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physics
thermodynamics

Heat Transfer 10th edition Jack Holman - Solutions

Rework Problem 3-64 with the surface marked at a constant 100ºC now absorbing a constant heat flux of 500 W/m2. Add nodes as necessary.Problem 3-64The two-dimensional solid shown in Figure P3-64 generates heat internally at the rate of 90 MW/m3. Using the numerical method calculate the
The tapered aluminum pin fin shown in Figure P3-72 is circular in cross section with a base diameter of 1 cm and a tip diameter of 0.5 cm. The base is maintained at 200ºC and loses heat by convection to the surroundings at Tˆž = 10ºC, h = 200 W/m2 · ºC. The tip is insulated. Assume
Write the nodal equations 1 through 7 for the symmetrical solid shown in Figure P3-73. ˆ†x = ˆ†y = 1 cm.Figure P3-73
Obtain the temperature for nodes 1 through 6 shown in Figure P3-74. ˆ†x = ˆ†y = 1 cm.
Write the nodal equations for nodes 1 through 9 shown in Figure P3-75.ˆ†x = ˆ†y = 1 cm.
Write the nodal equation for nodes 1 through 12 shown in Figure P3-76. Express the equations in a format for Gauss-Seidel iteration.
Sometimes a square grid is desired even for a circular system. Consider the quadrant of a circle shown in Figure P3-77 with r = 10 cm. ˆ†x = ˆ†y = 3 cm and k = 10W/m· ºC. Write the steady-state nodal equations for nodes 3 and 4. Make use of Tables 3-2 and 3-4.
A furnace of 70- by 60- by 90-cm inside dimensions is constructed of a material having a thermal conductivity of 0.5 Btu/h · ft · ºF. The wall thickness is 6 in. The inner and outer surface temperatures are 500 and 100ºF, respectively. Calculate the heat loss through the furnace wall.
The slanted intersection shown in Figure P3-81 involves materials A and B. Write steady state nodal equations for nodes 3, 4, 5, and 6 using Table 3-2(f and g) as a guide.
A horizontal plate, 25 by 50 cm, is maintained at a constant temperature of 78ºC and buried in a semi-infinite medium at a depth of 5 m. The medium has an isothermal surface maintained at 15ºC and a thermal conductivity of 2.8 W/m · ºC. Calculate the heat lost by the plate.
A cube 20 cm on a side is maintained at 80ºC and buried in a large medium at 10ºC with a thermal conductivity of 2.3 W/m· ºC. Calculate the heat lost by the cube. How does this compare with the heat that would be lost by a 20-cm-diameter sphere? Compare these heat transfers on a unit-volume
A long horizontal cylinder having a diameter of 10 cm is maintained at a temperature of 100ºC and centered in a 30-cm-thick slab of material for which k = W/m· ºC. The outside of the slab is at 20ºC. Calculate the heat lost by the cylinder per unit length.
A horizontal plate 20 by 150 cm is buried in a large medium at a depth of 2.0 m and maintained at 50ºC. The surface of the medium is at 10ºC and has k = 1.5W/m· ºC. Calculate the heat lost by the plate.
A thin disk 10 cm in diameter is maintained at 75ºC and placed on the surface of a large medium at 15ºC with k = 3 W/m· ºC. Calculate the heat conducted into the medium.
Repeat Problem 3-87 for a square 10 cm on a side. Compare the heat transfers on a per unit area basis. Problem 3-87 A thin disk 10 cm in diameter is maintained at 75ºC and placed on the surface of a large medium at 15ºC with k = 3 W/m· ºC. Calculate the heat conducted into the medium.
A hot steam pipe 10 cm in diameter is maintained at 200ºC and centered in a square mineral-fiber insulation 20 cm on a side. The outside surface temperature of the insulation is 35ºC. Calculate the heat lost by a 20-m length of pipe if the thermal conductivity of the insulation can be taken as 50
A cube 35 cm on each external side is constructed of fireclay brick. The wall thickness is 5.0 cm. The inner surface temperature is 500ºC, and the outer surface temperature is 80ºC. Compute the heat flow in watts.
A pipe having a diameter of 10 cm passes through the center of a concrete slab having a thickness of 70 cm. The surface temperature of the pipe is maintained at 100ºC by condensing steam while the outer surfaces of the concrete are at 24ºC. Calculate the heat lost by the pipe per meter of length.
