How should the control volume method be implemented at an interface between two materials with different thermal conductivities? Illustrate with a steady, one-dimensional example. Neglect contact resistance.
> It has been proposed that a highly concentrating solar collector such as the one that follows can be used to process materials economically when it is desirable to heat the material surface rapidly without significantly heating the bulk. In one such proc
> Hot flue gases from a combustion furnace flow through a chimney, which is 7 m tall and has a hollow cylindrical cross section with inner diameter di = 30 cm and outer diameter do = 50 cm. The flue gases flow with an average temperature of Tg = 300°C and
> For the geometry shown in the sketch, determine the layout of nodes and control volumes. Provide a scale drawing showing the problem geometry overlaid with the nodes and control volumes. Explain how to derive the energy balance equation for all the bound
> For one-dimensional conduction, why are the boundary control volumes half the size of the interior control volumes?
> How would the results of Problem 4.47 be modified if the problem were not axisymmetric?
> Determine the appropriate difference equations for an axisymmetric, steady, spherical geometry with volumetric heat generation. Explain how to solve the equations.
> Determine the difference equations applicable at the centerline and at the surface of an axisymmetric cylindrical geometry with volumetric heat generation and convection boundary condition. Assume steady- state conditions.
> A steel billet is to be heat treated by immersion in a molten salt bath. The billet is 5 cm square and 1 m long. Prior to immersion in the bath, the billet is at a uniform temperature of 20°C, The bath is 600°C, and the heat transfer coefficient at the b
> Using the information in Problem 1.22, estimate the ambient air temperature that could cause frostbite on a calm day on the ski slopes.
> A long concrete beam is to undergo a thermal test to determine its loss of strength in the event of a building fire. The beam cross section is triangular as shown in the sketch. Initially, the beam is at a uniform temperature of 20°C. At the s
> Derive the stability criterion for an inside-corner boundary control volume for two-dimensional steady conduction when a convection boundary condition exists.
> Derive Eq. (4.28).
> Derive the stability criterion for the explicit solution of two-dimensional transient conduction.
> Show that in the limit as / and the difference equation, Eq. (4.23), is equivalent to the two-dimensional version of the differential equation, Eq. (2.6).
> Repeat Problem 4.39 by considering, instead of the trapezoidal microchannel, the two geometries shown in the following figure (a) rectangular channel of dimension a = b = 200 mm, d = 200 mm, and p = 150 mm and the coolant flow heat transfer coefficient
> Discuss the advantages and disadvantages of using a large control volume.
> A novel method to cool high-powered microelectronic chips and chip modules is to etch the liquid coolant flow channels in silicon (Si) substrate itself [9–12]. A typical such arrangement with channels of trapezoidal cross section is sho
> Consider a band saw blade that is to cut steel bar stock. The blade thickness is 2 mm, its height is 20 mm, and it has penetrated the steel workpiece to a depth of 5 mm (see the accompanying sketch). Exposed surfaces of the blade are cooled by an ambient
> A long, steel beam with a rectangular cross section of 40 cm * 60 cm is mounted on an insulating wall as shown in the following sketch. The rod is heated by radiant heaters that maintain the top and bottom surfaces at 300°C. A stream of air at
> In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind-chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the r
> Determine (a) the temperature at the 16 equally spaced points shown in the accompanying sketch to an accuracy of three significant figures and (b) the rate of heat flow per meter thickness. Assume two-dimensional heat flow and k = 1 W/m K.
> The plate in Problem 4.34 gradually oxidizes over time so that the surface emissivity increases to 0.5. Calculate the resulting temperature in the plate, including radiation heat transfer to the surroundings at the same temperature as the ambient t
> A 1-cm-thick, 1-m-square steel plate is exposed to sunlight and absorbs a solar flux of 800 W/m2. The bottom of the plate is insulated, the edges are maintained at 20°C by water-cooled clamps, and the exposed face is cooled by a convection coefficie
> Repeat Problem 4.32 if the temperature distribution on the top surface of the bar varies sinusoidally from 40°C at the left edge to a maximum of 250°C in the center and back to 40°C at the right edge.
