1 Conduction ConceptestsContributions by: Janet deGrazia, Stephanie Bryant Department of Chemical and Biological Engineering University of Colorado Boulder, CO

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2 If the thickness of a plane wall is doubled from x to 2x while maintaining the same temperatures on each side of the wall, the conductive heat transfer rate (q) ___________. q x 2x increases decreases stays the same ANSWER: B. Decreases The heat transfer rate is proportional to temperature gradient. Since the temperature change across the wall is constant, but the distance that change occurs is increasing, the temperature gradient decreases. The smaller the gradient, the slower the heat transfer rate.

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3 A cone frustum is conducting heat at steady state with no generationA cone frustum is conducting heat at steady state with no generation. If the frustum is insulated as shown, the heat flux in is ______________ the heat flux out. heat flux in heat flux out Insulation equal to greater than less than ANSWER: B. greater than Heat rate must be conserved, but not heat flux. Since the area of the heat out is greater than the area of the heat in, the heat flux out is less than the heat flux in.

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4 A block of wood and a block of steel are placed flush against each other on one side. If temperature 1 (T1) is greater than temperature 2 (T2) during steady state, what temperature profile would result? T1 T2 A.B. C.D. T1 T2 Key: wood steel ANSWER: D. Wood will have significantly more resistance to conductivity than steel so less heat will be transferred through the material, i.e. the ΔT across the wood will be greater than the ΔT across the steel. Note: Realistically, it would be incorrect to assume no contact resistance, but it is probably more important to emphasize the difference in material resistances than contact resistance. The phrase “flush against each other” can justify the negligibility of the contact resistance. T1 T2 T1 T2

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5 A block of glass and a block of steel are placed flush against each other on one side. If Temperature 1 (T1) is greater than Temperature 2 (T2) during steady state, what temperature profile would result? T1 T2 A.B. C.D. T1 T2 Key: glass steel ANSWER: D. Glass will have significantly more resistance to conductivity than steel so less heat will be transferred through the material, i.e. the ΔT across the glass will be greater than the ΔT across the steel. T1 T2 T1 T2

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6 Air is cooling the top and bottom of an electronic chip uniformly generating heat. At steady state, what temperature profile will result within the chip? A. B. C. D. None of the above X T Airflow X T ANSWER: B. The maximum temperature will be furthest away from the cooling. Since both the top and bottom are being cooled, the maximum will occur in the center of the chip. Answer a) shows more conductive heat transfer than convective. Answer c) assumes that the other side of the base is insulated, which is not indicated X T

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7 An electronic chip is uniformly generating heatAn electronic chip is uniformly generating heat. The top of the chip is being cooled by flowing air while the base is insulated. If the chip has a height of L, the maximum temperature is at height _______________. L L/2 2L insulation Air L ANSWER: A. 0 The maximum temperature will be the point furthest away from cooling. Since all the cooling occurs at the top of the chip, the base will be at the maximum temperature.

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8 A pipe carrying hot water experiences an external cross flow of cold air. Given steady state conditions and no heat generation, the heat flux in to the pipe wall is _______________ the heat flux out of the pipe wall. Hot Water Cold Air Heat flux in Heat flux out equal to greater than less than unrelated to ANSWER: B. greater than In a steady state radial system heat rate will be conserved, not heat flux. Since the area of the heat in is less than the area of the heat out, the heat flux in is greater than the heat flux out.

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9 A concrete wall is insulated on one side and exposed to ambient air at 25°C on the other. If the air is constant and there is no heat generation in the wall, after an extended period of time, which of the following shows the temperature distribution? A. B. C. D. x y 25°C Insulation Concrete wall ANSWER: D. The system will be at a constant temperature if there is no generation. After a long time, the wall will need to be at ambient temperature, regardless of whether it was hotter or colder initially. A non-constant temperature profile would indicate heat transfer.

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10 A concrete wall is insulated on one side and initially at a temperature of 25°C. Given no generation and an ambient temperature of 100°C, the temperature distribution will look like ____________. A B. C D. ANSWER: B. If the ambient temperature is greater than the surface temperature, the concrete wall will be gaining heat. The closer a point is to the surface, the quicker it will heat up. Since there is no generation, the distribution will be linear.

