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mechanical engineering

Questions and Answers of Mechanical Engineering

The fin array of Problem 3.130 is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider
An isothermal silicon chip of width W = 20 mm on a side is soldered to an aluminum heat sink (k = 180 W/m ∙ K) of equivalent width. The heat sink has a base thickness of Lb = 3 mm and an array of
As seen in Problem 3.109, silicon carbide nanowires of diameter D = 15 nm can be grown onto a solid silicon carbide surface by carefully depositing droplets of catalyst liquid onto a flat silicon
As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable
In Problem 3.134, the prescribed value of ho = 1000 W/m2 ∙ K is large and characteristic of liquid cooling. In practice it would be far more preferable to use air cooling, for which a reasonable
As a means of enhancing heat transfer from high-performance logic chips, it is common to attach a heat sink to the chip surface in order to increase the surface area available for convection heat
Because of the large number of devices in today's PC chips, finned heat sinks are often used to maintain the chip at an acceptable operating temperature. Two fin designs are to be evaluated, both of
Consider design B of Problem 3.137. Over time, dust can collect in the fine grooves that separate the fins. Consider the buildup of a dust layer of thickness Ld, as shown in the sketch. Calculate and
A long rod of 20-mm diameter and a thermal conductivity of 1.5 W/m ∙ K has a uniform internal volumetric thermal energy generation of 10 6 W/m3. The rod is covered with an electrically insulating
An air heater consists of a steel tube (k = 20 W/m ∙ K), with inner and outer radii of r( = 13 mm and r2 = 16 mm, respectively, and eight integrally machined longitudinal fins, each of thickness t
Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular profile to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a
Heat is uniformly generated at the rate of 2 x 105 W/m 3 in a wall of thermal conductivity 25 W/m ∙ K and thickness 60 mm. The wall is exposed to convection on both sides, with different heat
Aluminum fins of triangular profile are attached to a plane wall whose surface temperature is 250°C. The fin base thickness is 2 mm, and its length is 6 mm. The system is in ambient air at a
An annular aluminum fin of rectangular profile is attached to a circular tube having an outside diameter of 25 mm and a surface temperature of 250°C. The fin is 1 mm thick and 10 mm long, and the
Annular aluminum fins of rectangular profile are attached to a circular tube having an outside diameter of 50 mm and an outer surface temperature of 200°C. The fins are 4 mm thick and 15 mm long.
It is proposed to air-cool the cylinders of a combustion chamber by joining an aluminum casing with annular fins (k = 240 W/m ∙ K) to the cylinder wall (k = 50 W/m ∙ K).The air is at 320K and the
Consider the air-cooled combustion cylinder of Problem 3.146, but instead of imposing a uniform heat flux at the inner surface, consider conditions for which the time-averaged temperature of the
Water is heated by submerging 50-mm diameter, thin-walled copper tubes in a tank and passing hot combustion gases (Tg = 750 K) through the tubes. To enhance heat transfer to the water, four straight
Consider the conditions of Problem 3.149, but now allow for a tube wall thickness of 5 mm (inner and outer diameters of 50 and 60 mm), a fin-to-tube thermal contact resistance of 10- 4 m2 ∙ K/W,
A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance
Consider the conditions of Example 3.12, except that the person is now exercising (in the air environment), which increases the metabolic heat generation rate by a factor of eight, to 5600 W/m3. At
In the method of separation of variables (Section 4.2) for two-dimensional, steady-state conduction, the separation constant λ2 in Equations 4.6 and 4.7 must be a positive constant. Show that a
A two-dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the exact solution for the heat equation presented in Section 4.2, calculate the temperature
Consider the two-dimensional rectangular plate of Problem 4.2 having a thermal conductivity of 50 W/m ∙ K. Beginning with the exact solution for the temperature distribution derive an expression
A two-dimensional rectangular plate is subjected to the boundary conditions shown. Derive an expression for the steady-state temperature distribution T(x, y).
