Heating and Cooling of Batch Processes

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-HEA-505

Heating and Cooling of Batch
Processes

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Process Engineering Guide:

Heating and Cooling of Batch
Processes

CONTENTS

SECTION

0

INTRODUCTION/PURPOSE

2

1

SCOPE

2

2

FIELD OF APPLICATION

2

3

DEFINITIONS

2

3.1 units

2

4

STATEMENT OF THE PROBLEM

3

5

DEVELOPMENT OF THE METHOD

6

5.1
5.2

Assumptions
Basic Equations

6
6

6

APPLICATION OF THE METHOD

10

6.1
6.2

Determining the Behavior of an Existing System
Specifying the Heat Transfer Duty for a New System

10
10


APPENDICES

A

DERIVATION OF THE EQUATIONS

12

B

WORKED EXAMPLES

26

FIGURES
1

CASES CONSIDERED

5

TABLES
1

DEFINITIONS OF FUNCTIONS

9

NOMENCLATURE

29

DOCUMENTS REFERRED TO IN THIS PROCESS
ENGINEERING GUIDE

30


0

INTRODUCTION/PURPOSE

This Process Engineering Guide is one of a series on heat transfer produced for
GBH Enterprises. It is intended to provide guidance in the estimation of batch
heating and cooling times and the design of heat transfer equipment for this
purpose.

1

SCOPE

This Guide gives methods for estimating the batch heating or cooling times for
cases where the heat transfer performance of the system may be estimated.
Alternatively, it may be used to specify the required heat transfer performance of
the system in order to meet a given heating or cooling time.
It does not give detailed advice on the estimation of heat transfer coefficients nor
the design or rating of heat transfer equipment. Information on this topic may be
found in other guides in the GBH Enterprises heat transfer series. Ratings are
usually performed with the aid of computer programs. See GBHE-PEG-HEA-502
for information on recommended computer programs.

2

FIELD OF APPLICATION

This Guide is intended for process engineers and plant operating personnel in
GBH Enterprises world-wide, who may be involved in the specification, design
or operation of batch equipment with cyclic variations in temperature, such as
batch reactors.

3

DEFINITIONS

For the purposes of this Guide, the following definitions apply:
LMTD

Logarithmic Mean Temperature Difference. The logarithmic mean
of two values X1 and X2 is given by:


3.1

Units

All equations in this guide are in coherent units. Base units for the SI system are
used.

However, the equations are, in general, equally valid if the individual terms are
expressed in any other coherent set of units.
A full list of symbols, with the appropriate base S.I. units, is given at the end of
the Guide.

4

STATEMENT OF THE PROBLEM

Heat exchangers for continuous processes are normally designed to meet a
specified set of process conditions. Although the required duty may vary during
the course of the plant operation, for example during start-up or to accommodate
changes in service fluid temperatures or equipment fouling, the conditions which
determine the exchanger size are usually obvious. These will be used for design,
and the resulting design checked against other conditions.
For a batch system, the problem is more difficult, as the process conditions and
the heat load are varying throughout the batch. It is thus not obvious what
conditions should be used to design the heat transfer system. Equally, given an
existing system, the estimation of the time required to make a given temperature
change is not obvious.
In order to design the heat transfer system, or to check that the given system is
adequate, the required duty as a function of the changing batch temperature
needs to be known. This Guide sets out a method of determining this. The results
are expressed as a series of equations giving the batch temperature, the heat
duty and the inlet and outlet temperatures for all the fluids which may be involved
as a function of time.

There are several different cases that can be identified. These lead to related,
but not identical, solutions. The different cases which need to be considered
depend on:
(a)

whether the batch fluid is:
(1)
(2)

circulated through an external exchanger; or

(3)

(b)

heated or cooled directly through a jacket and/or internal coils;

an intermediate fluid used in conjunction with a jacket and/or coils
and an external heat exchanger.

whether the service fluid is single phase (e.g. cooling water) or isothermal
two phase (e.g. condensing steam).

Figure 1 indicates the systems considered in this Guide. These are:
Case 1

Direct use of an isothermal service fluid, e.g. steam, in the vessel
jacket and/or internal coil.

Case 2

Direct use of a single phase service fluid, e.g. cooling water, in the
vessel jacket and/or internal coil.

Case 3

Recirculating the vessel contents through an external heat
exchanger, with an isothermal service fluid.

Case 4

Recirculating the vessel contents through an external heat
exchanger, with a single phase service fluid.

