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physics
thermodynamics

Heat Transfer 10th edition Jack Holman - Solutions

An insulating glass window is constructed of two 5-mm glass plates separated by an air layer having a thickness of 4 mm. The air layer may be considered stagnant so that pure conduction is involved. The convection coefficients for the inner and outer surfaces are 12 and 50 W/m2 ∙ ºC,
A wall consists of a 1-mm layer of copper, a 4-mm layer of 1 percent carbon steel, a 1-cm layer of asbestos sheet, and 10 cm of fiberglass blanket. Calculate the overall heat-transfer coefficient for this arrangement. If the two outside surfaces are at 10 and 150ºC, calculate each of the interface
A circumferential fin of rectangular profile has a thickness of 0.7 mm and is installed on a tube having a diameter of 3 cm that is maintained at a temperature of 200ºC. The length of the fin is 2 cm and the fin material is copper. Calculate the heat lost by the fin to a surrounding convection
A thin rod of length L has its two ends connected to two walls which are maintained at temperatures T1 and T2, respectively. The rod loses heat to the environment at T∞ by convection. Derive an expression (a) for the temperature distribution in the rod and (b) for the total heat lost by the rod.
A rod of length L has one end maintained at temperature T0 and is exposed to an environment of temperature T∞. An electrical heating element is placed in the rod so that heat is generated uniformly along the length at a rate . Derive an expression (a) for the temperature distribution in the rod
One end of a copper rod 30 cm long is firmly connected to a wall that is maintained at 200ºC. The other end is firmly connected to a wall that is maintained at 93ºC. Air is blown across the rod so that a heat-transfer coefficient of 17 W/m2 ∙ ºC is maintained. The diameter of the rod is 12.5
An aluminum rod 2.0 cm in diameter and 12 cm long protrudes from a wall that is maintained at 250ºC. The rod is exposed to an environment at 15ºC. The convection heat-transfer coefficient is 12 W/m2 · ºC. Calculate the heat lost by the rod.
Derive Equation (2-35) by integrating the convection heat loss from the rod of case 1 in Section 2-9.
One side of a copper block 4 cm thick is maintained at 175ºC. The other side is covered with a layer of fiberglass 1.5 cm thick. The outside of the fiberglass is maintained at 80ºC, and the total heat flow through the composite slab is 300 W. What is the area of the slab?
Derive Equation (2-36) by integrating the convection heat loss from the rod of case 3 in Section 2-9.
A long, thin copper rod 5 mm 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 20 W/m2 · ºC. Calculate the heat given up by the rod.
A very long copper rod [k = 372 W/m ∙ ºC] 2.5 cm in diameter has one end maintained at 90ºC. The rod is exposed to a fluid whose temperature is 40ºC. The heat-transfer coefficient is 3.5 W/m2 ∙ ºC. How much heat is lost by the rod?
An aluminum fin 1.5 mm thick is placed on a circular tube with 2.7-cm OD. The fin is 6 mm long. The tube wall is maintained at 150ºC, the environment temperature is 15ºC, and the convection heat-transfer coefficient is 20 W/m2 ∙ ºC. Calculate the heat lost by the fin.
A straight fin of rectangular profile has a thermal conductivity of 14 W/m ∙ ºC, thickness of 2.0 mm, and length of 23 mm. The base of the fin is maintained at a temperature of 220ºC while the fin is exposed to a convection environment at 23ºC with h = 25 W/m2 ∙ ºC. Calculate the heat lost
A circumferential fin of rectangular profile is constructed of a material having k = 55W/m ∙ ºC and is installed on a tube having a diameter of 3 cm. The length of fin is 3 cm and the thickness is 2 mm. If the fin is exposed to a convection environment at 20ºC with a convection coefficient of
The total efficiency for a finned surface may be defined as the ratio of the total heat transfer of the combined area of the surface and fins to the heat that would be transferred if this total area were maintained at the base temperature T0. Show that this efficiency can be calculated
A triangular fin of stainless steel (18% Cr, 8% Ni) is attached to a plane wall maintained at 460ºC. The fin thickness is 6.4 mm, and the length is 2.5 cm. The environment is at 93ºC, and the convection heat-transfer coefficient is 28 W/m2 ∙ ºC. Calculate the heat lost from the fin.
