CH ENGR 101B Chapter 7: Problem Sets
9 Jan 2019
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Problems
Flat Plate in Parallel Flow
7.1
Consider the following fluids at a film temperature of 300 K in parallel flow over a flat plate with velocity of 1
m/s: atmospheric air, water, engine oil, and mercury.
(a)
For each fluid, determine the velocity and thermal boundary layer thicknesses at a distance of 40 mm from the
leading edge.
(b)
For each of the prescribed fluids and on the same coordinates, plot the boundary layer thicknesses as a function
of distance from the leading edge to a plate length of 40 mm.
7.2
Engine oil at 100°C and a velocity of 0.1 m/s flows over both surfaces of a 1-m-long flat plate maintained at
20°C. Determine:
(a)
The velocity and thermal boundary layer thicknesses at the trailing edge.
(b)
The local heat flux and surface shear stress at the trailing edge.
(c)
The total drag force and rate of heat transfer per unit width of the plate.
(d)
Plot the boundary layer thicknesses and local values of the surface shear stress, convection coefficient, and heat
flux as a function of x for 0 ≤ x ≤ 1 m.
7.3
Consider a liquid metal (Pr ≪ 1), with free stream conditions u∞ and T∞, in parallel flow over an isothermal flat
plate at Ts. Assuming that u = u∞ throughout the thermal boundary layer, write the corresponding form of the
boundary layer energy equation. Applying appropriate initial (x = 0) and boundary conditions, solve this
equation for the boundary layer temperature field, T(x, y). Use the result to obtain an expression for the local
Nusselt number
Nux.
Hint:
This problem is analogous to one-dimensional heat transfer in a semi-infinite medium with a sudden
change in surface temperature.
7.4
Consider the velocity boundary layer profile for flow over a flat plate to be of the form u = C1 + C2y. Applying
appropriate boundary conditions, obtain an expression for the velocity profile in terms of the boundary layer
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thickness δ and the free stream velocity u∞. Using the integral form of the boundary layer momentum equation
(Appendix G), obtain expressions for the boundary layer thickness and the local friction coefficient, expressing
your result in terms of the local Reynolds number. Compare your results with those obtained from the exact
solution (Section 7.2.1) and the integral solution with a cubic profile (Appendix G).
7.5
Consider a steady, turbulent boundary layer on an isothermal flat plate of temperature Ts. The boundary layer is
“tripped” at the leading edge x = 0 by a fine wire. Assume constant physical properties and velocity and
temperature profiles of the form
()
(a)
From experiment it is known that the surface shear stress is related to the boundary layer thickness by an
expression of the form
()
Beginning with the momentum integral equation (Appendix G), show that
()
Determine the average friction coefficient .
(b)
Beginning with the energy integral equation, obtain an expression for the local Nusselt number Nux and use this
result to evaluate the average Nusselt number .
7.6
Consider flow over a flat plate for which it is desired to determine the average heat transfer coefficient over the
short span x1 to x2, , where (x2 − x1) ≪ 1.
Provide three different expressions that can be used to evaluate in terms of (a) the local coefficient at x = (x1 +
x
2)/2, (b) the local coefficients at x1 and x2, and (c) the average coefficients at x1 and x2. Indicate which of the
expressions is approximate. Considering whether the flow is laminar, turbulent, or mixed, indicate when it is
appropriate or inappropriate to use each of the equations.
7.7
Consider atmospheric air at 20°C and a velocity of 30 m/s flowing over both surfaces of a 1-m-long flat plate
that is maintained at 130°C. Determine the rate of heat transfer per unit width from the plate for values of the
critical Reynolds number corresponding to 105, 5 × 105, and 106.
7.8
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Consider laminar, parallel flow past an isothermal flat plate of length L, providing an average heat transfer
coefficient of . If the plate is divided into N smaller plates, each of length LN = L/N, determine an expression for
the ratio of the heat transfer coefficient averaged over the N plates to the heat transfer coefficient averaged over
the single plate, .
7.9
Repeat Problem 7.8 for the case when the boundary layer is tripped to a turbulent condition at its leading edge.
7.10
Consider a flat plate subject to parallel flow (top and bottom) characterized by u∞ = 5 m/s, T∞ = 20°C.
(a)
Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force
associated with an L = 2-m-long, w = 3-m-wide flat plate for airflow and surface temperatures of Ts = 30°C and
80°C.
(b)
Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force
associated with an L = 0.1-m-long, w = 0.1-m-wide flat plate for water flow and surface temperatures of Ts =
30°C and 80°C.
7.11
Consider two cases involving parallel flow of dry air at V = 1 m/s, T∞ = 45°C, and atmospheric pressure over an
isothermal plate at Ts = 20°C. In the first case, Rex,c = 5 × 105, while in the second case the flow is tripped to a
turbulent state at x = 0 m. At what x-location are the thermal boundary layer thicknesses of the two cases equal?
What are the local heat fluxes at this location for the two cases?
7.12
In Example 7.2, it was determined that the sixth segment of the flat plate has the maximum power requirement.
Which segment has the minimum power requirement if conditions are identical except the air temperature is
raised to T∞ = 190°C? What is this power requirement?
7.13
Consider water at 27°C in parallel flow over an isothermal, 1-m-long flat plate with a velocity of 2 m/s.
(a)
Plot the variation of the local heat transfer coefficient, hx(x), with distance along the plate for three flow
conditions corresponding to transition Reynolds numbers of (i) 5 × 105, (ii) 3 × 105, and (iii) 0 (the flow is fully
turbulent).
(b)
Plot the variation of the average heat transfer coefficient with distance for the three flow conditions of part (a).
(c)
What are the average heat transfer coefficients for the entire plate for the three flow conditions of part (a)?
7.14
Consider the photovoltaic solar panel of Example 3.3. The heat transfer coefficient should no longer be taken to
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