Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470917855
Author: Bergman, Theodore L./
Publisher: John Wiley & Sons Inc
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Chapter 4, Problem 4.44P
In a two-dimensional cylindrical configuration, the radial
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for all 0
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 ′′ qs (into the domain) to convection, at the location of the node (m, n). Write the steady-state, two- dimensional finite difference equation at this node.
Chapter 4 Solutions
Fundamentals of Heat and Mass Transfer
Ch. 4 - In the method of separation of variables (Section...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Consider the two-dimensional rectangular plate of...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Using the thermal resistance relations developed...Ch. 4 - Free convection heat transfer is sometimes...Ch. 4 - Consider Problem 4.5 for the case where the plate...Ch. 4 - Prob. 4.9PCh. 4 - Based on the dimensionless conduction heat rates...
Ch. 4 - Determine the heat transfer rate between two...Ch. 4 - A two-dimensional object is subjected to...Ch. 4 - An electrical heater 100 mm long and 5 mm in...Ch. 4 - Two parallel pipelines spaced 0.5 m apart are...Ch. 4 - A small water droplet of diameter D=100m and...Ch. 4 - A tube of diameter 50 mm having a surface...Ch. 4 - Pressurized steam at 450K flows through a long,...Ch. 4 - The temperature distribution in laser-irradiated...Ch. 4 - Hot water at 85°C flows through a thin-walled...Ch. 4 - A furnace of cubical shape, with external...Ch. 4 - Laser beams are used to thermally process...Ch. 4 - A double-glazed window consists of two sheets of...Ch. 4 - A pipeline, used for the transport of crude oil,...Ch. 4 - A long power transmission cable is buried at a...Ch. 4 - A small device is used to measure the surface...Ch. 4 - A cubical glass melting furnace has exterior...Ch. 4 - An aluminum heat sink (k=240W/mK), used to cool an...Ch. 4 - Hot water is transported from a cogeneration power...Ch. 4 - A long constantan wire of 1-mm diameter is butt...Ch. 4 - A hole of diameter D=0.25m is drilled through the...Ch. 4 - In Chapter 3 we that, whenever fins are attached...Ch. 4 - An igloo is built in the shape of a hemisphere,...Ch. 4 - Prob. 4.34PCh. 4 - An electronic device, in the form of a disk 20 mm...Ch. 4 - The elemental unit of an air heater consists of a...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Prob. 4.39PCh. 4 - Prob. 4.40PCh. 4 - One of the strengths of numerical methods is their...Ch. 4 - Determine expressionsfor...Ch. 4 - Consider heat transfer in a one-dimensional...Ch. 4 - In a two-dimensional cylindrical configuration,...Ch. 4 - Upper and lower surfaces of a bus bar are...Ch. 4 - Derive the nodal finite-difference equations for...Ch. 4 - Consider the nodal point 0 located on the boundary...Ch. 4 - Prob. 4.48PCh. 4 - Prob. 4.49PCh. 4 - Consider the network for a two-dimensional system...Ch. 4 - An ancient myth describes how a wooden ship was...Ch. 4 - Consider the square channel shown in the sketch...Ch. 4 - A long conducting rod of rectangular cross section...Ch. 4 - A flue passing hot exhaust gases has a square...Ch. 4 - Steady-state temperatures (K) at three nodal...Ch. 4 - Functionally graded materials are intentionally...Ch. 4 - Steady-state temperatures at selected nodal points...Ch. 4 - Consider an aluminum heat sink (k=240W/mK), such...Ch. 4 - Conduction within relatively complex geometries...Ch. 4 - Prob. 4.60PCh. 4 - The steady-state temperatures (°C) associated with...Ch. 4 - A steady-state, finite-difference analysis has...Ch. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Consider a two-dimensional. straight triangular...Ch. 4 - A common arrangement for heating a large surface...Ch. 4 - A long, solid cylinder of diameter D=25mm is...Ch. 4 - Consider Problem 4.69. An engineer desires to...Ch. 4 - Prob. 4.71PCh. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Refer to the two-dimensional rectangular plate of...Ch. 4 - The shape factor for conduction through the edge...Ch. 4 - Prob. 4.77PCh. 4 - A simplified representation for cooling in very...Ch. 4 - Prob. 4.84PCh. 4 - A long trapezoidal bar is subjected to uniform...Ch. 4 - Consider the system of Problem 4.54. The interior...Ch. 4 - A long furnace. constructed from refractory brick...Ch. 4 - A hot pipe is embedded eccentrically as shown in a...Ch. 4 - A hot liquid flows along a V-groove in a solid...Ch. 4 - Prob. 4S.5PCh. 4 - Hollow prismatic bars fabricated from plain carbon...