Consider a circumferential fin of rectangular profile as shown in Figure 2-12. Set up nodal equations for a fin of thickness t, heat transfer coefficient h, thermal conductivity k, and heat generation rate q as a function of radial coordinate r, taking increments of ˆ†r. Write the nodal
Set up a nodal equation for the geometry of Problem 2-123, using increments in the height of the truncated cone as the one-dimensional variable. Then work the problem with the numerical method and compare with the one-dimensional analytical solution. Problem 2-123 The truncated hollow cone shown in
Set up nodal equations for the geometry of Problem 2-122, using increments in an angle θ as the one-dimensional variable. Then work the problem using the numerical method and compare with the one-dimensional analytical solution.Problem 2-122The cylindrical segment shown in Figure P2-122 has a
A cube 30 cm on a side is buried in an infinite medium with a thermal conductivity of 1.8 W/m· ºC. The surface temperature of the cube is 30ºC while the temperature of the medium is 10ºC. Calculate the heat lost by the cube.
A thin horizontal disk having a diameter of 15 cm is maintained at a constant surface temperature of 87ºC and buried at a depth of 20 cm in a semi-infinite medium with an adiabatic surface. The thermal conductivity of the medium is 2.7 W/m· ºC and the temperature of the medium a large distance
A copper rod has an internal heater that maintains its surface temperature at 50ºC while it is buried vertically in a semi-infinite medium. The rod is 2 cm in diameter and 40 cm long and the isothermal surface of the medium is at 20ºC. Calculate the heat lost by the rod if the thermal
Rework Problem 2-122, using a numerical approach with five nodes operating in increments of the radial angle Î¸, and compare with the analytical results of Problem 2-122.Problem 2-122The cylindrical segment shown in Figure P2-122 has a thermal conductivity of 100 W/m · ºC. The inner and outer
A basement for a certain home is 4 × 5 m with a ceiling height of 3 m. The walls are concrete having a thickness of 10 cm. In the winter the convection coefficient on the inside is 10 W/m2 ∙ ºC and the soil on the outside has k = 1.7 W/m· ºC. Analyze this problem and determine an overall heat
A copper sphere initially at a uniform temperature T0 is immersed in a fluid. Electric heaters are placed in the fluid and controlled so that the temperature of the fluid follows a periodic variation given by T∞ − Tm = A sin ωτ where Tm = time-average mean fluid temperature A = amplitude of
A stainless-steel rod (18% Cr, 8% Ni) 6.4 mm in diameter is initially at a uniform temperature of 25ºC and is suddenly immersed in a liquid at 150ºC with h = 120 W/m2·ºC. Using the lumped-capacity method of analysis, calculate the time necessary for the rod temperature to reach 120ºC.
The plate in Problem 3-63 is initially uniform in temperature at 100ºC and suddenly exposed to the convection boundary. Select a value for ∆τ and calculate the nodal temperatures after 10 time increments. Take ρ = 7500 kg/m3 and c = 440 J/kg · ºC.
The solid shown in Problem 3-64 is initially uniform in temperature at 100ºC and suddenly exposed to the convection boundary and heat generation while the right face is kept at 100ºC. Select a value for ∆τ and calculate the nodal temperatures after 10 time increments. Take ρ = 7600 kg/m3 and
A steel rod 12.5 mm in diameter and 20 cm long has one end attached to a heat reservoir at 250ºC. The bar is initially maintained at this temperature throughout. It is then subjected to an airstream at 30ºC such that the convection heat-transfer coefficient is 35W/m2 · ºC. Estimate the time
The two-dimensional body of Figure 3-6 has the initial surface and internal temperatures as calculated. At time zero the 500ºC face is suddenly lowered to 30ºC. Taking ∆x = ∆y = 15 cm and α = 1.29 × 10−5 m2/s, calculate the temperatures at nodes 1, 2, 3, and 4 after 30 min. Perform the
A 5-cm-diameter copper sphere is initially at a uniform temperature of 200ºC. It is suddenly exposed to an environment at 20ºC having a heat-transfer coefficient h = 28W/m2·ºC. Using the lumped-capacity method of analysis, calculate the time necessary for the sphere temperature to reach 90ºC.
