Question

Who doesn’t love a good espresso? But as every great barista knows, the key is all about permeability. The ideal espresso double shot is 60 ml, extracted at a brewing temperature of 94 °C and a pressure of 9 bars, and takes 26 seconds to “pull”. The standard diameter of the brewhead (and therefore of the coffee “puck” through which the water flows) is 58 mm. The “puck” is typically 7/8” thick.

1. What are the hydraulic conductivity and permeability of the ideal espresso puck? Be mindful that fluid properties (density and viscosity) are dependent on temperature, so you’ll need to do some digging for the right values.

2. In the case of the overzealous home barista, a common mistake is to grind the coffee too coarse or too fine. Suppose all of the above parameters are the same, except that the result is a measly 25 ml espresso shot that takes 57 seconds to produce. What’s the permeability of that puck, and what should be changed to fix the problem?  

Answer 1

A) Answer: every calculation is in SI unit.

Viscosity of coffee, u= exp[1.141-22.42w-17.78w²- (973.1-17923w²)/(T+273.15)] for w= concentration = 0.5-0.8

Considering higher concentration w=0.8 and temperature 94° = 367 K,

we get u= 1.52

A= pi (58) sqmm = pi*(0.056)² sqm = ; p1-p2 = (9-1.01325) bar = 7.98675 bar = 798675 Pa.

Q= 60 ml = 6*10^-5 ; L= 7/8" = 8 " = 0.2032 m

Now, from darcy's law, we know, hydraulic conductivity k= Q*u*L / t*A* (P1-P2)

k = (6*10^-5 * 1.52 * 0.2032 ) / (26 * 3.141 * 0.003136 * 798675 ) = 9.06 x 10^-11 (Answer A.1)

density of coffee at 94 ° d= 951.843 kg/m³

Now, we know permeability = (K* u ) / (d*g) = (9.06 x 10^-11 x 1.52) / (951843 * 9.81 ) = 1.4748 x 10^-17 (Answer A.2)

B) permeability = (Q/t) * (u²*L / A * p1-p2 * d*g) = 6.34 x 10^-12 where Q= 2.5e-5, and t=57 sec

To fix the problem, pressure should be reduced.