# Entropy Generation of Forced Convection during Melting of Ice Slurry

## Abstract

**:**

## 1. Introduction

^{−4}–0.5 mm. The mass fraction of ice crystals in the ice slurry, which may be transported in practice, do not exceed 30%. As an environmentally neutral and natural heat carrier, ice slurry demonstrates a refrigeration potential commensurate to that of refrigerants. Ice slurry flow may be accompanied by phase segregation, which leads to a change in the mean density values and in the value of the dynamic coefficient of fluid viscosity, consequently leading to higher flow resistance values [1]. The heat transfer process in the ice slurry is accompanied by micro-convection of solid particles, resulting in an increase in thermal conduction coefficients and an enhancement of the heat transfer process. Ice slurries differ significantly from single-phase heat carriers in terms of their rheological properties. Ice slurries are non-Newtonian fluids [2,3,4,5,6,7,8,9,10]. The phenomena associated with flow resistance and heat transfer processes involving ice slurries have been extensively discussed in articles, such as [2,3,4,5,6,7,8,9,11,12]. A significant operational problem of ice slurry-fed systems is the elimination of phase segregation and ensuring a homogenous flow. According to [13,14], regardless of the ice content, homogenous flow of ethanol-based ice slurry is possible at a minimum flow velocity of 0.54 ms

^{−1}[13] and 0.75 ms

^{−1}[14] for 0.02 and 0.024 m diameter pipes, respectively. Flow velocity cut-off values for homogenous and heterogeneous flow in d = 0.02 m pipes for 10.5% ethanol-based ice slurry, calculated in [15] on the basis of a solid particle distribution model according to Kitanowski [4,16,17,18], for a 5% and 25% volume fraction of ice amount to 0.68 ms

^{−1}and 0.15 ms

^{−1}, respectively.

_{s}≥ 10% and the same flow velocity, ice slurry flow resistances may be smaller than the flow resistances of the carrier fluid, provided that the type of flow of the carrier fluid and of the ice slurry is different [3,19,20,21]. In the case of homogeneous flow, these phenomena can be explained by the effect of the absorption of turbulence kinetic energy by ice particles (flow laminarization) and the shifting of the moment of transition to the turbulent flow area by slurries with high mass fractions of ice and higher Reynolds number than in the case of the carrier liquid. For heterogeneous flow, which is favoured by low ice mass fractions and larger dimensions of solid particles, it is difficult to interact with the carrier fluid and solid particles that accumulate in the top part of the tube. This effect favors an increase in turbulence of the carrier liquid and an increase in flow resistance. The character of motion changes at lower speeds and lower Reynolds numbers.

_{ai}= 10.6% at −4.5 °C) [2,12,25]. On the other hand, there are some velocities at which for ice slurries with an ice content of x

_{s}> 10%–20%, the heat transfer coefficients are lower than those of the carrier fluid. In the turbulent flow area, the ice content is not as significant in terms of its impact on the heat transfer coefficient value, although the values of these coefficients are about 20%–30% higher than the heat transfer coefficients of the carrier fluids [2,12,25].

_{i}, w, x

_{s}parameters should be selected so that the ice slurry flow is homogenous and laminar, with heat transfer coefficients exceeding those of the carrier fluid. Laminar flow favors lower flow resistances, which are always higher for ice slurries in the turbulent flow area than the flow resistances of the carrier fluid with the same flow velocity.

_{h}= h/h

_{ref}) and the flow resistance increase ratio (${\epsilon}_{\mathsf{\Delta}p}=\mathsf{\Delta}p/\mathsf{\Delta}{p}_{ref}$). The mutual relationship between ε

_{h}and εΔ

_{p}is expressed as their quotient (ε

_{h−}

_{Δp}= ε

_{h}/εΔ

_{p}) [26], or the so-called overall enhancement efficiency, defined as ε

_{Nu−f}= (Nu/Nu

_{ref})/(f/f

_{ref})

