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Mathematical modeling of molten metal dispensing: A studyof a pneumatically actuated diaphragm-driven pump

Vivek Jairazbhoy a, Randy C. Stevenson b,*

a Climate Control, Automotive Components Holdings, Plymouth, MI 48170, USAb Chassis CAE, Automotive Components Holdings, Dearborn, MI 48120, USA

Received 1 June 2005; received in revised form 1 August 2006; accepted 2 November 2006Available online 19 January 2007

Abstract

This study describes a lumped-element model for a pneumatically actuated, diaphragm-driven pump. The pump wasdeveloped to dispense molten metal for automotive electronic circuit manufacturing. In place of a CFD analysis, whichproved to involve excessively long computations because of a large range of length scales, we created a lumped-elementrepresentation of the physics. The pump has five main region volumes: an air filling chamber, a solder reservoir, an adjust-able throttle valve, a main pumping chamber, and a dispensing nozzle or set of nozzles. The pumping process employs acompressed gas pulse, which fills the air chamber and deforms a diaphragm, which then displaces molten solder in the mainpumping chamber. The description of the dispensing process requires the coupling of three physical domains: pneumatic,

mechanical, and hydraulic. The fluid motion in the pump is modeled by creating lumped fluid elements from control vol-umes, some with deformable surfaces. The equations for the fluid elements are derived from Bernoulli’s equation, whenviscous effects are not important, or from the general integral momentum balance, when viscous effects are important.A parametric study describing the relationships between variables is presented, and experimental data is used to confirmthe validity of the model. 2006 Published by Elsevier Inc.

1. Introduction

We present a mathematical model of the operation of a pump designed for use as a molten solder dispenser

in electronic manufacturing applications. Fig. 1a shows a schematic diagram of the pump, an early conceptdrawing, and Fig. 1b shows a picture of a later actual working prototype. The pump operates by displacingmolten metal with a pneumatically actuated diaphragm. The resulting dispense characteristics, such as the dis-pense volume, and the dispense timing, are sensitive to numerous parameters, geometrical and operational. Arational selection of these parameters can be significantly aided, if the pump operation can be modeled.

0307-904X/$ - see front matter 2006 Published by Elsevier Inc.

doi:10.1016/j.apm.2006.11.005

* Corresponding author.E-mail addresses: [email protected] (V. Jairazbhoy), [email protected] (R.C. Stevenson).

Applied Mathematical Modelling 32 (2008) 141–169

www.elsevier.com/locate/apm

mailto:[email protected]:[email protected]:[email protected]:[email protected]

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Nomenclature

asonic characteristic (inverse time) for sonic flowthrough solenoid

asubsonic characteristic (inverse time) for subsonicflow through solenoid

c p heat capacity of filling gas at constantpressure

C S office discharge coefficient of solenoidDP viscous pressure drop at a port due to viscous lossd D diameter of diaphragmoV boundary of fluid deviceoV ports boundary of all ports of fluid deviceoV walls boundary of wall of fluid deviceE elastic modulus for diaphragm materialg an adjustable constant used to gauge the

jump condition in flow rates before andafter the diaphragm hit the stop

f body force field per unit massF p face of port p of fluid deviceF wall face of wall of fluid deviceu fluid velocity potentialU p average velocity potential over port p g acceleration due to gravityG N, G T frictional force (due to viscous losses) in

the nozzle and throttle volume regions,respectively

G

ret

effective ‘‘G ’’ for retraction phaseGwall force of fluid on wall through viscous andpressure forces

G R surface tension force in the nozzle regionh pq vertical height between port p and port qhN vertical height of nozzlehR vertical height of reservoirhT vertical height of throttlek D the effective spring constant for the dia-

phragmk in, k out adjustable dimensionless constants char-

acterizing the degree of loss of kinetic en-

ergy at a port due to viscous surface forcesK p kinetic energy factor for port p K drvN ; K

drvT effective ‘‘K ’’ for the driving phase for

the nozzle stream and throttle streamrespectively

K ret effective ‘‘K ’’ for retraction phasel pq characteristic length scale along flow from

port p to port ql N flow path length of nozzlel R flow path length of reservoirl T flow path length of throttle

l̂ unit vector along direction of centralstream line in lumped fluid element

M matrix of connection coefficientsM pq connection coefficient between port p and

port q M eff pq effective connection coefficient from port p

to port ql coefficient of viscosity for fluidn p normal vector to face F pnwall normal vector to wall faceN number of nozzlesN P number of ports for fluid devicem Poisson’s ratio for diaphragm material p pressure field of fluid

P p average value of pressure p over port pP 0 ambient pressureP F air pressure in air filling chamberP G gas pressure upstream of solenoid valveQ volume flow rate (used in two-port lumped

fluid element)Q p average flow rate over port prthresholdpressure ratio that marks that transition

from sonic to subsonic flowRo outer radius of throttleRi inner radius of throttle

RN radius of nozzlesq density of fluids coordinate along mean stream line in flow

in lumped fluid elementS (s) cross-sectional area of lumped fluid ele-

ment at position sS p area of port p of fluid devicer surface tension constant for fluidt timeT fluid stress fieldTvis viscous contribution to fluid stress field

T

s thickness of diaphragmH(Æ) unit step functionh contact angle for fluid–air–wall interfacev fluid velocity fieldvS velocity of moving fluid surfacev p velocity of fluid at port pV D(xD) volume displaced by diaphragm with cen-

ter displacement xDV Dispensed volume of fluid dispensedxD the displacement coordinate for the center

of the diaphragm

142 V. Jairazbhoy, R.C. Stevenson / Applied Mathematical Modelling 32 (2008) 141–169

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In addition, if the modeling simulations and corresponding data correspond, then one can say that one under-stands the operation of the pump, to an appreciable degree. The operational parameters include drivepressure, throttle aperture, and the diaphragm displacement. Geometrical properties include the pressure headfrom the reservoir, the shape of the arrangement of the interior volumes and surfaces of the pump, as well asthe length and diameter of the nozzles. Other design considerations include, the number of dispensing nozzles,material characteristics, and very importantly, a nitrogen inerting system to limit the oxidation of the solder atall times during the operation of the pump. Given the complexity of the system, and the desire to predict tran-sient behavior, a comprehensive CFD model proved intractable. Thus, we were led to pursue a lumped-ele-ment type of analysis for the dispense operation. However, our lumped parameter analysis of the fluid flow

excluded any detailed model of drop formation.

v gravitational potential functionz Z -coordinate of gravity field

Z p average value of z-coordinate over port p

Compressed

Gas

Solenoid

valve

Nozzles

ReservoirSolder

Throttle

Diaphragm

Stop

Diaphragm

FillingChamber

Nitrogen

Inerting

Manifold

Fig. 1a. Schematic of an early prototype molten solder dispenser.

Thermal Couple

Solder Reservoir

Main Chamber

Optical Probe for

Diaphragm

Displacement

Compressed Gas

Line

Nozzles

(underneath

and at center)

Pins for

Attachment to

Robot

Inlet for Inerting

Nitrogen

Throttle

Adjustment

Heating Coil

Diaphragm Stop

Adjustment

Nitrogen

Inerting Diffuser

Fig. 1b. Final prototype molten solder dispenser.

V. Jairazbhoy, R.C. Stevenson / Applied Mathematical Modelling 32 (2008) 141–169 143

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1.1. Previous work

The basic goal of the solder dispensing technology was to directly place solder onto substrates and deviceleads without the need for printing. The technology was directly aimed at fine-pitch device applications, andsolder bumping for flip-chip applications. The trend towards smaller dimensions was one of the main drivers

for this technology. The leads (say on a microprocessor device) may have a 400-lm pitch. Conventional solderprinting techniques are problematic at these dimensions, in part, because bridging may occur between theleads. Additionally, the promise of fluxless soldering, and reduced thermal stress, were also major consider-ations for pursuing the technology.

There were other efforts and techniques, similar to the effort discussed in this paper, which developed work-ing prototypes. For example, the MPM Corporation developed two ‘‘metal jetting’’ technologies [1,2]. Onetechnique involved an electrostatically guided stream of solder balls; a continuously dispensed stream of mol-ten solder was broken into a stream of balls through surface-tension induced instabilities. The stream of balls(50–300 lm in diameter) was directed onto electronic substrates with sub-micron placement accuracy. MPMalso developed a piezoelectrically actuated, single nozzle, ‘‘drop on demand’’ system, more akin to the systemdiscussed in this paper. This technique is essentially a copy of the inkjet dispensing scheme; it employs a cap-illary tube, one end of which is connected to a solder reservoir and the other end is a nozzle. The nozzle end

has a surrounding piezoelectric element, which, when electrically pulsed, rapidly ejects the solder. The dynam-ics of the pump described in this paper is analogous to that used in the ink jet process, although at a firstglance this may not be apparent. In both the inkjet scheme, and in our scheme, the dispensing technique relieson a balancing of fluid momentum at the two ends of a capillary tube. This balancing requires a connection of one end of the tube to a reservoir with a free surface. The other end of the tube is dominated by capillaryeffects (surface tension forces).

