Computational Fluid Mechanics and Heat Transfer Third Edition_6 doc

184 Chapter 4: Analysis of heat conduction and some steady one-dimensional problems the dimensionless functional equation for the local velocity (cf. Section 8.5). 4.7 A steam preheater consists of a thick, electrically conduct- ing, cylindrical shell insulated on the outside, with wet stream flowing down the middle. The inside heat transfer coefficient is highly variable, depending on the velocity, quality, and so on, but the flow temperature is constant. Heat is released at ˙ q J/m 3 s within the cylinder wall. Evaluate the temperature within the cylinder as a function of position. Plot Θ against ρ, where Θ is an appropriate dimensionless temperature and ρ = r/r o . Use ρ i = 2/3 and note that Bi will be the parameter of a family of solutions. On the basis of this plot, recommend criteria (in terms of Bi) for (a) replacing the convective bound- ary condition on the inside with a constant temperature condi- tion; (b) neglecting temperature variations within the cylinder. 4.8 Steam condenses on the inside of a small pipe, keeping it at a specified temperature, T i . The pipe is heated by electrical resistance at a rate ˙ q W/m 3 . The outside temperature is T ∞ and there is a natural convection heat transfer coefficient, h around the outside. (a) Derive an expression for the dimensionless expression temperature distribution, Θ = (T −T ∞ )/(T i −T ∞ ), as a function of the radius ratios, ρ = r/r o and ρ i = r i /r o ; a heat generation number, Γ = ˙ qr 2 o /k(T i − T ∞ ); and the Biot number. (b) Plot this result for the case ρ i = 2/3, Bi = 1, and for several values of Γ . (c) Discuss any interesting aspects of your result. 4.9 Solve Problem 2.5 if you have not already done so, putting it in dimensionless form before you begin. Then let the Biot numbers approach infinity in the solution. You should get the same solution we got in Example 2.5, using b.c.’s of the first kind. Do you? 4.10 Complete the algebra that is missing between eqns. (4.30) and eqn. (4.31b) and eqn. (4.41). 4.11 Complete the algebra that is missing between eqns. (4.30) and eqn. (4.31a) and eqn. (4.48). Problems 185 4.12 Obtain eqn. (4.50) from the general solution for a fin [eqn. (4.35)], using the b.c.’s T(x = 0) = T 0 and T(x = L) = T ∞ . Comment on the significance of the computation. 4.13 What is the minimum length, l, of a thermometer well neces- sary to ensure an error less than 0.5% of the difference between the pipe wall temperature and the temperature of fluid flowing in a pipe? The well consists of a tube with the end closed. It hasa2cmO.D. and a 1.88 cm I.D. The material is type 304 stainless steel. Assume that the fluid is steam at 260 ◦ C and that the heat transfer coefficient between the steam and the tube wall is 300 W/m 2 K. [3.44 cm.] 4.14 Thin fins with a 0.002 m by 0.02 m rectangular cross section and a thermal conductivity of 50 W/m·K protrude from a wall and have h  600 W/m 2 K and T 0 = 170 ◦ C. What is the heat flow rate into each fin and what is the effectiveness? T ∞ = 20 ◦ C. 4.15 A thin rod is anchored at a wall at T = T 0 on one end and is insulated at the other end. Plot the dimensionless temperature distribution in the rod as a function of dimensionless length: (a) if the rod is exposed to an environment at T ∞ through a heat transfer coefficient; (b) if the rod is insulated but heat is removed from the fin material at the unform rate − ˙ q = hP(T 0 − T ∞ )/A. Comment on the implications of the comparison. 4.16 A tube of outside diameter d o and inside diameter d i carries fluid at T = T 1 from one wall at temperature T 1 to another wall a distance L away, at T r . Outside the tube h o is negligible, and inside the tube h i is substantial. Treat the tube as a fin and plot the dimensionless temperature distribution in it as a function of dimensionless length. 4.17 (If you have had some applied mathematics beyond the usual two years of calculus, this problem will not be difficult.) The shape of the fin in Fig. 4.12 is changed so that A(x) = 2δ(x/L) 2 b instead of 2δ(x/L)b. Calculate the temperature distribution and the heat flux at the base. Plot the temperature distribution and fin thickness against x/L. Derive an expression for η f . 