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CONDUCTION	CONVECTION
HEAT EXCHANGE (COMBINING RESISTANCES)	HEAT EXCHANGER DESIGN
RADIATION VIEW FACTORS	RADIATIVE HEAT TRANSFER

Solutions will be made available via hyperlinks after the problems have been dealt with in tutorial sessions.

1. A heat rate of 3 kW is conducted through a section of an insulating material of cross sectional area 10 m² and of thickness 25 mm. If the inner (hot) surface temperature is 415°C and the thermal conductivity of the material is 0.2 W/m.K, what is the outer surface temperature?

 

 

[377.5°C]
 

2. One-dimensional, steady-state conduction without heat generation occurs in a plane wall of thickness 0.5 m and thermal conductivity 25 W/m.K. Determine the unknown quantities for each case in the table below, sketch the temperature distribution and show the direction of the heat flux.

Case	Inside temp (T1)	Outside temp (T2)	dT/dx (K/m)	Heat flux (W/m2)
1	400K	300K	?	?
2	100°C	?	-250	?
3	80°C	?	+200	?
4	?	-5°C	?	4000
5	30°C	?	?	-3000

[-200 K/m, +5000 W/m²; -25°C, 6250 W/m²; 180°C, -5000 W/m², 75°C, -160 K/m; 90°C, 120 K/m]
 

3. A plain slab is subjected to a temperature of 400 K on one side and 300 K on the other. The thermal conductivity of the material of the slab is a function of temperature given by 30exp(T/100) W/m.K. Find the temperature at the mid point within the slab.

 

 

[362 K]
 

4. A lagged steam pipe has a total outside diameter of 0.12 m, including a 20 mm thick layer of calcium silicate insulation on the outside. The inner and outer surfaces of the insulation are at temperatures of 800 K and 490 K respectively. Calculate the heat loss per unit length of pipe.  Thermal conductivity of calcium silicate = 0.07 W/m.K.

 

 

[336 W/m length]
[Click here for WORKED EXAMPLE]
 

5. A steam pipe 0.2 m OD is covered by two layers of insulation material, each 0.025 m thick, and the thermal conductivity of one material is three times that of the other.  Calculate the difference in the effective conductivity of the two layers for the situations:

(i) better insulating material on outside;
(ii) better insulating material on inside.

Assume that the temperatures of the two surfaces remain unaltered.

[(i) conducts 10% more heat than (ii)]
 

6. A scraped surface chiller, used for crystallising para-xylene, consists of long steel cylinders 0.3 m OD and walls 0.01 m thick containing rotating scrapers which remove the crystals. This cylinder is surrounded by a coolant at low temperature. As the scrapers wear, a layer of crystals build up on the inner surface hence new scrapers have to be installed when the chiller efficiency falls by 20%. What will be the thickness of the crystal layer when this point is reached?

ksteel	= 40 W/m.K
kp-xylene crystals	= 1.0 W/m.K

[60 mm]
 

7. A cylindrical cement kiln has an external shell of 0.02 m thick steel and OD of 2.0 m. The refractory temperature inside the kiln is 1400 K and the outer steel surface has a temperature of 400 K when there is a refractory lining 0.2 m thick. The outer surface temperature is used to determine the time between shutdowns for relining which occurs when this is 500 K. What is the running time for a refractory erosion rate of 1 mm per week? Assume that the heat loss remains unchanged.

ksteel	= 30 W/m.K
krefractory	= 1 W/m.K

[20 weeks]
 
8. If the setting of concrete is accompanied by the diffuse evolution of heat at the rate of 50 W/m³, determine the maximum temperature in a concrete slab used for foundations 1 m thick by 1 m deep, during the setting period. It may be assumed that steady state conditions prevail through the setting period and that the surface temperature of the slab is maintained at 280K. Thermal conductivity of wet concrete is 0.8 W/m.K.

 

 
 
 

[284 K]

Saturated steam at 800 kPa flows through a pipe of outside diameter 4.8 cm. The outside of the pipe is insulated with magnesia insulation, 5 cm thick; the thermal conductivity of the magnesia is 0.07 W/m.K. The outside pipe surface is at 170°C and the outside of the insulation is at 35°C. Calculate the rate of condensation of steam in a 30 m length of pipe.

