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5–2C How do numerical solution methods differ from analytical ones? What are the advantages and disadvantages of numerical and analytical methods
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5–3C What is the basis of the energy balance method? How does it differ from the formal finite difference method? For a specified nodal network, will these two methods result in the same or a different set of equations
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5–4C Consider a heat conduction problem that can be solved both analytically, by solving the governing differential equation and applying the boundary conditions, and numerically, by a software package available on your computer. Which approach would you use to solve this problem? Explain your reasoning
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5–5C Two engineers are to solve an actual heat transfer problem in a manufacturing facility. Engineer Amakes the necessary simplifying assumptions and solves the problem analytically, while engineer B solves it numerically using a powerful software package. Engineer A claims he solved the problem exactly and thus his results are better, while engineer B claims that he used a more realistic model and thus his results are better. To resolve the dispute, you are asked to solve the problem experimentally in a lab. Which engineer do you think the experiments will prove right? Explain.
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5–6C Define these terms used in the finite difference formulation: node, nodal network, volume element, nodal spacing, and difference equation
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5–7 Consider three consecutive nodes n 0 1, n, and n - 1 in a plane wall. Using the finite difference form of the first derivative at the midpoints, show that the finite difference form of the second derivative can be expressed as
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5–8 The finite difference formulation of steady twodimensional heat conduction in a medium with heat generation and constant thermal conductivity is given by
in rectangular coordinates. Modify this relation for the threedimensional case
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5–9 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of Dx. Using the finite difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux q · 0 at the left boundary (node 0) and convection at the right boundary (node 4) with a convection coefficient of h and an ambient temperature of T∞
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5–10 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, 4, and 5 with a uniform nodal spacing of Dx. Using the finite difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissivity of E and surrounding temperature of Tsurr
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5–11C Explain how the finite difference form of a heat conduction problem is obtained by the energy balance method
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5–12C In the energy balance formulation of the finite difference method, it is recommended that all heat transfer at the boundaries of the volume element be assumed to be into the volume element even for steady heat conduction. Is this a valid recommendation even though it seems to violate the conservation of energy principle
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5–13C How is an insulated boundary handled in the finite difference formulation of a problem? How does a symmetry line differ from an insulated boundary in the finite difference formulation
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5–14C How can a node on an insulated boundary be treated as an interior node in the finite difference formulation of a plane wall? Explain
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5–15C Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as
(a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable
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5–16 Consider steady heat conduction in a plane wall whose left surface (node 0) is maintained at 30°C while the right surface (node 8) is subjected to a heat flux of 800 W/m2. Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heat generation. Also obtain the finite difference formulation for the rate of heat transfer at the left boundary
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5–17 Consider steady heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux q · 0 at the left boundary (node 0) and convection at the right boundary (node 4) with a convection coefficient of h and an ambient temperature of T∞
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5–18 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, 4, and 5 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissivity of and surrounding temperature of Tsurr
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5–19 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, 4, and 5 with a uniform nodal spacing of Dx. The temperature at the right boundary (node 5) is specified. Using the energy balance approach, obtain the finite difference formulation of the boundary node 0 on the left boundary for the case of combined convection, radiation, and heat flux at the left boundary with an emissivity of , convection coefficient of h, ambient temperature of T∞, surrounding temperature of Tsurr, and uniform heat flux of q · 0. Also, obtain the finite difference formulation for the rate of heat transfer at the right boundary.
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5–20 Consider steady one-dimensional heat conduction in a composite plane wall consisting of two layers Aand B in perfect contact at the interface. The wall involves no heat generation. The nodal network of the medium consists of nodes 0, 1 (at the interface), and 2 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of this problem for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 2) with an emissivity of and surrounding temperature of Tsurr
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5–21 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0, 1, and 2 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of this problem for the case of specified heat flux q · 0 to the wall and convection at the left boundary (node 0) with a convection coefficient of h and ambient temperature of T∞, and radiation at the right boundary (node 2) with an emissivity of E and surrounding surface temperature of Tsurr.
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5–22 Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at T with a heat transfer coefficient of h. The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and2 (at the fin tip) with a uniform nodal spacing of x. Using the energy balance approach, obtain the finite difference formulation of this problem to determine T1 and T2 for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in °C.
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5–23 Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at T∞ with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of Tsurr The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of this problem to determine T1 and T2 for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in °C
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5–24 Consider a large uranium plate of thickness 5 cm and thermal conductivity k = 28 W/m · °C in which heat is generated uniformly at a constant rate of g = 6 x 105 W/m3. One side of the plate is insulated while the other side is subjected to convection to an environment at 30°C with a heat transfer coefficient of h = 60 W/m2 · °C. Considering six equally spaced nodes with a nodal spacing of 1 cm, (a) obtain the finite difference formulation of this problem and (b) determine the nodal temperatures under steady conditions by solving those equations
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5–25 Consider an aluminum alloy fin (k = 180 W/m · °C) of triangular cross section whose length is L = 5 cm, base thickness is b = 1 cm, and width w in the direction normal to the plane of paper is very large. The base of the fin is maintained at a temperature of T0 = 180°C. The fin is losing heat by convection to the ambient air at T∞ = 25°C with a heat transfer coefficient of h = 25 W/m2 · °C and by radiation to the surrounding surfaces at an average temperature of Tsurr = 290 K. Using the finite difference method with six equally spaced nodes along the fin in the x-direction, determine (a) the temperatures at the nodes and (b) the rate of heat transfer from the fin for w = 1 m. Take the emissivity of the fin surface to be 0.9 and assume steady one-dimensional heat transfer in the fin.
