牛奶污染下的板式换热器动态建模与仿真研究
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"该研究文章探讨了在牛奶污垢影响下的板式换热器的动态建模和仿真。通过详细的动态模型，对复杂的板式换热器进行了数学建模，结合了一个基于反应/质量传递机制的复杂污垢模型。所有影响牛奶热处理的重要因素都得到了正式量化。最终模型由一组偏微分、积分和代数方程组成。通过对动态优化问题的求解进行了参数估计分析。模拟结果与现有的实验工作相吻合。为了展示污垢行为的各个方面，考虑了三种具有复杂流体布置的不同配置。模拟结果为理解影响牛奶污垢的关键因素提供了重要见解。该研究发表于2000年的《化学工程科学》杂志，由Michael C. Georgiadis和Sandro Macchietto等人撰写。"
这篇论文详细阐述了在处理牛奶热处理过程中，如何利用动态建模和仿真技术来研究板式换热器的性能。动态建模是解决复杂系统如板式换热器在处理乳制品时所面临的挑战的关键，因为它可以捕捉到实时变化和过程中的非线性特性。文章中提到的复杂污垢模型考虑了化学反应和质量传递，这有助于理解和预测污垢形成的过程。
数学建模部分包括建立偏微分方程（PDEs）、积分方程和代数方程，以全面描述板式换热器的热动力学行为。这些方程组可以模拟流体流动、传热以及污垢积累对系统性能的影响。通过参数估计分析，研究人员能够确定模型参数，这通常涉及到优化问题的求解，以使模型预测与实际实验数据尽可能一致。
在实际应用中，他们模拟了三种不同的流动配置，这不仅展示了不同设计下污垢行为的差异，也为优化换热器设计提供了参考。通过对比分析，可以深入理解哪些因素（如流速、温度、材料性质等）对污垢形成和热效率有显著影响，从而有助于改进设备设计，减少污垢积累，提高换热效率。
这项工作为板式换热器在乳制品加工中的动态性能评估提供了一种科学方法，对于提升食品工业中的热交换效率和产品质量具有重要意义。此外，这种建模和仿真技术也可以扩展到其他可能存在污垢问题的工业应用中，例如在化工、能源和环境工程等领域。
Fig. 1. The protein reaction scheme used in the fouling model.
Table 1
Kinetic data for the reactions of b}lactoglobulin
E
N
(kJ/mol) k
No
(1/s) E
D
(kJ/mol) k
Do
(m3/kg s)
261 312 3.37]1037 1.36]1043
primarily due to the existence of both counter current
and co-current #ows, the strong energy interactions
between #uids in adjacent channels, and the great variety
of complex geometries that can be found in industrial
applications.
With the increasing trend towards improved predictive
accuracy of process models in this work we present
fundamental dynamic models to perform a comprehens-
ive study of milk fouling in PHEs as a process a!ected by
momentum, heat- and mass-transfer phenomena. Such
models, under speci"ed assumptions, may have predic-
tive power over wide range of operating conditions and
with the recent developments in modelling tools can be
used for on-line applications. Moreover, accurate predic-
tion of fouling dynamics for a given system would pave
the way for obtaining optimal operating policies and
design characteristics of industrial heat exchangers
(Fryer, 1989; De Jong, 1996).
2. Description of fouling model
The fouling model used in this work relies on the
b-lactoglobulin reaction scheme as shown in Fig. 1. It
was adopted from Toyoda and Fryer (1997) and was "rst
proposed by De Jong et al. (1992). When milk is heated
above 653C, b-lactoglobulin becomes thermally unstable
and it (i) unfolds in molecular denaturation exposing
reactive sulphydry (-SH) groups and (ii) polymerizes irre-
versibly to give insoluble particles in aggregation (De
Jong et al., 1992). The key step in studying fouling is to
capture the interrelationship between the chemical reac-
tions which give rise to deposition and the #uid mechan-
ics associated with the heat transfer equipment. The
reaction scheme is described as follows:
f Proteins react in both the bulk and the thermal bound-
ary layer in the milk. Native protein N is transformed
to denaturated protein D, in a "rst-order reaction. The
denaturated protein then reacts to give aggregated
protein A in a second order reaction.
f Mass transfer between the bulk and the thermal
boundary layer takes place for each protein.
f Only the aggregated protein is deposited on the wall.
The deposition rate is proportional to the concentra-
tion of aggregated protein in the thermal boundary
layer.
f The fouling resistance to heat transfer is proportional
to the thickness of the deposit.
The reaction rate constants are expressed in the com-
mon form as:
k"k
0
exp(!E/R¹)
The pre-exponential factors k
o
and the activation ener-
gies, E, for the two reactions are taken from De Jong et al.
(1992) and given in Table 1.
3. A thermal dynamic model of PHEs
A plate heat exchanger consists of a number of parallel
#ow channels formed by metal plates which are separ-
ated by gasket material around the perimeter of each
plate (see Fig. 2). Nozzles for the #ow of #uids extend
through the frames to the plate packages. Heat is trans-
ferred through these plates from one #uid to another.
For the purpose of modelling, consider a di!erential
element of the PHE in channel j (Fig. 3). The modelling
assumptions are de"ned as follows:
f No di!usion of heat is considered in the axial direc-
tion.
f The #owrate and temperature pro"les are uniform
across the channel and plate width.
f Each #uid is split equally between all related channels.
f Heat losses to the environment are negligible.
f The head and follower parts of the PHE are assumed
to be insulated and serve as adiabatic plates.
Based on the fundamental energy conservation law the
following equations describe the heat transfer between
process #uid or heating medium in a channel j and its
adjacent plates p
j
and p
j`1
:
A
x
p
j
Cp
j
A
L¹
j
Lt
#n
j
u
x
L¹
j
Lx
B
";
j
A
j
(¹
p
j~1
!¹
j
)#;
j
A
j
(¹
pj
!¹
j
). (1)
M.C. Georgiadis, S. Macchietto / Chemical Engineering Science 55 (2000) 1605}1619 1607
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