# Effective Condensing Dehumidification in a Rotary-Spray Honey Dehydrator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Heat and Mass Exchange in a Honey Dehydrator

- -
- The heat and mass transfer processes under consideration are transient;
- -
- In the initial phase, the heaters and air transfer heat to the honey changing its temperature—the heating phase;
- -
- Once the temperature difference between the inlet air and the honey has stabilised, the dominant process occurring between the honey and air is the mass exchange process—the drying phase.

- -
- The stabilisation of the mass flux of evaporating water is accompanied by the maximisation of the heat flux transferred by air to honey by convection and maximisation of the values of heat transfer coefficients;
- -
- Low mixer speeds corresponded to the maximum values of heat transfer and mass transfer coefficients;
- -
- Higher air flow velocities and higher heat flux densities are conducive to higher values of heat transfer coefficients;
- -
- In the case of mass transfer coefficients, a clear effect of the actual honey–air contact area (mixer speed) on the intensity of the mass exchange process was observed. The effect of air velocity on the mass transfer rate was significant until a certain limit of air velocity was reached due to the limited drying capability of the external cooling unit.
- -
- With the increase in the driving force of the mass transfer process (ΔX = X″(T
_{H}, A_{w}) − X(T_{a-}_{in}, φ_{a}_{-in})), a slight decrease in mass transfer coefficients was observed.

_{i}and C

_{i}coefficients in Formulas (1) and (2) are summarised in Table 1. The scope of validity of Formulas (1) and (2) included the following ranges of variation of the criterion numbers: 600 ≤ Re

_{p}≤ 2400, 2400 ≤ Re

_{H}≤ 11,000, 0.13 ≤ K ≤ 0.4, 1.8 ≤ Sc ≤ 5.8.

#### 2.2. Mathematical Model of the Heat and Mass Transfer Process for the Dehydrator

- The process is transient;
- The dry air mass flux is constant;
- The ambient temperature is constant;
- At the time τ, the output parameters of the air from the dehydrator tank are the input parameters for the cooling unit;
- At the time τ, the output parameters of the air from the cooling device are the input parameters for the dehydrator tank at the time τ + δτ;
- The effect of honey pump operation on the heat balance of the dehydrator tank is ignored;
- The heat exchange process of the heat pump with the environment is omitted.

_{H}whose temperature changes from the initial value T

_{τ}to the value of T

_{τ}

_{+}

_{δτ}, and whose specific enthalpy corresponds to the enthalpy of water vapour ${h}^{\u2033}=r+c{p}^{\u2033}\xb7{T}_{\tau +\delta \tau}$. The change in honey weight and the change in temperature are related to the change in the heat accumulated in honey δQ

_{H}. The heat sources in the dehydrator tank are air, which exchanges heat with honey by convection (${\dot{Q}}_{con}\xb7\delta \tau $), and electric heaters (${\dot{Q}}_{heat}\xb7\delta \tau $). Some of the heat supplied by the heaters and air is transferred, due to heat loss, to the environment (${\dot{Q}}_{L}^{\ast}\xb7\delta \tau =U\xb7{A}_{T}^{\ast}\xb7\left({T}_{\tau}-{T}_{amb}\right)\xb7\delta \tau $). In this case, the value of heat loss to the environment is determined by the Peclet relationship in which the surface area of the heat transfer (${A}_{T}^{\ast}$) corresponds to the surface area of the part of the tank filled with honey.

_{w}for multifloral honey can be calculated from relationship (7) [33]:

_{a}is determined from formula (8) [34]:

_{w}for drying air (X

_{a}) is calculated as $\left(\phi /100\right)\xb7{p}_{w}^{\u2033}$, and for saturated air above the surface of the honey (${X}_{a}^{\u2033}\left({T}_{\tau},{A}_{w}\right)$) as ${A}_{w}\xb7{p}_{w}^{\u2033}$.

