PROBLEM #1: The production of ethanol and the consumption of sugars from an enzymatic mixture (of several enzymes) that

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Problem 1 The Production Of Ethanol And The Consumption Of Sugars From An Enzymatic Mixture Of Several Enzymes That 1 (68.94 KiB) Viewed 19 times

Problem 1 The Production Of Ethanol And The Consumption Of Sugars From An Enzymatic Mixture Of Several Enzymes That 2 (59.78 KiB) Viewed 19 times

PROBLEM #1: The production of ethanol and the consumption of sugars from an enzymatic mixture (of several enzymes) that participate in the decomposition of fruits and that consume the sugars contained in them are being studied. For this, the fruit is taken and blended to make a reaction broth. Some spherical (filled) particles are immersed in the reaction broth that serve as a support for the enzymatic mixture. In principle, it is desired to know the relationship between the concentration of sugars and the radial position of the spherical particles in a steady state, assuming that only radial diffusion of the sugars occurs. Also assume that sugars are measured as glucose equivalent, since as glucose is consumed, one of the enzymes in the enzyme mixture produces more glucose from the sugars in the reaction broth so that the concentration of glucose in the reaction broth remains constant. For reaction purposes, the following (simplified) reaction is used, which is considered, for the study region, as an irreversible first-order reaction with respect to glucose: FIND: C6H12062CH3 - CH₂OH + 2C02 1- Find the differential model that describes the relationship between the glucose concentration and the radial position in the spherical particle, using the methodological and scientific principles seen in class. You must develop all the parts, from the diagram, definition of variables, to the constitutive relations, conditions of the model and development of the conservation law(s) necessary for the solution. 2- Integrate the differential model analytically and show that the final model is equivalent to the one described below: Where: C = mx senh(₁ k DABXr) k k -DABXRX √DEX COB ( √D XR) + DA B x sent (√x R) + KX RX sen (√(TR). cosh h > BX > XR DA, B DA, B DA, B DA, B =m-1 KXC₁XR²
3- From the model found in point 1 above, solve the differential equation by the Runge-Kutta numerical method of order 4 from r = 0 to r= 1.5 cm, with a step of 0.1 cm and an initial point at r = 0, C = 0.0247 mol/m³. 4- From the model obtained in point 2 above, calculate the value of the concentration of glucose from r≈ 0 to r = 1.5 cm, with a step of 0.1 cm. 5- Compare the results of both models, points 3 and 4 above, using a graph that contains the analytical model (with a continuous line point 2 and point 4) and the values found numerically (point 3). Calculate the error in the numerical solution (point 3) and the analytical solution (point 4). DATA: Mass diffusivity of glucose in the sphere material, DAR = 0,0001 m²/min Radius, R = 1,5 cm Reaction rate coefficient, k = 0,08 s1 Mass transfer coefficient, K = 0,1 m/s Glucose concentration in the medium surrounding the sphere, C =0,1 mol/m³ Important note: For each problem, you must develop each part of the approach and solution of the problem, as seen in class, that is: diagram, definition of the control volume, definition of variables (independent, dependent, fixed, parameters), model conditions, constitutive relations, application of the conservation law(s) necessary to find the solution, boundary conditions, simplification and solution of the differential equation (must show, at least, the most important steps for the solution of the differential equation), simplify the integral equation (if necessary), find the constants of integration, perform model checking (at least check dimensional consistency and test a boundary condition).