1 Introduction

Black grape juice being a flavourful and rejuvenating drink is a very popular beverage worldwide owing to its delicious taste, vivid colour and health benefits. A significant component of black grape juice is a group of phenolic compounds known as anthocyanin responsible for the deep purple colour of black grapes along with antioxidant and anti-inflammatory effects in the body [1, 2]. Anthocyanin and phenolic have also been linked with a reduced risk of coronary heart diseases, lower rates of some types of cancer, diabetes and hypertension [3,4,5,6]. Also, a study regarding consumption of black grape juice prior to exercise depicts enhanced endurance along with lowering muscle damages amongst athletes [7]. All these applications of black grape juice are possible due to specific components present in various parts of the grape with the skin containing anthocyanin, pro-anthocyanin and flavour compounds whereas the seed containing pro-anthocyanin and the pulp consisting of organic acids, sugars and flavour compounds [8]. Physico-chemical properties like density, thermal conductivity and heat capacity play an important role in the application of black grape juice.

Density of black grape juice is an essential physico-chemical property which can be vital to estimate its applications. Usually, high density grape juice is preferred for the production of grape syrups, concentrates and fermentation processes to produce wine [9]. Also, density impacts the operations of clarification and filtration during processing and may require additional steps in order to attain the targeted clarity [10]. Packaging and distribution of black grape juice can also be impacted by the density in order to maintain the quality and freshness during transport and storage stages [11]. Heat capacity is another physico-chemical property having its importance during thermal processing stages like pasteurization and sterilization as this property would determine the heat absorbance and related temperature rise leading to efficient and uniform heating during juice production [12]. Also maintaining high temperature related to higher heat capacity for reducing microorganism growth and longer shelf life of the juice. Thermal conductivity has similar impact on thermal processing activities like pasteurization, refrigeration and sterilization leading to quick and uniform heating leading to efficient processing and preservation of black grape juice [9, 13]. These above mentioned physico-chemical properties are essential for an effective juice manufacturing process. These properties are fundamental for optimization and calculations related to design, control and automation of the equipment and related processes. When the values of these properties are incorporated as a function of temperature and concentration, it not only helps in installation but also transformation of the process in order to maintain, predict and achieve enhanced quality of juice produced. This information is significant while designing food processing equipment like pumps, pipes, heat exchangers, filters and mixers [13,14,15,16,17,18]. Density additionally acts as a quality index [14].

Density, heat capacity and thermal conductivity are some of the properties that are affected by the amount of solid content and temperature. Thus, it is extremely essential to have the knowledge of these properties as a function of temperature and concentration [14]. Several research articles have previously demonstrated model fitting of experimental data with empirical equations that resemble closely with the trend followed by experimental values with respect to variations in temperature and concentration [19, 20]. Estimation of density and heat capacity for the Trebbiano Bianco grape variety depicts 0.2% and 2.8% error respectively for a range of 0–80°Brix concentrations at a specific temperature [19]. This incorporates pseudo sucrose and grape sugar components but fails to determine the temperature change and individual component contribution impacts on property prediction. Other experimental studies show an estimation of density of concentrated grape and sulphide grape juices over a range of concentrations only at 20 °C without elaborating the effect of temperature [21]. In a separate study, various existing and proposed models were fitted over a varying concentration and temperature ranges data for thermal conductivity [22]. The model equation proposed by the research group in this study correlates closely to the experimental data as compared to equations previously proposed by other researchers over a temperature range of 20 to 80 °C and 20 to 60% dry matter. The same group of authors propose a modelling equation for heat capacity for black grape juice depicting similar outcome on comparing the heat capacity values estimated by the proposed model and previously existing model [23]. The proposed model displays lower errors over a temperature range of 20 to 80 °C and 20 to 60% dry matter. A framework of density modelling for black grape juice over a temperature range of 20 to 80 °C and 20 to 60% dry matter [17]. These studies however fail to incorporate the physico-chemical properties of black grape juice for storage temperatures below 20 °C and processing variations below 20% dry matter. Additionally, the individual grape component contribution towards estimation of physico-chemical properties is not discussed.

