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Brewing Filter Coffee: Mathematical Model of Coffee Extraction - - PowerPoint PPT Presentation
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
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The Problem
Modelling Camp, 2016
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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
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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
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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
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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
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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
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Variables
Basic Variables: Cc := mc Vθ, Cg := mg V(1 − θ), where: Cc represents the concentration of the coffee in water Cg the concentration of the coffee granules θ is the porosity of the coffee
Modelling Camp, 2016
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Basic Equations
Equation of transport of coffee for constant density of water at a certain temperature: dCc dt = α(1 − θ)(Cg − Gλ)(S − Cc) − (vw · ∇Cc) Conservation of coffee granules d dt
- θCc + (1 − θ)Cg
- = 0
= ⇒ θCc + (1 − θ)Cg = (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
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Basic Equations
Equation of transport of coffee for constant density of water at a certain temperature: dCc dt = α(1 − θ)(Cg − Gλ)(S − Cc) − (vw · ∇Cc) Conservation of coffee granules d dt
- θCc + (1 − θ)Cg
- = 0
= ⇒ θCc + (1 − θ)Cg = (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
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Basic Equations
Equation describing the coffee concentration within the granules: dCg dt = −θα(Cg − Gλ)(S − Cc)
Modelling Camp, 2016
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Basic Equations
Equation describing the coffee concentration within the granules: dCg dt = −θα(Cg − Gλ)(S − Cc)
Modelling Camp, 2016
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Dimensionless System
Dimensionless system without advection: d Cc dt = B(1 − θ)G( Cg − λ)(1 − Cc) d Cg dt = −θ( Cg − λ)(1 − Cc), where Cc := Cc S , Cg := Cg G , t = t/T and B = G/S
Modelling Camp, 2016
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Dimensionless System
Dimensionless system without advection: d Cc dt = B(1 − θ)G( Cg − λ)(1 − Cc) d Cg dt = −θ( Cg − λ)(1 − Cc), where Cc := Cc S , Cg := Cg G , t = t/T and B = G/S
Modelling Camp, 2016
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Results for the concentrations
- Cc = B(1 − θ)
θ (1 − Cg)
- Cg = 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
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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
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Pressure-Velocity
Pressure: P = ρwgy H h(x) − 1
- + P0
Velocity: vy = −κ θµ ρwg H h(x) − 1
- Figure: Pressure Distribution.
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Rotating the Problem
Pressure: P = ρwgh−1y′ (H − x′ sin(φ) − h(x′) cos(φ))
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Rotating the Problem
Figure: Pressure distribution at inclination angle 45, 30, 60 respectively
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Mean-field Approximation
Average over coffee bed height:
- Cc = 1
h h Cc dz
- Cg = 1
h h Cg dz 1 h h (∇ · vwCc) dz = vw (Cc(h) − Cc(0)) = −vw Cc Mean-field approximation: 1 h h f(Cc, Cg) dz ≈ f( Cc, Cv)
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Average Cc and Cg
Average over volume using mean-field argument: h ∂Cc ∂t + ∇ · (Ccvw)
- dz =
h α(1 − θ)(Cg − Gλ)(S − Cc)dz ∂ ˆ Cc ∂t − vw ˆ Cc = α(1 − θ)(ˆ Cg − Gλ)(S − ˆ Cc) h ∂Cg ∂t dz = h −αθ(Cg − Gλ)(S − Cc)dz ∂ ˆ Cg ∂t = −αθ(ˆ Cg − Gλ)(S − ˆ Cc)
Modelling Camp, 2016
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Illustration of the solution with advection
Cc-blue curve, Cg red curve
Modelling Camp, 2016
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Brewing Contral Chart Comparison
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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
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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
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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
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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
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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
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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
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