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5–26 Reconsider Problem 5–25. Using EES (or other) software, investigate the effect of the fin base temperature on the fin tip temperature and the rate of heat transfer from the fin. Let the temperature at the fin base vary from 100°C to 200°C. Plot the fin tip temperature and the rate of heat transfer as a function of the fin base temperature, and discuss the results
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5–27 Consider a large plane wall of thickness L = 0.4 m, thermal conductivity k = 2.3 W/m · °C, and surface area A = 20 m2. The left side of the wall is maintained at a constant temperature of 80°C, while the right side loses heat by convection to the surrounding air at T∞ = 15°C with a heat transfer coefficient of h = 24 W/m2 · °C. Assuming steady onedimensional heat transfer and taking the nodal spacing to be 10cm, (a) obtain the finite difference formulation for all nodes, (b) determine the nodal temperatures by solving those equations, and (c) evaluate the rate of heat transfer through the wall
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5–28 Consider the base plate of a 800-Whousehold iron having a thickness of L = 0.6 cm, base area of A = 160 cm2, and thermal conductivity of k = 20 W/m · °C. The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heaters inside. When steady operating conditions are reached, the outer surface temperature of the plate is measured to be 85°C. Disregarding any heat loss through the upper part of the iron and taking the nodal spacing to be 0.2 cm, (a) obtain the finite difference formulation for the nodes and (b) determine the inner surface temperature of the plate by solving those equations.
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5–29 Consider a large plane wall of thickness L = 0.3 m, thermal conductivity k = 2.5 W/m · °C, and surface area A = 12 m2. The left side of the wall is subjected to a heat flux of q0 = 700 W/m2 while the temperature at that surface is measured to be T0 = 60°C. Assuming steady one-dimensional heat transfer and taking the nodal spacing to be 6 cm, (a) obtain the finite difference formulation for the six nodes and (b) determine the temperature of the other surface of the wall by solving those equations
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5–30E A large steel plate having a thickness of L = 5 in., thermal conductivity of k = 7.2 Btu/h · ft · °F, and an emissivity of E= 0.6 is lying on the ground. The exposed surface of the plate exchanges heat by convection with the ambient air at T∞= 80°F with an average heat transfer coefficient of h = 3.5Btu/h · ft2 · °F as well as by radiation with the open sky at an equivalent sky temperature of Tsky = 510 R. The ground temperature below a certain depth (say, 3 ft) is not affected by the weather conditions outside and remains fairly constant at 50°F at that location. The thermal conductivity of the soil can be taken to be ksoil = 0.49 Btu/h · ft · °F, and the steel plate can be assumed to be in perfect contact with the ground. Assuming steady one-dimensional heat transfer and taking the nodal spacings to be 1 in. in the plate and 0.6 ft in the ground, (a) obtain the finite difference formulation for all 11 nodes shown in Figure P5–30E and (b) determine the top and bottom surface temperatures of the plate by solving those equations
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5–31E Repeat Problem 5–30E by disregarding radiation heat transfer from the upper surface.
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5–32 Consider a stainless steel spoon (k = 15.1 W/m · C, E = 0.6) that is partially immersed in boiling water at 95°C in a kitchen at 25°C. The handle of the spoon has a cross section of about 0.2 cm x 1 cm and extends 18 cm in the air from the free surface of the water. The spoon loses heat by convection to the ambient air with an average heat transfer coefficient of h = 13 W/m2 · °C as well as by radiation to the surrounding surfaces at an average temperature of Tsurr = 295 K. Assuming steady one-dimensional heat transfer along the spoon and taking the nodal spacing to be 3 cm, (a) obtain the finite difference formulation for all nodes, (b) determine the temperature of the tip of the spoon by solving those equations, and (c) determine the rate of heat transfer from the exposed surfaces of the spoon
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5–33 Repeat Problem 5–32 using a nodal spacing of 1.5 cm.
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5–34 Reconsider Problem 5–33. Using EES (or other) software, investigate the effects of the thermal conductivity and the emissivity of the spoon material on the temperature at the spoon tip and the rate of heat transfer from the exposed surfaces of the spoon. Let the thermal conductivity vary from 10 W/m · °C to 400 W/m · °C, and the emissivity from 0.1 to 1.0. Plot the spoon tip temperature and the heat transfer rate as functions of thermal conductivity and emissivity, and discuss the results
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5–35 One side of a 2-m-high and 3-m-wide vertical plate at 130°C is to be cooled by attaching aluminum fins (k = 237 W/m · °C) of rectangular profile in an environment at 35°C. The fins are 2 cm long, 0.3 cm thick, and 0.4 cm apart. The heat transfer coefficient between the fins and the surrounding air for combined convection and radiation is estimated to be 30 W/m2 · °C. Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be 0.5 cm, determine (a) the finite difference formulation of this problem, (b) the nodal temperatures along the fin by solving these equations, (c) the rate of heat transfer from a single fin, and (d) the rate of heat transfer from the entire finned surface of the plate
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5–36 A hot surface at 100°C is to be cooled by attaching 3-cm-long, 0.25-cm-diameter aluminum pin fins (k = 237 W/m · °C) with a center-to-center distance of 0.6 cm. The temperature of the surrounding medium is 30°C, and the combined heat transfer coefficient on the surfaces is 35 W/m2 · °C. Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be 0.5 cm, determine (a) the finite difference formulation of this problem, (b) the nodal temperatures along the fin by solving these equations, (c) the rate of heat transfer from a single fin, and (d) the rate of heat transfer from a 1-m x 1-m section of the plate.