_{τ}

_{+}

_{δτ}at the next time step can be determined from Equation (3). The input air parameters at the beginning of each time period can be determined from the heat balance (9) and the balance (10) of the air moisture content for the cooling unit (Figure 1):

_{w}follows the amount of latent heat ${Q}_{r}$ exchanged in the evaporator of the cooling unit ($\delta {m}_{w}\xb7r={Q}_{r}$).

_{H}≈ T

_{tank}), and as heat losses to the environment $\left({\dot{Q}}_{L}\right)$. The heat loss to the environment ${\dot{Q}}_{L}$ in Equation (13) refers to the total surface area of the tank.

- Initial mass of honey in the tank, m
_{H}; - Initial temperature of honey and its moisture content, T, X;
- Initial temperature and relative humidity (of air), T
_{a}, φ_{a}, (X_{a}); - Ambient temperature, T
_{amb}.

_{k}= 54 °C), for which in the evaporation temperature range (−5; 15) °C, the operating characteristics were approximated by Equations (14) and (15).

_{o}) and temperature of air at the evaporator inlet (T

_{a-out,τ}) as well as degrees of humidity of air at the evaporator inlet (X

_{a-out,t}) and saturated air at the temperature of the external surface of the evaporator (T

_{w}

_{-out}≈ T

_{o})

_{H}), the following values were determined:

- Values ${h}_{a-in,\tau}$, A
_{w}, β, δm_{h}, α respectively from Equations (1), (2), (6), (7) and (11); - Honey temperature (Tτ + δτ) from Equations (3)–(5);
- Air enthalpy (H
_{a-out,τ}) at the dehydrator outlet at time τ from Equation (13); - RCJ values, ${\dot{Q}}_{o},{\dot{Q}}_{r}$, from Equations (14), (16) and (17), respectively;
- Actual mass of water δm removed from honey from relationship (12) and relationship $(\delta {m}_{w}\xb7r={Q}_{r}$);
- The degree of humidity of the air (${X}_{a-in,\tau +\delta \tau}$) returning to the dehydrator tank in time τ + δτ from Equation (10);
- Specific air enthalpy ${h}_{a-in,\tau +\delta \tau}$ from Equation (9);
- Temperature of the air (${T}_{a-in,\tau +\delta \tau}$) returning to the dehydrator tank in time τ + δτ from Equation (11).

## 3. Results and Discussion

#### 3.1. Experimental Verification of the Numerical Model of the Honey Dehydrator

- Mixer speeds: n = 120, 165, 270 rpm.
- Average air volume flow, its initial moisture content and initial temperature: ${\dot{V}}_{a}$= 0.01; 0.016, 0.024 m
^{3}s^{−1}, $24\%<\phi {h}_{a,\tau =0}<36\%$, $20\xb0\mathrm{C}T{h}_{a,\tau =0}36\xb0\mathrm{C}$. - Mass of honey, its initial moisture content and initial temperature: m
_{H}= 28 kg, X_{τ=0}≈ 0.24, $16.5\xb0\mathrm{C}T{h}_{\tau =0}24\xb0\mathrm{C}$. - Ambient temperature $16.5\xb0\mathrm{C}T{h}_{amb}20\xb0\mathrm{C}$.

^{3}s

^{−1}. Figure 3 shows the validation of the calculation results for a fixed air flow rate ${\dot{V}}_{a}$= 0.024 m

^{3}s

^{−1}and different mixer speeds. Figure 4 shows a comparison of the calculated volume of water removed from honey. Figure 2, Figure 3 and Figure 4 indicate that the proposed model correctly qualitatively and quantitatively describes the dehydration process of honey in the dehydrator.

