# Predictive Modeling of a Paradigm Mechanical Cooling Tower: I. Adjoint Sensitivity Model

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## Abstract

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## 1. Introduction

## 2. Results

_{a,out(Tidbit)}; (ii) outlet air temperature measured with the sensor called “Hobo”, which will be denoted as T

_{a,out(Hobo)}; (iii) outlet water temperature, which will be denoted as ${T}_{w,out}^{meas}$; (iv) outlet air relative humidity, which will be denoted as RH

^{meas}. Histogram plots of these 7668 measurement sets (each set containing measurements of T

_{a,out(Tidbit)}, T

_{a,out(Hobo)}, ${T}_{w,out}^{meas}$ and RH

^{meas}), together with statistical analyses of these measurements, are presented in Appendix A. These measured quantities provide the basis for the choosing the state functions underlying the mathematical modeling of the cooling tower, which is presented in Section 2.1. An accurate and efficient numerical method for solving the equations underlying the counter-flow cooling tower is also presented in this section. Section 2.2 presents the development of the cooling tower adjoint sensitivity model, along with the solution method for computing the adjoint state functions. The numerical accuracy of solving the equations underlying the adjoint sensitivity model will also be verified.

#### 2.1. Mathematical Model of the Counter-Flow Cooling Tower

- the air and/or water temperatures are uniform throughout each stream at any cross section;
- the cooling tower has uniform cross-sectional area;
- the heat and mass transfer occur solely in the direction normal to flows;
- the heat and mass transfer through tower walls to the environment is negligible;
- the heat transfer from the cooling tower fan and motor assembly to the air is negligible;
- the air and water vapor mix as ideal gasses;
- the flow between flat plates is unsaturated through the fill section.

- the water mass flow rates, denoted as ${m}_{w}^{(i)}\text{}(i=2,\dots ,50)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower;
- the water temperatures, denoted as ${T}_{w}^{(i)}\text{}(i=2,\dots ,50)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower;
- the air temperatures, denoted as ${T}_{a}^{(i)}\text{}(i=1,\dots ,49)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower; and
- the humidity ratios, denoted as .${\omega}^{(i)}\text{}(i=1,\dots ,49)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower.

- Liquid continuity equations:
- (i)
- Control Volume i = 1:$${N}_{1}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {m}_{w}^{(2)}-{m}_{w,in}+\frac{M({m}_{a},\mathit{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(2)}({T}_{w}^{(2)},\mathit{\alpha})}{{T}_{w}^{(2)}}-\frac{{\mathsf{\omega}}^{(1)}{P}_{atm}}{{T}_{a}^{(1)}(0.622+{\mathsf{\omega}}^{(1)})}\right]=0;$$
- (ii)
- Control Volumes i = 2,..., I−1:$${N}_{1}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {m}_{w}^{(i+1)}-{m}_{w}^{(i)}+\frac{M({m}_{a},\mathit{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(i+1)}({T}_{w}^{(i+1)},\mathit{\alpha})}{{T}_{w}^{(i+1)}}-\frac{{\mathsf{\omega}}^{(i)}{P}_{atm}}{{T}_{a}^{(i)}(0.622+{\mathsf{\omega}}^{(i)})}\right]=0;$$
- (iii)
- Control Volume i = I:$${N}_{1}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {m}_{w}^{(I+1)}-{m}_{w}^{(I)}+\frac{M({m}_{a},\mathit{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(I+1)}({T}_{w}^{(I+1)},\mathit{\alpha})}{{T}_{w}^{(I+1)}}-\frac{{\mathsf{\omega}}^{(I)}{P}_{atm}}{{T}_{a}^{(I)}(0.622+{\mathsf{\omega}}^{(I)})}\right]=0;$$