^{1/3}[27,28]. The use of the second principle of thermodynamics in design is based on an analysis of the entropy generation rate, which enables a quantitative determination of the irreversibility of the analyzed thermal and flow process. It is common practice to use the entropy generation rate for the thermo-dynamic assessment of thermal and flow processes and as an optimization criterion, for example, in the design of heat exchangers. Also used are complex criteria for the assessment of efficiency—a combination of the performance evaluation criteria and the entropy generation minimization (EGM) criterion [29]. In [30], the authors propose a method for the analysis of the entropy generation rate in the design of single-phase thermal processes, heat storage systems or various thermal cycles. The method proposed by Bejan [30] has been successfully used by other authors to perform a thermodynamic assessment of thermal processes during single-phase flows [31,32], multi-phase flows [33,34,35,36,37,38], heat exchange involving non-Newtonian fluids [32,39,40] and design of thermal systems and exchangers [40,41,42,43,44,45,46,47,48].

_{Nu−f}> 1. The innovative element presented in the paper was proposing a dimensionless relationship to identify the interdependency between the flow velocity and the mass fraction of ice for which the entropy generation rate was at its minimum level.

## 2. Entropy Generation Rate during Flow in a Straight Pipe

#### 2.1. Entropy Generation during the Melting of Ice Slurry; Flow with Phase Separation

_{IS}corresponded to the mean temperature of the ice slurry, ${T}_{IS}={\overline{T}}_{IS}$.

#### 2.2. Ice Slurry Flow and Heat Transfer in Pipes

^{*}and consistency index K

^{*}. For Bingham fluid and an arbitrary geometry of cross-section, these parameters are found from equations included in Table 1 [19].

_{i}in Equations (11) and (12) are shown in Table 2 [25]. Equations (7)–(12) are the result of own experimental studies carried out for ethanol ice slurry, for which measurement stations, the course of experiments, measurement accuracy and results were discussed in detail in studies [2,12,19,25,55]. The experimental scope of the applicability of Equations (7)–(12) is provided in Table 2.

#### 2.3. Results of Calculations

_{s}< 30%. Calculations were made for ice slurry flow through straight pipes with diameters d

_{i}= 0.01, 0.016, 0.02 m, mass flux of ice slurry $0.04\le {\dot{m}}_{IS}\le 0.62$ kgs

^{−1}and heat flux density $2\le \dot{q}\le 10$ kWm

^{−2}. Figure 2 presents the change in the total entropy generation rate as a function of the flow velocity and the mass share of ice during ice slurry flow through a d

_{i}= 0.016 m pipe with a constant heat flux density $\dot{q}=8$ kWm

^{−2}.

_{s}< 10% and the minimum value of the entropy generation rate is recorded in the turbulent flow area. This is related to the mutual relationship between entropy generation in thermal and flow processes (Figure 3).

^{−1}for an ice content greater than x

_{s}> 21.9%, the flow is laminar in nature, and the ice slurry flow resistance values are moderate, while the presence of solid particles favors the highest values of heat transfer coefficients [2,12]. In this case, entropy generation is determined by the irreversibility of the thermal processes. For low mass fractions of ice, obtaining suitably high values of heat transfer coefficients depends not only on the presence of solid particles, but also on the appropriate value of flow velocity. As a consequence, this leads to a change into the turbulent flow area, where, in turn, flow resistance values have a significant impact on the induction of the total entropy generation. It also needs to be noted that large mass fractions of ice (x

_{s}> 20%) are typically associated with a relatively large range of velocities, for which the total entropy generation rate is at its minimum. This results from the dominant impact of solid particles and phase change, but not flow velocity, on heat transfer coefficients [2]. The calculations made for various heat flux densities and pipe diameters made it possible to determine for specific mass fraction of ice, the flow velocities for which the total entropy generation rate was at its minimum.

_{s})

_{S}

_{min}for various heat flux densities and the flow of ice slurry through a pipe with a diameter of d

_{i}= 0.016 m. The values of w, x

_{s}, determined analytically, were validated using experimental values.