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5–37 Repeat Problem 5–36 using copper fins (k = 386 W/m ·°C) instead of aluminum ones.
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5–38 Two 3-m-long and 0.4-cm-thick cast iron (k = 52 W/m · °C, E= 0.8) steam pipes of outer diameter 10 cm are connected to each other through two 1-cm-thick flanges of outer diameter 20 cm, as shown in the figure. The steam flows inside the pipe at an average temperature of 200°C with a heat transfer coefficient of 180 W/m2 · °C. The outer surface of the pipe is exposed to convection with ambient air at 8°C with a heat transfer coefficient of 25 W/m2 · °C as well as radiation with the surrounding surfaces at an average temperature of Tsurr 290 K. Assuming steady one-dimensional heat conduction along the flanges and taking the nodal spacing to be 1 cm along the flange (a) obtain the finite difference formulation for all nodes, (b) determine the temperature at the tip of the flange by solving those equations, and (c) determine the rate of heat transfer from the exposed surfaces of the flange.
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5–39 Reconsider Problem 5–38. Using EES (or other) software, investigate the effects of the steam temperature and the outer heat transfer coefficient on the flange tip temperature and the rate of heat transfer from the exposed surfaces of the flange. Let the steam temperature vary from 150°C to 300°C and the heat transfer coefficient from 15 W/m2 · °C to 60 W/m2 · °C. Plot the flange tip temperature and the heat transfer rate as functions of steam temperature and heat transfer coefficient, and discuss the results
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5–40 Using EES (or other) software, solve these systems of algebraic equations.
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5–41 Using EES (or other) software, solve these systems of algebraic equations.
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5–42 Using EES (or other) software, solve these systems of algebraic equations.
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5–43C Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as
(a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable
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5–44C Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as
(a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable
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5–45C What is an irregular boundary? What is a practical way of handling irregular boundary surfaces with the finite difference method
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5–46 Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The temperatures at the selected nodes and the thermal conditions at the boundaries are as shown. The thermal conductivity of the body is k = 45 W/m · °C, and heat is generated in the body uniformly at a rate of g = 6 x 106 W/m3. Using the finite difference method with a mesh size of Dx = Dy = 5.0 cm, determine (a) the temperatures at nodes 1, 2, and 3 and (b) the rate of heat loss from the bottom surface through a 1-m-long section of the body.
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5–47 Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The measured temperatures at selected points of the outer surfaces are as shown. The thermal conductivity of the body is k = 45 W/m · °C, and there is no heat generation. Using the finite difference method with a mesh size of Dx = Dy = 2.0 cm, determine the temperatures at the indicated points in the medium. Hint: Take advantage of symmetry.
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5–48 Consider steady two-dimensional heat transfer in a long solid bar whose cross section is given in the figure. The measured temperatures at selected points of the outer surfaces are as shown. The thermal conductivity of the body is k = 20 W/m · °C, and there is no heat generation. Using the finite difference method with a mesh size of Dx = Dy = 1.0 cm, determine the temperatures at the indicated points in the medium.
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5–49 Starting with an energy balance on a volume element, obtain the steady two-dimensional finite difference equation for a general interior node in rectangular coordinates for T(x, y) for the case of variable thermal conductivity and uniform heat generation.
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5–50 Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The temperatures at the selected nodes and the thermal conditions on the boundaries are as shown. The thermal conductivity of the body is k = 180 W/m · °C, and heat is generated in the body uniformly at a rate of g = 107 W/m3. Using the finite difference method with a mesh size of Dx = Dy = 10 cm, determine (a) the temperatures at nodes 1, 2, 3, and 4 and (b) the rate of heat loss from the top surface through a 1-m-long section of the body.
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5–51 Reconsider Problem 5–50. Using EES (or other) software, investigate the effects of the thermal conductivity and the heat generation rate on the temperatures at nodes 1 and 3, and the rate of heat loss from the top surface. Let the thermal conductivity vary from 10 W/m · °C to 400 W/m · °C and the heat generation rate from 105 W/m3 to 108 W/m3. Plot the temperatures at nodes 1 and 3, and the rate of heat loss as functions of thermal conductivity and heat generation rate, and discuss the results
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5–52 Consider steady two-dimensional heat transfer in a long solid bar whose cross section is given in the figure. The measured temperatures at selected points on the outer surfaces are as shown. The thermal conductivity of the body is k = 20 W/m · °C, and there is no heat generation. Using the finite difference method with a mesh size of Dx = Dy = 1.0 cm, determine the temperatures at the indicated points in the medium. Hint: Take advantage of symmetry.