A two-dimensional rectangular plate is subjected to prescribed temperature boundary conditions on three sides and a uniform heat flux into the plate at the top surface. Using the general approach of
Using the thermal resistance relations developed in Chapter 3, determine shape factor expressions for the following geometries:(a) Plane wall, cylindrical shell, and spherical shell.(b) Isothermal
Consider Problem 4.5 for the case where the plate is of square cross section, W = L.(a) Derive an expression for the shape factor, Smax associated with the maximum top surface temperature, such that
Based on the dimensionless conduction heat rates for cases 12-15 in Table 4.1 b, find shape factors for the following objects having temperature T1, located at the surface of a semi-infinite medium
Radioactive wastes are temporarily stored in a spherical container, the center of which is buried a distance of 10 m below the earth's surface. The outside diameter of the container is 2 m, and 500 W
A pipeline, used for the transport of crude oil, is buried in the earth such that its centerline is a distance of 1.5 m below the surface. The pipe has an outer diameter of 0.5 m and is insulated
A long power transmission cable is buried at a depth (ground to cable centerline distance) of 2 m. The cable is encased in a thin-walled pipe of 0.1-m diameter, and to render the cable
An electrical heater 100 mm long and 5 mm in diameter is inserted into a hole drilled normal to the surface of a large block of material having a thermal conductivity of 5 W/m ∙ K. Estimate the
Two parallel pipelines spaced 0.5 m apart are buried in soil having a thermal conductivity of 0.5 W/m ∙ K. The pipes have outer diameters of 100 and 75 mm with surface temperatures of 175°C and
A tube of diameter 50 mm having a surface temperature of 85°C is embedded in the center plane of a concrete slab 0.1 m thick with upper and lower surfaces at 20°C. Using the appropriate tabulated
Pressurized steam at 450 K flows through a long, thin-walled pipe of 0.5-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.5 m on a side. The axis of the
Hot water at 85°C flows through a thin-walled copper tube of 30 mm diameter. The tube is enclosed by an eccentric cylindrical shell that is maintained at 35°C and has a diameter of 120 mm. The
A furnace of cubical shape, with external dimensions of 0.35 m, is constructed from a refractory brick (fireclay). If the wall thickness is 50 mm, the inner surface temperature is 600°C, and the
The temperature distribution in laser-irradiated materials is determined by the power, size, and shape of the laser beam, along with the properties of the material being irradiated. The beam shape is
Laser beams are used to thermally process materials in a wide range of applications. Often, the beam is scanned along the surface of the material in a desired pattern. Consider the laser heating
A cubical glass melting furnace has exterior dimensions of width W = 5 m on a side and is constructed from refractory brick of thickness L = 0.35 m and thermal conductivity k = 1.4 W/m. K. The sides
A hot fluid passes through circular channels of a cast iron platen (A) of thickness LA = 30 mm which is in poor contact with the cover plates (B) of thickness L B = 7.5 mm. The channels are of
A long constantan wire of 1-mm diameter is butt welded to the surface of a large copper block, forming a thermo-couple junction. The wire behaves as a fin, permitting heat to flow from the surface,
A hole of diameter D = 0.25 m is drilled through the center of a solid block of square cross section with w = 1m on a side. The hole is drilled along the length, 1 = 2 m, of the block, which has a
In Chapter 3 we assumed that, whenever fins are attached to a base material, the base temperature is unchanged. What in fact happens is that, if the temperature of the base material exceeds the fluid
An igloo is built in the shape of a hemisphere, with an inner radius of 1.8 m and walls of compacted snow that are 0.5 m thick. On the inside of the igloo the surface heat transfer coefficient is 6
Consider the thin integrated circuit (chip) of Problem 3.136. Instead of attaching the heat sink to the chip surface, an engineer suggests that sufficient cooling might be achieved by mounting the
An electronic device, in the form of a disk 20 mm in diameter, dissipates 100 W when mounted flush on a large aluminum alloy (2024) block whose temperature is maintained at 27Â°C. The mounting
An aluminum heat sink (k = 240 W/m ∙ K) used to cool an array of electronic chips consists of a square channel of inner width w = 25 mm, through which liquid flow may be assumed to maintain a
Hot water is transported from a cogeneration power station to commercial and industrial users through steel pipes of diameter D = 150 mm, with each pipe centered in concrete (k = 1.4 W/m ∙ K) of
The elemental unit of an air heater consists of a long circular rod of diameter D, which is encapsulated by a finned sleeve and in which thermal energy is generated by Ohmic heating. The N fins of