Another interesting technique was developed in Germany by Pac Tech-Packaging Technologies [3]. In thistechnology, preformed solid solder balls (80–760 lm diameters) are ejected from a capillary tube. As the ballsare ejected they are melted by a laser. What is interesting about this process is that the volume of the balls ispredetermined, and the laser not only melts the ball but can add extra heat to the balls in a controlled fashionso that they can better form intermetallic joints with the substrate.

We mention in passing that, in addition to the mechanics of the dispensing process, the solidificationprocess and formation of the intermetallic joints, which the dispensed solder makes with the substrate and/or device leads, must also be controlled. A discussion of the modeling of the joint formation can be foundin [4–6]. It appears to the authors of this paper that there is more in the literature regarding joint formationthan dispense modeling, probably because of the proprietary nature of the workings of the dispensersthemselves.

2. Description of the pump and its operation

The pump is constructed almost entirely of titanium, so that the pump walls are not chemically reactivewith the molten solder. The pumping action is executed with a diaphragm driven by a controlled (open loop)actuation of pneumatic (solenoid) valves. In some applications, the pump is required to simultaneously dis-pense 256 drops of solder (where each drop is typically about 200 lm in diameter) onto the 4 · 64 = 256 leadsof a microprocessor device. Nozzle arrays (64 nozzles in one array) were fabricated with silicon micro-machin-ing techniques. For dispensing a single drop, a single nozzle was employed. In this case, the single nozzles werefabricated in thin titanium plates with very fine drills, as small as several thousands of an inch in diameter. Thenozzles were typically 100 lm in diameter and 1000 lm long. The flow is, of course, highly viscous within thesechannels.

In preparation for dispensing, one must prime the pump; i.e., one must fill the pump entirely with liquidsolder, with virtually no air trapped inside. Because the dispense volumes are so small, small amounts of trapped air can lead to an uncontrollable compressibility. Once the pump has been primed, it is in the restingstate, and one may wonder why the pump does not then leak. There are no valves preventing the solder fromdraining out the nozzles at the bottom of the pump. Indeed, the pump does not leak because the surface ten-

sion of the solder–air interface in the nozzles supports the pressure head.


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One may divide that part of the pump, which contains the fluid, into four sections: the solder reservoir,main pump chamber, the adjustable throttle between the reservoir and the main pump chamber, and the dis-pensing nozzle or nozzles. These four sections of fluid interact during the dispense process, which proceeds asfollows: the diaphragm pushes on the fluid in main pump chamber; the fluid then has two paths to flow. Onepath goes from the diaphragm up through the adjustable throttle valve, and into the pump reservoir. The

other path from the diaphragm goes down to the nozzles. We have several control knobs to help tune the dis-pense process. We can control displacement and speed of the diaphragm, the pressure head in the reservoir,and the constriction of the throttle valve. The diaphragm displacement is limited by a ‘‘stop’’, which allows thediaphragm to move, typically, between one to two thousandths of an inch. Once the diaphragm hits the stop,the fluid may still flow through the throttle to the reservoir, or down through the nozzles, but only in onedirection. A proper selection of the throttle position will cause the fluid to flow up, creating a draw back mech-anism on the nozzle flow, which is adjustable and independent of the diaphragm drawback, which occurs wellafter each dispense. In the absence of the throttle-mitigated draw back, the pump may continue to dispense atlow velocities toward the end of the dispense cycle, resulting in misdirection and dribbling of the solder ema-nating from the nozzles. Conversely, excessive drawback can lead to high retraction velocities within the noz-zles, causing air to be drawn into the pump. The optimum control of drawback is therefore essential tosuccessful dispensing.

To be able to perform a succeeding pumping action, one must allow the diaphragm to relax to its restingposition. We do not include this part of the process in our analytical description, since no fluid is dispensed.

3. Analytical formulation

From this characterization of the dispense process, one can see that a description of the dispensing processrequires the analysis of coupled processes from at least three physical domains. There is

1. A pneumatic (air flow) process: the filling of the air chamber upstream of the diaphragm by opening andclosing a solenoid valve.

2. A mechanical process: the dynamic response of the diaphragm due the applied pressure from the air pulse

and back pressure of the fluid.3. A fluid (liquid) flow process: the flow of solder within the pump volume and out from the nozzles.

To this end, we provide a representation for the air pulse that drives the diaphragm, we use a simple Hook’sLaw model for lumped-element representation of the diaphragm, and we obtain an analytical representation, alumped-element formulation, of the physics in the four fluid regions of the pump.

We treat the driving function as an applied boundary condition, and the regions are connected togetherthrough the continuity of physical variables. A quantitative description of the pump behavior is then obtainedby simultaneous solution of the aforementioned analytical forms for a given set of design parameters.

Before we construct the various lumped elements for the volume regions of the pump, we give below a gen-eral derivation of the approximate dynamics of the fluid lumped elements.

3.1. Construction of lumped fluid elements – general treatment

In the next two sections we develop a formalism to allow us to construct lumped-element models for thefour regions volumes of the pump in which the molten metal flows. In Section 3.2 below, we develop a generalBernoulli lumped fluid element. This element will be used for the main chamber and reservoir region volumes.For these regions, the salient engineering approximation that we make is that the fluid dynamics of the moltensolder is well approximated by potential flow. We argue that the fluid motion in the main chamber and res-ervoir is essentially inviscid because the significant dimensions are relatively large (the total volume in eitherregion is very much larger than the volume displaced in a single pulse), and the velocities are small (no morethan a few millimeters per second.) The sudden change in magnitude and, in some cases, direction of the flowafter the diaphragm hits the stop could create local vortices that might have a minor impact on the subsequent

flow, but we neglect their impact in our analysis.


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In Section 3.3, we develop a fluid element applicable to flow in narrow tubes, where viscous effects areimportant. The treatment for these fluid elements is, in one sense, more general than that for the Bernoullifluid elements, in that we account for viscous effects. It is less general, in another sense, in that we must relyon specific known solutions to be able to complete the dynamics of the element. The reason that we need toinvoke these specific solutions is that there is no general solution for the velocity field within a control volume

for the Navier–Stokes equation, as far as the authors know, in terms of just data on the surface of the controlvolume.

3.2. General formulation for a Bernoulli lumped fluid element

We begin our construction of Bernoulli lumped fluid elements by following Stevenson [7]. We specify aninviscid flow of velocity v in a region V with boundary oV . The fluid has density q, pressure p, and gravita-tional body force density f = g. The boundary of the region volume, or the control volume, is comprised of a ‘‘wall’’, through which no fluid passes, and a series of N P ports, through which fluid may pass, as illustratedby Fig. 2. One may write for the boundary

oV ¼ oV ports oV walls ¼ N P

p ¼1

F p F wall: ð1ÞFor inviscid flow, the velocity v is derivable from a potential u.

v ¼ ru: ð2Þ

With conservation of mass for incompressible flow, i.e.,

r v ¼ 0; ð3Þ

the potential satisfies Laplace’s equation

r2u ¼ 0: ð4Þ

We may speak of the physics within the lump volume in terms of the language of bond graphs [8]. There is athrough variable, for us this variable is volume flow rate, and a cross-variable, for us the velocity potential.From Stevenson et al., we know that we may write an approximate integral form of Laplace’s equation as

U p ¼X N P

q

M pqQq; p ¼ 1; . . . ; N P : ð5Þ

f = g

1 1 1 1 1Face F : S ,Q , ,K Φ

Φ 2 2 2 2 2

Face F : S ,Q , ,K

N N N N N Face F : S ,Q , ,K

N Wall F F F F V ++++=∂ ...21

Port 1

Port 2

Port P N

n N

n1

n2

Region Volume V

Φ

Fig. 2. A fluid device with N p ports. Each port has a face F , an area S , and an outward pointing normal n. There is an average flow rate Qout of the port in the direction n. There is an average potential U (for Bernoulli flow), as well as a K constant that characterizes the kinetic

energy out of the port in terms of Q.


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Here, N P is the number of ports associated with the control volume, U p is the average of the potential u over athe face F p of port p, and Qq is the average volume flow rate out of the port p,

Qq

Z F p

dS

S p ðS p n p vÞ: ð6Þ

The symmetric matrix, M = {M pq}, is a set of connection coefficients, essentially geometrical quantities, thatrepresents a generalized impedance relation between the through and cross variables. For example, if the po-tential u represented a static electric potential, then the connection coefficients would represent an (inverse)capacitance matrix elements between potential and charge.