186 Chapter 4: Analysis of heat conduction and some steady one-dimensional problems 4.18 Work Problem 2.21, if you have not already done so, nondi- mensionalizing the problem before you attempt to solve it. It should now be much simpler. 4.19 One end of a copper rod 30 cm long is held at 200 ◦ C, and the other end is held at 93 ◦ C. The heat transfer coefficient in be- tween is 17 W/m 2 K (including both convection and radiation). If T ∞ =38 ◦ C and the diameter of the rod is 1.25 cm, what is the net heat removed by the air around the rod? [19.13 W.] 4.20 How much error will the insulated-tip assumption give rise to in the calculation of the heat flow into the fin in Example 4.8? 4.21 A straight cylindrical fin 0.6 cm in diameter and 6 cm long protrudes from a magnesium block at 300 ◦ C. Air at 35 ◦ Cis forced past the fin so that h is 130 W/m 2 K. Calculate the heat removed by the fin, considering the temperature depression of the root. 4.22 Work Problem 4.19 considering the temperature depression in both roots. To do this, find mL for the two fins with insulated tips that would give the same temperature gradient at each wall. Base the correction on these values of mL. 4.23 A fin of triangular axial section (cf. Fig. 4.12) 0.1 m in length and 0.02 m wide at its base is used to extend the surface area of a 0.5% carbon steel wall. If the wall is at 40 ◦ C and heated gas flows past at 200 ◦ C(h = 230 W/m 2 K), compute the heat removed by the fin per meter of breadth, b, of the fin. Neglect temperature distortion at the root. 4.24 Consider the concrete slab in Example 2.1. Suppose that the heat generation were to cease abruptly at time t = 0 and the slab were to start cooling back toward T w . Predict T = T w as a function of time, noting that the initial parabolic temperature profile can be nicely approximated as a sine function. (Without the sine approximation, this problem would require the series methods of Chapter 5.) 4.25 Steam condenses ina2cmI.D. thin-walled tube of 99% alu- minum at 10 atm pressure. There are circular fins of constant thickness, 3.5 cm in diameter, every 0.5 cm on the outside. The Problems 187 fins are 0.8 mm thick and the heat transfer coefficient from them h = 6 W/m 2 K (including both convection and radiation). What is the mass rate of condensation if the pipe is 1.5 m in length, the ambient temperature is 18 ◦ C, and h for condensa- tion is very large? [ ˙ m cond = 0.802 kg/hr.] 4.26 How long must a copper fin, 0.4 cm in diameter, be if the tem- perature of its insulated tip is to exceed the surrounding air temperature by 20% of (T 0 − T ∞ )? T air = 20 ◦ C and h = 28 W/m 2 K (including both convection and radiation). 4.27 A 2 cm ice cube sits on a shelf of aluminum rods, 3 mm in diam- eter, in a refrigerator at 10 ◦ C. How rapidly, in mm/min, does the ice cube melt through the wires if h at the surface of the wires is 10 W/m 2 K (including both convection and radiation). Be sure that you understand the physical mechanism before you make the calculation. Check your result experimentally. h sf = 333, 300 J/kg. 4.28 The highest heat flux that can be achieved in nucleate boil- ing (called q max —see the qualitative discussion in Section 9.1) depends upon ρ g , the saturated vapor density; h fg , the la- tent heat vaporization; σ , the surface tension; a characteristic length, l; and the gravity force per unit volume, g(ρ f − ρ g ), where ρ f is the saturated liquid density. Develop the dimen- sionless functional equation for q max in terms of dimension- less length. 4.29 You want to rig a handle for a door in the wall of a furnace. The door is at 160 ◦ C. You consider bending a 16 in. length of ¼ in. 0.5% carbon steel rod into a U-shape and welding the ends to the door. Surrounding air at 24 ◦ C will cool the handle (h = 12 W/m 2 K including both convection and radiation). What is the coolest temperature of the handle? How close to the door can you grasp it without being burned? How might you improve the handle? 4.30 A 14 cm long by 1 cm square brass rod is supplied with 25 W at its base. The other end is insulated. It is cooled by air at 20 ◦ C, with h = 68 W/m 2 K. Develop a dimensionless expression for Θ as a function of ε f and other known information. Calculate the base temperature. 188 Chapter 4: Analysis of heat conduction and some steady one-dimensional problems 4.31 A cylindrical fin has a constant imposed heat flux of q 1 at one end and q 2 at the other end, and it is cooled convectively along its length. Develop the dimensionless temperature distribu- tion in the fin. Specialize this result for q 2 = 0 and L →∞, and compare it with eqn. (4.50). 