Spray Simulation

Spray forming combines the metallurgical processes of metal casting and powder metallurgy to fabricate metal products with enhanced properties. This book provides an instruction to the various modelling and simulation techniques employed in spray forming, and shows how they are applied in process analysis and development.

     The author begins by deriving and describing the main models. He then presents their application in the simulation of the key features of spray forming. Wherever possible he discusses theoretical results with reference to experimental data. Building on the features of metal spray forming, he also derives common characteristic modelling features that may be useful in the simulation of related spray processes.

     The book is aimed at researchers and engineers working in process technology, chemical engineering and materials science.

Udo Fritsching received his Ph.D. from the University of Bremen, Germany, and is currently head of the Research Group at the Institute for Materials Science and apl. Professor at the University of Bremen. He is the author or coauthor of 160 scientific papers and has five patent applications pending.

Spray Simulation

Modelling and Numerical Simulation of Sprayforming Metals

Udo Fritsching

Universität Bremen

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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© Cambridge University Press 2004

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2004

Printed in the United Kingdom at the University Press, Cambridge

Typefaces Times 10/13 pt. and Helvetica System LATEX 2_e [TB]

A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data
Fritsching, Udo, 1959–
Spray simulation: modelling and numerical simulation of sprayforming metals / Udo Fritsching.
p. cm.
ISBN 0 521 82098 7
1. Metal spraying – Mathematical models. 2. Metal spraying – Computer simulation. I. Title.
TS655.F75 2003
6.71.7′34 – dc21 2003055191

ISBN 0 521 82098 7 hardback

The publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarentee that a site will remain live or that the content is or will remain appropriate.

Contents

Preface

This book describes the fundamentals and potentials of modelling and simulation of complex engineering processes, based on, as an example, simulation of the spray forming process of metals. The spray forming process, in this context, is a typical example of a complex technical spray process. Spray forming, basically, is a metallurgical process whereby near-net shaped preforms with outstanding material properties may be produced direct from a metal melt via atomization and consolidation of droplets. For proper analysis of this process, first successive physical submodels are derived and are then implemented into an integrated coupled process model. The theoretical effects predicted by each submodel are then discussed and are compared to experimental findings, where available, and are summarized under the heading ‘spray simulation’. The book should give engineering students and practising engineers in industry and universities a detailed introduction to this rapidly growing area of research and development.

     In order to develop an integral model for such technically complex processes as the spray forming of metals, it is essential that the model is broken down into a number of smaller steps. For spray forming, the key subprocesses are :

• atomization of the metal melt,
• dispersed multiphase flow in the spray,
• compaction of the spray and formation of the deposit.

These subprocesses may be further divided until a sequential (or parallel) series of unit operational tasks is derived. For these tasks, individual balances of momentum, heat and mass are to be performed to derive a fundamental model for each. In addition, some additional submodels need to be derived or applied. The general description of this modelling approach to the spray forming process is the fundamental aim of this book, which therefore :

• introduces a general modelling and simulation strategy for complex spray processes,
• reviews relevant technical contributions on spray form modelling and simulation, and
• analyses and discusses the physical behaviour of each subprocesses and materials in the spray forming process.

     This work is based on a number of investigations of spray forming carried out by researchers all over the world. Major contributions have been given from research projects conducted by the author’s research group on ‘multiphase flow, heat and mass transfer’ at the University of Bremen, the Foundation Institute for Material Science (IWT), as well as the Special Research Cooperation Project on spray forming SFB 372 at the University of Bremen. These projects have been funded, for example, by the Deutsche Forschungsgemeinschaft DFG, whose support is gratefully acknowledged. Several graduate and PhD students contributed to this project. I would like to thank all of them for their valuable contributions, especially Dr.-Ing. O. Ahrens, Dr.-Ing. D. Bergmann, Dr.-Ing. I. Gillandt, Dr.-Ing. U. Heck, Dipl.-Ing. M. Krauss, Dipl.-Ing. S. Markus, Dipl.-Ing. O. Meyer and Dr.-Ing. H. Zhang. Also, I would like to thank those guests whom I had the pleasure of hosting at the University of Bremen and who contributed to the development of this book, namely Professor Dr.-Ing. C. T. Crowe and Professor Dr. C. Cui. I acknowledge Professor Dr.-Ing. K. Bauckhage for initiating research in this field and thank him for his continuous support of research in spray forming at the University of Bremen.

I would like to thank my family, Karin and Anna, for their understanding and support.

