HEAT TRANSFER (2151909) B.E. Semester – V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1 TOPIC 2 [A] STEADY STATE HEAT CONDUCTION CLASS TUTORIAL 1. The walls of a refrigerated truck consist of 1.2 mm thick steel sheet (k=18 W/m-K) at the outer surface, 22 mm thick cork (k=0.04 W/m-K) on the inner surface. Consider Heat transfer coefficient of 5 W/m 2 -K (between inside air and inside surface) and Heat transfer coefficient of 30 W/m2-K (between outside air and outside surface). The temperatures at the inside and outside air are 0℃ & 35℃respectively. Calculate I. heat transfer rate II. steel-cork interface temp. 2. A storage chamber of interior dimensions 10 8 2.5 high has its inside maintained at a temperature of −20℃ whilst the outside is at 25℃. The walls and ceiling of the chamber have three layers made of 60 mm thick board ( = 0.2 − ⁄ ) on the inside 90 mm thick insulation ( = 0.04 − ⁄ ) at the mid 240 mm thick concrete ( = 1.8 − ⁄ ) on the outside Neglecting flow of heat through the floor, determine the rate at which heat can flow towards inside of the chamber. (D.S. Kumar, Example 3.13) 3. A steam pipe of 5 cm inside diameter and 6.5 cm outside diameter is insulated with a 2.75 cm radial thickness of high temperature insulation (k = 1.1 W/mK). The surface heat transfer coefficient for inside and outside surfaces are 4650 W/m 2 K and 11.5 W/m 2 K respectively. The thermal conductivity of the pipe material is 45 W/m K. If the steam temperature is 200°C and ambient air temperature 25°C, determine I. Heat loss per meter length of pipe II. Temperature at the interface III. Overall heat transfer coefficient 4. An 8 mm thick metal plate, having thermal conductivity 98.6 W m − K ⁄ is exposed to vapor at 100 ℃ on one side and cooling water at 30 ℃ on another side. The heat transfer coefficients are 14200 W m 2 K ⁄ on vapor side and 2325 W m 2 K ⁄ on water side. Determine the rate of heat transfer and drop in temperature on each side of the plate. Assume area of the plate as unity. (Summer 2014)(Similar to D.S. Kumar, Example 3.40)
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HEAT TRANSFER (2151909)
B.E. Semester – V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1
TOPIC 2
[A] STEADY STATE HEAT CONDUCTION
CLASS TUTORIAL
1. The walls of a refrigerated truck consist of 1.2 mm thick steel sheet (k=18 W/m-K) at
the outer surface, 22 mm thick cork (k=0.04 W/m-K) on the inner surface. Consider
Heat transfer coefficient of 5 W/m2-K (between inside air and inside surface) and
Heat transfer coefficient of 30 W/m2-K (between outside air and outside surface).
The temperatures at the inside and outside air are 0℃ & 35℃respectively. Calculate
I. heat transfer rate
II. steel-cork interface temp.
2. A storage chamber of interior dimensions 10 𝑚𝑋 8 𝑚𝑋 2.5 𝑚 high has its inside
maintained at a temperature of −20℃ whilst the outside is at 25℃. The walls and
ceiling of the chamber have three layers made of
60 mm thick board (𝑘 = 0.2 𝑊 𝑚 − 𝑑𝑒𝑔⁄ ) on the inside
90 mm thick insulation (𝑘 = 0.04 𝑊 𝑚 − 𝑑𝑒𝑔⁄ ) at the mid
240 mm thick concrete (𝑘 = 1.8 𝑊 𝑚 − 𝑑𝑒𝑔⁄ ) on the outside
Neglecting flow of heat through the floor, determine the rate at which heat can flow
towards inside of the chamber. (D.S. Kumar, Example 3.13)
3. A steam pipe of 5 cm inside diameter and 6.5 cm outside diameter is insulated with
a 2.75 cm radial thickness of high temperature insulation (k = 1.1 W/mK). The surface
heat transfer coefficient for inside and outside surfaces are 4650 W/m2K and 11.5
W/m2K respectively. The thermal conductivity of the pipe material is 45 W/m K. If
the steam temperature is 200°C and ambient air temperature 25°C, determine
I. Heat loss per meter length of pipe
II. Temperature at the interface
III. Overall heat transfer coefficient
