Air flows past a flat plate at a free-stream Mach number of M∞ = 0.8. The air pressure and temperature in the free-stream are p∞ = 1 atm. and T∞ = 350 K, respectively. The plate is maintained at a temperature of Tw = 300 K. Determine the heat flux on the plate at a location x = 0.02 m downstream

Question

Air flows past a flat plate at a free-stream Mach number of [latex]M_{\infty}=0.8[/latex]. The air pressure and temperature in the free-stream are [latex]p_{\infty}=1 ext { atm }[/latex]. and [latex]T_{\infty}=350 K[/latex], respectively. The plate is maintained at a temperature of [latex]T_{w}=300 K[/latex]. Determine the heat flux on the plate at a location x = 0.02 m downstream of the leading edge. Use the reference temperature method. What will happen if compressibility effects and viscous dissipation are ignored?

Accepted Answer

(Note that all temperatures are in Kelvin in this class of problems)

Step 1 The air properties at the free-stream condition are:

Density: [latex] ho_{\infty}=0.995 kg / m ^{3}[/latex]

Kinematic viscosity: [latex] u_{\infty}=20.92 imes 10^{-6} m ^{2} / s[/latex]

Specific heat: [latex]C_{p \infty}=1009 K / kg K[/latex]

Thermal conductivity: [latex]k_{\infty}=0.030 W / m K[/latex]

The Prandtl number of air may be taken as a constant at [latex]\operatorname{Pr}=0.7[/latex] .

Step 2 The speed of sound in air at the ambient temperature is based on [latex]\gamma=1.4[/latex] and R = 287 J/kg K.

[latex]a_{\infty}=\sqrt{1.4 imes 287 imes 350}=375 m / s[/latex]

The free-stream air velocity is then obtained as

[latex]U=M_{\infty} a_{\infty}=0.8 imes 375=300 m / s[/latex]

Step 3 The recovery factor (based on Eq. 17.153) is given by

[latex]r=\sqrt{\operatorname{Pr}}, 0.6 \leq \operatorname{Pr} \leq 15[/latex] (17.153)

[latex]r=\sqrt{\operatorname{Pr}}=\sqrt{0.7}=0.837[/latex]

assuming the flow to be laminar. The adiabatic wall temperature is then given by

[latex]T_{a w}=T_{\infty}+r \frac{U^{2}}{2 C_{p \infty}}=350+0.837 \frac{300^{2}}{2 imes 1009}=387.3 K[/latex]

The reference temperature is given by (using Eq. 17.154)

[latex]\frac{T^{*}}{T_{\infty}}=1+0.032 M_{\infty}^{2}+0.58\left[\frac{T_{w}}{T_{\infty}}-1 ight][/latex] (17.154)

[latex]\frac{T^{*}}{T_{\infty}}=1+0.032 imes 0.8^{2}+0.58\left[\frac{300}{350}-1 ight]=0.938[/latex] or [latex]T^{*}=0.938 imes 350=328.2 K[/latex]

Step 4 The air properties of interest at the reference temperature are:

Density: [latex] ho^{*}=1.068 kg / m ^{3}[/latex]

Dynamic viscosity: [latex]\mu^{*}=19.75 imes 10^{-6} kg / m s[/latex]

Step 5 We now calculate the Reynolds number using the properties at the reference temperature.

[latex]x^{\prime *}=\frac{ ho^{*} U x}{\mu^{*}}=\frac{1.068 imes 300 imes 0.02}{19.75 imes 10^{-6}}=3.245 imes 10^{5}[/latex]

The flow is laminar. The Stanton number is then given by (Eq. 17.156)

[latex]S t=\frac{0.332}{x^{* *\left(\frac{1}{2} ight)} \operatorname{Pr}^{*\left(\frac{2}{3} ight)}}[/latex] (17.156)

[latex]S t=\frac{0.332}{\sqrt{\left(3.245 imes 10^{5} ight) 0.7^{\frac{2}{3}}}}=7.39 imes 10^{-4}[/latex]

Step 6 The heat transfer coefficient is then given by

[latex]h=-\operatorname{St} ho_{\infty} U C_{p \infty}=-7.39 imes 10^{-4} imes 0.995 imes 300 imes 1009=-222.67 W / m ^{2} K[/latex]

Step 7 The local wall heat flux is then given by

[latex]q_{w, x=0.02 m}=h\left(T_{a w}-T_{w} ight)=-222.67(387.5-300)=-19483.6 W / m ^{2}[/latex]

Heat transfer is from air to wall.

Step 8 This may be compared with the incompressible boundary layer solution ignoring viscous dissipation. All the properties are based on the free-stream temperature. The various quantities of interest are (the reader may verify
these):

Local Reynolds number: [latex]R e_{x}=2.867 imes 10^{5}[/latex]

Stanton number: [latex]S t=7.864 imes 10^{-4}[/latex]

Heat transfer coefficient: [latex]h=236.9 W / m ^{2} K[/latex]

Local wall heat flux: [latex]q_{w, x=0.02 m }=-11840 W / m ^{2}[/latex]

There is thus a dramatic difference when compressibility effects are taken in to account, as they should be!

Question 17.18: Air flows past a flat plate at a free-stream Mach number of ...

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