Case 5

This is a special case of Case 4, where the products of the mass
flowrate and specific heat of the recirculating process fluid and the
service fluid are equal. This leads to certain of the equations in
Case 4 becoming indeterminate, requiring a different formulation.

Case 6

The use of an intermediate heat transfer fluid circulating through
the vessel jacket and/or coil and an external heat exchanger, with
an isothermal service fluid as the ultimate heat source/sink.

Case 7

The use of an intermediate heat transfer fluid circulating through
the vessel jacket and/or coil and an external heat exchanger, with a
single phase service fluid as the ultimate heat source/sink.


Case 8

This is a special case of Case 7, where the products of the mass
flowrate and specific heat of the intermediate fluid and the service
fluid are equal.

Some of these cases are covered in the book 'Process Heat Transfer' by D Q
Kern. However, this coverage is not as comprehensive as given here. Moreover,
there are errors in Kern's treatment.
FIGURE 1

CASES CONSIDERED


5

DEVELOPMENT OF THE METHOD

5.1

Assumptions

In order to simplify the problem, a number of assumptions are made. The validity
of some of these may open to question, particularly for systems whose properties
vary dramatically with temperature. The magnitude of any likely errors may be
checked by comparing the results of detailed heat transfer calculations at the
start and end of the batch with the estimated performance based on the
approach given in this Guide. In extreme cases it will be necessary to perform a
series of detailed calculations to obtain an accurate picture, but the methods in
this Guide may still be useful to obtain a simple over-view of the problem.
(a)

Heat transfer coefficients and specific heats remain constant throughout
the batch cycle.

(b)

The Logarithmic Mean Temperature Difference (LMTD) is used in
calculating heat exchanger performance. The F correction factor to the
LMTD to allow for variations from pure counter-current flow is assumed to
remain constant throughout the batch and is included in the overall heat
transfer coefficient.

(c)

The thermal masses of the external heat exchanger and the intermediate
circuit, if used, are negligible.

(d)

There are no time lags in the system, so that the instantaneous rate of
heat transfer between the vessel and the intermediate fluid is equal to that
between the intermediate fluid and the service fluid.


(e)

The thermal mass of the vessel can be included with the thermal mass of
its contents. (This implies also that all resistance to heat transfer in the
jacket/coils is on the service side.)

5.2

Basic Equations

The basic equations which describe the thermal performance of the system are
the same for all cases considered. They involve the use of certain intermediate
functions denoted by the letters B, E and D which are functions of the different
systems, but, provided that the assumptions listed above hold, these functions
are constant for any given system. Details of the derivations of these equations
are given in Appendix A, where appropriate sub-scripts are used for the different
systems.
The batch time is related to the initial and final temperatures of the batch by the
equation:

The thermal mass of the batch and vessel is given by:


The variation of batch temperature with time is given by:

where q is the batch temperature (K) at time s seconds.
The variation of heat load with time is given by:

Note:
For these equations, the heat load is positive if the vessel contents are being
cooled, and negative if they are being heated.
For cases where there is an intermediate fluid between the batch and the service
fluid, the temperature of this intermediate fluid entering the service (external)
exchanger is given by:

Other temperatures in the system may be derived by a heat balance as follows:
For a single phase service fluid, the outlet temperature is given by:


For the batch fluid circulated through an external exchanger, the outlet
temperature from the external exchanger is given by:

For an intermediate fluid circulated through an external exchanger, the outlet
temperature from the external exchanger is given by:

The functions B and D are functions of the flow rates and specific heats of the
various fluids and the heat transfer coefficients. They are defined for the various
cases in Table 1. In order to simplify the equations, in many cases further
intermediate functions E are also defined. Provided that the assumptions listed in
5.1 apply, these functions are constant for a given system.
The various terms in Table 1 are as follows:


TABLE 1

DEFINITIONS OF FUNCTIONS

6

APPLICATION OF THE METHOD

6.1

Determining the Behavior of an Existing System

If the system is completely defined in terms of the mechanical details of the
equipment and the flow rates and properties of the fluids, determination of the
batch time is straight forward:
(a)

Determine the performance of the heat transfer equipment at the start and
end of the temperature cycle. For an external heat exchanger, either used
directly on the process fluid or as part of an intermediate system, this can
usually be done using a suitable computer program, following the
recommendations of GBHE-PEG-HEA-502. For heat transfer between the
vessel contents and a jacket or coil, the best recommendations available
at present are given in the HTFS Design Report. The situation is rather


more complicated if there is an intermediate fluid between the vessel and
the service fluid, as in Cases 6 to 8, as the temperature of this fluid
entering the external exchanger is needed. It is necessary to adjust this
temperature until the heat duty between the vessel and the intermediate
fluid matches that between the intermediate fluid and the service fluid.
(b)

If the values of the heat transfer coefficients for the jacket/coil and the
external heat exchanger are reasonably constant over the cycle, calculate
the value of B for the appropriate case from Table 1. If the values are not
constant, go to (d).