Solution:

r₁ = 2.4 cm; r₂ = 7.4 cm; T₁ = 170°C; T₂ = 35°C

Q’ = m’ l_fg

m’ = 1582 ¸ 2040 J/g = 0.78 g/s

[Click here for WORKED EXAMPLE]

1. Derive the expression that relates the thermal cubic expansivity of an ideal gas to its absolute temperature.

 

 
 
 

[b = 1/T]
 

2. An unlagged pipe of outer diameter 15 cm carries steam the 20 metre distance along a road between boilerhouse and plant. If steam at 1 bar (gauge) is used, 3.0 grammes per second of condensate have formed by the time the pipe has reached the plant. If the outside air is at 15°C, what is its convective heat transfer coefficient? Assume that the resistances to heat transfer presented by convection inside the pipe and conduction through its walls are negligible in comparison.

 

 
 
 

[6.7 W/m²K]
 

3. Justify the above coefficient with a suitable correlation.

 

 
 
 

[solution]
 

4. A 30cm pipe carries crude oil from an off-shore drilling platform to a storage tanker. The length of the pipe is submerged in sea water at 7°C, the oil enters at 23°C and leaves at 17°C. Taking the physical properties of sea water to be the same as those for pure water, what is the predicted heat transfer coefficient of the sea, and what doubts do you have in this prediction? Again, assume that the resistances to heat transfer presented by convection inside the pipe and conduction through its walls are negligible in comparison.

 

 
 
 

[340 W/m²K]
 

5. Atmospheric air is blown at is 7.50 ´ 10^-4 m³/s through a tube 2.00 m long and 40.0 mm inner diameter, and heated from 290 to 310 K. Calculate the inside film coefficient by the most appropriate correlation.

 

 
 
 

[3.35 W/m²K]
 

6. Water is heated from 20.00°C to 50.00°C in the shell-side of a heat exchanger. Inside the exchanger, tubes with 19.05 mm outer diameter (d_o) are arranged in a horizontal bundle of 8 rows, on a tube pitch (p_t) of 23.81 mm. The equivalent diameter for the narrowest gap between tubes is given by:


The water at the narrowest gap between tubes has a velocity of 1.600 m/s and the outer surface of the tubes are maintained at a constant 100.0°C. Calculate the outside film coefficient by the most appropriate correlation, and hence find the heat flux.

[8.903 kW/m²K, 578.7 kW/m²]
 

7. Engine oil is stored in a sump below an internal combustion motor. To keep its viscosity low enough for easy pumping, but high enough for useful lubrication, the oil temperature is maintained at 310 K. This is done by diverting some of the exhaust fumes down a pipe of 12.7 mm outer diameter, which then passes horizontally through the sump. If the pipe has an average outer surface temperature of 353 K, and 4.5 kW of heat is needed to keep the oil temperature steady, how long must the pipe be?

 

 
 
 

[26 m]
 

8. The element inside an electric kettle has a circular cross-section of 11 mm diameter. If the element has a steady temperature of 120°C, what are the heat transfer coefficients when the kettle is newly filled (take tap water as 12°C) and just before the water starts to boil?

 

 
 
 

[1780 & 1531 W/m²K]
 

9. If the kettle element in the above question were 50 cm long, to what power output would it have to be rated?

Water is being heated from 50.0 to 90.0°C inside a tube of internal diameter 40.0 mm. The water flow rate is 3.00 ´ 10^-3 m³/s. Calculate the inside film coefficient by the most appropriate correlation.

SOLUTIONS

Heat transfer is by forced convection. The mean temperature of the water in contact with the tube must be calculated first. As no information on the film temperature is available, the overall temperatures will have to be used:

Average temperature of water is:

We can only therefore use a correlation that takes the fluid temperature as being the average of the inlet and outlet temperatures, as calculated above. The physical properties are found in Steam Tables.
 