_{s}were determined by calculating, at each measurement point, the entropy generation rate (Equation (6)) using the measured values of heat transfer coefficients and the flow resistance values [2,25]. Next, for the given values of x

_{s}, the measurement points were chosen at which $\dot{S}={\dot{S}}_{\mathrm{min}}$ and the non-monotonic course of the relationship w(x

_{s})

_{S}

_{min}was associated with an area of transition between laminar and turbulent flow. The plotted theoretical ${w}_{cal-\dot{S}\mathrm{min}}({X}_{V})$ curves provide a qualitatively correct description of the course of analogous curves obtained directly from the measuring points. The mean relative difference between the experimental and analytical flow velocities ${w}_{\dot{S}\mathrm{min}}({X}_{V})$ does not exceed 7.6%. The maximum difference between the measured velocities and the calculated velocities concerned the case of heat flux density $\dot{q}$ = 2 kW and was lower than 30%.

_{sC}, and w

_{C}found on the basis of the criterion for motion nature change valid for the ice slurry (Equation (13)) [54]:

_{s}, d

_{i}, w, $\dot{q}$) are independent, but they affect the heat transfer coefficients and flow resistance, and thus the entropy flux and its minimum value, in various ways. Charts similar to the one presented in Figure 5 allow the selection of parameters x

_{s}, (X

_{v}Table 1) w, d

_{i}, $\dot{q}$ in such a way that the entropy flux generated by the ice slurry is minimal. The results presented in Figure 5 and similar results for tube diameters d

_{i}= 0.01 m and d

_{i}= 0.02 m can be presented in the form of criteria relationship (14). Correlation (14) defines the relationship between the Reynolds number and dimensionless numbers taking into account the effect of geometric (K

_{qX}) and flow parameters (K

_{q}, K

_{qX}) on the generated minimal entropy flux.

_{IS}calculated as for Bingham’s fluid) and $\dot{q}{X}_{v}/r$.

_{s}, w both for $\dot{S}={\dot{S}}_{\mathrm{min}}$ and ε

_{Nu−f}= (Nu/Nu

_{pt})/(f/f

_{pt})

^{1/3}= 1, for ice slurry heat transfer during ice slurry flow through d

_{i}= 0.01 and 0.02 mpipes with constant heat flux density values of $\dot{q}$ = 5, 10 kWm

^{−2}. The range of x

_{s}and w values, for which ε

_{Nu−f}> 1, means that while the same power is required to transport the refrigerant, the use of ice slurry over the use of the carrier fluid as the refrigerant translates into better thermal efficiency. The results shown in Figure 7 suggest that the minimum entropy generation rate condition generally shifts the scope of ice slurry use towards higher flow velocities. The exception is low heat flux densities for the minimum pipe diameter under consideration, d

_{i}= 0.01 m, for which entropy generation minimization implies lower values of flow velocity with respect to the condition ε

_{Nu−f}> 1. Figure 2 and Figure 7 show that, regardless of the adopted criterion for the assessment of the heat exchange process involving ice slurry, the laminar flow range is especially preferred as regards the selection of x

_{s}and w. In design practice, the recommended flow velocities in pipes amount to w < 1 ms

^{−1}. This velocity range for the flow of ice slurry in pipes with d

_{i}< 0.02 m enables effective enhancement of the heat transfer process with the minimum entropy generation rates.

- ○
- During pipe flow, the lowest entropy generation rates were characteristic of small mass fractions of ice in the turbulent flow area and for the flow velocity of 1.5 < w < 2 ms
^{−1}. - ○
- In the laminar flow area, the lowest entropy generation rates corresponded to the highest analyzed mass fraction of ice x
_{s}= 30%. - ○
- Regardless of the share of solid particles, the minimum entropy generation rate criterion requires the application of high flow velocities, which for heat flux density values of $\dot{q}\ge $ 10 kWm
^{−2}are greater than w > 1 ms^{−1}.

## 3. Entropy Generation in Heat Exchangers Fed with Ice Slurry

#### 3.1. Air Cooler

_{a-out}= 1 bar) was assumed. On the other hand, air pressure at the inlet to the exchanger was p

_{a-out}= p

_{a-out}+ Δp

_{a}.