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5–53 Consider steady two-dimensional heat transfer in an L-shaped solid body whose cross section is given in the figure. The thermal conductivity of the body is k = 45 W/m · °C, and heat is generated in the body at a rate of g = 5 x 106 W/m3. The right surface of the body is insulated, and the bottom surface is maintained at a uniform temperature of 120°C. The entire top surface is subjected to convection with ambient air at T∞ = 30°C with a heat transfer coefficient of h = 55 W/m2 ·°C, and the left surface is subjected to heat flux at a uniform rate of q · L = 8000 W/m2. The nodal network of the problem consists of 13 equally spaced nodes with Dx = Dy = 1.5 cm. Five of the nodes are at the bottom surface and thus their temperatures are known. (a) Obtain the finite difference equations at the remaining eight nodes and (b) determine the nodal temperatures by solving those equations
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5–54E Consider steady two-dimensional heat transfer in a long solid bar of square cross section in which heat is generated uniformly at a rate of g = 0.19 x 105 Btu/h · ft3. The cross section of the bar is 0.4 ft = 0.4 ft in size, and its thermal conductivity is k = 16 Btu/h · ft · °F. All four sides of the bar are subjected to convection with the ambient air at T∞= 70°F with a heat transfer coefficient of h = 7.9 Btu/h · ft2 · °F. Using the finite difference method with a mesh size of Dx = Dy = 0.2 ft, determine (a) the temperatures at the nine nodes and (b) the rate of heat loss from the bar through a 1-ft-long section.
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5–55 Hot combustion gases of a furnace are flowing through a concrete chimney (k = 1.4 W/m · °C) of rectangular cross section. The flow section of the chimney is 20 cm x 40 cm, and the thickness of the wall is 10 cm. The average temperature of the hot gases in the chimney is T∞ = 280°C, and the average convection heat transfer coefficient inside the chimney is hi = 75 W/m2 · °C. The chimney is losing heat from its outer surface to the ambient air at To = 15°C by convection with a heat transfer coefficient of ho = 18 W/m2 · °C and to the sky by radiation. The emissivity of the outer surface of the wall is E=0.9, and the effective sky temperature is estimated to be 250 K. Using the finite difference method with Dx = Dy = 10 cm and taking full advantage of symmetry, (a) obtain the finite difference formulation of this problem for steady twodimensional heat transfer, (b) determine the temperatures at the nodal points of a cross section, and (c) evaluate the rate of heat loss for a 1-m-long section of the chimney
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5–56 Repeat Problem 5–55 by disregarding radiation heat transfer from the outer surfaces of the chimney
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5–57 Reconsider Problem 5–55. Using EES (or other) software, investigate the effects of hot-gas temperature and the outer surface emissivity on the temperatures at the outer corner of the wall and the middle of the inner surface of the right wall, and the rate of heat loss. Let the temperature of the hot gases vary from 200°C to 400°C and the emissivity from 0.1 to 1.0. Plot the temperatures and the rate of heat loss as functions of the temperature of the hot gases and the emissivity, and discuss the results
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5–58 Consider a long concrete dam (k = 0.6 W/m · °C, as = 0.7 m2/s) of triangular cross section whose exposed surface is subjected to solar heat flux of qs = 800 W/m2 and to convection and radiation to the environment at 25°C with a combined heat transfer coefficient of 30 W/m2 ·°C. The 2-m-high vertical section of the dam is subjected to convection by water at 15°C with a heat transfer coefficient of 150 W/m2 · °C, and heat transfer through the 2-m-long base is considered to be negligible. Using the finite difference method with a mesh size of Dx =Dy = 1 m and assuming steady two-dimensional heat transfer, determine the temperature of the top, middle, and bottom of the exposed surface of the dam.
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5–59E Consider steady two-dimensional heat transfer in a V-grooved solid body whose cross section is given in the figure. The top surfaces of the groove are maintained at 32°F while the bottom surface is maintained at 212°F. The side surfaces of the groove are insulated. Using the finite difference method with a mesh size of Dx = Dy = 1 ft and taking advantage of symmetry, determine the temperatures at the middle of the insulated surfaces.
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5–60 Reconsider Problem 5–59E. Using EES (or other) software, investigate the effects of the temperatures at the top and bottom surfaces on the temperature in the middle of the insulated surface. Let the temperatures at the top and bottom surfaces vary from 32°F to 212°F. Plot the temperature in the middle of the insulated surface as functions of the temperatures at the top and bottom surfaces, and discuss the results
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5–61 Consider a long solid bar whose thermal conductivity is k = 12 W/m · °C and whose cross section is given in the figure. The top surface of the bar is maintained at 50°C while the bottom surface is maintained at 120°C. The left surface is insulated and the remaining three surfaces are subjected to convection with ambient air at T∞= 25°C with a heat transfer coefficient of h = 30 W/m2 · °C. Using the finite difference method with a mesh size of Dx = Dy = 10 cm, (a) obtain the finite difference formulation of this problem for steady two dimensional heat transfer and (b) determine the unknown nodal temperatures by solving those equations.
