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heat transfer practice problems

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Problem sheet Steady state heat conduction Composite system
1. A steam boiler furnace is made of a layer of fireclay (k = 0.533 W/m K)12.5cm thick and a layer of red brick(k = 0.7 W/mK) 50cm thick. If the wall temperature inside the boiler furnace is 1100
C and that on the outside wall is 50
o
C, determine the amount of heat loss per square meter of the furnace wall. 2. The wall of a cold storage consists of three layers, an outer layer of ordinary brick of 25cm thick, a middle layer of cork,10cm thick, and inner layer of cement,6cm thick. The thermal conductivities of the materials are: k
brick
= 0.7 W/m
o
C, k
cork
= 0.043 W/m
o
C, k
cement
= 0.72 W/m
o
C. The temperature of the outer surface of the wall is 30
o
C, and the inner is -15
o
C. Calculate rate of heat transfer per unit area of the wall and interface temperatures. 3.The inner dimensions of a freezer cabinet are 60cm x 60cm x 50cm(height). The cabinet walls consist of two 2mm thick enameled sheet steel (k= 40W/mK) walls separated by a 4cm layer of fiberglass(k=0.049W/mK) insulation. The inside temperature is to be maintained at -15
o
C and the outside temperature on a hot summer day is 45
o
C.Calculate the rate of heat transfer assuming heat transfer coefficient of 10 W/m
2
K both on the inside and outside of the cabinet. Also calculate the outer surface temperature of the cabinet. 4. A wall of 0.5m thickness is to be constructed from a material which has an average thermal conductivity of 1.4W/mK. The wall is to be insulated with a material having an average thermal conducting of 0.35W/mK, so that the heat loss per square meter will not exceed 1450W. Assuming that the inner and outer surface temperatures are 1200
o
C and 15
o
C respectively, calculate the thickness of insulation required. 5. An exterior wall of a house may be approximated by a 10cm layer of common brick(k=0.7W/m
C) followed by a 3.75cm layer of gypsum plaster (k=0.48W/m
C).What thickness of loosely packed rock-wool insulation (k=0.065W/m
C) should be added to reduce the heat loss through the wall by 80%? 6. The door of a cold storage plant is made from two 6mm thick glass sheets(k = 0.75W/mK) separated by a uniform air gap(k = 0.02W/mK) of 2mm. The temperature of the air inside the room is -20
C and the ambient air temperature is 30
o
C. Assuming the heat transfer coefficient between glass and air to be 23.26 W/m
2
K, determine the rate of heat loss into the room per unit area of the door. Neglect convection effect in the air gap. 7.A hot steam pipe having an inside surface temperature of 250
C has an inside diameter of 8cm and a wall thickness of 5.5mm. It is covered with a 9cm layer of insulation having k=0.5 W/m
C, followed by a 4cm layer of insulation having k = 0.25 W/m
C. The outside temperature of insulation is 20
C. Calculate the heat lost per meter of length. Assume k=47 W/m
C for the pipe. 8.A multilayer cylindrical wall of a furnace is constructed of 4.5cm layer of insulating brick with thermal conductivity of 0.081W/mK followed by a 9cm layer of common brick with thermal conductivity of 0.0812W/mK . The inner wall temperature is 2500
o
C and outer wall temperature is 70
o
C .What is the heat loss through the wall, when the inner diameter of the furnace is 1.2m and the length of the wall is 1m? 9.A steel pipe having an I.D.52.50mm and an O.D.60.33mm and k=39.7kcal/hr m
o
C carries steam at 150
o
C . It is lagged with 12.7mm thick rock wool of thermal conductivity 0.049kcal/hr m
o
C and the surrounding air is at 20
o
C .If the heat transfer coefficient from the insulated pipe to the surrounding air is 6kcal/hr m
2o
C, what will be the (a) heat loss per meter length of pipe (b) temperature at the boundaries between the pipe wall and rock wool and (c) between the rock wool and surrounding air.