(c)

Calculate the heating or cooling time from Equation 5.1 and the variation
of heat load and temperatures with time, if required, from Equations 5.2 to
5.7.

(d)

If the overall coefficients calculated in (a) are shown to vary significantly
between the start and end of the batch, a rough estimate of the batch time
may be obtained by calculating the value of B for the mean conditions.
A more accurate estimate of the time can be obtained by performing a
series of heat transfer calculations for a range of batch temperatures
through the batch cycle. If the batch temperature is then plotted against
the reciprocal of the heat duty, the area under this graph will be the cycle
time.

6.2

Specifying the Heat Transfer Duty for a New System

Often, when designing a batch system, the desired time to heat or cool the
vessel contents is fixed, and it is required to specify the heat exchanger that will
enable this time to be achieved.
The suggested procedure for specifying the exchanger is as follows:
(a)

Determine the thermal mass of the vessel and contents, W, the required
cycle time, S, and the initial and final temperatures, θ0 and θS. Then, using
Equation 5.1 determine the required value of the function B.

(b)

Determine the mean temperature of the batch fluid, θm. As the batch fluid
temperature falls towards the service inlet temperature with an exponential
decay, as shown by Equation 5.2, the best value to use for this is that
corresponding to the LMTD between the start and end of the cycle:


(c)

Calculate the heat duty at this temperature from Equation 5.3, using
the value of B calculated in (a).

(d)

Calculate the other temperatures in the system from Equations 5.4
to 5.7 as appropriate.

(e)

These temperatures, together with the physical properties of the
fluids, define the required heat transfer duties, and enable the
exchangers to be designed using appropriate methods. See GBHEPEG-HEA-502 for recommendations on suitable computer
programs for the design of heat exchangers, or HTFS Design
Report for methods for the estimation of heat transfer to agitated
vessels.

(f)

Rate the designs at conditions corresponding to the start and finish
of the cycle and compare these calculations with the estimates
obtained assuming a constant value of B in the equations in 5.2. If
reasonable agreement is obtained, the process is complete.

(g)

If, due to changes in physical properties during the cycle, the
agreement is poor, it will be necessary to carry out detailed rating
calculations at a series of temperatures and estimate the cycle time
as described in 6.1.

(h)

If the estimated cycle time differs from the desired value, estimate a
new value of the heat duty at mean conditions by scaling the
original value in the ratio of estimated cycle time/desired cycle time,
and repeat from (d).


APPENDIX A

DERIVATION OF THE EQUATIONS

Note:
The equation numbering system in this section is such that similar equation
numbers are used for the same process in each case. This means that in some
cases, the numbering is not contiguous.
A1

CASE 1 - ISOTHERMAL FLUID IN JACKET OR COIL

For heat transfer in the jacket or coil, the heat duty is given by:

Note that If the vessel has both a jacket and coil, in general both the areas and
the coefficients of these will differ. However, as these items always occur as their
product a compound value may be used which is given by:

where the subscripts j and c refer to jacket and coil respectively.
The rate of change of temperature of the batch is given by:


Re-arranging and integrating, with the boundary conditions that the batch
temperature is θ0 at time zero and θS at time S, gives the batch time:

A2

CASE 2 - SINGLE PHASE FLUID IN JACKET OR COIL

For heat transfer in the jacket or coil, the heat duty is given by:

The heat duty is also related to the change in temperature of the service fluid:


Hence, substituting in Equation A2.2 gives:

The rate of change of temperature of the vessel contents is given by:

Rearranging and integrating with the boundary conditions that the batch

A3

CASE 3 - EXTERNAL HEAT EXCHANGER WITH ISOTHERMAL
SERVICE FLUID

The heat lost by the process fluid in the exchanger is:


Also the heat transferred in the exchanger is:

Equating these gives:


Rearranging and integrating with the boundary conditions that the batch


A4

CASE 4 - EXTERNAL HEAT EXCHANGER WITH SINGLE
PHASE SERVICE FLUID

The heat duty of the exchanger may be defined in three ways:
For the service fluid:

For the process fluid:

For the exchanger:

From Equations A4.2 and A4.3:


Hence, re-arranging and integrating with the boundary conditions that the batch

A5

CASE 5 - EXTERNAL HEAT EXCHANGER WITH SINGLE PHASE
SERVICE FLUID SPECIAL CASE WITH EQUAL FLOWING THERMAL
MASSES

If the flowing thermal masses of the process and service fluids, M.C and m.c are
equal, i.e. r = 1, the approach of Case 4 breaks down, as certain terms become
undefined.
As for Case 4, the heat duty of the exchanger may be defined in three different
ways:
For the service fluid:


For the process fluid

For the exchanger, the temperature difference is constant along the length, so
the log mean temperature difference becomes the temperature difference at
either end. Thus:

Hence, rearranging Equation A5.4 and substituting for θ2 in Equation A5.3:


Re-arranging and integrating with the boundary conditions that the batch


Note:
For Cases 6 - 8, the terms M and C refer to the intermediate fluid, whereas in
Cases 3 - 5 they refer to the process fluid.

A6

CASE 6 - INDIRECT SYSTEM WITH ISOTHERMAL SERVICE FLUID

Using assumption (d) in 5.1, the instantaneous heat loads on the vessel
jacket/coil and the external heat exchanger are the same, and will equal the heat
duty associated with the change in temperature of the intermediate fluid passing
through the jacket/coil and exchanger. The heat loads are:
For the jacket/coils:


For the intermediate fluid:


Re-arranging:

The rate of temperature change of the process batch is given by:



A7

CASE 7- INDIRECT SYSTEM WITH SINGLE PHASE SERVICE FLUID

through the jacket/coil and exchanger and that associated with the service fluid.
The heat loads are:


A8

CASE 8 - INDIRECT SYSTEM WITH SINGLE PHASE SERVICE FLUID
SPECIAL CASE WITH EQUAL FLOWING THERMAL MASSES

If the flowing thermal masses of the intermediate and service fluids, M.C and m.c
are equal, i.e. r = 1, the approach of Case 7 breaks down, as certain terms
become undefined.
through the jacket/coil and exchanger and that associated with the service fluid.
The heat loads are:



For the external heat exchanger, the temperature difference is constant along the
length, so the log mean temperature difference becomes the temperature
difference at either end.
Thus:

Eliminating T2 between Equations A8.3 and A8.4:


The rate of change of temperature of the batch fluid is given by:


APPENDIX B
B1

WORKED EXAMPLES

EXAMPLE 1

It is required to cool the contents of a vessel from 80ºC to 40ºC in 30 minutes, by
circulating the vessel contents through an external heat exchanger which is
cooled with cooling water. Specify the design conditions for this exchanger. The
relevant data are as follows:
Mass of batch fluid:
Mass of vessel:
Specific heat of batch fluid
Specific heat of vessel metal
Circulation rate of batch fluid
Cooling water inlet temperature
Cooling water flowrate
Cooling water specific heat

(a)

5000 kg
1000 kg
3500 J.kg-1K-1
500 J.kg-1K-1
15000 kg.h -1
21ºC
20000 kg.h-1
4817 J.kg-1K-1

Calculate the thermal mass of the vessel and contents:
W = (1000 x 500) + (5000 x 3500) = 1.8 x 107 (J.K-1)

(b)

Calculate the value of the intermediate B by re-arrangement of Equation
5.1:


B2

EXAMPLE 2
The contents of a jacketed vessel are to be cooled by circulating an
intermediate fluid between the jacket and an external heat exchanger,
cooled using cooling water. Estimate the time to cool the vessel contents
from 80ºC to 40ºC. Also, in order to check the assumed heat transfer
coefficients, estimate the temperatures of the various fluids at the start of
the cooling process. The relevant data are as follows:

This system corresponds to Case 7, as defined in Table 1 and Appendix A.


DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE
This Process Engineering Guide makes reference to the following documents:
PROCESS ENGINEERING GUIDES
GBHE-PEG-HEA-502

Computer Programs for the Thermal Design of Heat
Exchangers(referred to in Clause 1, 6.1 and 6.2)

OTHER DOCUMENTS
HTFS Design Guide

Heat Transfer to Newtonian and Non-Newtonian
Fluids in Agitated

Vessels

(referred to in Clause 1, 6.1 and 6.2)

Process Heat Transfer

D.Q. Kern. (referred to in Clause 4).


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