Heat Transfer

ChE 421:Unit Operations 2 Heat Transfer ConductionSaint Louis University , Baguio city 1

PROBLEMS:

1. The front slab of lead ( k = 35 W/m. K ) is kept at 110 0C and the back is kept at 500C. If the area of the slab is 0.4 m2 and it is 0.03 m thick, compute the heat transfer rate

Given: lead, Pb k = 35 W/ m. K

q

T1 = 110oC T2 = 50oC

0.03m

Required : q/A , q

Solution:


2. A copper slab ( k = 372 W/m .K ) is 3mm thick . it is protected from the corrosion on each side by a 2 mm thick layer of stainless steel ( k = 17 W/m .K ) . The temperature is 40oC on one side of this composite wall and 100oC on the other. Find the temperature distribution in the copper slab and the heat conducted through the wall.

Given: ksteel = 17 W/m .K kcopper = 372 W/m .K

Steel Copper Steel T1 = 40oC T2 T3 T4 = 10oC q

2mm 3mm 2mm

Required: T2, T3 , q/ A

Solution:

ChE 421:Unit Operations 2 Heat Transfer ConductionSaint Louis University, Baguio city 3

3. The flux q, is 6000 W/m2 at the surface of an electrical heater . The heater temperature is 120oC when it cooled by air at 70oC .What is the average convective heat transfer coeffiencient , h ? What will be the heater temperature if the power is reduced so that q is 2000 W/ m2?

Given : T = 120o C

AIR @ 70oC q /A = 6000 W/m2

Required : h, Telectrical heater , if q /A = 2000 W/m2

Solution :

ELECTRICAL HEATER


4. Consider a 0.8 m high and 1.5 m wide glass window with a thickness of 8 mm and a thermal conductivity of k = 0.78 W/m. oC . Determine the steady state rate of heat transfer through this glass window and the temperature of its inner surface for a day during which the room is maintained at 20oC while the temperature of the outdoors is – 10oC. Take the heat transfer coefficients on thec inner and outer surfaces of the window to be h1 = 10 W/m2 .oC and h2 = 40 W/m2 .oC, which includes the effects of radiation.

Given : glass W = 1.5 m k = 0 .78 W/m .oC

T1 = 20oC T2 = -10oC 0.8 m

h1 = 10 W/m2 .oC h2 = 40 W/m2 .oC

q


5. Consider a 0.8 m high and 1.5 m wide double pane window consisting of two 4 –mm thick layers of glass (k = 0.78 W/m .oC) separated by a 10 –mm wide stagnant air space (k = .026 W/m .0C ). Determine the steady rate of heat transfer through this double – pane window and the temperarure of its inner surface for a day during which the room is maintained at 20oC while the temperature of the outdoors is – 10oC. Take the heat transfer coefficients on the inner and outer surfaces of the window to be h1 = 10 W/m2 .oC and h2 = 40 W/m2 .oC, which includes the effects of radiation.

Given: glass air glass ( kglass = 0.78 W/m .oC ) ( kair = 0.026 W/m .oC) W = 1.5 m glass T1 = 200C 0.8m ti

T1 = 200C

4mm 10mm 4mm Required: q , ti

Solution:

q


Problems:1. A 3-m high and 5-m wide wall consist of long 16-cm x 22 –cm cross section horizontal bricks

(k= 0.72 W/ m. oC) seperated by 3- cm – thick plaster layers ( k = 0.22 W/m . oC ). There are also 2- cm – thick plaster layers on each side of the brick and a 3- cm thick rigid foam foam ( k = 0.026 W/m oC) on the inner side of the wall, as shown. The indoor and the outdoor temperatures are 20oC and h1 = 10 W/m.oC and h2 = 25 W/m. oC, repectively. Assume one – dimensional heat transfer through thec wall.