At 70.0°C,	m		= 400 ´ 10-6	kg/m.s
	k		= 662 ´ 10-3	W/m.K
	r	= 1 ¸ (0.1023 ´ 10-2)	= 977.5	kg/m3
	Pr		= 2.532	(dimensionless)

But before choosing the correlation, the Reynolds number needs to be known.

i.e turbulent

The length of the tube is unknown, hence the Nusselt correlation for turbulent flow cannot be used. Alternatively, there is the Dittus-Boelter correlation, if the tube is assumed to be long enough with respect to its internal diameter. Expect an accuracy of ± 20%.

where Re > 10 000, 0.7 > Pr > 16 700

Nu = 0.027 (233 365)^0.8 (2.532)^0.33 (1) = 722.69

If the tube surface temperature were to be given as 100°C, this opens avenues to several more calculations.

Firstly, we can confirm the assumptions made for the Dittus-Boelter prediction:

At 50.0°C,	r	= 1 ¸ (0.1012 ´ 10-2)	= 988.14229	kg/m3
	Water flowrate	= 988.14 ´ (3 ´ 10-3)	= 2.9644269	kg/s
At 70.0°C,	CP		= 4191	J/kg.K
	Heat load	= 2.964 ´ 4191(90-50)	= 496 956	W

i.e. assumption valid

At 100°C, m_W = 279 ´ 10^-6 kg/m.s

Nu = 0.027 (233 365)^0.8 (2.532)^0.33 (1.0517) = 760.04

Use this to give an iterative solution for the Nusselt correlation, with 12000 W/m²K as the first estimate:

= 0.036 (233365)^0.8 (2.532)^0.33 (0.73429) = 707.53

= 0.036 (233365)^0.8 (2.532)^0.33 (0.73330) = 706.57

= 0.036 (233365)^0.8 (2.532)^0.33 (0.73325) = 706.52

IN SUMMARY

Dittus-Boelter	Sieder-Tate (without viscosity correction)	Sieder-Tate (with viscosity correction)	Nusselt
10 636 W/m2K	11 961 W/m2K	12 579 W/m2K	11 639 W/m2K

1. The composite wall of an oven consists of three materials, two of which are of known thermal conductivity, k_A = 20 W/m.K, and k_C = 50 W/m.K, and of known thicknesses D x_A = 0.30 m and D x_C = 0.15 m. The third material B is sandwiched between materials A and C and is of thickness 0.15 m.

 

 
 
 

Under steady state conditions measurements reveal that the outer surface of material C is at a temperature of 20°C, the inner exposed surface of material A is at 600°C and the air temperature adjacent to this surface is 800°C with a convective heat transfer coefficient to this surface of 25 W/m²K.

Determine the thermal conductivity of material B

[1.5 W/m.K]
 

2. A composite wall separates combustion gases at 2600°C from a liquid at 100°C, with respective coefficients of 50 and 1000 W/m²K. The wall is composed of a layer of beryllium oxide 10 mm thick on the gas side and a slab of stainless steel 20 mm thick on the liquid side. The poor contact between the two gives rise to a resistance between the oxide and the steel of 0.05 m²K/W.

 

 
 
 

What is the heat flux through the composite?

Sketch the temperature distribution from the gas to the liquid.

Thermal conductivities;	Beryllium oxide	272 W/m.K
	Stainless steel	14.9 W/m.K.

[33 900 W/m²]
 

3. A composite wall of height H and of unit depth perpendicular to the page is insulated at its ends and is comprised of four different materials as shown in the diagram below.
(a) sketch the analogue electrical circuit of the system;

(b) consider a wall for which H = 3.0 m, H_B = H_C = 1.5 m, L₁ = L₃ = 0.05 m, L₂ = 0.10 m, k_A = k_D = 50 W/m.K, k_B = 10 W/m.K, and k_C = 1 W/m.K. Under conditions for which

T_¥,1 = 200°C, h₁ = 50 W/m²K,
T_¥,2 = 25°C, h₂ = 10 W/m²K. determine the rate of heat transfer through the wall. What are the interfacial temperatures T₁ and T₂?

[3745 W; 173.8, 151.1°C]
 

4. A 0.2 m diameter thin-walled steel pipe is used to transport saturated steam at an absolute pressure of 20 bar in a room for which the air temperature is 25°C and the convection heat transfer coefficient at the outer surface of the pipe is 20 W/m²K.

 

 
 
 

(a) Considering convection losses only determine the rate of heat loss per metre length of pipe.

(b) Recalculate the heat loss on the assumption that the pipe has been insulated with a 50 mm thick layer of magnesia (k = 0.28 W/m.K).