We note that mass conservation equation yields, with the averaging process over ports, the relationX N q

Qq ¼ 0: ð7Þ

We now add dynamics to our lump formalism in the form of Bernoulli’s equation

qou

ot þ

1

2qv2 þ p þ q gz ¼ 0: ð8Þ

To convert to a lumped approximation, we average Bernoulli’s equation over the face F pZ F p

dS

S p q

ou

ot þ

1

2qv2 þ p þ q gz ¼ 0

! q

oU p

ot þ

1

2qhv2i p þ P p þ q gZ p ¼ 0: ð9Þ

The quantities Z p and P p are, respectively, the average z-value, and the average pressure over the port p.Consider the quantity hv2i p. We typically do not have detailed information on the flow fields at the ports,

but we may postulate, or impose, a boundary condition here. We will specify that the velocity at the port isapproximately proportional to the average normal flow Q p out of the port

vj p ! ðQ p =S p Þn p ; so that 1

2qhv2i p !

1

2q

K p

S 2 p Q2 p : ð10Þ

The dimensionless constant K p is an adjustable, or empirically determined, constant that expresses our imper-fect knowledge of the actual flow at the port p. We expect K p to be less than or on the order of 1 (the average of the square is less than the square of the average).

If we now substitute the impedance relation Eq. (5), into the averaged Bernoulli equation, Eq. (9), weobtain the lumped-element relation that we have been seeking

qX N

q

d

dt ð M pqQqÞ þ

1

2q

K p

S 2 p Q2 p þ q gZ p þ P p ¼ 0; p ¼ 1; . . . ; N : ð11Þ

Here we have a set of coupled non-linear first order differential equations for the flow through each port. Wemust also include the mass conservation relation, Eq. (7), with this set.

3.3. General formulation for a viscous lumped element

The flow in the throttle and nozzles is different from that in the pump body and reservoir, since the fluidvelocities are significantly higher, and correspondingly the characteristic dimensions are smaller than that of the main chamber of the pumb body. Viscous effects could be important during certain phases of the pulse. Inaddition, in certain periods during the dispense process, the solder meniscus in the nozzles moves. Thus, ourgeneral treatment must consider moving boundaries as well as viscous effects.

Following the procedure outlined by Slattery [9], we begin by briefly deriving the generalized integralmomentum balance equations. Initially, the region volume, or control volume, has the same general geometryas employed for the fluid element illustrated in Fig. 2; we then specialize to control volumes with more tubular

shapes, more akin to the nozzle and throttle element, such as that illustrated in Fig. 3.


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The momentum balance equation is

o

ot ðqvÞ þ r ðqvvÞ ¼ ð r TÞT þ qf ; ð12Þ

where we will assume that the body force density f is derivable from a potential, and that the density is con-stant, i.e.,

f ¼ rv; where v ¼ g z: ð13Þ

We shall make the conventional assumption that the stress tensor T partitions into an isotropic pressure p andviscous stress Tvis

T ¼ p I þ Tvis: ð14Þ

We then integrate over our time-changing control volume V , and apply the divergence theorem.

Z V dV oot ðqvÞ ¼ Z oV dS n ðqvvÞ þ Z oV dS ðT nÞ þ Z oV dS nðqvÞ: ð15ÞHere, n is an outwardly pointing normal to the surface oV of V . Next, we transform the remaining volumeintegral above by applying the transport theorem to the momentum density, qv, i.e.Z

V

dV o

ot ðqvÞ ¼

d

dt

Z V

dV qv

Z oV

dS ðn vSÞqv: ð16Þ

The velocity vS is the velocity of the moving boundary. We combine the last two equations to yield thefollowing:

d

dt

Z V

dV qv ¼

Z oV

dS n ðv vSÞqv þ

Z oV

dS ðT nÞ þ

Z oV

dS nðqvÞ: ð17Þ

The reader is referred to Fig. 2 again to see that the boundary oV is partitioned into two types: ports and walls.In principle, the surface of the port could be moving; e.g., the bottom port of a nozzle is a moving free surface.We assume that walls do not move and, of course, no fluid crosses a wall. Thus, if there are a total of N P ports,

d

dt

Z V

dV qv ¼ X N P p ¼1

Z F p

dS n p ðv vSÞqv þX N P p ¼1

Z F p

dS ðT n p þ n p qvÞ

Z F wall

dS nwall ðv vSÞqv

þ

Z F wall

dS ðT nwall þ nwallqvÞ: ð18Þ

We now simplify the RHS of this equation. Consider the second term on the RHS. We assume that the viscousforces across the surface F p of a port are negligible

Tvis nj F p ! 0; ð19Þ

2 2 2 2 2: , , ,F P Q S Z

n1

1 1 1 1 1: , , ,F P Q S Z

n2

V, S wall

vS

v

12ˆl l

s

S(s)

⊥v

| |

Fig. 3. Model for control volume V used to derive viscous lumped elements.


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given that the port surfaces are generally perpendicular to the flow. HenceZ F p

dS ðT n p þ n p qvÞ ¼

Z F p

dS Tvis n p þ

Z F p

dS n p ð p þ qvÞ ¼

Z F p

dS n p ð p þ qvÞ: ð20Þ

For the third term on the RHS of Eq. (18), the normal component of the fluid velocity nwall Æ v vanishes at a

wall as does nwall Æ

vS . HenceZ F wall

dS nwall ðv vSÞqv ! 0: ð21Þ

For the fourth term on the RHS of Eq. (18), we substitute our expression for T, Eq. (14), and for v, Eq. (13), toobtain

Gwall

Z F wall

dS ðTvis nwall nwallð p þ q gz ÞÞ: ð22Þ

Here, G wall is the force that the fluid exerts on the bounding wall. Our simplified integral form then reduces to

d

dt Z V dV qv ¼ X N P

p ¼1 Z F p dS n p ðv vSÞqv X N P

p ¼1 Z F p dS n p ð p þ q gz Þ þ Gwall: ð23ÞThe equation above forms the basis for our integral momentum balances in the throttle and nozzles. Below wewill apply this equation to a more specific lump geometry than that of Fig. 2. As with the Bernoulli lumpedelements, one must solve Eq. (23) in conjunction with the mass balance, Eq. (7).

3.4. Application of the viscous integral momentum form to tubular elements

We now tailor the integral momentum equation, Eq. (23), more to our application, with the simple controlvolume of Fig. 3 in mind. This control volume has the basic shape of the throttle and nozzle volume regions.

First we consider the volume term of the LHS of Eq. (23), and examine Fig. 3; we define l12 l̂ to point fromport 1 to port 2, and positive flow is from port 1 to port 2. We wish to relate this volume integral to the average

volume flow rate Q. In general, one may split the velocity field, v, into parallel components, vk (along the axialdirection l̂ ), and perpendicular components, v?. We then assume that the perpendicular components approx-imately average to 0, i.e.,

d

dt

Z V

dV qv ¼ d

dt

Z V

dV qvk þ d

dt

Z V

dV qv?

zfflfflfflfflfflffl}|fflfflfflfflfflffl{ 0: ð24Þ

We then express the remaining integral above in terms of the average volume flow rate, as follows:

Z V dV vk ¼ Z l d sS ð sÞ Z SdS

S ð sÞ

vk zfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflffl ffl{ vkð sÞ̂l

¼ l12 Z ld s

l

S ð sÞvkð sÞ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Q

l̂ ¼ l12Ql̂ : ð25Þ

From mass conservation we know that Q = Q2 = Q1, where Q2 is positive, going out of the second port. Thevolume term of Eq. (24) then reduces to

d

dt

Z V

dV qv ! l̂ qdðl12QÞ

dt : ð26Þ

Next we consider the first surface term on the RHS of Eq. (23). We may write this term as

X2

p ¼1

Z F p

dS n p ðv vSÞqv ¼

qS 1

Z F 1

dS

S 1ðn1 v1v1Þ þ qS 2

Z F 2

dS

S 2ðn2 v2v2Þ; F 1 and F 2 fixed;

qS 1 Z F 1dS

S 1ðn1 v1v1Þ; F 1 fixed; F 2 moving with v2 ¼ vS:

8>>><>>>:

ð27Þ


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This term is a boundary kinetic energy term, similar to that encountered in the Bernoulli lumped-element for-mulation. As with the Bernoulli term, we generally do not know the details of the flows at the entrances andexits of our simple control volumes. We observe that we have an average over a quadratic dependenceon velocity, at each port. Therefore, much as we did for the Bernoulli lumped element, we postulate akinetic boundary condition (based on a possibly empirically determined constant.) We say that, if n p l̂ ¼

sgnðQ p Þ,Z F p

dS

S p n p ðv p v p Þ ! sgnðQ p Þ̂l

1

2 K p Q

2 p =S

2 p

; p ¼ 1; 2: ð28Þ

We write, with sgn(Q) = 1,X2 p ¼1

Z F p

dS n p ðv vSÞqv ¼ l̂ 1

2q

K 2ðvSÞ

S 2Q2

1

2q

K 1

S 1Q2

: ð29Þ

To complete our construction of a viscous fluid element, we move to the second surface term in Eq. (23) andexpress this surface integral as

