Formulation and validation of applied engineering equations for heat transfer processes in the food industry

Martin Gram Christensen

Research output: Book/ReportPh.D. thesis

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Abstract

The study is focused on convective heat transfer in the processing of solid foods, specifically with the scope to develop simple analytical calculation tools that can be incorporated into spreadsheet solutions. In areas of food engineering such as equipment manufacture the use of predictive calculations, modelling activities and simulations for improved design is employed to a high degree. In food manufacture the use process calculations are seldom applied. Even though, the calculation of thermal processes is not a challenging task in academia; this is not the case for food manufacture. However; the calculations need fundamental validation and a generality that ensures a wide application, thus also the development of simplified approximations and engineering equations have to be conducted in academia. The focus group for the utilization of the presented work is; food manufacture, authorities ensuring food safety standards and students pursuing a food engineering career but lacks full engineering training. The approach in this study is to identify possible simplifications to the complete Fourier series expansion [Fo-exp]. This is done through; a new method to non-iteratively find the Fourier exponents and lag factors needed in a 1st term approximation, expanding the use of the 1st term approximation to also cover low Fourier numbers [Fo], and investigating the input in the series expansion in terms of the determination of convective heat transfer coefficients. For the investigation it was crucial to establish a thorough understanding of the origin of both the standard [Fo-exp] solution and the criteria coupled with standard simplified solutions. A new description of the internal and external resistance to heat transfer has been suggested in form of a normalization of the Biot number [Bi]. The normalized Biot number [Binorm] enables a simple, monotonically increasing expression, used to determine the Fourier exponents and lag factors needed in the [Fo-exp] solution to the heat equation. The proposed method has a low prediction error and can be used as an alternative to iterative methods or the use of charts. Additionally, [Binorm] provides a rational investigation of the sensitivity of important parameters such as the thermal conductivity and the heat transfer coefficients [h]. For the calculation of the thermal history during convective heating and cooling of solids, a solution is proposed that can also handle the initial heating/cooling period (Fo<0.2). In the construction of the new procedure the residual between a 1st term [Fo-exp] and the complete [Fo-exp] was modelled without introducing new parameters, except one experimental constant. The combined procedure of the determination of Fourier exponents and lag factors have been used in excel calculations for the calculation of finite bodies. The developed method is validated with numerical solutions with comparable accuracy in two representative cases; cooling of packaged cream cheese and a three step processing of ham. In the study, three investigations into the measurement methods for convective heat transfer coefficients [hc] have been conducted. The [hc] for separate boundaries have been measured for a cooling operation, where also the influence of a present headspace was investigated. The contribution of the phenomena of boiling in the overall [h] to suspended particles was investigated in a new experimental setup. Experiments conducted at a comparison level emphasize that process control of vessel cooking should also include boiling rate instead of only using temperature. iii A study in fluid to particle heat transfer coefficients [hfp] have been conducted, where it is shown that potatoes can be used as a model food device for temperature measurements, in otherwise challenging environments. The method utilizes an observed gelatinization front in potatoes and inverse calculations of the thermal curve. Based on a literature search it has been experienced that the common rules acknowledged in all textbooks and papers on the subject have not been properly investigated in terms of induced uncertainties coupled with the common rules. This includes the use of the lumped capacitance method for [Bi<0.1], and the criteria that a 1st term approximation is adequate for [Fo>0.2]. Whereas it was possible to trace the origin of the [Fo>0.2] criterion, the [Bi<0.1] criterion for the lumped capacitance method were unsuccessful. However, the error accompanied by this assumption is now documented and I believe it should be stated along with the criteria in future textbooks. The analysis shows that for elementary geometries the criteria [Fo>0.2], in worst case, generate calculation errors of up to 1.8%. The most troubling is that the worst case is for infinite slabs, which are used in the construction of general geometries, such as the shape of a box, increasing the induced error to almost 6%. The highest errors were observed at [Bi] around 2. For food manufacture [Bi] around 2 are extremely common. The thesis presents an analysis and description of the [Fo-exp] to the heat equation, and also presents solutions to common challenges when calculations are conducted in food manufacture. The study provides a method where traditional processes can be calculated with a high precision by using an expanded 1st term approximation to the series expansion. This is an advantageous in terms of application in the industry where the solution can be incorporated into spreadsheet solutions. This feature is important in conducting process planning and scheduling, handling changes in products and processes and it is valuable in debottlenecking operations. It is wished that the proposed work could help facilitate that the use of rational engineering calculations are performed in food manufacture. It is also hoped that the solutions provided and the insight to the [Fo-exp] will become a part of the engineering training for food science students. And most important, that the study will find application in the food industry.
Original languageEnglish
PublisherTechnical University of Denmark
Number of pages223
ISBN (Print)87-93109-51-2
Publication statusPublished - 2014