(c) The costs associated with generating steam and installing the insulation are known to be £4 per 10⁹ J and £100 per metre length of pipe length respectively. If the steam line is to operate for 625 hours per month, how many months are needed to pay back the investment in the insulation?

[2355 W/m, 661 W/m, 6.6 months]
 

5. A storage tank consists of a cylindrical section that has a length of 2 m and an inner diameter of 1 m. The end sections are hemispherical. The tank is constructed from 20 mm thick glass (k = 1.4 W/m.K) and is exposed to ambient air for which the temperature is 300 K and the convection coefficient is 10 W/m²K. The tank is used to store heated oil that maintains the inner surface at a temperature of 400 K.

 

 
 
 

Determine the electrical power needed to supply a submerged heater in order to maintain the oil temperature.

[8700 W]
 

6. A hollow aluminium sphere (k = 237 W/m.K) with an electrical heater at the centre is used to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.15 and 0.18 m respectively and the testing is done under steady state conditions with the inner surface of the aluminium maintained at 250°C.

 

 
 
 

In a particular test a spherical insulating shell is cast on the outer surface of the aluminium sphere to a depth of 0.12 m. The system is in a room for which the air temperature is 20°C and the convection coefficient from the sphere is 30 W/m²K. The heater dissipates 80 W.

Determine the thermal conductivity of the insulation.

[0.0621 W/m.K]
 

7. A hot cylinder is insulated with a layer of insulation of conductivity k and loses heat to atmosphere from the surface of this insulation with a heat transfer coefficient h. Show that the heat loss from the cylinder will be a maximum when the outer surface of the insulation has a radius of k/h. Find a typical radius for a lagged domestic hot water pipe.

 

 
 
 

Determine the equivalent critical radius for the insulation of a sphere, and try to name an instance where insulating only as far as the critical radius is a GOOD idea.

[r = 2k/h]
 

8. Calculate the thickness of insulating material of thermal conductivity 0.07 W/m.K necessary to reduce the heat loss from a hot-water tank to 25% of the unlagged loss.

 

 
 
 

Assume that the tank is a perfect conductor and that the heat transfer coefficients at the surface due to convection are 3.5 W/m²K for the unlagged and 6.8 W/m²K for the lagged case. The dimensions of the tank may be assumed to be sufficiently large for the areas of the unlagged and lagged tank to be the same.

[70 mm]
 

9. North Sea oil enters the well head at 32°C and flows at a rate of 100kg/s down 10 km of 0.9m OD steel pipe 1.6 cm thick, coated on the outer surface with a 2 cm layer of protective material. Assuming that the average sea temperature is 5°C and a linear temperature profile throughout the length of the pipeline, what will be the exit temperature of the oil? What would this temperature be for a 100 km pipeline?

Thermal conductivity of steel	= 40 W/m.K
Thermal conductivity of coating	= 20 W/m.K
Heat transfer coefficient on outer surface	= 30 W/m2K
Heat transfer coefficient on inner surface	= 10 W/m2K
Specific heat capacity of oil	= 2.2 kJ/kg.K

[T = 15°C for 10 km, for 100km common sense is required.]
 

10. Butyl rubber cement solution is to be stored at 460 K in a domed storage vessel equipped with mechanical stirring to maintain a uniform temperature. The designers require to estimate the maximum heat duty of the steam heaters that will be required to maintain a full insulated tank at this temperature in the depth of winter when the air temperature will be 260 K.

 

 
 
 

The vessel may be regarded as a 10 m high cylinder with one hemispherical dome of internal diameter 10 m. The shell is an epoxy lined carbon steel insulated by mineral wool and weather-proofed by thin aluminium sheeting which may be regarded as a perfect conductor.

Use the following data to estimate the heat duty:

	wall thickness (m)	thermal conductivity (W/m.K)
epoxy coating	0.002	0.5
carbon steel	0.01	3.0
mineral wool	0.02	0.025

Heat transfer coefficient on outer surface = 15 W/m²K
Heat transfer coefficient on inner surface = 20 W/m²K

The contact coefficients between surfaces may be ignored and the vessel may be assumed to be perfectly insulated on the ground face.

