Download flow resistance R

ENTC 4350
COMPONENTS 2
THE BLOOD FLOW ANALOGY

To begin the analogy, let’s consider the
heart as a pump,
•
which produces a maximum pressure P
during systole.

This pressure induces
a flow of blood through
the aorta and the
arteries that lead from
the aorta to various
parts of the body.


Note that the arteries involved— the
brachiocephalic, the left common carotid, and
the left subuclavian—are of different
diameters.
This suggests that the flow of blood, in
milliliters per minute (ml/min), is different in
each artery, with the largest flow passing
through the largest artery.
•
We will symbolize the rate of blood flow with the
letterQ.
Pressure and Resistance

Being scientific types, we wish to go
beyond this mere observation and seek
a mathematical relationship between the
blood pressure, the arterial diameter,
and the flow of blood.

This relation, or equation, for blood flow
is analogous to what we shall call Ohm’s
law, and we will write it as:
P( pressure)
Q(m l / m in) 
R(resistance)


Here we have introduced the new symbols P
for pressure (in, say, millimeters of mercury, or
mm Hg) and R for flow resistance.
The term flow resistance denotes the
combined effects of the arterial diameter, the
frictional losses as the blood moves through
the artery, and so on.
•
It should be obvious that larger vessels offer less
resistance, in other words, ~ the ~irterial diameter d
goes up.

In fact, since R is inversely proportional
to the cross-sectional area (A) of the
artery.
• It is inversely proportional to the square of the
•
diameter (1/d2), because A = ¼pd2 (where r =
3.14).
The point to remember here is that the blood
flow increases with arterial pressure P and
decreases with flow resistance R.

The point is that this pressure drop is
caused by the flow resistance R of the
body.
• The heart uses energy to pump the blood
pressure from 4 to 20 mm Hg.
• This pressure pushes the blood through the body,
but it is gradually used up in the process, so that
the blood returns to the heart at its original
pressure of 4 mm Hg.
Rate of Energy Loss or Heat
Production

The heart, then, is a pump whose task is
to raise the blood pressure so that the
blood can flow through the body.
• The next questions we might ask are how is
the pressure drop related to the flow
resistance, and what is the rate of energy
loss, or heat production, as a function of flow
resistance?

The drop in pressure, which we write as
P = P1  P2, is equal to the product of
the rate of flow times the flow resistance
R, or:
P  Q  R

For a given flow, the pressure drop is
greatest across the greatest resistance.

The loss of energy law is a little more
complex, and here we must ask you to
accept the idea that the rate of energy loss is
proportional to the square of the flow, of Q2,
times the flow resistance R.
• We write this as:
W (rateof energyloss)  Q  R
2

We write this as:
W (rateof energyloss)  Q  R
2
where W is expressed in watts.
• Watts are often used as units of heat; for
example, electrical heaters are rated in watts.

The fact that the flow rate of blood is related to
heat production has been known since the
beginning of medicine.
•
•
Hippocrates noted that inflamed, infected, or injured
areas could be detected by their temperature; it was
higher than that of other areas of the body.
We know now that the body sends a higher flow of
blood plus necessary white cells and fibrinogen to
such injured areas in order to fight infection and
promote healing.
• This extra flow of blood is to a large extent responsible
for the higher temperature.

If the blood flow Q goes up, W must go
up much faster, since W goes up with
the square of the flow rate.
Power Output



Power is the rate at which work is done.
The heart is a pump, and as such it
obtains its energy from the blood supply
that is provided by the coronary arteries.
To permit an adequate flow of blood, we
must have, first an adequate blood
pressure P, and second, the necessary
flow Q.


If the arteries are blocked by fatty deposits, the
flow will be inadequate no matter how much
pressure is provided, i.e., even if hypertension
exists.
An even worse situation Occurs when the
heart cannot provide an adequate pressure, as
in fibrillation.
•
Even the best of arteries are no help in this case, and
you have some three minutes to get the heart going
again before it is too late.

It follows from all this that the rate at
which the heart (or any other organ) can
work—i.e., its rate of energy
production—is determined by the
product of P, the blood pressure, and Q,
the rate at which blood flows to the
organ.

