The Discrete Agglomeration Model: Equivalent Problems

In this paper we develop equivalent problems for the Discrete Agglomeration Model in the continuous context.


Introduction
Agglomeration of particles in a fluid environment (e.g., a chemical reactor or the atmosphere) is an integral part of many industrial processes (e.g., Goldberger [1]) and has been the subject of scientific investigation (e.g., Siegell [2]).A fundamental mathematical problem is the determination of the number of particles of each particle-type as a function of time for a system of particles that may agglutinate during two particle collisions.Little analytical work has been done for systems where particle-type requires several variables.Efforts have focused on particle size (or mass).This allows use of what is often called the coagulation equation which has been well studied in aerosol research (Drake [3]).Original work on this equation was done by Smoluchowski [4]) and it is also referred to as Smoluchowski's equation.The agglomeration equation is perhaps more descriptive since the term coagulation implies a process carried out until solidification whereas we focus on the agglomeration process; that is, on the determination of a time-varying particle-size distribution even if coagulation is never reached.
In his original work Smoluchowski considered the agglomeration equation in a discrete form.Later it was considered in a continuous form by Muller [5]).In either case, an initial particle-size distribution to specify the initial number of particles for each particle size is needed to complete the initial value problem (IVP).We refer to these as the Discrete Agglomeration Model and the Continuum Agglomeration Model respectively.Solution of either model yields an updated particle-size distribution giving number densities as time progresses.For various conditions, studies of these and more general models include Morganstern [6], Melzak [7], Mcleod [8], Marcus [9], White [10], Spouge [11], Treat [12], Mc-Laughlin, Lamb, and McBride [13], Moseley [14], and Moseley [15].
Let R be the real numbers, To develop the discrete model, assume that all particles are a multiple of a particle of smallest size (volume), say v  .Thus a particle made up of i smallest-sized particles has size i v  .In polymer chemistry, the particle is called an i-mer.The  ) that approximates the number of i-mers in the reactor at time t.Since there are an infinite number of sizes, initially, we take the state (or phase) space to be Assume the initial number density   As time passes, particles collide, agglutinations occur, and larger particles result.The net rate of increase in n i (t) with time, dn i /dt, is the rate of formation minus the rate of depletion (conservation of mass).For , is a doubly infinite array of real-valued functions of time either in , we establish no topology on  R .The resultant Discrete Agglomeration Model or Discrete Agglomeration Problem (DAP) is an IVP consisting of an infinite system of Ordinary Differential Equations (ODE's) each with an Initial Condition (IC) that may be written in scalar (componentwise) form as: i j,j i j j i i,j j j i j 1 IC's : n t n t I t , t t where for i = 1 the empty sum on the right hand side of ( 1) is assumed to be zero.The first sum in the scalar (componentwise) discrete agglomeration Equation ( 1) is the (average) rate of formation of i-mers by agglutinations of   i j -mers  with j-mers.The 1/2 avoids double counting.The second sum is the (average) rate of depletion of i-mers by the agglutinations of i-mers with all particle sizes.We model a stochastic process as deterministic.The physical system is often stationary so that each i, j K is time independent and the model is said to be autonomous.In a physical context, we require   0 0 i, j 1 i K t 0, n 0, and n 0 for i 1     .However, we will address DAP as a mathematical problem where we allow the initial number of particles 0 i n , the components of the kernel   i, j K t , and the components of the solution,   i n t , to be negative.The physical context will be a special case.
Smoluchowski found in the physical context that when where (and hence a metric and a topology).Equality of two vectors in p  requires the metric (the norm of their difference) to be zero.This is equivalent to both vectors being in p  and being componentwise equal.If (Naylor and Sell [17,p. 58]).To insure that 0 M n exists (even for negative initial conditions), we will require We are particularly interested in the time-varying kernel  which depends on time, but not on particle size.In the continuous context where . In the analytic context where . For any kernel, solution requires that both sides of (1) are continuous in the continuous context and analytic in the analytic context.
The i th depletion coefficient associated with 0 t I  and the distribution is defined formally by the infinite series The only direct dependence of (the Σ space) and satisfies (1) on I and (2), then it solves DAP on I.This formulation of DAP does not require mathematics beyond calculus and is often used by engineers and scientists.
For DAP with a time varying kernel, , in the analytic context, Moseley [14] established that the more general formula where or the physical context where The formula ( 6) satisfies (1) on I and (2) in the continuous context as well where we now allow However, since (6) was not derived using equivalent equation operations, uniqueness has not been proved rigorously for . Unless otherwise stated, for the rest of the paper, we focus on the continuous context.
Moseley [14] divided DAP into several problems which could be considered separately.Under certain conditions, a reasonably complicated change of (both the independent and dependent) variables transforms DAP with a time varying kernel (Moseley,[14]) into another IVP which Moseley later referred to as the Fundamental Agglomeration Problem (FAP).The solution process for FAP is fully documented in Moseley [15].For FAP, Moseley established existence and uniqueness for both the analytic and continuous contexts by using a sequential solution.To facilitate further progress, in this paper we develop equivalent problems for DAP in the continuous context.Analogs for the analytic context can be obtained.
To rearrange terms in infinite series we will need If all sums exist, we add all of the elements in in two different ways.Since we use them often, we will use  to mean "for all" and  to mean "there exists" (with apologies to the logicians).If y = n(t), we use any of n, n(t), y(t) and n(  ) to denote the function.Also, we denote the restriction of a function to a smaller domain by the same symbol.The context will make it clear.

