# Mathematical Model of Air Dryer Heat Pump Exchangers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Calculation Algorithm

- The exchanger works in a steady state (with constant flow rates and temperatures of air and thermodynamic medium).
- Heat losses to and from the environment are taken into account in the calculations.
- Structural elements do not absorb or generate thermal energy.
- At each point in the exchanger cross-section a uniform temperature distribution is assumed.
- The thermal resistance of each element is constant and uniform throughout its volume.
- The refrigerant in the exchangers only undergoes a phase-to-phase change, and subcooling and superheating are not considered.
- Heat transfer coefficients between fluids are independent of temperature, time and location.
- The mass flow of air and refrigerant at the inlet is the same as at the outlet, and the flow rate is evenly distributed through the exchanger throughout the volume.
- The air flow direction corresponds to the orientation of the fins.
- Thermal radiation is not taken into account in the calculations.
- The thermophysical properties of the fluids and the exchangers are constant.
- The resistance (thermal and hydraulic) of the water vapor condensing from the air on the cold elements of the evaporator is disregarded.

#### 2.1. Heat Transfer Coefficient from Air Side ${\alpha}_{p}$

#### 2.2. Heat Transfer Coefficient from Refrigerant Side ${\alpha}_{c}$ and Cooling Capacity of Exchanger $\dot{Q}$

#### 2.2.1. Condensation

#### 2.2.2. Evaporation

## 3. Validation of the Proposed Algorithm

#### 3.1. Verification Calculations

#### 3.2. Test Results

#### 3.3. Energy Verification of Results

## 4. Optimisation of Fan Operation

- develop a simulation model of the steady state heat exchanger using well-tested relationships;
- simulate for a variable flow velocity of the air;
- the flow velocity of the air is dependent on the drive and speed of the fan. This characteristic is available from the manufacturer or can be easily measured;
- work out the relationship between exchanger performance and power consumption of the fan drive on the basis of the above simulations;
- this makes it possible to optimize the control of the power consumption.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | surface area, ${\mathrm{m}}^{2}$ |

${c}_{p}$ | specific heat capacity at constant pressure, $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

$D=d$ | diameter, $\mathrm{m}$ |

$G$ | exchanger width, $\mathrm{m}$ |

$g$ | gravitational acceleration, $\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ |

$H$ | exchanger height, $\mathrm{m}$ |

${h}_{\dot{\mathrm{z}}}$ | weighted fin height, $\mathrm{m}$ |

$h$ | enthalpy, $\frac{\mathrm{kJ}}{\mathrm{kg}}$ |

${k}_{A}$ | heat transfer coefficient related to surface A, $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

$L$ | length of heat exchanger pipes, length of flow, $\mathrm{m}$ |

$l$ | length, characteristic dimension, $\mathrm{m}$ |

$\dot{M}$ | mass flux density, $\frac{\mathrm{kg}}{{\mathrm{m}}^{2}\xb7\mathrm{s}}$ |

${M}_{M}$ | molar mass, $\frac{\mathrm{kg}}{\mathrm{kmol}}$ |

$\dot{m}$ | mass flow, $\frac{\mathrm{kg}}{\mathrm{s}}$ |

${N}_{RZ}$ | number of rows |

$NTU$ | number of transfer units |

${n}_{r}$ | number of pipes |

${n}_{z}$ | number of injections |

${n}_{\dot{\mathrm{z}}}$ | number of fins |

${p}_{a}$ | atmospheric pressure, $\mathrm{Pa}$ |

${p}_{n}$ | saturation pressure, $\mathrm{Pa}$ |

${p}_{w}$ | partial pressure of water-vapour molecules in the air, $\mathrm{Pa}$ |

$\dot{Q}$ | heat transfer coefficient of the exchanger, $\mathrm{W}$ |

$\dot{q}$ | heat flux, $\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$ |

${R}_{z}$ | thermal resistance of pollutants, $\frac{{\mathrm{m}}^{2}\xb7\mathrm{K}}{\mathrm{W}}$ |