A 2.5-cm-diameter tube has circumferential fins of rectangular profile spaced at 9.5-mm increments along its length. The fins are constructed of aluminum and are 0.8 mm thick and 12.5 mm long. The tube wall temperature is maintained at 200ºC, and the environment temperature is 93ºC. The
A circumferential fin of rectangular profile surrounds a 2-cm-diameter tube. The length of the fin is 5 mm, and the thickness is 2.5 mm. The fin is constructed of mild steel. If air blows over the fin so that a heat-transfer coefficient of 25 W/m2 ∙ ºC is experienced and the temperatures of the
A plane wall is constructed of a material having a thermal conductivity that varies as the square of the temperature according to the relation k = k0(1 + βT2). Derive an expression for the heat transfer in such a wall.
A straight rectangular fin 2.0 cm thick and 14 cm 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 for convection is 20 W/m2 ∙ ºC. Calculate the heat lost from the fin per unit depth.
An aluminum fin 1.6 mm thick surrounds a tube 2.5 cm in diameter. The length of the fin is 12.5 mm. The tube-wall temperature is 200ºC, and the environment temperature is 20ºC. The heat transfer coefficient is 60W/m2 ∙ ºC. What is the heat lost by the fin?
Derive a differential equation (do not solve) for the temperature distribution in a straight triangular fin. For convenience, take the coordinate axis as shown in Figure P2-83 and assume one-dimensional heat flow.
A circumferential fin of rectangular profile is installed on a 10-cm-diameter tube maintained at 120ºC. The fin has a length of 15 cm and thickness of 2 mm. The fin is exposed to a convection environment at 23ºC with h = 60 W/m2 ∙ ºC and the fin conductivity is 120 W/m ∙ ºC. Calculate the
A long stainless-steel rod [k = 16 W/m ∙ ºC] has a square cross section 12.5 by 12.5 mm and has one end maintained at 250ºC. The heat-transfer coefficient is 40 W/m2 · ºC, and the environment temperature is 90ºC. Calculate the heat lost by the rod.
A straight fin of rectangular profile is constructed of duralumin (94% Al, 3% Cu) with a thickness of 2.1 mm. The fin is 17 mm long, and it is subjected to a convection environment with h = 75 W/m2 ∙ ºC. If the base temperature is 100ºC and the environment is at 30ºC, calculate the heat
A certain internal-combustion engine is air-cooled and has a cylinder constructed of cast iron [k = 35 Btu/h ∙ ft ∙ ºF]. The fins on the cylinder have a length of 5/8 in and thickness of 1/8 in. The convection coefficient is 12 Btu/h ∙ ft2 ∙ ºF. The cylinder diameter is 4 in. Calculate
A 1.5-mm-diameter stainless-steel rod [k = 19 W/m ∙ ºC] protrudes from a wall maintained at 45ºC. The rod is 12 mm long, and the convection coefficient is 500W/m2 ∙ ºC. The environment temperature is 20ºC. Calculate the temperature of the tip of the rod. Repeat the calculation for h = 200
An aluminum block is cast with an array of pin fins protruding like that shown in Figure 2-10d and subjected to room air at 20ºC. The convection coefficient between the pins and the surrounding air may be assumed to be h = 13.2 W/m2 ∙ ºC. The pin diameters are 2 mm and their length is 25 mm.
A steel tube having k = 46W/m · ºC 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 1500W/m2 · ºCon the inside surface, while a second fluid flows across the outside of the tube producing a
A finned tube is constructed as shown in Figure 2-10b. Eight fins are installed as shown and the construction material is aluminum. The base temperature of the fins may be assumed to be 100ºC and they are subjected to a convection environment at 30ºC with h = 15 W/m2 ∙ ºC. The longitudinal
Circumferential fins of rectangular profile are constructed of aluminum and attached to a copper tube having a diameter of 25 mm and maintained at 100ºC. The length of the fins is 2 cm and thickness is 2 mm. The arrangement is exposed to a convection environment at 30ºC with h = 15 W/m2 · ºC.