_{H}) and drying air temperature (T

_{a-in}) for the dehydrator parameters n = 120 rpm, ${\dot{V}}_{a}$ = 0.024 m

^{3}s

^{−1}is presented in Figure 5. This confirms the nature of the temperature distributions obtained in the simulation and in the experiment. However, it should be noted that the calculated temperature distribution indicates faster stabilization of heat and mass transfer conditions. This may be due to the adopted model of a heat pump with constant evaporation temperature and differences in obtained values of heat and mass transfer coefficients. The mean relative difference between calculated and measured temperature was 5% for the honey and 2.2% for the air. The standard deviations of the sample for differences in calculated (T

_{cal}) and measured (T

_{exp}) values for honey and air temperatures are, respectively, ${\sigma}_{{T}_{H}}$= 0.74; ${\sigma}_{{T}_{a}}$= 0.44. The sums of squares of the differences of calculated and measured values for honey and air temperatures are, respectively, ${S}_{{T}_{H}}$= 3.47; ${S}_{{T}_{a}}$= 0.93. When validating the calculation model, the time step of δτ was adopted, corresponding to the time step in which the measurements were made (δτ = 900 s).

^{3}s

^{−1}). Therefore, adopting the time step δτ = 900 s in the calculations for all variants of the calculations allows us to determine the moisture content of the dried honey with satisfactory accuracy.

#### 3.2. Selection of Optimal Operating Parameters for the Cooling Unit

_{k}= 54 °C). The calculations were designed to minimise the drying time of the honey to commercial honey moisture values below 20% (X

_{H}< 20%). Calculations were performed for the measurement cases described in Section 3. The results of the calculations are presented in Figure 6.

_{o}) of the cooling unit.

_{o}= 0 °C and T

_{o}= 12 °C (Figure 6), it can be noted that higher values of evaporation temperatures correspond to a greater reduction in drying time in the case of an increase in air volume flow and a reduction in the rotational speed of the mixers. A more significant effect of evaporation temperature on drying time is observed for low heat and mass transfer coefficients in the dehydrator tank. In the evaporation temperature range 0 ≤ T

_{o}≤ 15 °C, the maximum change in drying time for conditions n = 120 rpm, ${\dot{V}}_{a}$= 0.011 m

^{3}s

^{−1}is 13%, while for conditions n = 120 rpm, ${\dot{V}}_{a}$= 0.024 m

^{3}s

^{−1}− 8%. The enlarged markers in Figure 6 indicate the optimal evaporation temperature values of the cooling unit and the minimum time for the honey to reach a moisture content of X

_{H}< 20%. Analysis of the calculation results indicates that in the case of large values of the air flow rate (higher thermal load of the cooling device), it is advisable to use higher evaporation temperatures, which correspond to higher cooling capacities (case n = 120 rpm, ${\dot{V}}_{a}$ = 0.024 m

^{3}s

^{−1}, T

_{opt}= 10–12 °C). In this case, large air flows receive large amounts of water with less driving force of the mass exchange process (smaller differences ΔX = X”(T

_{H}, A

_{w}) − X(T

_{a-}

_{in}, φ

_{a}

_{-in})). Since air is also the source of heat in the dehydrator tank, it is more important in this case to raise the air temperature in the condenser of the cooling unit. As the sensible and latent heat fluxes given off by the air in the cooling unit decrease, due to the poorer heat and mass transfer conditions in the dehydrator (e.g., the cases of n = 270 rpm, ${\dot{V}}_{a}$= 0.023 m

^{3}s

^{−1}, T

_{opt}= 5 °C and n = 120 rpm, ${\dot{V}}_{a}$= 0.011 m

^{3}s

^{−1}, T

_{opt}= 0–3 °C), the optimal evaporation temperatures due to the drying time shift towards values lower than 5 °C. Less favourable conditions of heat and especially mass exchange in the dehydrator are compensated in this case by the lower humidity of the air at the inlet to the dehydrator tank (outlet from the cooling device) and drying at a higher difference of partial pressures of water vapour in honey and air. Hence, lower evaporation temperatures associated with increased drying of the circulating air are optimal.