A fireproof safe is constructed of loosely packed asbestos contained between thin sheets of stainless steel. The safe is built in the form of a cube with inside and outside dimensions of 0.5 and 1.0 m. If the safe is initially uniform in temperature at 30ºC and the outside is suddenly exposed to a
A large slab of brick [k = 1.07W/m · ºC, α = 5.4 × 10−7m2/s] is initially at a uniform temperature of 20ºC. One surface is suddenly exposed to a uniform heat flux of 4500 W/m2. Calculate and plot the surface temperature as a function of time. Also calculate the heat flux through the plane
A granite sphere having a diameter of 15 cm and initially at a uniform temperature of 120ºC is suddenly exposed to a convection environment with h = 350 W/m2 · ºC and T = 30ºC. Calculate the temperature at a radius of 4.5 cm after 21 min and the energy removed from the sphere in this time. Take
The solid of Problem 3-74 is initially uniform in temperature at 100ºC, but suddenly the two surfaces are lowered to 0 and 40ºC. If the solid has k = 20 W/m·ºC and α = 5 × 10−6 m2/s, find the steady-state temperature of each node and the nodal temperatures after 1 min.
A copper sphere having a diameter of 3.0 cm is initially at a uniform temperature of 50ºC. It is suddenly exposed to an airstream of 10ºC with h = 15 W/m2·ºC. How long does it take the sphere temperature to drop to 25ºC?
Apply the lumped-capacity criterion of Equation (4-6) [h(V/A)/k < 0.1] to each of the geometries treated with the Heisler charts. Approximately what percent error would result for each geometry in the value of θ/θ0 if a lumped capacity is assumed for the conditions of Equation (4-6)?
Because of symmetry, the temperature gradient ∂T/∂x at the centerline of an infinite plate will be zero when both sides are subjected to the same boundary condition in a cooling process. This may be interpreted that a half plate will act like a plate with one side insulated (∂T/∂x = 0), and
Rework Problem 4-135 for the surface temperature suddenly lowered to 25ºC. This is equivalent to h→∞. Problem 4-135
Rework Problem 4-135, assuming the plate behaves as a lumped capacity. Problem 4-135
Rework Problem 4-135, assuming the aluminum plate behaves as a semi-infinite solid with the desired temperature occurring at x = 5 cm. Perform the same kind of calculation for Problem 4-136. Problem 4-135
A concrete driveway having a thickness of 18 cm attains an essentially uniform temperature of 30ºC on a warm November day in Texas. A "blue norther" arrives, which suddenly subjects the driveway to a convection boundary with h = 23 W/m2 · ºC and T∞ = 0ºC. How long will it take for the surface
An aluminum sphere, 5.0 cm in diameter, is initially at a uniform temperature of 50ºC. It is suddenly exposed to an outer-space radiation environment at 0 K (no convection). Assuming the surface of aluminum is blackened and lumped-capacity analysis applies, calculate the time required for the
A 5-lb roast initially at 70ºF is placed in an oven at 350ºF. Assuming that the heat-transfer coefficient is 2.5 Btu/h · ft2 · ºF and that the thermal properties of the roast may be approximated by those of water, estimate the time required for the center of the roast to attain a temperature
Oranges with a diameter of about 3 in are to be cooled from room temperature of 25ºC to 3ºC using an air-convection environment with h = 45 W/m2 · ºC and T∞ = 0ºC. Assuming that the oranges have the properties of water at 10ºC, calculate the time required for the cooling and the total
A cold-storage building 16 × 35 m is built on a concrete slab having a thickness of 15 cm, which is placed on a suitable insulating material in contact with the ground. During the start-up period the interior of the building is exposed to convection air with h = 20 W/m2 · ºC and T = −15ºC.
A 2.0-mm-thick sheet of polyethylene covers a 10-cm-thick slab of high-density particle board that is perfectly insulated on the back side. The assembly is initially uniform in temperature at 20ºC. If the outer surface of the plastic is suddenly exposed to a constant heat flux of 1300W/m2,
An aluminum can having a volume of about 350 cm3 contains beer at 1ºC. Using a lumped-capacity analysis, estimate the time required for the contents to warm to 15ºC when the can is placed in a room at 20ºC with a convection coefficient of 15 W/m2·ºC. Assume beer has the same properties as
Free convection in air at atmospheric pressure is found to experience a convection heat-transfer coefficient that varies as h = A(∆T)n, where ∆T is the temperature difference between the surface and the surrounding air, A is a constant, and n is some exponent. You are to devise a way to
A 12-mm-diameter aluminum sphere is heated to a uniform temperature of 400ºC and then suddenly subjected to room air at 20ºC with a convection heat-transfer coefficient of 10 W/m2·ºC. Calculate the time for the center temperature of the sphere to reach 200ºC.
