John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 48
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on an isothermal, flat heater is much larger than δt . The small part of thevelocity b.l. inside the thermal b.l. is approximately u/u∞ =332 y/δ = 2 φ(y/δt ). Derive Nux for this case based on thisvelocity profile.6.22Plot the ratio of h(x)laminar to h(x)turbulent against Rex in therange of Rex that might be either laminar or turbulent. Whatdoes the plot suggest about heat transfer design?6.23Water at 7◦ C flows at 0.38 m/s across the top of a 0.207 m-long,thin copper plate. Methanol at 87◦ C flows across the bottom ofthe same plate, at the same speed but in the opposite direction.Make the obvious first guess as to the temperature at which toevaluate physical properties. Then plot the plate temperatureas a function of position.
(Do not bother to correct the physicalproperties in this problem, but note Problem 6.24.)6.24Work Problem 6.23 taking full account of property variations.6.25If the wall temperature in Example 6.6 (with a uniform qw =420 W/m2 ) were instead fixed at its average value of 76◦ C, whatwould the average wall heat flux be?6.26A cold, 20 mph westerly wind at 20◦ F cools a rectangular building, 35 ft by 35 ft by 22 ft high, with a flat roof. The outer wallsare at 27◦ F. Find the heat loss, conservatively assuming thatthe east and west faces have the same h as the north, south,and top faces.
Estimate U for the walls.6.27A 2 ft-square slab of mild steel leaves a forging operation0.25 in. thick at 1000◦ C. It is laid flat on an insulating bed and27◦ C air is blown over it at 30 m/s. How long will it take to coolto 200◦ C. (State your assumptions about property evaluation.)6.28Do Problem 6.27 numerically, recalculating properties at successive points. If you did Problem 6.27, compare results.6.29Plot Tw against x for the situation described in Example 6.10.6.30Consider the plate in Example 6.10. Suppose that instead ofspecifying qw = 1000 W/m2 , we specified Tw = 200◦ C.
Plotqw against x for this case.Problems3356.31A thin metal sheet separates air at 44◦ C, flowing at 48 m/s,from water at 4◦ C, flowing at 0.2 m/s. Both fluids start at aleading edge and move in the same direction. Plot Tplate and qas a function of x up to x = 0.1 m.6.32A mixture of 60% glycerin and 40% water flows over a 1-mlong flat plate. The glycerin is at 20◦ C and the plate is at 40◦ .A thermocouple 1 mm above the trailing edge records 35◦ C.What is u∞ , and what is u at the thermocouple?6.33What is the maximum h that can be achieved in laminar flowover a 5 m plate, based on data from Table A.3? What physicalcircumstances give this result?6.34A 17◦ C sheet of water, ∆1 m thick and moving at a constantspeed u∞ m/s, impacts a horizontal plate at 45◦ , turns, andflows along it.
Develop a dimensionless equation for the thickness ∆2 at a distance L from the point of impact. Assume thatδ ∆2 . Evaluate the result for u∞ = 1 m/s, ∆1 = 0.01 m, andL = 0.1 m, in water at 27◦ C.6.35A good approximation to the temperature dependence of µ ingases is given by the Sutherland formula:µ=µrefTTref1.5Tref + S,T +Swhere the reference state can be chosen anywhere.
Use datafor air at two points to evaluate S for air. Use this value topredict a third point. (T and Tref are expressed in kelvin.)6.36We have derived a steady-state continuity equation in Section 6.3.Now derive the time-dependent, compressible, three-dimensionalversion of the equation:∂ρ =0+ ∇ · (ρ u)∂tTo do this, paraphrase the development of equation (2.10), requiring that mass be conserved instead of energy.6.37Various considerations show that the smallest-scale motionsin a turbulent flow have no preferred spatial orientation atChapter 6: Laminar and turbulent boundary layers336large enough values of Re. Moreover, these small eddies areresponsible for most of the viscous dissipation of kinetic energy.
The dissipation rate, ε (W/kg), may be regarded as giveninformation about the small-scale motion, since it is set by thelarger-scale motion. Both ε and ν are governing parameters ofthe small-scale motion.a. Find the characteristic length and velocity scales of thesmall-scale motion. These are called the Kolmogorov scalesof the flow.b. Compute Re for the small-scale motion and interpret theresult.c.
The Kolmogorov length scale characterizes the smallestmotions found in a turbulent flow. If ε is 10 W/kg andthe mean free path is 7 × 10−8 m, show that turbulentmotion is a continuum phenomenon and thus is properlygoverned by the equations of this chapter.6.38The temperature outside is 35◦ F, but with the wind chill it’s−15◦ F. And you forgot your hat. If you go outdoors for long,are you in danger of freezing your ears?6.39To heat the airflow in a wind tunnel, an experimenter uses anarray of electrically heated, horizontal Nichrome V strips.