## 4. Conclusions

_{o}= 0–3 °C, for ${\dot{V}}_{a}$ = 0.011 m

^{3}s

^{−1}for mixer speed n = 120 rpm. For moderate heat and mass transfer conditions in the dehydrator tank, the optimal evaporation temperature value of the cooling unit was T

_{o}≈ 5 °C for ${\dot{V}}_{a}$ = 0.023 m

^{3}s

^{−1}, n = 165–270 rpm. For large heat fluxes exchanged by air in the dehydrator tank, the optimum evaporation temperature range was T

_{o}≈ 10–12 °C. For fixed mixer speed and air flow rates, optimal values of evaporation temperatures allow for an 8–13% reduction in honey drying time. However, proper selection of the rotational speed of mixers and the air flow rate, as well as the evaporation temperature of the cooling unit, can reduce the honey drying time by up to 64%.

## 5. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | surface area (m^{2}) |

A_{w} | water activity of honey, (-) |

B_{i} | coefficient in Equation (1), (-) |

C_{i} | coefficient in Equation (2), (-) |

cp | specific heat, (Jkg^{−1}K^{−1}) |

d_{H} | hydraulic diameter, d_{h} = 4V/A_{V}, (m), |

d_{m} | diameter of mixer, (m) |

D | water vapor diffusion coefficient in air, (m^{2}s^{−1}) |

h | specific enthalpy, (Jkg^{−1}) |

h″ | specific enthalpy of water vapour at the temperature T, (Jkg^{−1}) |

H | entalpy, (J) |

HBP | hight pressure compressors |

K | $\mathrm{phase}\mathrm{transition}\mathrm{number},K=\frac{\Delta {\mathit{h}}_{\mathit{v}}}{\left({\mathit{T}}_{\mathit{a}-\mathit{i}\mathit{n}}-{\mathit{T}}_{\mathit{H}}\right)\xb7\mathit{c}{\mathit{p}}_{\mathit{a}}}$, (-) |

$\dot{\mathit{L}}$ | compressor power input, (W) |

m | mass, (kg) |

${\dot{\mathit{m}}}_{\mathit{a}}$ | dry air mass flow rate, (kg) |

n | rotational speed of the mixer (rpm) |

Nu | $\mathrm{Nusselt}\mathrm{number},\frac{\mathit{\alpha}\xb7{\mathit{d}}_{\mathit{h}}}{{\mathit{\lambda}}_{\mathit{a}}}$ |

p_{a} | atmospheric pressure, (Pa) |

p_{w} | partial pressure of water vapour, (Pa) |

p″ | water vapour pressure in air at the state of saturation, (Pa) |

Pr | r– heat of water evaporation at 0 °C, (Jkg^{−1}) |

Prandtl number | |

Re_{a} | $\mathrm{air}\mathrm{Reynolds}\mathrm{number},\mathit{R}{\mathit{e}}_{\mathit{a}}=\frac{{\mathit{u}}_{\mathit{a}}\xb7{\mathit{d}}_{\mathit{h}}\xb7{\mathit{\rho}}_{\mathit{a}}}{{\mathit{\mu}}_{\mathit{a}}}$ |

Re_{H} | $\mathrm{honey}\mathrm{Reynolds}\mathrm{number},\mathit{R}{\mathit{e}}_{\mathit{H}}=\frac{\mathit{n}\xb7{\mathit{d}}_{\mathit{m}}\xb7{\mathit{\rho}}_{\mathit{H}}}{4{\mathit{\mu}}_{\mathit{H}}}$ |

RCJ | degree of process openness, (-) |

Q | amount of heat, (J) |

$\dot{\mathit{Q}}$ | heat flow, (W) |

${\dot{\mathit{Q}}}_{\mathit{r}}$ | latent heat flux exchanged in the evaporator of the cooling unit, (W) |

u_{a} | $\mathrm{air}\mathrm{flow}\mathrm{velocity},\left({\mathrm{ms}}^{-1}\right),{\mathit{u}}_{\mathit{a}}=\frac{{\dot{\mathit{V}}}_{\mathit{a}}}{{\mathit{A}}_{\mathit{c}\mathit{s}}}$ |

S | $\mathrm{sum}\mathrm{of}\mathrm{squared}\mathrm{differences}\mathrm{between}\mathrm{calculated}\mathrm{and}\mathrm{measured}\mathrm{values},\mathit{S}=\frac{1}{\mathit{N}}{\sum}_{\mathit{i}=1}^{\mathit{N}}{\left({\mathit{x}}_{\mathit{c}\mathit{a}\mathit{l}}-{\mathit{x}}_{\mathit{e}\mathit{x}\mathit{p}}\right)}^{2}$ |