A 4-cm-diameter copper sphere is initially at a uniform temperature of 200ºC. It is suddenly exposed to a convection environment at 30ºC with h = 20 W/m2·ºC. Calculate the time necessary for the center of the sphere to reach a temperature of 80ºC.
When a sine-wave temperature distribution is impressed on the surface of a semi-infinite solid, the temperature distribution in the solid is given bywhere Tx,Ï„ = temperature at depth x and time Ï„ after start of temperature wave at surface Tm = mean surface temperature n =
Using the temperature distribution of Problem 4-18, show that the time lag between maximum points in the temperature wave at the surface and at a depth x is given byProblem 4-18 When a sine-wave temperature distribution is impressed on the surface of a semi-infinite solid, the temperature
An infinite plate having a thickness of 2.5 cm is initially at a temperature of 150ºC, and the surface temperature is suddenly lowered to 30ºC. The thermal diffusivity of the material is 1.8 × 10−6 m2/s. Calculate the center-plate temperature after 1 min by summing the first four nonzero terms
A thick concrete wall having a uniform temperature of 54ºC is suddenly subjected to an airstream at 10ºC. The heat-transfer coefficient is 10 W/m2·ºC. Calculate the temperature in the concrete slab at a depth of 7 cm after 30 min.
A very large slab of copper is initially at a temperature of 300ºC. The surface temperature is suddenly lowered to 35ºC. What is the temperature at a depth of 7.5 cm 4 min after the surface temperature is changed?
On a hot summer day a concrete driveway may reach a temperature of 50ºC. Suppose that a stream of water is directed on the driveway so that the surface temperature is suddenly lowered to 10ºC. How long will it take to cool the concrete to 25ºC at a depth of 5 cm from the surface?
A semi-infinite slab of copper is exposed to a constant heat flux at the surface of 0.5 MW/m2. Assume that the slab is in a vacuum, so that there is no convection at the surface. What is the surface temperature after 5 min if the initial temperature of the slab is 20ºC? What is the temperature at
A large slab of copper is initially at a uniform temperature of 90ºC. Its surface temperature is suddenly lowered to 30ºC. Calculate the heat-transfer rate through a plane 7.5 cm from the surface 10 s after the surface temperature is lowered.
A large slab of aluminum at a uniform temperature of 30ºC is suddenly exposed to a constant surface heat flux of 15 kW/m2. What is the temperature at a depth of 2.5 cm after 2 min?
What error would result from using the first four terms of Equation (4-3) to compute the temperature at τ = 0 and x = L?
An aluminum sphere having a diameter of 5.6 cm is initially at a uniform temperature of 355ºC and is suddenly exposed to a convection environment at T = 23ºC with a convection heat transfer coefficient of 78W/m2·ºC. Calculate the time for the center of the sphere to cool to a temperature of
A large thick layer of ice is initially at a uniform temperature of −20ºC. If the surface temperature is suddenly raised to −1ºC, calculate the time required for the temperature at a depth of 1.5 cm to reach −11ºC. The properties of ice are ρ = 57 lbm/ft3, cp = 0.46 Btu/lbm · ºF, k =
A large slab of concrete (stone 1-2-4 mix) is suddenly exposed to a constant radiant heat flux of 900W/m2. The slab is initially uniform in temperature at 20ºC. Calculate the temperature at a depth of 10 cm in the slab after a time of 9 h.
A very thick plate of stainless steel (18% Cr, 8% Ni) at a uniform temperature of 300ºC has its surface temperature suddenly lowered to 100ºC. Calculate the time required for the temperature at a depth of 3 cm to attain a value of 200ºC.
A large slab has properties of common building brick and is heated to a uniform temperature of 40ºC. The surface is suddenly exposed to a convection environment at 2ºC with h = 25 W/m2 · ºC. Calculate the time for the temperature to reach 20ºC at a depth of 8 cm.