[74 kW]
 

11. A refrigerated room is to be constructed from brick 0.1m thick, insulated externally by cork that is protected by wood 0.02m thick. Estimate the thickness of cork that would be necessary to prevent ice forming on the outside walls when the temperature of the inside wall is -10°C and the ambient temperature is 20°C.

1. Water flows at a rate of 2.0 kg/s through a 40 mm internal diameter tube of length 15.0 m. The water enters the tube at 25°C and the surface temperature of the tube is constant at 90°C.

 

 
 
 

Calculate the temperature at which the water leaves the tube and the corresponding heat transfer rate to the water.

Physical property data for water are given in Steam Tables. Assume a mean water temperature of 50°C and check your assumption.

[79.5°C, 455.6 kW]
 

2. A cooling coil, consisting of a single length of tubing through which water is circulated, is provided in a reaction vessel, the contents of which are kept uniformly at 360 K by means of a stirrer. The inlet and outlet temperatures of the cooling water are 280 K and 320 K respectively. What would the outlet water temperature become if the length of the cooling coil were increased 5 times? Assume the overall heat transfer coefficient to be constant over the length of the tube and independent of the water temperature.

 

 
 
 

[357.5 K]
 

3. Under what conditions, if any, would you recommend the use of co-current rather than counter-current flow in a single pass heat exchanger?

 

 
 
 

A double pipe heat exchanger is used to cool a process liquid from 410 K to 350 K, with cooling water flowing through the annulus. The cooling water enters at 285 K at a rate of 0.18 kg/s and the process liquid flows at a rate of 0.23 kg/s. If the thermal resistance of the pipe wall can be neglected, and the mean diameter of the pipe is 1.5 cm, estimate the length of the heat exchanger required, assuming the flow to be counter-current. Compare your answer with that required if co-current flow were used.

Film coefficient of heat transfer for process liquid	= 2.3 kW/m2K
Film coefficient of heat transfer for water	= 6.2 kW/m2K
Specific heat of process liquid	= 2.3 kJ/kg.K
Specific heat of water	= 4.18 kJ/kg.K

[5.5 m; 6.7 m]
 

4. When designing a multi-pass heat exchanger the log mean temperature difference calculated on the basis of true counterflow has to be corrected. Explain why this is necessary and indicate the weaknesses of the method usually adopted.

 

 
 
 

The feed to a distillation column is to be pre-heated from 290 K to 370 K in a 1-2 heat exchanger using the bottom product from the column as the heating medium. If the bottom product leaves the column at 415 K and is to be cooled to 365 K, determine whether an exchanger having 100 tubes, of mean diameter 2.5 cm and arranged in two passes, will be suitable. The feed is passed at a rate of 5 kg/s through the tubes and the tube length is 3 m.
 

Overall coefficient of heat transfer	= 1.5 kW/m2K
Specific heat of feed	= 3.8 kJ/kg.K

Comment upon:

1. the reliability of the design calculation carried out, bearing in mind the value of F obtained;
2. the length to diameter ratio for the tubes used in the available heat exchanger.


[Satisfactory]
 

5. It is desired to warm an oil of specific heat 2.0 kJ/kg.K from 300 K to 325 K by passing it through a tubular heat exchanger with metal tubes of inner diameter 10 mm. Along the outside of the tubes flows water, inlet temperature 372 K, and outlet temperature 361 K.

 

 
 
 

The overall heat transfer coefficient from water to oil, based on the inside area of the tubes, may be assumed constant at 230 W/m²K, and 75 g/s of oil is to be passed through each tube. The oil is to make two passes through the heater. The water makes one pass along the outside of the tubes. Calculate the length of the tubes required.

[5.09 m]
 

6. A heat exchanger is required to cool continuously 20 kg/s of warm water from 360 K to 335 K by means of 25 kg/s of cold water, inlet temperature 300 K. Assuming that the water velocities are such as to give an overall efficient of heat transfer of 2 kW/m²K, assumed constant, calculate the total area of surface required:
1. in a counterflow heat exchanger, i.e. one in which the hot and cold fluids flow in opposite directions;
1. in a multi-pass heat exchanger, with the cold water making two passes through the tubes, and the hot water making one pass along the outside of the tubes.
In case (b) assume that the hot-water flows in the same direction as the inlet cold water, and that its temperature over any cross-section is uniform.   [27.95 m²; 30.06 m²]  
7. A heat exchanger contains 450 tubes, each being 2 cm in diameter and 3 m long. Untreated pressurised water passes through the tubes where it is heated from 410 K to 500 K by saturated steam condensing at 40 bar (gauge) outside the tubes. The water flows at a total rate of 37.5 kg/s. Initially, the heat exchanger was capable of attaining the water outlet temperature of 500 K, but as time went on, this exit temperature dropped, and finally stabilised at 470 K. If this loss of performance is assumed to be due to scale formation on the inside of the tubes, estimate the thermal resistance of this scale, assuming the tubes to be thin-walled and of negligible thermal resistance.

