1. 1
Reducing Blood Alcohol Concentration to Driving Level
Barbara Kaufmann
Yifang Wang
Zhuoming Li
Executive Summary
This project was about creating a model to best describe the time needed for a man or woman to
reduce blood alcohol level to driving level, which is 1g/liter. The simplest model was the one
that involved only one system, the blood system. This system was considered as a batch system.
The reaction was assumed to follow the first order reaction. Then this model was analyzed by
applying an overall mass balance and a molar balance on alcohol. The results from this model
indicated that this model is not the best to describe the time needed for a man or woman to
reduce blood alcohol to driving level. A better model was the one that involved two systems,
which were the GI system and blood system. These two systems were considered as
continuously-stirred tank reactor (CSTR) systems and were analyzed by calculating
dimensionless groups, overall mass balance and molar balance on alcohol. The results indicated
that it was the better model to describe the project topic.
Introduction
This project is to study the relationship between Blood Alcohol Concentration and time after
drinking. A batch model was used to calculate the time needed to reduce Blood Alcohol

2. 2
Concentration within a human body to accepted driving level after 40 gr of alcohol, which is
about 2*12’’ of cans of 5.5% beer, or 10’’ of 13% wine, was quickly assimilated by his or her
blood system. The batch model assumed all the alcohol was absorbed quickly and spread out
evenly into the blood system. As per the instructions, the first model established was a 1st order
model with a simple constant reaction rate. This result was an approximate value of time that
was not close enough to the authentic case. However, though the 1st order model, the basic
concept of how the Blood Alcohol Concentration was changing with time was obtained. Based
upon the highly oversimplified 1st order prototype, the GI tract system (into which 40 gr of
alcohol enters) was added into the model. After the GI tract was added, the whole system was
composed of two CSTR systems. It worked as the alcohol flows from the GI tract system into the
Blood Fluid system and eventually is eliminated from the human body. The most important
parameters for this project to be concerned about are: the amount of moles of alcohol that enters
the human body, the concentration of alcohol in blood, and the reaction rate of the alcohol that is
reduced in the blood. After those key parameters were found, the numbers were plugged into the
models step by step. First, all the dimensionless groups were defined to start the basic analysis
for all the parameters, and then the overall mass and molar component balances were solved to
actually figure the connection among the core parameters. Then, all the equations and unknowns
were listed out, and eventually the time needed was computed after plugging in the numbers.
Analysis
Dimensional Analysis
The parameters thought to be relevant and their basic units are listed in Table 1.
Table 1. Parameters used in dimensional analysis.
Parameter (core parameters Parameter Description Basic Units (t is time, n is

3. 3
bolded) moles, l is length)
kblood Reaction constant 1/t
KGI Flow rate constant 1/t
CGI Concentration of alcohol in
gastrointestinal system
n/l3
Cblood Concentration of alcohol in
blood
n/l3
Tau Time t
VGI Volume of GI l3
Vblood Volume of blood l3
nGI Moles of alcohol in GI n
nblood Moles of alcohol in blood n
Balance Equations
The simplified process described in Part B involved one system, the blood. It was a batch
process.
Equation 1: Simplified Overall Mass Balance
d/dt ( 𝜌blood*Vblood) = 0
Equation 2: Simplified Alcohol Mole Balance
d/dt (Cblood*Vblood) = -kblood*Cblood*Vblood

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The two-compartment process described in Part C involved two systems, the GI and the blood. It
was a well-mixed flow process.
Equation 3: GI Overall Mass Balance
d/dt ( 𝜌GI*VGI) = -KGI * VGI* 𝜌GI
Equation 4: Blood Overall Mass Balance
d/dt ( 𝜌blood*Vblood) = KGI * VGI* 𝜌GI
Equation 5: GI Alcohol Mole Balance
d/dt(CGI*VGI) = -KGI*(CGI-Cblood)*VGI
Equation 6: Blood Alcohol Mole Balance
d/dt (Cblood*Vblood) = -kblood*Cblood*Vblood +KGI*(CGI-Cblood)*VGI
Parameter Values and Boundary Conditions
For a 70 kg man, Vblood = 5.25 L, and for a 60 kg woman, Vblood = 4.5 L [1]. kblood = 0.2/h, and
KGI = 1.5/min, as given in the problem statement. For a man, VGI = (0.022L/ kg body)*70kg
body = 1.54 L. For a woman, VGI = (0.022L/ kg body)*60kg body = 1.32 L. These values were
assumed constant. The concentration of alcohol is (mass/V) * (1/molar mass). Boundary
conditions were as follows in Table 2.
Table 2. Initial and final conditions for balance equation parameters for each model and gender.
Model Parameter Initial/Final Gender Value
Simplified Cblood Initial Male (40g/5.25
L)*(mol/46.07g) =
0.1654 M
Both Cblood Final Both (1g/L)*(mol/46.07g