A 2-cm-diameter glass rod 6 cm long [k = 0.8 W/m ∙ ºC] has a base temperature of 100ºC and is exposed to an air convection environment at 20ºC. The temperature at the tip of the rod is measured as 35ºC. What is the convection heat-transfer coefficient? How much heat is lost by the rod?
A straight rectangular fin has a length of 2.5 cm and a thickness of 1.5 mm. The thermal conductivity is 55 W/m ∙ ºC, and it is exposed to a convection environment at 20ºC and h = 500 W/m2 ∙ ºC. Calculate the maximum possible heat loss for a base temperature of 200ºC. What is the actual
A straight rectangular fin has a length of 3.5 cm and a thickness of 1.4 mm. The thermal conductivity is 55 W/m ∙ ºC. The fin is exposed to a convection environment at 20ºC and h = 500 W/m2 ∙ ºC. Calculate the maximum possible heat loss for a base temperature of 150ºC. What is the actual
A circumferential fin of rectangular profile is constructed of 1 percent carbon steel and attached to a circular tube maintained at 150ºC. The diameter of the tube is 5 cm, and the length is also 5 cm with a thickness of 2 mm. The surrounding air is maintained at 20ºC and the convection
A circumferential fin of rectangular profile is constructed of aluminum and surrounds a 3-cm-diameter tube. The fin is 2 cm long and 1mmthick. The tube wall temperature is 200ºC, and the fin is exposed to a fluid at 20ºC with a convection heat-transfer coefficient of 80 W/m2 ∙ ºC. Calculate
A 1.0-cm-diameter steel rod [k = 20 W/m ∙ ºC] is 20 cm long. It has one end maintained at 50ºC and the other at 100ºC. It is exposed to a convection environment at 20ºC with h = 50 W/m2 · ºC. Calculate the temperature at the center of the rod.
A circumferential fin of rectangular profile is constructed of copper and surrounds a tube having a diameter of 1.25 cm. The fin length is 6 mm and its thickness is 0.3 mm. The fin is exposed to a convection environment at 20ºC with h = 55 W/m2 ∙ ºC and the fin base temperature is 100ºC.
A straight rectangular fin of steel (1% C) is 2 cm thick and 17 cm 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/m2 · ºC. Calculate the heat lost from the fin per unit depth
Beginning with the separation-of-variables solutions for λ2 = 0 and λ2 < 0 [Equations (3-9) and (3-10)], show that it is not possible to satisfy the boundary conditions for the constant temperature at y = H with either of these two forms of solution. That is, show that, in order to satisfy the
Two long cylinders 8.0 and 3.0cmin diameter are completely surrounded by a medium with k = 1.4 W/m · ºC. The distance between centers is 10 cm, and the cylinders are maintained at 200 and 35ºC. Calculate the heat-transfer rate per unit length.
A groundwater heat pump is a refrigeration device that rejects heat to the ground through buried pipes instead of to the local atmosphere. The heat rejection rate for such a machine at an Oklahoma location is to be 22 kW in a location where the ground temperature at depth is 17ºC. The thermal
Professional chefs claim that gas stove burners are superior to electric burners because of the more uniform heating afforded by the gas flame and combustion products around the bottom of a cooking pan. Advocates of electric stoves note the lack of combustion products to pollute the air in the
A small building 5 m wide by 7 m long by 3 m high (inside dimensions) is mounted on a flat concrete slab having a thickness of 15 cm. The walls of the building are constructed of concrete also, with a thickness of 7 cm. The inside of the building is used for cold storage at −20ºC and the outside
A 10-cm-diameter sphere maintained at 30ºC is buried in the earth at a place where k = 1.2 W/m · ºC. The depth to the centerline is 24 cm, and the earth surface temperature is 0ºC. Calculate the heat lost by the sphere.
A 20-cm-diameter sphere is totally enclosed by a large mass of glass wool. A heater inside the sphere maintains its outer surface temperature at 170ºC while the temperature at the outer edge of the glass wool is 20ºC. How much power must be supplied to the heater to maintain equilibrium
A large spherical storage tank, 2 m in diameter, is buried in the earth at a location where the thermal conductivity is 1.5 W/m · ºC. The tank is used for the storage of an ice mixture at 0ºC, and the ambient temperature of the earth is 20ºC. Calculate the heat loss from the tank.
