Coker A.K. Fortran Programs for Chemical Process Design, Analysis, and Simulation
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Heat Transfer 657
I II III IV
dq' = Mc dt
_ WhCh(T ' _ T2 ) _ UAAtLMTD
(8-141)
dO dO
The log mean temperature difference,
AtLMTD
is"
AtLMTD ---- T1 -- T2 (8-142)
(w,-t /
In T2-t
Equating III and IV in Equation 8-141 and rearranging gives:
WhC h
(Tl - T 2 ) Tl -- T 2
UA
ln/Wl-t /
T 2 -t
(8-143)
Equation 8-143 becomes"
(T~ -t / _ UA
In Tz-t WhC h
(8-144)
T 1 -
t
UA/WhC h
= e (8-145)
T: -t
Rearranging Equation 8-145 gives"
Y l
-t
T 2 - t + UA/WnC" (8-146)
e
where
K l - eUA/WhCh (8-147)
Equating II and III in Equation 8-141 and substituting Equation 8-146
in Equation 8-141 gives"
( { W t}/
Mc dt _W C h T - t+ ~ (8-148)
dO
h 1
Kl
658 Fortran Programs for Chemical Process Design
K l
-1]
: WhC h i~ I
(Z I -t)
(8-149)
t~ OWhCh/Kl -1/
I,- dt = So d0 (8-150)
TI - t Mc K~
Rearranging and integrating Equation 8-150 from tl to
t 2
while the time
passes from 0 to 0 gives:
t,)/WhCh//K, 1)0
In Ti-t~ - Mc K,
(8-~5~)
where A = heat transfer surface
c = specific heat of batch liquid
C h = heating medium specific heat
M = weight of batch liquid
TI = heating medium temperature
tl = initial batch temperature
t 2 =
final batch temperature
U = overall heat transfer coefficient
W h = heating medium flow rate
0 = time
Batch Cooling" Non-Isothermal Cooling Medium
When cooling a batch with internal coil and a non-isothermal cool-
ing medium, the following equation can be applied.
dq' = -Mc d__TT _ WcC~ (t 2 _ t, ) - UAAtLM~D (8-152)
dO dO
UA/W~C~
where K 2 = e
and
/ C )(
tl _WcCc
In
-
Mc K 2
(8-~53)
where A = heat transfer surface
c = specific heat of liquid
Heat Transfer 659
Cc = coolant specific heat
M = weight of batch liquid
T~ = initial batch temperature
T 2 = final batch temperature
t~ = initial coolant temperature
U = overall heat transfer coefficient
Wc = coolant flow rate
8 = time
Batch Heating Through An External Heat
Exchanger, Isothermal Heating Medium
Figure 8-17 illustrates the arrangement in which the fluid in the tank
is heated by an external heat exchanger. The heating medium is isother-
mal, therefore any type of exchanger with steam in the shell and tubes
can be used. The variable temperature out of the exchanger, t',will dif-
fer from the variable tank temperature, t. An energy balance around the
tank and the heat exchanger gives:
M, t
1
t
/
t::
N h Ch
Figure
8-17. Batch heating through an external heat exchanger, isothermal heat-
ing medium.