A large block having the properties of chrome brick at 200ºC is at a uniform temperature of 30ºC when it is suddenly exposed to a surface heat flux of 3 × 104 W/m2. Calculate the temperature at a depth of 3 cm after a time of 10 min. What is the surface temperature at this time?
A slab of copper having a thickness of 3.0 cm is initially at 300ºC. It is suddenly exposed to a convection environment on the top surface at 80ºC while the bottom surface is insulated. In 6 min the surface temperature drops to 140ºC. Calculate the value of the convection heat-transfer
A large slab of aluminum has a thickness of 10 cm and is initially uniform in temperature at 400ºC. Suddenly it is exposed to a convection environment at 90ºC with h = 1400 W/m2 · ºC. How long does it take the centerline temperature to drop to 180ºC?
A horizontal copper plate 10 cm thick is initially uniform in temperature at 250ºC. The bottom surface of the plate is insulated. The top surface is suddenly exposed to a fluid stream at 80ºC. After 6 min the surface temperature has dropped to 150ºC. Calculate the convection heat-transfer
A large slab of aluminum has a thickness of 10 cm and is initially uniform in temperature at 400ºC. It is then suddenly exposed to a convection environment at 90ºC with h = 1400 W/m2 · ºC. How long does it take the center to cool to 180ºC?
A plate of stainless steel (18% Cr, 8% Ni) has a thickness of 3.0 cm and is initially uniform in temperature at 500ºC. The plate is suddenly exposed to a convection environment on both sides at 40ºC with h = 150W/m2·ºC. Calculate the times for the center and face temperatures to reach 120ºC.
A steel cylinder 10 cm in diameter and 10 cm long is initially at 300ºC. It is suddenly immersed in an oil bath that is maintained at 40ºC, with h = 280 W/m2 · ºC. Find (a) the temperature at the center of the solid after 2 min and (b) the temperature at the center of one of the regular faces
Derive an expression for the heat flux per unit area at depth x and time τ when a semi-infinite solid is suddenly exposed to an instantaneous energy pulse at the surface of strength Q0/A.
Buildings of various constructions exhibit different responses to thermal changes in climate conditions. Consider a 10-cm-thick wall of normal weight structural concrete (c = 0.9 kJ/kg · ºC) suddenly exposed to a "blue norther" at−10ºCwith a convection coefficient of 65 W/m2 · ºC. The wall
A semi-infinite solid of aluminum is coated with a special chemical material that reacts suddenly to ultraviolet radiation and releases energy in the amount of 1.0 MJ/m2. If the solid is initially uniform in temperature at 20ºC, calculate the temperature at a depth of 2.3 cm after 1.8 s.
A semi-infinite solid of stainless steel (18% Cr, 8% Ni) is initially at a uniform temperature of 0ºC. The surface is pulsed with a laser with 10 MJ/m2 instantaneous energy. Calculate the temperature at the surface and depth of 1 cm after a time of 3 s.
What strength pulse would be necessary to produce the same temperature effect at a depth of 1.2 cm as that experienced at a depth of 1.0 cm?
A semi-infinite solid of aluminum is to be pulsed with a laser at the surface such that a temperature of 600ºC will be attained at a depth of 2 mm, 0.2 s after the pulse. The solid is initially at 30ºC. Calculate the strength of pulse required, expressed in MJ/m2.
A slab of polycrystalline aluminum oxide is to be pulsed with a laser to produce a temperature of 900ºC at a depth of 0.2 mm after a time of 0.2 s. The solid is initially at 40ºC. Calculate the strength of pulse required expressed in MJ/m2.
A 20 by 20 cm slab of copper 5 cm thick at a uniform temperature of 260ºC suddenly has its surface temperature lowered to 35ºC. Using the concepts of thermal resistance and capacitance and the lumped-capacity analysis, find the time at which the center temperature becomes 90ºC; ρ = 8900 kg/m3,
Repeat Problem 4-49 for window glass. Problem 4-49 A slab of polycrystalline aluminum oxide is to be pulsed with a laser to produce a temperature of 900ºC at a depth of 0.2 mm after a time of 0.2 s. The solid is initially at 40ºC. Calculate the strength of pulse required expressed in MJ/m2.
