The Lambert Function Should Be in the Engineering Mathematical Toolbox

Abstract

Discovered well over two centuries ago and little used for long, the Lambert function has emerged in an increasing number of science and engineering applications in the last couple of decades. Here we present case studies relevant to the diverse interests of chemical engineers. We show how the Lambert function can be used for both analysis and computation. While some of these studies expound on prior literature results, the rest are new. We conjecture that if this tool becomes more widely known, many more instances of application will appear. Therefore, given its simplicity and usefulness, we would reasonably argue that the Lambert function should be included in the standard mathematical toolbox of chemical engineers.

1. Introduction

The engineering mathematical toolbox includes a number of familiar elementary functions, such as the power, exponential, logarithm, and trigonometric functions. Because these functions appear widely in all of engineering, expression of the solution of an engineering problem in terms of such functions has tangible benefits in at least two ways. One is computation. This benefit has been realized for centuries, as many lengthy computations would typically be reduced to simpler computations in terms of known functions whose values would be pre-calculated and tabulated (Grier, 2005). Another benefit, perhaps far more important in our computer-enabled-computation era,² is communication and analysis of the behavior exhibited by the solution of an engineering problem. Indeed, properties of elementary functions, such as continuity, differentiability, oscillation, or boundedness, are well understood and may be used to analyze properties of entities dependent on them. For example, numerical solution of initial-value problems for ordinary or partial differential equations is as simple as programmable from scratch on an electronic spreadsheet in a few minutes, yet analysis of boundedness or oscillations of the solution can be facilitated by elementary functions.

In recent years, diverse research by many groups (such as Brkić and Praks (2018); Chen (2019); Cranmer (2004); Fries and Dreyer (2008); Lehtonen (2016); Morales (2005); More (2006); Rollmann and Spindler (2015); Schnell and Mendoza (1997); Valluri et al. (2000) for a partial list) including ours (Nikolaou, 2020a, 2020b, 2020c), has come across an elementary function that has been known for well over two centuries (Lambert, 1758), yet it had long remained underappreciated, until it came to the forefront in 1996 (Corless et al., 1996): The Lambert function, defined simply as all solutions x (real or complex) of the algebraic equation

$$x{e}^x=w$$ (1)

for a given w. In their seminal publication, Corless et al. (1996) demonstrated that even though it may not be immediately obvious, the Lambert function naturally appears in the solution of a wide range of problems in science and engineering. The purpose of this publication is to draw attention to a number of problems of interest to chemical engineers, for which the Lambert function can help provide a powerful and elegant solution, in the context of helping both compute and analyze the properties of the solution. For some of these problems solution in terms of the Lambert function has already been on record in literature, while for the rest we believe that the propositions in this paper are new. It is conjectured that once the chemical engineering community becomes widely aware of the Lambert function, more cases, beyond sporadic instances, are likely to appear that accept a Lambert-based solution. In fact, a main message of this publication is to alert the chemical engineering community about such instances being worth recognizing, since recognition that a problem conforms to a pattern amenable to treatment by a powerful tool has well known value.³ The cases presented here are not meant to represent an exhaustive list. Rather, they purport to show the diversity of relevant problems that, we believe, chemical engineers would find interesting.

In the rest of the paper, we first provide a basic background on the Lambert function, emphasizing elements that may not be easily available elsewhere in literature. Then, we present a number of case studies in which the Lambert function has something interesting to offer. Finally, we discuss the future potential of this interesting tool.

2. Background

2.1. Definition and basic properties of the Lambert function

The multiple solutions of eqn. (1) are denoted by $W_k(w)$, indexed by the integer k in $(-\infty ,+\infty )$. Real solutions of eqn. (1) for real values of w can be visualized in Figure 1. It can be shown rigorously (Corless et al., 1996) and understood intuitively from Figure 1 that for w ≥ 0 a unique real solution exists for each real w. This solution is typically denoted by $W_0(w)$ or just $W(w)$. The same function $W_0(w)$ also provides a real solution in the range $-\frac{1}{e}\le w<0$, for which a second real solution can be shown to exist, provided by the function $W_{-1}(w)$, as shown in Figure 1 as well. For clarity, the behavior of $W_0(w)$ and $W_{-1}(w)$ for all real w is summarized in Figure 2. Note that $W_{-1}(w)$ is complex-valued for real w outside the interval $\left[-\frac{1}{e},0\right]$, as is $W_0(w)$ for $w<-1/e$.

Figure 1.

Real solutions of eqn. (1) as intersection of the functions $w{e}^{-x}$ and x for several values of w (left) and as function of w (right). Comparison with the natural logarithm (intersections at horizontal line at 1) is also shown on the left.

Figure 2.

Real and imaginary parts of Lambert functions $W_k(w)$. Note that $\mathrm{Re}\left[W_{-k}(w)\right]=\mathrm{Re}\left[W_{k-1}(w)\right],\text{Im}\left[W_{-k}(w)\right]=-\text{Im}\left[W_{k-1}(w)\right]$ for $w<0$ and $\mathrm{Re}\left[W_{-k}(w)\right]=\mathrm{Re}\left[W_k(w)\right],\text{Im}\left[W_{-k}(w)\right]=-\text{Im}\left[W_k(w)\right]$ for $w>0$.

In all cases other than $W_0$ and $W_{-1}$, all $W_k(w)$ are complex-valued and come in conjugate pairs, as can be immediately inferred from eqn. (1). The indexing of these pairs follows the convention

$$W_{-k}(w)=\left\{\begin{array}{cc}\overline{W_{k-1}(w)},& w<0\\ \overline{W_k(w)},& w>0\end{array}\right\}$$ (2)

Following this convention, Figure 3 shows the real and imaginary parts for a collection of $W_k(w)$ for $k\ge 0$. Corresponding values of $W_k(w)$ for $k\le -2$ need not be shown because of eqn. (2).