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5–62 Consider a 5-m-long constantan block (k = 23 W/m ·°C) 30 cm high and 50 cm wide. The block is completely submerged in iced water at 0°C that is well stirred, and the heat transfer coefficient is so high that the temperatures on both sides of the block can be taken to be 0°C. The bottom surface of the bar is covered with a low-conductivity material so that heat transfer through the bottom surface is negligible. The top surface of the block is heated uniformly by a 6-kW resistance heater. Using the finite difference method with a mesh size of Dx = Dy = 10 cm and taking advantage of symmetry, (a) obtain the finite difference formulation of this problem for steady two-dimensional heat transfer, (b) determine the unknown nodal temperatures by solving those equations, and (c)determine the rate of heat transfer from the block to the iced water.
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5–63C How does the finite difference formulation of a transient heat conduction problem differ from that of a steady heat conduction problem? What does the term
represent in the transient finite difference formulation
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5–64C What are the two basic methods of solution of transient problems based on finite differencing? How do heat transfer terms in the energy balance formulation differ in the two methods
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5–65C The explicit finite difference formulation of a general interior node for transient heat conduction in a plane wall is given by
Obtain the finite difference formulation for the steady case by simplifying the relation above.
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5–66C The explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by
Obtain the finite difference formulation for the steady case by simplifying the relation above
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5–67C Is there any limitation on the size of the time step Dt in the solution of transient heat conduction problems using (a) the explicit method and (b) the implicit method
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5–68C Express the general stability criterion for the explicit method of solution of transient heat conduction problems
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5–69C Consider transient one-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall are at specified temperatures, express the stability criterion for this problem in its simplest form
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5–70C Consider transient one-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall are subjected to specified heat flux, express the stability criterion for this problem in its simplest form
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5–71C Consider transient two-dimensional heat conduction in a rectangular region that is to be solved by the explicit method. If all boundaries of the region are either insulated or at specified temperatures, express the stability criterion for this problem in its simplest form
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5–72C The implicit method is unconditionally stable and thus any value of time step Dt can be used in the solution of transient heat conduction problems. To minimize the computation time, someone suggests using a very large value of Dt since there is no danger of instability. Do you agree with this suggestion? Explain
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5–73 Consider transient heat conduction in a plane wall whose left surface (node 0) is maintained at 50°C while the right surface (node 6) is subjected to a solar heat flux of 600 W/m2. The wall is initially at a uniform temperature of 50°C. Express the explicit finite difference formulation of the boundary nodes 0 and 6 for the case of no heat generation. Also, obtain the finite difference formulation for the total amount of heat transfer at the left boundary during the first three time steps
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5–74 Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of Dx. The wall is initially at a specified temperature. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary nodes for the case of uniform heat flux q · 0 at the left boundary (node 0) and convection at the right boundary (node 4) with a convection coefficient of h and an ambient temperature of T∞. Do not simplify
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5–75 Repeat Problem 5–74 for the case of implicit formulation
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5–76 Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, 4, and 5 with a uniform nodal spacing of Dx.The wall is initially at a specified temperature. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissivity of and surrounding temperature of Tsurr
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5–77 Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of Dx. The wall is initially at a specified temperature. The temperature at the right boundary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary node 0 for the case of combined convection, radiation, and heat flux at the left boundary with an emissivity of , convection coefficient of h, ambient temperature of T∞, surrounding temperature of Tsurr, and uniform heat flux of q · 0 toward the wall. Also, obtain the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps.
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5–78 Starting with an energy balance on a volume element, obtain the two-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T(x, y, t) for the case of constant thermal conductivity and no heat generation
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5–79 Starting with an energy balance on a volume element, obtain the two-dimensional transient implicit finite difference equation for a general interior node in rectangular coordinates for T(x, y, t) for the case of constant thermal conductivity and no heat generation
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5–80 Starting with an energy balance on a disk volume element, derive the one-dimensional transient explicit finite difference equation for a general interior node for T(z, t) in a cylinder whose side surface is insulated for the case of constant thermal conductivity with uniform heat generation
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5–81 Consider one-dimensional transient heat conduction in a composite plane wall that consists of two layers A and B with perfect contact at the interface. The wall involves no heat generation and initially is at a specified temperature. The nodal network of the medium consists of nodes 0, 1 (at the interface), and 2 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 2) with an emissivity of E and surrounding temperature of Tsurr.
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5–82 Consider transient one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at T∞ with a heat transfer coefficient of h and by radiation to the surrounding surfaces at an average temperature of Tsurr. The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of a specified temperature at the fin base and negligible heat transfer at the fin tip
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5–83 Repeat Problem 5–82 for the case of implicit formulation.
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5–84 Consider a large uranium plate of thickness L = 8 cm, thermal conductivity k = 28 W/m · °C, and thermal diffusivity a=12.5 x 10-6 m2/s that is initially at a uniform temperature of 100°C. Heat is generated uniformly in the plate at a constant rate of g = 106 W/m3. At time t = 0, the left side of the plate is insulated while the other side is subjected to convection with an environment at T∞= 20°C with a heat transfer coefficient of h = 35 W/m2 · °C. Using the explicit finite difference approach with a uniform nodal spacing of Dx = 2 cm, determine (a) the temperature distribution in the plate after 5 min and (b) how long it will take for steady conditions to be reached in the plate.
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5–85 Reconsider Problem 5–84. Using EES (or other) software, investigate the effect of the cooling time on the temperatures of the left and right sides of the plate. Let the time vary from 5 min to 60 min. Plot the temperatures at the left and right surfaces as a function of time, and discuss the results.
