Serth Problems

7/27/2019 Serth Problems

1/7

2 / 78 C O NV E CT IV E H E AT T R A N S FE R

(c) The prime surface area per meter of tube length.(d) The weighted efficiency of the finned surface.(e) The rate of heat transfer from the fins per meter of tube length.(f) The rate of heat transfer from the prime surface per meter of tube length.

(g) The total rate of heat transfer per meter of tube length.

Ans. (a) 86.7%. (b) 0.183828 m2. (c) 0.056398 m2. (d) 89.8%.(e) 14,344 W. (f) 5076 W. (g) 19,420 W.

(2.22) Annular steel fins (k= 56.7 W/m K) are attached to a steel tube that is 30 mm in externaldiameter. The fins are 2 mm thick and 15 mm long. The tube wall temperature is 350 K andthe surrounding fluid temperature is 450 K with a heat-transfer coefficient of 75 W/m2 K.

There are 200 fins per meter of tube length. Calculate:(a) The fin efficiency.(b) The fin surface area per meter of tube length.(c) The prime surface area per meter of the tube length.

(d) The weighted efficiency of the finned surface.(e) The rate of heat transfer per meter of tube length.

Ans. (a) 85%. (b) 0.9248 m2. (c) 0.0565 m2. (d) 86%. (e) 6330 W.

(2.23) A finned heat exchanger tube is made of aluminum alloy (k= 186 W/m K) and contains125 annular fins per meter of tube length. The bare tube between fins has an OD of 50 mm.

The fins are 4 mm thick and extend 15 mm beyond the external surface of the tube. Theouter surface of the tube will be at 200C and the tube will be exposed to a fluid at 20C witha heat-transfer coefficient of 40 W/m2 K. Calculate:

(a) The rate of heat transfer per meter of tube length for a plain (un-finned) tube.(b) The fin efficiency.(c) The fin and prime surface areas per meter of tube length.(d) The weighted efficiency of the finned surface.(e) The rate of heat transfer per meter of tube length for a finned tube.(f) If the cost per unit length of finned tubing is 25% greater than for plain tubing, determine

whether plain or finned tubing is more economical for this ser vice.

Ans. (a) 1130 W. (b) 98.6% (c) 0.8946 m2 and 0.07854 m2.(d) 98.7%. (e) 6920W.

(2.24) A stream of ethylene glycol having a flow rate of 1.6 kg/s is to be cooled from 350 to 310 Kby pumping it through a 3-cm ID tube, the wall of which will be maintained at a temperatureof 300 K. What length of tubing will be required?

Ans. 49m.

(2.25) A small steam superheater will be made of -in. schedule 80 stainless steel pipe which willbe exposed to hot flue gas from a boiler. 50 lbm/h of saturated steam at 320F will enterthe pipe and be heated to 380F. Assuming that the pipe wall temperature will vary from375F at the steam inlet to 415F at the outlet, calculate the length of pipe required for thesuperheater.

Ans. 6.1 ft.


2/7

C O NV E CT I VE H E AT T R A N S FE R 2 / 79

(2.26) Suppose that the steam superheater of Problem 2.25 has a length of 10 ft and the walltemperature is constant at 415F over the entire length of the pipe. Using the value of theheat-transfer coefficient calculated in Problem 2.24, estimate the temperature of the steamleaving the superheater.

Ans. 405F.

(2.27) A gas heater shown below in cross-section consists of a square sheet metal duct insulatedon the outside. A 5 cm OD steel pipe passes through the center of the duct and the pipe wallis maintained at 250C by condensing steam flowing through the pipe. Air at 20C will befed to the heater at a rate of 0.35 kg/s.(a) Calculate the equivalent diameter for the heater.(b) Calculate the length of the heater required to heat the air stream to 60C.

Ans. (a) De =0.1083 m. (b) 8.1 m.

Insulated onall four sides

Condensing steam

15cm

15cmAir

(2.28) Water at 20C with a mass flow rate of 1.0 kg/s enters an annulus formed by an inner pipehaving an OD of 2.5 cm and an outer pipe having an ID of 10 cm. The wall temperature ofthe inner pipe varies from 80C at the inlet to 100C at the outlet. The outer surface of theouter pipe is well insulated. Calculate the length of the annulus required to heat the waterto 75C.

Ans. 75.3 m.

(2.29) A stream of ethylene glycol having a flow rate of 4000 lb/h is to be heated from 17C to 37Cby passing it through a circular annular heater. The outer pipe of the annulus will have an

ID of 5 cm and the inner pipe will have an OD of 4 cm. The exterior surface of the outerpipe will be well insulated, while the inner pipe wall temperature will vary from 67C at theentrance to 87C at the exit. The required length of the heater is desired. Assuming as afirst approximation thatL= 20 m, calculate a second approximation for the length.

Ans. 20.8 m.