Figure 3.

Real and imaginary parts of Lambert functions $W_k(w)$. Note that $\mathrm{Re}\left[W_{-k}(w)\right]=\mathrm{Re}\left[W_{k-1}(w)\right]$ and $\text{Im}\left[W_{-k}(w)\right]=-\text{Im}\left[W_{k-1}(w)\right]$ for $w<0;\mathrm{Re}\left[W_{-k}(w)\right]=\mathrm{Re}\left[W_k(w)\right]$ and $\text{Im}\left[W_{-k}(w)\right]=-\text{Im}\left[W_k(w)\right]$ for $w>0$. The real part of $W_{-1}$ (shown in Figure 2) is not shown here, to avoid cluttering.

Calculus with the Lambert function is relatively simple, with interesting patterns. For example,

$$\frac{dW}{dw}=\frac{W(w)}{w(1+W(w))}=\frac{1}{w+e^{W(w)}},w\ne -\frac{1}{e}$$ (3)

$$\int W(w)dw=wW(w)-w+e^{W(w)}+c=w\left(W(w)-1+\frac{1}{w(w)}\right)+c$$ (4)

A collection of numerous additional properties of the Lambert function can be found in Corless et al. (1996) and references therein.

2.2. Software for lambert

Routines for evaluation of the Lambert function are readily included in standard software packages such as Maple (LambertW), Mathematica (ProductLog), Matlab (lambertw), or Python (lambertw). Mathematica and Maple can also generate symbolic solutions of corresponding problems solvable by the Lambert function, and can symbolically detect many instances (although certainly not all, as shown in section 3.7) for which elementary manipulations can convert equations such as $ax+\ln x=b,x=w{e}^{-ax}$, or $x=w{e}^{-a{x}^b}$ to eqn. (1). In fact, a theme of the cases that follow is how to quickly spot such manipulations.

3. The Lambert function in action

3.1. Continuous-flow tank dynamics

Given the continuous-flow liquid storage tank shown in Figure 4, it is of interest to determine the height, h of the liquid over time, t.

Figure 4.

Continuous-flow liquid storage tank.

Mass balance around the tank immediately yields

$$\frac{dh}{dt}=\frac{1}{A}\left(F_i-K\sqrt{h(t)}\right)$$ (5)

where A is the cross-sectional area; F_i is the inlet volumetric flow rate, held constant; and F is the outlet volumetric flow rate, assumed to be equal to $K\sqrt{h(t)}$. Solution of eqn. (5) through integration by separation of variables (APPENDIX A) yields

$$\sqrt{h(t)}-\sqrt{h(0)}+\frac{F_i}{K}\ln |\frac{\sqrt{h(t)}-\frac{F_i}{K}}{\sqrt{h(0)}-\frac{F_i}{K}}=-\frac{K}{2A}t$$ (6)

The algebraic equation shown above is implicit in h(t) Yet, use of the Lambert function easily produces the explicit analytical solution

$$h(t)=h_s{\left(1+W\left[\left(\sqrt{\frac{h(0)}{h_s}}-1\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}-1-\frac{t}{\tau }\right)\right]\right)}^2$$ (7)

where $h_S\stackrel{\text{def}}{=}{\left(\frac{F_i}{K}\right)}^2$ is the steady-state value of the liquid height and $\tau \stackrel{\text{def}}{=}\frac{2A{F}_i}{K^2}$ is the time constant of the system. Figure 5 shows $\frac{h(t)}{h_s}$ for a number of initial conditions, following eqn. (7). Corresponding solutions for the approximate solution

$$\frac{h_L(t)}{h_s}=1+\left(\frac{h(0)}{h_s}-1\right)\exp \left(-\frac{t}{\tau }\right)$$ (8)

for the counterpart of eqn. (5) obtained by linearization around h_s using Taylor series, is also shown in Figure 5, for comparison.

Figure 5.

Dimensionless height, $\frac{h}{h_s}$ as a function of dimensionless time, $\frac{t}{{\tau }^{\prime }}$ for the explicit analytical solution of eqn. (5) in terms of the Lambert function, eqn. (7). The analytical solution of the linear approximation of eqn. (5) is also shown.

Note that the existence and uniqueness of the solution of eqn. (5) can be easily examined via eqn. (7): Based on section 2.1, a real solution exists for eqn. (7) if and only if (iff)

$$-\frac{1}{e}\le \left(\sqrt{\frac{h(0)}{h_s}}-1\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}-1-\frac{t}{\tau }\right)$$ (9)

which is easily established (APPENDIX B). Further, it is also easily established (APPENDIX B) that eqn. (7) uniquely involves the Lambert function W₀ alone.

3.2. The Colebrook equation for friction factor

The Colebrook equation (Colebrook, 1939)

$$\frac{1}{\sqrt{f}}=-2\log \left(\frac{2.51}{\mathrm{Re}}\frac{1}{\sqrt{f}}+\frac{ϵ}{3.71D}\right)$$ (10)

has long been a standard model for estimating the friction factor, f in turbulent flows in a pipe, given values for the Reynolds number (Re) in the range $4000<\mathrm{Re}<{10}^8$ and for the relative roughness of the inner surface of the pipe $\left({\epsilon }^{\ast }=\frac{ϵ}{D}\right)$ in the range $0<{\epsilon }^{\ast }<0.05$ (Baehr & Stephan, 2006; Bergman et al., 2011; Bird et al., 2002; Cengel, 1997; Cussler, 2009; Economides et al., 2013; Kessler & Greenkorn, 2019; Kreith et al., 2010; Moss, 2007; Welty et al., 2007). Eqn. (10) is implicit in f, a fact that has prompted explicit approximations of various kinds (Zigrang & Sylvester, 1982) by a number of investigators (Brkić, 2010, and references therein). Recently, sporadic results have appeared (Brkić & Praks, 2018; More, 2006; Rollmann & Spindler, 2015) which, rather independently of one another, recognize that the Colebrook equation accepts the simple explicit analytical solution

$$f=\frac{1}{{\left(-0.1074{\epsilon }^{\ast }\text{Re}+0.8686W\left[0.4587\mathrm{Re}\exp \left(0.1236{\epsilon }^{\ast }\text{Re}\right)\right]\right)}^2}$$ (11)

in terms of the Lambert function W Eqn. (11) is shown in Figure 6, a 3D counterpart of the classic 2D diagram for the Moody/Fanning friction factor (Moody, 1944) included in standard textbooks on transport phenomena.