Mathematical Problem Solving
Often, a mathematical problem is specified by giving a condition (or conditions) (e.g., an algebraic equation or an ODE with an initial condition) on elements in a Σ set (the designated set where we look for solutions, e.g., If the  set is a vector space, we say Σ space.A problem is (set-theoretically) well-posed if it has exactly one solution in its  set.(In this paper, we will not consider continuity with respect to problem parameters.)A well-developed model of dynamics using an IVP is well-posed (exactly one event happens).As mod-elers, we expect our models to be well-posed.As mathematicians, we require rigorous proof.Often, we solve equations by using equivalent equation operations to isolate the unknown(s).This yields uniqueness, and, as all steps are reversible, existence.(Squaring both sides of an equation is not an equivalent equation operation and may lead to extraneous roots.)For linear ODE's, we may guess the form of a solution and prove existence and uniqueness by using the linear theory.For nonlinear problems, we may prove existence by substituting back into the equation.Uniqueness then becomes an issue. Let in the  space   1 C I,R .Thus, as is usually done, we require solutions to (8) to not only exist, but to also have continuous derivatives.We also require

 
f C I U,   R where U  R and the range of y(t) is in U for y(t) in the  space.Placing these additional constraints avoids dealing with pathology, but narrows the space where a known solution is to be shown to be unique.There may be (pathological) solutions to (8) where the derivative exists, but is not continuous.Also, as is usually done, we allow I to vary.If we show that there exists a solution for some I, then we say that we have local existence on I.
as a separate condition for solution, we may incorporate it into the Σ space.We refer to DAP with the Σ space as the Scalar Discrete Agglomeration Problem (SDAP).
Obviously, this may be formulated in an analytic context as well.
Recalling the constraint , instead of  R , we may choose the state space as 1 vs   R  which has a norm (and hence a metric and a topology).A solution on I is then a time-varying infinite-dimensional "state vector" . Later we will choose an appropriate  space and write DAP in vector form.We refer to this formulation of DAP as the Vector Discrete Agglomeration Problem (VDAP).As with SDAP, VDAP may be in the continuous or analytic context.If SDAP is well-posed, and its solution is in the (smaller)  space for VDAP, then SDAP and VDAP are equivalent except for the space where local uniqueness is proved.That is, by choosing a smaller  space, VDAP requires proving local uniqueness in a smaller space than does SDAP.If we do not worry about pathology, and redefine the  space for SDAP to be the same as for VDAP, the two problems are equivalent.The question is: How do we choose an appropriate (smaller)  space?But first we consider an equivalent scalar problem and p  spaces.