- ○
- constant thermal efficiency values $\dot{Q}=$ 20; 40 kW,
- ○
- constant air flow stream ${\dot{m}}_{a}=$ 2.2 kgs
^{−1}, - ○
- constant inlet air temperature T
_{a-in}= 20 °C, - ○
- various ice slurry mass flux values 0.3 $\le \dot{m}\le $ 1.25 kgs
^{−1}, - ○
- various mass fractions of ice 5 ≤ x
_{s}≤ 30%, - ○
- constant number of pipes and feeds in the heat exchanger made of d
_{i}= 0.01 m pipes.

^{−1}). This results in a significant increase in entropy generation associated with ice slurry and air-side flow resistances. The Bejan number for a 40 kW cooler efficiency changed from 0.04 to 0.43 for the mass fractions of ice of x

_{s}= 30% and x

_{s}= 5%, respectively. Figure 8b shows how the mass share of ice and the mass flux of the ice slurry affect the total entropy generation rates in the air coolers (non-complete ice melting scenario). Regardless of the mass flux, the minimum entropy generation rates were obtained for the maximum mass share of ice x

_{s}= 30%. For a constant mass flux of the ice slurry, the dominant entropy generation component is the entropy generated by the air-side heat transfer the air-side heat transfer process. Lower mass fractions correspond to higher mean temperatures of the ice slurry. Therefore, the receipt of the same heat flux is conditional upon a larger change in the specific enthalpy of the refrigerant. The entropy generation rate as a function of the mass flux is not a monotonic function. This has been illustrated by Figure 9a.

_{s}= 30% indicate that for the discussed structure of the exchanger, function $\mathsf{\Delta}\dot{S}=f(\dot{m},{x}_{s}=30\%)$ has its minimum value at the point corresponding to the mass flux of $\dot{m}=$0.83 kgs

^{−1}. At the point of the minimum of function $\mathsf{\Delta}\dot{S}$, the Bejan number was 0.2, varying in the entire discussed mass flux range between 0.07 ≤ Be ≤ 0.28. The results presented in Figure 9a refer to various surface areas and therefore also to various heat flux densities. The flow velocity curve makes it possible to determine the ice slurry flow velocity in heat exchanger pipes for particular mass fluxes.

_{i}= 0.01 m pipe. Just as in the case of flow through a straight pipe, in the discussed heat exchanger higher heat flux densities also corresponded to higher optimum flow velocities. The comparison (Table 4) indicates that when designing a heat exchanger, it is possible to determine the operational parameters of the cooler which meet the $\dot{S}$~${\dot{S}}_{\mathrm{min}}(d,\dot{q})$ condition by finding the ice slurry flow velocity using Equation (14).

#### 3.2. Fluid Cooler

_{i}for a rectangular cross-section, included in Table 2.

- ○
- constant thermal efficiency values $\dot{Q}=$20; 30, 40 kW,
- ○
- a change in the temperature of the fluid cooled in the exchanger between T
_{f−in}= 12, 20 °C and T_{f−in}= 2, 5 °C (T_{f}= 10, 15 K), - ○
- various ice slurry mass flux values 0.3$\le \dot{m}\le $12 kgs
^{−1}, - ○
- various mass fractions of ice 5 ≤ x
_{s}≤ 30%, - ○
- two geometrical configurations of the exchanger: n
_{p}= 6, n_{s}= 1; n_{p}= 12, n_{s}= 2.

_{p}= 6, n

_{s}= 1). In this calculation variant, the constant mass flux condition implies variable values of changes in the ice slurry’s specific enthalpy and various degrees of ice melting. Just as in the case of the air cooler, regardless of the mass flux, minimum entropy generation rates were recorded for the maximum mass share of ice x

_{s}= 30%. For a constant mass flux of the ice slurry, the dominant component of entropy generation is that generated by the ice slurry-side heat transfer process. In a plate heat exchanger, the entropy generation due to flow resistances is determined by the flow of the ice slurry. In the calculations performed by the authors, the flow was of a laminar nature and the greatest flow resistances were generated for an ice slurry with the mass share of ice of x

_{s}= 30%, for which the dynamic plastic viscosity coefficient and limit shear stress were highest. The highest Bejan number of 0.15 corresponded to x

_{s}= 30%, $\dot{m}$ = 1.87 kgs

^{−1}.