- Liquid energy balance equations:
- (i)
- Control Volume i = 1:$$\begin{array}{ll}\hfill {N}_{2}^{(1)}({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}& )\triangleq {m}_{w,in}{h}_{f}({T}_{w,in},\mathit{\alpha})-({T}_{w}^{(2)}-{T}_{a}^{(1)})H({m}_{a},\mathit{\alpha})\hfill \\ & -{m}_{w}^{(2)}{h}_{f}^{(2)}({T}_{w}^{(2)},\mathit{\alpha})-({m}_{w,in}-{m}_{w}^{(2)}){h}_{g,w}^{(2)}({T}_{w}^{(2)},\mathit{\alpha})=0;\hfill \end{array}$$
- (ii)
- Control Volumes i = 2,…, I−1:$$\begin{array}{l}{N}_{2}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {m}_{w}^{(i)}{h}_{f}^{(i)}({T}_{w}^{(i)},\mathit{\alpha})-({T}_{w}^{(i+1)}-{T}_{a}^{(i)})H({m}_{a},\mathit{\alpha})\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}-{m}_{w}^{(i+1)}{h}_{f}^{(i+1)}({T}_{w}^{(i+1)},\mathit{\alpha})-({m}_{w}^{(i)}-{m}_{w}^{(i+1)}){h}_{g,w}^{(i+1)}({T}_{w}^{(i+1)},\mathit{\alpha})=0;\end{array}$$
- (iii)
- Control Volume i = I:$$\begin{array}{l}{N}_{2}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {m}_{w}^{(I)}{h}_{f}^{(I)}({T}_{w}^{(I)},\mathit{\alpha})-({T}_{w}^{(I+1)}-{T}_{a}^{(I)})H({m}_{a},\mathit{\alpha})\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}-{m}_{w}^{(I+1)}{h}_{f}^{(I+1)}({T}_{w}^{(I+1)},\mathit{\alpha})-({m}_{w}^{(I)}-{m}_{w}^{(I+1)}){h}_{g,w}^{(I+1)}({T}_{w}^{(I+1)},\mathit{\alpha})\text{}=0;\end{array}$$

- Water vapor continuity equations:
- (i)
- Control Volume i = 1:$${N}_{3}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {\mathsf{\omega}}^{(2)}-{\mathsf{\omega}}^{(1)}+\frac{{m}_{w.in}-{m}_{w}^{(2)}}{\left|{m}_{a}\right|}=0;$$
- (ii)
- Control Volumes i = 2,..., I−1:$${N}_{3}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {\mathsf{\omega}}^{(i+1)}-{\mathsf{\omega}}^{(i)}+\frac{{m}_{w}^{(i)}-{m}_{w}^{(i+1)}}{\left|{m}_{a}\right|}=0\text{};$$
- (iii)
- Control Volume i = I:$${N}_{3}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq {\mathsf{\omega}}_{in}-{\mathsf{\omega}}^{(I)}+\frac{{m}_{w}^{(I)}-{m}_{w}^{(I+1)}}{\left|{m}_{a}\right|}=0\text{};$$

- The air/water vapor energy balance equations:
- (i)
- Control Volume i = 1:$$\begin{array}{l}{N}_{4}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq ({T}_{a}^{(2)}-{T}_{a}^{(1)}){C}_{p}^{(1)}(\frac{{T}_{a}^{(1)}+273.15}{2},\mathit{\alpha})-{\mathsf{\omega}}^{(1)}{h}_{g,a}^{(1)}({T}_{a}^{(1)},\mathit{\alpha})\\ \text{\hspace{1em}}+\frac{({T}_{w}^{(2)}-{T}_{a}^{(1)})H({m}_{a},\mathit{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w,in}-{m}_{w}^{(2)}){h}_{g,w}^{(2)}({T}_{w}^{(2)},\mathit{\alpha})}{\left|{m}_{a}\right|}+{\mathsf{\omega}}^{(2)}{h}_{g,a}^{(2)}({T}_{a}^{(2)},\mathit{\alpha})=0\text{};\end{array}$$
- (ii)
- Control Volumes i = 2,..., I−1:$$\begin{array}{l}{N}_{4}^{(i)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq ({T}_{a}^{(i+1)}-{T}_{a}^{(i)}){C}_{p}^{(i)}(\frac{{T}_{a}^{(i)}+273.15}{2},\mathit{\alpha})-{\mathsf{\omega}}^{(i)}{h}_{g,a}^{(i)}({T}_{a}^{(i)},\mathit{\alpha})\\ \text{}+\frac{({T}_{w}^{(i+1)}-{T}_{a}^{(i)})H({m}_{a},\mathit{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(i)}-{m}_{w}^{(i+1)}){h}_{g,w}^{(i+1)}({T}_{w}^{(i+1)},\mathit{\alpha})}{\left|{m}_{a}\right|}+{\mathsf{\omega}}^{(i+1)}{h}_{g,a}^{(i+1)}({T}_{a}^{(i+1)},\mathit{\alpha})=0\text{};\end{array}$$
- (iii)
- Control Volume i = I:$$\begin{array}{l}{N}_{4}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega};\mathit{\alpha}\right)\triangleq ({T}_{a,in}-{T}_{a}^{(I)}){C}_{p}{}^{(I)}(\frac{{T}_{a}^{(I)}+273.15}{2},\mathit{\alpha})-{\mathsf{\omega}}^{(I)}{h}_{g,a}^{(I)}({T}_{a}^{(I)},\mathit{\alpha})\\ \text{}+\frac{({T}_{w}^{(I+1)}-{T}_{a}^{(I)})H({m}_{a},\mathit{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(I)}-{m}_{w}^{(I+1)}){h}_{g,w}^{(I+1)}({T}_{w}^{(I+1)},\mathit{\alpha})}{\left|{m}_{a}\right|}+{\mathsf{\omega}}_{in}{h}_{g,a}({T}_{a,in},\mathit{\alpha})=0\text{}.\end{array}$$