Several other studies have been carried out for varying fruit juices like grape juice concentrates [24], passion fruit [13], clarified pineapple juice [25], malus floribunda juice [14], clarified apple juice [15] and orange juice [16] for similar physico-chemical properties by determining several degree of polynomial equations. But none of them estimate how contribution of individual components of grape affects the property values. Thus, in order to understand how an individual component impacts the property estimation it is important to incorporate the structure of the component. This is exactly done by the use of the group contribution method and therefore becomes the primary objective of this study. In this method, the property to be estimated for a compound becomes a function of parameters that are structure dependent. These in turn are determined by the summation of the product of the number of times each group occurs in a molecule and the group's contribution. This approach is quick and does not require huge computational resources [26]. Further, the temperature and concentration dependence can also be correlated with the physico-chemical properties though such frameworks. Therefore the two fold objectives i.e. structural variation of black grape components and variation in process parameters (e.g. temperature and concentration) make the group contribution led parameters undertaken in this work as novel and robust in nature. Several research articles regarding the estimation of critical properties [27,28,29], density [30,31,32,33], heat capacity [34,35,36] and thermal conductivity [37,38,39] of ionic liquids and organic compounds have gained immense interest in the past few years.

Herein, group contribution method (GCM) led formulations are adopted to establish a-priori modelling framework of density, thermal conductivity and heat capacity. Cohesion based equation of state is implemented within this framework to model density of black grape juice. Geometric similitude concept in conjunction with cubic equation of state is implemented in GCM framework to estimate the thermal conductivity. The models can accommodate the variation of composition and temperature of the black grape juice and thus open the gateway of understanding their impact on food product processing, packaging and storage.

2 Computational methodology

There is a key constraint of component identification in developing the a-priori framework of physico-chemical properties. Since, most of the literature is concerned about reporting the composition of various class of compounds such as sugars, organic acids, vitamins, carotenoids, phenolic and volatile compounds, we select Malbec V. Vinifera specie Alphonse Lavallee variety of black grape juice whose compound wise specifications are detailed in literature [40]. This specie has three of flavanols, six of anthocyanins, two of stilbenes and several other food entities. The details of each component along with the molecular structure are given the Table S1 of supplementary material. Once the molecular structures are identified, critical properties are estimated first with the help of group contribution parameters. Later these critical properties are used to estimate the density and thermal conductivity of the juice. A separate group contribution methodology has been adopted to estimate the heat capacity and will be described in the subsequent sections. The overall framework of calculation has been depicted in Fig. 1.

Fig. 1
figure 1

Overall framework of computational methodology

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2.1 Estimation of critical property

Modified Lydersen-Joback-Ried method lays down the formulae for the calculation of normal boiling temperature (Tb), critical temperature (Tc), critical pressure (Pc) and critical volume (Vc) [27,28,29, 41]. Additionally, Rudkin's model is employed to estimate the acentric factor [41, 42]. As a starting procedure, each component mentioned in the Table S1 of supplementary material is divided into functional moieties. Such denominations are mentioned in Table S2 of supplementary material. Thereafter, the group contribution parameters given in Table S3 of supplementary material are used for the calculation of critical properties and acentric factor [27]. The corresponding equations of modified Lydersen-Joback-Ried method and Rudikin’s model are elaborated in Table 1. Thereafter, Lee-Kesler mixing rules are applied to calculate the critical properties of black grape juice concentrate. The corresponding equations are summarized in Table 2. It is important to note that experimental data of black grape juice includes considerable amount of water which will be treated separately.

Table 1 Normal boiling point, critical properties, acentric factor [27, 29, 41, 42]
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Table 2 Lee-Kesler mixing rule [27, 44]
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2.2 Framework of density