X2 p ¼1 Z F p dS n

p ð p þ q gz Þ ¼ S 1 Z F 1dS

S 1 n1ð p þ q gz Þ þ S 2 Z F 2dS

S 2 n2ð p þ q gz Þ

! l̂ ðS 2 P 2 S 1 P 1Þ þ l̂ q g ðS 2Z 2 S 1Z 1Þ: ð30Þ

Finally, we consider the wall term Gwall, which is the force that the fluid exerts on the wall through viscous andpressure forces. We consider only those components along our direction of flow l̂ , in which case, we pick off primarily viscous forces, since l̂ 12 tends to be orthogonal to the normal vector, the direction of the pressureforces. So we write

l̂ Gwall G ðQÞ; ð31aÞ

where G (Q) is the force of the wall on the fluid . We allow for the dependence of G on fluid velocity through adependence of G on Q. Note, also, that a provision must be made for G to switch signs, if the flow direction

changes direction. Thus, G (Q) must have the property thatG ðQÞ ¼ sgnðQÞjG ðQÞj: ð31bÞ

We now collect all terms together, and ‘‘dot’’ all terms in the vector equation, Eq. (23) by l̂ , to remove thevector dependence. The dynamic equation for our tubular viscous lumped element is then

q d

dt ðl12QÞ ¼

1

2q

K 1

S 1

K 2

S 2

Q2 þ q g ðS 1Z 1 S 2Z 2Þ G ðQÞ þ ðS 1 P 1 S 2 P 2Þ: ð32Þ

As with the Bernoulli equation, the dynamical equation above is not dependent on our chosen reference port.If one changes the sign of Q, and also interchanges the port indices, the equation remains invariant.

3.5. Lossy port condition for viscous lumped element

In our application, we have viscous lumped elements, such as the throttle, which adjoin ‘‘Bernoulli’’ lumpedelements such as the main chamber and the reservoir. However, in our formulation so far, we have notaccounted for the pressure losses that can occur at a port for flow from a constricted region to one more open.We will do this accounting by constructing a simple viscous lumped-element model of the transition region atthe port and then affix this element to the viscous lumped element (see Fig. 4).

We will assume in the transition region, that if flow is from the viscous region to the Bernoulli region, thenwe have an adjustable loss (likely, almost total loss) of the kinetic energy. This loss leads to a pressure change.If the flow is in the reverse direction, we then assume little loss, again adjustable. We then combine the tran-sition region with the viscous region, through the continuity of pressure, to form a single region. We shrink

this combined region back to the size of the original viscous region.


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Referring to Fig. 4, and using Eq. (32), we write the lumped equation for the transition region

q ddt

l0Qð Þ ¼ 12q K

002

S 002 K

002

S 002

Q2 þ q g S 02Z

02 S

002 Z

002

G 0ðQÞ þ ðS 02 P

02 S

002 P

002Þ: ð33Þ

The model that we use for the port loss, as mentioned above, is to assume a pressure drop, DP viscous, that arisesfrom the viscous forces originating locally at the port wall. We then say,

D P viscous ¼G 0ðQÞ

S 2!

1

2q K loss2 ðQÞ

Q2

S 22: ð34Þ

We define,

K loss2 ðQÞ þk

out; Q > 0;k in; Q < 0;

!ideally 1; Q > 0;0; Q < 0;

¼ HðQÞ: ð35ÞNext, we write the corresponding equation for the lower viscous region

q d

dt ðlQÞ ¼

1

2q

K 1

S 1

K 02S 02

Q2 þ q g ðS 1Z 1 S

02Z

02Þ G ðQÞ þ ðS 1 P 1 S

02 P

02Þ ð36Þ

We wish to eliminate the intermediate pressure P 02 from the equations for the transition region, Eq. (33), andthe lower viscous region, Eq. (36). This elimination is easily accomplished, if we just add the two equationstogether; we obtain

q d

dt ðl þ l0ÞQ½ ¼

1

2q

K 1

S 1

K 002S 002

K loss2 ðQÞ

S 2

Q2 þ q g ðS 1Z 1 S

002Z

002Þ G ðQÞ þ ðS 1 P 1 S

002 P

002Þ: ð37Þ

We then shrink the transition layer, so that

l0 ! 0; so that S 002 ! S 2; Z 002 ! Z 2; K

002 ! K 2; and P

002 ! P 2: ð38Þ

We find the final form for the viscous lumped element, with a modified kinetic boundary condition to accountfor port loss, is

q d

dt ðl12QÞ ¼

1

2q

K 1

S 1

K 2 þ K loss2 ðQÞ

S 2 Q2 þ q g ðS 1Z 1 S 2Z 2Þ G ðQÞ þ ðS 1 P 1 S 2 P 2Þ: ð39Þ

Fig. 4. Model used to develop boundary condition for a port that transitions between a viscous region and a Bernoulli region.


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3.6. Comparison of viscous and Bernoulli fluid elements

The similarity in form between the viscous fluid element equation, and that of the Bernoulli, which wereproduce below

qX N

q

d

dt ð M pqQqÞ ¼

1

2q

K p

S 2 p Q

2

p q gZ p P p ; p ¼ 1;. . .

; N ; ð40Þ

is apparent from inspection. However, we can make the similarity even more apparent, if we consider applyingthe Bernoulli lumped-element equation to a two-port device, with a port 1 and a port 2. We consider port 1 thereference port; in this case, we subtract the Bernoulli lumped equation for port 1

q d

dt ð M 11Q1 þ M 12Q2Þ ¼

1

2q

K p

S 2 p Q21 q gZ 1 P 1; ð41Þ

from that of port 2,

q d

dt ð M 21Q1 þ M 22Q2Þ ¼

1

2q

K p

S 2 p Q22 q gZ 2 P 2; ð42Þ

we use the fact that Q1 + Q2 = 0, define, Q2 Q, in addition, define

M eff 12 M 11 2 M 12 þ M 22 ¼ trðMÞ 2 M 12 ¼ M eff 21 : ð43Þ

We arrive at the Bernoulli lumped-element equation for a two-port element,

q d

dt ð M eff 12 QÞ ¼

1

2q

K 1

S 21

K 2

S 22

!Q2 þ q g ðZ 1 Z 2Þ þ ð P 1 P 2Þ: ð44Þ

Note that the equation is invariant, if we let Q ! Q, and switch port labels. We see that the Bernoulli for-mulation and the viscous formulation give rise to similar lumped-element representations, which one mightexpect, but, of course, the Bernoulli element does not contain the viscous force contribution. The similarityis closer if one identifies the connection coefficients, the M ’s, as essentially inverse characteristic length scales,

approximately the length along the center fluid stream line, divided by an area M eff 12 $ l=S : ð45Þ

Of course, this relationship is essentially that found for the inverse of the capacitance between two chargedplates in electrostatics.

4. Derivation of lumped elements for the pump volume regions

In the next six subsections, we present a representation for the driving pneumatic pulse, and give specificrepresentations of the five lumped-element volume regions for the pump.

4.1. Pneumatic driving force

In this section, we provide an analytical function that describes the time dependence of the build-up of thepressure, P F(t), within the air filling chamber, by flow through a solenoid valve. This pressure drives the dis-placement of the diaphragm (see Fig. 5a); this pressure function provides a boundary condition for the lumpedelement of the diaphragm.

The formation of the transient pressure pulse is well described by the filling of a hot chamber with a coldgas through an orifice. The filling of the chamber proceeds as follows: If, at the beginning of the pulse, thepressure ratio (the pressure downstream of the orifice divided by the pressure upstream of the orifice, P F/P G, is less than rthreshold = 0.526 [10]), we initially have choked (sonic) flow through the solenoid valve. Asthe chamber fills the downstream pressure rises, and, if the pulse is long enough, as is generally the case inour application, the pressure ratio, P F/P G, rises above rthreshold. The flow then becomes subsonic through

the orifice. If the filling process were to continue indefinitely, the downstream pressure would rise enough


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for the flow in the orifice to become incompressible. In our application, the pulse is not long enough for theincompressible state to be reached. If the initial pressure ratio is greater than rthreshold, then the ‘‘choked con-dition’’ never prevails; only the subsonic analysis is considered.

Since the volume displacement of the diaphragm is small compared with the volume of the pneumaticchamber, the chamber volume may be assumed constant during the filling operation. Simultaneous massand energy balances for the chamber yield a relationship for the pressure transient in terms of the mass flow

rate through the orifice, and the incoming gas temperature. The controlling resistance to the air flow lies in thesolenoid. Andersen [10] reports formulas for the mass flow rates through the orifice in terms of the drivingpressures, and gas temperature, for both subsonic and sonic flow in the orifice. These formulas are substitutedinto the aforementioned pressure transient relationship. The resulting equations that define the pressure tran-sient in the chamber for subsonic and sonic flow in the orifice are summarized below for convenience.

