Heat and Mass Transfer Assignment 6 1. Consider steady, parallel flow of atmospheric air over a flat plate. The air has a temperature
and free stream velocity of 300 K and 25 m/s. =1, 10 and 100 mm from the (a) Evaluate the boundary layer thickness at distance of x =1, leading edge. If a second plate were installed parallel to and at a distance of 3 mm from the first plate, what is the distance from the leading edge at which boundary layer merger would occur? (b) Evaluate the surface shear stress and the y-velocity component at the outer edge of the
boundary layer for the single plate at x = 1, 10 and 100 mm. (c) Comment on the validity of the boundary layer approximation. 0
2. A flat plate of width 1 m is maintained at a uniform surface temperature of T s=150 C by
using independently controlled, heat-generating rectangular modules of thickness a=10 mm and length b=50 mm. Each module is insulated from its neighbours, as well as on its 0
backside. Atmospheric air at 25 C flows over the plate at a velocity of 30 m/s. The 3
thermophysical properties of the module are k =5.2 =5.2 W/m.K, C p=320 J/kg.K and ρ=2300 kg/m . 3
(a) Find the required power generation q (W/m ), in a module positioned at a distance 700
mm from the leading edge. (b) Find the maximum temperature T max max in the heat-generating module.
3. The roof of a refrigerated truck compartment is of composite construction, consisting of a
layer of foamed urethane insulation ( t 2=50 mm, ki=0.026 W/m.K) sandwiched between aluminium alloy panels (t 1=5 mm, k p=180 W/m.K). The length and width of the roof are L=10 0
m and W =3.5 =3.5 m, respectively, and the temperature of the inner surface is T s,i s,i = -10 C. Consider conditions for which the truck is moving at a speed of V = 105 km/h, the air temperature is T ∞
32
=
0
2
C, and the solar irradiation is Gs= 750 W/m . Turbulent flow may be
assumed over the entire length of the roof. (a) For equivalent values of the solar absorptivity and the emmisivity of the outer surface
(αs=ε= 0.5), estimate the average temperature T s,o s,o of the outer surface. What is the corresponding heat load imposed on t he refrigeration system? (b) A special finish (αs=0.15, ε= 0.8) may be applied to the outer surface. What effect would
such an application have on the surface temperature and heat load? (c) If, with αs=ε= 0.5, the roof is not insulated ( t 2=0), what are the corresponding values of
surface temperature and heat load.
4. You have been asked to determine the feasibility of using an impinging jet in a soldering operation for electronic assemblies. The schematic illustrates the use of a single, round nozzle to direct high velocity, hot air to a location where a surface mount joint is to be formed.
For your study, consider a round nozzle with a diameter of 1 mm located a distance of 2 mm from the region of the surface mount, which has a diameter of 2.5 mm. (a)For an air jet velocity of 70 m/s and a temperature of 500°C, estimate the average convection coefficient over the area of the surface mount. (b)Assume that the surface mount region on the printed circuit board (PCB) can be modelled as a semi-infinite medium, which is initially at a uniform temperature of 25°C and suddenly experiences convective heating by the jet. Estimate the time required for the surface to reach 183°C. The thermo physical properties of a typical solder are . 5. The cylindrical chamber of a pebble bed nuclear reactor is of length and diameter . The chamber is filled with spherical uranium oxide pellets of core diameter . Each pellet generates thermal energy in its core at a rate of and is coated with a layer of non-heat –generating graphite, which is of uniform thickness , to form a pebble. The uranium oxide and graphite each have a thermal conductivity of . The packed bed has a porosity of . Pressurized helium at 40 bars is used to absorb the thermal energy from the pebbles. The helium enters the packed bed at with a velocity of 3.2 m/s. The properties of the helium may be assumed to be are
(a) For a desired overall thermal energy transfer rate of , determine the mean outlet temperature of the helium leaving the bed, and the amount of thermal energy generated by each pellet . (b)The amount of energy generated by the fuel decreases if a maximum operating temperature of approximately is exceeded. Determine the maximum internal temperature of the hottest pellet in the packed bed. For Reynolds numbers in the range , Equation 7.81 may be replaced by . 6. Latent heat capsules consist of a thin-walled spherical shell within which a solid-liquid, phase-change material (PCM) of melting point and latent heat of fusion is enclosed. As shown schematically, the capsules may be packed in a cylindrical vessel through which there is fluid flow. If the PCM is in its solid state and , heat is transferred from the fluid to the capsules and latent energy is stored in the PCM as it melts. Conversely, if the PCM is a liquid and , energy is released from the PCM as it freezes and heat is transferred to the fluid. In either situation, all of the capsules within the packed bed would remain at through much of the phase change process, in which case the fluid outlet temperature would remain at a fixed va lue .
