Page 1 Chapter 2 Heat Conduction Equation 2-1 Chapter 2 HEAT CONDUCTION EQUATION Introduction 2-1C Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point. Temperature, on the other hand, is a scalar quantity. 2-2C The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is two- dimensional if heat tranfer in the third dimension is negligible. 2-3C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the drink will change with time during heating. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the bottom surface. 2-4C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the potato will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the potato. 2-5C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one- dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the egg will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the egg. Page 2 Chapter 2 Heat Conduction Equation 2-1 Chapter 2 HEAT CONDUCTION EQUATION Introduction 2-1C Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point. Temperature, on the other hand, is a scalar quantity. 2-2C The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is two- dimensional if heat tranfer in the third dimension is negligible. 2-3C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the drink will change with time during heating. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the bottom surface. 2-4C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the potato will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the potato. 2-5C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one- dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the egg will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the egg. Chapter 2 Heat Conduction Equation 2-2 2-6C Heat transfer to a hot dog can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the hot dog will change with time during cooking. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the hot dog. Heat transfer in a very long hot dog could be considered to be one-dimensional in preliminary calculations. 2-7C Heat transfer to a roast beef in an oven would be transient since the temperature at any point within the roast will change with time during cooking. Also, by approximating the roast as a spherical object, this heat transfer process can be modeled as one-dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction because of symmetry about the center point. 2-8C Heat loss from a hot water tank in a house to the surrounding medium can be considered to be a steady heat transfer problem. Also, it can be considered to be two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction.) 2-9C Yes, the heat flux vector at a point P on an isothermal surface of a medium has to be perpendicular to the surface at that point. 2-10C Isotropic materials have the same properties in all directions, and we do not need to be concerned about the variation of properties with direction for such materials. The properties of anisotropic materials such as the fibrous or composite materials, however, may change with direction. 2-11C In heat conduction analysis, the conversion of electrical, chemical, or nuclear energy into heat (or thermal) energy in solids is called heat generation. 2-12C The phrase “thermal energy generation” is equivalent to “heat generation,” and they are used interchangeably. They imply the conversion of some other form of energy into thermal energy. The phrase “energy generation,” however, is vague since the form of energy generated is not clear. 2-13 Heat transfer through the walls, door, and the top and bottom sections of an oven is transient in nature since the thermal conditions in the kitchen and the oven, in general, change with time. However, we would analyze this problem as a steady heat transfer problem under the worst anticipated conditions such as the highest temperature setting for the oven, and the anticipated lowest temperature in the kitchen (the so called “design” conditions). If the heating element of the oven is large enough to keep the oven at the desired temperature setting under the presumed worst conditions, then it is large enough to do so under all conditions by cycling on and off. Heat transfer from the oven is three-dimensional in nature since heat will be entering through all six sides of the oven. However, heat transfer through any wall or floor takes place in the direction normal to the surface, and thus it can be analyzed as being one-dimensional. Therefore, this problem can be simplified greatly by considering the heat transfer as being one- dimensional at each of the four sides as well as the top and bottom sections, and then by adding the calculated values of heat transfers at each surface. 