 

 
 
 

What temperature must the saturated steam be raised to so that the water exit temperature returns to its required value? Is this a good idea?

Specific heat of water = 4.18 kJ/kg K

[0.000376 m²K/W; 307°C]
 

8. A liquid has its temperature raised from 15°C to 65°C by passing it through a heater in which the heating surface is maintained at 125°C by condensing steam. If the overall coefficient of heat transfer can be assumed to vary as the velocity raised to the power 0.8, what will be the exit temperature of the liquid if its velocity is doubled? Assume liquid thermal properties are temperature-independent over the range concerned.

 

 
 
 

[60.15°C]
 

9. A process liquid is heated in the tubes of a shell and tube heat exchanger, where vapour condenses outside the tube. For a fixed flowrate of vapour, the following values of the fluid velocity and the overall coefficient of heat transfer were measured:

v (m/s)	1.000	1.139	1.322	1.564	1.898	2.378
U (W/m2K)	2174	2326	2500	2688	2924	3195

Regard the tubes as thin-walled and of negligible thermal resistance. Use the above data to estimate the film heat transfer coefficient of heat transfer of the condensing vapour, and the liquid film heat transfer coefficient for a velocity of 0.8 m/s. State any assumptions made. (This form of data investigation is called a WILSON PLOT).

[h_o = 6015 W/m²K; h_i = 2866 W/m²K]
 

10. Air is being pre-heated by passing it through tubes, which have their temperature held constant at 110°C by electrical heating. Normally, the air enters at 15°C and leaves at 80°C. On a particular day, the air requirement is increased by 50%. Can the heater deliver the increased flow of air without the exit air temperature dropping below 75°C?

 

 
 
 

Assume that the thermal properties of air are independent of temperature over the range concerned. State any other assumptions made.

[Yes]
 

11. In a multi-pass heat exchanger why is it necessary to correct the log mean temperature difference based on true counterflow? Outline briefly the method usually adopted.

 

 
 
 

A 1-4 heat exchanger (i.e. one shell-side pass, four tube-side passes) is to be designed to heat liquid from 300 K to 370 K. The heating medium enters the shell-side at 430 K and leaves at 380 K. If 10 kg/s of cold liquid is supplied, find the heat transfer area required and suggest a suitable tube arrangement.

If the hot fluid outlet temperature drops to 360 K, is the design method still valid? If not, why not?

Overall coefficient to heat transfer	= 1.3 kW/m2K
Specific heat of both liquids	= 3.8 kJ/kg K

[34.2 m², No]
 

12. The feed to a distillation column is to be pre-heated from 400 K to 510 K by passing it through the tubes of a shell and tube heat exchanger at a rate of 30 kg/s. Saturated steam at 50 bar (absolute) is to condense outside the tubes. A heat exchanger, containing 500 tubes of 2 cm diameter and 3 m length, is available. Assuming the tubes to be thin-walled and of negligible thermal resistance, determine whether this exchanger would be suitable ("rating" the exchanger). The overall coefficient of heat transfer, on a clean basis U_C, is 3.2 kW/m²K and the fouling resistance on the tube side is 0.00037 m²K/W. It may be assumed that the condensing steam is clean.

 

 
 
 

If the heat exchanger cannot meet the conditions required, by how much must the steam temperature be raised to meet the design specification?