Given:

Foam Plaster

1.5 cm

h1 = 10 W/m. oC 22 cm , h2 = 25 W/m. oC T1 = 200C T2 = -100C

1.5 cm

3 2 16 cm 2 Required: q total

Solution:

Brick


2. Steam at 320oC flows in a cast iron pipe ( k = 80 Wm oC) whose inner and outer diameters are 5 cm and 5.5 cm , respectively . The pipe is covered with 3 cm thick glass wool insulation with k = 0.05 W/m 0C. Heat is lost to the surrounding at 50C by natural convection and radiation , with a combined heat transfer coefficient inside the pipe to be h2 = 18 W/ m2

0C. taking the heat transfer coefficient of h1 = 60 W/m 0C ,determine the rate of heat loss from the steam per unit length of the pipe . Also determine the temperature drops across the pipe shell and the insulation.

Given: 50C , h1 = 60 W/m 0C

glass wool k = 80 Wm oC

k = 0.05 W/m 0C h2 = 18 W/ m2 0C ,

Required: q , ∆Tpipe , ∆Tinsulation

Solution:

Pipe

320oC


CRITICAL RADIUS OF INSULATION , rc

o Outer radius that provides maximum rate of heat transfer

CYLINDER:

SPHERE: where: r2 < rc – increases HT rate r2 = rc - reaches the maximum value of HT rate r2 >rc – decreases HT rate

Problem:

A 5 –mm diameter spherical ball at 50oC is covered by a 1mm thick plastic insulation ( k = 0.13 W/ moC). The ball is exposed and radiation heat transfer coefficient of 20 W/m2 .oC. Determine the plastic insulation on the ball will help or hurt heat transfer from the ball?

Given: plastic insulation

1-mm

Required:

Solution:

ri=kh

rc=2kh


UNSTEADY STATE CONDUCTION

o Time varying conductiono Average temperature changes quite rapidly with time.

BIOT NUMBER:

FOURIER NUMBER :

A 7.5 cm diameter orange originally at 26oC is placed in a refrigerator where the air temperature is 2oC and the average convective heat transfer coefficient over the surface of the orange is 57 W/m2. K. estimate the time required for the time temperature of the orange to reach 4oC?

Given: d = 7.5 cm ( orange ) Tf = 2oC ( air ) Ta = 26oC , Tb = 4oC , h = 57 W/m2 .K

Required: tT

Solution:

Bi=hSk

(slab) Bi=hrm

k(cylinder )/ sphere¿

Fo=αt /S2(slab) Fo=αt /rm2(cylinder /sphere)

ChE 421:Unit Operations 2 Heat Transfer Principles of Heat Flow in FluidsSaint Louis University , Baguio city 1

Problems:1. A double pipe heat exchanger is constructed of a copper ( k = 380 W/m. oC ) inner tube of

internal tube diameter Di = 1.2cm and external diameter Do = 1.6 cm and an outer tube of diameter 3.0 cm .The convection heat transfer coefficient is reported to be hi = 700 W/m2 .oC on the inner surface of the tube and ho = 1400 W/m2 .oC on its inner surface. For a fouling factor Rf,i = 0.0005m2 oC/W on the tube side and Rf,o = 0.0002m2. oC/W on the shell side, determine (a) the thermal resistance of the heat exchanger per unit length and (b) the overall heat transfer coefficients Ui and Uo based on the inner and outer surface areas of the tube respectively.

Given: double pipe HE

Required: (a) R total (b) Ui and Uo Solution:


2. Steam in condenser of a steam power plant is to be condensed of a temperature of 50 oC (hfg = 2305 KJ/Kg) with coolimg water (Cp = 4180 J/Kg . oC) from a nearby lake, which enters the tubes of the condsenser at 18oC and leaves at 27oC. The surface area of the tubes is 58m2, and the overall heat transfer coefficient is 2400 W/m2. oC. Determine the mass flow rate of the cooling water needed and the rate of condensation of the steam in the condenser?