660 Fortran Programs for Chemical Process Design
I II III
dq'
dt
= Mc
dt
dO
-- W hC h (t'
- t)
- UAAtLMTD
Heat Heat enter- Transfer
accumu- ing the rate in the
lation in batch by external
the batch recirculation exchanger
The log mean temperature difference,
AtLMTD
is"
At
LMTD =
(T,- ti)-(T ~
-t')
In( T~- - t' )
Tj - t'
(8-154)
(8-155)
t p ~ t I
In( T~- - tl )
T I - t'
Equating II and III in Equation 8-154 gives:
WhCh(t" --
t)
- UAAtLMTD
That is"
WhC h (t' - t) - UA
(t'- t)
,n{T, t;}
T l -
t'
Rearranging Equation 8-158 gives
(T~-t)_ UA
In ~i_t ' WhCh
Equation 8-159 can be expressed as"
T l - t
- e UA/w~C~
(T~ - t')
(8-156)
(8-157)
(8-158)
(8-159)
(8-160)
Heat Transfer 661
where
UA/WhC h
K3=e
T l -
t
= K 3(Tj
- t')
(8-161)
Therefore
T l -t]
t'-
T l -
K3
(8-162)
Equating I and II in Equation 8-154
dt
Mc
= WhC h
(t' - t) (8-163)
dO
Substituting Equation 8-162 into Equation 8-163 and rearranging gives:
Me . dt fT,_(T,-t/t_t
W h C h dO
K 3
(8-164)
_ (K 3 - 1)(T, - t)
K3
(8-165)
8_,66
St, Tj - t - K 3 Mc
Rearranging and integrating Equation 8-166 from tj to
t 2
while the time
passes from 0 to 0 gives:
IT,-tl)-/K3 -'//WcCh/0
In TI- t2 K 3 Mc
(8-167)
where A = heat transfer surface
c = specific heat of batch liquid
C h = heating medium specific heat
M = weight of batch liquid
T~- heating medium temperature
t~ = initial batch temperature
662 Fortran Programs for Chemical Process Design
t 2 =
final batch temperature
U = overall heat transfer coefficient
W h = heating medium flow rate
0 = time
Batch Cooling Through an External Heat
Exchanger, Isothermal Cooling Medium
When cooling a batch with an external heat exchanger and an iso-
thermal cooling medium, the equation is:
ln/W' t') /WcCc//K4'/~ Mc K4
UA / W~C~.
where
K 4 =
e
A = heat transfer surface
c = specific heat of liquid
C c = coolant specific heat
M = weight of batch liquid
T 1 = initial batch temperature
T 2 - final batch temperature
t 1 = initial coolant temperature
U = overall heat transfer coefficient
W c = coolant flow rate
0 = time
(8-168)
Batch Cooling: External Heat Exchanger (Countercurrent
Flow), Non-isothermal Cooling Medium
When cooling a batch with an external heat exchanger and a non-
isothermal cooling medium, the following equation can be used.
T~ - t I M
ln(T 2 _t])-( Ks-
WbWcCc /0
K5WcC-~ - ~VbC
(8-169)
where K s = exp{ UA(
1AVbC -
1/WcC c) }
A = heat transfer surface
c - specific heat of liquid
C c = coolant specific heat
M = weight of batch liquid
Heat Transfer 663
T~ = initial batch temperature
T 2 = final batch temperature
tl = initial coolant temperature
U = overall heat transfer coefficient
W b = batch flow rate
W c = coolant flow rate
0 = time
Batch Heating: External Heat Exchanger
and Non-isothermal Heating Medium
When heating a batch with an external heat exchanger and non-
isothermal heating, the following equation applies:
/ WbWhCh /0
In Ti-i~ - M K6WhC h-~Vbc
where K 6 = exp{UA(1/WbC
- l[V~rhCh) }
A = heat transfer surface
c - specific heat of batch liquid
C h - heating medium specific heat
M = weight of batch liquid
T~ = heating medium temperature
t~- initial batch temperature
t 2 = final batch temperature
U = overall heat transfer coefficient
W b = batch flow rate
W h = heating medium flow rate
0 = time
Batch Heating: External Heat Exchanger (1-2 Multipass
Heat Exchangers), Non-isothermal Heating Medium
The procedure used for batch heating with external 1-2 multipass
heat exchangers with non-isothermal heating media involves using the
same heat balance as defined by the following equation:
dq'
de
= Mc
I
dt
dO
II III IV
-- WbC(t'-
t)-
WhC h
(T,- T2)- UAAtLMTD
(8-171)
664
Fortran Programs for Chemical Process Design
Equating I and
II
in Equation 8-171
dt
Mc
= WbC (t'
- t) (8 - 172)
dO
Rearranging Equation 8-172 gives:
M
dt
t' = t + ~. (8-173)
W b
dO
The parameter S can be defined by
t, t /M//1 /at
S - = (8-174)
T~ - t Wb T ! - t dO
The parameter R can be defined by equating II and Ill in Equation 171
R - T~ - T 2
= WbC (8-175)
t' -
t W hC h
Rearranging Equation 8-174 gives:
i~_ ~ dt
__ __
S. Wb i~d ~
(8-176)
T~-t M
Integrating from t~ to
t2, as
the time passes from 0 to 0 yields:
/_t,/(s Wb 0
In T' t2 - M
f I -
(8-177)
and
S .__
2(K 7 - 1)
K7{R + 1+ (R 2 + 1) 0.5 } -{R + 1-(R2+ 1) 0.5 }
where K 7 = exp{ (UA/WbC)(R 2 + 1)~
A = heat transfer surface
c = specific heat of batch liquid
C h = heating medium specific heat
M = weight of batch liquid
T~ = initial temperature of heating medium
t~ = initial batch temperature
t 2 = final batch temperature
(8-178)
Heat Transfer 665
U = overall heat transfer coefficient
W b = batch flow rate
W h = heating medium flow rate
0 = time
Batch Heating and Cooling: External Heat Exchanger
(2-4 Multipass Heat Exchangers)
The same procedure can be used as in 1-2 multipass heat exchangers.