Specific heat of tube-side fluid = 3.6 kJ/kg.K

[No, 15.4 deg C]
 

13. Derive, from first principles, the relationship that gives the cross-sectional area "S" of shell-side flow perpendicular to the tube bundle as:

Show, with some numerical example as justification, why the above can be approximated to:

[solution]

1. An aluminium kettle has a 20 cm diameter and a 1.2 mm thick bottom. The heat load for boiling 0.833 grams per second of water at 100°C is 1883 W. Find the flame temperature and the kettle base temperature for the following values of the heat transfer coefficients:

 

 
 
 

h_i (boiling water) = 4000 W/m²K; h_o (gas flame) = 40 W/m²K

SOLUTION:


U = 10⁶ ¸ 25256 = 39.6 W/m²K

Thus U is not much smaller than h_o; the outside resistance is the controlling resistance.
Q’ = 1883 W
A = p ´ 10^-2 m². .
Q’ = UA (T₀ - T₃)

T₃ = 100°C
T₀ = 1615°C

Also Q’ = h_oA (T₀ - T₁)

T₁ = 115°C

From lectures:
(T₁ - T₂) = 0.35 deg K
T₂ = 114.7°C

Alternatively, as a check: Q’ = h_iA (T₂ - T₃)

T₂ = 115°C
 

2. Water at 80°C flows through a copper pipe d_i = 1.8 cm, d_o = 2.0 cm, k = 380 W/m.K. The surrounding air is at 20°C. Correlations have been used to predict the values of the coefficients as:

h_i = 8000 W/m²K; h_o = 15 W/m²K.

Calculate the rate of heat loss per unit length of pipe.

SOLUTION:
The individual thermal resistances can be calculated:


Hence, R_o is controlling, and we can write:

or U_o = h_o = 15 W/m²K
= 15 ´p´ 0.020 ´ (80 - 20) = 56.6 W
 

3. The pipe in Example 2 is lagged with a 1 cm thickness of magnesia insulation [k = 0.06 W/m.K]. The predicted outside transfer coefficient is h_o = 10 W/m²K. What is the percentage reduction in the rate of heat loss?

 

 
 
 

SOLUTION:
We have four resistances in series:


(neglecting two small resistances)




U_o = 3.02 W/m²K (cf .15 W/m²K for unlagged pipe)

For 1 metre length:
Q’ = 3.02 ´ (p´ 0.04 ´ 1) ´ 60 = 22.8 W

% Reduction in heat loss = [100 ´ (55.6 - 22.8)] ¸ 55.6 = 59%
 

4. A stream of oil, with specific heat 2.4 kJ/kg.K, flowing at 1000 kg/hr is to be cooled from 130 to 80°C in a double pipe heat exchanger, using 1200 kg/hr water at 20°C as the cooling medium. The predicted overall heat transfer coefficient, based on the outside diameter of the insider pipe, is 360 W/m²K. Find the heat transfer area required if the flow is to be (a) co-current and (b) counter-current.

 

 
 
 

SOLUTION:
(a)



Q’ = m’_A C_PA (T_A1 - T_A2) = m’_B C_PB (T_B2 - T_B1)
= 1000 ´ 2400 ´ 50 = 1200 ´ 4200 ´ (T_B2 - 20)
T_B2 = 44°C


(b) Q’ and T_B2 will be the same



Note: counter-current flow requires less area for the same terminal temperature.
 

5. A double pipe heat exchanger contains a 6m long tube of 5cm OD. It is being used to condense 100 kg/hr of saturated steam at atmospheric pressure. Cooling water enters the tubes at 25°C and leaves at 38°C. Calculate the water flow rate m’_W (kg/s) and the observed overall heat transfer coefficient.

SOLUTION:   One phase is at constant temperature, so although this type of exchanger does not provide true counter-current or co-current flow, we may still use:   Q’ = UA DT_LM   The rate at which the condensing steam gives up heat is given by the energy balance equation: Q’ = m’_S l_fg

where the enthalpy of condensation of steam, l_fg, at 1 atm is 2260 kJ/kg (from steam tables).

  Q’ = (100 ¸ 3600) ´ 2260 = 62.8 kW   Heat is transferred into the cooling water at the same rate:   Q’ = m’_W C_P (t₁-t₂)
62.8 = m’_W ´ 4.18 ´ (38 - 25)
m’_W = 1.16 kg/s   Area of outside of tubes:
A_o = p ´ 0.05 ´ 6 = 0.3p m²
  Note that the arithmetic mean (= 68.5) could have been used in this case.