Cite this

Christensen, Martin Gram. / Formulation and validation of applied engineering equations for heat transfer processes in the food industry. Technical University of Denmark, 2014. 223 p.
@phdthesis{b5e6a449127e49d4ab68c2b1311abcdf,
title = "Formulation and validation of applied engineering equations for heat transfer processes in the food industry",
abstract = "The study is focused on convective heat transfer in the processing of solid foods, specifically with the scope to develop simple analytical calculation tools that can be incorporated into spreadsheet solutions. In areas of food engineering such as equipment manufacture the use of predictive calculations, modelling activities and simulations for improved design is employed to a high degree. In food manufacture the use process calculations are seldom applied. Even though, the calculation of thermal processes is not a challenging task in academia; this is not the case for food manufacture. However; the calculations need fundamental validation and a generality that ensures a wide application, thus also the development of simplified approximations and engineering equations have to be conducted in academia. The focus group for the utilization of the presented work is; food manufacture, authorities ensuring food safety standards and students pursuing a food engineering career but lacks full engineering training. The approach in this study is to identify possible simplifications to the complete Fourier series expansion [Fo-exp]. This is done through; a new method to non-iteratively find the Fourier exponents and lag factors needed in a 1st term approximation, expanding the use of the 1st term approximation to also cover low Fourier numbers [Fo], and investigating the input in the series expansion in terms of the determination of convective heat transfer coefficients. For the investigation it was crucial to establish a thorough understanding of the origin of both the standard [Fo-exp] solution and the criteria coupled with standard simplified solutions. A new description of the internal and external resistance to heat transfer has been suggested in form of a normalization of the Biot number [Bi]. The normalized Biot number [Binorm] enables a simple, monotonically increasing expression, used to determine the Fourier exponents and lag factors needed in the [Fo-exp] solution to the heat equation. The proposed method has a low prediction error and can be used as an alternative to iterative methods or the use of charts. Additionally, [Binorm] provides a rational investigation of the sensitivity of important parameters such as the thermal conductivity and the heat transfer coefficients [h]. For the calculation of the thermal history during convective heating and cooling of solids, a solution is proposed that can also handle the initial heating/cooling period (Fo<0.2). In the construction of the new procedure the residual between a 1st term [Fo-exp] and the complete [Fo-exp] was modelled without introducing new parameters, except one experimental constant. The combined procedure of the determination of Fourier exponents and lag factors have been used in excel calculations for the calculation of finite bodies. The developed method is validated with numerical solutions with comparable