The equation for batch heating with external 2-4 multipass heat
exchangers and non-isothermal heating media is
In T~ - t~ _ )0
(8-177)
where
2(K - 1)[1 + {(1- S)(1- RS)} ~
S - s (8-179)
(K s - 1)(R + 1)+ (K s + 1)(R: + 1) o.5
and
UA
K 8 - exp
2WbC
(R 2 + 1)~ }
(8-180)
Because S cannot be expressed explicitly, Equation 8-179 can be solved
only by trial and error, assuming different values of S until an equality
is attained. The equations for batch heating and cooling are the same as
those developed for the 1-2 multipass exchangers, except that the value
of S from Equation 8-179 replaces the value of S from Equation 8-178.
PROBLEMS AND SOLUTIONS
Problem 8-1
Kerosene leaves the bottom of a distillation column at 390~ and will
be cooled to 200~ by crude oil from storage at 100~ and heated to
170~ Calculate the number of shell passes required, the F-factor, and
the corrected LMTD.
Hot kerosene inlet temperature = 390~
Hot kerosene outlet temperature = 200~
666 Fortran Programs for Chemical Process Design
Crude oil inlet temperature
= 100~
Crude oil outlet temperature
- 170~
Solution
The computer program PROG81 calculates the required number of
shell passes, the F-factor, the Log Mean Temperature Difference
(LMTD), and the Corrected Mean Temperature Difference (CMTD).
Table 8-21 shows the input data and computer output of the problem.
The number of shells required is 1, the F-factor is 0.8917, the LMTD is
152.2~ and the corrected LMTD is 135.7~
Problem 8-2
Acetone (s = 0.79) at 250~ is to be sent to storage at 100~ and at a
rate of 60,000 lb/hr. The heat will be received by 185,000 lb/hr of 100
percent acetic acid (s = 1.07) coming from storage at 90~ and heated to
150~ Pressure drops of 10.0 psi are available for both fluids. Assum-
ing that the fouling factor on the tube side is 0.001 and that on the shell
side is 0.003, calculate the heat transfer coefficients for the tube and
shell sides, the overall heat transfer coefficient for the exchanger, out-
side area of unit, and the heat transferred.
Available for the service are a large number of 1-2 exchangers hav-
ing 21 1/4 in. ID shells with 270 tubes 3/4 in. OD, 14 BWG, 16 ft long
Table 8-21
Input Data and Computer Output for the Corrected Mean
Temperature Difference in a Shell and Tube Heat Exchanger
DATA81.DAT
390.0 200.0
I00.0 170.0
i.
THE CORRECTED LMTD IN A SHELL AND TUBE HEAT EXCHANGER
HOT FLUID INLET TEMPERATURE, oF:
HOT FLUID OUTLET TEMPERATURE, oF:
COLD FLUID INLET TEMPERATURE, oF:
COLD FLUID OUTLET TEMPERATURE, oF:
NUMBER OF SERIES EXCHANGER SHELLS:
THE NUMBER OF SHELLS REQUIRED:
THE F- FACTOR:
THE LOG MEAN TEMPERATURE DIFFERENCE,oF:
THE CORRECTED LMTD, oF:
__-
390.000
200.000
I00.000
170.000
i.
i.
.8917
152.1959
135.7112