Figure 6.

Friction factor f from analytical solution of the Colebrook equation.

3.3. The continuous-cycling Ziegler-Nichols method for controller tuning

The continuous-cycling method, developed in the influential paper by Ziegler and Nichols (1942), has long been a standard technique for tuning industrial proportional-integral-derivative (PID) controllers, using experimental data collected for the purpose (Seborg et al., 2017). The method relies on first performing an experiment that brings to continuous cycling a stable process controlled by a P-controller whose gain, K_c (Figure 7) is gradually increased to a critical value, K_cu known as the ultimate gain. For a typical process modeled by the first-order-plus-time-delay transfer function $G(s)=\frac{K{e}^{-\theta s}}{\tau s+1}$ (Figure 7) analysis of how the closed-loop process is brought to continuous cycling makes use of the closed-loop poles, namely the roots of the characteristic equation $1+G(s)C(s)=0$. For the preceding transfer function G(S) the closed-loop poles are solutions of the algebraic equation (Bellman & Cooke, 1963)

$$\tau s+1+K{K}_c{e}^{-\theta s}=0.$$ (12)

Figure 7.

Block diagram of closed-loop system in continuous-cycling method

At this point a standard approach to assessing the solutions of eqn. (12) is by replacing $e^{-\theta s}$ by a first-order Padé approximation (Seborg et al., 2017)

$$e^{-\theta s}\approx \frac{1-\frac{\theta }{2}s}{1+\frac{\theta }{2}s}$$ (13)

to get the second-order polynomial equation

$$\frac{\theta }{2}\tau s^2+\left(\frac{\theta }{2}+\tau -K_{c}K\frac{\theta }{2}\right)s+1+K_{c}K\approx 0$$ (14)

which can be easily analyzed. Thus, the closed-loop behavior will switch from overdamped (non-oscillatory) to underdamped (oscillatory) when the discriminant of eqn. (14) crosses the 0 value, namely

$${\left(\tau +\frac{\theta }{2}-\frac{\theta K{K}_C}{2}\right)}^2-4\frac{\theta }{2}\tau \left(1+K_{c}K\right)=0$$ (15)

resulting in the controller gain

$$K_{c,o}\approx \frac{1}{K}\left(1+\frac{6-4\sqrt{\frac{\theta }{\tau }+2}}{\frac{\theta }{\tau }}\right)$$ (16)

where the appropriate one of the two roots of eqn. (15) was selected. Further, the closed loop will be brought to continuous oscillations when the controller gain is such that both roots of eqn. (14) are purely imaginary, namely

$$\frac{\theta }{2}+\tau -K_{c}K\frac{\theta }{2}\approx 0$$ (17)

resulting in the ultimate gain

$$K_{c,u}\approx \frac{1}{K}\left(1+\frac{2}{\theta /\tau }\right)$$ (18)

and corresponding ultimate period of oscillations

$$P_u\approx \pi \theta \sqrt{\frac{1}{\left(1+\frac{\theta }{\tau }\right)}}$$ (19)

Closer inspection of eqn. (12) reveals that it accepts the following solution in terms of the Lambert function, as pointed out by Corless et al. (1996) concerning analysis of the dynamics of delay-differential equations via Laplace transforms (Hwang & Cheng, 2005; Ivanoviene & Rimas, 2013):

$$p_k=\frac{1}{\tau }\left(-1+\frac{W_k\left(-K{K}_c\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)\right)}{\frac{\theta }{\tau }}\right),k=-1,0,1,2,\dots$$ (20)

The above eqn. (20) can be used with properties of Lambert functions discussed in section 2.1 to derive the counterparts of eqns. (16) and (18), as follows:

To start with, the closed loop is stable iff the real parts of all poles p_k in eqn. (20) are negative. To that end, inspection of the relative placement of $\mathrm{Re}\left[W_k(w)\right]$ with respect to each other in Figure 3 immediately implies that if $\mathrm{Re}\left[p_{-1}\right]<0$ then $\mathrm{Re}\left[p_k\right]<\mathrm{Re}\left[p_{-1}\right]<0$ for all $k\ge 1$, because $p_k=a+b{W}_k$ with $b>0$, according to eqn. (20). Therefore, if $\mathrm{Re}\left[p_{-1}\right]<0$, all modes corresponding to $p_k,k\ge 1$, will decay exponentially faster than the mode corresponding to $p_{-1}$. Furthermore, Figure 3 also indicates that $\mathrm{Re}\left[p_{-1}\right]<p_0$ for $w>0$ and $p_{-1}<p_0$ for $-\frac{1}{e}\le w<0$.