accuracy in two representative cases; cooling of packaged cream cheese and a three step processing of ham. In the study, three investigations into the measurement methods for convective heat transfer coefficients [hc] have been conducted. The [hc] for separate boundaries have been measured for a cooling operation, where also the influence of a present headspace was investigated. The contribution of the phenomena of boiling in the overall [h] to suspended particles was investigated in a new experimental setup. Experiments conducted at a comparison level emphasize that process control of vessel cooking should also include boiling rate instead of only using temperature. iii A study in fluid to particle heat transfer coefficients [hfp] have been conducted, where it is shown that potatoes can be used as a model food device for temperature measurements, in otherwise challenging environments. The method utilizes an observed gelatinization front in potatoes and inverse calculations of the thermal curve. Based on a literature search it has been experienced that the common rules acknowledged in all textbooks and papers on the subject have not been properly investigated in terms of induced uncertainties coupled with the common rules. This includes the use of the lumped capacitance method for [Bi<0.1], and the criteria that a 1st term approximation is adequate for [Fo>0.2]. Whereas it was possible to trace the origin of the [Fo>0.2] criterion, the [Bi<0.1] criterion for the lumped capacitance method were unsuccessful. However, the error accompanied by this assumption is now documented and I believe it should be stated along with the criteria in future textbooks. The analysis shows that for elementary geometries the criteria [Fo>0.2], in worst case, generate calculation errors of up to 1.8{\%}. The most troubling is that the worst case is for infinite slabs, which are used in the construction of general geometries, such as the shape of a box, increasing the induced error to almost 6{\%}. The highest errors were observed at [Bi] around 2. For food manufacture [Bi] around 2 are extremely common. The thesis presents an analysis and description of the [Fo-exp] to the heat equation, and also presents solutions to common challenges when calculations are conducted in food manufacture. The study provides a method where traditional processes can be calculated with a high precision by using an expanded 1st term approximation to the series expansion. This is an advantageous in terms of application in the industry where the solution can be incorporated into spreadsheet solutions. This feature is important in conducting process planning and scheduling, handling changes in products and processes and it is valuable in debottlenecking operations. It is wished that the proposed work could help facilitate that the use of rational engineering calculations are performed in food manufacture. It is also hoped that the solutions provided and the insight to the [Fo-exp] will become a part of the engineering training for food science students. And most important, that the study will find application in the food industry.",
author = "Christensen, {Martin Gram}",
year = "2014",
language = "English",
isbn = "87-93109-51-2",
publisher = "Technical University of Denmark",