Equivalent Scalar Problems
Again assume for i (11) which also map 0 I to For these functions, as with we may now write (1) as the system of ODE's Initially, we assume and investigate Proof.Sums, products, and compositions of continuous functions involving ℓ 1 are continuous.■ Detailed ε-δ proofs follow proofs in an elementary real analysis course.All functions map to R. We must choose n n satisfies a Lipschitz condition (Bartle [18,p. 161]) and hence is continuous We investigate continuity and differentiability in ℓ p in more detail in the next section.
. Previewing the next section, we define the function spaces as the componentwise continuous functions that have codomain ℓ 1 , and claim that Proof.Sums, products, and compositions of continuous functions involving 1   are continuous.■ We now show that in the continuous context if , then SDAP given by ( 12) and (2) with the Σ space   is equivalent to the infinite system of scalar (componentwise) Voltera integral equations where and not just that the integral in (6) exists.)We refer to this problem as the Integral Scalar Discrete Agglomeration Problem (ISDAP) in the continuous context.A formulation in the analytic context can also be established.Theorem 2.3.In the continuous context, a distribution . We have by the definition of a solution of SDAP, that , that ( 13) is satisfied on I, and that (2) is satisfied.Since both sides of ( 13) are continuous, we may integrate from t 0 to t I  to obtain Applying the initial condition we obtain (13).Simi- . Substituting in t 0 we obtain (2).
. Differentiating we see that (11) is satisfied.■ For the scalar Equation (2.1), it is the integral formulation that is used to obtain existence (Picard iterations) and uniqueness using a Lipschitz condition.If we choose to specify as the  space for both prob-lems, the problems remain equivalent as any solution to (13) in . That is, there are no solutions to (13) in . These results can also be established in the analytic context.

Continuity and Differentiability for ℓ p Spaces
Since p  has a norm (and hence a metric) we have a topology on the subspace p  of  R .Many of the limit laws can be extended to p  .For example, if that is, given 0, 0 Hence we can define the function spaces , the range is restricted to the set B whereas, for , it is allowed to be in the larger set C. Since . However, p  has a norm (and hence a metric and a topology), but  R does not.(We could establish a topology for  R , but this is not necessary if the system states are all in p  .)We will use for functions that are componentwise continuous with codomain p  and write ; that is, we use the same symbol for the restriction of a function to a smaller domain.
We give necessary and sufficient conditions for    n to be in is a vector space, by our previous comments (or any normed linear space), then the triangle inequality Since the composition of continuous functions (to and However, Although not sufficient individually for   : I . However, all of these do force   , , and we can choose N sufficiently large so that , it may be easier to check that for each t I and Following the standard proof for products, we also have Theorem 2.8.If . We define integration componentwise.Following Theorem 2.6, we have Theorem 2.10.If Theorem 2.11.

   
Proof.The first containment follows from Theorem 2.10.The remaining proofs are straight forward and often similar to the proof of Theorem 2.4.■ Theorem 2.12 (Fundamental Theorem of Calculus) Note that the indefinite integral requires an arbitrary constant vector.

Kernels, State Spaces and Σ Spaces
In the analytic context with an analytic kernel, He then obtained the explicit formula (6) for the (analytic) solution when A(t) is analytic.He did not rigorously isolate the unknown so he established global existence by showing that the solution given by the formula ( 6) was in the Σ space, checking the initial conditions (2), and then substituting the formula into (1).Since global existence holds, local uniqueness implies global uniqueness.
The problem of interest is to extend Moseley's results for the analytic context to the continuous context.The solution given by ( 6) remains the same except that we now only require Global existence may be obtained as before.However, local uniqueness is not as easy as it was in the analytic context.McLaughlin, Lamb, and McBride [13] provided local existence and uniqueness for a Continuum Agglomeration Model of linear fragmentation with coagulation as a perturbation using semigroup theory.Spouge [11] provided a local existence theorem in the physical case, but not uniqueness.The standard procedure in Brauer and Noel [20] for a finite dimensional system requires a Lipschitz condition on the right hand side to obtain local existence and local uniqueness.In this paper, we provide preliminaries for using a Lipschitz condition to prove uniqueness in the continuous context by giving equivalent problems in scalar and vector form for DAP with K(t) in a larger collection than and write . Furthermore, when , and Proof.Compositions of continuous functions (in 1  ) are continuous so that if . (See Corollary 2.2.) ■ In the continuous context we wish conditions on . Then the convergence and continuity condition on need not be explicitly stated for the Σ space or as a condition for solution (except as required for interpreting (1)).We begin with three classes of kernels: , and