_{p}= 12, n

_{s}= 2) and a constant mass share of ice x

_{s}= 30%. Function $\mathsf{\Delta}\dot{S}=f(\dot{m},{x}_{s}=30\%)$ demonstrates a minimum which shifts towards higher mass flux values as thermal efficiency increases. At the point marking the minimum of function $\mathsf{\Delta}\dot{S}$, the Bejan number was 0.27, varying in the entire discussed range of mass flux changes between 0.02–0.93. Note that in the calculations performed, the minimum entropy generation rate (due to the value of Re

_{K}= 380−530) was recorded for laminar flow, but for w = 1.8−2.3 ms

^{−1}. These flow velocity values are significantly higher than the velocity values used in typical plate exchangers (w < 0.5 ms

^{−1}). The application of a flow velocity of w < 1 ms

^{−1}leads to an increase in the entropy generation rate by at least 14%–67%, depending on thermal efficiency.

^{−1}), lower entropy generation rates are recorded in an ethanol-fed exchanger than in an ice slurry-fed exchanger. If ethanol is used, however, the required surface area of the exchanger is at least 60% higher than in the case of an ice slurry fed exchanger.

## 4. Conclusions

_{s}) for different tube diameters and heat flux densities for which the entropy flux generated during the melting process in the conditions of forced convection of ice slurry is minimal. The original aspect presented in the paper is the description of the set of optimal parameters for w, x

_{s}, the criteria relationship between the Reynolds number and dimensionless numbers taking into account the effect of geometric and flow parameters on the generated entropy flux.

_{Nu−f}> 1 more significantly limits the velocity of ice slurry flow from above than the minimum entropy flux criterion.

^{−1}).

## Funding

## Conflicts of Interest

## Nomenclature

A | heat transfer surface, m^{2} |

Be | Bejan number, $\mathrm{Be}=\frac{\mathsf{\Delta}{\dot{S}}_{\mathsf{\Delta}p}}{\mathsf{\Delta}\dot{S}}$ |

B_{i} | coefficients in Equations (11) and (12) (Table 2) |

c_{f} | Fanning friction factor, |

C_{i} | coefficients in Equations (15) and (16) (Table 3) |

c_{p} | specific heat, Jkg^{−1}K^{−1} |

d | diameter, m |

$\dot{G}$ | mass flux density, kgm^{−2}s^{−1} |

Gz | Graetz number, $\mathrm{Gz}={\mathrm{Re}}_{K}{\mathrm{Pr}}_{B}\frac{{d}_{h}}{l}$ |

He | Hedström number, $\mathrm{He}=\frac{{\rho}_{IS}{\tau}_{p}{d}_{h}^{2}}{{\mu}_{p}^{2}}$ |

i | specific enthalpy, Jkg^{−1} |

K^{*} | consistency index |

K_{F} | phase change number, ${K}_{F}=\frac{r}{{c}_{p-IS}\left({T}_{w}-{T}_{IS}\right)}$ |

l | length, m |

$\dot{m}$ | mass flux, kgs^{−1} |

n^{*} | flow behavior index |

n_{p} | plate number |

n_{s} | number of feeds |

Nu | Nusselt number, $\mathrm{Nu}=\frac{\alpha \hspace{0.17em}{d}_{h}}{\lambda}$ |

p | pressure, Pa |

P | circumference, m |

PCM | phase change materials |

$\dot{q}$ | heat flux density, Wm^{−2} |

$\dot{Q}$ | heat flux, W |

R | individual gas constant, Jkg^{−1}K^{−1} |

Re_{IS} | Reynolds number for Bingham fluid, ${\mathrm{Re}}_{IS}=\frac{{\rho}_{IS}w{d}_{h}}{{\mu}_{p}}$ |

Re_{K} | generalized Reynolds number according to Kozicki, ${\mathrm{Re}}_{K}=\frac{{\rho}_{IS}{w}^{2-n*}{d}_{h}^{n*}}{{8}^{n*-1}{K}^{*}}$ |

s | specific entropy, Jkg^{−1}K^{−1} |

${\dot{S}}_{l}$ | entropy generation rate referred to a unit of length, WK^{−1}m^{−1} |