}

Formulation and validation of applied engineering equations for heat transfer processes in the food industry. / Christensen, Martin Gram.

Technical University of Denmark, 2014. 223 p.

Research output: Book/ReportPh.D. thesis

TY - BOOK

T1 - Formulation and validation of applied engineering equations for heat transfer processes in the food industry

AU - Christensen, Martin Gram

PY - 2014

Y1 - 2014

N2 - The study is focused on convective heat transfer in the processing of solid foods, specifically with the scope to develop simple analytical calculation tools that can be incorporated into spreadsheet solutions. In areas of food engineering such as equipment manufacture the use of predictive calculations, modelling activities and simulations for improved design is employed to a high degree. In food manufacture the use process calculations are seldom applied. Even though, the calculation of thermal processes is not a challenging task in academia; this is not the case for food manufacture. However; the calculations need fundamental validation and a generality that ensures a wide application, thus also the development of simplified approximations and engineering equations have to be conducted in academia. The focus group for the utilization of the presented work is; food manufacture, authorities ensuring food safety standards and students pursuing a food engineering career but lacks full engineering training. The approach in this study is to identify possible simplifications to the complete Fourier series expansion [Fo-exp]. This is done through; a new method to non-iteratively find the Fourier exponents and lag factors needed in a 1st term approximation, expanding the use of the 1st term approximation to also cover low Fourier numbers [Fo], and investigating the input in the series expansion in terms of the determination of convective heat transfer coefficients. For the investigation it was crucial to establish a thorough understanding of the origin of both the standard [Fo-exp] solution and the criteria coupled with standard simplified solutions. A new description of the internal and external resistance to heat transfer has been suggested in form of a normalization of the Biot number [Bi]. The normalized Biot number [Binorm] enables a simple, monotonically increasing expression, used to determine the Fourier exponents and lag factors needed in the [Fo-exp] solution to the heat equation. The proposed method has a low prediction error and can be used as an alternative to iterative methods or the use of charts. Additionally, [Binorm] provides a rational investigation of the sensitivity of important parameters such as the thermal conductivity and the heat transfer coefficients [h]. For the calculation of the thermal history during convective heating and cooling of solids, a solution is proposed that can also handle the initial heating/cooling period (Fo<0.2). In the construction of the new procedure the residual between a 1st term [Fo-exp] and the complete [Fo-exp] was modelled without introducing new parameters, except one experimental constant. The combined procedure of the determination of Fourier exponents and lag factors have been used in excel calculations for the calculation of finite bodies. The developed method is validated with numerical solutions with comparable accuracy in two representative cases; cooling of packaged cream cheese and a three step processing of ham. In the study, three investigations into the measurement methods for convective heat transfer coefficients [hc] have been conducted. The [hc] for separate boundaries have been measured for a cooling operation, where also the influence of a present headspace was investigated. The contribution of the phenomena of boiling in the overall [h] to suspended particles was investigated in a new experimental setup. Experiments conducted at a comparison level emphasize that process control of vessel cooking should also include boiling rate instead of only using temperature. iii A study in fluid to particle heat transfer coefficients [hfp] have been conducted, where it is shown that potatoes can be used as a model food device for temperature measurements, in otherwise challenging environments. The method utilizes an observed gelatinization front in potatoes and inverse calculations of the thermal curve. Based on a literature search it has been experienced that the common rules acknowledged in all textbooks and papers on the subject have not been properly investigated in terms of induced uncertainties coupled with the common rules. This includes the use of the lumped capacitance method for [Bi<0.1], and the criteria that a 1st term approximation is adequate for [Fo>0.2]. Whereas it was possible to trace the origin of the [Fo>0.2] criterion, the [Bi<0.1] criterion for the lumped capacitance method were unsuccessful. However, the error accompanied by this assumption is now documented and I believe it should be stated along with the criteria in future textbooks. The analysis shows that for elementary geometries the criteria [Fo>0.2], in worst case, generate calculation errors of up to 1.8%. The most troubling is that the worst case is for infinite slabs, which are used in the construction of general geometries, such as the shape of a box, increasing the induced error to almost 6%. The highest errors were observed at [Bi] around 2. For food manufacture [Bi] around 2 are extremely common. The thesis presents an analysis and description of the [Fo-exp] to the heat equation, and also presents solutions to common challenges when calculations are conducted in food manufacture. The study provides a method where traditional processes can be calculated with a high precision by using an expanded 1st term approximation to the series expansion. This is an advantageous in terms of application in the industry where the solution can be incorporated into spreadsheet solutions. This feature is important in conducting process planning and scheduling, handling changes in products and processes and it is valuable in debottlenecking operations. It is wished that the proposed work could help facilitate that the use of rational engineering calculations are performed in food manufacture. It is also hoped that the solutions provided and the insight to the [Fo-exp] will become a part of the engineering training for food science students. And most important, that the study will find application in the food industry.