Thestrips are perpendicular to the flow. They are 20 cm long, verythin, 2.54 cm wide (in the flow direction), with the flat sidesparallel to the flow. They are spaced vertically, each 1 cm abovethe next. Air at 1 atm and 20◦ C passes over them at 10 m/s.a. How much power must each strip deliver to raise the meantemperature of the airstream to 30◦ C?b. What is the heat flux if the electrical heating in the stripsis uniformly distributed?c.
What are the average and maximum temperatures of thestrips?6.40An airflow sensor consists of a 5 cm long, heated copper slugthat is smoothly embedded 10 cm from the leading edge ofa flat plate. The overall length of the plate is 15 cm, and thewidth of the plate and the slug are both 10 cm. The slug iselectrically heated by an internal heating element, but, owingProblems337to its high thermal conductivity, the slug has an essentiallyuniform temperature along its airside surface.
The heater’scontroller adjusts its power to keep the slug surface at a fixedtemperature. The air velocity is found from measurementsof the slug temperature, the air temperature, and the heatingpower needed to hold the slug at the set temperature.a. If the air is at 280 K, the slug is at 300 K, and the heaterpower is 5.0 W, find the airspeed assuming the flow islaminar. Hint: For x1 /x0 = 1.5 x1x0−1/3√x −1/2 1 − (x0 /x)3/4dx = 1.0035 x0b. Suppose that a disturbance trips the boundary layer nearthe leading edge, causing it to become turbulent over thewhole plate.
The air speed, air temperature, and the slug’sset-point temperature remain the same. Make a very roughestimate of the heater power that the controller now delivers, without doing a lot of analysis.6.41Equation (6.64) gives Nux for a flat plate with an unheatedstarting length. This equation may be derived using the integral energy equation [eqn. (6.47)], modelling the velocity andtemperature profiles with eqns. (6.29) and (6.50), respectively,and taking δ(x) from eqn. (6.31a). Equation (6.52) is again obtained; however, in this case, φ = δt /δ is a function of x forx > x0 .
Derive eqn. (6.64) by starting with eqn. (6.52), neglecting the term 3φ3 /280, and replacing δt by φδ. After somemanipulation, you will obtainx134 d 3φ + φ3 =3 dx14 PrShow that its solution isφ3 = Cx −3/4 +1314 Prfor an unknown constant C. Then apply an appropriate initialcondition and the definition of qw and Nux to obtain eqn. (6.64).Chapter 6: Laminar and turbulent boundary layers338References[6.1] S. Juhasz.
Notes on Applied Mechanics Reviews – ReferativnyiZhurnal Mekhanika exhibit at XIII IUTAM, Moscow 1972. Appl.Mech. Rev., 26(2):145–160, 1973.[6.2] F.M. White. Viscous Fluid Flow. McGraw-Hill, Inc., New York, 2ndedition, 1991.[6.3] H. Schlichting. Boundary-Layer Theory. (trans. J. Kestin). McGrawHill Book Company, New York, 6th edition, 1968.[6.4] C. L. Tien and J. H. Lienhard. Statistical Thermodynamics.
Hemisphere Publishing Corp., Washington, D.C., rev. edition, 1978.[6.5] S. W. Churchill and H. Ozoe. Correlations for laminar forced convection in flow over an isothermal flat plate and in developing andfully developed flow in an isothermal tube. J. Heat Trans., Trans.ASME, Ser. C, 95:78, 1973.[6.6] O. Reynolds. On the extent and action of the heating surface forsteam boilers. Proc.
Manchester Lit. Phil. Soc., 14:7–12, 1874.[6.7] J.A. Schetz. Foundations of Boundary Layer Theory for Momentum,Heat, and Mass Transfer. Prentice-Hall, Inc., Englewood Cliffs, NJ,1984.[6.8] P. S. Granville. A modified Van Driest formula for the mixing lengthof turbulent boundary layers in pressure gradients. J. Fluids Engr.,111(1):94–97, 1989.[6.9] P. S. Granville. A near-wall eddy viscosity formula for turbulentboundary layers in pressure gradients suitable for momentum,heat, or mass transfer. J. Fluids Engr., 112(2):240–243, 1990.[6.10] F. M. White. A new integral method for analyzing the turbulentboundary layer with arbitrary pressure gradient. J.
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