$\dot{S}$ | entropy generation rate, WK^{−1} |

T | temperature, K |

$\overline{T}$ | mean temperature, K |

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

w | mean velocity, ms^{−1} |

$\dot{W}$ | thermal capacity, WK^{−1} |

x_{s} | mass fraction of ice, % |

${X}_{v}$ | volume fraction of ice, % |

Greek symbols | |

ν | specific volume, m^{3}kg^{−1} |

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

ε | enhancement efficiency |

η | exchanger efficiency |

λ | thermal conduction coefficient, Wm^{−1}K^{−1} |

μ_{p} | dynamic plastic viscosity, Pas |

ρ | density, kgm^{−3} |

τ_{p} | yield shear stress, Pa |

τ_{w} | shear stress at pipe wall, Pa |

Subscripts | |

a | air |

ai | carrier fluid |

calc | calculated value |

crit, C | critical value |

et | ethanol |

F | fluid |

h | hydraulic |

H | heat transfer |

i | internal, ordinal number |

in | inlet |

IS | ice slurry |

l | length |

L | laminar |

o | baseline value |

out | outlet |

ref | reference value |

S | ice, ice crystal |

T | turbulent |

w | wall |

Δp | flow resistance |

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**Figure 2.**Total entropy generation rate as a function of the mass fraction of ice and ice slurry flow velocity: d

_{i}= 0.016 m, $\dot{q}=8$ kWm

^{−2}.

**Figure 3.**Relationship between entropy generation rates ${\dot{S}}_{H},\hspace{0.17em}{\dot{S}}_{\mathsf{\Delta}p},\hspace{0.17em}\dot{S}$, for d

_{i}= 0.016 m, $\dot{q}=8$ kWm

^{−2}: (

**a**) x

_{s}= 11.7%; (

**b**) x

_{s}= 26.6%.

**Figure 4.**Effect of tube diameter on the generated entropy flux: (

**a**) $\dot{q}$= 2 kW, effect of the mass fraction of ice; (

**b**) x

_{s}= 11.7%, effect of heat flux density.

**Figure 5.**The calculated and experimentally determined parameters w and x

_{s}, for which the ice slurry entropy generation rate is at its minimum, d

_{i}= 0.016 m, 2 kWm

^{−2}$\le \dot{q}\le $ 8 kWm

^{−2}.

**Figure 6.**A comparison of the values calculated on the basis of correlation (13) with the determined Re

_{IS}and $\dot{q}{X}_{v}/r$, for which $\dot{S}={\dot{S}}_{\mathrm{min}}$.

**Figure 7.**Curves $w{({x}_{s})}_{\dot{S}\mathrm{min}}$, $w{({X}_{v})}_{\epsilon Nu-f=1}$: (

**a**) d

_{i}= 0.01 m; (

**b**) d

_{i}= 0.02 m.

**Figure 8.**Entropy generation rate in a lamelled air cooler: (

**a**) $\mathsf{\Delta}\dot{S}(\dot{Q},\hspace{0.17em}{x}_{s})$, ${\dot{m}}_{a}=$ 2.2; 3.2 kgs

^{−1}; (

**b**) $\mathsf{\Delta}\dot{S}(\dot{m},\hspace{0.17em}{x}_{s})$, ${\dot{m}}_{a}=$ 2.2 kgs

^{−1}, $\dot{Q}$ = 20 kW.

**Figure 9.**Minimum entropy generation rate $\mathsf{\Delta}{\dot{S}}_{\mathrm{min}}(\dot{m})$, x

_{s}= 30%: (

**a**) lamelled air cooler $\dot{Q}$ = 20 kW; (

**b**) ice water cooler $\dot{Q}$ = 20–40 kW.

**Figure 10.**Entropy generation rate in a plate heat exchanger, milk cooler $\dot{Q}$ = 30 kW: (

**a**) $\mathsf{\Delta}\dot{S}({\dot{m}}_{f},\hspace{0.17em}{x}_{s})$, n

_{s}= 1, 2, n

_{p}= 6, 12; (

**b**) $\mathsf{\Delta}\dot{S}(\dot{m},\hspace{0.17em}{x}_{s})$, n

_{s}= 1, n

_{p}= 6.