2-14E The power consumed by the resistance wire of an iron is given. The heat generation and the heat flux are to be determined. Assumptions Heat is generated uniformly in the resistance wire. Analysis A 1000 W iron will convert electrical energy into heat in the wire at a rate of 1000 W. Therefore, the rate of heat generation in a resistance wire is simply equal to the power rating of a resistance heater. Then the rate of heat generation in the wire per unit volume is determined by dividing the total rate of heat generation by the volume of the wire to be q = 1000 W L = 15 in D = 0.08 in Page 3 Chapter 2 Heat Conduction Equation 2-1 Chapter 2 HEAT CONDUCTION EQUATION Introduction 2-1C Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point. Temperature, on the other hand, is a scalar quantity. 2-2C The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is two- dimensional if heat tranfer in the third dimension is negligible. 2-3C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the drink will change with time during heating. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the bottom surface. 2-4C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the potato will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the potato. 2-5C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one- dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the egg will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the egg. Chapter 2 Heat Conduction Equation 2-2 2-6C Heat transfer to a hot dog can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the hot dog will change with time during cooking. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the hot dog. Heat transfer in a very long hot dog could be considered to be one-dimensional in preliminary calculations. 2-7C Heat transfer to a roast beef in an oven would be transient since the temperature at any point within the roast will change with time during cooking. Also, by approximating the roast as a spherical object, this heat transfer process can be modeled as one-dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction because of symmetry about the center point. 2-8C Heat loss from a hot water tank in a house to the surrounding medium can be considered to be a steady heat transfer problem. Also, it can be considered to be two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction.) 2-9C Yes, the heat flux vector at a point P on an isothermal surface of a medium has to be perpendicular to the surface at that point. 2-10C Isotropic materials have the same properties in all directions, and we do not need to be concerned about the variation of properties with direction for such materials. The properties of anisotropic materials such as the fibrous or composite materials, however, may change with direction. 2-11C In heat conduction analysis, the conversion of electrical, chemical, or nuclear energy into heat (or thermal) energy in solids is called heat generation. 2-12C The phrase “thermal energy generation” is equivalent to “heat generation,” and they are used interchangeably. They imply the conversion of some other form of energy into thermal energy. The phrase “energy generation,” however, is vague since the form of energy generated is not clear. 2-13 Heat transfer through the walls, door, and the top and bottom sections of an oven is transient in nature since the thermal conditions in the kitchen and the oven, in general, change with time. However, we would analyze this problem as a steady heat transfer problem under the worst anticipated conditions such as the highest temperature setting for the oven, and the anticipated lowest temperature in the kitchen (the so called “design” conditions). If the heating element of the oven is large enough to keep the oven at the desired temperature setting under the presumed worst conditions, then it is large enough to do so under all conditions by cycling on and off. Heat transfer from the oven is three-dimensional in nature since heat will be entering through all six sides of the oven. However, heat transfer through any wall or floor takes place in the direction normal to the surface, and thus it can be analyzed as being one-dimensional. Therefore, this problem can be simplified greatly by considering the heat transfer as being one- dimensional at each of the four sides as well as the top and bottom sections, and then by adding the calculated values of heat transfers at each surface. 2-14E The power consumed by the resistance wire of an iron is given. The heat generation and the heat flux are to be determined. Assumptions Heat is generated uniformly in the resistance wire. Analysis A 1000 W iron will convert electrical energy into heat in the wire at a rate of 1000 W. Therefore, the rate of heat generation in a resistance wire is simply equal to the power rating of a resistance heater. Then the rate of heat generation in the wire per unit volume is determined by dividing the total rate of heat generation by the volume of the wire to be q = 1000 W L = 15 in D = 0.08 in Chapter 2 Heat Conduction Equation 2-3 3 7 ft Btu/h 10 7.820 ? ? ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ]( 4 / ft) 12 / 08 . 0 ( [ W 1000 ) 4 / ( 2 2 wire L D G V G g ? ? ? Similarly, heat flux on the outer surface of the wire as a result of this heat generation is determined by dividing the total rate of heat generation by the surface area of the wire to be 2 5 ft Btu/h 10 1.303 ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ( ft) 12 / 08 . 0 ( W 1000 wire ? ?DL G A G q ? ? ? Discussion Note that heat generation is expressed per unit volume in Btu/h ?ft 3 whereas heat flux is expressed per unit surface area in Btu/h ?ft 2 . Page 4 Chapter 2 Heat Conduction Equation 2-1 Chapter 2 HEAT CONDUCTION EQUATION Introduction 2-1C Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point. Temperature, on the other hand, is a scalar quantity. 2-2C The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is two- dimensional if heat tranfer in the third dimension is negligible. 2-3C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the drink will change with time during heating. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the bottom surface. 2-4C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the potato will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the potato. 2-5C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one- dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the egg will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the egg. Chapter 2 Heat Conduction Equation 2-2 2-6C Heat transfer to a hot dog can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the hot dog will change with time during cooking. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the hot dog. Heat transfer in a very long hot dog could be considered to be one-dimensional in preliminary calculations. 2-7C Heat transfer to a roast beef in an oven would be transient since the temperature at any point within the roast will change with time during cooking. Also, by approximating the roast as a spherical object, this heat transfer process can be modeled as one-dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction because of symmetry about the center point. 2-8C Heat loss from a hot water tank in a house to the surrounding medium can be considered to be a steady heat transfer problem. Also, it can be considered to be two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction.) 2-9C Yes, the heat flux vector at a point P on an isothermal surface of a medium has to be perpendicular to the surface at that point. 2-10C Isotropic materials have the same properties in all directions, and we do not need to be concerned about the variation of properties with direction for such materials. The properties of anisotropic materials such as the fibrous or composite materials, however, may change with direction. 2-11C In heat conduction analysis, the conversion of electrical, chemical, or nuclear energy into heat (or thermal) energy in solids is called heat generation. 2-12C The phrase “thermal energy generation” is equivalent to “heat generation,” and they are used interchangeably. They imply the conversion of some other form of energy into thermal energy. The phrase “energy generation,” however, is vague since the form of energy generated is not clear. 2-13 Heat transfer through the walls, door, and the top and bottom sections of an oven is transient in nature since the thermal conditions in the kitchen and the oven, in general, change with time. However, we would analyze this problem as a steady heat transfer problem under the worst anticipated conditions such as the highest temperature setting for the oven, and the anticipated lowest temperature in the kitchen (the so called “design” conditions). If the heating element of the oven is large enough to keep the oven at the desired temperature setting under the presumed worst conditions, then it is large enough to do so under all conditions by cycling on and off. Heat transfer from the oven is three-dimensional in nature since heat will be entering through all six sides of the oven. However, heat transfer through any wall or floor takes place in the direction normal to the surface, and thus it can be analyzed as being one-dimensional. Therefore, this problem can be simplified greatly by considering the heat transfer as being one- dimensional at each of the four sides as well as the top and bottom sections, and then by adding the calculated values of heat transfers at each surface. 