John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 27
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Section 6.5).4.5Water vapor condenses on a cold pipe and drips off the bottomin regularly spaced nodes as sketched in Fig. 3.9. The wavelength of these nodes, λ, depends on the liquid-vapor densitydifference, ρf − ρg , the surface tension, σ , and the gravity, g.Find how λ varies with its dependent variables.4.6A thick film flows down a vertical wall. The local film velocityat any distance from the wall depends on that distance, gravity,the liquid kinematic viscosity, and the film thickness. Obtain184Chapter 4: Analysis of heat conduction and some steady one-dimensional problemsthe dimensionless functional equation for the local velocity (cf.Section 8.5).4.7A steam preheater consists of a thick, electrically conducting, cylindrical shell insulated on the outside, with wet streamflowing down the middle.
The inside heat transfer coefficientis highly variable, depending on the velocity, quality, and soon, but the flow temperature is constant. Heat is released atq̇ J/m3 s within the cylinder wall. Evaluate the temperaturewithin the cylinder as a function of position. Plot Θ againstρ, where Θ is an appropriate dimensionless temperature andρ = r /ro . Use ρi = 2/3 and note that Bi will be the parameterof a family of solutions. On the basis of this plot, recommendcriteria (in terms of Bi) for (a) replacing the convective boundary condition on the inside with a constant temperature condition; (b) neglecting temperature variations within the cylinder.4.8Steam condenses on the inside of a small pipe, keeping it ata specified temperature, Ti . The pipe is heated by electricalresistance at a rate q̇ W/m3 .
The outside temperature is T∞ andthere is a natural convection heat transfer coefficient, h aroundthe outside. (a) Derive an expression for the dimensionlessexpression temperature distribution, Θ = (T − T∞ )/(Ti − T∞ ),as a function of the radius ratios, ρ = r /ro and ρi = ri /ro ;a heat generation number, Γ = q̇ro2 /k(Ti − T∞ ); and the Biotnumber.
(b) Plot this result for the case ρi = 2/3, Bi = 1, andfor several values of Γ . (c) Discuss any interesting aspects ofyour result.4.9Solve Problem 2.5 if you have not already done so, puttingit in dimensionless form before you begin. Then let the Biotnumbers approach infinity in the solution.
You should get thesame solution we got in Example 2.5, using b.c.’s of the firstkind. Do you?4.10Complete the algebra that is missing between eqns. (4.30) andeqn. (4.31b) and eqn. (4.41).4.11Complete the algebra that is missing between eqns. (4.30) andeqn. (4.31a) and eqn.
(4.48).Problems1854.12Obtain eqn. (4.50) from the general solution for a fin [eqn. (4.35)],using the b.c.’s T (x = 0) = T0 and T (x = L) = T∞ . Commenton the significance of the computation.4.13What is the minimum length, l, of a thermometer well necessary to ensure an error less than 0.5% of the difference betweenthe pipe wall temperature and the temperature of fluid flowingin a pipe? The well consists of a tube with the end closed. Ithas a 2 cm O.D. and a 1.88 cm I.D. The material is type 304stainless steel.
Assume that the fluid is steam at 260◦ C andthat the heat transfer coefficient between the steam and thetube wall is 300 W/m2 K. [3.44 cm.]4.14Thin fins with a 0.002 m by 0.02 m rectangular cross sectionand a thermal conductivity of 50 W/m·K protrude from a walland have h 600 W/m2 K and T0 = 170◦ C. What is the heatflow rate into each fin and what is the effectiveness? T∞ =20◦ C.4.15A thin rod is anchored at a wall at T = T0 on one end and isinsulated at the other end. Plot the dimensionless temperaturedistribution in the rod as a function of dimensionless length:(a) if the rod is exposed to an environment at T∞ through aheat transfer coefficient; (b) if the rod is insulated but heat isremoved from the fin material at the unform rate −q̇ = hP (T0 −T∞ )/A.
Comment on the implications of the comparison.4.16A tube of outside diameter do and inside diameter di carriesfluid at T = T1 from one wall at temperature T1 to anotherwall a distance L away, at Tr . Outside the tube ho is negligible,and inside the tube hi is substantial. Treat the tube as a finand plot the dimensionless temperature distribution in it as afunction of dimensionless length.4.17(If you have had some applied mathematics beyond the usualtwo years of calculus, this problem will not be difficult.) Theshape of the fin in Fig. 4.12 is changed so that A(x) = 2δ(x/L)2 binstead of 2δ(x/L)b.
Calculate the temperature distributionand the heat flux at the base. Plot the temperature distributionand fin thickness against x/L. Derive an expression for ηf .186Chapter 4: Analysis of heat conduction and some steady one-dimensional problems4.18Work Problem 2.21, if you have not already done so, nondimensionalizing the problem before you attempt to solve it. Itshould now be much simpler.4.19One end of a copper rod 30 cm long is held at 200◦ C, and theother end is held at 93◦ C.