Sc | $\mathrm{Schmidt}\mathrm{number}\mathit{S}\mathit{c}=\frac{{\mathit{\mu}}_{\mathit{a}}\xb7\Delta \mathit{X}\xb7{10}^{3}}{{\mathit{D}}_{\mathit{a}}\xb7{\mathit{\rho}}_{\mathit{a}}}$ |

Sh | $\mathrm{Sherwood}\mathrm{number},\frac{\mathit{\beta}\xb7{\mathit{d}}_{\mathit{h}}}{{\mathit{D}}_{\mathit{a}}\xb7{\mathit{\rho}}_{\mathit{a}}}$ |

SMER | specific moisture extraction, (kg kWh^{−1}) |

U | overall heat transfer coefficient, (Wm^{−2}K^{−1}) |

V | volume occupied by air in the dehydrator tank, m^{3} |

$\dot{\mathit{V}}$ | flow rate, (m^{3}s^{−1}) |

T | temperature, (°C) |

T_{τ} | temperature of honey in time τ, (°C) |

TXV | type of thermostatic expansion valve |

X | absolute humidity in air, (kgkg^{−1}) |

X″ | absolute humidity in air at the state of saturation, (kgkg^{−1}) |

X_{H} | mass fraction of water in honey*100%, (%) |

X_{τ} | mass fraction of water in honey in time, (kgkg^{−1}) |

ΔX | difference in the absolute humidity, ΔX = X″(T_{τ}, A_{w}) − X(T_{a-in}, φ_{a-in}), (kgkg^{−1}) |

Symbols | |

α | heat transfer coefficient, (Wm^{−2}K^{−1}) |

β | mass transfer coefficient, (kgm^{−2}s^{−1})δ, |

Δ | increase |

δm_{H} | the mass of water downloaded by the air from the honey over time τ, (kg) |

δm_{w} | mass of water condensed from the air in the evaporator of the refrigerating device |

over time τ, (kg) | |

δτ | time interval, (s) |

ε | directional coefficient of conversion, (kJ kg^{−1}) |

φ | relative humidity, (%) |

λ | thermal conductivity, Wm^{−1}K^{−1} |

μ | dynamic viscosity coefficient, Pas |

ρ | density, (kgm^{−3}) |

σ | standard deviation for differences between calculated and measured values |

τ | time, (s) |

Indices | |

* | filled with honey |

″ | state of saturation |

- | mean value |

τ = 0 | for initial conditions |

τ + δτ | in time τ + δτ, (s) |

a | air |

amb | ambient temperature |

cal | calculated value |

con | convection |

CS | horizontal cross-section of the dehydrator tank |

exp | measured value |

H | honey |

h | hydraulic |

heat | heaters |

in | inlet |

k | condensation |

L | heat loss due to penetration and accumulation of heat in the tank |

o | evaporation |

opt | optimal value |

out | outlet |

r | latent heat |

s | system |

tank | tank |

T_{o} | at evaporation temperature T_{o} |

w | water |

w-out | external heat exchange surface of the evaporator |

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**Figure 1.**Schematic diagram of a closed-loop honey dehydrator: 1—honey sprayer; 2—mixer; 3—heat pump; 4—evaporator; 5—condenser; 6—electric heaters; 7—honey pump system; 8—air supply line; 9—air return line.

**Figure 2.**Comparison of the calculated and experimental values of honey moisture as a function of time X

_{H}(τ), for n = 120 rpm.

**Figure 3.**Comparison of the calculated and experimental values of honey moisture as a function of time X

_{H}(τ), for ${\dot{V}}_{a}$ ≈ 0.023–0.024 m

^{3}s

^{−1}.