Cohesion based equation of state (EoS) has been proven successful in the estimation of density of neoteric liquids such as ionic liquid and deep eutectic solvent [33, 43, 44]. Herein, a formulation of such framework i.e., predictive Soave–Redlich–Kwong (PSRK) equation of state with NSM1 alpha function model (NMPRNSM1) [44,45,46] has been used to estimate the density of black grape juice concentrate. The respective equations are described in Table 3. The critical temperature and acentric factor are calculated from Lee-Kesler mixing rule equations. Since, the water is a crucial component in the black grape juice, the estimated density of juice is obtained by [47, 48]

$$\frac{1}{{\rho_{m} }} = \frac{{w_{s} }}{{\rho_{s} }} + \frac{{(1 - w_{s} )}}{{\rho_{b} }} - w_{s} (1 - w_{s} )\left[ {\frac{1}{{\rho_{s} }} + \frac{1}{{\rho_{b} }}} \right]B_{ij}$$
(1)

where ρm, ρs, ρb, ws and Bij represent density of mixture, water and juice concentrate, water weight fraction and correlated coefficient respectively. However, parameter δ2 and Bij are obtained regressing the experimental data in genetic algorithm (GA) formulation. This is obtained by minimizing the following objective function:

$$f(\delta_{2} ,B_{ij} ) = \sum\limits_{i = 1}^{n} {\left| {\frac{{\left( {\rho_{i,\exp t} - \rho_{i,pred} } \right)}}{{\rho_{i,\exp t} }}} \right|}$$
(2)

where ρi,expt and ρi,pred are experimental and estimated values of density. The error is estimated by absolute average deviation (%), given by,

$$\% AAD = \left| {\frac{{X_{expt} - X_{pred} }}{{X_{expt} }}} \right| \times 100$$
(3)

where Xexpt and Xpred represent the values of experimental and estimated thermophysical properties respectively (herein density).

Table 3 Density model- Predictive Soave − Redlich − Kwong (PSRK) equation of state with NSM1 alpha function (NMPRNSM1) model [33, 43, 45]
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2.3 Framework of thermal conductivity

Geometric similitude concept in conjunction with the Redlich-Kwong EoS has been employed to compute the thermal conductivity of the black grape juice concentrate [49, 50]. The respective equations are elaborated in Table 4. Like the density formulation, water has been treated separately in the computation of thermal conductivity and the thermal conductivity of the black grape juice has been calculated by the following equation [13].

$$k = \sum\limits_{i} {y_{i} k_{i} }$$
(4)

where ki = individual component thermal conductivity and yi = mole fraction of component. Genetic algorithm formulation has been employed to estimate the e1-7 and β1-2 after minimizing the following objective function.

$$f(e_{1 - 7} ,\beta_{1 - 2} ) = \sum\limits_{i = 1}^{n} {\left| {\frac{{\left( {k_{i,\exp t} - k_{i,pred} } \right)}}{{k_{i,\exp t} }}} \right|}$$
(5)

where ki,expt and ki,pred are experimental and estimated values of thermal conductivity.

Table 4 Thermal Conductivity model using the Redlich − Kwong Cubic Equation of State[49, 50, 59]
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2.4 Framework of heat capacity

A group contribution methodology of estimating heat capacity of liquids has been documented in the literature [26, 51]. The framework is contrary to the critical property led approach of density and thermal conductivity. Hence, the group contribution parameters for the estimation of heat capacity have been given Table S4 of supplementary material. The characteristic equation of heat capacity framework is given by equation [35, 42].

$$C_{p}^{l} \left( T \right) = C_{p0}^{l} \left( T \right) + \sum\limits_{l} {N_{i} } C_{p1 - i}^{l} \left( T \right) + w\sum\limits_{j} {M_{j} C_{p2 - j}^{l} \left( T \right)} + z\sum\limits_{k} {O_{k} C_{p3 - k}^{l} \left( T \right)}$$
(6)
$$C_{{p,q^{th} level - i,j,k}}^{l} \left( T \right) = a_{q - i,j,k} + b_{q - i,j,k} \left( \frac{T}{100} \right) + d_{q - i,j,k} \left( \frac{T}{100} \right)^{2}$$
(7)

where Clp0(T) = additional adjustable parameter (zeroth level group contribution); Cp1-i (T) = first level group contribution of type i; Cp2-j (T) = second level group contribution of type j; Cp3-k (T) = third level group contribution of type k; Ni, Mj, and Ok = number of occurrences of individual groups in a compound; aq-i, j, or k, b, and d = adjustable parameters for the temperature dependence of zeroth, first, second and third level group contributions. The respective levels are elaborated in the literature[35]. However, as per the classification of functional groups, first level group contributions are sufficient to estimate the heat capacity of black grape juice concentrate. Hence, the modified equations will be

$$C_{p}^{l} \left( T \right) = C_{p0}^{l} \left( T \right) + N_{i} a_{1 - i,j,k} + N_{i} b_{1 - i,j,k} \left( \frac{T}{100} \right) + N_{i} d_{1 - i,j,k} \left( \frac{T}{100} \right)^{2}$$
(8)

