Surface Roughness R a — Definition, Calculation, and Typical Casting Values: 23-Ind-B2 Manufacturing Processes, May 2018, Q2

Question 2: Surface Roughness R_a — Definition, Calculation, and Typical Casting Values (20 marks: 10/10)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Defining and Calculating R_a

Definition. The arithmetic average roughness $R_a$ (also called the centre-line average, CLA) is the average of the absolute values of the surface profile's deviations from its mean line, taken over the evaluation (sampling) length: $R_a=\dfrac{1}{L}\displaystyle\int_0^L |y(x)|\,dx$, or, for a profile sampled at $n$ equally spaced points as in the figure below, $R_a=\dfrac{1}{n}\displaystyle\sum_{i=1}^{n}|y_i|$, where $y_i$ is the signed deviation of point $i$ from the mean line and the mean line itself is positioned so the areas above and below it are equal.

Given. A surface profile sampled at 24 equally spaced points, with deviations from the mean line (peak values above the line, valley values below it) read directly off the source figure.

Find. The arithmetic average roughness $R_a$ of the profile.

[Figure not reproduced: Reconstructed surface-profile trace: 24 equally spaced sample points, alternating peaks above and valleys below the mean line, with the deviation magnitude at each point matching the source figure. See the official exam paper.]

Approach. Read the signed deviation $y_i$ at each of the 24 equally spaced sample points off the mean line, take the absolute value of each, sum them, and divide by the point count $n$.

List the signed deviations. Following the profile left to right: peak run $+1,+4,+6,+8,+5,+2$; valley run $-1,-4,-1$; single peak $+2$; valley run $-1,-4,-7,-4,-1$; peak run $+2,+5,+8,+5,+2$; valley run $-1,-4,-1$; single peak $+1$ — 24 points in all.
Sum the absolute deviations. $\sum|y_i|=(1+4+6+8+5+2)+(1+4+1)+2+(1+4+7+4+1)+(2+5+8+5+2)+(1+4+1)+1=26+6+2+17+22+6+1=80$.
Divide by the point count. $R_a=\dfrac{\sum|y_i|}{n}=\dfrac{80}{24}=\boxed{3.33}$ (in the units of the source figure — treated here as micrometres, $\mu\text{m}$, the standard SI unit for $R_a$; see the check note below).

Check: the source figure gives the profile deviations as bare numbers with no stated unit. Surface roughness $R_a$ is conventionally reported in micrometres ($\mu\text{m}$) or microinches; the calculation above is unit-agnostic (it reproduces whatever unit the raw deviation values are actually in) and $\mu\text{m}$ is used here as the standard engineering convention, consistent with the typical-value ranges quoted in part (ii).

Question 2(i) — final result
Quantity	Value
Sample points, $n$	24
Sum of absolute deviations, $\sum\|y_i\|$	80
Arithmetic average roughness, $R_a$	3.33 $\mu\text{m}$

(ii) Typical R_a of Sand Casting vs. Investment Casting

Sand casting produces a comparatively rough as-cast surface, typically $R_a\approx12.5$–$25\ \mu\text{m}$, because the coarse sand grains that form the mold cavity wall leave their impression directly on the casting surface, and mold-metal reaction/burn-in at the interface roughens it further. Investment (lost-wax) casting produces a much finer surface, typically $R_a\approx1.5$–$3\ \mu\text{m}$ — roughly an order of magnitude smoother — because the mold cavity is formed against a smooth wax pattern and a fine ceramic slurry coating, so the mold wall itself has far less inherent surface texture to transfer to the casting.

Topic. Arithmetic-average (CLA) surface roughness $R_a$: definition, discrete calculation, and its dependence on the casting process used to form the mold cavity.

Key relations. $R_a=\dfrac{1}{n}\sum|y_i|$ for a discretely sampled profile (equivalently $\dfrac{1}{L}\int|y(x)|\,dx$ for a continuous trace); a smoother mold-cavity surface (finer aggregate/pattern material) directly produces a lower as-cast $R_a$.

Why this works. $R_a$ averages the magnitude of surface texture without regard to sign, which is why deviations above and below the mean line both contribute positively — it measures how "textured" the surface is, not whether the texture nets out to zero (which it always does, by definition of the mean line). Casting-process surface finish is fundamentally a mold-cavity-wall replication problem: any texture on the mold wall (sand-grain size, pattern smoothness) is stamped directly onto the metal.

Common pitfall. Averaging the signed deviations instead of their absolute values — by construction, the mean line makes the signed average exactly zero, which would give the physically meaningless result $R_a=0$ for every profile.

Source. Groover, Fundamentals of Modern Manufacturing, 6th ed., Ch. 5 (surface texture and measurement, $R_a$ definition) and Ch. 11 (comparative surface finish/tolerances by casting process).

Question 2 of 7: Surface Roughness R a — Definition, Calculation, and Typical Casting Values

Question 2: Surface Roughness Ra — Definition, Calculation, and Typical Casting Values (20 marks: 10/10)

(i) Defining and Calculating Ra

(ii) Typical Ra of Sand Casting vs. Investment Casting

Question 2: Surface Roughness R_a — Definition, Calculation, and Typical Casting Values (20 marks: 10/10)

(i) Defining and Calculating R_a

(ii) Typical R_a of Sand Casting vs. Investment Casting