John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 15
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It has an 8 cm diameter cavity containing boilingwater (hinside is very high) which is vented to the atmosphere.What is Q through the shell?2.19A slab is insulated on one side and exposed to a surrounding temperature, T∞ , through a heat transfer coefficient on theother. There is nonuniform heat generation in the slab suchthat q̇ =[A (W/m4 )][x (m)], where x = 0 at the insulated walland x = L at the cooled wall. Derive the temperature distribution in the slab.2.20800 W/m3 of heat is generated within a 10 cm diameter nickelsteel sphere for which k = 10 W/m·K. The environment is at20◦ C and there is a natural convection heat transfer coefficientof 10 W/m2 K around the outside of the sphere. What is itscenter temperature at the steady state? [21.37◦ C.]2.21An outside pipe is insulated and we measure its temperaturewith a thermocouple.
The pipe serves as an electrical resistance heater, and q̇ is known from resistance and current mea-Problems91surements. The inside of the pipe is cooled by the flow of liquid with a known bulk temperature. Evaluate the heat transfercoefficient, h, in terms of known information. The pipe dimensions and properties are known. [Hint: Remember that h is notknown and we cannot use a boundary condition of the thirdkind at the inner wall to get T (r ).]2.22Consider the hot water heater in Problem 1.11. Suppose that itis insulated with 2 cm of a material for which k = 0.12 W/m·K,and suppose that h = 16 W/m2 K.
Find (a) the time constantT for the tank, neglecting the casing and insulation; (b) theinitial rate of cooling in ◦ C/h; (c) the time required for the waterto cool from its initial temperature of 75◦ C to 40◦ C; (d) thepercentage of additional heat loss that would result if an outercasing for the insulation were held on by eight steel rods, 1 cmin diameter, between the inner and outer casings.2.23A slab of thickness L is subjected to a constant heat flux, q1 , onthe left side. The right-hand side if cooled convectively by anenvironment at T∞ .
(a) Develop a dimensionless equation forthe temperature of the slab. (b) Present dimensionless equation for the left- and right-hand wall temperatures as well. (c)If the wall is firebrick, 10 cm thick, q1 is 400 W/m2 , h = 20W/m2 K, and T∞ = 20◦ C, compute the lefthand and righthandtemperatures.2.24Heat flows steadily through a stainless steel wall of thicknessLss = 0.06 m, with a variable thermal conductivity of kss = 1.67 +0.0143 T(◦ C).
It is partially insulated on the right side with glasswool of thickness Lgw = 0.1 m, with a thermal conductivityof kgw = 0.04. The temperature on the left-hand side of thestainless stell is 400◦ Cand on the right-hand side if the glasswool is 100◦ C. Evaluate q and Ti .2.25Rework Problem 1.29 with a heat transfer coefficient, ho = 40W/m2 K on the outside (i.e., on the cold side).2.26A scientist proposes an experiment for the space shuttle inwhich he provides underwater illumination in a large tank ofwater at 20◦ C, using a 3 cm diameter spherical light bulb. Whatis the maximum wattage of the bulb in zero gravity that willnot boil the water?92Chapter 2: Heat conduction, thermal resistance, and the overall heat transfer coefficient2.27A cylindrical shell is made of two layers– an inner one withinner radius = ri and outer radius = rc and an outer one withinner radius = rc and outer radius = ro .
There is a contactresistance, hc , between the shells. The materials are different,and T1 (r = ri ) = Ti and T2 (r = ro ) = To . Derive an expressionfor the inner temperature of the outer shell (T2c ).2.28A 1 kW commercial electric heating rod, 8 mm in diameter and0.3 m long, is to be used in a highly corrosive gaseous environment.
Therefore, it has to be provided with a cylindrical sheathof fireclay. The gas flows by at 120◦ C, and h is 230 W/m2 K outside the sheath. The surface of the heating rod cannot exceed800◦ C. Set the maximum sheath thickness and the outer temperature of the fireclay. [Hint: use heat flux and temperatureboundary conditions to get the temperature distribution.
Thenuse the additional convective boundary condition to obtain thesheath thickness.]2.29A very small diameter, electrically insulated heating wire runsdown the center of a 7.5 mm diameter rod of type 304 stainless steel. The outside is cooled by natural convection (h = 6.7W/m2 K) in room air at 22◦ C. If the wire releases 12 W/m, plotTrod vs. radial position in the rod and give the outside temperature of the rod. (Stop and consider carefully the boundaryconditions for this problem.)2.30A contact resistance experiment involves pressing two slabs ofdifferent materials together, putting a known heat flux throughthem, and measuring the outside temperatures of each slab.Write the general expression for hc in terms of known quantities.