4. An 8 mm thick metal plate, having thermal conductivity 98.6 W m − K⁄ is exposed to
vapor at 100℃ on one side and cooling water at 30℃ on another side. The heat
transfer coefficients are 14200 W m2K⁄ on vapor side and 2325 W m2K⁄ on water
side. Determine the rate of heat transfer and drop in temperature on each side of the
plate. Assume area of the plate as unity.(Summer 2014)(Similar to D.S. Kumar,
Example 3.40)
HEAT TRANSFER (2151909)
B.E. Semester – V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 2
5. A steel tube of 5 cm inner diameter and 8 cm outer diameter(𝑘 = 16 𝑊 𝑚𝐾⁄ ) , is
covered with an insulation of 3 cm thickness(𝑘 = 0.3 𝑊 𝑚𝐾⁄ ). A hot gas at350℃, ℎ =
400 𝑊 𝑚2𝐾⁄ flows. Calculate the heat loss from the tube for 20 meter length. Also
calculate the temperature at the interface of insulation and steel. Outside air
temperature is at30℃,ℎ = 60 𝑊 𝑚2𝐾⁄ . (May-2012) (Similar to Mahesh Rathod,
Example 3.29)
6. A refrigeration suction line having outer diameter 30 mm is required to be thermally
insulated. The outside air convective heat transfer coefficient is 12 𝑊 𝑚2𝐾⁄ . The
thermal conductivity of the insulating material is0.3 𝑊 𝑚𝐾⁄ . Determine:
I. Whether the insulation will be effective
II. Estimate the maximum value of thermal conductivity of insulating material to
reduce heat transfer
III. The thickness of cork insulation to reduce the heat transfer to 20% (k=0.04
W/m oC)(Summer 2013)
ASSIGNMENT
Theory
1. Derive an expression for three dimensional time dependent heat conduction with
internal heat generation and constant thermal conductivity in cartesian
2. Define total emissive power (Eb) and intensity of radiation (Ib). Show that Eb = π×Ib
(DEC-15, MAY-17, MAY-18)
3. (a) Explain Wien’s displacement law of radiation.
(b) Explain Kirchoff’s law of radiation.
(DEC-16, MAY-17, NOV-17)
4. What is Radiosity (J)? Show that the net radiant energy leaving the surface is given
by
(DEC-16)
5. What is radiation shield? Show that presence of n number of radiation shields reduces
the radiation heat transfer by a factor of (n+1).
(MAY-18)
HEAT TRANSFER (2151909)
B.E. Semester V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 10
Examples
1. Consider two large parallel plates, one at temperature at 727 °C with emissivity 0.8
and other at 227 °C with emissivity 0.4. An aluminium radiation shield with an
emissivity of 0.05 on both sides is placed between two plates. Calculate reduction in
heat transfer rate between two plates as a result of shield.
(MAY-17)
2. An enclosure measures 1.5m* 1.5m with a height of 2m under steady state conditions,
the wall and ceiling are maintained at 525 K and floor is at 400K. Determine net
radiation to floor. Take emissivity of ceiling and wall = 0.85 and emissivity of floor =
0.75
(NOV-17)
ASSIGNMENT - CONVECTION HEAT TRANSFER (2151909)
B.E. Semester – V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1
CHAPTER – 3 CONVECTION Theory
1. Discuss and define:
a) Natural and Forced Convection.
b) Mean Film Temperature & Bulk Mean Temperature.
2. State the general equation for the rate of heat transfer by convection and hence define the coefficient of heat transfer. What are the various factors on which the value of this coefficient depends?
3. Define and discuss the following dimensionless numbers:
a) Reynolds Number
b) Prandtl Number
c) Grashoff Number
d) Nusselt Number.
4. Derive the generalized co-relation for natural convection by using Buckingham’s π theorem method.
5. Derive the generalized co-relation for forced convection by using Buckingham’s π theorem method.
6. Derive the momentum equation for hydrodynamic boundary layer over a flat plate.
7. Define and discuss the hydrodynamic and thermal boundary layers over a flat plate. Show the thickness of these layers for different Prandtl numbers.