Given: Steam, 50oC

27oC

18oC Water

50oC

Required: mc and rate of condensation

Solution:


3. A double – pipe parallel – flow heat exchanger is to heat water (Cp = 4180 J/Kg .oC) from 25oC to 60oC at a rate of 0.2 kg/s. The heating is to be accomplished by geothermal water (Cp = 4310 J/Kg . oC) b available at 140oC at a mass flow raste of 0.3 kg/s. The inner tube is thin –walled and has a diameter of 0.8cm. if the overall heat transfer coefficient of the heat exchanger is 550 W/m2. oC, determine the length of the heat exchanger required to achieve the desired heating.

Given: water , Cp = 4180 J/Kg . oC

Tca = 25oC , m = 0.2 kg/s

Geot. water

Tha = 140oC T hb = ? m = 0.3 kg/s Cp = 4310 J/Kg oC

Tcb = 60oC

Required: L

Solution


4. A double – pipe counter flow heat exchanger is to cool ethylene glycol (Cp = 2560 J/Kg oC) flowing at a rate of 3.5 Kg/s from 80oC to 40oC by water (Cp = 4180 J/Kg oC) that enters at 20oC and leaves at 55oC . The overall heat transfer coefficient based on the inner surface area of the tube is 250 W/m2 .oC, Determine (a) the rate of heat transfer , (b) the mass flow rate of water, and (c) the heat transfer surface area on the inner side of the tube.

Cold water, 20oC Given:

Hot glycol , 80oC

3.5 kg/s 40 oC

Required: (a) q (b) m water (c) Ai

Solution:


CONDUCTION WITH HEAT GENERATION

where: = heat generation per unit volume

An electric current of 200 Amperes is passed through a stainless steel wire with radius of 0.001268m. The wire is 0.91 m long with electrical resistance of 0.126 ohm. The outer surface temperature is held at 422.1K , k = 22.5 W/m.K (a) Find the center temperature(b) Find the temperature between the surface and the center.

Given: r1 =0.001268 , T1 = 422.1 K I = 200 A 4 R = .126 ohm r = 0 ,To

L = .91 m k = 22.5 W/m.K

Required: (a) To

(b) T

Solution:

T−T o=−r2

4kT−T 1=

−❑4K

(r 2−r12 )

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids without Phase Change Saint Louis University , Baguio city 1

Problems:12.8 From Mc Cabe , Smith and Hariott (7th ed.)

Water must be heated from 15 to 50oC in a simple double –pipe exchanger at a rate of 3500 kg/h. The water is flowing inside the inner tube with steam condensing at 110oC on the outside. The tube wall is so thin that the wall resistance may be neglected. Assume that the steam film coefficient ho is 11kW/m2 .oC. What is the length of the shortest heat exchanger that will heat the water to the desired temperature? Average properties of water are as follows:

ρ= 993 kg/m3 k = 0.61 W/m . oC µ = 0.78 cP Cp = 4.19 J/g . oC

Given: Steam @ 110oC , ho = 112 kW/m2. oC

Water

Ta = 15oC Tb = 50oC m = 3500 kg/h

Required: Lmin

Solution:

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids without Phase Change Saint Louis University, Baguio city 2

12.15 Mc Cabe , Smith and Harriott (7th ed.) Water at 15oC is flowing at right angles across a heated 25-mm- OD cylinder, the

surface temperature of which is 120oC. The approach velocity of the water is 1 m/s. (a)What is heat flux in killowatts per square meter, from the surface of the cylinder to the water?(b) What would be the flux if the cylinder were replaced by a 25- mm – OD sphere, also with a surface temperature of 120OC?

(a) Given: Water @ 15oC ,v = 1 m/s

120OC

Required:q/A

Solution:

OD = 25 mm


(b) 120oC OD sphere = 25 mm

Water@ 15oC

Required: q/A

Solution:


12.21 Mc Cabe, Smith and Harriott (Seventh edition) Air is 1 atm and 20oC flows at 10m/s past a horizontal flat plate which has a surface temperature of 80oC. The length of the plate is 1.6 m. (a) calculate the average heat flux in W/m2 and the local heat flux at the discharge end of the plate. (b) What is the thickness of the thermal boundary layer at the end of the plate?