An aluminum bar has a diameter of 11 cm and is initially uniform in temperature at 300ºC. If it is suddenly exposed to a convection environment at 50ºC with h = 1200 W/m2 · ºC, how long does it take the center temperature to cool to 80ºC? Also calculate the heat loss per unit length.
A fused-quartz sphere has a thermal diffusivity of 9.5 × 10−7 m2/s, a diameter of 2.5 cm, and a thermal conductivity of 1.52 W/m· ºC. The sphere is initially at a uniform temperature of 25ºC and is suddenly subjected to a convection environment at 200ºC. The convection heat-transfer
Lead shot may be manufactured by dropping molten-lead droplets into water. Assuming that the droplets have the properties of solid lead at 300ºC, calculate the time for the center temperature to reach 120ºC when the water is at 100ºC with h = 5000 W/m2 · ºC, d = 1.5 mm.
A steel sphere 10 cm in diameter is suddenly immersed in a tank of oil at 10ºC. The initial temperature of the sphere is 220ºC; h = 5000 W/m2 · ºC. How long will it take the center of the sphere to cool to 120ºC?
A boy decides to place his glass marbles in an oven at 200ºC. The diameter of the marbles is 15 mm. After a while he takes them from the oven and places them in room air at 20ºC to cool. The convection heat-transfer coefficient is approximately 14 W/m2 · ºC. Calculate the time the boy must wait
A lead sphere with d = 1.5 mm and initial temperature of 200ºC is suddenly exposed to a convection environment at 100ºC and h = 5000 W/m2 · ºC. Calculate the time for the center temperature to reach 120ºC.
A long steel bar 5 by 10 cm is initially maintained at a uniform temperature of 250ºC. It is suddenly subjected to a change such that the environment temperature is lowered to 35ºC. Assuming a heat-transfer coefficient of 23 W/m2 · ºC, use a numerical method to estimate the time required for
A steel bar 2.5 cm square and 7.5 cm long is initially at a temperature of 250ºC. It is immersed in a tank of oil maintained at 30ºC. The heat-transfer coefficient is 570 W/m2 · ºC. Calculate the temperature in the center of the bar after 3 min.
A cube of aluminum 10 cm on each side is initially at a temperature of 300ºC and is immersed in a fluid at 100ºC. The heat-transfer coefficient is 900 W/m2 · ºC. Calculate the temperature at the center of one face after 1 min.
A piece of aluminum weighing 6 kg and initially at a temperature of 300ºC is suddenly immersed in a fluid at 20ºC. The convection heat-transfer coefficient is 58W/m2 · ºC. Taking the aluminum as a sphere having the same weight as that given, estimate the time required to cool the aluminum to
A short concrete cylinder 15 cm in diameter and 30 cm long is initially at 25ºC. It is allowed to cool in an atmospheric environment in which the temperature is 0ºC. Calculate the time required for the center temperature to reach 10ºC if the heat-transfer coefficient is 17 W/m2 · ºC.
Assume that node m in Problem 3-39 occurs along a circular rod having a diameter of 2 cm with ∆x = 1 cm. The material is glass with k = 0.8 W/m· ºC, ρ = 2700 kg/m3, c = 0.84 kJ/kg · ºC. The convection surrounding condition is h = 50W/m2 · ºC and T∞ = 35ºC. Write the transient nodal
A 4.0-cm cube of aluminum is initially at 450ºC and is suddenly exposed to a convection environment at 100ºC with h = 120 W/m2 · ºC. How long does it take the cube to cool to 250ºC?
A cube of aluminum 11 cm on each side is initially at a temperature of 400ºC. It is suddenly immersed in a tank of oil maintained at 85ºC. The convection coefficient is 1100 W/m2 · ºC. Calculate the temperature at the center of one face after a time of 1 min.
An aluminum cube 5 cm on a side is initially at a uniform temperature of 100ºC and is suddenly exposed to room air at 25ºC. The convection heat-transfer coefficient is 20 W/m2 · ºC. Calculate the time required for the geometric center temperature to reach 50ºC.
A stainless steel cylinder (18% Cr, 8% Ni) is heated to a uniform temperature of 200ºC and then allowed to cool in an environment where the air temperature is maintained constant at 30ºC. The convection heat-transfer coefficient may be taken as 200 W/m2 · ºC. The cylinder has a diameter of 10

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