> In the long, 30-cm-square bar shown in the accompanying sketch, the left face is maintained at 40°C and the top face is maintained at 250°C. The right face is in contact with a fluid at 40°C through a heat transfer coeffi
> The horizontal cross section of an industrial chimney is shown in the accompanying sketch. Flue gases maintain the interior surface of the chimney at 300°C, and the outside is exposed to an ambient temperature of 0°C through a heat
> Determine the temperature at the four nodes shown in the sketch. Assume steady conditions and two- dimensional heat conduction. The four faces of the square shape are at different temperatures as shown.
> Give an example of a practical problem in which the variation of thermal conductivity with temperature is significant and for which a numerical solution is therefore the only viable solution method.
> Develop a reasonable layout of nodes and control volumes for the geometry shown in the sketch. Provide a scale drawing showing the problem geometry overlaid with the nodes and control volumes. Identify each type of control volume used.
> Develop a reasonable layout of nodes and control volumes for the geometry shown in the sketch. Provide a scale drawing showing the problem geometry overlaid with the nodes and control volumes.
> In an experimental set up in a laboratory, a long cylinder with a 5-cm diameter, and an electrical resistance heater inside its entire length is cooled with water flowing crosswise over the cylinder at 25°C and a velocity of 0.8 m/s. For these flow condi
> A long cylindrical rod, 8 cm in diameter, is initially at a uniform temperature of 20°C. At time t = 0, the rod is exposed to an ambient temperature of 400°C through a heat transfer coefficient of 20 W/m2 K. The thermal conductivity of the rod is 0.8 W/m
> An interior wall of a cold furnace, initially at 0°C, is suddenly exposed to a radiant flux of 15 kW/m2 when the furnace is brought on line. The outer sur- face of the wall is exposed to ambient air at 20°C through a heat transfer c
> To more accurately model the energy input from the sun, suppose the absorbed flux in Problem 4.23 is given by where t is in hours and qabs is in W/m2. (This time variation of qabs gives the same total heat input to the wall as in Problem 4.23, i.e., 2000
> Repeat the numerical calculations of Problem 4.23 (Trombe wall) in order to obtain results to graph the temperature distribution in the Trombe wall after 2, 4, 6, and 8 hours of exposure.
> A Trombe wall is a masonry wall often used in passive solar homes to store solar energy. Suppose that such a wall, fabricated from 20-cm-thick solid concrete blocks / is initially at 15°C in equilibrium with the room in which it is located. I
> A 3-m-long steel rod / / is initially at 20°C and is insulated completely except for its end faces. One end is suddenly exposed to the flow of combustion gases at 1000°C through a heat transfer coefficient of 250 W/m2 K an
> Equation (4.16) is often called the fully-implicit form of the one-dimensional transient conduction difference equation because all quantities in the equation, except for the temperatures in the energy storage term, are evaluated at the new time step, m
> What are the advantages and disadvantages of using explicit and implicit difference equations?
> What is the physical significance of the statement that the temperature of each node is just the average of its neighbors if there is no heat generation [with reference to Eq. (4.3)]?
> Consider one-dimensional transient conduction with a convection boundary condition in which the ambient temperature near the surface is a function of time. Determine the energy balance equation for the boundary control volume. How would the solution meth
> A high-speed computer is located in a temperature- controlled room at 26°C. When the machine is operating, its internal heat generation rate is estimated to be 800 W. The external surface temperature of the computer is to be maintained below 85°C. The he
> The weight of the insulation in a spacecraft may be more important than the space required. Show analytically that the lightest insulation for a plane wall with a specified thermal resistance is the insulation that has the smallest product of density tim
> Determine the largest permissible time step for a one-dimensional transient conduction problem to be solved by an explicit method if the node spacing is 1 mm and the material is (a) carbon steel 1C, (b) window glass. Explain the difference in the two r
> Show that in the limit as the difference equation, Eq. (4.13), is equivalent to the differential equation, Eq. (2.5).