The solid shown in Figure P3-15 has the upper surface, including the half-cylinder cutout, maintained at 100ºC. At a large depth in the solid the temperature is 300 K; k = 1 W/m· ºC. What is the heat transfer at the surface for the region where L = 30 cm and D = 10 cm?
In certain locales, power transmission is made by means of underground cables. In one example an 8.0-cm-diameter cable is buried at a depth of 1.3 m, and the resistance of the cable is 1.1 × 10−4 Ω/m. The surface temperature of the ground is 25ºC, and k = 1.2 W/m· ºC for earth. Calculate the
Two long, eccentric cylinders having diameters of 20 and 5 cm, respectively, are maintained at 100 and 20ºC and separated by a material with k = 2.5 W/m · ºC. The distance between centers is 5.5 cm. Calculate the heat transfer per unit length between the cylinders.
Two pipes are buried in the earth and maintained at temperatures of 200 and 100ºC. The diameters are 9 and 18 cm, and the distance between centers is 40 cm. Calculate the heat-transfer rate per unit length if the thermal conductivity of earth at this location is 1.1 W/m· ºC.
Write out the first four nonzero terms of the series solutions given in Equation (3-20). What percentage error results from using only these four terms at y = H and x = W/2? Equation (3-20)
A hot sphere having a diameter of 1.5 m is maintained at 300ºC and buried in a material with k = 1.2 W/m· ºC and outside surface temperature of 30ºC. The depth of the centerline of the sphere is 3.75 m. Calculate the heat loss.
A scheme is devised to measure the thermal conductivity of soil by immersing a long electrically heated rod in the ground in a vertical position. For design purposes, the rod is taken as 2.5 cm in diameter with a length of 1 m. To avoid improper alteration of the soil, the maximum surface
Two pipes are buried in an insulating material having k = 0.8 W/m· ºC. One pipe is 10 cm in diameter and carries a hot fluid at 300ºC while the other pipe is 2.8 cm in diameter and carries a cool fluid at 15ºC. The pipes are parallel and separated by a distance of 12 cm on centers. Calculate
At a certain location the thermal conductivity of the earth is 1.5 W/m· ºC. At this location an isothermal sphere having a temperature of 5ºC and a diameter of 2.0 m is buried at a centerline depth of 5.0 m. The earth temperature is 25ºC. Calculate the heat gained by the sphere.
Two parallel pipes are buried very deep in the earth at a location where they are in contact with a rock formation having k = 3.5 W/m· ºC. One pipe has a diameter of 20 cm and carries a hot fluid at 120ºC while the other pipe has a diameter of 40 cm and carries a cooler fluid at 20ºC. The
Steam pipes are sometimes carelessly buried in the earth without insulation. Consider a 10-cm pipe carrying steam at 150ºC buried at a depth of 23 cm to centerline. The buried length is 100 m. Assuming that the earth thermal conductivity is 1.2W/m· ºC and the surface temperature is 15ºC,
A hot steam pipe, 5 cm in diameter and carrying steam at 150ºC, is placed in the center of a 15-cm-thick slab of lightweight structural concrete. The outside of the concrete slab is exposed to a convection environment that maintains the top and bottom of the sheet at 20ºC. Calculate the heat lost
Seven 1.0-cm-diameter tubes carrying steam at 100ºC are buried in a semi-infinite medium having a thermal conductivity of 1.2 W/m· ºC and surface temperature of 25ºC. The depth to the centerline of the tubes is 5 cm and the spacing between centers is 3 cm. Calculate the heat lost per unit
Two parallel pipes 5 cm and 10 cm in diameter are totally surrounded by loosely packed asbestos. The distance between centers for the pipes is 20 cm. One pipe carries steam at 110ºC while the other carries chilled water at 3ºC. Calculate the heat lost by the hot pipe per unit length.