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)= 0.02171 M
Simplified Cblood Initial Female (40g/4.5
L)*(mol/46.07g) =
0.1929 M
Two-compartment Cblood Initial Both 0 M
Two-compartment CGI Initial Male (40g/1.54 L)*(mol/
46.07g) = 0.5638 M
Two-compartment CGI Initial Female (40g/1.32 L)*(mol/
46.07g) = 0.6350 M
Results
Dimensionless Groups
Pi1: kblood*nblood
a * Vblood
b * Tauc [=] t-1 * na * l3b * tc = t0 * n0 * l0
n: a = 0
l: 3b = 0, b = 0
t: -1+c = 0, c = 1
Pi1 = kblood*Tau
Similarly, Pi2 = KGI*Tau
Pi3: CGI * nblood
a * Vblood
b * Tauc [=] n*l-3 * na * l3b * tc = n0 * l0 * t0
n: 1+a = 0, a = -1

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l: -3+3b = 0, b = 1
t: c = 0
Pi3 = CGI * Vblood / nblood
Similarly, Pi4 = Cblood * Vblood / nblood
Pi5: VGI * nblood
a * Vblood
b * tc [=] l3 * na * l3b * tc = l0 * t0 * n0
n: a = 0
l: 3+3b = 0, b = -1
t: c = 0
Pi5 = VGI / Vblood
Pi6: nGI * nblood
a * Vblood
b * tc [=] n*na * l3b * tc = n0 * l0 * t0
n: 1+a = 0, a = -1
t: c = 0
l: 3b = 0, b = 0
Pi6 = nGI / nblood
Balance Equations
As stated, blood and GI volumes, kblood, and KGI were assumed constant. Both the simplified and
two-compartment models assumed alcohol degrades in the blood only as a first-order reaction.
Solving Equation 1 demonstrated that the total density of the simplified blood system is constant:
Vblood*d 𝜌blood / dt = 0, d 𝜌blood / dt = 0. Solving Equation 2 resulted in an expression for the
concentration of alcohol in the simplified blood system as a function of time:
Vblood*dCblood / dt = -0.2/h * Cblood*Vblood
∫
𝐶𝐶𝐶𝐶𝐶𝐶
𝐶𝐶𝐶𝐶𝐶𝐶,0
𝐶𝐶𝐶𝐶𝐶𝐶𝐶 / 𝐶𝐶𝐶𝐶𝐶𝐶 = ∫
𝐶
0
− 0.2/𝐶 𝐶𝐶
ln(Cblood / Cblood,0) = -0.2/h *t
Cblood = Cblood,0*exp(-0.2*t), where t is in hours
Substituting the target value, 0.0217 M, for Cblood allowed calculation of the time:

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Man: t = ln(0.0217 M / 0.1654 M) / -0.2 , tman = 10.15 h
Woman: t = ln(0.0217 M / 0.1929 M) / -0.2 , twoman = 10.92 h
Thus, for the simplified model, a man must wait 10.2 hours to drive, and a woman must wait
10.9 hours.
Solving Equations 3 and 4 provided expressions for density as functions of time:
d 𝜌GI / dt = -1.5/min * 𝜌GI , solve by separation of variables as demonstrated for Equation 2
𝜌GI = 𝜌GI,0*exp(-1.5*t), where t is in minutes
Substituting the Equation 3 solution into Equation 4:
Vblood * d 𝜌blood / dt = 1.5/min * VGI* 𝜌GI,0*exp(-1.5*t), solve by separation of variables
𝜌blood = 𝜌blood,0 - (VGI / Vblood )* 𝜌GI,0 *exp(-1.5*t), where t is in minutes
These functions indicated the total density of the GI decreases while that of the blood increases
over time. This is reasonable since material is flowing from the GI to the blood, yet the total
volumes of each were assumed constant.
Equations 5 and 6 were re-arranged to express the rate of change of concentration:
dCGI / dt = -1.5/min*(CGI-Cblood)
Woman: dCblood / dt = -0.015/min * Cblood + 0.44/min*(CGI - Cblood)
Man: dCblood / dt = -0.0175/min * Cblood + 0.44/min*(CGI - Cblood)
Equations 5 and 6 were coupled for both genders; that is, each equation depended on the other.
When a Laplace transform was applied to these two ordinary differential equations, they became
a system of algebraic equations The following expressions for Cblood were determined, where t is
in minutes:
Cblood,woman = (.2794*(-0.3844e-2*exp(-0.3844e-2*t)+1.951*exp(-1.951*t)))/(1.947)
Cblood,man = (.2481*(-0.1353e-1*exp(-0.1353e-1*t)+1.944*exp(-1.944*t)))/(1.93)
0.02171 M was substituted for Cblood , and the times for a woman and a man to drive were
solved: twoman = 1.298 min, tman = 1.218 min.