For sonic flow into the hot filling chamber at temperature T F and an initial pressure P G (on the upstreamside of the solenoid valve), we have the pressure P F in the chamber versus time given by

P Fðt Þ

P G¼

P 0

P Gþ ðasonict Þ; ð46aÞ

where

asonic 0:231c p S SC S

V F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT Fmmol R

r and P

F

P G6 r threshold: ð46bÞ

The filling gas is characterized by a heat capacity at constant pressure, c p, and a molecular weight mmol; thesolenoid valve has the phenomenological parameters S S, the effective cross-sectional area, and C S, the orificedischarge coefficient.

When the pressure in the filling chamber P F/P G increases above rthreshold, the flow through the solenoidvalve becomes subsonic, then the pressure in the filling chamber versus time becomes

P Fðt Þ

P G¼ sin arcsin

P 0

P G

þ ðasubsonict Þ

; ð47aÞ

where

asubsonic ¼0:24c p S SC S

V F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT Fmmol

R

r ; and

P F

P G> r threshold: ð47bÞ

Once the flow becomes incompressible, Eq. (47a) is no longer applicable. In particular, Eq. (47a) can no longerbe used when the argument of the sine function exceeds p/2. However, the relationship is accurate for therange of pulse times used in this application.

4.2. Lumped element for the diaphragm

The diaphragm is acted upon by the pressure difference across its two faces, as illustrated in Fig. 5b. Thereis the driving air pressure from the filling chamber P F on one side, and the back pressure of the fluid P D

against the diaphragm on the other. We choose to model the equation of motion for the diaphragm in terms

Solenoid Valve

PG C S

PF

V F

S S

Air Filling

Chamber

Diaphragm

Fig. 5a. Parameters used to model the compressed air pulse pressure.


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of the static displacement xD of the center of the diaphragm, essentially Hook’s Law. We will ignore the inertiaof the diaphragm in its dynamics. We know that Roark [11] for a thin, clamped diaphragm of area, S D, thecenter displacement, in term of the pressure differential across the diaphragm, is

k D xD ¼ ð P F P DÞS D; ð48Þ

where

k D ¼ E

ð1 m2Þ16s3

d 2Dð49Þ

The constants E and m, are the modulus of elasticity and Poisson’s ratio, respectively, of the diaphragm mate-rial (in our case titanium.) The constants s and d D are the diaphragm thickness (on the order of .010 in.) anddiameter (on the order of 1.00 in.), respectively. Again, using the formulas provided by Roark [11], we canrelate the center displacement, xD, to the volume, V D(xD), displaced by the diaphragm as

V D ð xDÞ ¼ 1=3S D xD: ð50Þ

The volume flow rate generated by the diaphragm is then

QD ¼ d

dt

V D ð xÞ ¼ 1=3S Dd xD

dt

; or d xD

dt

¼ 3QD

S D

: ð51Þ

The maximum displacement of the diaphragm is DxD.

4.3. Lumped element for the main chamber

We next apply the Bernoulli lumped element, developed in Sections 3.2 and 3.5, to the flow of solder in themain body. There are two distinct phases of pump operation that need to be analyzed when describing thedynamics of the flow in the pump as a whole.

A first phase, which we will call the driving phase, begins when the solenoid valve opens to allow com-pressed gas into the pneumatic chamber upstream of the diaphragm. As the chamber fills, the diaphragmdeforms, only slightly, due to pressure build-up on the pneumatic side; the diaphragm displaces molten solderin the body of the pump. During this phase, the flow in the pump body splits into two streams, one leads to thereservoir, and the other to the dispensing nozzles, as shown in Fig. 5c. The solder meniscus in the nozzles, seeFig. 5f , begins its displacement from its resting position at the top entrance to the nozzles. The fluid motion istherefore generally downward in the nozzles, and generally upward in the throttle and the reservoir. The firstphase persists until the diaphragm hits the stop, which controls its maximum displacement; its motion haltsvirtually instantaneously. If we ignore the possibility of cavitation, then this occurrence has the effect of instan-taneously coupling the two streams, which were previously independent.

In a second phase, which we will call the retraction phase, and which begins after the diaphragm hits thestop, the capillary forces in the nozzles work against inertial, gravitational, and viscous forces to eventuallyforce the solder upward, until the solder interfaces rise to the top of the channels. A single effective streamis sufficient to describe the flow in the pump body during this period. The flow associated with this streamcould be either downward or upward immediately after the diaphragm impact, depending on the throttle

and nozzle parameters.

Main

Chamber

Diaphragm

Stop

Diaphragm Parameters

k D, S D

x D

S D

PF

P D

∆ x D

Fig. 5b. Parameters used to model diaphragm physics.


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In light of the preceding discussion, we will develop lumped elements for the main chamber that describethe transient evolution of the system during the two distinct phases.

4.4. Flow in the main chamber – driving phase

We apply the two-port lumped Bernoulli equation, Eq. (44), to the flow in the main chamber. During thedriving phase of the dispense process we shall ignore the mutual interactions between the ports correspondingto the throttle entrance and the nozzle entrances. As discussed earlier, the treatment is equivalent to appor-tioning the pump body into two two-port devices, each beginning at the diaphragm port D, which we useas the reference port. In our treatment, we divide the diaphragm into two effective ports, port Dtop and portDbot. The line that divides the diaphragm diameter d D between its ‘‘top’’ part and its ‘‘bot’’ is not necessarily inthe middle of the diaphragm, but we will represent it as so. The resulting balances, in the driving phase, are asfollows. For the path from the diaphragm to the bottom of the throttle, l DT,

Diaphragm (port Dtop) to Throttle (port T bot)

q d

dt ½ M eff DTQT ¼

1

2q

K topD

ðS topD Þ2

K botT

S 2T

!Q2T þ q g ðZ

topD Z

botT Þ þ ð P D P

botT Þ: ð52Þ

For the path from the diaphragm to the nozzles, lDNtop , we treat all of the nozzles as identical and independent,so that they combine into one effective port with total flow QtotalN ¼ NQN, and total port area S

totalN ¼ NS N; thus

Diaphragm (port Dbot) to Nozzles (port N top):

q d

dt ½ M eff DNð NQNÞ ¼

1

2q

K botD

ðS bot

D Þ

2

K topN

ð NS NÞ

2 !ð NQNÞ2 þ q g ðZ botD Z

topN Þ þ ð P D P

topN Þ: ð53Þ

We have chosen positive QT to be up, diaphragm to throttle, and positive QN to be down, diaphragm tonozzles.

Even though the diaphragm is not necessarily divided evenly, and in fact the division dynamically shifts, theeffect on the overall dynamics is negligible. We argue that since K topD , K

botT , K

botD , and K

topN are all of order 1, and

that, typically

S T S botD ; and NS N S

topD : ð54Þ

then

K topD

ðS topD Þ

2

K botT

S 2T

; and K botD

ðS botD Þ

2

K topN

ð NS NÞ2 : ð55Þ

, , ,top top top

D D D DP S K Z

, , ,bot bot bot

D D D DP S K Z

( ) ( ), , , ,top top top N N N N N P N S K Z N Q

Port D, S D Q D

h DT

Port T bot

Port N top

, , , ,bot bot bot

T T T T T P S K Z Q

l DT

top DN l

l NT

d D

top DN h

h NT

Fig. 5c. Parameters used to characterize lumped element for the main pump chamber.


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Thus, we will ignore the terms K topD =ðS topD Þ

2, and K botD =ðS

botD Þ

2. In addition, if we let

Z topD Z

botT ¼ ðhDT d D=4Þ; ð56aÞ

and

Z botD Z topN ¼ ðhDNtop d D=4Þ; ð56bÞ

then we may rewrite the lump equations for the driving phase in the main chamber as

q d

dt ½ M eff DTQT ¼

1

2q

K botT

S 2T

!Q2T q g ðhDT d D=4Þ þ ð P D P

botT Þ; ð57Þ

and

q d

dt ð NM eff DNQNÞ ¼

1

2q

K topN

S 2N

!Q2N þ q g ðhDNtop d D=4Þ þ ð P D P

topN Þ: ð58Þ

We perform one last manipulation on the equations above, to put them in a form that will be a bit more usefulfor us. We let

S T M eff DT ¼ lDT ¼ const: ð59Þ

Similarly, we let

ð NS NÞ M eff DT ¼ lDNtop ¼ const: ð60Þ

The parameters l DT and lDNtop are characteristic length scales in the pump body associated with the streamsfrom the diaphragm to the throttle, and from the diaphragm to the nozzle, respectively.

Our new equations are then

qlDT

S T

d

dt QT ¼

1

2q

K botT

S 2T

!Q2T q g ðhDT d D=4Þ þ ð P D P

botT Þ; ð61Þ

and

qlDNtop

S N

d

dt QN ¼

1

2q

K topN

S 2N

!Q2N þ q g ðhDNtop d D=4Þ þ ð P D P

topN Þ: ð62Þ

One can see from these equations, with the attendant approximations used to derive them, that we have re-moved reference to the detailed position of dividing line across the diaphragm for the two flows in the mainchamber of the pump, and thus effectively decoupled the two streams.