(2.30) 100 lbm/h of oil at 130F is needed for a process modification. The oil will be available at70F, so a plant engineer has designed a heater consisting of a 1-in. ID tube, 10 ft long, the

wall of which will be maintained at 215F by condensing steam. Will the heater work asrequired? Properties of the oil may be assumed constant at the following values:

CP = 0.49 Btu/lbm F = 1.42lbm/ft h

k = 0.0825 Btu/h ft F = 55 lbm/ft3


3/7


(2.31) A stream of Freon-12 with a flow rate of 4000 lb/h is to be heated from 20C to 30C foruse in a chemical processing operation. Available for this service is an annular heater 3.05 min length. The outer pipe of the annulus has an ID of 5 cm and an O.D. of 5.5 cm. The innerpipe has an ID of 3.75 cm and an OD of 4 cm. When in operation, the walls of both pipes aremaintained at a constant temperature that can be set to any value between 30C and 70C.

Will the heater be suitable for this service?

(2.32) A feed stream to a chemical reactor consists of 0.2 kg/s of ammonia vapor at 300 K. Priorto entering the reactor, the ammonia is preheated by passing it through a rectangular duct

whose walls are heated to 500 K by hot process waste gas. The duct cross-section is 9 cmby 20 cm, and it is 2.5-m long. At what temperature does the ammonia enter the reactor?

(2.33) An air preheater is required to heat 0.2 kg/s of process air from 15Cto115C. Thepreheaterwill be constructed from rectangular ducting having a cross-section of 7.5 cm by 15 cm. Theair will flow inside the duct and the duct walls will be maintained at 250C by hot flue gas.

What length of ducting will be required?

(2.34) A cylindrical storage tank with a diameter of 4 m and a length of 10 m will hold a fluid whosetemperature must be maintained at 347 K. In order to size the heater required for the tank,the following worst-case scenario is considered:

Ambient air temperature=20CWind speed= 20m/sTank wall temperature= 347K

What size (kW) heater will be needed?

Ans. 400 kW.

(2.35) A viscous liquid is to be pumped between two buildings at a chemical plant in an above-ground pipe that has an OD of 22 cm and is 110 m long. To facilitate pumping, the liquid

will be heated to 40C in order to reduce its viscosity. The liquid flow rate will be 20 kg/sand the specific heat of the liquid is 1300 J/kg K. Determine the temperature drop that theliquid will experience over the length of the pipe under the following worst-case conditions:

Ambient air temperature=10CWind speed= 14m/sPipe surface temperature= 37C

Ans. 5.7C.

(2.36) A surge tank to be used in a chemical process is spherical in shape with a diameter of 10 ft.The tank will hold a liquid that must be maintained at 180F by means of a heating unit. Thefollowing parameters have been established for design purposes:

Ambient air temperature= 20FWind speed= 20 miles/hr.Tank wall temperature= 180F

(a) Estimate the duty (Btu/h) that the heater must supply.(b) Comment on the probable accuracy of your estimate.

Ans. (a) 88,500 Btu/h.

(2.37) A duct is being designed to transport waste gas from a processing unit to a pollution con-trol device. The duct will be 5 ft high, 6 ft wide, and 100 ft long. In order to determine


4/7


whether the duct will need to be insulated, the rate of heat loss to the environment mustbe estimated. Based on the following design conditions, compute the rate of heat loss fromthe duct.

Average duct surface temperature= 250FAmbient air temperature= 10F

Wind speed= 20 miles/hr.

Ans. 1.4 106 Btu/h.

(2.38) A pipeline at a chemical complex has an OD of 10 cm, an ID of 9.4 cm, and is covered witha layer of insulation (k= 0.055 W/m K) that is 4-cm thick. A process liquid at 50C flows inthe pipeline with a heat-transfer coefficient of 400 W/m K. The pipeline is exposed to theenvironment on a day when the air temperature is 20C and the wind speed is 15 m/s.

What is the rate of heat loss from the pipeline per meter of length?

Ans. 40 W.

(2.39) Consider again the graphite heat exchanger of Problem 2.12. What air velocity is requiredto achieve the stated heat - transfer coefficient of 100 W/m2 K?

(2.40) An energy recovery system is being considered to preheat process air using hot flue gas.The process requires air at 80C. In the proposed energy recovery system, air at 25C willenter a rectangular duct with a flow rate of 2 kg/s. The duct dimensions are 1 m 2 m 6 mlong. The duct walls will be maintained at a temperature of 250C by hot flue gas flowingover the outside of the duct. Estimate the air temperature that will be achieved with thissystem and thereby determine whether or not the energy recovery system will satisfy theprocess requirement.

(2.41) The surface temperature of the electronic chip shown below is 75C and it is surroundedby ambient air at 25C. Calculate the rate of heat loss by natural convection from the uppersurface of the chip.

Ans. 0.178 W.

16 mm

16 mm Chip carrier

(2.42) A petrochemical storage tank is cylindrical in shape with a diameter of 6 m and a heightof 10 m. The surface temperature of the tank is 10C when, on a calm clear night, the airtemperature drops rapidly to 10C. Estimate the rate of heat loss from the tank underthese conditions.


