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18.2 The fuel plates in a nuclear reactor are 4 ft long and stacked with a 1/2-in. gap between them. The heat flux along the plate surfaces varies sinusoidally according to the equation...where α = 250 Btu/h ft2, β = 1500 Btu/h ft2, x is the distance from the leading edge of the plates, and L is the total plate length. If air at 120°F, 80 psi, flowing at a mass velocity of 6000 lbm/h ft2, is used to cool the plates, prepare plots showing
a. the heat flux vs. x
b. the mean air temperature vs. x
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18.3 In a thermal heat sink the heat flux variation along the axis of a cooling passage is approximated as...where x is measured along the passage axis and L is its total length.A large installation involves a stack of plates with a 3-mm air space between them. The flow passages are 1.22 m long, and the heat flux in the plates varies according to the above equation where a = 900 W/m2 and v = 2500 W/m2. Air enters at 100°C with a mass velocity (the product of ρV) of 7.5 kg/s · m2. The surface coefficient along the flow passage can be considered constant with a value of 56W/m2 · K.Generate a plot of heat flux, mean air temperature, and plate surface temperature as functions of x. Where does the maximum surface temperature occur and what is its value?
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18.4 Glycerin flows parallel to a flat plate measuring 2 ft by 2 ft with a velocity of 10 fps. Determine values for the mean convective heat-transfer coefficient and the associated drag force imposed on the plate for glycerin temperatures of 350°F, 50°F, and 180°F. What heat flux will result, in each case, if the plate temperature is 50°F above that of the glycerin?
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18.5 Nitrogen at 100°F and 1 atm flows at a velocity of 100 fps. A flat plate 6 in. wide, at a temperature of 200°F, is aligned parallel to the direction of flow. At a position 4 ft from the leading edge, determine the following (a) δ; (b) δt; (c) Cfx; (d) Cfl; (e) hx; (f) h; (g) total drag force; (h) total heat transfer.
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18.6 A plane surface 25 cm wide has its temperature maintained at 80°C. Atmospheric air at 25°C flows parallel to the surface with a velocity of 2.8 m/s. Using the results of boundarylayer analysis, determine the following for a 1-m long plate:
a. the mean coefficient of skin friction, CfL
b. the total drag force exerted on the plate by the air flow
c. the total heat-transfer rate from the plate to the air stream
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18.7 An engineered tissue system consists of a flat plate of cell mass immobilized on a scaffold measuring 5 cm in length, and is 0.5 cm thick. The bottom face of the scaffold is exposed to water and organic nutrients. The top face is exposed to flowing O2 gas to provide O2 for aerobic respiration. At present the specific oxygen consumption of the tissue mass is 0.5 mmol O2/cm3 cells-hr, and from respiration energetics, the energy released by respiration is 468 J/mmol O2 consumed. We are interested in using the flowing O2 gas at 1 atm to control the temperature at the surface of the tissue scaffold. The properties of O2 gas at 300 K are ρ = 1.3 kg/m3, CP = 920 J/kg · K, μ = 2.06 × 10–5 kg/m sec, and k = 0.027 W/m · K. We are interested in determining the O2 flow rate necessary to keep the surface temperature within 10°C of the flowing gas temperature (i.e., surface temperature below 310 K or 37°C).
a. What is the Prandtl number (Pr) for the flowing fluid?
b. Based on the process energy balance and heat-transfer, what is the required heat transfer coefficient, h?
c. What is the Nusselt number, Nu?
d. The mean heat-transfer coefficient, h, is an integral averaged value obtained by integrating the local values of h(x) from x = 0 to x = L. If the O2 flow is assumed to be laminar (for flat plate, Re < 2 × 105), what is the needed convective heat-transfer correlation?
e. What are the required values for Reynolds number (Re) and fluid velocity, v∞?