> A heat sink made of an array of fins that have a straight, rectangular cross section is used to cool an electronic micro-chip module, as schematically shown in the figure. The heat sink (base and fins) is made of copper, and each fin is 3 mm thick and
> Light-emitting diodes or LEDs are currently perhaps the most energy-efficient lighting systems. Finned- surface heat sinks are used to cool high-intensity LED lighting that are used for spot and/or track lighting systems. A typical circular pin-fin heat
> A turbine blade 5 cm long with a cross-sectional area A = 4.5 cm2 and a perimeter P = 12 cm is made of a high-alloy steel (k = 25 W/m K) . The temperature of the blade attachment point is 500°C, and the blade is exposed to combustion gases at
> How would you include contact resistance between the two materials in Problem 4.12? Derive the appropriate difference equations.
> How would you treat a radiation heat transfer boundary condition for a one-dimensional steady problem? Develop the difference equation for a control volume near the boundary, and explain how to solve the entire system of difference equations. Assume that
> Consider a pin fin with variable conductivity k(T), constant cross-sectional area Ac and constant perimeter, P. Develop the difference equations for steady one-dimensional conduction in the fin, and suggest a method for solving the equations. The fin is
> Show that in the limit as /the difference equation for one-dimensional steady conduction with heat generation, Eq. (4.2), is equivalent to the differential equation, Eq. (2.24).
> A cryogenic fluid is stored in a 0.3-m-diameter spherical container in still air. If the convection heat transfer coefficient between the outer surface of the container and the air is 6.8 W/m2 K, the temperature of the air is 27°C, and the temperature of
> The heat transfer coefficients for the flow of 26.6°C air over a sphere of 1.25 cm in diameter are measured by observing the temperature-time history of a copper ball the same dimension. The temperature of the copper ball was measured by two
> Green coffee beans, after harvesting, are dried and roasted in a fluidized-bed roaster. This type of roaster has hot air, typically at 250°C, and the air blows through a screen or perforated plate above which the beans float in an agitated suspension (fl
> In a metal wire manufacturing facility, continuously drawn copper wire with a 2.5 mm diameter is annealed by heating it from its initial temperature of 45°C to 400°C in a soaking oven. The oven inside air temperature is 700°C with an average heat transf
> A 0.6-cm-diameter mild steel rod a t 38°C is suddenly immersed in a liquid at 93°C with /Determine the time required for the rod to warm to 88°C.
> In a ball-bearing production facility, steel balls that are each of 15 mm in diameter are annealed by first heating them to 870°C and then slowly cooling in air to 125°C. If the cooling air stream temperature is 60°C, and it has a convective heat transfe
> It is a well-known physiological phenomenon that some materials feel cooler to the touch than others. This is important in the design of instruments for the use of operators, especially in a space station. Experiments conducted at NASA have shown that di
> A large billet of steel initially at 260°C is placed in a radiant furnace where the surface temperature is held at 1200°C. Assuming the billet to be infinite in extent, compute the temperature at point P (see the accompanying sketch
> Estimate the depth in moist soil at which the annual temperature variation is 10% of that at the surface.
> Surface hardening of metallic machine components, such as ball and roller bearings, is carried out in a heat treatment process where the surface temperature is increased to a desirable level without altering the internal temperature substantially. In one
> A thermocouple is made up of a spherical bead bimetallic junction at the end of two very thin wires of different materials. For example, in a type-T thermocouple, a copper wire and a constantan wire, with wire diameters ranging from 0.5 mm to 0.025 mm, a
> The heat transfer coefficient for a gas flowing over a thin flat plate 3-m long and 0.3-m wide varies with distance from the leading edge according to If the plate temperature is 170°C and the gas temperature is 30°C, calculate (a)
> In the experimental determination of the heat transfer coefficient between a heated steel ball and crushed mineral solids, a series of 1.5% carbon steel balls were heated to a temperature of 700°C and the center temperature-time history of each was measu
> An egg, which for the purposes of this problem is assumed to be a 5-cm-diameter sphere having the thermal properties of water, is initially at a temperature of 4°C. It is immersed in boiling water at 100°C for 15 min. The heat transfer coefficient from t
> A monster turnip (assumed spherical) weighing in at 0.45 kg is dropped into a cauldron of water boiling at atmospheric pressure. If the initial temperature of the turnip is 17°C, how long does it take to reach 92°C at the center?