Given: Air @ 1 atm, 20oC, V = 10m/s

Tw =80oC

L = 1.6 m

Required: (a) Average heat flux , local heat flux (b) thickness of boundary layer

Solution:


Estimate the heat loss from a vertical wall exposed to nitrogen @ 1 atm and 4oC. The wall is 1.8 m high and 2.45 m wide. It is maintained at 50oC.

Given: N2 @ 1 atm and 4oC

1.8 m

2.45 m Required: q

Solution:

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids With Phase Change Saint Louis University , Baguio city 1

CHAPTER 13 : HEAT TRANSFER TO FLUIDS WITH PHASE CHANGE

oBOILING: Pool boiling – absence of bulk fluid flow

heating

Flow boiling or forced convection boiling

heating

Types : Sub-cooled boiling Saturated boiling

80oC 100oC

Figure 1 Figure 2

Note: Figure 1 : T main body of liquid < Tsat Figure 2 : T main body Tsat

Sub-cooled water 107oC

Saturatedwater 107oC


Boiling Curve Max. critical heat flux Burn out point C

bubbles collapsed E 1 2 3 4 B

q,boiling W/m2

A bubbles rised to the free surface

D Leidenfrost point

5 30 120 ∆T excess = Ts – Tsat ,oC

For boiling regime:1. Natural convection boiling 2. Nucleate Boiling

100oC 100oC

heating heating

3. Transition Boiling 4. Film Boiling

100oC 100oC

Heating heating

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids With Phase Change Saint Louis University, Baguio city 3

NUCLEATE BOILING

Rohsenow equation:qnucleate=µ l λ ¿¿

where: q nucleate = nucleate boiling heat flux, W/m2

µl = liquid viscosity, kg/m . s λ = enthalpy of vaporization, J/Kg ρl = liquid density, kg/m3

ρ v = vapor density , kg/m3

σ = surface tension of liquid vapor interface, N/m Cp = liquid specific heat, J/kg. or Ts = surface temperature of heater, oC Tsat = saturation temperature of heater, oC Csf, n = constants Prl =liquid Prandtl number PEAK HEAT FLUX

Kutate ladze and Zuber equation: qmax=Crλ ¿¿

MINIMUM HEAT FLUX

Zuber equation:qmin=0.09 ρv λ¿¿

FILM BOILING

Brownley equation:

q film=C film¿¿ where : Kv=¿ thermal conductivity of vapor , W/m oC C film = 0.62 ( for horizontal cylinder ) = 0.67 ( for spheres )


Examples:

1. Water is to be boiled @ atmospheric pressure mechanically polished stainless steel pan placed on a top of a unit. The inner surface of the bottom of the bottom of the pan is maintained @ 108oC. If inner diameter of the pan is 3cm , determine (a) the rate of heat transfer to the water (b) the rate of evaporation of water.

Given: P = I atm 100oC

D = 30 cm

heating

Required: (a) q (b) mevaporation

Solution:


108oC

2. Water in a tank is to be boiled at sea level by a 1-cm-diameter nickel plated steel heating element equipped with electrical resistance wires inside. Determine the maximum heat flux that can be attained in the nucleate boiling regime and the surface temperature of the heater surface in that case.

Given:

d = 1cm

heating element

heating

Required: q max ,Ts

Solution:


Ts

3. Water is boiled at atmospheric pressure by a horizontal polished copper heating element of D = 5 mm immersed in water. If the surface temperature of the wire is 350oC,determine the rate of heat transfer from the wire to the water per unit length of the wire.

Given:

100oC

D= 5 mm Heating element

Vapor film

Required: q film

Solution:


Ts = 350oC

4. Water is to be boiled at atmospheric pressure in a mechanically polished steel pan placed on the heating unit. The inner surface of the bottom of the pan is maintained at 110oC. If the diameter of the bottom of the pan is 25 cm , determine the (a) the rate of heat transfer to the water and (b) the rate of evaporation.