2
10.A steel pipe (k= 44W/mK) of 5.08cm I.D and 7.62cm O.D is covered with a 2.54cm layer of asbestos insulation(k=0.208W/mK).The inside surface of the pipe receives heat from the hot gas at a temperature of 316
o
C with the heat transfer coefficient 284W/m
2
K,while the outer surface of the insulation is exposed to the ambient air at 38
o
C with the heat transfer coefficient of 17W/m
2
K. Calculate the heat loss to ambient air for 3m length of the pipe and also calculate the interface temperatures.
Extended surfaces Long fins
1.
One end of a very long aluminum rod is connected to a wall at 140ºC,while the other end protrudes into a room whose air temperature is 15ºC.The rod is 3mm in diameter and the heat transfer coefficient between the rod surface and environment is 300W/m
2
K. Calculate the total heat dissipated by the rod taking its thermal conductivity as 150W/mK. 2.
In an experiment to determine the thermal conductivity of a long solid 2.5cm diameter rod, its base is placed in a furnace with a large portion of it projecting into the room air at 22ºC. After steady state conditions prevail, the temperature at two points,10cm apart, are found to be 110ºC and 85ºC respectively. The convective heat transfer coefficient between the rod surface and the surrounding air is 28.4W/m
2
K.Determine the thermal conductivity of the rod material. 3.
In an experiment to determine the thermal conductivity of a long solid 20mm diameter rod,its base is placed in a furnace with a large portion of it projecting out. The temperature at two points 100mm apart are found to be 125ºC and 91ºC respectively. The heat transfer coefficient over the surface of the rod is estimated to be 17.45W/m
2
K.Determine the thermal conductivity of the rod material. 4.
A long,thin copper rod 5mm in diameter is exposed to an environment at 20ºC.The base temperature of the rod is 120ºC. The heat transfer coefficient between the rod and the environment is 20W/m
2
K.Calculate the heat given up by the rod. The thermal conductivity of the rod material is 372 W/mK. 5.
A very long copper rod(k = 372 W/mK) 2.5cm in diameter is exposed to an environment at 40ºC. The base temperature of the rod is 90ºC. The heat transfer coefficient is 3.5W/m
2
K.How much heat is lost by the rod?
Convection from the fin end
1.
A 1 m long,5cm diameter cylinder placed in an atmosphere at 40ºC is provided with 12 longitudinal straight fins(k= 75.6W/mK).The fins are 0.8mm thick and protrude 2.5cm from the cylinder surface. The heat transfer coefficient from the cylinder and fins to the ambient air is 23.25W/m
2
K.Calculate the rate of heat transfer, if the surface temperature is 150ºC. 2.
A 1 m long,5cm diameter cylinder placed in an atmosphere at 45ºC is provided with 10 longitudinal straight fins(k= 120W/mK).The fins are 0.76mm thick and protrude 1.27cm from the cylinder surface. The heat transfer coefficient from the cylinder and fins to the ambient air is 17W/m
2
K.Calculate the rate of heat transfer and the temperature at the end of fins, if the surface temperature is 150ºC.
Fin with an insulated tip
1.
A turbine bladé 6cm long and having a cross sectional area 4.65cm2 and perimeter 12cm is made of stainless steel(k=23.3W/mK).The temperature at the root is 500ºC. The blade is exposed to a hot gas at 870ºC. The heat transfer coefficient between the blade surface and the gas is 442 W/m2K. Determine the temperature distribution and the rate of heat flow at the root of the blade. Assume tip of the blade to be insulated. 2.
Compare the temperature distribution in a pin fin having a diameter of 2cm and length 10cm and exposed to a convection environment with h = 25W/m2K for three fin materials: copper(k = 385W/mK),stainless steel (k = 17W/mK), and a glass (k=0.8W/mK).Also calculate the relative heat flows and the fin efficiencies with respect to the copper fin.
3
3.
An aluminum fin (k=200W/mK) 3mm thick and 7.5cm long protrudes from a wall at 300ºC. The ambient temperature is 50ºC with h = 10W/m
2
K.Calculate the heat loss from the fin per unit depth of the material. Also calculate its efficiency and effectiveness. 4.