I, B A I , R :
A C I , and B I , , then for all kernels in these three classes, if . However, for clarity, we proceed class by class.
, then for all i, j N we have is the first moment of the solution and ρ is the mass density.Treat [12] suggested (as have others) on a physical basis, that these and other moments, possibly all moments, should exist (converge and be continuous).We will take our Σ space as a subspace of we view a solution as a time-varying infinite-dimensional and is in f n is Lipschitz continuous and hence in (by Theorem 2.15).By Corollary 2.9

Weierstrass M-Test and Local Uniform Boundedness
Let . We briefly review the Weierstrass M-test and succeeding theorems on absolute and uniform convergence (Kaplan [21, pp. 436-444]).This should be familiar to engineers and scientists.We then consider a fourth class of kernels.Let ).
We say that K(t) is locally uniformly bounded in time and size.Theorem 2.17 (Weierstrass M-Test Extended).Let (converge absolutely t J   and are uniformly convergent on J so that theyare in   C J,R .Since t 1 was arbitrary, t J   and are uniformly convergent on J so that they are in C(J,R).Since t 1 was arbitrary, to satisfy a stronger local uniform boundedness condition in time.We have It can be shown (similar to Moseley [14] in the analytic context) that in the continuous (physical) context, . By Theorem 2.17, , then, again by Theorem 2.17, Everything else is straight forward or follows in a manner similar to the proof of Theorem 2.4.■ We show that if . For i  N , and Then . However, we do not have . We may (or may not) be able to prove this with a further extension of the Weirstrauss M-test.Instead we let Hence by Theorem 2.17, Then by Theorem 2.16, If we can show that   I   all map to 1   so these functions are all in   , from (2.12) we have Since t 1 was arbitrary,   , by Theorem 2.17, for fixed   where we have used (7).Hence we have that . By using Theorem 2.1 and above, where the derivative and equality are in 1  .That is, we now require the derivative to be defined with respect to the norm topology, , and equality as equality in 1  .For VP, we take our Σ space as where for (the Σ space) to be a solution of (21), we require i   N , that (12) holds; that is, integration is componentwise.Equality is in 1   .We refer to this problem as the Integral Vector Problem (IVP) For both problems we have chosen the Σ space to be      1) Provide a rigorous derivation of ( 6) that provides (existence and) uniqueness.

Summary and Future Work
2) Develop and use a Lipschitz condition for:   i f t, ;K n in the scalar problems SDAP and ISDAP.3) Extend the (existence and) uniqueness results for FAP in the continuous context to obtain a unique sequential solution to DAP.
4) Develop and use a Lipschitz condition for   t, ;K f n in the vector problems VDAP1 and VDAP2.We have provided preliminaries for the development of a Lipschitz condition for VDAP1 and VDAP2.However, all four alternatives appear to be worthwhile.
R I is a finite, infinite, or semi-infinite open interval}, and for o the largest time interval of interest.We indicate this by the extended interval notation

I
Int t ,I  we consider as a possible Σ space (i.e., the designated space where we look for solutions) either  cw I,  R A for the analytic context or   1 cw I,  R Cfor the continuous context where

M
n in (4) and the infinite sum in (1.1) motivate consideration of the Banach spaces

1 I
  with respect to the norm topology, we write be in   p C I, , we need its range to be in p  , the zeroth moment of the sequence.In the physical context, i n 0  so that, mass of the particles (which should not change) where  D R ) as the Σ space in the analytic context.

For
[14] used the following problem solving procedure.He first established local uniqueness in obtained the explicit formula(6) for the (analytic) solution.He did not rigorously isolate the unknown so he established global existence by showing that the solution given by the formula (6) was in the Σ space, checking the initial conditions, and then substituting the formula into(1).Since global existence holds, local uniqueness in the analytic context implies global uniquenessΣ space, then SDAP, ISDAP, VDAP1, and IVDAP2 are all equivalent in the continuous context if they have the same problem parameters we need not specify this condition separately.For the time varying kernel, the solution given by (6) is in  1 LUB I,  R C, where   A t C I,  R in the continuous context.However, we have not shown (local) uniqueness in the con-tinuous context.To do this we have (at least) four choices: ∞.They are componentwise continuous.