Thereafter, the heat capacity of the black grape juice is obtained by the following mixing rule where water is treated separately likewise density and thermal conductivity [13]

$$Cp = \sum\limits_{i} {y_{i} Cp_{i} }$$
(9)

where Cpi = individual component heat capacity and yi = mole fraction of component. The above-mentioned framework introduces Clp0(T) which will be obtained from the GA formulation minimizing the following objective function.

$$f(C_{p0}^{l} ) = \sum\limits_{i = 1}^{n} {\left| {\frac{{\left( {C_{pi,expt} - C_{pi,pred} } \right)}}{{C_{pi,expt} }}} \right|}$$
(10)

where Cpi,expt and Cpi,pred are the experimental and estimated values of thermal conductivity.

2.5 Genetic algorithm

Genetic algorithm (GA) has been coupled with the respective algorithm of density, thermal conductivity and heat capacity to estimate the model parameters. This metaheuristic algorithm first developed by John Holland and related collaborative group of researchers [52], origins from Darwin's theory of natural selection and introduces genetic operators like selection, crossover and mutation to solve engineering problems. The introduction of numerous random alterations facilitates the optimization where traditional gradient based algorithms fail to generate results for highly complex optimization problems. The global minima of the solution are obtained by minimizing the objective function constructed from a set of experimental data and the corresponding model estimations. Such objective functions for density, thermal conductivity and heat capacity have been given in Eqs. (2), (5) and (10) respectively. However, the details of GA are elaborated in literature [48]

3 Results and discussion

The objective of this study is to establish the a-priori modelling framework of physico-chemical properties of black grape juice. The a-priori nature compels to enroute through the identification of molecular components of black grape juice. Such detailed experimental dataset for the Malbec V. Vinifera specie Alphonse Lavallee variety of Black grape juice covers the temperature range from 274 to 339 K and concentration range of 13.6–45°Brix [24]. The frameworks are having various adjustable parameters which can only be estimated through the minimization of objective function incorporating experimental and predicted dataset. To estimate the parameters related to density, thermal conductivity and heat capacity, first experimental dataset at each concentration and for all reported temperatures are correlated with the estimated properties in GA formalism. Therefore, the models are built considering the temperature range from 274 to 339 K and concentration range of 13.6–45°Brix. However, being a-priori nature, they can be widely used to other temperature and concentration as well. Table 5 represents the estimated %AAD of 1.43 and 3.95 for density and thermal conductivity respectively within the temperature range of 274 to 339 K and 13.6°Brix. Also heat capacity estimation at 13.6°Brix over a temperature range of 274 to 319 K results 1.12%AAD represented in Table 5. The %AADs represented in Table 5 are of the black grape juice considering the water content. The modelling framework considers physico-chemical properties of water separately and it is added as weighted molar sum of each component in the juice. Since, the black grapes and other types of grapes are having numerous compounds and they affect the physicochemical properties, the accurate experimental values may not be available over all the temperature and concentration studied. Hence, group contribution led approach has been adopted. Despite the wider application of group contribution approach in the engineering calculations, it is still an approximate method to estimate the properties. On the contrary, water is a widely studied compound and the physico-chemical properties of such temperature and concentration are well documented. Because of the accurate availability, the route of estimation of water through group contribution pathway has been avoided and instead the physico-chemical properties of the juice has been calculated with weighted molar sum of each component of pure pulp and water using Eqs. 1, 4 and 9. However, for the other concentrations i.e., 21–45°Brix and within the above-mentioned respective temperature range of physicochemical properties, the %AADs are provided in Table 6. The estimated %AADs for density, thermal conductivity and heat capacity for all concentrations are less than 2%, 5% and 10% respectively. Acceptable accuracy of the estimated properties led to determine the estimated parameters of the selected physico-chemical properties.

