Chapter 15, Problem 10P

To determine

Find the power requirements of air resistance as a function of air speed and temperature in kilowatts and horsepower.

Explanation of Solution

Given data:

The value of drag coefficient $C_{\text{d}}$ is 0.4.

The width of car is 74.4 inches.

The height of car is 57.4 inches.

The air speed varies from the range of $15\frac{\text{m}}{\text{s}}<V<35\frac{\text{m}}{\text{s}}$.

The temperature varies from the range of $0°\text{C}<T<45°\text{C}$.

Formula used:

Express the drag coefficient by the equation,

$C_{\text{d}}=\frac{F_{\text{d}}}{\frac{1}{2}\rho V^{2}A}$ (1)

Here,

$F_{\text{d}}$ is the drag force,

$\rho$ is the air density,

$V$ is the air speed,

$A$ is the frontal area.

Express the relation for power consumption in order to overcome the air resistance,

$P=F_{\text{d}}V$ (2)

Calculate the area by multiplying the factor $0.85$ with width and height.

Therefore, express the area as below.

$A=0.85\times \text{width×height}$ (3)

Express the formula to calculate the air density as below,

${\rho }_{\text{air}}=\frac{p}{RT}$ (4)

Here,

$p$ is the atmosphere pressure of air $\left(101325\text{Pa}\right)$,

$R$ is the ideal gas constant $\left(287.05\frac{\text{J}}{\text{kg}\cdot {\text{K}}^{\text{-1}}}\right)$

$T$ is the temperature.

Calculation:

Rearrange equation (1) to find $F_{\text{d}}$.

$F_{\text{d}}=\frac{1}{2}{C}_{\text{d}}\rho V^{2}A$ (5)

Substitute equation (4) in (5).

$F_{\text{d}}=\frac{1}{2RT}{C}_{\text{d}}p{V}^{2}A$ (6)

Substitute equation (3) in (6).

$\begin{array}{c}{F}_{\text{d}}=\frac{0.85}{2RT}{C}_{\text{d}}p{V}^2\times \text{width×height}\\ \text{=}\frac{0.425}{RT}{C}_{\text{d}}p{V}^2\times \text{width×height}\end{array}$ (7)

Substitute equation (7) in equation (2).

$\begin{array}{c}P=\frac{0.425}{RT}{C}_{\text{d}}p{V}^2\times \text{width×height}\times V\\ =\frac{0.425}{RT}{C}_{\text{d}}p\times \text{width×height}\times V^3\text{W}\\ =\frac{0.000425}{RT}{C}_{\text{d}}p\times \text{width×height}\times V^3\text{kW}\end{array}$ (8)

Summarize the steps to find the power consumption as a function of air speed and temperature using MATLAB as follows:

• Input the given values.
• Use equation (1) to find the power in watts.
• Use the temperature value range from $0°\text{C}<T<45°\text{C}$ and air speed from $15\frac{\text{m}}{\text{s}}<V<35\frac{\text{m}}{\text{s}}$ to find the different values of power.
• Divide the equation (1) by 1000 to find the power in kilowatts.
• Find the power in horsepower using the relation 1horpower=1.341 kilowatts.
• Print the values in the form of table.

In the M-file editor, type the code as follows and save the file named “air speed” as .m file and run the code.

C_d=0.4;

width=74.4*0.0254;            % convert inches into meter

height=57.4*0.0254;           % convert inches into meter

p=101325;

R=287.05;

V=15:5:35;

T=0:5:45;

for i=1:1:5

    for j=1:1:10

        P_kw(i,j)=(0.000425*C_d*p*width*height*V(i)^3)/(R*(T(j)+273));

        P_hp(i,j)=1.341*P_kw(i,j);

    end

end

table_kw=[V',P_kw];

table_hp=[V',P_hp];

fprintf('\n---------------------------------------------------------------------------------------------------------------\n');

fprintf('\t\t      \t\t\t\t\t\t\t\t\t\t\t\t\t\t Ambient temperature (C) \n');

fprintf('\t\tCar speed -------------------------------------------------------------------------------------------------\n');

fprintf('\t(m/s)\t\t0\t\t5\t\t\t10\t\t\t15\t\t20\t\t25\t\t30\t\t\t35\t\t40\t\t\t45\n');

fprintf('--------------------------------------------------------------------------------------------------------------------\n');

disp(table_kw);

fprintf('-----------------------------------------------------------------------------------------------------------------------------\n');

fprintf('\n---------------------------------------------------------------------------------------------------------------\n');

fprintf('\t\t      \t\t\t\t\t\t\t\t\t\t\t\t\t\t Ambient temperature (C) \n');

fprintf('\t\tCar speed --------------------------------------------------------------------------------------------------------\n');

fprintf('\t(m/s)\t\t0\t\t5\t\t\t10\t\t\t15\t\t20\t\t25\t\t30\t\t\t35\t\t40\t\t\t45\n');

fprintf('--------------------------------------------------------------------------------------------------------------------------\n');

disp(table_hp);

fprintf('-----------------------------------------------------------------------------------------------------------------------------\n');

In the command window of the MATLAB, the output will be displayed as follows:

Therefore, the power consumption as a function of air speed and temperature in terms of kilowatts and horsepower is calculated.

Conclusion:

Thus, the power consumption as a function of air speed and temperature in terms of kilowatts and horsepower is calculated using MATLAB.