Now, the closed loop exhibits non-oscillatory behavior if it has at least a dominant real pole. As explained in the previous paragraph, such a pole can only arise from W₀ and/or W₋₁ According to the discussion in section 2.1, a real pole arises from W₋₁ iff

$$-\frac{1}{e}\le -K{K}_c\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)<0$$ (21)

and from W₀ iff

$$-\frac{1}{e}\le -K{K}_c\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)$$ (22)

respectively. Combination of eqns. (21) and (22) implies that a real pole arises from W₀ and W₋₁ if $K{K}_c\le \frac{\tau }{\theta }\exp \left(-1-\frac{\theta }{\tau }\right)$, which yields

$$K_{c,o}=\frac{\tau }{K\theta }\exp \left(-1-\frac{\theta }{\tau }\right)$$ (23)

The above eqn. (23) is the exact counterpart of the approximate eqn. (16), as illustrated in Figure 8.

Figure 8.

P-controller gain, K_c, for transition from stable non-oscillatory to stable oscillatory to unstable oscillatory behavior (at $K_{c,o}$ and $K_{c,u}$, respectively) for the feedback system of Figure 7. The exact and approximate formulas refer to eqns. (23) and (16) for $K_{c,o}$, and to eqns. (24) and (18) for K_c,u respectively.

Continuing the analysis, eqns. (21) and (22) imply that the closed loop exhibits oscillatory behavior when $-\frac{1}{e}>-K{K}_c\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)$, in which case complex poles will correspond to $W_{-k},k=-1,1,2,\dots$. The closed loop will exhibit sustained oscillations when $\mathrm{Re}\left[p_{-1}\right]=0$, with $\mathrm{Re}\left[p_k\right]<0$ for all $k\ge 1$ as already discussed. Therefore, the ultimate gain, $K_{c,u}$, satisfies the equation

$$\mathrm{Re}\left[W_{-1}\left(-K{K}_{c,u}\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)\right)\right]-\frac{\theta }{\tau }=0$$ (24)

with corresponding ultimate period of oscillations

$$P_u=2\pi \frac{1}{\left|\mathrm{Im}\left[p_{-1}\right]\right|}=2\pi \frac{\theta }{\left|\text{Im}\left[w_{-1}\left(-K{K}_{c,u}\frac{\theta }{\tau }\exp \left(\frac{\theta }{\tau }\right)\right)\right]\right|}$$ (25)

The above eqns. (24) and (25) are the exact counterparts of the approximate eqns. (18) and (19). While eqn. (24) is implicit in K_c,u it can be easily used to show $K{K}_{c,u}$ as a function of $\theta /\tau$ (Figure 8). Similarly, eqn. (25) can be presented as shown in Figure 9.

Figure 9.

Ultimate period, P_u, at transition of the feedback system of Figure 7 from stable oscillatory to unstable oscillatory behavior. The exact and approximate formulas refer to eqns. (25) and (19), respectively.

Finally, for comparison, note that the transition from stable oscillations to unstable oscillations, as K_c increases to K_c,u and beyond, can be analyzed numerically by setting $s=i\omega$ in eqn. (12), equating real and imaginary parts of the resulting equation, and solving the resulting two algebraic equations numerically.

3.4. Dynamics of epidemics

In its simplest form, a basic model that captures the spread of an infectious disease in a fixed-size population first separates the population into three distinct groups: susceptible to the infection (S), infectious (I), and removed from the infectious group by recovery or death (R). Given this model structure, the dynamics of the spread of the infection in a population of fixed size is captured by the celebrated SIR ordinary differential equations (ODEs) (Kermack & McKendrick, 1927)

$$s^{\prime }(t)=-\beta s(t)i(t)$$ (26)

$$i^{\prime }(t)=\beta s(t)i(t)-\gamma i(t)\stackrel{\text{def}}{=}\beta \left(s(t)-\frac{1}{R_0}\right)i(t)$$ (27)

$$r^{\prime }(t)=\gamma i(t)$$ (28)

where s,i,r are the population fractions of the S, I, R groups, respectively, satisfying the constraint $s(t)+r(t)+i(t)=1$; and prime denotes time derivative (Anderson et al., 1992; Brauer, 2017; Keeling & Rohani, 2008; Murray, 2002). Consistent with the importance of the SIR ODEs, the basic reproductive ratio, $R_0\stackrel{\text{def}}{=}\beta /\gamma$, in eqn. (27) is widely considered “one of the most critical epidemiological parameters” (Keeling & Rohani, 2008) and has even become a household name in the recent coronavirus epidemic (Wang et al., 2020). For the epidemic to spread it is necessary that $R_0>1$, whereas countermeasures aim at reducing R₀ to the extent possible.

Lambert functions appear in multiple instances when analyzing the SIR model (Nikolaou, 2020a, 2020b, 2020c). To start with, it can be shown (Hethcote, 2000) that the model is stable for initial conditions in [0,1], with asymptotically converging long-term values $s_{\infty },i_{\infty }=0$, and $r_{\infty }=1-s_{\infty }$. Based on that, one can easily show (Kermack & McKendrick, 1927) that eventually

$$\frac{1}{R_0}\ln s_{\infty }-s_{\infty }+1=0$$ (29)

for a spreading epidemic. The above eqn. (29) is typically followed by a comment such as “This equation is transcendental and hence an exact solution is not possible.” (Keeling & Rohani, 2008, p. 22). However, a solution can be easily obtained in terms of the Lambert function as

$$r_{\infty }=1-s_{\infty }=1+\frac{W\left[-R_0\exp \left(-R_0\right)\right]}{R_0}$$ (30)

as illustrated in Figure 10. The importance of $r_{\infty }$ is that it refers to the total fraction of the population that will have been infected by the end of the epidemic for a given $R_0$ particularly in comparison with the value of $r(0)=1-s(0)$ that confers to the population herd immunity at time 0 by ensuring that $s(0)<\frac{1}{R_0}$ in eqn. (27) (Keeling & Rohani, 2008). Note that the well known Lambert function identity $W\left[x{e}^x\right]=x$ for $x=R_0$ does not apply in eqn. (30), because $x<-1$.

Figure 10.