> A steel sphere with a diameter of 7.6 cm is to be hardened by first heating it to a uniform temperature of 870°C and then quenching it in a large bath of water at a temperature of 38°C. The following data apply: Calculate (a) the t
> Estimate the time required to heat the center of a 1.5-kg roast in a 163°C oven to 77°C. State your assumptions carefully and compare your results with cooking instructions in a standard cookbook.
> Ball bearings are to be hardened by quenching them in a water bath at a temperature of 37°C. You are asked to devise a continuous process in which the balls roll from a soaking oven at a uniform temperature of 870°C into the water,
> Consider a heat treatment process in which steel rods with a 10 cm diameter at an initial temperature of 600°C are inserted into an oil bath at 25°C. Assuming a convection coefficient of 400 W/m2. K between the oil and the rod, estimate how long it take
> Repeat Problem 3.31(a), but assume that the billet is only 1.2 m long with the average heat transfer coefficient at both ends equal to 136 W/m2 K.
> A long, 0.6-m-OD 347 stainless steel (k = 14 W/m K) cylindrical billet at 16°C room temperature is placed in an oven where the temperature is 260°C. If the average heat transfer coefficient is 170 W/m2 K, (a) estimate the time required for the center tem
> A solid lead cylinder 0.6 m in diameter and 0.6 m long, initially at a uniform temperature of 121°C, is dropped into a 21°C liquid bath in which the heat transfer coefficient / 1135 W/m2 K. Plot the temperature-time history of the c
> The heat transfer rate from hot air by convection at 100°C flowing over one side of a flat plate with dimensions 0.1 m * 0.5 m is determined to be 125 W when the surface of the plate is kept at 30°C. What is the average convection heat transfer coefficie
> Quenching is a rapid cooling process by which many metallic alloys are hardened. Brass (a copper alloy: 70 Cu, 30 Zn) plates that are 4 mm thick are quenched in a water bath at a heat treating plant. If a brass plate is initially at 450°C, determine the
> A mild-steel cylindrical billet 25 cm in diameter is to be raised to a minimum temperature of 760°C by passing it through a 6-m long strip-type furnace. If the furnace gases are at 1538°C and the overall heat transfer coefficient on the outside of the bi
> A long wooden rod at 38°C with a 2.5-cm-OD is placed into an airstream at 600°C. The heat transfer coefficient between the rod and air is 28.4 W/m2 K. If the ignition temperature of the wood is 427°C, r = 800 kg/m3, k = 0.173 W/m K, and c = 25
> A long copper cylinder 0.6 m in diameter and initially at a uniform temperature of 38°C is placed in a water bath at 93°C. Assuming that the heat transfer coefficient between the copper and the water is 1248 W/m2 K, calculate the time required to heat
> A stainless steel cylindrical billet (k = 14.4 W/m K, ( 5 3.9 * 10-6 m2/s ) is heated to 593°C preparatory to a forming process. If the minimum temperature permissible for forming is 482°C, how long can the billet be exposed to air
> In a curing process for plastic and other polymer- based materials, a plastic sheet, which initially is 25°C, is placed on a hot metal surface that is maintained at 250°C. If the back (or opposite) side of the 5-mm-thick plastic film is considered to be
> A frozen-food company freezes its spinach by first compressing it into large slabs and then exposing the slab of spinach to a low-temperature cooling medium. The large slab of compressed spinach is initially at a uniform temperature of 21°C; it mus
> In the inspection of a sample of meat intended for human consumption, it was found that certain undesirable organisms were present. To make the meat safe for consumption, it is ordered that the meat be kept at a temperature of at least 121°C for a perio
> A 2.5-cm-thick sheet of plastic initially at 21°C is placed between two heated steel plates that are maintained at 138°C. The plastic is to be heated just long enough for its mid plane temperature to reach 132°C. If the thermal conductivity of the p
