Institutional Digital Repository (Indian Institute of Technology Kharagpu)

Paul, Jayanta

THEORETICAL AND NUMERICAL ASPECTS OF THE POPULATION BALANCE
EQUATIONS
JAYANTA PAUL (ROLL: 15MA90J04)
SUPERVISOR: PROF. JITENDRA KUMAR
DEPARTMENT OF MATHEMATICS, IIT KHARAGPUR
ABSTRACT
In the motion of particles in a dynamical structure, the spontaneous interactions between
them result in the change of size, shape, porosity, volume, etc., of the particles. Therefore, the
number distribution or density of particles differs with time. Thus, a mathematical model
known as the population balance equation (PBE) is necessitated to represent this variation.
This thesis deals with the PBEs corresponding to the coagulation and fragmentation
processes. The PBE representing pure fragmentation is classified into two classes: linear and
non-linear processes, whereas the PBE representing the coagulation process is purely
non-linear. So, in this thesis, we aim to prove the mathematical existence theory involving a
considerable class of practically oriented kernels for the non-linear fragmentation process.
Meanwhile, establishing the existence result, it is noticed that the exact solution can only be
derived for very simple problems. In this part of the study, we develop two new and efficient
numerical schemes based on finite volume methods to approximate the non-linear
fragmentation equation. The development of the weighted schemes is concluded by
demonstrating a detailed convergence and consistency analysis. Moreover, we extend the
one-dimensional models to solve multivariate problems. Several numerical examples are
presented to validate the proficiency and accuracy of the developed schemes.
Population balance models incorporating spatially inhomogeneous coagulation and
condensation processes play a significant role in modeling many real-life applications. It is
necessary to mention that the creation of dust particles (or particles with nearly zero
size/volume) takes place in several industrial sectors. So, the system is set up with
coagulation kernels having singularity at the coordinate axes. In this context, we prove the
mathematical existence-uniqueness of a continuous solution involving the singular
coagulation kernel.
Finally, the existence-uniqueness of a steady-state solution to the coagulation equation with
source and efflux terms is addressed. The coagulation kernel considered in this work has
singularity on both coordinate axes. Further, an explicit form of the equilibrium solutions to
the problems with sources and effluxes for the constant and product kernels have been
derived.
Keywords: Population balance equations; Coagulation; Fragmentation; Existence of
solutions; Uniqueness of solutions; Finite volume scheme; Volume conservation; Number
preservation; Consistency; Convergence; Multi-dimensional model.


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CLIMATE CHANGE AND AGRICULTURE: AN ASSESSMENT OF FARMERS’ 
ADAPTATION IN THE FOOTHILLS OF WEST BENGAL, INDIA 
 
ABSTRACT 
Climate change is likely to significantly impact the Indian agricultural sector and the lives of 
millions of farmers. It is expected that with the implementation of adaptation measures, the 
vulnerability of this sector could be reduced. In praxis, however, not all farm-level adaptations 
undertaken by farmers are equally effective, and some could be maladaptive. Also, farmers 
might not perceive climate change accurately and therefore lack in adopting adaptation 
strategies. In this context, the present study examines farmers’ perceptions of climate change, 
their adaptation strategies, factors influencing the adoption of adaptation strategies, and 
potential maladaptive outcomes in the Indian Eastern Himalayan foothills of West Bengal by 
employing various statistical as well qualitative data analysis techniques. By interviewing 300 
farming households using a pre-tested semi-structured schedule, this study obtained 
information on farmers’ perceptions of and adaptation to climate change and other data 
pertaining to various socioeconomic-demographic aspects of households. Further, by involving 
51 stakeholders and eliciting information from the scientific literature, this study determined 
several potential maladaptive outcomes. Besides, secondary data on temperature and 
precipitation are collected from the Climatic Research Unit (CRU) to examine how farmers’ 
perceptions correspond with instrumental records. Results show that most farmers’ perceptions 
aligned with the increasing summer temperature and decreasing monsoon precipitation. 
Farmers are responding to climate change impacts by implementing at least one and up to seven 
adaptation measures. Farmers preferred agroforestry, a shift from cereals to low water-intensive 
commercials, irrigation, and intensification of winter crops as the most efficient. There is, 
however, a misalignment between the perceived efficiency of adaptation measures and their 
scale of adoption. Farmers’ perceptions of pest infestation, satisfaction with farming, soil 
characteristics, farm size, remittances, and access to credit were found to be positively and 
significantly influencing the adaptation indices, while open-mindedness towards changing 
farming practices and crop-raiding by elephants are found to be negatively and significantly 
associated with adaptation indices. Besides, the study determined several potentially 
maladaptive outcomes of the adopted strategies, such as a decline in groundwater levels, 
increased greenhouse gas emissions from irrigation systems using diesel pumps, resistance 
among pests, increased investment costs, financial loss, weakening of social ties as a result of 
changes in cultural practices, and so on. However, the study identified potential maladaptive 
outcomes based on qualitative interviews with various stakeholders, which limits understanding 
of the actual extent of maladaptation in this region. Therefore, future research might 
concentrate on quantifying the maladaptive outcomes and provide empirical evidence of 
maladaptation. 
Keywords 
Climate Change; Farmers; Perception; Adaptation; Maladaptation; Eastern-Himalayan foothills 


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