Question 1. Estimating the time of a victim’s death during homicide investigations is a complex problem that cannot be solved by analysising simple equations or functions of one variable. However, many mathematical texts examine time of death estimation based around analysis of Newton’s Law of Cooling. Such analysis is based on implicit simplifying assumptions that: the only dependent variable of interest in determining the time of death is the victim’s body temperature, T(t); the victim’s baseline body temperature when alive, T0, is known; and the air temperature of the victim’s surroundings, Ts, is constant. Here we will examine such a problem.
(a) Assume that immediately following death, a victim’s body begins to cool from a standard healthy body temperature of 37◦ Celsius. Further, assume that experimental work has determined that the rate constant in Newton’s Law of Cooling for a human body is approximately k = 0.1947 when time t is measured in hours. Determine a function derived from Newton’s Law of Cooling, T(t), that models the temperature of a victim’s body t hours after death, assuming that the temperature of the body’s surroundings is a constant 15.5◦ Celsius.

(b) If the temperature of the victim’s body is now 22.2◦, how long ago was their time of death?

(c) If the victim’s body temperature at death had instead been 36.3◦ Celsius (within the range of normal body temperatures for a healthy adult), what time of death would be estimated via a Newton’s Law of Cooling model? By what duration does this estimate differ to the time that you determined in part (b)?

(d) In reality, how might the modelling assumptions made to address this problem be violated?

The diffusion of a molecule in a tissue is studied by measuring the uptake of labeled protein into the tissue of thickness L =149.4 um. Initially, there is no labeled protein in the tissue. At t=0, the tissue is placed in a solution with a molecular concentration of C1=1.2 uM, so the surface concentration at x=0 is maintained at C1. Assume the tissue can be treated as a semi-infinite medium. Surface area of the tissue is A = 93.9 cm2. Calculate the uptake of the labeled protein M by the tissue over 100 s.Please give your answer with a unit of pmol. Assuming the diffusion coefficient is known of D=1*10-9 cm2/s

Problem 2. : A superheated steam at 20 bars and 500^oC is fed to a turbine to generate electricity with a capacity of output of work at 500 kJ/kg. The existing steam is a saturated vapor at 2 bars. The surrounding environment temperature is 25^oC. The outer surface temperature of the turbine is 150^oC 1.) Clearly state your assumptions and draw the process flow diagram with labels ; 2.) Calculate the amount of heat exchanged between the system and surrounding ; 3.) Calculate the internal, external and total entropy generation for the turbine in kJ/kg-K and total entropy generation of the whole processes (turbine + surrounding environment) .

What is the equation for boil-up rate?

Fluid X is flowing in a circular pipe with a constant velocity ν (m/s). The fluid is cooled by a jacket kept at constant temperature, Tj. Fluid velocity is plug shaped , in other words uniform at radial positions. Assume that temperature is uniform in radial positions because of turbulent flow conditions; ρ and Cp of the fluid are constants. The inlet temperature (at z=0) is constant and uniform at To (To>Tj). Assume that thermal conduction of heat along the z axis is small relative to convection. Heat transfer film coeffiecient h is given as 40 (J/m2.°C.s).

Δz

Cooling jacket, Tj

v, To, ρ, Cp
R

z z=0

a. Perform an unsteady state energy balance using shell balance technique and obtain a PDE model. Do not solve.

b. Perform a steady state energy balance using shell balance technique to obtain an ODE model . Solve the model to find the steady state temperature distribution as a function of axial position z. Take Tref=0.

c. Given that; at t=0, To=90°C; ν=0.5 m/s; h=40 J/(m2.°C.s); R=0.1m; ρ=100 kg/m3; Tj=10 °C and Cp=10J/(kg°C); find the temperature value at z=1 m? (Ans.= 26 °C)

Predict trends in glass transition and melting temperatures. Why are these properties considered in the design of consumer products like plastic cups and ice trays?

explain population growth in terms of ode

Change in concentration of salt in a reactor can be modeled
by the following equation y’ = ty+y1/2
Initial concentration of salt at time (t= 0 hours) is 1g/l, y(0)=
1, find the concentration of salt after 0.2 hour y(0.2)= ? by
using Euler, Heun and RK4 methods. Use h= 0.1

4. (20) A stoichiometric mixture of CO and O₂ enters a reactor at 100 ^oC. CO₂, CO and O₂ leave the reactor in chemical equilibrium at 1527 ^oC and 2 atm. Calculate the partial pressures of CO₂, CO and O₂ in the products.

CO + ½ O₂ = CO₂:                  ΔG^o_373K = -250.8 kJ

CO + ½ O₂ = CO₂:                  ΔG^o_1800K = -127.4 kJ

The reaction rate constants for an investigation of ascorbic acid deterioration in a multivitamin mix during storage at different temperatures were:

(i) Determine the activation energy at 70 oC

(ii) Determine the rate constant at 80 oC

(iii) Suppose the temperature increased from 20 oC to 70 oC, determine the sensitivity of the reaction rate to this rise is temperature.

(iv) At what time would the ascorbic acid deteriorate by 50%?

A binary mixture of benzene and toluene will be separated by continuous distillation. The mixture consists of 20% mol of benzene and 80% mol of toluene at the bubble point of 91ºC. The binary mixture is available at 100 mol h-1 and the separation should deliver a distillate product that has a concentration of 90% mol of benzene and that contains half of the benzene in the feed stream. a) Plot the equilibrium line for benzene. b) Determine the minimum reflux ratio. b) Determine the minimum number of stages required to achieve this separation. Note: You may require looking for properties of components◢

volatilities of components need to looked up

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