$r$ | heat of vaporisation, $\frac{\mathrm{J}}{\mathrm{kg}}$ |

${S}_{l}$ | spacing longitudinal |

${S}_{q}$ | spacing transversal |

${S}_{z}$ | spacing diagonally |

$t$ | temperature, $\xb0\mathrm{C}$ |

$\dot{W}$ | heat capacity of medium flux, $\frac{\mathrm{W}}{\mathrm{K}}$ |

$w$ | velocity, $\frac{\mathrm{m}}{\mathrm{s}}$ |

${X}_{p}$ | moisture content in the air, $\frac{{\mathrm{kg}}_{{\mathrm{H}}_{2}\mathrm{O}}}{{\mathrm{kg}}_{\mathrm{dry}}}$ |

${x}_{c}$ | steam dryness fraction |

Greek Symbols: | |

$\mathsf{\alpha}$ | heat transfer coefficient, $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

$\mathsf{\delta}$ | thickness, $\mathrm{m}$ |

${\mathsf{\epsilon}}_{\dot{\mathrm{z}}}$ | fins efficiency |

$\mathsf{\lambda}$ | thermal conductivity, $\mathrm{W}/\left(\mathrm{m}\xb7\mathrm{K}\right)$ |

$\mathsf{\mu}$ | dynamic viscosity, $\mathrm{kg}/\left(\mathrm{m}\xb7\mathrm{s}\right)$ |

$\mathsf{\rho}$ | density, $\mathrm{kg}/\left(\mathrm{m}\xb7\mathrm{s}\right)$ |

$\mathsf{\sigma}$ | surface tension, $\mathrm{N}/\mathrm{m}$ |

${\mathsf{\phi}}_{\mathrm{p}}$ | relative air humidity, $\%$ |

$\mathsf{\xi}$ | fin’s pitch, $\mathrm{m}$ |

Subscripts: | |

1 | start/inlet value |

$2$ | end/outlet value |

$c$ | refrigerant |

$cz$ | frontal area of exchanger |

$k$ | condensation |

$lat$ | latent |

$m$ | average value |

$o$ | evaporation |

$p$ | air |

$r$ | pipe |

$s$ | wall |

w | inner surface |

$z$ | outer surface |

$\dot{\mathrm{z}}$ | fin |

$\prime $ | liquid phase in a saturated state |

$\u2033$ | gas phase in a saturated state |

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**Figure 2.**Pipe arrangement in the exchanger: (

**a**) in-line; (

**b**) staggered for $\frac{{S}_{l}}{{d}_{z}}\ge 0.5\ast {\left(2\ast \frac{{S}_{q}}{{d}_{z}}+1\right)}^{0.5}$; (

**c**) staggered for $\frac{{S}_{l}}{{d}_{z}}<0.5\ast {\left(2\ast \frac{{S}_{q}}{{d}_{z}}+1\right)}^{0.5}$ [16].

**Figure 3.**Schematic picture of different elementary fins: (

**a**) rectangular where ${S}_{q}>{S}_{l}$; (

**b**) hexagonal where ${S}_{l}\ge 0.5\ast {S}_{q}$; (

**c**) hexagonal where ${S}_{l}<0.5\ast {S}_{q}$ [16].

**Figure 4.**Location of measuring points (1–6) and components (from the left): evaporator (

**a**), condenser (

**b**) and fan (

**c**).

**Figure 5.**Flow characteristics through (from left) the evaporator and the condenser and their geometry.

**Table 1.**K parameter values from Equation (8) [21].

Type Arrangement and Number of Rows | K Parameter |
---|---|

In-line arrangement, 1–3 rows | 0.20 |

In-line arrangement, above 4 rows | 0.22 |

Staggered arrangement, 2 rows | 0.33 |

Staggered arrangement, 3 rows | 0.36 |

Staggered arrangement, above 4 rows | 0.38 |

**Table 2.**Data of exchangers [21].

Evaporator | |||||||||||

$G$ | $H$ | ${d}_{z}$ | ${d}_{w}$ | ${S}_{q}$ | ${S}_{l}$ | $\xi $ | ${\delta}_{\dot{\mathrm{z}}}$ | ${n}_{r}$ | ${n}_{z}$ | ${N}_{rz}$ | ${\dot{Q}}_{0}$ |

0.25 | 0.2415 | 0.008 | 0.0072 | 0.0125 | 0.0217 | 0.003 | 0.0002 | 9 | 9 | 2 | 1270 |