Of course, the mass balance is still enforced. We must take the volume flow rate of the diaphragm QD to benegative, for flow into main chamber. Conservation of mass for the main chamber is then

QD þ QT þ NQN ¼ 0: ð63Þ

4.5. Flow in the main chamber – retraction phase

The flow in the pump body after the diaphragm is halted is significantly different from the flow during for-ward diaphragm motion. It becomes a single two-port device, with fluid moving from the nozzles to the throt-tle or vice versa. The change in flow pattern implies that we need a single characteristic length scalerepresenting the interaction between the two ports. We shall treat the nozzles as the reference port. Eq.(44) then implies that

Nozzle (port N top) to Throttle (port T bot):

q d

dt

ð M eff NTQTÞ ¼1

2

q K

topN

ð NS NÞ2

K botT

S 2

T !Q2T þ q g ðZ

topN Z

botT Þ þ ð P

topN P

botT Þ ð64Þ


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We define

lNT S T M eff NT; and hNT Z

botT Z

topN ; ð65Þ

we arrive at the desired result

qlNT

S T

ddt

QT ¼12q

K topN

ð NS NÞ2

K botT

S 2T

!Q2T q ghNT þ ð P

topN P

botT Þ: ð66Þ

The mass balance equation for the retraction phase in the main chamber is

þ QT þ NQN ¼ 0: ð67Þ

4.6. The lumped element for the throttle

We treat the throttle as an annular conduit of a known inner radius, Ri , outer radius, Ro, and height hT (seeFig. 5d). The amount of throttle is adjusted by moving a thin, slightly tapered, rod into the throttle neck, effec-tively increasing or decreasing the radius of the inner cylinder.

If we now employ our general lumped model for a viscous tubular element, but with two lossy ports, we canwrite the equation of motion for the throttle as

Throttle (port T bot) to Throttle (port T top):

q lT

S T

d

dt QT ¼

1

2q

K botT þ K botlossT ðQTÞ

S 2T

!Q2T

1

2q

K topT þ K

toplossT ðQTÞ

S 2T

!Q2T q ghT

G TðQTÞ

S T

þ ð P botT P topT Þ: ð68Þ

We have added a second loss term to Eq. (39) for the bottom throttle port; but, because the positive flow QT isinto the bottom port, instead of out of the port, we must account for a sign change in QT in the argument of K botlossT .

As a model for the frictional term G T (QT), we employ the standard steady state treatment of flow in anannulus Slattery [9, p. 76]. In this case

G TðQTÞ ¼8

pl

lT

R2ok

QT; ð69aÞ

where

k ¼ 1 j4 þ 1 j2lnðjÞ

; and j ¼ Ri R0

: ð69bÞ

Ri

hT =lT

Ro

Porttop

T top

T P , S T ,

top

T K ,

top

T Z

Portbot

T bot

T P , S T ,

top

T K ,

bot

T Z

Viscosity µ

Fig. 5d. Parameters used to model the annular throttle element.


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4.7. The lumped element for the reservoir

We apply the lumped Bernoulli equation, Eq. (44) to the reservoir. In this case, the top of the reservoir is afree surface.

We have (see Fig. 5e)Throttle (port T top) to Reservoir (port R):

q ddt

ð M eff TRQRÞ ¼12q

K topT

S 2T

K R

S 2R

!Q2R þ q g ðZ

topT Z RÞ þ ð P

topT P RÞ: ð70Þ

Instead of using the volume flow rate QR of the second port, we shall this time use the flow rate QT. However,we have chosen QT to be positive flowing out of the main chamber, so we must attach a negative sign to QT inthe mass balance for the reservoir

QT þ QR ¼ 0: ð71Þ

In a similar manner to our previous approximations for the main chamber lumped element, we argue that

S 2R S 2T; or that K

topT =S

2T K R=S

2R; ð72Þ

for our reservoir lumped element. In addition, we specify that

hR Z R Z topT ; and lTR S T M

eff TR: ð73Þ

In addition, at the top surface of the reservoir, the total pressure P R is the sum of that due to the surface ten-sion force, and the reference atmospheric pressure, P 0, or possibly an applied ‘‘bias’’ pressure P B P 0. Weneglect the pressure arising from the surface tension force since the area of the top surface is relatively large.Thus, we take, P R = (P B P 0) + P 0. We let l R be the characteristic length scale for the reservoir. Thus, ourlumped equation for the reservoir is

qdðlRQTÞ

dt ¼

1

2q

K topT

S TQ2T q ghR þ ð P

topT P BÞ: ð74Þ

We make one additional approximation. We observe that the free surface of the reservoir will move impercep-

tibly during any phase of the dispense; i.e., we will assume that d l R/dt ! 0. In this case, our final form for thereservoir Bernoulli lumped element is

q lR

S T

dQTdt

¼1

2q

K topT

S 2T

!Q2T q ghR þ ð P

topT P BÞ: ð75Þ

This equation is valid for either phase of the dispense process.

4.8. The lumped element for a nozzle

Fig. 5f gives the basic picture of the flow of solder in the nozzles. The surface tension in the solder causes ameniscus to form around a moving contact line. At this contact line, there is always an upward force on the

column of solder above.

h R = l R

PT , S R , K R , Z R ,Port R

Port T top top

T P , S T ,top

T K ,top

T Z

l R

Fig. 5e. Parameters used in the description of the lumped element for the reservoir.


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The treatment of the flow of the solder in the nozzles, which have length, l N, and diameter DN = 2RN,requires some finesse. We wish to describe the dispensing process accurately enough to predict the amountof solder dispensed, but at the same time we do not wish to tackle the very difficult dynamical problem of the instabilities that must develop to break the solder from the nozzles. (There are many papers on this topic.Several references include Chaudhary [13–15], Jeggers [16,17].)

With this philosophy in mind, we can qualitatively describe the motion of the solder column through thechannels as follows: During the driving phase, the solder meniscus in any particular nozzle moves downwardfrom the top entrance of the nozzle, as the diaphragm begins to move forward. One port for the nozzle is thusa free surface and a moving boundary. The meniscus quickly reaches the nozzle exit, and solder dispensing

begins. The exiting solder stream necks, and detaches roughly two nozzle diameters beyond the exit. The dia-phragm hits the stop after approximately a millisecond or two time span, and the retraction phase begins; butthe downward fluid flow in the nozzle does not necessarily cease at this time. The amount of solder dispensedis counted (with some arbitrariness) as the amount of fluid that passes through a nozzle exit up to that timethat the velocity of the solder column vanishes. After the column stops, the flow in the nozzles soon reverses,the meniscus moves upward from the nozzle exit to the top. This retraction occurs under the action of surfacetension forces in the absence of the driving diaphragm motion.

One may estimate the maximum error associated with the definition of the term, ‘‘amount of solder dis-pensed’’. From high speed video footage, the quantity of residual solder that draws back into the nozzles isat most that of a conical sliver attached to the nozzle exit (see Fig. 5f (iii)) of height hsliver 2DN. For nozzles100 lm in diameter, an estimate of the maximum error is given by, Estimated Dispense Error 1

3p

D2N

4 hsliver !

0:13 mm3

, i.e. roughly 8–20% of the volume dispensed for data reported in this work. This estimate is actuallyquite conservative. In practice, the error is generally less than the estimate, since the conical sliver at break off is normally based in the interior of the nozzle by about one nozzle diameter, rather than at its exit.

We again apply our equation for the tubular viscous lumped element, with a lossy boundary condition onthe top port, and moving boundary on the bottom port. Our reference port is the top of the nozzle. Hence, ourlumped element for the nozzles is (refer to Fig. 5f )

q d

dt ðlNQNÞ ¼

1

2q

K topN þ K

lossN ðQNÞ

S N

Q2N þ S Nq ghN G NðQNÞ þ G R þ S Nð P

topN P 0Þ: ð76Þ

We take the kinetic energy boundary term to vanish for the bottom port, since it is a moving boundary. Thereis the viscous force G N (QN), the sign of which depends of the direction of flow. This term will oppose the flow,

regardless of the flow direction, since it depends directly on QN. We have Slattery [9, p. 77]

2

residual

N

h

D≈

i) Meniscus

Resting Position

N l

R N

h N = l N

ii) Meniscus

Moving in channel in

the driving and

retraction phases

Porttop

N

Q N , S N ,top

N K ,top

N P ,top

N Z

θ

σ

Contact

Line

Portbot

N

S N , 0bot

N K = , 0P ,bot

N Z

Viscosity µ

Surface

Tension σ

Contact

Angle θ

µ

N D

iii) Dispense phase

Fig. 5f. Three configurations of the nozzle meniscus and the parameters used to model the annular nozzle element.