Total fraction of a population infected by the end of an epidemic, $r_{\infty }$, as a function of the basic reproductive ratio $R_0\stackrel{\text{def}}{=}\frac{\beta }{\gamma }$, according to eqn. (30). The epidemic spreads for $R_0>1$, whereas it is contained for $R_0<1$.

An additional outcome from analysis of eqns. (26)-(28) is the value of $R_0$ that would result in a certain maximum of the fraction of the infectious group, $r^{\ast }$, during a spreading epidemic (Figure 11). Adjustment of $R_0$ by public policy has been the central focus among efforts to “Flatten the Curve”, an aim that became a household name in the latest coronavirus epidemic (Dong et al., 2020).

Figure 11.

Profiles of s, i, r of a population through an epidemic according to the SIR model, eqns. (26)-(28), for R₀ = 2.

It can be easily shown (APPENDIX C) that i* and the corresponding $R_0$ are related as

$$\frac{1}{R_0}=\exp \left(-R_0+R_0{i}^{\ast }+1\right)$$ (31)

and the above eqn. (31) admits a solution in terms of the Lambert function $W_{-1}$ as

$$R_0=\frac{W_{-1}\left(\frac{i^{\ast }-1}{e}\right)}{i^{\ast }-1}$$ (32)

Finally, it can be shown (APPENDIX D) that an explicit dependence of s on i can be established based on eqns. (26) and (27), as

$$s(i)=\left\{\begin{array}{cc}-\frac{W_{-1}\left[-R_0\exp \left(-R_0+i{R}_0\right)\right]}{R_0},& i\text{from}0\text{to}{i}^{\ast }\\ -\frac{W_0\left[-R_0\exp \left(-R_0+i{R}_0\right)\right]}{R_0},& i\text{from}{i}^{\ast }\text{to}0\end{array}\right\}$$ (33)

Note that the above eqn. (33) indicates two branches for s(i) each followed by $W_{-1}$ and $W_0$ respectively, as $\{s,i\}$ changes from $\{1-ϵ,ϵ\}\approx \{1,0\}$ through $\left\{s^{\ast },i^{\ast }\right\}$ to $\left\{s_{\infty },0\right\}$. Also note that eqn. (33) implies that real solutions exist iff

$$i<1-\frac{1+\ln R_0}{R_0}<1$$ (-34)

which is compatible with eqn. (31) solved for r* in terms of R₀

3.5. Logarithmic mean expressions in heat and mass transfer

The logarithmic mean expression $\left(x_1-x_2\right)/\ln \left(x_1/x_2\right)$ is well established in various heat or mass transfer calculations. Classic examples include the log-mean temperature difference in heat exchangers or the log-mean radius difference in pipe insulation (Kern, 1950; McCabe et al., 1967) and the log-mean concentration difference in absorption columns (Bird et al., 2002). When a value, a can be determined for $\left(x_1-x_2\right)/\ln \left(x_1/x_2\right)$, the Lambert function can be immediately used to solve for x₁ in terms of x₂ (or vice versa, due to symmetry) as

$$x_1=-aW\left(-\frac{x_2}{a}\exp \left[-\frac{x_2}{a}\right]\right)$$ (35)

Note that for x₁ in eqn. (35) to be real, the inequality

$$-\frac{1}{e}\le -\frac{x_2}{a}\exp \left[-\frac{x_2}{a}\right]$$ (36)

must be satisfied (section 2.1), which is always true, because ${\min }_{x_2}\left(-\frac{x_2}{a}\exp \left[-\frac{x_2}{a}\right]\right)=-\frac{1}{e}$.

Also note that if $x_2\le 0$, a unique solution exists, while for $x_2>0$ a choice has to be made whether w in eqn. (35) refers to $W_0$ or $W_{-1}$ based on comparison of x₁to a.

A discussion of applying the Lambert function to heat exchanger calculations and comparison with approximations of the log-mean temperature difference in presented in (Chen, 2019).

3.6. High circulation rate implies a well mixed system

Differences between animals and humans in the pharmacokinetics of drug elimination from the body (Shargel & Yu, 2016) make in vitro testing both necessary and valuable (Doern, 2014; Gloede, 2010; Tamma, 2012). Drugs injected to humans typically follow a concentration profile approximated by exponential decline after each injection. For drug tests such as assessment of the effect of an antibiotic on bacteria, a system involving a hollow-fiber cartridge (Figure 14) is widely used to simulate the desired pharmacokinetic behavior in vitro (Bulitta et al., 2019; Yoon, 2016). Bacteria are restrained in the hollow fibers while exposed to nutrients and antibiotics flowing in solution through the cartridge. To ensure that all bacteria are exposed to the same concentration of an antibiotic at each time, a high enough internal circulation flow rate F_HF is used. This essentially creates a well mixed system comprising the central vessel and hollow-fiber cartridge. While it is intuitively easy to understand the need for high F_HF, a quantitative assessment of appropriate values of F_HF that ensure a well mixed system requires further analysis. We present such analysis for a single drug A next. Extension to two or more drugs is discussed in Kesisoglou et al. (2020).

Figure 14.

A hollow-fiber system for in vitro simulation of clinically relevant pharmacokinetics of drugs. The volumetric flow rate F_A is pure broth. The mass flow rate f_A0 refers to impulse bolus injection as $f_{A0}(t)=M_A\delta (t)$. With appropriate selection of $\left\{V,F_A,M_A\right\}$ the system simulates exponential-decline drug elimination pharmacokinetics corresponding to the half-life of the tested drug.

