Brewing Filter Coffee: Mathematical Model of Coffee Extraction
Modelling Camp, ICMS March 24, 2016
Modelling Camp, 2016
The Problem
Modelling Camp, 2016
Outline
I
Examining the concentration of granules vs coffee in solution.
I
Model the flow through the coffee-bed.
I
Simplify model of extraction with advection in the filter.
I
Exciting Results!
Modelling Camp, 2016
Outline
I
Examining the concentration of granules vs coffee in solution.
I
Model the flow through the coffee-bed.
I
Simplify model of extraction with advection in the filter.
I
Exciting Results!
Modelling Camp, 2016
Outline
I
Examining the concentration of granules vs coffee in solution.
I
Model the flow through the coffee-bed.
I
Simplify model of extraction with advection in the filter.
I
Exciting Results!
Modelling Camp, 2016
Outline
I
Examining the concentration of granules vs coffee in solution.
I
Model the flow through the coffee-bed.
I
Simplify model of extraction with advection in the filter.
I
Exciting Results!
Modelling Camp, 2016
Outline
I
Examining the concentration of granules vs coffee in solution.
I
Model the flow through the coffee-bed.
I
Simplify model of extraction with advection in the filter.
I
Exciting Results!
Modelling Camp, 2016
Variables
Basic Variables: Cc :=
mg mc , Cg := , Vθ 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
Basic Equations Equation of transport of coffee for constant density of water at a certain temperature: dCc = α(1 − θ)(Cg − Gλ)(S − Cc ) − (vw · ∇Cc ) dt Conservation of coffee granules d θCc + (1 − θ)Cg = 0 dt =⇒ 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
Basic Equations Equation of transport of coffee for constant density of water at a certain temperature: dCc = α(1 − θ)(Cg − Gλ)(S − Cc ) − (vw · ∇Cc ) dt Conservation of coffee granules d θCc + (1 − θ)Cg = 0 dt =⇒ 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
Basic Equations
Equation describing the coffee concentration within the granules: dCg = −θα(Cg − Gλ)(S − Cc ) dt
Modelling Camp, 2016
Basic Equations
Equation describing the coffee concentration within the granules: dCg = −θα(Cg − Gλ)(S − Cc ) dt
Modelling Camp, 2016
Dimensionless System
Dimensionless system without advection: ec dC e g − λ)(1 − C ec ) = B(1 − θ)G(C dt eg dC e g − λ)(1 − C e c ), = −θ(C dt e c := Cc , C e g := Cg , et = t/T and B = G/S where C S G
Modelling Camp, 2016
Dimensionless System
Dimensionless system without advection: ec dC e g − λ)(1 − C ec ) = B(1 − θ)G(C dt eg dC e g − λ)(1 − C e c ), = −θ(C dt e c := Cc , C e g := Cg , et = t/T and B = G/S where C S G
Modelling Camp, 2016
Results for the concentrations e c = B(1 − θ) (1 − C eg ) C θ −jet e g = λθ + (1 − λ)(θ − B(1 − θ))e , C θ + (1 − λ)(1 − θ)Be−jet where ej = 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) k q = − ∇P µ
Figure: x = Lu , y = h(u)v . Modelling Camp, 2016
Pressure-Velocity Pressure: P = ρw gy
H − 1 + P0 h(x)
−κ Velocity: vy = ρw g θµ
Modelling Camp, 2016
H −1 h(x)
Rotating the Problem
Pressure: P = ρw gh−1 y 0 (H − x 0 sin(φ) − h(x 0 ) 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:
1 h
Z
ec = 1 C h
Z
eg = 1 C h
Z
h
Cc dz 0 h
Cg dz 0
h
(∇ · vw Cc ) dz = vw (Cc (h) − Cc (0)) 0
ec = −vw C Mean-field approximation: 1 h Modelling Camp, 2016
Z
h
ec , C ev ) f (Cc , Cg )dz ≈ f (C 0
e c and C eg Average C Average over volume using mean-field argument: Z 0
h
Z h ∂Cc + ∇ · (Cc vw ) dz = α(1 − θ)(Cg − Gλ)(S − Cc )dz ∂t 0 ˆc ∂C ˆ c = α(1 − θ)(C ˆ g − Gλ)(S − C ˆc) − vw C ∂t
Z 0
Modelling Camp, 2016
h
Z h ∂Cg −αθ(Cg − Gλ)(S − Cc )dz dz = ∂t 0 ˆg ∂C ˆ g − Gλ)(S − C ˆc) = −αθ(C ∂t
Illustration of the solution with advection Cc -blue curve, Cg red curve
Modelling Camp, 2016
Brewing Contral Chart Comparison
Modelling Camp, 2016
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
Conclusions and future research
I
We developed a basic model, which for a given geometry of the coffee bed predicts quality of the coffee
I
More coffee is extracted at the top of the filter rather than at the bottom due to the lower pressure and lower velocity
I
An decrease in the angle of inclination of the filter leads to an increase in the concentration of coffee in the solution
I
Our model predicts the height of the coffee bed along the filter should be in the range 0.8 < h < 1 cm
I
Straightforward extensions: 3D axisymmetric model, variable h
I
Further improvements: consider the process of a coffee bed deformation and chemical impact
Modelling Camp, 2016