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5–86 Consider a house whose south wall consists of a 30-cmthick Trombe wall whose thermal conductivity is k = 0.70 W/m · °C and whose thermal diffusivity is a=0.44 x 10-6 m2/s. The variations of the ambient temperature Tout and the solar heat flux q · solar incident on a south-facing vertical surface throughout the day for a typical day in February are given in the table in 3-h intervals. The Trombe wall has single glazing with an absorptivity-transmissivity product of K=0.76 (that is, 76 percent of the solar energy incident is absorbed by the exposed surface of the Trombe wall), and the average combined heat transfer coefficient for heat loss from the Trombe wall to the ambient is determined to be hout = 3.4 W/m2 · °C. The interior of the house is maintained at Tin = 20°C at all times, and the heat transfer coefficient at the interior surface of the Trombe wall is hin 0 9.1 W/m2 · °C. Also, the vents on the Trombe wall are kept closed, and thus the only heat transfer between the air in the house and the Trombe wall is through the
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5–87 Consider two-dimensional transient heat transfer in an L-shaped solid bar that is initially at a uniform temperature of 140°C and whose cross section is given in the figure. The thermal conductivity and diffusivity of the body are k = 15 W/m ·°C and a=3.2 x 10-6 m2/s, respectively, and heat is generated in the body at a rate of g =2 x 107 W/m3. The right surface of the body is insulated, and the bottom surface is maintained at a uniform temperature of 140°C at all times. At time t = 0, the entire top surface is subjected to convection with ambient air at T∞= 25°C with a heat transfer coefficient of h = 80 W/m2 · °C, and the left surface is subjected to uniform heat flux at a rate of q · L = 8000 W/m2. The nodal network of the problem consists of 13 equally spaced nodes with Dx = Dy = 1.5 cm. Using the explicit method, determine the temperature at the top corner (node 3) of the body after 2, 5, and 30 min.
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5–88 Reconsider Problem 5–87. Using EES (or other) software, plot the temperature at the top corner as a function of heating time varies from 2 min to 30 min, and discuss the results.
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5–89 Consider a long solid bar (k = 28 W/m · °C and a= 12 x 10"6 m2/s) of square cross section that is initially at a uniform temperature of 20°C. The cross section of the bar is 20 cm x 20 cm in size, and heat is generated in it uniformly at a rate of g = 8 x 105 W/m3. All four sides of the bar are subjected to convection to the ambient air at T∞= 30°C with a heat transfer coefficient of h = 45 W/m2 · °C. Using the explicit finite difference method with a mesh size of Dx =Dy = 10 cm, determine the centerline temperature of the bar (a) after 10 min and (b) after steady conditions are established.
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5–90E Consider a house whose windows are made of 0.375-in.-thick glass (k = 0.48 Btu/h · ft · °F and a=4.2 x 10-6 ft2/s). Initially, the entire house, including the walls and the windows, is at the outdoor temperature of To = 35°F. It is observed that the windows are fogged because the indoor temperature is below the dew-point temperature of 54°F. Now the heater is turned on and the air temperature in the house is raised to Ti = 72°F at a rate of 2°F rise per minute. The heat transfer coefficients at the inner and outer surfaces of the wall can be taken to be hi = 1.2 and ho = 2.6 Btu/h · ft2 · °F, respectively, and the outdoor temperature can be assumed to remain constant. Using the explicit finite difference method with a mesh size of Dx = 0.125 in., determine how long it will take for the fog on the windows to clear up (i.e., for the inner surface temperature of the window glass to reach 54°F)
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5–91 A common annoyance in cars in winter months is the formation of fog on the glass surfaces that blocks the view. Apractical way of solving this problem is to blow hot air or to attach electric resistance heaters to the inner surfaces. Consider the rear window of a car that consists of a 0.4-cm-thick glass (k = 0.84 W/m · °C and a=0.39 x 10-6 m2/s). Strip heater wires of negligible thickness are attached to the inner surface of the glass, 4 cm apart. Each wire generates heat at a rate of 10 W/m length. Initially the entire car, including its windows, is at the outdoor temperature of To = -3°C. The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be hi = 6 and ho = 20 W/m2 · °C, respectively. Using the explicit finite difference method with a mesh size of Dx = 0.2 cm along the thickness and Dy = 1 cm in the direction normal to the heater wires, determine the temperature distribution throughout the glass 15 min after the strip heaters are turned on. Also, determine the temperature distribution when steady conditions are reached.