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- Three (3) bricks, specifically A, B, and C were arranged horizontally in such a way that it can be illustrated as a sandwich panel. Consider the system to be in series and in the order of Brick A, Brick B and Brick C. The outside surface temperature of Brick A is 1,500℃ and 150 ℃ for the outside surface of Brick C. The thermal conductivities for Brick A, Brick B and Brick C, are 2 ?/? °? , 0.50 ?/? °? , 60 ?/? °?. The thickness of Brick A and Brick C are 50 cm and 22 cm. The rate of heat transfer per unit area is 1,000 ?/?2 . Determine the following: The thickness of Brick B in the unit of mm. Assume that all the conditions were retain except that the thickness of Brick B was increased to 800 mm, what is the new value for the rate of heat transfer per unit area in ???/ℎ? . ??2 please explain the principles to solve thisarrow_forwardTo avoid overheating, natural convection analysis is to beconsidered. If the minimum allowable heat transfer due tonatural convection is 30 W, find the maximum angle at whichthe TV can hang off the wall (0 < θ < 60). Some temperatureswere collected; Back panel surface: 45 °C, ambient: 32 °C. A typical 55-inch TV screen measures (122 cm x 69 cm). please answer and show all detailed calculationsarrow_forwardHi, kindly solve this problem and show the solution. Thank youarrow_forward
- 1- The energy transferred from the anterior chamber of the eye through the cornea varies considerably depending on whether a contact lens is worn. Treat the eye as a spherical system and assume the system to be at steady state. The convection coefficient ho is unchanged with and without the contact lens in place. The cornea and the lens cover one-third of the spherical surface area. T he T h Anterior chamber Contact lens Cornea are as follows: Values of the parameters representing this situation r 10.2mm, r 12.7 mm, r3= 16.5 mm, Teoj= 37°C, Teoo = 21°C, ki = 0.35 W/m.K, k2 0.80 W/m.K, h 12 W/m2.K, ho 12 W/m2.K. (a) Construct the thermal circuits, labeling all potential and flows form the systems excluding the contact lens and including the contact lens. Write resistance elements in terms of appropriate parameters (b) Determine the heat loss from the interior chamber with and without the contact lens in place (c) Discuss the implication of your results.arrow_forwardConsider the square channel shown in the sketch operating under steady state condition. The inner surface of the channel is at a uniform temperature of 600 K and the outer surface is at a uniform temperature of 300 K. From a symmetrical elemental of the channel, a two-dimensional grid has been constructed as in the right figure below. The points are spaced by equal distance. Tout = 300 K k = 1 W/m-K T = 600 K (a) The heat transfer from inside to outside is only by conduction across the channel wall. Beginning with properly defined control volumes, derive the finite difference equations for locations 123. You can also use (n, m) to represent row and column. For example, location Dis (3, 3), location is (3,1), and location 3 is (3,5). (hint: I have already put a control volume around this locations with dashed boarder.) (b) Please use excel to construct the tables of temperatures and finite difference. Solve for the temperatures of each locations. Print out the tables in the spread…arrow_forward2. Consider the temperature distributions associated with a dx differential control volume within the one-dimensional plane walls shown below. T(x,00) T\x,00) * dx * dx (a) (Б) Tx,1) T(x,1) * dx dx (c) (d) (a) Steady-state conditions exist. Is thermal energy being generated within the differential control volume? If so, is the generation rate positive or negative? (b) Steady-state conditions exist as in part (a). Is the volumetric generation rate positive or negative within the differential control volume? (c) Steady-state conditions do not exist, and there is no volumetric thermal energy generation. Is the temperature of the material in the differential control volume increasing or decreasing with time? (d) Transient conditions exist as in part (c). Is the temperature increasing or decreasing with time?arrow_forward
- 2. The slab shown is embedded in insulating materials on five sides, while the front face experiences convection off its face. Heat is generated inside the material by an exothermic reaction equal to 1.0 kW/m'. The thermal conductivity of the slab is 0.2 W/mk. a. Simplify the heat conduction equation and integrate the resulting ID steady form of to find the temperature distribution of the slab, T(x). b. Present the temperature of the front and back faces of the slab. n-20- 10 cm IT- 25°C) 100 cm 100 cmarrow_forwardA solid circular rode of 40 cm long and 4 cm diameter is subjected to a uniform temperature at the circular surface of one of its ends of 100 °C. The surface of other end are subjected to free convection with surrounding air with a convection coefficient of h = 10 W/(m². °C) and air temperature of 30 °C. Lateral surface is insulated. Use three linear elements to find the temperature along the rod, and the energy entered at the end of constant temperature. The thermal conductivity of the rod is K= 60 W/(m. °C K)arrow_forwardQ2/ A thermopane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20°C from outside ambient air at -10°C. The convection coefficient associated with the inner (room-side) surface is 10 W/m? K. If the convection coeficient associated with the outer (ambient) air is 80 W/m? K, what is the heat loss through a window that is 0.8 m long by 0.5 m wide? Neglect radiation,[ kglass = 1.4 W/m-K: kair-0.024S W/m-K ] Window, 0.8 m x 0.5 m Glass 7+7+7-1 Air! Toi= 20 °C h; = 10 W/m2 K Air11 L=0.007 m To = -10 °C ho = 80 W/m2-K Airarrow_forward
- A long cylinder of radius, ro = 2.8cm, thermal conductivity k = 225 S mk thermal diffusivity 1.10 kJ and initial temperature, T = 200°C is kgK suddenly exposed to a convective environment at To 65°C with a convection coefficient, a = 6.82x105 m², specific heat, c h=510- W Watch your units on this one! m²K Determine the temperature at the center of the cylinder after t 130s, time, of exposure to the convective environment in degrees Celsius. 97.14C 95.30 198.08 70.94arrow_forwardConsider *arrow_forwardTransient Heat Conduction Cooking a Thanksgiving turkey is an art form and, if your skills in the kitchen are like mine, it is sometimes more of a mystical, elusive art form. Thankfully, science also has much to contribute in the kitchen as well as the laboratory. Let us consider the change in temperature of a common, 20-lb holiday fowl as it is cooked in a convection oven. To simplify the analysis, let's assume the bird can be modeled as a uniform sphere of radius 7.0 in. with a specific heat of 3.53 kJ/kg-K. Moreover, the turkey will be assumed to have a uniform temperature, T, throughout that will change with time as it is cooked according to the following relationship: 。 + (To - T∞)ept T(t) = T∞ + where To is the initial temperature of the turkey, T∞, is the oven temperature, V is the volume of the turkey, As is the surface area of the turkey, and h is the convection coefficient for the scenario which is 11.3 W/m²-K. If the oven is set to 325 °F and the initial temperature of the…arrow_forward
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