Mass balance for drug A around the central vessel, which maintains enough stirring for good mixing, immediately yields

$$V\frac{d{C}_A}{dt}=f_{A0}(t)-F_{\text{HF}}{C}_A(t)+F_{\text{HF}}{C}_A(t-\theta )-F_A{C}_A(t)$$ (37)

where $\theta \stackrel{\text{def}}{=}\frac{V_{\text{HF}}}{F_{\text{HF}}}$ is the time needed for liquid drawn from the central vessel to circulate in plug flow at rate F_HF through the hollow-fiber cartridge volume V_HF Taking Laplace transforms of the above eqn. (37) and solving for ${\tilde{C}}_A(s)$ yields

$${\tilde{C}}_A(s)=\frac{\frac{{\tilde{f}}_{A0}(s)}{F_{\text{HF}}}+\frac{V}{F_{\text{HF}}}{C}_A(0)}{-\exp (-\theta s)+\frac{V}{F_{\text{HF}}}s+1+\frac{F_A}{F_{\text{HF}}}}$$ (38)

where tilde denotes the Laplace transform of a corresponding function of time. The above eqn. (38) implies that $C_A(t)$ will include a weighted sum of exponentials of the form $e^{p_{n}t}$ where p_n are the poles of ${\tilde{C}}_A(s)$, namely in the set of the roots of the transcendental algebraic equation of s

$$-\exp (-\theta s)+\frac{V}{F_{\text{HF}}}s+1+\frac{F_A}{F_{\text{HF}}}=0$$ (39)

It can be easily shown that the solution of the above eqn. (39) can be expressed in terms of the Lambert function as

$$p_n=-\frac{F_A}{V}\left(1+r-r\frac{V}{V_{\text{HF}}}{W}_n\left(\frac{V_{\text{HF}}}{V}\exp \left[\left(\frac{1}{r}+1\right)\frac{V_{\text{HF}}}{V}\right]\right)\right)\stackrel{\text{def}}{=}-\frac{F_A}{V}{q}_n$$ (40)

where

$$r\stackrel{\text{def}}{=}\frac{F_{\text{HF}}}{F_A}$$ (41)

and W_n is the Lambert function of order $n=-1,0,1,2,\dots .$

Based on the properties of Lambert functions (section 2.1) it is straightforward to show that $\mathrm{Re}\left[p_n\right]<0$ for $n=-1,0,1,2,\dots$, therefore $C_A(t)$ remains bounded at all times.

In addition, all p_n are complex, except p₀

More importantly, Figure 15 shows that for comparable (of similar order of magnitude) values of V_HF and V if the ratio r is high enough, the real parts of all complex-valued p_n are much larger than the magnitude of p₀ suggesting that the corresponding terms $\exp \left(p_{n}t\right),n\ne 0$ will decay much faster than $\exp \left(p_{0}t\right)$, thus quickly becoming negligible. Therefore, for values of r greater than about 30 the combined system comprising the central vessel and the hollow-fiber cartridge will be well mixed, with an effective single-vessel volume of

$$V_A=V+V_{\text{HF}}=\frac{V}{q_0}$$ (42)

as shown in Figure 15.

Figure 15.

Real and imaginary parts (top and bottom, respectively) of modes q_n in eqn. (40) in terms of the circulation ratio $r\stackrel{\text{def}}{=}{F}_{HF}/F_A$. Note that for r greater than about 30, the real part of $q_n,n=-1,0,1,2,\dots$ suggests that the decay of all $\exp \left(-\frac{F_A}{V}{q}_{n}t\right),n\ne 0$ is so much faster than the decay of $\exp \left(-\frac{F_A}{V}{q}_{0}t\right)\approx \exp \left(-\frac{F_A}{V}t\right)$ as to be negligible. Furthermore, for r greater than about 30, the imaginary part of $q_n,n=-1,1,2,\dots$ suggests that no appreciable oscillations are going to appear in $\exp \left(-\frac{F_A}{V}{q}_{n}t\right),n\ne 0$, as all oscillation frequencies $\frac{\left|Im\left[q_n\right]\right|}{2\pi },n=-1,1,2,\mathrm{...}$ are comparable to the exponential decay rates $Re\left[q_n\right]$.

Numerical integration of eqn. (37) in dimensionless form is shown in Figure 16, which confirms the assessment about reasonable values of r made in Figure 15.

Figure 16.

Response of C_A to impulse bolus $f_{A0}(t)=M_A\delta (t)$, adjusted to yield $C_A(0)=1$.

3.7. Calculating the h-index

In his highly cited publication, Hirsch (2005) developed the widely used h-index to quantify the output of science researchers, based on the distribution of citations of their publications. Extensive empirical evidence collected by Hirsch (2005) suggests that the number of publications y with citations $N_c(y)$ is well approximated by the function

$$N_c(y)=N_0\exp \left[-{\left(\frac{y}{y_0}\right)}^{\beta }\right]$$ (43)

where $N_0>0,0<y_0<N_0,0<\beta \le 1$ are parameters of the above stretched exponential (Weibull) distribution (Bertoli-Barsotti & Lando, 2017). Based on the above, the h-index is simply the value h where the lines $N_c(y)$ and y intersect, namely

$$h=N_0\exp \left[-{\left(\frac{h}{y_0}\right)}^{\beta }\right]$$ (44)

as shown in Figure 17.

Figure 17.

Graphical definition of the dimensionless h-factor $\frac{h}{y_0}$ at the intersection of $\frac{\left(y/y_0\right)}{\left(N_0/y_0\right)}$ and exp $\left[-{\left(\frac{y}{y_0}\right)}^{\beta }\right]$ for several values of the parameters $\beta$ and $\frac{N_0}{y_0}$ as suggested by eqn. (44).

It can be shown (APPENDIX E) that the algebraic equation used to define the h-index implicitly in eqn. (44) has the following explicit analytical solution in terms of the Lambert function:

$$h=y_0{\left(\frac{1}{\beta }W\left[\beta {\left(\frac{N_0}{y_0}\right)}^{\beta }\right]\right)}^{\frac{1}{\beta }}$$ (45)

as shown in Figure 18.

