  Recent cosmological observations show a statistically significant tension in
the estimated values of the cosmological parameters within the standard
$\Lambda$CDM framework. In a recent study, Li and Shafieloo introduced a simple
Phenomenological Emergent Dark Energy (PEDE) model, which possesses the same
number of parameters as that of the $\Lambda$CDM model. Their research
highlighted this model as a viable alternative to $\Lambda$CDM, capable of
alleviating the Hubble tension and explaining the late-time cosmic
acceleration. Following this, we consider a series of PEDE-type models where a
new parameter $b$ is introduced in the dark energy expression that
distinguishes one model from the other and is designated as bPEDE models. The
PEDE and $\Lambda$CDM models were the special cases of the bPEDE model. In
contrast to the PEDE model, the bPEDE model demonstrates the presence of dark
energy in the past while indicating its absence in the asymptotic future.
Confronting these models with the observational Hubble data (OHD) shows that a
series of bPEDE models fit the data better than the PEDE model and the standard
$\Lambda$CDM model. Notably, the Hubble constant ($H_0$) value computed using
the best-fit bPEDE models closely aligns with the CMBR prediction. It
significantly deviates from the local measurement at a significance level of
approximately $3.4\sigma$ for the model independent OHD data combination. The
outcome suggests reconsidering the systematic uncertainties associated with the
local measurement. The best-fit bPEDE models predict the deceleration to
acceleration transition at a redshift $z_T \sim 0.78$ which is in close
agreement with the $\Lambda$CDM prediction. The age of the universe predicted
by the bPEDE model is $\sim 14$ Gyr, slightly higher than the age predicted by
the $\Lambda$CDM model. The statefinder trajectory reveals a quintessence
nature of dark energy.
