Question

Assuming the transition to turbulence for flow over a flat plate happens at a Reynolds number of 5x105, determine the following for air at 300 K and engine oil at 380 K. Assume the free stream velocity is 3 m/s. a. The distance from the leading edge at which the transition will occur b. Expressions for the momentum and thermal boundary layer thicknesses as a function of x for a laminar boundary layer c. Which fluid has the higher heat transfer

Answer 1

Given:

Assuming the transition to turbulence for flow over a flat plate happens at a Reynolds number of 5x105, determine the following for air at 300 K and engine oil at 380 K. Assume the free stream velocity is 3 m/s.

To Find:

a. The distance from the leading edge at which the transition will occur.

b. Expressions for the momentum and thermal boundary layer thicknesses as a function of x for a laminar boundary layer

c. Which fluid has a higher heat transfer

Calculation:

The transition from the lamina to turbulent begins when the critical Reynolds

number reaches
`5* 10^5`

`(a). \;\text{Rex}_(cr)=5 * 10^5\\\\(\rho\;vx)/(\mu)=5 * 10^5\\\text{density of of air at}\;300K=1.16 (kg)/(m\cdot s)\\\text{viscosity of of air at}\;300K=1.846 * 10^(-5) (kg)/(m\cdot s) \\v=3m/s\\\Rightarrow x=(5* 10^5 * 1.846 * 10^(-5) )/(1.16 * 3) =2.652 \;m \;\text{for air}\\(\text{similarly for engine oil at 380 K for given}\; \rho \;\text{and} \;\mu)\\`

`(b).\; \text{For the lamina boundary layer momentum boundary layer thickness is given by}:\\(\delta)/(x) =(5)/(√(R_e))\;\;\;\;\quad\text{for}\; R_e <5 * 10^5\\\\\text{for thermal boundary layer}\\\delta _t=\frac{\delta}{{P_r}^{(1)/(3)}}\quad\quad \text{where} \;P_r=(C_p\mu)/(K)\\\Rightarrow \delta_t=\frac{5x}{√(R_e){P_r}^{(1)/(3)}}`
`(c). (\delta)/(\delta_t)={P_r}^{(r)/(3)}\\\text{For air} \;P_r \;\text{equivalent 1 hence both momentum and heat dissipate with the same rate for oil}\; \\P_r >>1 \text{heat diffuse very slowly}\\\text{So heat transfer rate will be high for air.}\\\text{Convective heat transfer coefficient will be high for engine oil.}`