from equation given below:
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N.B: The temperature drop across the steel interior wall is only 1.4 K because the thermal
resistance of the wall is small compared to the resistance of the brick, across which the
temperature drop is many times larger.
A3
Solution:
The heat lost by convection = Q = hA(T1 – T2)
Fig. A3
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A5
The energy balance provides (Fig. A5) heat received convection by
Sheet 1 = heat radiation exchange between sheet 1 and 2.
= heat convected by sheet 2.
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A7
The data and equivalent circuit are shown in Fig A7.
Fig. A7. Composite wall.
Using equation:
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Note: The contact drops and drop in the metal plate are very small. The insulation resistances
and outside convection are the controlling resistances.
A9
The heat is lost from the surface of the device by convection without fin:
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Fig. A9
A11
Using equation:
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A13
Considering A
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A15
Film temperature = (– 20 + 20)/2 = 0°C
As density and kinematic viscosities will vary with pressure, dynamic viscosity is read from
tables.
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A17
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A19
(a) The radiation energy emitted by an area A streaming through a differential solid angle 𝑑𝜔 =
𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅ in any direction is given by
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