An aluminum fin(k=200W/mK) 1.5mm thick is placed on a circular tube with 2.7cm OD. The fin is 6mm long. The tube wall is maintained at 150ºC. The environment temperature is 15ºC, and the heat transfer coefficient is 20 W/m
2
K.Calculate the heat lost by the fin per unit depth of material. 5.
A straight fin of rectangular profile has a thermal conductivity of 14 W/mºC, thickness 2mm and length of 23mm.The base of the fin is maintained at a temperature of 220ºC,while the fin is exposed to a convection environment at 23ºCwith heat transfer coefficient 25W/m
2
ºC. Calculate the heat lost per meter of fin depth. 6.
A straight rectangular fin 2cm thick and 14cm long is constructed of steel and placed on the outside of a wall maintained at 200ºC. The environment temperature is 15ºC, and the heat transfer coefficient is 20 W/m2ºC. Calculate the heat lost by the fin per unit depth. 7.
A straight fin of steel(1%C)(k= 43W/mºC) is 2.6cm thick and 17cm long. It is placed on the outside of a wall which is maintained at 230ºC. The surrounding air temperature is 25ºC,and the convection heat transfer coefficient is 23 W/m
2
ºC.Calculate the heat lost from the fin per unit depth and the fin efficiency.
Circumferential fin
1.
A circumferential fin of rectangular profile has a thickness of 0.7mm and is installed on a tube having a diameter of 3cm that is maintained at a temperature of 200ºC. The length of the fin is 2cm and the fin material is copper. Calculate the heat lost by the fin to a surrounding convection environment at 100ºC with a convection heat transfer coefficient of 524 W/m
2
ºC. 2.
A circumferential fin of rectangular profile is constructed of a material having thermal conductivity 55 W/mºC, and is installed on a tube having diameter of 3cm. The length of the fin is 3cm and the thickness is 2mm. If the fin is exposed to a convection environment at 20ºC with a convection coefficient of 68 W/m
2
ºC and the tube wall temperature is 100ºC.Calculate the heat lost by the fin. 3.
A 2.5cm diameter tube has circumferential fins of rectangular profile spaced at 9.5mm increments along its length. The fins are constructed of aluminum (k=200W/mºC) and are 0.8mm thick and 12.5mm long. The tube wall temperature is maintained at 200ºC,and the environment temperature is 93ºC. The heat transfer coefficient is 110W/m2ºC.Calculate the heat loss from the tube per meter of length. 4.
A circumferential fin of rectangular profile surrounds 2cm diameter tube. The length of the fin is 5mm,and the thickness is 2.5mm. the fin is constructed of mild steel. If air blows over the fin sothat a heat transfer coefficient of 25W/m2ºC is experienced and the temperature of the base and air are 260 and 93ºC respectively, calculate the heat transfer from the fin. 5.
An aluminum fin (k=200W/mºC)1.6mm thick surrounds a tube 2.5cm in diameter. The length of the fin is 12.5mm. The tube wall temperature is 200ºC,and the environment temperature is 20ºC. The heat transfer coefficient is 60W/m
2
ºC. Calculate the heat lost by the fin. 6.
A circumferential fin of rectangular profile is installed on a 10cm diameter tube maintained at 120ºC. The fin has a length of 15cm and thickness of 2mm. The fin is exposed to a convection environment at 23ºC, with heat transfer coefficient 60W/m
2
ºC. and the fin conductivity is 120W/mºC. Calculate the heat lost by the fin. 7.
Circumferential aluminium fin (k=200W/mK) of rectangular profile (1.5cm wide x 1mm thick) are fitted onto a 2.5cm diameter tube. The fin base temperature is 170ºC and the ambient fluid temperature is 25ºC. Calculate the heat loss per fin. The heat transfer coefficient may be taken as 130W/m
2
K.
4
Unsteady state heat conduction (Transient Heat conduction) Lumped Heat Capacity method of analysis:(systems with negligible internal resistance)
1.