This relationship may be written as:
W  PQ

where W is also expressed in watts.
Summary of Relationships
1. Given a blood pressure P and an artery with flow
resistance R, the flow of blood will be Q = P/R.
2. As blood is pushed through the body, the pressure P
drops at a rate given by P = QR.
The greater the blood flow or the greater the arterial
resistance, the larger will he the pressure drop.
3. The heat produced in the tissue by the flow of blood is
given by W = Q2R.
If the blood flow doubles, the rate of heat generation
goes up by a factor of four.
4. The power that the heart can deliver is given by the
product PQ, which we called W (power output) or W =
PQ.
ELECTRICAL RELATIONSHIPS
AND OHM’S LAW

To transfer the knowledge expressed in the
previous relationships to the world of electricity,
we need only note that with electricity,
•
•
•
the unit of pressure will be V (volts) instead of P (mm
Hg),
the unit of flow will be I (amps) instead of Q (mI/min),
and
best of all, the symbol for resistance. R, will remain
the same and its unit will acquire a name. “ohms.”

We can rewrite all our fluid equations so
they can be used with electricity just by
substituting symbols:
• P = V,
• Q = I, and
• R = R.

The following are the fundamental relationships that
you will be using throughout this book.
•
I suggest that you memorize these few rules.
•
The first and foremost electrical relationship that we will focus
upon is Ohm’s law:
•
V
I
R
where
• I = amps (current),
• V = volts (voltage), and
• R = ohms (resistance)

Ohm’s Law is the most fundamental law of
electricity.
•
It may also be written as:
V  I  R

where V means the difference in voltage
between two points, or V2  V1.
•
You might just as well commit Ohm’s law to memory;
• we will be using it over and over again.

The next relationships are our electrical
power equations.
• The first is the power loss equation:
W  I R
2
• where W, expressed in watts, is the power
loss (through heat production) and I and R
mean the same as in Ohm’s law.

The second is the power output
equation:
W V  I
• where W means the rate of work output,
which may also be expressed in watts, and I
and V are as before.
• A watt, as a unit of power, is used for any kind
of power,
• whether it is the power output of an organ or a
machine,
• the rate at which energy is lost through heat, or even
electrical power.
• A watt is a watt, regardless of its source.

It thus provides us with a concept for
comparing different kinds of energy:
• for example, an electrical motor may require
•
•
so many watts of electrical power (1000 watts
or 1 kilowatt),
it may put out so many watts of mechanical
power (say, 750 watts, or about I
horsepower), and,
in running, it may dissipate so many watts of
heat (250 watts).

For electricity, it is convenient to
remember watts = amps x volts (W = IV).

It should be kept in mind that engineers
make a distinction between power
(watts) and energy.
• Power is the rate of energy loss or production:
• It may be expressed as energy per unit time.

If we multiply power times a unit of time
(say, seconds), we cancel out the unit of
time, leaving units of energy (A/B x B =
A).
• Thus, a common measure of energy is the
watt-second (watts  seconds),
•
You will see this unit again when we discuss the
defibrillator.

Energy is always conserved:
• What goes in, must come out, and vice versa.
• Energy, being conserved, is never really lost
or produced, but when we speak of energy
loss or energy production, we mean its loss or
production for useful purposes.

The purpose of the heart as a pump is to
move blood, and we spoke of the heat
produced by the flowing blood as energy
lost.
• It is energy lost from the body, and it is lost in
that it can no longer serve to move more
blood, but this energy is in no way destroyed.
Summary of Basic Units


A voIt is a measure of electrical force or
pressure, and it is defined as the difference in
electrical potential between two points a and b.
An ampere, or amp, is a measure of the flow of
electricity, or current, and is defined as the
number of electrical charges flowing through a
conductor per unit time.
•
Physicists think of current as a flow of electrons, which
are small, electrically charged particles.

An ohm is a measure of the resistance to
such a flow of electricity, and this
resistance is property of the material
that the current flows through.
• Certain materials, like metal, are good
conductors, whereas others, like plastics, are
poor conductors, or nonconductors.
• Nonconductors are called insulators.

Now that we have told you what these
terms mean,
• You can forget all these definitions.
• All you need to remember is Ohm ‘s Law, I = V/R,
and the formulas relating to it.

We are concerned with using these
concepts, not their meaning.

In discussing the definition of volt, we use the
terms positive and negative in connection with
the terminals on a battery.
•
You will often see these marked as + and - .
• The designations positive and ‘negative, which were
originally introduced by Benjamin Franklin (believe it or
not) are arbitrary conventions;
•
•
It does not make any difference which terminal is called
positive and which one negative, as long as you keep
them straight.
If you get them mixed up, you may get what is called a
short circuit, which, among other adverse side effects, is
demonstrated by a lot of sparks.
SERIES AND PARALLEL
Resistance


The figure shows a typical freeway traffic situation
where four lanes are squeezed into two by
construction.
The distribution of automobiles is shown by the black
dots.
•
The density of the dots is a result of the local resistances, R
that the different road segments offer to traffic.
•
We have marked the resistance of the four-lane road segment
R1 and the construction area as R2.