4.32 A thin metal cylinder of radius r o serves as an electrical re- sistance heater. The temperature along an axial line in one side is kept at T 1 . Another line, θ 2 radians away, is kept at T 2 . Develop dimensionless expressions for the temperature distributions in the two sections. 4.33 Heat transfer is augmented, in a particular heat exchanger, with a field of 0.007 m diameter fins protruding 0.02 m into a flow. The fins are arranged in a hexagonal array, with a mini- mum spacing of 1.8 cm. The fins are bronze, and h f around the fins is 168 W/m 2 K. On the wall itself, h w is only 54 W/m 2 K. Calculate h eff for the wall with its fins. (h eff = Q wall divided by A wall and [T wall −T ∞ ].) 4.34 Evaluate d(tanh x)/dx. 4.35 An engineer seeks to study the effect of temperature on the curing of concrete by controlling the temperature of curing in the following way. A sample slab of thickness L is subjected to a heat flux, q w , on one side, and it is cooled to temperature T 1 on the other. Derive a dimensionless expression for the steady temperature in the slab. Plot the expression and offer a criterion for neglecting the internal heat generation in the slab. 4.36 Develop the dimensionless temperature distribution in a spher- ical shell with the inside wall kept at one temperature and the outside wall at a second temperature. Reduce your solution to the limiting cases in which r outside  r inside and in which r outside is very close to r inside . Discuss these limits. 4.37 Does the temperature distribution during steady heat transfer in an object with b.c.’s of only the first kind depend on k? Explain. 4.38 A long, 0.005 m diameter duralumin rod is wrapped with an electrical resistor over 3 cm of its length. The resistor imparts Problems 189 a surface flux of 40 kW/m 2 . Evaluate the temperature of the rod in either side of the heated section if h = 150 W/m 2 K around the unheated rod, and T ambient = 27 ◦ C. 4.39 The heat transfer coefficient between a cool surface and a satu- rated vapor, when the vapor condenses in a film on the surface, depends on the liquid density and specific heat, the tempera- ture difference, the buoyant force per unit volume (g[ρ f −ρ g ]), the latent heat, the liquid conductivity and the kinematic vis- cosity, and the position (x) on the cooler. Develop the dimen- sionless functional equation for h. 4.40 A duralumin pipe through a cold room hasa4cmI.D. and a 5 cm O.D. It carries water that sometimes sits stationary. It is proposed to put electric heating rings around the pipe to protect it against freezing during cold periods of −7 ◦ C. The heat transfer coefficient outside the pipe is 9 W/m 2 K (including both convection and radiation). Neglect the presence of the water in the conduction calculation, and determine how far apart the heaters would have to be if they brought the pipe temperature to 40 ◦ C locally. How much heat do they require? 4.41 The specific entropy of an ideal gas depends on its specific heat at constant pressure, its temperature and pressure, the ideal gas constant and reference values of the temperature and pressure. Obtain the dimensionless functional equation for the specific entropy and compare it with the known equation. 4.42 A large freezer’s door has a 2.5 cm thick layer of insulation (k in = 0.04 W/m 2 K) covered on the inside, outside, and edges with a continuous aluminum skin 3.2 mm thick (k Al = 165 W/m 2 K). The door closes against a nonconducting seal 1 cm wide. Heat gain through the door can result from conduction straight through the insulation and skins (normal to the plane of the door) and from conduction in the aluminum skin only, going from the skin outside, around the edge skin, and to the inside skin. The heat transfer coefficients to the inside, h i , and outside, h o , are each 12 W/m 2 K, accounting for both con- vection and radiation. The temperature outside the freezer is 25 ◦ C, and the temperature inside is −15 ◦ C. a. If the door is 1 m wide, estimate the one-dimensional heat gain through the door, neglecting any conduction around 190 Chapter 4: Analysis of heat conduction and some steady one-dimensional problems the edges of the skin. Your answer will be in watts per meter of door height. b. Now estimate the heat gain by conduction around the edges of the door, assuming that the insulation is per- fectly adiabatic so that all heat flows through the skin. This answer will also be per meter of door height. 