Condenser | |||||||||||

$G$ | $H$ | ${d}_{z}$ | ${d}_{w}$ | ${S}_{q}$ | ${S}_{l}$ | $\xi $ | ${\delta}_{\dot{\mathrm{z}}}$ | ${n}_{r}$ | ${n}_{z}$ | ${N}_{rz}$ | ${\dot{Q}}_{0}$ |

0.25 | 0.245 | 0.007 | 0.0062 | 0.0125 | 0.0217 | 0.003 | 0.0002 | 9 | 9 | 4 | 3300 |

**Table 3.**Properties of the refrigerant and humid air [38].

Refrigerant | Air | ||||
---|---|---|---|---|---|

Parameter | $\mathrm{For}\text{}{t}_{k}$ | $\mathrm{For}\text{}{t}_{o}$ | Parameter | $\mathrm{For}\text{}25.1,\xb0\mathrm{C}$$;\text{}57.2,\%$ | $\mathrm{For}\text{}19.4,\left[\xb0\mathrm{C}\right]$$,\text{}72.1,\%$ |

${\rho}_{c}^{\prime}$ | $467.00$ | $520.43$ | ${\rho}_{p}$ | $1.18$ | $1.22$ |

${\rho}_{c}^{\u2033}$ | $31.56$ | $11.18$ | ${c}_{pp}$ | $1009.11$ | $1008.86$ |

${\mu}_{c}^{\prime}$ | $9.90\times {10}^{-5}$ | $1.33\times {10}^{-4}$ | ${\lambda}_{p}$ | $0.026$ | $0.025$ |

${\mu}_{c}^{\u2033}$ | $9.29\times {10}^{-6}$ | $7.45\times {10}^{-6}$ | ${\mu}_{p}$ | $18.42\times {10}^{-6}$ | $17.95\times {10}^{-6}$ |

${\lambda}_{c}^{\prime}$ | $8.92\times {10}^{-2}$ | $1.08\times {10}^{-1}$ | $P{r}_{p}$ | $0.72$ | $0.72$ |

${\lambda}_{c}^{\u2033}$ | $2.25\times {10}^{-2}$ | $1.70\times {10}^{-2}$ | |||

${c}_{pc}^{\prime}$ | $2849.85$ | $2466.35$ | |||

${c}_{pc}^{\prime \prime}$ | $2341.91$ | $1813.19$ | |||

$P{r}_{c}^{\prime}$ | $3.16$ | $3.04$ | |||

$P{r}_{c}^{\u2033}$ | $0.97$ | $0.80$ | |||

$r$ | $3.04\times {10}^{5}$ | $3.70\times {10}^{5}$ | |||

${M}_{m}$ | - | $44.10$ | |||

$\sigma $ | - | $7.02\times {10}^{-3}$ | |||

${p}_{n}$ | - | $4.92\times {10}^{5}$ | |||

${p}_{kr}$ | - | $4.25\times {10}^{6}$ |

${w}_{o}$ | $R{e}_{p}$ | $N{u}_{p}$ | $RCJ$ | ${\alpha}_{p\xi}$ | ${\epsilon}_{\dot{\mathrm{z}}}$ | ${M}_{0}$ | $Re{\prime}_{c}$ | ${\alpha}_{c0}$ | ${C}_{wo}$ | ${R}_{MS}$ | $R{{e}^{\prime}}_{o}$ |

9.23 | 4730.10 | 35.48 | 1.75 | 202.13 | 0.86 | 13.2 | 253.45 | 45.05 | 6.629 | 20.70 | 713.96 |

${C}_{p}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ | ${\dot{W}}_{p}$ | $f\left({\dot{q}}_{w}\right)$ | ${\alpha}_{0}$ | ${k}_{Aw}$ | ${\dot{Q}}_{r}$ | ${t}_{p2}$ | ${\phi}_{p2}$ |

0.01 | 0.02 | 0.0008 | 0.02 | 60.58 | 390.72 | 17,151.7 | 2113.63 | 818.2 | 1745.83 | 20.63 | 66.2 |

Iterations | Input Data | Output Data | Criterion | ||||
---|---|---|---|---|---|---|---|

