Parametric Evaluation of Condensate Water Yield from Plain Finned Tube Heat Exchangers in Atmospheric Water Generation

Abstract

This article presents the moisture harvesting capability of plain finned tube heat exchangers, used in active atmospheric water generation systems. The study reviews the data and correlations of heat transfer for such heat exchangers in open literature and embed suitable correlations for predicting the yield of water when exposed to humid air. Set of results are obtained at the inlet and exit of an experimental setup at varying inlet conditions at constant air flowrate and heat exchanger geometry. The heat exchanger is analyzed using a characteristic unit cell where computational fluid dynamics is used to predict temperature drop across it, using COMSOL Multiphysics, whereas the repetitive incremental routines have been carried out using MATLAB. The flowchart is verified by the experimental data. The water product as a result of cooling of the heat exchanger, often considered parasitic for effectiveness of the heat exchanger in sensible cooling applications, is given prime importance, which increases the heat exchanger’s water harvesting capability. Psychrometric conditions at the inlet under an interval of dry bulb temperature of 20–40 °C and relative humidity of 20–70%, with rows ranging 1 through 4, fin density of 8, 10, 12 and 14 fins per inch and forced convection corresponding to 100 through 2000 cfm, have been analyzed. The results are specified as plots of various variables against the yield of water on a per unit frontal area basis. The increase in water yield is, respectively, highest and lowest for increasing number of rows and increasing fin density.

Appendix

The use of linear regression with weighted least squares and bisquare robust fit has been made. The function was buit-in for MATLAB for the curve fitting tool, however a detailed working is presented here and was chosen based on the result of CFD for unit cell which essentially meant one row. The dcomentation of the tool provided sufficient information for examining the fit.

By Eq. (19), one can devise a vector form for the chosen interval of inlet conditions, having 121 sample points denoted by $n$, replacing ${\dot{m}}_{w}$ by the vector ${{\varvec{M}}}_{{\varvec{w}}}$

$$\begin{aligned}{\left\{{{\varvec{M}}}_{{\varvec{w}}}\right\}}_{i}&= {P}_{1}{\left\{{\varvec{R}}{{\varvec{H}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{i}^{2}+{P}_{2}{\left\{{\varvec{D}}{\varvec{B}}{{\varvec{T}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{{\varvec{i}}}^{2}+{P}_{3}{\left\{{\varvec{R}}{{\varvec{H}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{i}\\& \quad +{P}_{4}{\left\{{\varvec{D}}{\varvec{B}}{{\varvec{T}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{i}+{P}_{5} {\left\{{\varvec{R}}{{\varvec{H}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{i} {\left\{{\varvec{D}}{\varvec{B}}{{\varvec{T}}}_{{\varvec{i}}{\varvec{n}}}\right\}}_{i}+{P}_{6}\end{aligned}$$

(A1)

$${\left\{{{\varvec{M}}}_{{\varvec{w}}}\right\}}_{i}= {P}_{1}{x}_{1i}+{P}_{2}{x}_{2i}+{P}_{3}{x}_{3i}+{P}_{4}{x}_{4i}+{P}_{5} {x}_{5i}+{P}_{6}$$

(A2)

$$\forall 1\le i\le n$$

where Eqs. (A1) and (A 2) are equivalent. This model can be represented by estimated values of coefficients in a vector ${\varvec{\beta}}$ and error vector ${\varvec{\epsilon}}$ in Eq. (A 3).

$${{\varvec{M}}}_{{\varvec{w}}}= {\varvec{X}}{\varvec{\beta}}+{\varvec{\epsilon}}$$

(A3)

where ${\varvec{X}}$ is a 121 by 6 matrix of values of $DB{T}_{in}$ and $R{H}_{in}$ given by Equation

$${\varvec{X}}= \left[\begin{array}{cccc}{x}_{1i}& \cdots & {x}_{5i}& 1\\ \vdots & \ddots & \vdots & \vdots \\ {x}_{1n}& \cdots & {x}_{5n}& 1\end{array}\right]$$