Given:

100oC

D = 25 cm

Required: (a) q (b) ms

Solution:


110oC

5. Water is to be boiled at atmospheric pressure on a 3 – cm diameter mechanically polished steel heater. Determine the maximum heat flux that can be attained in the nucleate boiling regime and the surface in that case.

Given:

100oC

D = 3 cm

Heating Required: q max

Solution:

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids With Phase Change Saint Louis University, Baguio city

Ts

o CONDENSATION HEAT TRANSFER

Condensation occur when the temperature of a vapor is reduced below its saturation temperature.

2 distinct form of condensation

Film condensation – condensate wets the surface and forms a liquid film on the surface that slides down under the influence of gravity.

Dropwise condensation – the condensed vapor forms droplets on the surface instead of continuous film and the surface is covered by countless droplets of varying diameters.

FLOW REGIME Re = 0

Laminar( wave free)

Re = 30 Laminar (wavy )

Re = 1800

Turbulent


Vertical Plate or Tube

For wave free laminar ( Re<30 )

hvert=0.943[ k f ρ f2 g λ¿

∆ ¿Lµf ¿]

14

λ¿= λ+0.68Cpl(Tsat−Ts)

ℜ= 4 g3V l

2 [ k l

3h4 ]

3

hvert=1.47k l Re

−13 ( g

V l2 )13

For laminar wavy (30<Re<1800 )

h=Rek l

1.08ℜ1.22−5.2 ( gV l2 )13

ℜ=4.81+[ 3.70 Lk l(Tsat−T s)µl λ

¿ ( gV l2 )13 ]0.820

For Turbulent (Re > 1800 )

h=Rek l

8750+58Pr−.50−Re.75−253 ( g

V l2 )13

ℜ=[ 0.690 Lk l Pr.50(Tsat−T s)

µl λ¿ ( g

V l2 )13−151P r

.50+253]43


Inclined Plates

Ǿ

Condensate vapor

Inclined plate

Horizontal Tubes and Spheres

h=0.729¿¿

h=0.729¿¿

where: N = number of tubes

hverthhori

=1.29( DL )

1k


Problems:

1.Saturated steam at atmospheric condenses an a 2-m high and 3-m wide vertical plate and that is maintained at 80oC by circulating cooling water through the other side.Determine the ff:(a) the rate of heat transfer by condensation to the plate (b) the rate at which the condensate drops off the plate at the bottom.

Given: 3m

Ts = 80oC

2m

condensate


Solution:


2. What would your answer be to the preceding problem if the plate were tilted 30o from the vertical?

Given:

plate

Ǿ

30o


Solution:


4. Repeat the preceding problem for the case of 12 horizontal tube arranged in a rectangular array of 3 tubes high and 4 tubes wide

Given:

3 tubes

4 tubes


Solution:


3. The condenser of a steam power plant operates at 40oC and a pressure of 7.38 kPa. Steam at this pressure condenses on the surface of horizontal pipes through which cooling water circulates. The outer diameter of the pipe is 3cm, and the outer surfaces of the pipes are maintained at 30oC.Determine the ff:

(a) the rate of heat transfer to the cooling water circulating in the pipes.(b) the rate of condensation of steam per unit length of a horizontal pipe.

Given: Steam, 40oC Cooling water


Solution:

ChE 421:Unit Operations 2 Heat Transfer Conduction Saint Louis University, Baguio city

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids Without Phase ChangeSaint Louis University, Baguio city

ChE 421:Unit Operations 2 Heat Transfer Heat Transfer to Fluids With Phase Change Saint Louis University, Baguio city

ChE 421:Unit Operations 2 Heat Transfer Principles of Heat Flow in FluidsSaint Louis University , Baguio city

Heat Transfer

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