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G NðQNÞ ¼8

pl

lNS N

R4N

QN: ð77Þ

There is also the capillary force G R, or the surface tension force Batchelor [12],

G R ¼ 2p RNr cosðhÞ: ð78Þ

The parameter r is the surface tension. The surface tension force is applied along the contact line of the solid– liquid–air interface. (See Fig. 5f ) Because the nozzle walls are non-wetting, the contact angle is larger than 90;the force is always directed upwardly

One may take the term dl N/dt as the velocity of the meniscus. This moving boundary term then takes aform that is very similar to the kinetic energy term. Since

dlNdt

QN !Q2NS N

: ð79Þ

If we now collect all the terms together, we have for the nozzle lumped element

q lN

S N

d

dt QN ¼ q

1

S 2

N

Q2N þ1

2q

K topN þ K

lossN ðQNÞ

S 2

N !Q2N þ q ghN þ

G R G NðQNÞ

S Nþ ð P topN P 0Þ: ð80Þ

To connect the nozzle volume flow rate QN to the dispensed volume, we employ the mass balance equation forthe nozzles. By definition, the volume rate QN in the nozzles is

QN ¼dV N

dt ; ð81Þ

We then say that the volume dispensed, V Dispensed, is that volume that exits all nozzles during the time whenthe variable l N is greater than the length of the nozzle length LN to the time when the volume flow rate QNvanishes (changes from positive to negative)

V Dispensed ¼ Z time when QN¼0

time when lN> LN

dt ð NQNÞ: ð82Þ

4.9. Assembly of lumped elements for the driving phase

We assemble our equations for the driving phase of the pumping action. We have two paths, one from thediaphragm to the reservoir, D ! R, where we will thus assemble Eqs. (61), (68), and (75), and a second path,D ! N, from the diaphragm to the end of the nozzles, where we assemble Eqs. (62) and (80). To assemble theequations means to eliminate the unknown intermediate pressures and kinetic boundary terms between thelumps. We affect this assembly by just adding the equations together so that the intermediate pressure andkinetic terms pair-wise cancel. However, the port loss terms survive. We then obtain two equations, one equa-tion for the flow QT, and another for the flow QN. The resultant equation for the flow QT is

q lDRS T

ddt

QT ¼ 12q K

drvT ðQTÞ

S 2TQ2T q ghDR G

TðQTÞS T

þ ð P D P BÞ; ð83aÞ

where we define,

lDR lDT þ lT þ lR; ð83bÞ

hDR hDT d D=4 þ hT þ hR;

K drvT ðQTÞ K botlossT ðQTÞ þ K

toplossT ðQTÞ: ð83cÞ

If assemble Eqs. (62) and (80), we find the resultant equation for the flow QN, for the flow path D ! N ,

qlDN

S N

d

dt

QN ¼1

2

q K drvN ðQNÞ

S 2

N !Q2N þ q ghND þ

G R G NðQNÞ

S N

þ ð P D P 0Þ; ð84aÞ


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where we define,

lDN lDNtop þ lN; ð84bÞ

hND hN þ ðhDNtop d D=4Þ; and ð84cÞ

K drvN ðQNÞ K lossN ðQNÞ 2: ð84dÞ

These equations are driven by the diaphragm pressure,P D, which is subsequently driven by the air pulse gaugepressure P F, i.e., from Eq. (48a)

P D ¼ P F k D xD=S D: ð85Þ

To close the set of equations for the driving phase, one must include the relationship between xD, and the flowrate QD from Eq. (51). Hence,

d xDdt

¼ 3QDS D

¼ 3

S DðQT þ NQNÞ: ð86Þ

For the driving phase, we have a system of three coupled first order non-linear differential equations, Eqs.(83a), (84a), and (86) for the three unknowns, QT, QN, and xD.

We remark that the equations for the individual flows are quite transparent in their meaning. They lookvery much like that for our general tubular viscous element. Under the assumption of independent flows,one up through the throttle, and the other down through the nozzles, one may an essence write down theseequations for the flows QT, and QN, by inspection, using an effective cumulative path length, say l DR, forexample. One includes the pressure boundary conditions at the beginning and end of the path, as well asany viscous losses along the path, and finally a pressure head that results from the vertical change in heightfrom the beginning port to the ending port.

4.10. Assembly of lumped elements for the retraction phase

To assemble the equations for the retraction phase, we add the dynamical equations along the path N ! R,using Eqs. (80), (66), (68), and (75). In addition, we must employ the continuity condition for the retractionphase, namely

QT ¼ NQN; QD ¼ 0: ð87aÞ

The assembled dynamic equation is then,

qlNR

S N

d

dt QN ¼

1

2q K retðQNÞQ

2N þ q ghNR G

retðQNÞ þ P B: ð88aÞ

As with the dynamical equations derived for the driving phase, we see that the final dynamical equation in theretraction phase formally resembles a single lumped element, but with effective values for the various dynamicquantities that depend on the entire fluid path, starting from the reference port. We have,

lNR þ

NS N

S T ðlNT þ lT þ lTRÞ; ð88bÞ

and

hNR þ hNT þ hT þ hR: ð88cÞ

We will ignore the variations of l N and hN, with respect to the much larger quantities of l NR and hNR. In addi-tion, there is an effective ‘‘K ’’ constant for the retraction phase of

K retðQNÞ K

toplossT ðQNÞ K

botlossT ðQNÞ

ðS T= N Þ2

!þ

K lossN ðQNÞ

S 2N

2

S 2N

!; ð88dÞ

and an effective ‘‘G ’’ quantity for the retraction phase,


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G retðQNÞ G TðQNÞ

S T= N þ

G NðQNÞ

S N

G R

S N: ð88eÞ

With this last equation, we have completed the assembly of all the equations necessary to describe the dispenseprocess.

5. Solution methodology

The initial value problem of solder dispensing, defined by Eqs. (83), (84), (86), and (88), were solved on aUNIX workstation with an implicit Adams–Moulton scheme available in the IMSL Library. Given materialproperties and geometric information pertaining to the pump, nozzles, diaphragm, and throttle, the modeltakes the drive pressure and diaphragm stop location as input and predicts the time dependence of followingquantities:

1. The pressure fill of the pneumatic chamber upstream of the diaphragm.2. The diaphragm motion.3. The solder interface location, including the retraction phase.

4. The solder interface velocity, including the retraction phase.5. The solder velocity in the throttle, including the retraction phase.6. The amount of solder dispensed.

The numerical solution proceeded in two time intervals, that of the driving phase, and that of the retractionphase. Each phase has its own set of equations, as we have detailed above. We need to link these equationsacross the flow discontinuities caused by the diaphragm’s abrupt stop. Appendix A shows that the volumeflow rate in the nozzles just after the diaphragm hits the stop, QþN, is related to the volume flow rate in thenozzles just before the diaphragm hits the stop, QN, by the relation

QþN ¼ QN gQ

D; ðA:5Þ

where Q

D is the diaphragm volume flow rate, just before the diaphragm hits the stop. The equation above al-lows us to determine the starting volume flow rate for the nozzle flow in the retraction, given the nozzle anddiaphragm flows just before the diaphragm hits the stop. In the equation above, there is a constant g, which inprinciple depends on ratios of intrinsic length scales. However, in the numerical simulations that we per-formed, we adjusted the constant, experimentally.

6. Results

6.1. Parametric study

We verified our dynamical model by comparing predictions from the model, applied to a prototype dis-penser, with dispensing data gathered from the same dispenser. Typical parameters pertaining to that dis-penser are given in Table 1. The results of the study are summarized in Figs. 6–12.

Fig. 6 shows the calculated pressure transient in the pneumatic chamber, and the calculated diaphragm dis-placement transient through a single dispensing pulse. The stop is placed at a diaphragm displacement of approximately .04 mm (1.5 thousandths of an inch), and the diaphragm takes a little over 3 ms to reachthe stop.

Fig. 7 shows the solder interface position in the nozzles as the pulse progresses, and Fig. 8 shows the veloc-ity transient of the fluid in the nozzles. The interface accelerates slowly at first (while the pressure in the pneu-matic chamber builds). Rapid acceleration follows until the diaphragm hits the stop. In the featured run, theinterface takes roughly 4 ms to reach the channel exit. A rapid change in velocity is experienced when the dia-phragm hits the stop. Since the throttle opening in the featured run is small, the nozzle flow is dominant overthe throttle flow just prior to impact. The opposing momenta of the two streams results in a significant reduc-

tion in the nozzle flow, and a change in direction of the throttle flow, i.e. solder instantaneously begins to move


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from the reservoir back into the body of the pump. Hence, the solder continues to dispense after the dia-phragm has stopped moving. However, the capillary forces in the non-wetting nozzles are strong enough toovercome gravity, and the momentum of the flow, and act to retard the flow. Eventually, the flow rate throughthe nozzles reaches zero, and subsequently becomes negative, i.e., the interface begins to retract. The retractinginterface achieves a maximum velocity of about 0.3 m/s before the interface reaches the top of the nozzles, atwhich point the capillary forces reduce dramatically (since the diameter of the pump body is much larger thanthat of the nozzles), and the interface motion ceases. The impact of throttle cross-section on the peak retrac-tion velocity is discussed later in this section.