1. Without reference to tables or charts calculate the factors F₁₂ and F₂₁ for each of the following configurations;



[0.5, 0.3714; 1, 0.4142]

 
2. A cube of 0.6 m side has a disc of 0.2 m diameter placed in the centre of the base. Determine the view factors between;
1. the disc and the inside of the cube
2. the inside of the cube and the disc
3. that part of the cube base not covered by the disc and the inside surface
4. the inside surface of the cube and that part of the base not covered by the disc

 
[1, 0.0175, 1, 0.1825]
 
3. Two surfaces at right angles one 0.3 ´ 0.6 m and the other 0.3 ´ 0.7 m share a common short edge. Making use of the appropriate chart determine the two view factors of the surfaces relative to each other.

 

 
 
 

Each long surface is divided into two equal parts by a line perpendicular to its longer edge. Determine the view factors of the two surfaces remote from the common edge relative to each other.

[0.1326, 0.1547; 0.0280, 0.0326]
 

4. A ring of 0.8 m outside and 0.4 m inside diameter lies 0.5 m above a smaller ring of 0.4 m outside and 0.2 m inside diameter. The two discs are concentric and parallel. Determine the view factor of the larger ring with reference to the smaller one.

Determine view factors F₁₂ and F₂₁ for the following geometries:

1. Sphere (1) of inside a cubical box (2) of side length L equal to sphere diameter D

 

 
 
 



2. Diagonal partition (1) within a long square duct (2 and 3)

 

 
 
 



3. End (1 or 3) and inner side (2) of a circular cylinder of equal length L and diameter D.

(a) As the sphere is convex, no radiation will reach one part of its surface from another, i.e. all radiation from the sphere goes to the box
F₁₂ = 0

Rule of reciprocity: A₁F₁₂ = A₂F₂₁

Surface areas are p D² for the sphere and 6D² for the box:

i.e. 52.36% of the radiation from the box goes to the sphere. The other 47.64% travels between different faces of the box.

(b) Rule of summation: F₁₁ + F₁₂ + F₁₃ = 1
Surface 1 is perfectly flat: F₁₁ = 0
By symmetry F₁₂ = F₁₃
F₁₂ = 0.5 = F₁₃

Rule of reciprocity: A₁F₁₂ = A₂F₂₁

(c) Using formulae for coaxial parallel discs:

Surface 1 is perfectly flat: F₁₁ = 0

Rule of summation: F₁₁ + F₁₂ + F₁₃ = 1
F₁₂ = 1 - 0 - 0.17157 = 0.82843

Rule of reciprocity: A₁F₁₂ = A₂F₂₁

For circular cylinder, areas are 0.25p D² for surface 1 and p D² for surface 2:

1. Two parallel square plates, each with sides measuring 2.00 metres, are separated by a distance of one metre. The plates may be considered as black bodies at temperatures of 500°C and 1000°C respectively. The plates are placed in a large room at 20.0°C. Ignoring heat transfer from the rear of the plates, determine the net rate of radiative heat transfer to or from each plate and the net rate of heat transfer to the surroundings.

 

 
 
 

[561 kW from hot plate, 167 kW to cold plate, 394 kW to surroundings]
 

2. A circular plate of 1m diameter is heat treated by mounting it opposite a hemi-spherical dome of the same diameter. The temperature of the plate is 400°C and that of the dome is 850°C. Both surfaces may be assumed to be perfectly black. The distance between the plate and the bottom of the dome is 0.50 m. The surfaces are mounted in large surroundings at 15°C. Ignoring heat transfer from the rear of the plate, determine the net rate of radiative heat transfer to the plate.

 

 
 
 

[18 kW]
 

3. Two 2 m square black parallel plates are placed a metre apart and the sides of the resulting box enclosed by well insulated walls. The upper plate is at 1000°C and the lower one at 500°C.  Determine the mean temperature of the side walls.

 

 
 
 

It is found that in fact there is a heat loss through the side walls of 10 kW/m². Determine the mean temperature of the insulated walls for this case and the net amount of heat transferred to the lower plate.

[832°C, 771°C, 412 kW]
 

4. A vessel for producing steam contains water at 100°C and is mounted on insulated walls 1.00 m above a heater. Both the heater and the vessel have diameters of 1.50 metres and the heater is at 1500°C.