A 40 x 40 cm copper slab 5mm thick at a uniform temperature of 250ºC suddenly has its surface temperature lowered at 30ºC. Find the time at which the slab temperature becomes 90ºC;
= 9000kg/m
3
; Cp = 0.38 kJ/kg K; k =370 W/mK and h = 90 W/m
2
K 2.
A 2mm thick copper plate at 400
C is suddenly dipped into water at 20
C. Calculate the time required for the plate to reach a temperature of 40
C, taking h = 93 W/m
2
K. Data:
= 8800kg/m
3
; Cp = 0.381 kJ/kg K; area = 30cm x 30cm; k = 370W/mK. 3.
A stainless steel rod of OD 1 cm srcinally at a temperature of 320ºC is suddenly immersed in a liquid at 120ºC for which the convective heat transfer coefficient is 100 W/m
2
K. Determine the time required for the rod to reach a temperature of 200ºC. Properties of stainless steel :
= 7800kg/m
3
; Cp = 460J/kg K; k = 40 W/mK 4.
(i)A copper wire 0.8mm diameter at 150
C is suddenly dipped into water at 35
C. If the heat transfer coefficient is 85.5 W/m
2
K,calculate the time required to cool the wire to 95
C. Data:
= 9000kg/m
3
; Cp = 0.38 kJ/kg K; k = 373 W/m K. (ii) If the same wire is placed in air instead of water what will be the time required to cool it to 95
C. h= 11.65 W/m
2
K? 5.
A metallic rod 12.5mm diameter at 90
C is cooled to 35
C in 105 s by placing it in air stream at 30
C. Calculate the heat transfer coefficient for air. Data(for rod): mass = 0.1kg; Cp = 350J/kg K; surface area = 0.004m
2
. 6.
Aluminum sphere weighing 5.5 kg and initially at a temperature of 290ºC is suddenly immersed in a fluid at 15ºC. The convective heat transfer coefficient is 58W/m
2
K. Calculate the time required to cool the aluminum to 95ºC, using lumped heat capacity method of analysis. Properties of aluminum:
= 2700kg/m
3
; Cp = 900J/kg K; k = 205 W/mK. 7.
Aluminum sphere weighing 7 kg and initially at a temperature of 260ºC is suddenly immersed in a fluid at 10ºC. The convective heat transfer coefficient is 50W/m
2
K. Calculate the time required to cool the aluminum to 90ºC, using lumped heat capacity method of analysis. Properties of aluminum:
= 2700kg/m
3
; Cp = 900J/kg K; k = 205 W/mK. 8.
A steel ball 5cm in diameter and initially at a uniform temperature of 450ºC is suddenly placed in a controlled environment in which the temperature is maintained at 100ºC. The heat transfer coefficient is 10 W/m2 K. Calculate the time required for the ball to attain a temperature of 150ºC.Properties of steel:
= 7800kg/m
3
; Cp = 0.46 kJ/kg K; k = 35 W/mK. 9.
The average heat transfer coefficient for flow of 100ºC air over a flat plate is measured by observing the temperature time history of a 3 cm thick copper slab exposed to 100ºC air. In one test run, the initial temperature of the plate was 210ºC, and in 5 min the temperature decreased by 40ºC.Calculate the heat transfer coefficient. Properties of copper:
= 9000kg/m
3
; Cp = 0.38 kJ/kg K; k =370 W/mK
Systems with negligible surface resistance: 1.
A large plate,50cm thick, initially has a uniform temperature of 40
C. If the plate is suddenly raised to and maintained at a temperature of 440
C,calculate (i)
The temperature at a depth of 20cm from the surface after 30s (ii)
The instantaneous heat flow across the above plane per m
2
after 30s (iii)
The heat flow across the above plane after 30s (iv)
The temperature at the central plane after 30s.
Data: k = 6 W/m K; α = 2.5 x 10
-6
m
2
/s
2.
A large wall 62.8cm thick has uniform temperature of 50
Cinitially. If the temperature is suddenly raised to and maintained at 550
C,find (i)
The temperature at a depth of 15.7cm from the surface after 5h (ii)
The instantaneous heat flow across the above plane per m
2
after 5h

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