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5–92 Repeat Problem 5–91 using the implicit method with a time step of 1 min
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5–93 The roof of a house consists of a 15-cm-thick concrete slab (k = 1.4 W/m · °C and a=0.69 x 10-6 m2/s) that is 20 m wide and 20 m long. One evening at 6 PM, the slab is observed to be at a uniform temperature of 18°C. The average ambient air and the night sky temperatures for the entire night are predicted to be 6°C and 260 K, respectively. The convection heat transfer coefficients at the inner and outer surfaces of the roof can be taken to be hi = 5 and ho = 12 W/m2 · °C, respectively. The house and the interior surfaces of the walls and the floor are maintained at a constant temperature of 20°C during the night, and the emissivity of both surfaces of the concrete roof is 0.9. Considering both radiation and convection heat transfers and using the explicit finite difference method with a time step of Dt = 5 min and a mesh size of Dx = 3 cm, determine the temperatures of the inner and outer surfaces of the roof at 6 AM. Also, determine the average rate of heat transfer through the roof during that night
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5–94 Consider a refrigerator whose outer dimensions are 1.80 m x 0.8 m x 0.7 m. The walls of the refrigerator are constructed of 3-cm-thick urethane insulation (k = 0.026 W/m ·°C and a=0.36 x 10-6 m2/s) sandwiched between two layers of sheet metal with negligible thickness. The refrigerated space is maintained at 3°C and the average heat transfer coefficients at the inner and outer surfaces of the wall are 6 W/m2 · °C and 9 W/m2 · °C, respectively. Heat transfer through the bottom surface of the refrigerator is negligible. The kitchen temperature remains constant at about 25°C. Initially, the refrigerator contains 15 kg of food items at an average specific heat of 3.6 kJ/kg · °C. Now a malfunction occurs and the refrigerator stops running for 6 h as a result. Assuming the temperature of the contents of the refrigerator, including the air inside, rises uniformly during this period, predict the temperature inside the refrigerator after 6 h when the repairman arrives. Use the explicit finite difference method with a time step of Dt = 1 min and a mesh size of Dx = 1 cm and disregard corner effects (i.e., assume one-dimensional heat transfer in the walls)
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5–95 Reconsider Problem 5–94. Using EES (or other) software, plot the temperature inside the refrigerator as a function of heating time as time varies from 1 h to 10 h, and discuss the results.
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5–96C Why do the results obtained using a numerical method differ from the exact results obtained analytically? What are the causes of this difference
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5–97C What is the cause of the discretization error? How does the global discretization error differ from the local discretization error
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5–98C Can the global (accumulated) discretization error be less than the local error during a step? Explain
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5–99C How is the finite difference formulation for the first derivative related to the Taylor series expansion of the solution function
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5–100C Explain why the local discretization error of the finite difference method is proportional to the square of the step size. Also explain why the global discretization error is proportional to the step size itself
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5–101C What causes the round-off error? What kind of calculations are most susceptible to round-off error
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5–102C What happens to the discretization and the roundoff errors as the step size is decreased
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5–103C Suggest some practical ways of reducing the round-off error
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5–104C What is a practical way of checking if the round-off error has been significant in calculations
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5–105C What is a practical way of checking if the discretization error has been significant in calculations?
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5–106 Starting with an energy balance on the volume element, obtain the steady three-dimensional finite difference equation for a general interior node in rectangular coordinates for T(x, y, z) for the case of constant thermal conductivity and uniform heat generation
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5–107 Starting with an energy balance on the volume element, obtain the three-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T(x, y, z, t) for the case of constant thermal conductivity and no heat generation
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5–108 Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1, 2, and 3 with a uniform nodal spacing of Dx. The temperature at the left boundary (node 0) is specified. Using the energy balance approach, obtain the finite difference formulation of boundary node 3 at the right boundary for the case of combined convection and radiation with an emissivity of , convection coefficient of h, ambient temperature of T∞, and surrounding temperature of Tsurr. Also, obtain the finite difference formulation for the rate of heat transfer at the left boundary.
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5–109 Consider one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0, 1, and 2 with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of specified heat flux q · 0 and convection at the left boundary (node 0) with a convection coefficient of h and ambient temperature of T∞, and radiation at the right boundary (node 2) with an emissivity of and surrounding temperature of Tsurr
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5–110 Repeat Problem 5–109 for the case of implicit formulation
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5–111 Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection with the ambient air at T∞ (in °C) with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of Tsurr (in K). The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Dx. Using the energy balance approach, obtain the finite difference formulation of this problem for the case of a specified temperature at the fin base and convection and radiation heat transfer at the fin tip
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5–112 Starting with an energy balance on the volume element, obtain the two-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T(x, y, t) for the case of constant thermal conductivity and uniform heat generation
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5–113 Starting with an energy balance on a disk volume element, derive the one-dimensional transient implicit finite difference equation for a general interior node for T(z, t) in a cylinder whose side surface is subjected to convection with a convection coefficient of h and an ambient temperature of T∞ for the case of constant thermal conductivity with uniform heat generation
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5–114E The roof of a house consists of a 5-in.-thick concrete slab (k = 0.81 Btu/h · ft · °F and a=7.4 x 10-6 ft2/s) that is 45 ft wide and 55 ft long. One evening at 6 PM, the slab is observed to be at a uniform temperature of 70°F. The ambient air temperature is predicted to be at about 50°F from 6 PM to 10 PM, 42°F from 10 PM to 2 AM, and 38°F from 2 AM to 6 AM, while the night sky temperature is expected to be about 445 R for the entire night. The convection heat transfer coefficients at the inner and outer surfaces of the roof can be taken to be hi = 0.9 and ho = 2.1 Btu/h · ft2 · °F, respectively. The house and the interior surfaces of the walls and the floor are maintained at a constant temperature of 70°F during the night, and the emissivity of both surfaces of the concrete roof is 0.9. Considering both radiation and convection heat transfers and using the explicit finite difference method with a mesh size of Dx = 1 in. and a time step of Dt = 5 min, determine the temperatures of the inner and outer surfaces of the roof at 6 AM. Also, determine the average rate of heat transfer through the roof during that night
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5–115 Solar radiation incident on a large body of clean water (k = 0.61 W/m · °C and a=0.15 x 10-6 m2/s) such as a lake, a river, or a pond is mostly absorbed by water, and the amount of absorption varies with depth. For solar radiation incident at a 45° angle on a 1-m-deep large pond whose bottom surface is black (zero reflectivity), for example, 2.8 percent of the solar energy is reflected back to the atmosphere, 37.9 percent is absorbed by the bottom surface, and the remaining 59.3 percent is absorbed by the water body. If the pond is considered to be four layers of equal thickness (0.25 m in this case), it can be shown that 47.3 percent of the incident solar energy is absorbed by the top layer, 6.1 percent by the upper mid layer, 3.6 percent by the lower mid layer, and 2.4 percent by the bottom layer [for more information see Çengel and Özi¸sik, Solar Energy, 33, no. 6 (1984), pp. 581–591]. The radiation absorbed by the water can be treated conveniently as heat generation in the heat transfer analysis of the pond. Consider a large 1-m-deep pond that is initially at a uniform temperature of 15°C throughout. Solar energy is incident on the pond surface at 45°at an average rate of 500 W/m2 for a period of 4 h. Assuming no convection currents in the water and using the explicit finite difference method with a mesh size of Dx = 0.25 m and a time step of Dt = 15 min, determine the temperature distribution in the pond under the most favorable conditions (i.e., no heat losses from the top or bottom surfaces of the pond). The solar energy absorbed by the bottom surface of the pond can be treated as a heat flux to the water at that surface in this case.