2-14E The power consumed by the resistance wire of an iron is given. The heat generation and the heat flux are to be determined. Assumptions Heat is generated uniformly in the resistance wire. Analysis A 1000 W iron will convert electrical energy into heat in the wire at a rate of 1000 W. Therefore, the rate of heat generation in a resistance wire is simply equal to the power rating of a resistance heater. Then the rate of heat generation in the wire per unit volume is determined by dividing the total rate of heat generation by the volume of the wire to be q = 1000 W L = 15 in D = 0.08 in Chapter 2 Heat Conduction Equation 2-3 3 7 ft Btu/h 10 7.820 ? ? ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ]( 4 / ft) 12 / 08 . 0 ( [ W 1000 ) 4 / ( 2 2 wire L D G V G g ? ? ? Similarly, heat flux on the outer surface of the wire as a result of this heat generation is determined by dividing the total rate of heat generation by the surface area of the wire to be 2 5 ft Btu/h 10 1.303 ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ( ft) 12 / 08 . 0 ( W 1000 wire ? ?DL G A G q ? ? ? Discussion Note that heat generation is expressed per unit volume in Btu/h ?ft 3 whereas heat flux is expressed per unit surface area in Btu/h ?ft 2 . Chapter 2 Heat Conduction Equation 2-4 2-15E "GIVEN" E_dot=1000 "[W]" L=15 "[in]" "D=0.08 [in], parameter to be varied" "ANALYSIS" g_dot=E_dot/V_wire*Convert(W, Btu/h) V_wire=pi*D^2/4*L*Convert(in^3, ft^3) q_dot=E_dot/A_wire*Convert(W, Btu/h) A_wire=pi*D*L*Convert(in^2, ft^2) D [in] q [Btu/h.ft 2 ] 0.02 521370 0.04 260685 0.06 173790 0.08 130342 0.1 104274 0.12 86895 0.14 74481 0.16 65171 0.18 57930 0.2 52137 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 550000 D [in] q [Btu/h-ft 2 ] Page 5 Chapter 2 Heat Conduction Equation 2-1 Chapter 2 HEAT CONDUCTION EQUATION Introduction 2-1C Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point. Temperature, on the other hand, is a scalar quantity. 2-2C The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is two- dimensional if heat tranfer in the third dimension is negligible. 2-3C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the drink will change with time during heating. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the bottom surface. 2-4C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the potato will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the potato. 2-5C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one- dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about the center point. This would be a transient heat transfer process since the temperature at any point within the egg will change with time during cooking. Also, we would use the spherical coordinate system to solve this problem since the entire outer surface of a spherical body can be described by a constant value of the radius in spherical coordinates. We would place the origin at the center of the egg. Chapter 2 Heat Conduction Equation 2-2 2-6C Heat transfer to a hot dog can be modeled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the hot dog will change with time during cooking. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the center line, possibly at the center of the hot dog. Heat transfer in a very long hot dog could be considered to be one-dimensional in preliminary calculations. 2-7C Heat transfer to a roast beef in an oven would be transient since the temperature at any point within the roast will change with time during cooking. Also, by approximating the roast as a spherical object, this heat transfer process can be modeled as one-dimensional since temperature differences (and thus heat transfer) will primarily exist in the radial direction because of symmetry about the center point. 2-8C Heat loss from a hot water tank in a house to the surrounding medium can be considered to be a steady heat transfer problem. Also, it can be considered to be two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the azimuthal direction.) 2-9C Yes, the heat flux vector at a point P on an isothermal surface of a medium has to be perpendicular to the surface at that point. 2-10C Isotropic materials have the same properties in all directions, and we do not need to be concerned about the variation of properties with direction for such materials. The properties of anisotropic materials such as the fibrous or composite materials, however, may change with direction. 2-11C In heat conduction analysis, the conversion of electrical, chemical, or nuclear energy into heat (or thermal) energy in solids is called heat generation. 2-12C The phrase “thermal energy generation” is equivalent to “heat generation,” and they are used interchangeably. They imply the conversion of some other form of energy into thermal energy. The phrase “energy generation,” however, is vague since the form of energy generated is not clear. 