- (a)
- Write Equations (1)–(13) in vector form as:$$N\left(u\right)=0,$$$$N\triangleq {\left({N}_{1}^{(1)},..,{N}_{1}^{(I)},\dots ,{N}_{4}^{(1)},\dots ,{N}_{4}^{(I)}\right)}^{\u2020},\text{\hspace{1em}}u\triangleq {\left({m}_{w},\text{}{T}_{w},\text{}{T}_{a},\mathsf{\omega}\right)}^{\u2020}$$
- (b)
- Set the initial guess, ${u}_{0}$, to be the inlet boundary conditions;
- (c)
- Steps d through g, below, constitute the outer iteration loop; for $n=0,1,2,\dots $, iterate over the following steps until convergence:
- (d)
- Inner iteration loop: for $m=1,2,\dots ,$ use the iterative GMRES linear solver with the Modified Incomplete Cholesky (MIC) preconditioner, with restarts, to solve, until convergence, the following system to compute the vector $\delta u$:$$J\left({u}_{n}\right)\delta u=-N\left({u}_{n}\right),$$

^{2}), where m is the iteration number within the GMRES solver. To reduce this computational cost, the GMRES solver is configured to run with the restart feature. The optimized value for the restart frequency is 10 for this specific application. The MIC preconditioner can speed up the convergence of the GMRES solver using the parameters OMEGA and LVFILL [9] in the modified incomplete factorization methods for the MIC preconditioner; for this application the following values were found to be optimal: OMEGA = 0.000000001 and LVFILL = 1. The Jacobian is not updated inside the sparse GMRES solver. The default convergence of GMRES is tested with the following criterion [9]:

^{th}-iteration of the GMRES solver, $\delta {u}^{(m)}$ is the solution of Equation (17) at m

^{th}-iteration, and $\zeta $ denotes the stopping test value for the GMRES solver.

- (e)
- Set$${u}_{n+1}={u}_{n}+\delta u,$$
- (f)
- Test for convergence of the outer loop until the error in the solution is less than a specified maximum value. For solving Equations (2)–(13), the following error criterion has been used:$$error=\mathrm{max}\left(\frac{\left|\delta {m}_{w}^{(i)}\right|}{{m}_{w}^{(i)}},\frac{\left|\delta {T}_{w}^{(i)}\right|}{{T}_{w}^{(i)}},\frac{\left|\delta {T}_{a}^{(i)}\right|}{{T}_{a}^{(i)}},\frac{\left|\delta {\omega}^{(i)}\right|}{{\omega}^{(i)}}\right)<{10}^{-6}$$
- (g)
- Set $n=n+1$ and go to step d.

- (a)
- the vector ${m}_{w}\triangleq {\left[{m}_{w}^{(2)},\dots ,{m}_{w}^{(I+1)}\right]}^{\u2020}$ of water mass flow rates at the exit of each control volume i, $(i=1,\dots ,49)$;
- (b)
- the vector ${T}_{w}\triangleq {\left[{T}_{w}^{(2)},\dots ,{T}_{w}^{(I+1)}\right]}^{\u2020}$ of water temperatures at the exit of each control volume i, $(i=1,\dots ,49)$;
- (c)
- the vector ${T}_{a}\triangleq {\left[{T}_{a}^{(1)},\dots ,{T}_{a}^{(I)}\right]}^{\u2020}$ of air temperatures at the exit of each control volume i, $(i=1,\dots ,49)$;
- (d)
- the vector $RH\triangleq {\left[R{H}^{(1)},\dots ,R{H}^{(I)}\right]}^{\u2020}$, having components of the air relative humidity at the exit of each control volume i, $(i=1,\dots ,49)$.

#### 2.2. Development of the Cooling Tower Adjoint Sensitivity Model, with Solution Verification

- consider a small perturbation $\delta {\alpha}_{j}$ in the model parameter ${\alpha}_{j}$;
- re-compute the perturbed response $R\left({\alpha}_{j}^{0}+\delta {\alpha}_{j}\right)$, where ${\alpha}_{j}^{0}$ denotes the unperturbed parameter value;
- use the finite difference formula:$${S}_{j}^{FD}\cong \frac{R\left({\alpha}_{j}^{0}+\delta {\alpha}_{j}\right)-R\left({\alpha}_{j}^{0}\right)}{\delta {\alpha}_{j}}+O{\left(\delta {\alpha}_{j}\right)}^{2}$$
- use the approximate equality between Equations (41) and (40) to obtain independently the respective values of the adjoint function(s) being verified.