The heat transfer coefficient in between is 17 W/m2 K (including both convection and radiation).If T∞ = 38◦ C and the diameter of the rod is 1.25 cm, what isthe net heat removed by the air around the rod? [19.13 W.]4.20How much error will the insulated-tip assumption give rise toin the calculation of the heat flow into the fin in Example 4.8?4.21A straight cylindrical fin 0.6 cm in diameter and 6 cm longprotrudes from a magnesium block at 300◦ C. Air at 35◦ C isforced past the fin so that h is 130 W/m2 K.
Calculate the heatremoved by the fin, considering the temperature depression ofthe root.4.22Work Problem 4.19 considering the temperature depression inboth roots. To do this, find mL for the two fins with insulatedtips that would give the same temperature gradient at eachwall. Base the correction on these values of mL.4.23A fin of triangular axial section (cf.
Fig. 4.12) 0.1 m in lengthand 0.02 m wide at its base is used to extend the surface areaof a 0.5% carbon steel wall. If the wall is at 40◦ C and heatedgas flows past at 200◦ C (h = 230 W/m2 K), compute the heatremoved by the fin per meter of breadth, b, of the fin. Neglecttemperature distortion at the root.4.24Consider the concrete slab in Example 2.1. Suppose that theheat generation were to cease abruptly at time t = 0 and theslab were to start cooling back toward Tw . Predict T = Tw as afunction of time, noting that the initial parabolic temperatureprofile can be nicely approximated as a sine function. (Withoutthe sine approximation, this problem would require the seriesmethods of Chapter 5.)4.25Steam condenses in a 2 cm I.D.
thin-walled tube of 99% aluminum at 10 atm pressure. There are circular fins of constantthickness, 3.5 cm in diameter, every 0.5 cm on the outside. TheProblems187fins are 0.8 mm thick and the heat transfer coefficient fromthem h = 6 W/m2 K (including both convection and radiation).What is the mass rate of condensation if the pipe is 1.5 m inlength, the ambient temperature is 18◦ C, and h for condensation is very large? [ṁcond = 0.802 kg/hr.]4.26How long must a copper fin, 0.4 cm in diameter, be if the temperature of its insulated tip is to exceed the surrounding airtemperature by 20% of (T0 − T∞ )? Tair = 20◦ C and h = 28W/m2 K (including both convection and radiation).4.27A 2 cm ice cube sits on a shelf of widely spaced aluminumrods, 3 mm in diameter, in a refrigerator at 10◦ C.
How rapidly,in mm/min, do the rods melt their way through the ice cubeif h at the surface of the rods is 10 W/m2 K (including bothconvection and radiation). Be sure that you understand thephysical mechanism before you make the calculation. Checkyour result experimentally. hsf = 333, 300 J/kg.4.28The highest heat flux that can be achieved in nucleate boiling (called qmax —see the qualitative discussion in Section 9.1)depends upon ρg , the saturated vapor density; hfg , the latent heat vaporization; σ , the surface tension; a characteristiclength, l; and the gravity force per unit volume, g(ρf − ρg ),where ρf is the saturated liquid density.
Develop the dimensionless functional equation for qmax in terms of dimensionless length.4.29You want to rig a handle for a door in the wall of a furnace.The door is at 160◦ C. You consider bending a 40 cm lengthof 6.35 mm diam. 0.5% carbon steel rod into a U-shape andwelding the ends to the door. Surrounding air at 24◦ C willcool the handle (h = 12 W/m2 K including both convection andradiation). What is the coolest temperature of the handle? Howclose to the door can you grasp the handle without gettingburned if Tburn = 65◦ C? How might you improve the design?4.30A 14 cm long by 1 cm square brass rod is supplied with 25 W atits base. The other end is insulated.
It is cooled by air at 20◦ C,with h = 68 W/m2 K. Develop a dimensionless expression forΘ as a function of εf and other known information. Calculatethe base temperature.188Chapter 4: Analysis of heat conduction and some steady one-dimensional problems4.31A cylindrical fin has a constant imposed heat flux of q1 at oneend and q2 at the other end, and it is cooled convectively alongits length. Develop the dimensionless temperature distribution in the fin. Specialize this result for q2 = 0 and L → ∞, andcompare it with eqn.
(4.50).4.32A thin metal cylinder of radius ro serves as an electrical resistance heater. The temperature along an axial line in oneside is kept at T1 . Another line, θ2 radians away, is kept atT2 . Develop dimensionless expressions for the temperaturedistributions in the two sections.4.33Heat transfer is augmented, in a particular heat exchanger,with a field of 0.007 m diameter fins protruding 0.02 m into aflow. The fins are arranged in a hexagonal array, with a minimum spacing of 1.8 cm. The fins are bronze, and hf aroundthe fins is 168 W/m2 K.
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