Table 5 Experimental and estimated dataset of physico-chemical properties at 13.6°Brix
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Table 6 Concentration impact on estimated physico-chemical properties
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Figure 2 represents the density of black grape juice over the temperature range from 274 to 339 K and concentration range of 13.6–45°Brix. Herein, 35 experimental density data points are compared with the estimated densities over the same temperature and concentration range. Temperature and concentration variations show distinct effects on density. It is evident from Fig. 2 that with the increase in temperature for the same concentration, a significant decrease in value of density is observed. However, with increase of concentration at the same temperature, a significant increase in density values is observed. The reason for the temperature effect can be subjected to an increase of volume owing to its linear dependence on temperature thereby a decrease in the value of density. For the case of concentration effect an increase in density can be attributed to an increase in the total mass of the black grape juice concentrate [24]. The similar trend has been presented previously for passion fruit juice [13], clear grape juice [18], apple juice [15], malus floribunda juice [14], grape juice [53], mango pulp[54], Thai seedless guava juice [55], peach juice [56] and orange juice [57].

Fig. 2
figure 2

Experimental and estimated density values at different concentrations (°Brix)

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With the estimation of the e1-7 and β1-2 parameters, the thermal conductivity of black grape juice over the stated temperature range and 13.6–45°Brix concentrations are represented in Fig. 3. The impact of variations of temperature and concentration on thermal conductivity are depicted in Fig. 3. Herein, 40 experimental thermal conductivity data points are compared with the estimated thermal conductivities over the same temperature and concentration range. A directly proportional relationship is observed between temperature and thermal conductivity values. This effect of temperature is due to an increase in kinetic energies of particles at higher temperatures which in turn causes their robust movements thereby leading to greater collisions. These collisions intensify the thermal energy transfer resulting in higher thermal conductivity values. Also, a decrease in thermal conductivity values with an increase in concentration can be credited to the disruption in the alignment of solvent molecules due to addition of more solute particles at higher concentrations and thereby hindering the heat transfer by interfering the flow of heat as solute particles act as obstacles. These thermal conductivity variations are analogous to those presented for passion fruit [13], orange juice [16, 57], grape juice [17, 22, 53], mango pulp [54], apple juice [15] and Thai seedless guava juice [55] indicating a decrease and increase in thermal conductivity values with an increase in concentration and temperature respectively.

Fig. 3
figure 3

Experimental and estimated thermal conductivity values at different concentrations (°Brix)

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The estimated and experimental heat capacity of black grape juice over the temperature range of 274 to 319 K and 13.6–45°Brix concentrations with the estimated Clp0(T) parameters are represented in Fig. 4. Herein, 30 experimental heat capacity data points are compared with the estimated heat capacities over the same temperature and concentration range. Distinct effects on heat capacity estimation with varying temperature and concentration are represented in Fig. 4. It is observed that when graphs of heat capacity vs temperature are plotted for different concentrations between the experimental and estimated values the nature of the trendline does not converge well. However, it is also evident that the heat capacity values tend to increase slightly with the increase in temperature and this can be attributed to an increase the ability of atoms or molecules of the components to store higher level of thermal energy, thereby resulting in increase heat capacity values. On the other hand, a decrease in heat capacity values is observed with an increase with concentration owing to a decrease in the solvent composition which in this case is water having a higher individual heat capacity as compared to various solute components. Similar results have been previously obtained for passion fruit juice [13], grape juice [23, 53], apple juice [15], mango pulp [54] and orange juice [57] depicting a slight increase and decrease in heat capacity values with increasing temperature and concentration respectively.