A long cylinder has its surface maintained at 135ºC and is buried in a material having a thermal conductivity of 15.5 W/m· ºC. The diameter of the cylinder is 3 cm and the depth to its centerline is 5 cm. The surface temperature of the material is 46ºC. Calculate the heat lost by the cylinder
A horizontal pipe having a surface temperature of 67ºC and diameter of 25 cm is buried at a depth of 1.2 m in the earth at a location where k = 1.8W/m ∙ ºC. The earth surface temperature is 15ºC. Calculate the heat lost by the pipe per unit length.
A 2.5-m-diameter sphere contains a mixture of ice and water at 0ºC and is buried in a semi-infinite medium having a thermal conductivity of 0.2 W/m· ºC. The top surface of the medium is isothermal at 30ºC and the sphere centerline is at a depth of 8.5 m. Calculate the heat lost by the sphere.
An electric heater in the form of a 50- by-100-cm plate is laid on top of a semi-infinite insulating material having a thermal conductivity of 0.74W/m· ºC. The heater plate is maintained at a constant temperature of 120ºC over all its surface, and the temperature of the insulating material a
A thin isothermal disk, having a diameter of 1.8 cm, is maintained at 40ºC and buried in a semi-infinite medium at a depth of 2 cm. The medium has a thermal conductivity of 0.8 W/m· ºC and its surface is maintained at 15ºC. Calculate the heat lost by the disk.
Two parallel pipes, each having a diameter of 5 cm, carry steam at 120ºC and chilled water at 5ºC, respectively, and are buried in an infinite medium of fiberglass blanket (k = 0.04 W/m· ºC). Plot the heat transfer between the pipes per unit length as a function of the centerline spacing
A small furnace has inside dimensions of 60 by 70 by 80 cm with a wall thickness of 5 cm. Calculate the overall shape factor for this geometry.
A 15-cm-diameter steam pipe at 150ºC is buried in the earth near a 5-cm pipe carrying chilled water at 5ºC. The distance between centers is 15 cm and the thermal conductivity of the earth at this location may be taken as 0.7 W/m· ºC. Calculate the heat lost by the steam pipe per unit length.
Derive an equation equivalent to Equation (3-24) for an interior node in a three-dimensional heat-flow problem.Equation (3-24)
Derive an equation equivalent to Equation (3-24) for an interior node in a one-dimensional heat-flow problem.Equation (3-24)
Derive an equation equivalent to Equation (3-25) for a one-dimensional convection boundary condition.Equation (3-25)
Considering the one-dimensional fin problems of Chapter 2, show that a nodal equation for nodes along the fin in the Figure P3-39 may be expressed asFigure P3-39
A 6.0-cm-diameter pipe whose surface temperature is maintained at 210ºC passes through the center of a concrete slab 45 cm thick. The outer surface temperatures of the slab are maintained at 15ºC. Using the flux plot, estimate the heat loss from the pipe per unit length. Also work using Table 3-1.
Show that the nodal equation corresponding to an insulated wall shown in Figure P3-40 isTm,n+1 + Tm,nˆ’1 + 2Tmˆ’1,n ˆ’ 4Tm,n = 0Figure P3-40
Derive the equation in Table 3-2f.
Derive an expression for the equation of a boundary node subjected to a constant heat flux from the environment. Use the nomenclature of Figure 3-7.Figure 3-7Nomenclature for nodal equation with convective boundary condition.
An aluminum rod 2.5 cm in diameter and 15 cm long protrudes from a wall maintained at 300ºC. The environment temperature is 38ºC. The heat-transfer coefficient is 17 W/m2 · ºC. Using a numerical technique in accordance with the result of Problem 3-39, obtain values for the
For the wall in Problem 3-6 a material with k = 1.4 W/m· ºC is used. The inner and outer wall temperatures are 650 and 150ºC, respectively. Using a numerical technique, calculate the heat flow through the wall. Problem 3-6 A heavy-wall tube of Monel, 2.5-cm ID and 5-cm OD, is covered with a
Repeat Problem 3-48, assuming that the outer wall is exposed to an environment at 38ºC and that the convection heat-transfer coefficient is 17 W/m2 · ºC. Assume that the inner surface temperature is maintained at 650ºC. Problem 3-48 For the wall in Problem 3-6 a material with k = 1.4 W/m· ºC
A 2.5-cm-diameter pipe carrying condensing steam at 101 kPa passes through the center of an infinite plate having a thickness of 5 cm. The plate is exposed to room air at 27ºC with a convection coefficient of 5.1 W/m2 ∙ ºC on both sides. The plate is composed of an insulation material having k
In the section illustrated in Figure P3-51 the surface 1-4-7 is insulated. The convection heat transfer coefficient at surface 1-2-3 is 28 W/m2 · ºC. The thermal conductivity of the solid material is 5.2 W/m· ºC. Using the numerical technique, compute the temperatures at
In Figure P3-53, calculate the temperatures at points 1, 2, 3, and 4 using the numerical method.