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18.8 A 0.2-m-wide plate, 0.8 m long, is placed on the bottom of a shallow tank. The plate is heated, and maintained at a constant surface temperature of 60°C. Liquid water 12 cm deep flows over the flat plate with bulk volumetric flow rate of 2.4 10_3 m3/s. The bulk liquid water temperature may be assumed to remain constant at 20°C as it flows along the length of the plate.Thermophysical properties of water are given in the table below.
| Temperature (°C) | Density, ρ.kg/m3 | Viscosity, μ(kg/m-sec) | Heat Capacity, Cp (J/kg · K) | Thermal Conductivity, k (W/m•K) |
| 20 | 998.2 | 993 × 10–6 | 4282 | 0.597 |
| 40 | 992.2 | 658 × 10–6 | 4175 | 0.663 |
| 60 | 983.2 | 472 × 10–6 | 4181 | 0.658 |
a. What are the values of ReL and Pr for this convective heattransfer process? Is the flow laminar or turbulent at the end of the plate?
b. What is the local heat flux at a distance 0.5 m from the leading edge of the plate?
c. What is the total heat-transfer rate from the plate surface?
d. What are the thicknesses of the hydrodynamic and thermal boundary layers at the end of the plate?
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18.9 Simplified relations for natural convection in air are of the form...where α, β are constants; L is a significant length, in ft; DT is Ts T∞, in °F; and h is the convective heat-transfer coefficient, Btu/h ft2 °F. Determine the values for a and b for the plane vertical wall, using the equation from Problem 19.14.Problem 19.14An engineered tissue system consists of a flat plate of cell mass immobilized on a scaffold measuring 5 cm in length, and is 0.5 cm thick. The bottom face of the scaffold is exposed to water and organic nutrients. The top face is exposed to flowing O2 gas to provide O2 for aerobic respiration. At present the specific oxygen consumption of the tissue mass is 0.5 mmol O2/cm3 cells-hr, and from respiration energetics, the energy released by respiration is 468 J/mmol O2 consumed. We are interested in using the flowing O2 gas at 1 atm to control the temperature at the surface of the tissue scaffold. The properties of O2 gas at 300 K are ρ = 1.3 kg/m3, CP = 920 J/kg · K, μ = 2.06 × 10–5 kg/m sec, and k = 0.027 W/m · K. We are interested in determining the O2 flow rate necessary to keep the surface temperature within 10°C of the flowing gas temperature (i.e., surface temperature below 310 K or 37°C).
a. What is the Prandtl number (Pr) for the flowing fluid?
b. Based on the process energy balance and heat-transfer, what is the required heat transfer coefficient, h?
c. What is the Nusselt number, Nu?
d. The mean heat-transfer coefficient, h, is an integral averaged value obtained by integrating the local values of h(x) from x = 0 to x = L. If the O2 flow is assumed to be laminar (for flat plate, Re<2 × 105), what is the needed convective heat-transfer correlation?
e. What are the required values for Reynolds number (Re) and fluid velocity, v∞?
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18.10 Using the appropriate integral formulas for flow parallel to a flat surface with a constant free-stream velocity, develop expressions for the local Nusselt number in terms of Rex and Pr for velocity and temperature profiles of the form...
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18.11 Shown in the figure is the case of a fluid flowing parallel to a flat plate, where for a distance X from the leading edge, the plate and fluid are at the same temperature. For values of x > X, the plate is maintained at a constant temperature, Ts, where Ts > T∞. Assuming a cubic profile for both the hydrodynamic and the thermal boundary layers, show that the ratio of the thickness, x, is expressed as...Also show that the local Nusselt number can be expressed as...
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18.12 For the case of a turbulent boundary layer on a flat plate, the velocity profile has been shown to follow closely the form...Assuming a temperature profile of the same form—that is,...and assuming that δ = δ1, use the integral relation for the boundary layer to solve for hx and Nux. The temperature gradient at the surface may be considered similar to the velocity gradient at y = 0 given by equation (13-26).
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18.13 Water, at 60°F, enters a 1-in.-ID tube that is used to cool a nuclear reactor. The water flow rate is 30 gal/min. Determine the total heat transfer and the exiting water temperature for a 15-ft-long tube if the tube surface temperature is a constant value of 300°F. Compare the answer obtained using the Reynolds and Colburn analogies.
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18.14 Water at 60°F enters a 1-in.-ID tube that is used to cool a nuclear reactor. The water flow rate is 30 gal/min. Determine the total heat transfer, the exiting water temperature, and the wall temperature at the exit of a 15-ft long tube if the tube wall condition is one of uniform heat flux of 500 Btu/hr ft2.
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