5/7


(2.43) A horizontal elevated pipeline in a chemical plant has an OD of 10 cm, an ID of 9.4 cm, and iscovered with a layer of magnesia insulation that is 4-cm thick. A process liquid at 50C flowsin the pipeline with a heat-transfer coefficient of 400 W/m2 K. The pipeline is exposed tothe environment on a calm night when the air temperature is 5C.(a) Make a reasonable guess for the temperature of the exterior surface of the insulation

and use it to calculate the heat-transfer coefficient between the insulation and theambient air.

(b) Use the result of part (a) together with the other information given in the problem tocalculate the rate of heat loss per meter of pipe length.

(c) Use the result of part (b) to calculate the temperature of the exterior surface of theinsulation and compare it with the value that you assumed in part (a).

(2.44) A spherical storage tank has a diameter of 5 m. The temperature of the exterior surface ofthe tank is 10C when, on a calm clear night, the air temperature drops rapidly to 10C.Estimate the rate of heat loss from the tank under these conditions.

Ans. 5550 W.

(2.45) At a convective boundary such as the one atx=B in the sketch below, heat is transferredbetween a solid and a fluid. In order for a finite temperature to exist at the solidfluidinterface, the rate at which heat is transferred to the interface by conduction through thesolid must equal the rate at which heat is transferred away from the interface by convectionin the fluid. The reason is that the interface has no volume and no mass, and so has zeroheat capacity. The boundary condition at x=L is thus:

qconduction = qconvection

k dTdx

= h(T T)

h(T T) + kdT

dx= 0

Expressions for both T and dT/dx(obtained by integration of the conduction equation) aresubstituted into this equation to obtain a relationship between the constants of integration.

B

x

Fluid

h

T

Insulated

Insulated

Insulated

Solid

The rectangular solid shown above is insulated on all sides except the one atx=B, whichis exposed to a fluid at temperature T with heat-transfer coefficient, h. Heat is generatedwithin the solid at a rate per unit volume given byq=x, where is a constant.(a) Assuming constant thermal conductivity, derive an expression for the steady-state

temperature distribution, T(x), in the solid.


6/7


(b) Calculate the temperature of the insulated boundary atx= 0 for the following parametervalues:

T = 20C k = 1.5 W/m K = 50 W/m4

B= 2 m h = 30 W/m2 K

Ans. (a) T(x)=T+B2

2 h+

6 k(B3 x3). (b) 67.8C.

(2.46) Consider the rectangular solid shown below that is insulated on four sides. The side atx=Bis held at a fixed temperature, Tw, while the side atx= 0 is exposed to a fluid at temperatureT. Heat is generated within the wall at a rate per unit volume given byq=x, where isa constant. The thermal conductivity of the solid may be assumed constant.(a) Formulate an appropriate set of boundary conditions for this configuration.(b) Use the conduction equation to derive an expression for the steady-state heat flux, qx,

in the solid.

B

x

Fluid

h

T

Tw

Insulated

Insulated

Solid

Ans. (b) qx=x2

2+

h{k(TTw)B3/6}

(k+B h).

(2.47) A long solid cylindrical rod of radius R contains a heat source that generates heat per unitvolume at a rate q=r, where is a constant and r is radial position measured from thecenterline. The rod is completely surrounded by a fluid at temperature T with heat-transfercoefficient, h. Assuming heat flow only in the radial direction:

(a) Formulate an appropriate set of boundary conditions to be used with the heatconduction equation.(b) Derive an expression for the steady-state temperature profile, T(r), in the rod. State

any assumptions that you make in your derivation.(c) Obtain an expression for the maximum temperature in the rod at steady state.

Ans. (b) T(r)=T+R2

3 h+

(R3 r3)

9 k.

(2.48) Repeat Problem 2.47 for a heat source of the form q=/r.

Ans. (b) T(r)=T+/h+ (/k)(R r).

(2.49) A long hollow cylinder has inner and outer radii R1 and R2, respectively. Heat is generatedin the cylinder wall at a uniform rate, , per unit volume. The outer surface of the cylinderis well insulated. A fluid flows through the inside of the cylinder to provide cooling. The


7/7


fluid temperature is T and the heat-transfer coefficient is h. Assuming constant thermalconductivity, k, derive an expression for the steady-state temperature profile, T(r), in thecylinder wall.

Ans. T(r)=T+(R2

2R2

1)

2 hR1

+(R2

1 r2)

4 k+

R22ln(r/R1)

2 k.

(2.50) A solid sphere of radius R is immersed in a fluid with temperature T and heat-transfercoefficienth. Heat is generated within the sphere at a rate per unit volume given byq=r,

where is a constant and r is radial position measured from the center of the sphere.Assuming constant thermal conductivity, k:(a) Derive an expression for the steady-state temperature profile, T(r), in the sphere.(b) Calculate the maximum temperature in the sphere under steady-state conditions for

the following parameter values:

R= 1.5 m = 100W/m4

h = 100 W/m2

Kk = 0.5 W/m K T = 20

C

Ans. (a) T(r)=T+R2

4 h+

(R3 r3)

12 k. (b) 76.8C.

Documents

Serth Problems