**Figure 4.**Comparison of the calculated and experimental values of the amount of water removed from honey as a function of time m

_{w}(τ), for ${\dot{V}}_{a}$ ≈ 0.023–0.024 m

^{3}s

^{−1}.

**Figure 5.**Comparison of the calculated and measured values of the temperature of honey and air, n = 120 rpm, ${\dot{V}}_{a}$≈ 0.024 m

^{3}s

^{−1}.

**Figure 6.**Drying time ${\tau}_{{X}_{H}<20\%}\left({T}_{o}\right)$ and optimum evaporation temperature values T

_{o-opt}corresponding to ${\tau}_{{X}_{H}<20\%}={\tau}_{min}$.

**Figure 7.**SMER coefficient dependence of dehydrator operating conditions, $SMER({T}_{o},n,{\dot{V}}_{a})$.

**Table 1.**Values of the Bi and Ci coefficients in Equations (1) and (2) [17].

Nu (1) | B_{1} | B_{2} | B_{3} | B_{4} | ||

transient | steady state | transient | steady state | |||

140.8 | 601.9 | 0.4203 | −0.4718 | −0.6313 | 0 | |

Sh (2) | C_{1} | C_{2} | C_{3} | C_{4} | ||

transient | steady state | transient | steady state | |||

0.8479 | 1.299 | 0.9728 | −0.4072 | 0.3713 | 0 |

Calc. Variant | n = 120 rpm | n = 165 rpm | n = 270 rpm | ||
---|---|---|---|---|---|

${\dot{\mathit{V}}}_{\mathit{a}}=0.011{\mathbf{m}}^{3}{\mathbf{s}}^{-1}$ | ${\dot{\mathit{V}}}_{\mathit{a}}=0.016{\mathbf{m}}^{3}{\mathbf{s}}^{-1}$ | ${\dot{\mathit{V}}}_{\mathit{a}}=0.024{\mathbf{m}}^{3}{\mathbf{s}}^{-1}$ | ${\dot{\mathit{V}}}_{\mathit{a}}=0.023{\mathbf{m}}^{3}{\mathbf{s}}^{-1}$ | ||

max. err (%) | 7.7 | 0.6 | 2.4 | 2.1 | 1.9 |

avg. err (%) | 2 | 0.02 | 0.2 | 0.03 | 1.1 |

${\sigma}_{{X}_{H}}$ | 0.36 | 0.10 | 0.25 | 0.30 | 0.26 |

${S}_{{X}_{H}}$ | 0.46 | 0.013 | 0.062 | 0.084 | 0.24 |

Time Step δτ (s) | 100 | 300 | 900 | 1200 |
---|---|---|---|---|

Calculation Variant | X_{H} (%) | |||

n$=120\mathrm{rpm},{\dot{V}}_{a}$ = 0.024 m^{3}s^{−1} | 17.44 | 17.44 | 17.46 | 17.47 |

n$=270\mathrm{rpm},{\dot{V}}_{a}$ = 0.023 m^{3}s^{−1} | 19.22 | 19.24 | 19.29 | 19.39 |

n$=120\mathrm{rpm},{\dot{V}}_{a}$ = 0.016 m^{3}s^{−1} | 19.46 | 19.42 | 19.17 | 18.86 |

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## Share and Cite

**MDPI and ACS Style**

Morawski, M.; Malec, M.; Niezgoda-Żelasko, B.
Effective Condensing Dehumidification in a Rotary-Spray Honey Dehydrator. *Energies* **2022**, *15*, 100.
https://doi.org/10.3390/en15010100

**AMA Style**

Morawski M, Malec M, Niezgoda-Żelasko B.
Effective Condensing Dehumidification in a Rotary-Spray Honey Dehydrator. *Energies*. 2022; 15(1):100.
https://doi.org/10.3390/en15010100

**Chicago/Turabian Style**

Morawski, Marcin, Marcin Malec, and Beata Niezgoda-Żelasko.
2022. "Effective Condensing Dehumidification in a Rotary-Spray Honey Dehydrator" *Energies* 15, no. 1: 100.
https://doi.org/10.3390/en15010100