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5–116 Reconsider Problem 5–115. The absorption of solar radiation in that case can be expressed more accurately as a fourth-degree polynomial as
where qs is the solar flux incident on the surface of the pond in W/m2 and x is the distance from the free surface of the pond in m. Solve Problem 5–115 using this relation for the absorption of solar radiation
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5–117 A hot surface at 120°C is to be cooled by attaching 8 cm long, 0.8 cm in diameter aluminum pin fins (k = 237 W/m · °C and a=97.1 x 10-6 m2/s) to it with a center-tocenter distance of 1.6 cm. The temperature of the surrounding medium is 15°C, and the heat transfer coefficient on the surfaces is 35 W/m2 · °C. Initially, the fins are at a uniform temperature of 30°C, and at time t = 0, the temperature of the hot surface is raised to 120°C. Assuming one-dimensional heat conduction along the fin and taking the nodal spacing to be Dx = 2 cm and a time step to be At = 0.5 s, determine the nodal temperatures after 5 min by using the explicit finite difference method. Also, determine how long it will take for steady conditions to be reached.
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5–118E Consider a large plane wall of thickness L = 0.3 ft and thermal conductivity k ? 1.2 Btu/h · ft · °F in space. The wall is covered with a material having an emissivity of 0.80 and a solar absorptivity of as = 0.45. The inner surface of the wall is maintained at 520 R at all times, while the outer surface is exposed to solar radiation that is incident at a rate of qs = 300 Btu/h · ft2. The outer surface is also losing heat by radiation to deep space at 0 R. Using a uniform nodal spacing of Dx = 0.1 ft, (a) obtain the finite difference formulation for steady one-dimensional heat conduction and (b) determine the nodal temperatures by solving those equations.
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5–119 Frozen food items can be defrosted by simply leaving them on the counter, but it takes too long. The process can be speeded up considerably for flat items such as steaks by placing them on a large piece of highly conducting metal, called the defrosting plate, which serves as a fin. The increased surface area enhances heat transfer and thus reduces the defrosting time. Consider two 1.5-cm-thick frozen steaks at -18°C that resemble a 15-cm-diameter circular object when placed next to each other. The steaks are now placed on a 1-cm-thick blackanodized circular aluminum defrosting plate (k = 237 W/m ·°C, a=97.1 x 10-6 m2/s, and E= 0.90) whose outer diameter is 30 cm. The properties of the frozen steaks are P=970 kg/m3, Cp = 1.55 kJ/kg · °C, k = 1.40 W/m · °C, a=0.93 x 10-6 m2/s, and E= 0.95, and the heat of fusion is hif = 187 kJ/kg. The steaks can be considered to be defrosted when their average temperature is 0°C and all of the ice in the steaks is melted. Initially, the defrosting plate is at the room temperature of 20°C, and the wooden countertop it is placed on can be treated as insulation. Also, the surrounding surfaces can be taken to be at the same temperature as the ambient air, and the convection heat transfer coefficient for all exposed surfaces can be taken to be 12 W/m2 · °C. Heat transfer from the lateral surfaces of the steaks and the defrosting plate can be neglected. Assuming one-dimensional heat conduction in both the steaks and the defrosting plate and using the explicit finite difference method, determine how long it will take to defrost the steaks. Use four nodes with a nodal spacing of Dx = 0.5 cm for the steaks, and three nodes with a nodal spacing of Dr = 3.75 cm for the exposed portion of the defrosting plate. Also, use a time step of Dt = 5 s. Hint: First, determine the total amount of heat transfer needed to defrost the steaks, and then determine how long it will take to transfer that much heat.
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5–120 Repeat Problem 5–119 for a copper defrosting plate using a time step of Dt = 3 s.
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