2-13 Heat transfer through the walls, door, and the top and bottom sections of an oven is transient in nature since the thermal conditions in the kitchen and the oven, in general, change with time. However, we would analyze this problem as a steady heat transfer problem under the worst anticipated conditions such as the highest temperature setting for the oven, and the anticipated lowest temperature in the kitchen (the so called “design” conditions). If the heating element of the oven is large enough to keep the oven at the desired temperature setting under the presumed worst conditions, then it is large enough to do so under all conditions by cycling on and off. Heat transfer from the oven is three-dimensional in nature since heat will be entering through all six sides of the oven. However, heat transfer through any wall or floor takes place in the direction normal to the surface, and thus it can be analyzed as being one-dimensional. Therefore, this problem can be simplified greatly by considering the heat transfer as being one- dimensional at each of the four sides as well as the top and bottom sections, and then by adding the calculated values of heat transfers at each surface. 2-14E The power consumed by the resistance wire of an iron is given. The heat generation and the heat flux are to be determined. Assumptions Heat is generated uniformly in the resistance wire. Analysis A 1000 W iron will convert electrical energy into heat in the wire at a rate of 1000 W. Therefore, the rate of heat generation in a resistance wire is simply equal to the power rating of a resistance heater. Then the rate of heat generation in the wire per unit volume is determined by dividing the total rate of heat generation by the volume of the wire to be q = 1000 W L = 15 in D = 0.08 in Chapter 2 Heat Conduction Equation 2-3 3 7 ft Btu/h 10 7.820 ? ? ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ]( 4 / ft) 12 / 08 . 0 ( [ W 1000 ) 4 / ( 2 2 wire L D G V G g ? ? ? Similarly, heat flux on the outer surface of the wire as a result of this heat generation is determined by dividing the total rate of heat generation by the surface area of the wire to be 2 5 ft Btu/h 10 1.303 ? ? ? ? ? ? ? ? ? ? ? ? W 1 Btu/h 412 . 3 ft) 12 / 15 ( ft) 12 / 08 . 0 ( W 1000 wire ? ?DL G A G q ? ? ? Discussion Note that heat generation is expressed per unit volume in Btu/h ?ft 3 whereas heat flux is expressed per unit surface area in Btu/h ?ft 2 . Chapter 2 Heat Conduction Equation 2-4 2-15E "GIVEN" E_dot=1000 "[W]" L=15 "[in]" "D=0.08 [in], parameter to be varied" "ANALYSIS" g_dot=E_dot/V_wire*Convert(W, Btu/h) V_wire=pi*D^2/4*L*Convert(in^3, ft^3) q_dot=E_dot/A_wire*Convert(W, Btu/h) A_wire=pi*D*L*Convert(in^2, ft^2) D [in] q [Btu/h.ft 2 ] 0.02 521370 0.04 260685 0.06 173790 0.08 130342 0.1 104274 0.12 86895 0.14 74481 0.16 65171 0.18 57930 0.2 52137 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 550000 D [in] q [Btu/h-ft 2 ] Chapter 2 Heat Conduction Equation 2-5 2-16 The rate of heat generation per unit volume in the uranium rods is given. The total rate of heat generation in each rod is to be determined. Assumptions Heat is generated uniformly in the uranium rods. Analysis The total rate of heat generation in the rod is determined by multiplying the rate of heat generation per unit volume by the volume of the rod ? ? ?( / ) ( ) ( . / ]( G gV g D L ? ? ? ? ? ? rod 3 2 5 W / m [ m) m) 1.374 10 W = ? ? 2 7 4 7 10 0 05 4 1 137.4 kW 2-17 The variation of the absorption of solar energy in a solar pond with depth is given. A relation for the total rate of heat generation in a water layer at the top of the pond is to be determined. Assumptions Absorption of solar radiation by water is modeled as heat generation. Analysis The total rate of heat generation in a water layer of surface area A and thickness L at the top of the pond is determined by integration to be b ) e (1 g A bL 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? L bx L x bx V b e g A Adx e g dV g G 0 0 0 0 ) ( 2-18 The rate of heat generation per unit volume in a stainless steel plate is given. The heat flux on the surface of the plate is to be determined. Assumptions Heat is generated uniformly in steel plate. Analysis We consider a unit surface area of 1 m 2 . The total rate of heat generation in this section of the plate is ? ? ?( ) ( )( G gV g A L ? ? ? ? ? ? ? plate 3 2 5 W / m m )(0.03 m) 1.5 10 W 5 10 1 6 Noting that this heat will be dissipated from both sides of the plate, the heat flux on either surface of the plate becomes 2 W/m 75,000 ? ? ? ? ? 2 5 plate m 1 2 W 10 5 . 1 A G q ? ? Heat Conduction Equation 2-19 The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation is ? ? ? ? ? 2 2 1 T x g k T t ? ? ? . Here T is the temperature, x is the space variable, g is the heat generation per unit volume, k is the thermal conductivity, ? is the thermal diffusivity, and t is the time. 2-20 The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation is t T k g r T r r r ? ? ? ? ? ? ? 1 1 ? ? ? ? ? ? ? ? ? . Here T is the temperature, r is the space g = 7 ?10 7 W/m 3 L = 1 m D = 5 cm g LRead More

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