Heat Transfer Tutorial 2 (with Solutions)

1. It is required to heat a process stream flowing at 5kg/s with heat capacity 2 kJ/kg/K from 60oC to 100oC. What heat duty will this represent?

Q = G x Cp x DT = 5 x 2 x (100-60) = 400kW 

Note the units because Cp is in kJ/kg/K

Two alternative heating fluids are available, both at 150oC :

For each case determine the flowrate of heating medium required. At what temperature would you expect the condensed steam to leave the heat exchanger?

For steam Q = G x L 
L = 2100 kJ/kg
400 = G x 2100 so G = 0.19 kg/s
Since all the energy comes from latent heat not cooling, the condensed 
steam should leave at the same temperature as it entered, i.e. 150C.

In practice it will be slightly cooled.

For gas Q = G x Cp x DT 
using the maximum amount of temperature change available
400 = G x 1.5 x (150-100)
G = 400 / (1.5 x 50) = 5.33 kg/s

2. A small countercurrent heat exchanger operates with the following stream temperatures:

cold stream in 20oC ; cold stream out 100oC
hot stream in 120oC ; hot stream out 70oC

The unit has total area for heat transfer of 1 m²and overall heat transfer coefficient of 500W/m²/K. What is the rate of energy transfer ?

For countercurrent operation:
`cold' end driving force will be (70-20) = 50 deg C
`hot' end will be (120-100) = 20 deg C

Log mean is (50-20) / ln (50/20) = 30 / 0.916 = 32.7 deg C

Q = U A (Theta) = 1 x 500 x 32.7 = 16,370 W

3. Estimate the film coefficients for flow of an organic liquid of density 800 kg/m³, viscosity 0.0008 kg/m/s and thermal conductivity 0.2 W/m/K which flows at a mean velocity of 1 m/s:

Re = u d ro / mu = 1 x 0.04 x 800 / 0.0008 = 40,000
To power 0.8 = 4804
To power 0.6 = 577

Nu (inside) = 0.046 x 4804 = 221
Nu (outside) = 0.24 x 577 = 138.5

Nu = h d / k so h = Nu k / d = Nu x 0.2/0.04 = 5 Nu

hi = 5 x 221 = 1105 W/m2/K
ho = 5 x 138.5 = 692 W/m2/K

Estimate the overall heat transfer coefficient if heat is to be transfered between these two fluid streams.
It will be reasonable to neglect wall resistance, so
1/U = 1/1105 + 1/692 = 0.00235
So U = 425.5 W/m2/K

4. It is required to preheat the feed stream to a chemical reactor from 20oC to 80oC . The stream flowrate is 20 kg/s of material having a specific heat capacity of 4 kJ/kg/K.

From these figure we can calculate the required heat duty:

Q = G Cp DT = 20 x 4 x (80-20) = 4800 kW

Two streams of hot fluid are available which could be used to exchange heat with the above feed stream. Hot stream (1) is pressurised water at 110oC, available at a flowrate of 30 kg/s and has a heat capacity 4.2kJ/kg/K. Hot stream (2) is a light oil at 220oC available at a flowrate of 10 kg/s and has a heat capacity of 3 kJ/kg/K.

We can work out the outlet temperature for each case.
Again Q = G Cp DT = 4800

Water:
4800 = 30 x 4.2 x DT so DT ~ 38 deg C 
Water will come out at (110-38) = 72C

Oil:
4800 = 10 x 3 x DT so DT = 160 deg C
Oil will come out at (220-160) = 80C

The expected overall heat transfer coefficients are 500W/m²/K in an exchanger with water, and 200W/m²/K with oil.

Which heating fluid would you recommend to be used, on the basis of the above information? Briefly describe any other factors which might be relevant to the choice between these two alternatives.

The main factor is going to be the the size of the exchanger, so work out
heat transfer areas.

Water
Driving forces are:
Cold end (72-20) = 52
Hot end (150-80) = 70
Log mean = 60.55

Q = U A theta  
Since duty is in kW make U = 0.5 kW/m2/K
4800 = 0.5 x A x 61 
So A = 157m2

Oil
Cold end 80-20 = 60
Hot end 220-80 = 140
Log mean = 94.42

U = 0.2 kw/m2/K
4800 = 0.2 x A x 94.4
A = 254m2

Both are big units but this is nearly twice the size of the water exchanger.

5. The following experimental measurements were taken for the operation of a heat exchanger.

Hot side: Cold side:
flowrate 3.6 kg/s 2.9 kg/s
inlet temperature 200oC 100oC
outlet temperature 150oC 170oC
fluid specific heat capacity 3 kJ/kg/K 2.5 kJ/kJ/K

Dimensions : 200 tubes, each 25mm diameter and 2 m long.

Expected overall heat transfer coefficient 480W/m²/K. Fully countercurrent operation.

Discuss critically.

We are not asked to design a new heat excahnger, but to 
evaluate the performance of an existing unit. 

(i) We have enough information to evaluate the heat dutiues on both sides of
the unit, which should in theory be the same...
Q lost on hot side = 3.6 x 3 x (200-150) = 540 kW
Q gained on cold side = 2.9 x 2.5 x (170-100) = 507.5 kW

The cold side gains less heat than the hot side loses. This makes sense 
since there will inevitably be losses to the ouside. These amount
to 32.5kW or 6% of the total heat supplied. They might be reduced
by better insulation of the unit.

(ii) Since we have the area, temperature driving force and heat duty we
can calculate the actual heat transfer coefficient and compare it with
the `expected' one. This is the most sensible comparison to make
as this is the least certain item in designing any heat exchanger.

The two driving forces are (200-170) = 30 deg C 
and (150-100) = 50 deg C
Log mean is 39.15 deg C

Each tube has an area of 0.025 x pi x 2 = 0.157 m2
There are 200 so total area = 31.4 m2

U = Q / (A Theta) = 507.5 / 31.4 / 39.15 = 0.414 kW/m2/K

This doesn't compare too badly with the design figure of
0.48 kW/m2/K. However, it is low which is actually unexpected, since
when people are designing with uncertain figures they tend to
err on the safe side. It is a reasonable bet that the designer 
who estimated 480 w/m2/K actually believed that the real figure
was significantly higher! Unless this was wrong, the exchanger
is underpreforming and should probably be cleaned.


6. A stream of 10 kg/s of organic material with heat capacity 2 kJ/kg/K is to be heated from 80oC to 120oC . It is proposed to use steam at 3.5 bara as the heating medium. The steam condenses at 140oC and has latent heat of 2732 kJ/kg.

a) Determine the heat duty of the unit.
b) What flowrate of steam will be required?
c) Determine the Log mean temperature driving force
d) If the expected overall heat transfer coefficient is 400W/m²/K determine the area required.
e) Estimate the annual cost of heating the stream, given an 8000 hour working year, steam costing £3 per GJ and an annualised capital cost of £500 per m² of heat exchanger.