AB - The study is focused on convective heat transfer in the processing of solid foods, specifically with the scope to develop simple analytical calculation tools that can be incorporated into spreadsheet solutions. In areas of food engineering such as equipment manufacture the use of predictive calculations, modelling activities and simulations for improved design is employed to a high degree. In food manufacture the use process calculations are seldom applied. Even though, the calculation of thermal processes is not a challenging task in academia; this is not the case for food manufacture. However; the calculations need fundamental validation and a generality that ensures a wide application, thus also the development of simplified approximations and engineering equations have to be conducted in academia. The focus group for the utilization of the presented work is; food manufacture, authorities ensuring food safety standards and students pursuing a food engineering career but lacks full engineering training. The approach in this study is to identify possible simplifications to the complete Fourier series expansion [Fo-exp]. This is done through; a new method to non-iteratively find the Fourier exponents and lag factors needed in a 1st term approximation, expanding the use of the 1st term approximation to also cover low Fourier numbers [Fo], and investigating the input in the series expansion in terms of the determination of convective heat transfer coefficients. For the investigation it was crucial to establish a thorough understanding of the origin of both the standard [Fo-exp] solution and the criteria coupled with standard simplified solutions. A new description of the internal and external resistance to heat transfer has been suggested in form of a normalization of the Biot number [Bi]. The normalized Biot number [Binorm] enables a simple, monotonically increasing expression, used to determine the Fourier exponents and lag factors needed in the [Fo-exp] solution to the heat equation. The proposed method has a low prediction error and can be used as an alternative to iterative methods or the use of charts. Additionally, [Binorm] provides a rational investigation of the sensitivity of important parameters such as the thermal conductivity and the heat transfer coefficients [h]. For the calculation of the thermal history during convective heating and cooling of solids, a solution is proposed that can also handle the initial heating/cooling period (Fo<0.2). In the construction of the new procedure the residual between a 1st term [Fo-exp] and the complete [Fo-exp] was modelled without introducing new parameters, except one experimental constant. The combined procedure of the determination of Fourier exponents and lag factors have been used in excel calculations for the calculation of finite bodies. The developed method is validated with numerical solutions with comparable accuracy in two representative cases; cooling of packaged cream cheese and a three step processing of ham. In the study, three investigations into the measurement methods for convective heat transfer coefficients [hc] have been conducted. The [hc] for separate boundaries have been measured for a cooling operation, where also the influence of a present headspace was investigated. The contribution of the phenomena of boiling in the overall [h] to suspended particles was investigated in a new experimental setup. Experiments conducted at a comparison level emphasize that process control of vessel cooking should also include boiling rate instead of only using temperature. iii A study in fluid to particle heat transfer coefficients [hfp] have been conducted, where it is shown that potatoes can be used as a model food device for temperature measurements, in otherwise challenging environments. The method utilizes an observed gelatinization front in potatoes and inverse calculations of the thermal curve. Based on a literature search it has been experienced that the common rules acknowledged in all textbooks and papers on the subject have not been properly investigated in terms of induced uncertainties coupled with the common rules. This includes the use of the lumped capacitance method for [Bi<0.1], and the criteria that a 1st term approximation is adequate for [Fo>0.2]. Whereas it was possible to trace the origin of the [Fo>0.2] criterion, the [Bi<0.1] criterion for the lumped capacitance method were unsuccessful. However, the error accompanied by this assumption is now documented and I believe it should be stated along with the criteria in future textbooks. The analysis shows that for elementary geometries the criteria [Fo>0.2], in worst case, generate calculation errors of up to 1.8%. The most troubling is that the worst case is for infinite slabs, which are used in the construction of general geometries, such as the shape of a box, increasing the induced error to almost 6%. The highest errors were observed at [Bi] around 2. For food manufacture [Bi] around 2 are extremely common. The thesis presents an analysis and description of the [Fo-exp] to the heat equation, and also presents solutions to common challenges when calculations are conducted in food manufacture. The study provides a method where traditional processes can be calculated with a high precision by using an expanded 1st term approximation to the series expansion. This is an advantageous in terms of application in the industry where the solution can be incorporated into spreadsheet solutions. This feature is important in conducting process planning and scheduling, handling changes in products and processes and it is valuable in debottlenecking operations. It is wished that the proposed work could help facilitate that the use of rational engineering calculations are performed in food manufacture. It is also hoped that the solutions provided and the insight to the [Fo-exp] will become a part of the engineering training for food science students. And most important, that the study will find application in the food industry.

M3 - Ph.D. thesis

SN - 87-93109-51-2

BT - Formulation and validation of applied engineering equations for heat transfer processes in the food industry

PB - Technical University of Denmark

ER -