Then calculate hc if the slabs are 2 cm thick copper and1.5 cm thick aluminum, if q is 30,000 W/m2 , and if the twotemperatures are 15◦ C and 22.1◦ C.2.31A student working heat transfer problems late at night needsa cup of hot cocoa to stay awake. She puts milk in a pan on anelectric stove and seeks to heat it as rapidly as she can, withoutburning the milk, by turning the stove on high and stirring themilk continuously. Explain how this works using an analogouselectric circuit. Is it possible to bring the entire bulk of the milkup to the burn temperature without burning part of it?Problems932.32A small, spherical hot air balloon, 10 m in diameter, weighs130 kg with a small gondola and one passenger. How muchfuel must be consumed (in kJ/h) if it is to hover at low altitudein still 27◦ C air? (houtside = 215 W/m2 K, as the result of naturalconvection.)2.33A slab of mild steel, 4 cm thick, is held at 1,000◦ C on the backside.
The front side is approximately black and radiates toblack surroundings at 100◦ C. What is the temperature of thefront side?2.34With reference to Fig. 2.3, develop an empirical equation fork(T ) for ammonia vapor. Then imagine a hot surface at Twparallel with a cool horizontal surface at a distance H below it.Develop equations for T (x) and q. Compute q if Tw = 350◦ C,Tcool = −5◦ C, and H = 0.15 m.2.35A type 316 stainless steel pipe has a 6 cm inside diameter andan 8 cm outside diameter with a 2 mm layer of 85% magnesiainsulation around it.
Liquid at 112◦ C flows inside, so hi = 346W/m2 K. The air around the pipe is at 20◦ C, and h0 = 6 W/m2 K.Calculate U based on the inside area. Sketch the equivalentelectrical circuit, showing all known temperatures. Discussthe results.2.36Two highly reflecting, horizontal plates are spaced 0.0005 mapart. The upper one is kept at 1000◦ C and the lower one at200◦ C. There is air in between. Neglect radiation and computethe heat flux and the midpoint temperature in the air.
Use apower-law fit of the form k = a(T ◦ C)b to represent the air datain Table A.6.2.37A 0.1 m thick slab with k = 3.4 W/m·K is held at 100◦ C on theleft side. The right side is cooled with air at 20◦ C through aheat transfer coefficient, and h = (5.1 W/m2 (K)−5/4 )(Twall −T∞ )1/4 .
Find q and Twall on the right.2.38Heat is generated at 54,000 W/m3 in a 0.16 m diameter sphere.The sphere is cooled by natural convection with fluid at 0◦ C,and h = [2 + 6(Tsurface − T∞ )1/4 ] W/m2 K, ksphere = 9 W/m·K.Find the surface temperature and center temperature of thesphere.94Chapter 2: Heat conduction, thermal resistance, and the overall heat transfer coefficient2.39Layers of equal thickness of spruce and pitch pine are laminated to make an insulating material. How should the laminations be oriented in a temperature gradient to achieve the besteffect?2.40The resistances of a thick cylindrical layer of insulation mustbe increased. Will Q be lowered more by a small increase ofthe outside diameter or by the same decrease in the insidediameter?2.41You are in charge of energy conservation at your plant. Thereis a 300 m run of 6 in. O.D.
pipe carrying steam at 250◦ C. Thecompany requires that any insulation must pay for itself inone year. The thermal resistances are such that the surface ofthe pipe will stay close to 250◦ C in air at 25◦ C when h = 10W/m2 K. Calculate the annual energy savings in kW·h that willresult if a 1 in layer of 85% magnesia insulation is added. Ifenergy is worth 6 cents per kW·h and insulation costs $75 perinstalled linear meter, will the insulation pay for itself in oneyear?2.42An exterior wall of a wood-frame house is typically composed,from outside to inside, of a layer of wooden siding, a layerglass fiber insulation, and a layer of gypsum wall board.
Standard glass fiber insulation has a thickness of 3.5 inch and aconductivity of 0.038 W/m·K. Gypsum wall board is normally0.50 inch thick with a conductivity of 0.17 W/m·K, and the siding can be assumed to be 1.0 inch thick with a conductivity of0.10 W/m·K.a. Find the overall thermal resistance of such a wall (in K/W)if it has an area of 400 ft2 .b. Convection and radiation processes on the inside and outside of the wall introduce more thermal resistance. Assuming that the effective outside heat transfer coefficient(accounting for both convection and radiation) is ho = 20W/m2 K and that for the inside is hi = 10 W/m2 K, determine the total thermal resistance for heat loss from theindoors to the outdoors. Also obtain an overall heat transfer coefficient, U , in W/m2 K.Problems95c.
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