(A4)

$$\forall 1\le i<n$$

For method of least squares in linear regression, the matrix form for estimated parameters can be given by Eq. (A 5) as elements of vector $\widehat{{\varvec{\beta}}}$

$$\widehat{{\varvec{\beta}}}={\left({{\varvec{X}}}^{T}{\varvec{X}}\right)}^{-1}{{\varvec{X}}}^{T}{{\varvec{M}}}_{{\varvec{w}}}$$

(A5)

The estimated value for the variable of interest can then be evaluated from Eq. (A 6) where $H$ denotes the hat matrix.

$${\widehat{{\varvec{M}}}}_{{\varvec{w}}}={\varvec{X}}\widehat{{\varvec{\beta}}}={\varvec{H}}{{\varvec{M}}}_{{\varvec{w}}}$$

(A6)

Matrix of residuals can be found out by the hat matrix directly by Eq. (A 7).

$${\varvec{r}}=\left(1-{\varvec{H}}\right){{\varvec{M}}}_{{\varvec{w}}}$$

(A7)

Since we have used robust regression technique, a weighted estimate is required. The vector of weights ${\varvec{W}}$ is not given to the routine rather loop in bisquare program will calculate and assume suitable values. The system of weigthed least squares can be given by Eq. (A 8) in vector form. The revised form of estimated parameter vector ${\widehat{{\varvec{\beta}}}}_{{\varvec{W}}}$ can be given in Eq. (A 9).

$${\varvec{s}}={\sum }_{i=1}^{n}{\left\{{\varvec{W}}\right\}}_{{\varvec{i}}}{\left({\left\{{{\varvec{M}}}_{{\varvec{w}}}\right\}}_{{\varvec{i}}}-{\left\{{\widehat{{\varvec{M}}}}_{{\varvec{w}}}\right\}}_{{\varvec{i}}}\right)}^{2}$$

(A8)

$${\widehat{{\varvec{\beta}}}}_{{\varvec{W}}}={\left({{\varvec{X}}}^{T}{\varvec{W}}{\varvec{X}}\right)}^{-1}{{\varvec{X}}}^{T}{\varvec{W}}{{\varvec{M}}}_{{\varvec{w}}}$$

(A9)

For bisquare robust fit, adjusted residuals are required after the model has been fitted by weighted least square method. The adjusted residuals are given in Eq. (A 10) in which $h$ is the leverage for the residual points.

$$r{}_{adj}=r/\sqrt{1-h}$$

(A10)

Standardized adjusted residuals attain the form in Eq. (A 11) where the tuning factor $K$ assumes a value of 4.685 and ${V}_{r}$ is the robust variance calculated by median absolute deviation ($MAD)$ as in Eq. (A 12). The values are stored in a vector ${\varvec{U}}$ of length $n$.

$${r}_{std,adj}={r}_{adj}/K {V}_{r}$$

(A11)

$${V}_{r}=MAD/0.6745$$

(A12)

$${\left\{{\varvec{U}}\right\}}_{{\varvec{i}}}={{r}_{std,adj}|}_{i}$$

(A13)

The robust weights are stored in a vector ${{\varvec{W}}}_{{\varvec{r}}}$ which is a piecewise function of standard adjusted residual as in Eq. (A 14).

$${\left\{{{\varvec{W}}}_{{\varvec{r}}}\right\}}_{i}=\left\{\begin{array}{ll}{\left(1-{\left\{{\varvec{U}}\right\}}_{i}^{2}\right)}^{2},&\quad {|\left\{{\varvec{U}}\right\}}_{i}|<1\\ 0,&\quad {|\left\{{\varvec{U}}\right\}}_{i}|\ge 1\end{array}\right.$$

(A14)

If the fit converges, the finalized fit is obtained with the parameters in vector ${\widehat{{\varvec{\beta}}}}_{{\varvec{W}}}$ otherwise, the loop is repeated from the initial fit to the least square method without weights.