Figure 18.

Dimensionless h-factor $\frac{h}{y_0}$ as a function of $\frac{N_0}{y_0}$ for several values of the parameter $\beta$.

It is of interest that Mathematica fails to immediately produce an analytical solution of the above eqn. (44) as a response to the command

$$\text{Solve}\left[\text{y}==\text{N}0\ast \text{Exp}\left[-{(\text{y}/\text{y}0)}^{\wedge }\text{beta}\right],\text{y}\right]$$ (46)

It is also of interest that equation [13] in Hirsch (2005) is used for the implicit introduction of the factor $\alpha \stackrel{\text{def}}{=}\frac{y_0}{h}$, as “…determined by the equation [13]…”

$$“\alpha \exp \left[{\alpha }^{-\beta }\right]=\frac{a}{I(\beta )}”$$ (47)

where, according to Hirsch (2005), $I(\beta )\stackrel{\text{def}}{=}{\int }_0^{\infty }z\exp \left(-z^{\beta }\right)dz$ and $a\ge 2$. The above eqn. (47) can be solved using again the Lambert function in the exact same way as in eqn. (44), to yield

$$\alpha ={\left(-\frac{1}{\beta }W\left[-\beta {\left(\frac{a}{I(\beta )}\right)}^{-\beta }\right]\right)}^{-\frac{1}{\beta }}$$ (48)

Note that for W in the above eqn. (48) to be real-valued it must be

$$-\frac{1}{e}<-\beta {\left(\frac{a}{I(\beta )}\right)}^{-\beta }$$ (49)

as explained in section 2.1 (Figure 1). Given that $I(\beta )=\text{Γ}\left(1+\frac{1}{\beta }\right)$, the range of feasible values for $\beta$ in (0,1) implied by eqn. (49) can be easily determined, as shown in Figure 19. The two boundary lines in that Figure correspond to

$$\beta =1$$ (50)

and

$$a=\beta \exp \left(\frac{1}{\beta }\right)\text{Γ}\left(1+\frac{1}{\beta }\right)$$ (51)

Figure 19.

Feasible region of the parameters (a, β)as dictated by eqn. (49), with 0 < β < 1. Three isolated points identified by Hirsch (2005) are also shown.

Note that Hirsch (2005), after establishing that “$a=2$ can safely be assumed to be a lower bound” for a based on a convexity argument for $N_c(y)$, comments “I find empirically that a ranges between 3 and 5.” Figure 19 immediately indicates a tighter lower bound, $a_{\min }=2.7$ at the point where the inequality in eqn. (49) becomes an equality for $\beta =1$:

$$a_{\min }=\frac{e}{I(1)}=\frac{e}{\text{Γ}(2)}=e$$ (52)

Finally, Hirsch (2005) further comments that (1,3) is a feasible point and that “For β = 0.5 the lowest possible value of a is 3.70; … For β = 2/3, the smallest possible a is a = 3.24.” Figure 19, capturing the constraints on (a,β) according to eqns. (50) and (51), confirms that these three points indeed lie on the boundary of the feasible region for (a,β) derived with the help of the Lambert function.

4. Conclusions

Discovered well over two centuries ago, the Lambert function is emerging at an accelerating pace in a number of applications of interest to chemical engineers. It is a relatively simple elementary function with interesting properties and can facilitate analysis and computation in relevant problems, as demonstrated here through a number of case studies. We conjecture that if this tool becomes more widely known, many more instances of application will appear. Therefore, given its simplicity and usefulness, we would reasonably argue that the Lambert function should be included in the standard mathematical toolbox of chemical engineers, and possibly of others.

Figure 12.

Basic reproductive ratio R₀ corresponding to the maximum infectious fraction i* according to eqn. (32). The second solution of eqn. (31) shown in the plot violates the constraint R₀ > 1 required for spread of an epidemic, therefore it is rejected.

Figure 13.

Dependence of s on i from the beginning $(s\approx 1,i\approx 0)$ to the end $\left(s=s_{\infty },i=0\right)$ of an epidemic, according to eqn. (33). Notice the two branches of each path of $s(i)$, followed by $W_{-1}$ and $W_0$ which are joined at the points $\left\{i^{\ast },s^{\ast }\right\}$. The gray line marks the set of points $\left\{i^{\ast },s^{\ast }\right\}$.

Highlights:

• A number of literature and new cases studies show the merits of the Lambert function for analysis and computation.

• Once widely appreciated, the Lambert function is likely to appear in a number of relevant applications.

• A simple and powerful tool, the Lambert function belongs in the engineering mathematical toolbox

APPENDIX A. Integration of the ordinary differential equation in eqn. (5)

Eqn (5) implies $\int \frac{1}{F_i-K\sqrt{h}}dh=\int \frac{1}{A}dt$ Setting $y\stackrel{\text{def}}{=}\sqrt{h}$ yields

$$\int \frac{dh}{F_i-K\sqrt{h}}=\frac{-2y}{K}+\frac{-2F_i}{K^2}\text{ln}|y-\frac{F_i}{K}|$$ (A-1)

Taking the initial condition into consideration, Eq. (A-1) yields

$$\frac{-2\left(y\right(t)-y(0\left)\right)}{K}+\frac{-2F_i}{K^2}\text{ln}\left|\frac{y\left(t\right)-F_i/K}{y\left(0\right)-F_i/K}\right|=\frac{1}{A}t$$ (A-2)

which is eqn. (6).

APPENDIX B. Uniqueness of the solution of eqn. (5)

For $\frac{h(0)}{h_s}\ge 1$, eqn. (9) is trivially satisfied, because the right side is positive. Therefore, eqn. (6) has one real solution involving $W_0$ for this case.