**Figure 11.**Entropy generation rate in a plate heat exchanger (milk cooler) $\dot{Q}$ = 30 kW fed with a 30% ice slurry or 15% ethanol: (

**a**) a comparison of the efficiency η of plate exchangers (

**b**) a comparison of the surface areas of heat exchangers A

_{et}/A

_{IS}.

Properties | Formula | Comments |
---|---|---|

Enthalpy of ice slurry [51,52] | ${i}_{IS}=\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){i}_{s}+\left(1-{\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){i}_{a}$ | - |

Enthalpy of carrier liquid [51,52] | ${i}_{a}=\mathsf{\Delta}{i}_{mix}+{c}_{pa}\left(T-273.15\right)$ | T_{ref} = 273.15 K, i_{a} = i_{a}(x_{a},T) [51] |

Enthalpy of ice [52] | ${i}_{s}=-r+{c}_{ps}\left(T-273.15\right)$ | r = 332.4 kJ kg^{−1} |

Mean specific heat of ice slurry [51] | ${c}_{pIS}=\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){c}_{ps}+\left(1-{\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){c}_{pa}$ | c_{pa} [53], c_{ps} [51] |

Heat conductivity of ice slurry λ_{IS,w=0} [22] | ${\lambda}_{\mathrm{IS},\mathrm{w}=0}={\lambda}_{\mathrm{a}}\cdot \left[\frac{2\cdot {\lambda}_{\mathrm{a}}+{\lambda}_{\mathrm{s}}-2\cdot \left({\scriptscriptstyle \raisebox{1ex}{${X}_{v}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)\left({\lambda}_{\mathrm{a}}-{\lambda}_{\mathrm{s}}\right)}{2\cdot {\lambda}_{\mathrm{a}}+{\lambda}_{\mathrm{s}}+\left({\scriptscriptstyle \raisebox{1ex}{${X}_{v}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)\left({\lambda}_{\mathrm{a}}-{\lambda}_{\mathrm{s}}\right)}\right]$ | λ_{a} [53], λ_{s} [51] |

Ice slurry density | ${\rho}_{IS}=\left({\scriptscriptstyle \raisebox{1ex}{${X}_{\mathrm{V}}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){\rho}_{s}+\left(1-{\scriptscriptstyle \raisebox{1ex}{${X}_{\mathrm{V}}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right){\rho}_{a}$ | ρ_{a} [53], ρ_{s} [51] |

Yield shear stress of ice slurry for x_{ai} = 10.6%; d_{s} = 0.1–0.15 mm [3] | $\begin{array}{l}{\tau}_{p}=0.013-1.4284\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)+73.453{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{2}\\ -394.64{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{3}+835.82{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{4}\end{array}$ | - |

Plastic viscosity of ice slurry for x_{ai} = 10.6%; d_{s} = 0.1–0.15 mm [3] | $\begin{array}{l}{\mu}_{p}=0.0035+0.0644\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)-0.7394{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{2}\\ +5.6963{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{3}-19.759{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{4}+26.732{\left({\scriptscriptstyle \raisebox{1ex}{${x}_{s}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}\right)}^{5}\end{array}$ | - |

Parameter K* | $\begin{array}{l}{K}^{*}={\left(c{\mu}_{p}\right)}^{n*}{\tau}_{w}^{1+dn*/c}\\ {\left[\frac{c}{c+d}{\tau}_{w}^{1+d/c}-\frac{c}{d}{\tau}_{w}^{d/c}{\tau}_{p}+\frac{{c}^{2}}{d\left(c+d\right)}{\tau}_{p}^{1+d/c}\right]}^{-n*}\end{array}$ | ε_{B}= τ_{p}/τ_{w}c,d [54] |

Parameter n* | ${n}^{*}=\frac{c\left[\frac{1-{\epsilon}_{B}^{1+d/c}}{c+d}-\frac{{\epsilon}_{B}\left(1-{\epsilon}_{B}^{d/c}\right)}{d}\right]}{1-{\epsilon}_{B}-d\left[\frac{1-{\epsilon}_{B}^{1+d/c}}{c+d}-\frac{{\epsilon}_{B}\left(1-{\epsilon}_{B}^{d/c}\right)}{d}\right]}$ | |