${\dot{\mathit{Q}}}_{o}$ | ${\mathit{t}}_{\mathit{z}}$ | ${\dot{\mathit{Q}}}_{2}$ | ${\mathit{t}}_{\mathit{p}2}$ | ${\mathit{\phi}}_{\mathit{p}2}$ | ${\mathit{t}}_{\mathit{z}}$ | ||

1 | 1270 | 2.5 | 1745 | 20.6 | 66.2 | 12.1 | 27.2 |

2 | 1745 | 12.1 | 1431 | 20.8 | 68.7 | 11.6 | 22.0 |

3 | 1431 | 11.6 | 1532 | 20.6 | 68.9 | 11.6 | 6.6 |

4 | 1532 | 11.6 | 1508 | 20.7 | 68.7 | 11.7 | 1.6 |

5 | 1508 | 11.7 | 1512 | 20.7 | 68.7 | 11.6 | 0.3 |

${w}_{o}$ | $R{e}_{p}$ | $N{u}_{p}$ | ${\alpha}_{p}$ | ${\epsilon}_{\dot{\mathrm{z}}}$ | $C$ | ${C}_{1}$ | ${C}_{2}$ |

7.20 | 3424.79 | 34.56 | 123.42 | 0.88 | 40,492.20 | 0.001 | $4.41\times {10}^{-5}$ |

${C}_{3}$ | ${C}_{4}$ | ${\dot{W}}_{p}$ | $f\left({\dot{Q}}_{r}\right)$ | ${\alpha}_{k}$ | ${k}_{Aw}$ | ${t}_{p2}$ | ${\phi}_{p2}$ |

0.0008 | 4666.03 | 222.86 | 1862.23 | 1841.99 | 647.51 | 29.02 | 41.73 |

Parameter | Symbol | Value | Enthalpy for the Point |
---|---|---|---|

Evaporation pressure | ${p}_{c}$ | 13.2 | - |

Condensation pressure | ${p}_{e}$ | 4.1 | - |

Air flow velocity | ${w}_{p}$ | 3.1 | - |

Air temperature at points 1 and 2 | ${t}_{p1}$ | 25.1 | 55.2 |

Air temperature at points 3 and 4 | ${\phi}_{p1}$ | 59.2 | |

Air temperature at points 5 and 6 | ${t}_{p2}$ | 19.4 | 45.1 |

Air relative humidity at points 1 and 2 | ${\phi}_{p2}$ | 72.1 | |

Air relative humidity at points 3 and 4 | ${t}_{p3}$ | 30.9 | 59.4 |

Air relative humidity at points 5 and 6 | ${\phi}_{p3}$ | 39.8 | |

Heat pump—point A | ${t}_{cA}$ | 3.4 | 480.6 |

Heat pump—point B | ${t}_{cB}$ | 58.3 | 557.4 |

Heat pump—point C | ${t}_{cC}$ | 41.2 | 211.6 |

Heat pump—point D | ${t}_{cD}$ | 1.7 | 211.6 |

Parameter | Theoretical | Empirical | Relative Error, % |
---|---|---|---|

Air temperature at points 3 and 4 | 20.7 | 19.4 | 6.28 |

Air temperature at points 5 and 6 | 68.7 | 72.1 | 4.95 |

Air relative humidity at points 3 and 4 | 29.0 | 30.9 | 6.55 |

Air relative humidity at points 5 and 6 | 41.7 | 39.8 | 4.56 |

Condenser capacity | 1862 | 1798 | 3.44 |

Evaporator capacity | 1512 | 1399 | 7.47 |

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**MDPI and ACS Style**

Mołczan, T.; Cyklis, P.
Mathematical Model of Air Dryer Heat Pump Exchangers. *Energies* **2022**, *15*, 7092.
https://doi.org/10.3390/en15197092

**AMA Style**

Mołczan T, Cyklis P.
Mathematical Model of Air Dryer Heat Pump Exchangers. *Energies*. 2022; 15(19):7092.
https://doi.org/10.3390/en15197092

**Chicago/Turabian Style**

Mołczan, Tomasz, and Piotr Cyklis.
2022. "Mathematical Model of Air Dryer Heat Pump Exchangers" *Energies* 15, no. 19: 7092.
https://doi.org/10.3390/en15197092