For a small heat source attached to a large substrate, the spreading resistance associated with multidimensional conduction in the substrate may be approximated by the expression where Ar =
Consider nodal configuration 2 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations.(a) The horizontal boundary of the internal comer is
Consider nodal configuration 3 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations.(a) The boundary is insulated. Explain how Equation 4.42
Consider nodal configuration 4 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The upper boundary of the external corner is
One of the strengths of numerical methods is their ability to handle complex boundary conditions. In the sketch, the boundary condition changes from specified heat flux, if: (into the domain), to
Consider heat transfer in a one-dimensional (radial) cylindrical coordinate system under steady-state conditions with volumetric heat generation.(a) Derive the finite-difference equation for any
In a two-dimensional cylindrical configuration the radial (∆r) and angular (∆∅) spacing's of the nodes are uniform. The boundary at r = ri is of uniform temperature Ti. The boundaries in the
Upper and lower surfaces of a bus bar are convectively cooled by air at T∞ with hu ≠ h1. The sides are cooled by maintaining contact with heat sinks at To, through a thermal contact resistance of
Derive the nodal finite-difference equations for the following configurations.(a) Node m, n on a diagonal boundary subjected to convection with a fluid at T∞ and a heat transfer coefficient h.
Consider the nodal point 0 located on the boundary between materials of thermal conductivity kA and kB.Derive the finite-difference equation, assuming no internal generation.
Consider the two-dimensional grid (∆x = ∆y) representing steady-state conditions with no internal volumetric generation for a system with thermal conductivity k. One of the boundaries is
Consider a one-dimensional fin of uniform cross-sectional area, insulated at its tip, x = L. (See Table 3.4. case B). The temperature at the base of the fin Tb and of the adjoining fluid T∞, as
Consider the network for a two-dimensional system without internal volumetric generation having nodal temperatures shown below. If the grid space is 125 mm and the thermal conductivity of the
Consider the square channel shown in the sketch operating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to
Steady-state temperatures (K) at three nodal points of a long rectangular rod are as shown. The rod experiences a uniform volumetric generation rate of 5 X 107 W/m3 and has a thermal conductivity of
Steady-state temperatures at selected nodal points of the symmetrical section of a flow channel are known to be T2 = 95.47Â°C, T3 = 117.3Â°C, T5 = 79.79Â°C, T6 = 77.29Â°C, T8 = 87.28Â°C,
Consider an aluminum heat sink (k = 240 W/m ∙ K), such as that shown schematically in Problem 4.28. The inner and outer widths of the square channel are w = 20mm and W = 40mm, respectively, and an
The steady-state temperatures (oC) associated with selected nodal points of a two-dimensional system having a thermal conductivity of 1.5 W/m ∙ K are shown on the accompanying grid.(a) Determine
A steady-state, finite-difference analysis has been performed on a cylindrical fin with a diameter of 12 mm and a thermal conductivity of 15 W/m ∙ K. The convection process is characterized by a
A long bar of rectangular cross section is 60 mm by 90 mm on a side and has a thermal conductivity of 1W/m ∙ K. One surface is exposed to a convection process with air at 100Â°C and a convection
Consider two-dimensional, steady-state conduction in a square cross section with prescribed surface temperatures(a) Determine the temperatures at nodes 1, 2, 3, and 4. Estimate the midpoint
Consider a long bar of square cross section (0.8 m to the side) and of thermal conductivity 2 W/m ∙ K. Three of these sides are maintained at a uniform temperature of 300°C. The fourth side is
A long conducting rod of rectangular cross section (20mm X 30mm) and thermal conductivity k = 20 W/m ∙ K experiences uniform heat generation at a rate q = 5 X 107 W/m3 , while its surfaces are
A flue passing hot exhaust gases has a square cross section, 300 mm to a side. The walls are constructed of refractory brick 150 mm thick with a thermal conductivity of 0.85 W/m ∙ K. Calculate the
Consider the system of Problem 4.54. The interior surface is exposed to hot gases at 350°C with a convection coefficient of 100 W/m2 ∙ K, while the exterior surface experiences convection with air
A common arrangement for heating a large surface area is to move warm air through rectangular ducts below the surface. The ducts are square and located midway between the top and bottom surfaces that
Consider the gas turbine cooling scheme of Example 4.4. In Problem 3.23, advantages associated with applying a thermal barrier coating (TBC) to the exterior surface of a turbine blade are described.