Fig. 4
figure 4

Experimental and estimated heat capacity values at different concentrations (°Brix)

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Herein, we have adopted a strategy of estimating the physico-chemical properties at each concentration separately. This strategy is adopted since heat capacity and thermal conductivity contains large number of adjustable parameters as well as there is significant contribution of water in the black grape juice, however, we have tried to divide the dataset in various combinations of training and testing. The training dataset would estimate the parameters followed by the testing of parameters with respect to the other experimental dataset. However, this methodology is producing higher deviations because of the above-mentioned reasons. Hence, the parameters we obtain at various concentrations are given in Table S5 and S6 of supplementary material. Later the parameters are nonlinearly regressed with respect to the concentration. This method has produced a set of nonlinear equations for each parameter. They are depicted in Table 7. Hence, the parameters for any unknown concentration of black grape juice can be estimated from the nonlinear equations. Such nonlinear equations are a balanced approach for the estimation of physico-chemical properties of black grape juice and can be extended to other types of juices.

Table 7 Genetic algorithm defined physicochemical property estimation equations
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3.1 Comparison of established framework with the literature

There are some modelling aspects of physico-chemical properties of grape juice reported in literature. A series of experimental measurement of density, viscosity and coefficient of thermal expansion are performed varying the temperature from 293–353 K and 22.9–70.6°Brix concentration [18]. The obtained dataset is modelled through various orders of polynomials incorporating Levenburg-Marquardt formulation. The fitted density is later used to estimate the coefficient of thermal expansion and rheological models determine the viscosity. Such estimations are performed using Microsoft Excel software. Microsoft Excel driven spread sheets are also used to estimate the density, thermal conductivity, dynamic viscosity, thermal diffusivity and thermal capacity using wide range of experimental grape juice data varying the dry matter content of grape juice and the temperature [22]. Rheological models are used to fit the rheological data of Merlot grape juice over 13.6–45°Brix [58]. The statistical analysis over experimental measurement of density, thermal conductivity and heat capacity has been performed in Minitab 15 software. Similar set of analysis has been carried out for the Cabernet Sauvignon grape juice [53]. Although a limited number of modelling study available for grape juice, there is no modelling framework is reported specifically for black grape juice. Further the modelling studies are merely curve fitting heavily relies the experimental dataset. Herein, our developed approach based on the molecular structure stands alone. It connects the molecular structure to the physico-chemical properties of black grape juice. Once it is established, it will navigate the attention of researchers to look into the exact composition of various species of grapes and in broader sense of other types of fruits. Thus the focus of having experimental driven correlative model will change towards the structure driven a-priori modelling.

3.2 Limitation of modelling framework

Despite the satisfactory estimation of physico-chemical properties and creation of a series of correlative formulations for the black grape juice, the framework has its own drawbacks. The primary drawback is the framework is dependent on the accurate identification of components of black grape juice and their compositions. Very limited number of literatures are available in this regard. Such detailed compositions may not be available for other types of grapes as well as similar methodology cannot be extrapolated for other types of fruits because of the non-availability of list of all compounds and literatures. The group contribution parameters have wider applications and therefore, the values are not exact in all cases. However, the accountability in various applications leads to the acceptance in multiple engineering applications. Additionally, water plays a key role in the juice formulations. However, available physico-chemical properties of water are added to the estimated properties of other components. This strategy may not work for other variation of grapes as well as other types of fruit juices. Despite all these limitations, this approach successfully correlates the structural configuration of fruit components with the physico-chemical properties and thus opens a new window of modelling in juice processing.

4 Conclusion

The predictive formulation in the group contribution methodology successfully creates framework for the estimation of density, thermal conductivity and heat capacity for Malbec Vitis Vinifera species of black grape juice. The model parameters are estimated and validated over a common concentration range of 13.6–45°Brix for stated physico-chemical properties. Deviations less than 2% AAD for density and 5% AAD for thermal conductivity for a temperature range of 274–339 K whereas below 10% AAD for heat capacity over 274–319 K temperature range are observed when compared to the experimental dataset. Non-linear equations are generated to predict essential property parameters for density, thermal conductivity and heat capacity estimation. The frameworks take variation of temperature, molecular structure and concentration into the consideration. Hence, these frameworks can be viewed as initiation of new era in the process optimization, equipment designing and packaging applications during various stages of grape juice production process without the need of actually carrying out costly, time and resource consuming experiments. This study demonstrates the use of group contribution method and genetic algorithm for fruit juice property estimation and could be used as the basis for other food matrices.