Brewing Filter Coffee: Mathematical Model of Coffee Extraction - PowerPoint PPT Presentation

Brewing Filter Coffee: Mathematical Model of Coffee Extraction Modelling Camp, ICMS March 24, 2016

Modelling Camp, 2016

The Problem Modelling Camp, 2016

Outline ◮ Examining the concentration of granules vs coffee in solution. ◮ Model the flow through the coffee-bed. ◮ Simplify model of extraction with advection in the filter. ◮ Exciting Results! Modelling Camp, 2016

Outline ◮ Examining the concentration of granules vs coffee in solution. ◮ Model the flow through the coffee-bed. ◮ Simplify model of extraction with advection in the filter. ◮ Exciting Results! Modelling Camp, 2016

Outline ◮ Examining the concentration of granules vs coffee in solution. ◮ Model the flow through the coffee-bed. ◮ Simplify model of extraction with advection in the filter. ◮ Exciting Results! Modelling Camp, 2016

Outline ◮ Examining the concentration of granules vs coffee in solution. ◮ Model the flow through the coffee-bed. ◮ Simplify model of extraction with advection in the filter. ◮ Exciting Results! Modelling Camp, 2016

Outline ◮ Examining the concentration of granules vs coffee in solution. ◮ Model the flow through the coffee-bed. ◮ Simplify model of extraction with advection in the filter. ◮ Exciting Results! Modelling Camp, 2016

Variables Basic Variables: m g C c := m c V θ, C g := V ( 1 − θ ) , where: C c represents the concentration of the coffee in water C g the concentration of the coffee granules θ is the porosity of the coffee Modelling Camp, 2016

Basic Equations Equation of transport of coffee for constant density of water at a certain temperature: dC c = α ( 1 − θ )( C g − G λ )( S − C c ) − ( v w · ∇ C c ) dt Conservation of coffee granules � � d θ C c + ( 1 − θ ) C g = 0 dt = ⇒ θ C c + ( 1 − θ ) C g = ( 1 − θ ) G G is the starting concentration of granules, and S is the maximum concentration of dissolved coffee, α is the extraction rate. Modelling Camp, 2016

Basic Equations Equation of transport of coffee for constant density of water at a certain temperature: dC c = α ( 1 − θ )( C g − G λ )( S − C c ) − ( v w · ∇ C c ) dt Conservation of coffee granules � � d θ C c + ( 1 − θ ) C g = 0 dt = ⇒ θ C c + ( 1 − θ ) C g = ( 1 − θ ) G G is the starting concentration of granules, and S is the maximum concentration of dissolved coffee, α is the extraction rate. Modelling Camp, 2016

Basic Equations Equation describing the coffee concentration within the granules: dC g = − θα ( C g − G λ )( S − C c ) dt Modelling Camp, 2016

Basic Equations Equation describing the coffee concentration within the granules: dC g = − θα ( C g − G λ )( S − C c ) dt Modelling Camp, 2016

Dimensionless System Dimensionless system without advection: d � C c = B ( 1 − θ ) G ( � C g − λ )( 1 − � C c ) dt d � C g = − θ ( � C g − λ )( 1 − � C c ) , dt C c := C c C g := C g where � , � G , � t = t / T and B = G / S S Modelling Camp, 2016

Dimensionless System Dimensionless system without advection: d � C c = B ( 1 − θ ) G ( � C g − λ )( 1 − � C c ) dt d � C g = − θ ( � C g − λ )( 1 − � C c ) , dt C c := C c C g := C g where � , � G , � t = t / T and B = G / S S Modelling Camp, 2016

Results for the concentrations C c = B ( 1 − θ ) � ( 1 − � C g ) θ C g = λθ + ( 1 − λ )( θ − B ( 1 − θ )) e − j � t � , θ + ( 1 − λ )( 1 − θ ) Be − j � t where � j = θ − ( 1 − λ ) B ( 1 − θ ) Modelling Camp, 2016

Flow Through the Coffee-Bed Darcy’s law describes the flow of water through the coffee (porous medium) q = − k µ ∇ P Figure: x = Lu , y = h ( u ) v . Modelling Camp, 2016

Pressure-Velocity � H � Pressure: P = ρ w gy h ( x ) − 1 + P 0 � H � Velocity: v y = − κ θµ ρ w g h ( x ) − 1 Modelling Camp, 2016 Figure: Pressure Distribution.

Rotating the Problem Pressure: P = ρ w gh − 1 y ′ ( H − x ′ sin ( φ ) − h ( x ′ ) cos ( φ )) Modelling Camp, 2016

Rotating the Problem Figure: Pressure distribution at inclination angle 45, 30, 60 respectively Modelling Camp, 2016

Mean-field Approximation Average over coffee bed height: � h C c = 1 � C c dz h 0 � h C g = 1 � C g dz h 0 � h 1 ( ∇ · v w C c ) dz = v w ( C c ( h ) − C c ( 0 )) h 0 = − v w � C c Mean-field approximation: � h 1 dz ≈ f ( � C c , � f ( C c , C g ) C v ) h 0 Modelling Camp, 2016

Average � C c and � C g Average over volume using mean-field argument: � h � ∂ C c � � h + ∇ · ( C c v w ) dz = α ( 1 − θ )( C g − G λ )( S − C c ) dz ∂ t 0 0 ∂ ˆ C c − v w ˆ C c = α ( 1 − θ )(ˆ C g − G λ )( S − ˆ C c ) ∂ t � h � h ∂ C g ∂ t dz = − αθ ( C g − G λ )( S − C c ) dz 0 0 ∂ ˆ C g = − αθ (ˆ C g − G λ )( S − ˆ C c ) ∂ t Modelling Camp, 2016

Illustration of the solution with advection C c -blue curve, C g red curve Modelling Camp, 2016

Brewing Contral Chart Comparison Modelling Camp, 2016

Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016

Conclusions and future research ◮ We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee ◮ More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity ◮ An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution ◮ Our model predicts the height of the coffee bed along the filter should be in the range 0 . 8 < h < 1 cm ◮ Straightforward extensions: 3D axisymmetric model, variable h ◮ Further improvements: consider the process of a coffee bed deformation and chemical impact Modelling Camp, 2016