4.43 A thermocouple epoxied onto a high conductivity surface is in- tended to measure the surface temperature. The thermocou- ple consists of two each bare, 0.51 mm diameter wires. One wire is made of Chromel (Ni-10% Cr with k cr = 17 W/m·K) and the other of constantan (Ni-45% Cu with k cn = 23 W/m·K). The ends of the wires are welded together to create a measuring junction having has dimensions of D w by 2D w . The wires ex- tend perpendicularly away from the surface and do not touch one another. A layer of epoxy (k ep = 0.5 W/m·K separates the thermocouple junction from the surface by 0.2 mm. Air at 20 ◦ C surrounds the wires. The heat transfer coefficient be- tween each wire and the surroundings is h = 28 W/m 2 K, in- cluding both convection and radiation. If the thermocouple reads T tc = 40 ◦ C, estimate the actual temperature T s of the surface and suggest a better arrangement of the wires. 4.44 The resistor leads in Example 4.10 were assumed to be “in- finitely long” fins. What is the minimum length they each must have if they are to be modelled this way? What are the effec- tiveness, ε f , and efficiency, η f , of the wires? References [4.1] V. L. Streeter and E.B. Wylie. Fluid Mechanics. McGraw-Hill Book Company, New York, 7th edition, 1979. Chapter 4. [4.2] E. Buckingham. Phy. Rev., 4:345, 1914. [4.3] E. Buckingham. Model experiments and the forms of empirical equa- tions. Trans. ASME, 37:263–296, 1915. [4.4] Lord Rayleigh, John Wm. Strutt. The principle of similitude. Nature, 95:66–68, 1915. References 191 [4.5] J. O. Farlow, C. V. Thompson, and D. E. Rosner. Plates of the dinosaur stegosaurus: Forced convection heat loss fins? Science, 192(4244): 1123–1125 and cover, 1976. [4.6] D. K. Hennecke and E. M. Sparrow. Local heat sink on a convectively cooled surface—application to temperature measurement error. Int. J. Heat Mass Transfer, 13:287–304, 1970. [4.7] P. J. Schneider. Conduction Heat Transfer. Addison-Wesley Publish- ing Co., Inc., Reading, Mass., 1955. [4.8] A. D. Kraus, A. Aziz, and J.R. Welty. Extended Surface Heat Transfer. John Wiley & Sons, Inc., New York, 2001. 5. Transient and multidimensional heat conduction When I was a lad, winter was really cold. It would get so cold that if you went outside with a cup of hot coffee it would freeze. I mean it would freeze fast. That cup of hot coffee would freeze so fast that it would still be hot after it froze. Now that’s cold! Old North-woods tall-tale 5.1 Introduction James Watt, of course, did not invent the steam engine. What he did do was to eliminate a destructive transient heating and cooling process that wasted a great amount of energy. By 1763, the great puffing engines of Savery and Newcomen had been used for over half a century to pump the water out of Cornish mines and to do other tasks. In that year the young instrument maker, Watt, was called upon to renovate the Newcomen en- gine model at the University of Glasgow. The Glasgow engine was then being used as a demonstration in the course on natural philosophy. Watt did much more than just renovate the machine—he first recognized, and eventually eliminated, its major shortcoming. The cylinder of Newcomen’s engine was cold when steam entered it and nudged the piston outward. A great deal of steam was wastefully condensed on the cylinder walls until they were warm enough to accom- modate it. When the cylinder was filled, the steam valve was closed and jets of water were activated inside the cylinder to cool it again and con- dense the steam. This created a powerful vacuum, which sucked the piston back in on its working stroke. First, Watt tried to eliminate the wasteful initial condensation of steam by insulating the cylinder. But that simply reduced the vacuum and cut the power of the working stroke. 193 [...]