> In the vulcanization of tires, the carcass is placed into a jig and steam at 149°C is admitted suddenly to both sides. If the tire thickness is 2.5 cm, the initial temperature is 21°C, the heat transfer coefficient between the tire
> Water at a temperature of 77°C is to be evaporated slowly in a vessel. The water is in a low-pressure container surrounded by steam as shown in the sketch below. The steam is condensing at 107°C. The overall heat transfer coefficien
> Dielectric heating, also known as RF or high frequency heating, is a process in which a high-frequency alternating electric field or microwave electromagnetic radiation heats a dielectric material. An important application of this phenomenon is in the he
> High-strength steel is required for use in building structures and equipment (e.g., cranes). It is produced by heat treating quench-hardened steel in a process called tempering that reduces brittleness and imparts toughness. In a production facility, all
> A wire of perimeter P and cross-sectional area A emerges from a die at a temperature T (above the ambient temperature) and with a velocity U. Determine the temperature distribution along the wire in the steady state if the exposed length down-
> A small aluminum sphere of diameter D, initially at a uniform temperature To, is immersed in a liquid whose temperature, T(, varies sinusoidally according to If the heat transfer coefficient between the fluid and the sphere, /is constant and the system c
> A 1.4-kg aluminum household iron has a 500-W heating element. The surface area is 0.046 m2. The ambient temperature is 21°C, and the surface heat transfer coefficient is 11 W/m2 K. How long after the iron is plugged in does its temperature rea
> A large, 2.54-cm.-thick copper plate is placed between two air streams. The heat transfer coefficient on one side is 28 W/m2 K and on the other side is 57 W/m2 K. If the temperature of both streams is suddenly changed from 38°C to 93°C, determine how lo
> A thin-wall jacketed tank heated by condensing steam at one atmosphere contains 91 kg of agitated water. The heat transfer area of the jacket is 0.9 m2 and the over- all heat transfer coefficient U = 227 W/m2 K based on that area. Determine the heating t
> A thin-wall cylindrical vessel (1 m in diameter) is filled to a depth of 1.2 m with water at an initial temperature of 15°C. The water is well stirred by a mechanical agitator. Estimate the time required to heat the water to 50°C if
> A copper wire, 0.8-mm OD, 5 cm long, is placed in an air stream whose temperature rises at a rate given by where t is the time in seconds. If the initial temperature of the wire is 10°C, deter- mine its temperature after 2s, 10s, and 1 min. T
> A spherical stainless steel vessel at 93°C contains 45 kg of water initially at the same temperature. If the entire system is suddenly immersed in ice water, determine (a) the time required for the water in the vessel to cool to 16°C, and (b) the temper
> A thermocouple (0.8-mm-diameter wire) used to measure the temperature of the quiescent gas in a furnace gives a reading of 165°C. It is known, how- ever, that the rate of radiant heat flow per meter length from the hotter furnace walls to the
> In the heat treating process in a sheet-metal manufacturing plant, brass plates are annealed by first heating them in an industrial furnace oven the inside of which is maintained at a constant uniform temperature of 550°C. If the surface of the brass pla
> A spherical shell satellite (3-m-OD, 1.25-cm-thick stainless steel walls) re-enters the atmosphere from outer space. If its original temperature is 38°C, the effective average temperature of the atmosphere is 1093°C, and the effective heat transfer coeff
> Consider a flat plate or a plane wall with a thickness L and a long cylinder of radius ro. Both of these are made of materials such that they can be treated as lumped capacitances (Bi < 0.1) . Show that in each case, the characteristic length lc, defin
