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. Refinery Process Stream Purification Refinery Process Catalysts Troubleshooting Refinery Process Catalyst Start-Up / Shutdown Activation Reduction In-situ Ex-situ Sulfiding Specializing in Refinery Process Catalyst Performance Evaluation Heat & Mass Balance Analysis Catalyst Remaining Life Determination Catalyst Deactivation Assessment Catalyst Performance Characterization Refining & Gas Processing & Petrochemical Industries Catalysts / Process Technology - Hydrogen Catalysts / Process Technology – Ammonia Catalyst Process Technology - Methanol Catalysts / process Technology – Petrochemicals Specializing in the Development & Commercialization of New Technology in the Refining & Petrochemical Industries Web Site: www.GBHEnterprises.com

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)

(b)

whether the batch fluid is: (1)

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

(2)

circulated through an external exchanger; or

(3)

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 temperature is θ0 at time zero and θS at time S, gives the batch time:

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 temperature is θ0 at time zero and θS at time S, gives the batch time:


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 temperature is θ0 at time zero and θS at time S, gives the batch time:

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 temperature is θ0 at time zero and θS at time S, gives the batch time:


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

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 and that associated with the service fluid. The heat loads are: For the jacket/coils:




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. 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 and that associated with the service fluid. The heat loads are:


For the jacket/coils:

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).



Heating and Cooling of Batch Processes

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