... following b.c.’s and i.c.: T (−L, t > 0) = T (L, t > 0) = T1 and T (x, t = 0) = Ti (5.25) In fully dimensionless form, eqn (5.24) and eqn (5.25) are ∂Θ ∂2Θ = 2 ∂ξ ∂Fo (5.26) 204 Transient and multidimensional heat conduction §5.3 and Θ(0, Fo) = Θ(2, Fo) = 0 and Θ(ξ, 0) = 1 (5.27) where we have nondimensionalized the problem in accordance with eqn (5.4), using Θ ≡ (T − T1 )/(Ti − T1 ) and Fo ≡ αt/L2 ;... ) , (Ti − T1 ) ξ≡ x , L and Bi ≡ hL , k and we write Θ = fn (ξ, Bi, Π4 ) (5.1) One possible candidate for Π4 , which is independent of the other three, is Π4 ≡ Fo = αt/L2 (5.2) where Fo is the Fourier number Another candidate that we use later is x Π4 ≡ ζ = √ αt ξ this is exactly √ Fo (5.3) If the problem involved only b.c.’s of the first kind, the heat transfer coefficient, h and hence the Biot number—would... for hundreds more problems, and any reader who is faced with a complex heat conduction calculation should consult the literature before trying to solve it An excellent place to begin is Carslaw and Jaeger’s comprehensive treatise on heat conduction [5.6] Example 5.3 A 1 mm diameter Nichrome (20% Ni, 80% Cr) wire is simultaneously being used as an electric resistance heater and as a resistance thermometer... liquid flow The laboratory workers who operate it are attempting to measure the boiling heat transfer coefficient, h, by supplying an alternating current and measuring the difference between the average temperature of the heater, Tav , and the liquid temperature, T∞ They get h = 30, 000 W/m2 K at a wire temperature of 100◦ C and are delighted with such a high value Then a colleague suggests that h is so high... Switzer and Lienhard (see, e.g [5.8]), who gave response curves in the form Tmax − Tav = fn (Bi, ψ) Tav − T∞ (5.41) where the left-hand side is the dimensionless range of the temperature oscillation, and ψ = ωδ2 /α, where δ is a characteristic length [see Problem 5.56] Because this problem is common and the solution is not widely available, we include the curves for flat plates and cylinders in Fig 5.11 and. .. 5.10 The heat removal from suddenly-cooled bodies as a function of h and time 213 214 Transient and multidimensional heat conduction §5.4 Solution After 1 hr, or 3600 s: Fo = αt 2 = ro k ρc = 20◦ C 3600 s (0.05 m)2 (0.603 J/m·s·K)(3600 s) (997.6 kg/m3 )(4180 J/kg·K)(0.0025 m2 ) = 0.208 Furthermore, Bi−1 = (hro /k)−1 = [6(0.05)/0.603]−1 = 2.01 Therefore, we read from Fig 5.9 in the upper left-hand corner:... which two lumped-thermal-capacity systems are connected in series Such an arrangement is shown in Fig 5.5 Heat is transferred through two slabs with an interfacial resistance, h−1 between c them We shall require that hc L1 /k1 , hc L2 /k2 , and hL2 /k2 are all much 200 Transient and multidimensional heat conduction §5.2 Figure 5.4 Response of a thermometer to a linearly increasing ambient temperature... transient heat transfer processes are a dominant concern in the design of food storage units We therefore turn our attention, first, to an analysis of unsteady heat transfer, beginning with a more detailed consideration of the lumpedcapacity system that we looked at in Section 1.3 5.2 Lumped-capacity solutions We begin by looking briefly at the dimensional analysis of transient conduction in general and of... of λn — and thus of Bi — for ˆ ˆ slabs, cylinders, and spheres (e.g., for a slab Dn = An sin λn λn ) These functions can be used to plot Φ(Fo, Bi) once and for all Such curves are given in Fig 5.10 The quantity Φ has a close relationship to the mean temperature of a body at any time, T (t) Specifically, the energy lost as heat by time t determines the difference between the initial temperature and the... Temperature deviation at the surface of a flat plate heated with alternating current 217 Figure 5.12 Temperature deviation at the surface of a cylinder heated with alternating current 218 Transient and multidimensional heat conduction §5.5 In the present case: 30, 000(0.0005) h radius = = 1.09 k 13.8 [2π (60)](0.0005)2 ωr 2 = 27.5 = 0.00000343 α Bi = and from the chart for cylinders, Fig 5.12, we find . Thompson, and D. E. Rosner. Plates of the dinosaur stegosaurus: Forced convection heat loss fins? Science, 192(4244): 1123–1125 and cover, 19 76. [4 .6] D. K. Hennecke and E. M. Sparrow. Local heat sink. wall and have h  60 0 W/m 2 K and T 0 = 170 ◦ C. What is the heat flow rate into each fin and what is the effectiveness? T ∞ = 20 ◦ C. 4.15 A thin rod is anchored at a wall at T = T 0 on one end and. distribution and the heat flux at the base. Plot the temperature distribution and fin thickness against x/L. Derive an expression for η f . 1 86 Chapter 4: Analysis of heat conduction and some steady