Examples
1. Calculate the rate of heat loss from a human body which may be considered as vertical cylinder 30 cm in diameter and 175 cm high in still air at 15°C. The skin temperature is 35°C and emissivity at the skin surface is 0.4. Neglect sweating and effect of clothing. Use Nu = 0.13 (Gr Pr)0.33. [Ans: 208.61 W] [D.S. Kumar 11.15]
2. A nuclear reactor with its core constructed of parallel vertical plates 2.25 m high and 1.5 wide has been designed on free convection heating of liquid bismuth. Metallurgical considerations limit the maximum surface temperature of the plate to 975°C and the lowest allowable temperature of bismuth is 325°C. Estimate the maximum possible heat dissipation from both sides of each plate. The appropriate correlation for the convection coefficient is Nu = 0.13 (Gr Pr)1/3. Where, the different parameters are evaluated as the mean film temperature.
[Ans: 153 MW] [D.S. Kumar;11.6]
3. A sheet metal air duct carries air-conditioned air at an average temperature of 10°C. The duct size is 320mm x 200mm and length of the duct exposed to the surrounding air at 30°C is 15m long. Find the heat gain by the air in the duct. Assume 200mm side is vertical and top surface of the duct is insulated. Use the following correlation: Nu = 0.6 (Gr Pr)0.25 for vertical surface, and Nu = 0.27 (Gr Pr)0.25 for horizontal surface. [Ans: 7772.9W] [R. K. Rajput; 8.5]
ASSIGNMENT - CONVECTION HEAT TRANSFER (2151909)
B.E. Semester – V Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 2
4. A gas pipe is kept in an atmosphere of 20°C. The radius of pipe is 3.75cm and is lagged with insulation thickness of 2.5cm. The emissivity of the surface is 0.9.The length of pipe is 6 m. Surface temperature ts=80°C. Calculate (i) The total heat loss from pipe (ii) The overall heat transfer coefficient (iii) The heat transfer co efficient due to only radiation. For convection use co-relation, Nu =0.53(Gr.Pr)1/4.
[Ans: 1863.24W; 13.1797 W/m2-K; 6.9378 W/m2-K] [GTU – DEC 2013]
5. A spherical heater of 20cm diameter and at 60°C is immersed in a tank of water at 20°C. Determine the value of convective heat transfer coefficient & heat transfer rate by natural convection. For a sphere, the general correlation is Nu = 2 + 0.43 (Ra)0.25.
6. Air at 30°C and at atmospheric pressure flows over a flat plate at a speed of 1.9 m/sec. If the plate is maintained at 90°C, calculate the heat transfer per unit width of the plate; assuming the length of the plate along the flow of air is 2 meters. Where
local Nusselt number is given by,𝑁𝑢𝑥 = 0.332 (𝑅𝑒𝑥)1
2⁄ (𝑃𝑟𝑥)1
3⁄ [Ans: 917.16 W]
7. Air at 30°C and at atmospheric pressure is flowing over a flat plate with velocity 4 m/sec. If the plate is 2m long and the wall temperature is 70°C. Calculate the following at a location 2m from leading edge.
a. Hydrodynamic boundary layer thickness. b. Local heat transfer co-efficient.
[Ans: 13.90mm; 2.9169W/m2K]
8. Air at 27°C and 1 atm flows over a flat plate at a velocity of 3 m/s. Calculate the boundary layer thickness at distances of 25 and 45 cm from the leading edge of the plate. Calculate the mass flow which enters the boundary layer between x = 25cm and x = 45cm. The viscosity of air at 27°C is 1.85E-05 kg/m-s. Assume parabolic velocity distribution and unit depth in z-direction.
[Ans: 5.31mm; 7.125; 0.004 kg/s] [4.17; P. K. NAG]
9. Water at 50°C enters 1.5 cm diameter and 3 m long tube with a velocity of 1.5 m/s.
The tube wall is maintained at 100°C. Calculate the heat transfer coefficient and total
amount of heat transferred if the water exit temperature is 70°C. The relevant
properties of water are Pr = 3.15, ρ = 990 kg/m3, ν = 0.517 × 10-6 m2/s, Cp = 4184
J/kg-K, kf = 0.65 W/m-K. Use following correlation,
𝑁𝑢𝐷 = 0.023(𝑅𝑒𝐷)0.8(𝑃𝑟)0.4 Use following properties of fluid at required temperature,