A long bar of rectangular cross section, 0.4 m x 0.6 m on a side and having a thermal conductivity of 1.5 W/m ∙ K is subjected to the boundary conditions shown below. Two of the sides are
The top surface of a plate, including its grooves, is maintained at a uniform temperature of T, = 200Â°C. The lower surface is at T2 = 20Â°C, the thermal conductivity is 15 W/m ∙ K, and the
Refer to the two-dimensional rectangular plate of Problem 4.2. Using an appropriate numerical method with ∆x = ∆y = 0.25 m, determine the temperature at the midpoint (1, 0.5).
A long trapezoidal bar is subjected to uniform temperatures on two surfaces, while the remaining surfaces are well insulated. If the thermal conductivity of the material is 20 W/m ∙ K, estimate the
The shape factor for conduction through the edge of adjoining walls for which D > L/5, where D and L are the wall depth and thickness, respectively, the two-dimensional symmetrical element of the
The diagonal of a long triangular bar is well insulated, while sides of equivalent length are maintained at uniform temperatures Ta and Tb.(a) Establish a nodal network consisting of five nodes along
A straight fin of uniform cross section is fabricated from a material of thermal conductivity k = 5W/m ∙ K, thickness w = 20 mm, and length L = 200 mm. The fin is very long in the direction normal
Consider the long rectangular bar of Problem 4.50 with the prescribed boundary conditions.(a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/Temperature
Consider the long rectangular rod of Problem 4.53, which experiences uniform heat generation while its surfaces are maintained at a fixed temperature.(a) Using the finite-element method of FEHT,
Consider the symmetrical section of the flow channel of Problem 4.46, with the prescribed values of q, k, T∞,i, T∞,0, hi and ho. Use the finite-element method of FEHT to obtain the following
The hot-film heat flux gage shown schematically may be used to determine the convection coefficient of an adjoining fluid stream by measuring the electric power dissipation per unit area, P"e (W/m2),
A semiconductor industry roadmap for microlithography processing requires that a 300-mm-diameter silicon wafer be maintained at a steady-state temperature of 140°C to within a uniformity of 0.1 Dc.
A straight fin of uniform cross section is fabricated from a material of thermal conductivity 50 W/m. K, thickness w = 6 mm, and length L = 48 mm, and is very long in the direction normal to the
A rod of 10-mm diameter and 250-mm length has one end maintained at 100°C. The surface of the rod experiences free convection with the ambient air at 25°C and a convection coefficient that depends
A thin metallic foil of thickness 0.25 mm with a pattern of extremely small holes serves as an acceleration grid to control the electrical potential of an ion beam. Such a grid is used in a chemical
Small diameter electrical heating elements dissipating 50 W/m (length normal to the sketch) are used to heat a ceramic plate of thermal conductivity 2 W/m ∙ K. The upper surface of the plate is
A simplified representation for cooling in very large-scale integration (VLSI) of microelectronics is shown in the sketch. A silicon chip is mounted in a dielectric substrate, and one surface of the
Electronic devices dissipating electrical power can be cooled by conduction to a heat sink. The lower surface of the sink is cooled, and the spacing of the devices ws, the width of the device W d'
A major problem in packaging very large-scale integrated (VLSI) circuits concerns cooling of the circuit elements. The problem results from increasing levels of power dissipation within a chip, as
A heat sink for cooling computer chips is fabricated from copper (k s = 400 W/m' K), with machined micro channels passing a cooling fluid for which T∞ = 25°C and h = 30,000 W/m2 ∙ K. The lower
A plate (k = 10W/m ∙ K) is stiffened by a series of longitudinal ribs having a rectangular cross section with length L = 8 mm and width w = 4 mm. The base of the plate is maintained at a uniform
The bottom half of an I-beam providing support for a furnace roof extends into the heating zone. The web is well insulated, while the flange surfaces experience convection with hot gases at T∞. =

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