Fig. 9 shows the time dependence of the volume of solder that has passed through the nozzle exit. The vol-ume dispensed through the entire pulse, represented by the value at the end of the curve in Fig. 9, is about

1.25 mm3.

Table 1Typical parametric values for model study

Length of nozzle l N 1.0 mmDiameter of nozzle DN 80 lmNumber of nozzles N 256Diameter of diaphragm d D 20 mm

Diaphragm ‘‘Spring Constant’’ k D 2.564 N/mDiaphragm stop distance DxD 38.1 lmThrottle length l T 8.0 mmThrottle gap width R0 – Ri 130 lmCharacteristic length (diaphragm to throttle) l DT 15 mmCharacteristic length (diaphragm to nozzle) l DN 58 mmCharacteristic length (nozzle to throttle) l NT 66 mmVertical height (nozzle to diaphragm) hDNtop 51 mmSolder density q 8000 kg/m3

Solder viscosity l .002 kg/m/sSolder contact angle cos(h) 0.94Reservoir height hR 15 mmFilling chamber temperature T F 523 KGas pulse driving pressure P G 4.116–5 Pa

P r e s s u r e ( N / m 2 )

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

7.00E+05

8.00E+05

0 1 2 3 4 5

Time (ms)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Pressure

DiaphragmDisplacement

Di s pl a c em en t

( mi l s– t h o u s an d t h

s of ani n ch )

Fig. 6. Pressure and diaphragm displacement transients.


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Fig. 10 shows the effect of widening the throttle opening. The upward fluid momentum in the pump due toflow through the throttle becomes significantly larger since the throttle presents a smaller resistance to the flowthan in the previous run. Thus, after the diaphragm hits the stop, dispensing ceases, and the solder interface inthe nozzles begins to retract rapidly. The velocity in the nozzles is actually negative immediately after the dia-phragm hits the stop. When the interface reaches the top of the nozzles, the fluid velocity is greater than 1 m/s.Hence, there is a greater propensity to draw gas into the pump from the nozzles if the throttle is opened wide.

Fig. 11 shows the velocity transients of the flow through the throttle for two cases in which the solder levelsin the reservoir differ, in this case by 15 mm. To exercise the predictive capability of our formalism, we studiedhow the ‘‘quantity of solder dispensed’’ varies as the ‘‘height of solder in the reservoir’’ is changed. As the sol-der height in the reservoir increases, the upward flow through the throttle reduces (since a larger quantity of

solder must be moved), while the flow through the nozzles remains relatively unchanged. Due to the increase

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14

Time (ms)

I n t e r f a c e P o s i t i o n i n N o

z z l e s ( m m ) Interface Position

Fig. 7. Interface position transient of fluid in nozzles.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14

Time (ms)

V e l o c i t y i n N o

z z l e s ( m / s )

Nozzle

Fig. 8. Velocity transient of fluid in nozzles.


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in mass in the reservoir, and thus an increase in the inertia of the solder, the diaphragm takes longer to hit thestop and, hence, the total quantity of solder dispensed increases.

Fig. 12 shows the general trend of increasing dispensed solder volume as the solder height increases.

6.2. Comparison of data and simulations

To verify the fidelity of our dispensing model, experimental data was collected using the prototype appa-ratus described in Section 2. Measurements were extracted from high speed video of the dispensing operationlinked to the recorded diaphragm motion by means of a trigger, and the operating parameters were varied tocover a range of data. The model parameters, i.e., the characteristic length scales (the l ’s, particularly l DR, l DN,and l NR), and g, were determined by fitting the model predictions of

1. The dispensed solder volume.2. The time at which the interface reached the nozzle exit.

3. The time at which the interface began receding.

0 2 4 6 8 10 12 14

Time (ms)

Solder Volume

V o l u m

e ( m m 3 )

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 9. Volume-dispensed transient.

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

0 0.5 1 1.5 2 2.5 3

Time (ms)

V e

l o c i t y i n N o z z l e s ( m / s )

Nozzle Velocity (Wide Throttle)

Fig. 10. Nozzle velocity for wide throttle.


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Figs. 13–15 show dispensed-solder transients for three different throttle openings ranging from ‘‘tight’’, orrelatively small, to ‘‘wide’’, or relatively large.

Fig. 13 presents the results for the ‘‘intermediate throttle’’ case. The experimental displaced solder volume(1.17 mm3) compares favorably with the computed value (1.14 mm3). The time at which the interface reachesthe nozzle exit (3.2 ms), and the time at which the interface begins receding (5.8 ms) also compare well withthe computed values (3.2 and 5.7 ms respectively.) The comparison of the experimental and predicted values of these characteristic times is an excellent indicator of the ability of the model to capture the basic physics of thepump’s operation.

Fig. 14 is a similar comparison for the ‘‘tight’’ throttle case. Fluid issues from the nozzle and disappearsfrom the field of view after about 4 ms, after which volume comparisons are not possible. Prior to this time,however, the volume displaced matches well with data. Also, the time at which the interface reaches the nozzleexit (3 ms) compares favorably with the computed value.

Fig. 15 presents the results for the ‘‘wide’’ throttle case. The experimental data here appear somewhat

scattered. However, the computed time taken for the interface to reach the channel exit is within 10% of

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

90 95 100 105 110

Static Head (mm)

V o l u m e

D i s p e n s e d ( m m 3 )

Solder Volume Dispensed

Fig. 12. Solder volume dispensed vs. static head.

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

Time (ms)

T h r o t t l e V e l o c i t y

( m / s )

Static Head = 90 mm

Static Head = 110 mm

Fig. 11. Impact of static head on throttle velocity.


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the experimental value, the computed time at which the interface begins receding is within 20% of the exper-imental value, while the computed maximum solder volume displaced is within 30% of the experimentalvalue.

We remark that, due to the difficulties associated with the dimension measurement of tiny drops from videophotographs, the accuracy of the experimental data is somewhat limited. Hence, the aforementioned compar-isons seem to indicate that the data are well represented by the model, and that the model can serve as a valu-able aid while determining the best operating conditions for solder dispensing. In particular, dispensing is mostconsistent when the solder flow stops abruptly immediately after the diaphragm hits the stop. If, instead, thesolder continues to ‘‘dribble’’ from the nozzles (as would happen if the throttle setting were too tight), occa-sional scatter of the droplets can result. Conversely, if the throttle is opened too wide, the peak retraction

velocity of the interface may be too large and air may be drawn into the pump at the end of the pulse.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 1210

Time (ms)

V o l u m e D i s p e n s

e d

( m m 3 )

Solder exits

nozzle

Solder retracts

Model Simulation

Experimental Data

Throttle Area = 0.954 mm2

Fig. 13. Experimental verification – intermediate throttle position.

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8

V o l u m e D i s

p e n s e d ( m m 3 )

Model Simulation

Experimental


Time (ms)

Fig. 14. Experimental verification – tight throttle position.


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7. Conclusions

A mathematical model, employing fluid lumped elements, that describes molten solder dispensing from apneumatically actuated, diaphragm-driven pump was developed. The model is multi-physical in that capturesthe filling of the pneumatic drive chamber, the equation of motion of the diaphragm, and the flow of solder in

the pump. The simulation capability enables predictive ‘‘tuning’’ of the pump. The model indicates that thesystem parameters should be chosen such that a desired amount of solder is first dispensed, followed soonafter by the diaphragm hitting the stop. The throttle opening should be selected to ensure that the solder flowis near zero immediately after the diaphragm hits the stop. The interface will then break cleanly, and theretracting solder proceeds back up the nozzles with a small velocity and acceleration, reducing the propensityto draw gas into the pump.

Appendix A. Relation of flows across the discontinuity of diaphragm stop

Here we display the continuity conditions of the flow variables at the point in time when the diaphragm hitsthe diaphragm stop. At this time there is a step discontinuity in the flow driven by the diaphragm resulting in

step changes in the flow at the other two ports. Let us focus again on the main chamber, which has three ports,the diaphragm port D, the bottom throttle port T, and the Nozzle ports N. If one integrates Eq. (11),

qX N

q

d

dt ð M pqQqÞ þ

1

2q

K p

S 2 p Q2 p þ q gZ p þ P p ¼ 0; p ¼ 1; . . . ; N ðA:1Þ

over a vanishing small time interval around the time of the diaphragm impact, the only terms that survive are:

Port D: M DDDQD þ M DTDQT þ M DNDQN þ

Z ee

dtP D=q ¼ 0; ðA:2aÞ

Port T: M TDDQD þ M TTDQT þ M TNDQN ¼ 0; and ðA:2bÞ

Port N: M NDDQD þ M NTDQT þ M NNDQN ¼ 0: ðA:3cÞ

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

0.00E+00 2.00E+00 4.00E+00 6.00E+00 8.00E+00

Time (ms)

Model Simulation

Experimental Data


V o l u m e D i s p e n s e d ( m m 3 )

Fig. 15. Experimental verification – wide throttle position.


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Here, DQ p is the change in flow rate across any flow discontinuity at por

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