In order to keep the price of this book affordable, it has been decided to reproduce all figures in black/white. All coloured plots and pictures can be found and downloaded by interested readers from the author’s homepage. Some of the spray simulation programs used in this book may also be downloaded from this web page. The URL is : www.iwt-bremen.de/vt/MPS/

Nomenclature

A	area	m2
ai	coefficients
a	temperature conductivity	m2/s
bi	coefficients
cd	resistance (drag) coefficient
cp	specific heat capacity	kJ/kg K
c1, c2, c, cT	constants of turbulence model
Dk	dissipation
d, D	diameter	m
d3.2	Sauter mean diameter, SMD	m
dmax	maximum spread diameter	m
Ec	Eckart number
F	force	N
Ff	volume ratio, filling function
fr	coefficient of friction, normalized resistance
fs,l	solid or liquid content
f	frequency	1/s
f	distribution density of particles
G	coefficient for interparticulate forces
G	number of solid fragments
g	gravity constant	m/s2
ġ	growth rate	m/s
GMR	mass flow rate ratio gas/metal
H, h	height	m
H	enthalpy	kJ
h	specific enthalpy	kJ/kg
hf	film thickness	m
hl	ligament height	m
I, K	Bessel function
J	nucleation rate	1/s
kS	empirical constant
k	turbulent kinetic energy	m2/s2
kp	compaction rate
kB	Boltzmann constant	J/K
L	length	m
Lh	latent heat of fusion	kJ/kg
LT	dissipation length scale	m
La	Laplace number
l	length, distance to nozzle	m
M	fragmentation number
Ṁ	mass flow rate	kg/s
Ma	Mach number
m	mass	kg
m	mode
ṁ	mass flux	kg/m2 s
Nu	Nusselt number
N, n	number concentration, particle number	1/m3
Oh	Ohnesorge number
P	number of collisions
Pe	Peclet number
p	pressure	Pa
p	microporosity function
qr	probability density function	1/m
	heat flow rate	W
	heat flux	W/m2
r	radial coordinate	m
r0.5	half-width radius	m
R	gas constant	kJ/kg K
RL	Lagrangian time correlation coefficient
Re	Reynolds number
Real	real part
S	source/sink
Sha	Shannon entropy
St	Stokes number
Ste	Stefan number
s	path	m
T	temperature	K
T*	Stefan number
Ṫ	cooling rate (velocity)	K/s
ΔT	temperature difference	K
ΔT	undercooling	K
t	time	s
u, v, w	velocity components	m/s
V	volume	m3
v	velocity of solidification front	m/s
W, F, G, Q	matrix
We	Weber number
x, y, z	plane Cartesian coordinates	m
xK	length of supersonic core	m
xs	mean distance between solid fragments	m
Z*	splashing number
z	distance atomizer – substrate	m
z, r, θ	cylindrical coordinates	m, m, o
αG	gas nozzle inclination angle	o
αf, αg	volumetric content of gas, liquid
αspray	spray inclination angle	°
α	heat transfer coefficient	W/m2 K
Γ	diffusivity
ΓS	Gamma function
γ	solid–liquid surface tension	N/m
δ	excitation wavelength	m
δ	width of gas jets	m
ε	dissipation rate of turbulent kinetic energy	m2/s3
εS	radiation emissivity
ηS	amplitude function of perturbation
ηab, ηB	amplitude of surface waves
Θcol	impact angle	°
θ	contact angle	°
θ	modified temperature	K
κ	isentropic exponent
κ0	surface curvature	1/m2
λ	heat conductivity	W/m K
λ0	reference heat conductivity	W/m K
λd	wavelength	m
λe	solidification coefficient
μ	dynamic viscosity	kg/m s
ν	kinematic viscosity	m2/s
νm	molar volume	m3/mol
ξg	boundary layer coefficient
ξ, η	dimensionless coordinates
ρ	density	kg/m3
σl	surface tension	N/m
σd	logarithmic standard deviation
σh, σ, σk	constants of turbulence model
σS	Stefan–Boltzmann constant	W/m2 K4
σt	relative turbulence intensity
τ	shear stress	N/m2
τp	relaxation time	s
τT	eddy lifetime	s
τu	passing time through eddy	s
τv	interaction time	s
Φ	transport variable
Φ	velocity potential
Φ	impact angle	°
ϕ	velocity number
χ	impact parameter	m
Ψ	stream function
ψf	function of fluid density
Ω	collision function
ω	growth rate	1/s

Indices