For $\frac{h(0)}{h_s}<1$, eqn. (9) is equivalent to

$$1\ge \left(1-\sqrt{\frac{h(0)}{h_s}}\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}\right)\exp \left(-\frac{t}{\tau }\right)$$ (B-1)

which is also trivially satisfied, because $\left(1-\sqrt{\frac{h(0)}{h_s}}\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}\right)\le 1$ and $\exp \left(-\frac{t}{\tau }\right)\le 1$ for all $t\ge 0$.

For this case we have

$$-\frac{1}{e}\le \left(\sqrt{\frac{h(0)}{h_s}}-1\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}-1-\frac{t}{\tau }\right)<0$$ (B-2)

which implies that eqn. (6) has two solutions involving $W_0$ and $W_{-1}$. However, W₋₁ is rejected, because

$$1+W_{-1}\left[\left(\sqrt{\frac{h(0)}{h_s}}-1\right)\exp \left(\sqrt{\frac{h(0)}{h_s}}-1-\frac{t}{\tau }\right)\right]<0$$ (B-3)

(Figure 1) implying $h(t)<0$ in eqn. (7), which is not acceptable.

APPENDIX C. Derivation of eqns. (31) and (32)

Eqns. (26) and (28) yield

$$\frac{ds}{dr}=-\frac{\beta }{\gamma }s\Rightarrow s=s(0)\exp \left(-r{R}_0\right)$$ (C-1)

Now, according to eqn. (27), the maximum of i (t) occurs at $i^{\prime }\left(t^{\ast }\right)=0$ which implies $s^{\ast }=\frac{1}{R_0}$. Substituting into eqn. (C-1) for $s(0)\approx 1$ yields

$$\frac{1}{R_0}=\exp \left(-r^{\ast }{R}_0\right)=\exp \left(-\left(1-i^{\ast }-\frac{1}{R_0}\right)R_0\right)=\exp \left(-R_0+R_0{i}^{\ast }+1\right)\Rightarrow$$ (C-2)

$$R_0\left(i^{\ast }-1\right)\exp \left[R_0\left(i^{\ast }-1\right)\right]=\frac{i^{\ast }-1}{e}$$ (C-3)

which is of the form of eqn. (1) with $\chi \stackrel{\text{def}}{=}{R}_0\left(i^{\ast }-1\right)$ and $W\stackrel{\text{def}}{=}\frac{i^{\ast }-1}{e}$.

Because

$$-1<\frac{i^{\ast }-1}{e}<0$$

there are two possible real solutions to eqn. (C-3), as shown in Figure 1. The solution corresponding to W₀ is not acceptable, because it violates the constraint R₀ > 1 Indeed, according to eqn. (C-3), that solution would be

$$R_0=\frac{w_0\left(\frac{i^{\ast }-1}{e}\right)}{i^{\ast }-1}<1$$

which cannot hold for a spreading epidemic (Figure 10).

APPENDIX D. Derivation of eqn. (33)

Combining eqns. (26) and (27) with initial conditions $i(0)=ϵ\approx 0$ and $s(0)=1-ϵ\approx 1$ and solving the resulting ODE by separation of variables yields

$$\begin{gathered} \frac{ds}{di}=-\frac{\beta s}{\beta s-\gamma }\Rightarrow i=1-s+\frac{1}{R_0}\ln s\Rightarrow \\ -R_0\exp \left(-R_0+R_{0}i\right)=-\left(R_{0}s\right)\exp \left(-R_{0}s\right) \end{gathered}$$ (D-1)

which is of the form of eqn. (1) with $x\stackrel{\text{def}}{=}{R}_{0}S$ and $w\stackrel{\text{def}}{=}-R_0\exp \left(-R_0+R_{0}i\right)<0$. With

$$-\frac{1}{e}<-R_0\exp \left(-R_0+R_{0}i\right)$$ (D-2)

the above eqn. (D-1) accepts two possible real solutions in terms of W₀and W₋₁ and no real solution otherwise, as shown in Figure 1. The same Figure indicates that the two solutions are

$$s(i)=\left\{\begin{array}{cc}-\frac{W_{-1}\left[-R_0\exp \left(-R_0+i{R}_0\right)\right]}{R_0},& i\text{from}0\text{to}{i}^{\ast }\\ -\frac{W_0\left[-R_0\exp \left(-R_0+i{R}_0\right)\right]}{R_0},& i\text{from}{i}^{\ast }\text{to}0\end{array}\right\}$$ (D-3)

which is eqn. (33).

APPENDIX E. Derivation of eqn. (45)

Eqn. (44) implies

$$\ln y=\ln N_0-{\left(\frac{y}{y_0}\right)}^{\beta }$$ (E-1)

Setting

$$z\stackrel{\text{def}}{=}{\left(\frac{y}{y_0}\right)}^{\beta }\Rightarrow \ln z=\beta \ln y-\beta \ln y_0$$ (E-2)

and substituting In y from the above eqn. (E-2) into eqn. (E-1) yields

$$\begin{gathered} \frac{1}{\beta }\ln z+\ln y_0=\ln N_0-z\iff \\ (\beta z)\exp (\beta z)=\beta {\left(\frac{N_0}{y_0}\right)}^{\beta } \end{gathered}$$ (E-3)

which is of the form of eqn. (1). Because $\beta {\left(\frac{N_0}{y_0}\right)}^{\beta }>0$, eqn (E-3) implies

$$\beta z=W\left(\beta {\left(\frac{N_0}{y_0}\right)}^{\beta }\right)\Rightarrow h=y_0{\left(\frac{1}{\beta }W\left(\beta {\left(\frac{N_0}{y_0}\right)}^{\beta }\right)\right)}^{\frac{1}{\beta }}$$ (E-4)

which is eqn. (45).