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11 A concrete wall is insulated on one side and initially at a temperature of 25°C. Given no generation and an ambient temperature of 0°C, the temperature distribution will look like ____________. A B. C D. ANSWER: A. If the ambient temperature is less than the surface temperature, the concrete wall is losing energy. The closer a point is to the surface, the quicker it will decrease in temperature. Since there is no generation, the distribution will be linear.

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12 Two windows are of equal overall depth, but one is single pane and the other a double pane with an air gap. If the interior temperature is greater than the exterior, the interior window surface temperature of the single pane will be __________ than for the double pane. Air gap lower than the same as higher than Interior Double pane Answer: A. Lower The air gap between the two panes of the double pane acts like an insulator. Convection and conduction rely on some medium to carry the heat and stagnant air is a poor conductor of heat when compared to glass. When the thermal energy from the interior contacts the single pane, that energy will be quickly carried to the exterior. With the double pane, there will be more resistance, less heat will transfer. Exterior Single pane

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13 A composite wall consists of wood and steelA composite wall consists of wood and steel. The bond between the wood and steel is not perfect, resulting in small air gaps at the interface. If temperature 1 (T1) is greater than temperature 2 (T2), at steady state what temperature distribution could result? T1 T2 T1 T2 T1 T2 A. B.C. D E. Answer: C The wood will have more resistance than the steel so the ΔT across the wood will be greater than that for steel; the slope of the temperature distribution will be greater. Since there is also contact resistance, there will be an associated temperature drop between the wood and the steel. T1 T2 T1 T2 wood steel

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14 A rectangular fin is dissipating heat, where the temperature of the base, Tbase, is greater than the ambient temperature, Tsurroundings. The maximum heat transfer from a fin occurs when the temperature at the tip of the fin is equal to ______________. Fin tip Fin base Tbase ½ Tbase T surroundings ½ Tsurroundings Answer: C. When the fin tip is at the ambient temperature, the temperature gradient across the fin, from the base to the tip, is greatest. This means more heat transfer along the length of the fin.

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15 In designing an effective fin to remove heat, which statement is correct?Thermal conductivity of the fin material should be as low as possible. Ratio of cross-sectional area to perimeter of the fin should be as high as possible. The length of the fin should be very long. Fins should be used when convective heat transfer coefficients are high. None of the above. ANSWER: E. None of the above Lowering the thermal conductivity of the fin material would decrease the conduction through the fin and decrease the heat transfer. The effectiveness would decrease. In heat transfer through fins, conductive heat transfer is moving through the cross-sectional area while convective heat transfer is moving through the perimeter. An effective fin maximizes the convective heat transfer relative to the conductive. For that to happen, the perimeter needs to be sufficiently large relative to the cross-sectional area, or the opposite of B. Increasing the length of a fin has diminishing returns so there are better approaches to increasing effectiveness. A high convective heat transfer coefficient indicates that heat is easily removed off a surface. In that sense, fins would be unnecessary. It is much better to use fins when convective heat transfer coefficients are low

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16 To approximate a fin as infinitely long, it must be at __________ of maximum heat transfer.100% 75% 99% 90% ANSWER: C. 99% This is a definition question largely. From a reasoning standpoint, an infinitely long fin should have maximum possible heat transfer so the answer cannot be A, or else it would not be an approximation. B would be rather low to approximate as maximum heat transfer. Designing a fin to be more than 90% maximum heat transfer should not be a problem.