7. In a paper mill drying process, a sheet of paper slurry (water –fibre mixture) has a linear velocity of 5 m/s as it is rolled. Radiant heaters maintain a sheet temperature of , as evaporation occurs to dry, ambient air at above and below the sheet.
(a) What is the evaporative flux at a distance of from the leading edge of the roll? What is the corresponding value of the radiant flux (irradiation, G) that must be supplied to the sheet to maintain its temperature at ? The sheet has an absorptivity of
8. Motile bacteria are equipped with flagella that are rotated by tiny, biological electrochemical engines which, in turn, propel the bacteria through a host liquid. Consider a nominally spherical Escherichia coli bacterium that is of diameter . The bacterium is in a water-based solution at containing a nutrient which is characterized by a binary diffusion coefficient of and a food energy value of . There is a nutrient density difference between the fluid and the shell of the bacterium of . Assuming a propulsion efficiency of , determine the maximum speed of the E. coli . Report your answer in body diameters per second.
9. In a home furnace humidification system, water droplets of diameter D are discharged in a direction opposing the motion of warm air emerging from the heater. The air is humidified by evaporation from the droplets, and the excess water is collected on a splash plate, from which it is routed to a drain.
Consider conditions for which air enters the heater at a temperature and relative humidity of respectively, and leaves the heater at a temperature of . The droplet diameter is and the relative velocity between the droplets and the heated air is . During the time-of-flight, the change in droplet diameter may be neglected and the droplet temperature may be assumed to remain at . What is the rate of evaporation from a single droplet? 10. To enhance heat transfer from a silicon chip of width on a side, a copper pin fin is brazed to the surface of the chip. The pin length and diameter are and , respectively, and atmospheric air at and is in cross flow over the pin. The surface of the chip, and hence the base of the pin, are maintained at a temperature of
(a) Assuming the chip to have a negligible effect on flow over the pin, what is the average convection coefficient for the surface of the pin? (b) Neglecting radiation and assuming the convection coefficient at the pin tip to equal that calculated in part (a), determine the pin heat transfer rate. (c) Neglecting radiation and assuming the convection coefficient at the exposed chip surface to equal that calculated in part (a), determine the total rate of heat transfer from the chip. 11. In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness and width cooled as it transits the distance L between two rollers at a velocity V . In this problem, we consider cooling of an aluminum alloy (2024-T6) by an air stream moving at a velocity in counter flow over the top surface of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface.
(a) By applying conservation of energy to a differential control surface of length dx , which either moves with the sheet or is stationary and through which the sheet passes, derive a differential equation that governs the temperature distribution along the sheet. Because of the low emissivity of the aluminum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, and properties of the sheet the local convection coefficient associated with the counter flow, and the air temperature. For a known temperature of the sheet ( ) at the onset of cooling and a negligible effect of the sheet velocity on boundary layer development, solve the equation to obtain an expression for the outlet temperature . (b) For , , , , , , and , what is the outlet temperature ?
12. Highly reflective aluminium coatings may be formed on the surface of a substrate by impacting the surface with molten drops of aluminium. The droplets are discharged from an injector, proceed through an inert gas (helium), and must still be in a molten state at the time of impact.
Consider conditions for which droplets with a diameter, velocity, and initial temperature of respectively, traverse a stagnant layer of atmospheric helium that is at a temperature of . What is the maximum allowable thickness of the helium layer needed to ensure that the temperature of droplets impacting the substrate is greater than or equal to the melting point of aluminum Properties of the molten aluminum may be approximated as ,
13. An air duct heater consists of an aligned array of electrical heating elements in which the longitudinal and transverse pitches are .There are 3 rows of elements in the flow direction ( ) and 4 elements per row . Atmospheric air with an upstream velocity of 12 m/s and a temperature of , moves in cross flow over the elements, which have a diameter of 12 mm, a length of 250 mm, and are maintained at a surface temperature of 350. (a) Determine the total heat transfer to the air and the temperature of the air leaving the duct heater.
(b) Determine the pressure drop across the element bank and the fan power requirement.
14. A cryogenic probe is used to treat cancerous skin tissue the probe consists of single round jet of diameter that issues from a nozzle concentrically situated within a larger, enclosed cylindrical tube of diameter . The wall thickness of the AISI302 stainless steel probe is and the separation distance between the nozzle and the inner surface of the probe is H=5mm. Assuming the cancerous tissue to be semi-infinite medium and far from the probe location, determine the surface temperature . Neglect the contact resistance between the probe and the tissue. Cold nitrogen exits the jet at =100K . (Hint= due to the probe walls the jet is confined and behaves as if it were one in an array such as in fi g.)