8. 8
Discussion
9 key parameters and 3 basic units were gained from the information provided, therefore 6 (9-3)
dimensionless groups were found: Pi1 = kblood*Tau, Pi2 = KGIk*Tau, Pi3 = CGI * Vblood / nblood, Pi4
= Cblood * Vblood / nblood, Pi5 = VGI / Vblood, Pi6 = nGI / nblood. According to the Buckingham Theory,
when there are six dimensionless groups, each one of them is a function of other parameters.
Thus, the following expressions were obtained:
1.The Concentration of alcohol in the simplified blood systemas a function of time:
Cblood = Cblood,0*exp(-0.2*t)
2.Density as functions of time:
𝜌GI = 𝜌GI,0*exp(-1.5*t)
3. The rate of change of concentration:
dCGI / dt = -1.5/min*(CGI-Cblood)
4.Cblood as function of time:
Cblood,woman = (.2794*(-0.3844e-2*exp(-0.3844e-2*t)+1.951*exp(-1.951*t)))/(1.947)
Cblood,man = (.2481*(-0.1353e-1*exp(-0.1353e-1*t)+1.944*exp(-1.944*t)))/(1.93)
For the 1st order model, the time was calculated to be 10.15 h for men and 10.92 h for women to
reduce Blood Alcohol Concentration to 1gr/lit. These numbers underestimated the rate of
reaction in a human’s body, while the values of 1.218 min for men and 1.298 min for women
were calculated using the model with the GI tract. These results clearly overestimated the ability
of a human to reduce the Blood Alcohol Concentration within their body.

9. 9
Summary
In order to create a model to best describe the time needed for a man or woman to reduce alcohol
concentration to driving level, which is 1g/liter, two models involving different process systems
were analysed. The simpler model involved the blood system only and it was considered as the
batch system. The control volume was the human body and this was considered as constant.
Since the person drank 40g alcohol quickly, it was assumed that the initial concentration of
alcohol was 0.1654 M for men and 0.1929 M for women. Applying an overall mass balance to
this model demonstrated that the density was constant. Applying a molar balance on alcohol and
assuming the reaction in the blood followed a first order reaction (k=0.021/hr) resulted in a
function that described how the concentration of alcohol in the blood changes in terms of time.
Substituting the target value of alcohol concentration (0.0217M) into this model had these
results: (1) the man with average mass (70kg) needed 10.15 hours to reduce alcohol
concentration from 0.165 M to 0.217 M. (2) the woman with average mass (60kg) needed about
11 hours to have the alcohol concentration reduced to driving level (0.217M).
This model is highly oversimplified and does not really apply. A better model is one that consists
of two compartments: the GI system and the blood system. Once the person drank 40g alcohol
into his or her body, the alcohol entered the GI first and transferred to the blood due to the
concentration difference between the GI and the blood. The rate for that was K=1.5/min. Also,
the reaction occurred only in the blood system. To begin with, the dimensional analysis was
performed before applying balance equations. As listed in the analysis part, there were a total of
nine parameters thought to be important to the system. Finally, there were six dimensionless
groups, which were shown in the result part. After the dimensional analysis was finished, the GI
system was analyzed by applying the overall mass balance. It was considered as a well mixed
CTSR system and the volume of the GI was assumed to be constant so that the density decreased
in the GI and increased in the blood from the overall mass balance. But when the molar balance

10. 10
on alcohol was analyzed, the concentration in the balanced equation was the concentration
between the GI and the blood. In addition, the initial alcohol concentration in the GI for a woman
was assumed to be 0.635M, while that for a man was assumed to be 0.564 M, and there was only
outflow in this system. Also, the outflow of the GI system was considered as the inflow of the
blood system. The rate in the blood was still 0.2/hr. By applying an overall mass balance and
molar balance on alcohol, the results were: (1) the woman (60kg) needed 1.298 min to have the
initial alcohol concentration (0.635M) reduced to the target value (0.0217M). (2) the man (70kg)
needed 1.218min to have the initial alcohol concentration (0.564M) reduced to the target value
(0.0217M).
In reality, the average time for people who drank to drive is about 45 mins [2]. So the results
from the second model were expected to be close to 45 mins. However, the theoretical results
were 1.298 mins for a woman (60kg) to reduce alcohol concentration from 0.635M to 0.027M
and 1.218 mins for a man (70kg), which had a large percent error by comparing the average time
in reality. In general, there were two reasons for that: (1) the first model was oversimplified. (2)
the second model underestimated the time due to several assumptions being made in the
calculations. Specifically, the possible reasons are shown below: (1) arithmetic error (2) the
metabolism rate of alcohol was not still 0.2/hr when the alcohol was transfered from the GI(3)
the GI volume and blood volume were not equal (4) the second model was not the right model to
describe the time needed for a man or woman to have alcohol concentration reduced to target
value.

11. 11
References
[1] Estimated blood volume calculator [Online]. Medscape, 1998. Available:
http://reference.medscape.com/calculator/estimated-blood-volume
[2] The truth about blood alcohol content [Online]. DrinkingAndDriving.org, 2008. Available:
http://www.drinkinganddriving.org/lessons/the-truth-about-bac.html