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17 A nuclear fuel rod is uniformly generating heat inside a cylinderA nuclear fuel rod is uniformly generating heat inside a cylinder. If metal fins are added to the outer surface for cooling and the system is at steady state, which statement is correct? Heat flux at the outer surface will increase. Temperature at the center of the fuel will decrease Heat transfer rate will increase Both B and C are correct None of the above Generated Heat Removed Heat Answer: C. Heat transfer rate will increase In general, adding fins increases the heat transfer rate so statement C is correct. Statement A is incorrect because the heat transfer rate does not scale directly with surface area. Since adding fins will significantly increase surface area, the heat flux will actually decrease with fins. Statement B is incorrect since the heat generating source has not changed Added fin

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18 Two finned surfaces are geometrically identicalTwo finned surfaces are geometrically identical. Surface 1 has a convective heat transfer coefficient (h1) twice that of Surface 2 (h2). Surface 1 is __________ efficient and _________ effective than Surface 2. h1 = 2 h2 more, more more, less less, more less, less equally, equally ANSWER: D. less, less Mathematically, both fin efficiency and fin effectiveness are inversely proportional to the heat transfer coefficient. By definition, efficiency is the amount of heat transferred with fins divided by the maximum possible heat transfer with fins. An increased heat transfer coefficient would increase the actual heat transferred, but the maximum possible heat transfer would also increase. Without adjusting the fluid flow through the fins, the actual increase in heat transfer to the fluid cannot scale directly with the theoretical maximum heat transfer. Effectiveness is defined as the heat transferred with fins divided by the maximum possible heat transfer without fins. Again, the actual heat transferred cannot scale exactly with the maximum possible. Surface 1 Surface 2

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19 A warm stainless steel plate of uniform temperature is suddenly exposed to a quiescent fluid at room temperature. If heat loss is due to free convection, what could the temperature distribution be after a short period of time? B. C D. Answer: A. Free convection does not remove a significant amount of heat relative to other routes of heat transfer. Thus, the changes in temperature due to free convection will be less dramatic. Answer A has the least amount of change of the available figures. Figure B would only be seen if no cooling had occurred or after an extended period of time where everything reached steady state Figure C could be possible if the temperature difference between the plate and the fluid was large, but “warm” and “room temperature” imply a lesser temperature difference Figure D would be the profile if the plate were room temperature and the fluid was warm

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20 A hot sphere is placed in a cold water bathA hot sphere is placed in a cold water bath. The temperature at the center of the sphere will be close to the temperature at the surface of the sphere when the material has a _____________________. high thermal conductivity high density low heat capacity large surface area to volume ratio sphere Answer: A. high thermal conductivity This question is asking about the conditions of assuming lumped capacitance, when an object has a uniform temperature throughout. This is determined through the Biot number, which needs to be very small to assume lumped capacitance. The Biot number is h*Lc/k where h is coefficient for convective heat transfer, Lc is a characteristic length defined as volume/surface area, and k is thermal conductivity. To keep the Biot number small would require either a small convective heat transfer coefficient or characteristic length or large thermal conductivity. -Effects of density on heat transfer would depend on the material and, overall, have minimal effects on the heat transfer -Low heat capacity more dictates the total time for the sphere to reach bath temperature as opposed to the temperature profile during that time period -A large surface area to volume ratio would increase the characteristic length and increase the Biot number, which is the opposite of the desired temperature profile. bath

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21 Object A and object B are initially at the same temperature and suddenly placed in identical cooler environments. At the same instant in time, the two objects have the following temperature distributions. If the convective heat transfer coefficient is the same for both, object ___ has a higher Fourier number. B.C. Not enough information Answer: B The Fourier number is the ratio of heat conduction rate to the thermal energy storage rate. A high Fourier number would imply either a rapid rate of heat conduction or slow rate of thermal energy storage. Since there is no generation, the maximum temperature in either object would be expected to decrease over time. The object with a rapid rate of heat conduction would have a larger decrease in maximum temperature, in a given time period. The temperature distribution of B shows this.

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22 Consider two solid copper spheres, 1 and 2Consider two solid copper spheres, 1 and 2. Sphere 1 has twice the diameter of sphere 2. If both spheres are initially at 0 °C and placed in quiescent air at 25 °C, which sphere will reach air temperature first? Sphere 1 Sphere 2 Both spheres will reach temperature at the same time ANSWER: B. Sphere 2 Maximizing the heat transfer rate requires minimizing the volume to surface area ratio.


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For Sphere 1, doubling the diameter would significantly increase the volume with a lesser increase in surface area; volume is radius cubed while surface area is only squared, which would decrease the heat transfer rate.

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