The composite strip in Figure P3-55 is exposed to the convection environment at 300ºC and h = 40 W/m2 · ºC. The material properties are kA = 20 W/m· ºC, kB = 1.2 W/m· ºC, and kC = 0.5 W/m· ºC. The strip is mounted on a plate maintained at the
The fin shown in Figure P3-56 has a base maintained at 300ºC and is exposed to the convection environment indicated. Calculate the steady-state temperatures of the nodes shown and the heat loss if k = 1.0 W/m· ºC.
Calculate the steady-state temperatures for nodes 1 to 16 in Figure P3-57. Assume symmetry.
Calculate the steady-state temperatures for nodes 1 to 9 in Figure P3-58.
Calculate the steady-state temperatures for nodes 1 to 6 in Figure P3-59.
A heavy-wall tube of Monel, 2.5-cm ID and 5-cm OD, is covered with a 2.5-cm layer of glass wool. The inside tube temperature is 300ºC, and the temperature at the outside of the insulation is 40ºC. How much heat is lost per foot of length? Take k = 11 Btu/h · ft · ºF for Monel.
Calculate the temperatures for the nodes indicated in Figure P3-60. The entire outer surface is exposed to the convection environment and the entire inner surface is at a constant temperature of 300ºC. Properties for materials A and B are given in the figure.
A rod having a diameter of 2 cm and a length of 10 cm has one end maintained at 200ºC and is exposed to a convection environment at 25ºC with h = 40 W/m2 · ºC. The rod generates heat internally at the rate of 50 MW/m3 and the thermal conductivity is 35 W/m· ºC.
Calculate the steady-state temperatures of the nodes in Figure P3-62. The entire outer surface is exposed to the convection environment at 20ºC and the entire inner surface is constant at 500ºC. Assume k = 0.2 W/m· ºC.
Calculate the steady-state temperatures for the nodes indicated in Figure P3-63.
The two-dimensional solid shown in Figure P3-64 generates heat internally at the rate of 90 MW/m3. Using the numerical method calculate the steady-state nodal temperatures for k = 20 W/m· ºC.
Two parallel disks having equal diameters of 30 cm are maintained at 120ºC and 34ºC. The disks are spaced a distance of 80 cm apart, on centers, and immersed in a conducting medium having k = 3.4 W/m· ºC. Assuming that the disks exchange heat only on the sides facing each other, calculate the
A tube has diameters of 4 mm and 5 mm and a thermal conductivity 20 W/m· ºC. Heat is generated uniformly in the tube at a rate of 500 MW/m3 and the outside surface temperature is maintained at 100ºC. The inside surface may be assumed to be insulated. Divide the tube wall into four nodes and
Repeat Problem 3-67 with the inside of the tube exposed to a convection condition with h = 40 W/m2 · ºC. Check with an analytical calculation.Problem 3-67
Rework Problem 3-57 with the surface absorbing a constant heat flux of 300 W/m2 instead of the convection boundary condition. The bottom surface still remains at 200ºC.Problem 3-57Calculate the steady-state temperatures for nodes 1 to 16 in Figure P3-57. Assume symmetry.
Rework Problem 3-60 with the inner surface absorbing a constant heat flux of 300W/m2 instead of being maintained at a constant temperature of 300ºC.Problem 3-60Calculate the temperatures for the nodes indicated in Figure P3-60. The entire outer surface is exposed to the convection environment and

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