${X}_{v}$ | ${X}_{v}={\left(1+\frac{1-{x}_{s}}{{x}_{s}}\frac{{\rho}_{s}}{{\rho}_{a}}\right)}^{-1}$ | - |

Cross Section | Type of Flow | Parameter | Value | Applicability Range |
---|---|---|---|---|

Pipe $0.01\le {d}_{i}\le 0.02$ | laminar | B_{1} | 2.52 | 3% < x_{s} < 30%w _{m} > 0.1 ms^{−1}200 < Re _{K} < 2100 |

B_{2} | 0.11 | |||

B_{3} | −0.10 | |||

B_{4} | −0.35 | |||

B_{5} | 0.052 | |||

turbulent | B_{1} | 0.0096 | 3% < x_{s} < 30%w _{m} <4.5 ms^{−1}2100 < Re _{K} < 11,000 | |

B_{2} | 0.70 | |||

B_{4} | −0.10 | |||

Rectangular and slit cross-section $0.0055\le {d}_{h}\le 0.012$ | laminar | B_{1} | 3.66 | 5.6% < x_{s} <3 0%w _{m} > 0.5 ms^{−1}30 < Re _{K} < 2300 |

B_{2} | 0.16 | |||

B_{3} | −0.28 | |||

B_{4} | −0.12 | |||

B_{5} | 0.16 | |||

turbulent | B_{1} | 0.0032 | 3% <x_{s} < 30%w _{m} < 3.1 ms^{−1}1900 < Re _{K} < 6000 | |

B_{2} | 0.86 |

Parameter | Value |
---|---|

C_{1} | ${C}_{1}={c}_{11}{\left(\raisebox{1ex}{${d}_{i}$}\!\left/ \!\raisebox{-1ex}{${d}_{0}$}\right.\right)}^{2}+{c}_{12}\left(\raisebox{1ex}{${d}_{i}$}\!\left/ \!\raisebox{-1ex}{${d}_{0}$}\right.\right)+{c}_{13}$; c_{11} = −2292.0231, c_{12} = 10,483.068, c_{13} = −4879.7335 |

C_{2} | ${C}_{2}={c}_{21}{\left(\raisebox{1ex}{${d}_{i}$}\!\left/ \!\raisebox{-1ex}{${d}_{0}$}\right.\right)}^{2}+{c}_{22}\left(\raisebox{1ex}{${d}_{i}$}\!\left/ \!\raisebox{-1ex}{${d}_{0}$}\right.\right)+{c}_{33}$; c_{21} = −0.0708, c_{22} = 0.2925, c_{23} = 0.6583 |

C_{3} | 11.3766 |

C_{4} | 1.15317 |

C_{5} | 52.0511 |

${\dot{q}}_{o}$ | 10,000 |

${\dot{G}}_{o}$ | 380 |

d_{io} | 0.01 |

**Table 4.**Optimum velocity values according to Equation (14) and for $\mathsf{\Delta}{\dot{S}}_{\mathrm{min}}$ from Equation (23) d

_{i}= 0.01, x

_{s}= 30%.

q | w_{tube} | w_{HE} | Be_{tube} | Be_{HE} |
---|---|---|---|---|

8 | 0.93 | 0.72 | 0.08 | 0.08 |

10 | 1.19 | 1.23 | 0.125 | 0.14 |

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**MDPI and ACS Style**

Niezgoda-Żelasko, B.
Entropy Generation of Forced Convection during Melting of Ice Slurry. *Entropy* **2019**, *21*, 514.
https://doi.org/10.3390/e21050514

**AMA Style**

Niezgoda-Żelasko B.
Entropy Generation of Forced Convection during Melting of Ice Slurry. *Entropy*. 2019; 21(5):514.
https://doi.org/10.3390/e21050514

**Chicago/Turabian Style**

Niezgoda-Żelasko, Beata.
2019. "Entropy Generation of Forced Convection during Melting of Ice Slurry" *Entropy* 21, no. 5: 514.
https://doi.org/10.3390/e21050514