Anyone who has driven on the freeway
knows that the resistance R2 is greater
than R1, and that the flow of vehicles
throughout this entire section of highway
will be controlled primarily by R2.

To write this in mathematical form, we
define I as vehicles passing R2 per
minute,V as vehicles per mile of
highway, and R as the resistance to
vehicle flow.
• We can now write:
V
I
R1  R2

This simply means that the flow of vehicles per minute is
equal to the number of vehicles per mile divided by the
sum (total) of the different resistances (in some units or
other).
•
•

The point is that if R2 equals some large number, say
kW,
while R1 is a much smaller number, say, 1,
•
10
then we can write I = VIR2. because 1000 + 1  1000.
•
You should have no trouble recognizing the equation I = V/R by
now.)
This confirms our intuitive hunch that the traffic flow will
be controlled primarily by the greatest resistance, R2.



The pressure drop across R2, in terms of the
values of V (vehicles per mile) on the upstream
and downstream side of R2. is large.
The energy loss at the bottleneck—which may
be described by W = PR—appears as heat,
wasted gasoline, lost tempers, crying children,
and banged fenders.
Obviously. W will go up as both I2 and R go
up.
The Voltage Divider—Resistors
in Series

Two resistors and a battery are shown in
a series circuit below.
• The word series simply means that the
current flows through the resistors one after
the other.

The battery can be thought of as a pump with
an output pressure of V volts.
•
The pressure of the returning electricity will be taken
as zero.
• We know that:
V  V1  V2
•
Because the individual drops in pressure (V1 and V2) must
equal the total pressure drop, which is the difference
between the output of the battery (V) and the returning
pressure to the battery (zero).

Also
•

V
I
R1  R2
because the electrical situation is analogous to the previous
traffic flow example.
The flow of current is impeded by the resistors just as
the flow of traffic was impeded by the construction
bottleneck.
•
The small resistor (R1) corresponds to the normal resistance
of the four-lane highway, and the larger resistance (R2)
corresponds to the two-lane detour.

If R2 = 10 * R1, intuition tells us that the
voltage pressure drop across R2 will be
greater than that across R1.
• Since the same current flows through each
resistor,
V1  I  R1
V2  I  R2


Since the current is the same through
both resistors, I, is the same in both
equations.
The current is same throughout the
circuit and is given by the equation:
V
I
R1  R2

Substituting in the equations for voltages
yields:
V
VR1
V1  I  R1  V1 
 R1 
R1  R2
R1  R2
V
VR2
V2  I  R2  V2 
 R2 
R1  R2
R1  R2
•
This the voltage divider equation.
• The voltage across any resistor in a series circuit is the
supply voltage (V) times the resistor divided all the
resistors in the series circuit.

Note that the largest drop in pressure
occurs across the largest resistance in
the circuit.
THE CURRENT DIVIDER—
RESISTORS IN PARALLEL

Let’s look at the external iliac and
femoral arteries and show both the deep
femoral and femoral arteries as they
really exist.

The flow in the femoral artery (Q1)
parallels that in the deep femoral artery
(Q2).
• The term parallel flow implies that the blood
flows through both arteries at the same time.

Note that the same pressure P is applied to
both arteries at the same time.
•
So,
P
Q1 
R1
and
P
Q2 
R2

If the femoral artery is partially blocked by a
clot, R1 will go up.
•
This means that Q1 will go down because:
P
Q1 
R1

To keep total flow Q = Q1 + Q2 constant, Q2
must go up.
•
This is called collateral circulation.

Suppose we want to calculate Q1 and Q2 when
we know Q, R1, and R2.
P
P
P
Q   Q1 
 Q2 
R
R1
R2

We noted that Q = Q1 + Q2, so
P P P
 
R R1 R2

Note that,
P P P
1 1
1
 
  
R R1 R2
R R1 R2
• This is the equation for parallel resistances.
• Note the differences between resistances in series.
R  R1  R2
1 1
1
 
R R1 R2
1
R2
R1


R R1 R2 R1 R2
1 R2  R1

R
R1 R2
1
1
R1 R2


1 / R R2  R1 R2  R1
R1 R2